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29 Steel Design Guide Vertical Bracing Connections— Analysis and Design 29 Steel Design Guide Vertical Bracing Connections— Analysis and Design Larry S. Muir, P.E. AISC Atlanta, GA William A. Thornton, Ph.D., P.E. Cives Steel Corporation Roswell, Georgia A MERICAN INSTITUT E OF S T E E L CONS T RUCT I O N AISC © 2014 by American Institute of Steel Construction All rights reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher. The AISC logo is a registered trademark of AISC. The information presented in this publication has been prepared in accordance with recognized engineering principles and is for general information only. While it is believed to be accurate, this information should not be used or relied upon for any specific application without competent professional examination and verification of its accuracy, suitability and applicability by a licensed professional engineer, designer or architect. The publication of the material contained herein is not intended as a representation or warranty on the part of the American Institute of Steel Construction or of any other person named herein, that this information is suitable for any general or particular use or of freedom from infringement of any patent or patents. Anyone making use of this information assumes all liability arising from such use. Caution must be exercised when relying upon other specifications and codes developed by other bodies and incorporated by reference herein since such material may be modified or amended from time to time subsequent to the printing of this edition. The Institute bears no responsibility for such material other than to refer to it and incorporate it by reference at the time of the initial publication of this edition. Printed in the United States of America Authors Larry S. Muir, P.E. is the Director of Technical Assistance in the AISC Steel Solutions Center. He is a member of both the AISC Committee on Specifications and the Committee on Manuals. William A. Thornton, Ph.D., P.E. is a corporate consultant to Cives Corporation in Roswell, GA. He was Chairman of the AISC Committee on Manuals for over 25 years and still serves on the Committee. He is also a member of the AISC Committee on Specifications and its task committee on Connections. Acknowledgments The authors wish to acknowledge the support provided by Cives Steel Company during the development of this Design Guide and to thank the American Institute of Steel Construction for funding the preparation of this Guide. The ASCE Committee on Design of Steel Building Structures assisted in the development of Appendix D. They would also like to thank the following people for assistance in the review of this Design Guide. Their comments and suggestions have been invaluable. Leigh Arber Scott Armbrust Bill Baker Charlie Carter Carol Drucker Cindi Duncan Lanny Flynn Scott Goodrich Pat Hassett Steve Herlache Steve Hofmeister Larry Kloiber Bill Lindley Margaret Matthew Ron Meng Chuck Page Bill Pulyer Ralph Richard Dave Ricker Tom Schlafly Bill Scott Bill Segui Victor Shneur Gary Violette Ron Yeager Preface This Design Guide provides guidance for the design of braced frame bracing connections based on structural principles and adhering to the 2010 AISC Specification for Structural Steel Buildings and the 14th Edition AISC Steel Construction Manual. The content expands on the discussion provided in Part 13 of the Steel Construction Manual. The design examples are intended to provide a complete design of the selected bracing connection types, including all limit state checks. Both load and resistance factor design and allowable stress design methods are employed in the design examples. i ii TABLE OF CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . 1 1.1 1.2 5.3 OBJECTIVE AND SCOPE . . . . . . . . . . . . . . . 1 DESIGN PHILOSOPHY . . . . . . . . . . . . . . . . . 1 5.4 CHAPTER 2 COMMON BRACING SYSTEMS . . . . 3 2.1 2.2 ANALYSIS CONSIDERATIONS . . . . . . . . . . . 3 CHEVRON BRACED FRAMES (CENTER TYPE) . . . . . . . . . . . . . . . . . . . . . 3 5.5 5.6 CHAPTER 3 BRACE-TO-GUSSET CONNECTION ARRANGEMENTS . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 3.2 3.3 3.4 3.5 3.6 5.7 SMALL WIDE-FLANGE BRACES . . . . . . . . . 9 LARGE WIDE-FLANGE BRACES . . . . . . . . . 9 ANGLE AND WT-BRACES . . . . . . . . . . . . . 11 CHANNEL BRACES . . . . . . . . . . . . . . . . . . 11 FLAT BAR BRACES . . . . . . . . . . . . . . . . . . 11 HSS BRACES . . . . . . . . . . . . . . . . . . . . . . . 11 5.8 5.9 5.10 CHAPTER 4 DISTRIBUTION OF FORCES . . . . . . 17 4.1 4.2 4.3 OVERVIEW OF COMMON METHODS . . . . 17 4.1.1 Corner Connections . . . . . . . . . . . . . . 17 4.1.2 Central or Chevron Connections . . . . . . 21 4.1.3 Comparison of Designs—Uniform Force Method vs. Parallel Force Method . . . . . 22 THE UNIFORM FORCE METHOD . . . . . . . . . . 24 4.2.1 The Uniform Force Method— General Case . . . . . . . . . . . . . . . . . . . 24 4.2.2 Nonconcentric Brace Force— Special Case 1 . . . . . . . . . . . . . . . . . . 29 4.2.3 Reduced Vertical Brace Shear Force in Beam-to-Column Connection— Special Case 2 . . . . . . . . . . . . . . . . . . 31 4.2.4 No Gusset-to-Column Connection— Special Case 3 . . . . . . . . . . . . . . . . . . 32 4.2.5 Nonorthogonal Corner Connections . . . . . . 33 4.2.6 Effect of Frame Distortion . . . . . . . . . . . . . . 34 BRACING CONNECTIONS TO COLUMN BASE PLATES . . . . . . . . . . . . . . . . . . . . . . 37 5.11 5.12 CHAPTER 6 DESIGN OF BRACING CONNECTIONS FOR SEISMIC RESISTANCE . . . . . . . . . . . . . 291 6.1 5.2 COMPARISON BETWEEN HIGH-SEISMIC DUCTILE DESIGN AND ORDINARY LOWSEISMIC DESIGN . . . . . . . . . . . . . . . . . . 291 Example 6.1a High-Seismic Design in Accordance with the AISC Specification and the AISC Seismic Provisions . . . . . . 292 Example 6.1b Bracing Connections for Systems not Specifically Detailed for Seismic Resistance (R=3) . . . . . . . . . . . . 321 APPENDIX A. DERIVATION AND GENERALIZATION OF THE UNIFORM FORCE METHOD . . . . . . . . . . . . . . . . . . . . . . 347 CHAPTER 5 DESIGN EXAMPLES . . . . . . . . . . . . . 43 5.1 CORNER CONNECTION-TO-COLUMN FLANGE: UNIFORM FORCE METHOD SPECIAL CASE 2 . . . . . . . . . . . . . . . . . . . . 84 CORNER CONNECTION-TO-COLUMN FLANGE WITH GUSSET CONNECTED TO BEAM ONLY: UNIFORM FORCE METHOD SPECIAL CASE 3 . . . . . . . . . . . . . . . . . . . . 98 CORNER CONNECTION-TO-COLUMN WEB: GENERAL UNIFORM FORCE METHOD . . 118 CORNER CONNECTION-TO-COLUMN WEB: UNIFORM FORCE METHOD SPECIAL CASE 1 . . . . . . . . . . . . . . . . . . . 147 CORNER CONNECTION-TO-COLUMN WEB: UNIFORM FORCE METHOD SPECIAL CASE 2 . . . . . . . . . . . . . . . . . . . 163 CORNER CONNECTION-TO-COLUMN WEB WITH GUSSET CONNECTED TO BEAM ONLY: UNIFORM FORCE METHOD SPECIAL CASE 3 . . . . . . . . . . . . . . . . . . . 178 CHEVRON BRACE CONNECTION . . . . . . 189 NONORTHOGONAL BRACING CONNECTION . . . . . . . . . . . . . 205 TRUSS CONNECTION . . . . . . . . . . . . . . . 237 BRACE-TO-COLUMN BASE PLATE CONNECTION . . . . . . . . . . . . . . . . . . . . . 268 5.12.1 Strong-Axis Case . . . . . . . . . . . . . . . 268 5.12.2 Weak-Axis Case . . . . . . . . . . . . . . . 284 CORNER CONNECTION-TO-COLUMN FLANGE: GENERAL UNIFORM FORCE METHOD . . . . . . . . . . . . . . . . . . . . . . . . . 43 CORNER CONNECTION-TO-COLUMN FLANGE: UNIFORM FORCE METHOD SPECIAL CASE 1 . . . . . . . . . . . . . . . . . . . . 78 A.1 iii GENERAL METHOD . . . . . . . . . . . . . . . . 347 A.1.1 Beam Control Point . . . . . . . . . . . . . 347 A.1.2 Gusset Control Point . . . . . . . . . . . . . 348 A.1.3 Determination of Forces . . . . . . . . . . 349 A.1.4 Accounting for the Beam Reaction . . . 350 Example A.1 Vertical Brace-to-Column Web Connection Using an Extended Single Plate . . . . . . . . . . . . . . . . . . . . . . . 351 C.4 C.5 APPENDIX B. USE OF THE DIRECTIONAL STRENGTH INCREASE FOR FILLET WELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 APPENDIX D. TRANSFER FORCES . . . . . . . . . . . . 383 D.1 APPENDIX C. BUCKLING OF GUSSET PLATES . . . . . . . . . . . . . . . . . . . . . . . 374 C.1 C.2 C.3 AN APPROACH TO GUSSET PLATE FREE EDGE BUCKLING USING STATICALLY ADMISSIBLE FORCES (THE ADMISSIBLE FORCE MAINTENANCE METHOD) . . . . . 380 APPLICATION OF THE FREE EDGE APPROACH TO EXAMPLE 6.1a . . . . . . . . . 382 D.2 BUCKLING AS A STRENGTH LIMIT STATE— THE LINE OF ACTION METHOD . . . . . . . 374 BUCKLING AS A HIGH-CYCLE FATIGUE LIMIT STATE—GUSSET PLATE EDGE BUCKLING . . . . . . . . . . . . . . . . . . . . . . . 374 BUCKLING AS A LOW-CYCLE FATIGUE LIMIT STATE—GUSSET PLATE FREE EDGE BUCKLING . . . . . . . . . . . . . . . . . . . . . . . 377 D.3 D.4 THE EFFECT OF CONNECTION CONFIGURATION ON THE TRANSFER FORCE . . . . . . . . . . . . . . . . . 383 PRESENTATION OF TRANSFER FORCES IN DESIGN DOCUMENTS . . . . . . . . . . . . 384 ADDITIONAL CONSIDERATIONS . . . . . . . 386 EFFECTS OF MODELING ASSUMPTIONS ON TRANSFER FORCES . . . . . . . . . . . . . 387 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 iv Chapter 1 Introduction 1.1 OBJECTIVE AND SCOPE This Design Guide illustrates a method for the design of braced frame bracing connections based on structural principles, and presents the design basis and complete design examples illustrating the design of: 1. All orthogonal and nonorthogonal connections involving a brace, a beam and a column (corner type) 2. Connections involving a beam or column and one or two braces, such as chevron or K-bracing, and eccentric braces (center type) 3. Connections of braces to columns at column base plates (base type) 4. Both nonseismic and seismic situations are covered 1.2 DESIGN PHILOSOPHY All structural design, except for that which is based directly on physical testing, is based either explicitly or implicitly on the principle known as the lower bound theorem of limit analysis. This theorem is important because it allows structural engineers to be confident that 1) their assumptions about the internal force field will not over-predict the strength of an indeterminate structure, and 2) different methodologies for determining an admissible force field, while they may vary significantly in their predictions of the available strength, are nonetheless all valid. This theorem, which was first proven in the form given in the following in the 1950s (Baker et al., 1956), states that: Given:An admissible internal force field (i.e., a distribution of internal forces in equilibrium with the applied load) Given:Satisfaction of all applicable limit states Then:The external load in equilibrium with the internal force field is less than, or at most equal to, the connection capacity. The lower bound theorem is applicable to ductile limit states, and most connection limit states have some ductility. For instance, bolts in shear undergo significant shear deformation, on the order of a in. for a w-in.-diameter bolt, before fracture. Limit states such as block shear and net shear can accommodate significant distortion of the material before fracture. Plate or column buckling, while generally conceived as a nonductile limit state, is in a sense a ductile limit state; when a plate or column buckles, it does not become incapable of supporting any load, but rather will continue to support the buckling load as long as any excess load can be distributed to other components of the structural system. This phenomenon can be observed in the laboratory when a displacement-type testing machine is used. If a force-type machine is used, the load will increase continuously, and kinking and complete collapse will occur. Actually all structural design relies on the validity of the lower bound theorem. For instance, if a building is modeled by a frame analysis computer program, a certain distribution of column loads will result. This distribution is dependent on thousands of assumptions. Shear connections are assumed not to carry any moment at all, and moment connections are assumed to maintain the angle between members. Neither assumption is true. Therefore, the column design loads at the footings will sometimes be drastically different from the actual loads, if these loads were measured. Some columns will be designed for loads smaller than the true load, and some will be designed for larger loads. Because of the lower bound theorem, this is not a concern. Ductility can also be provided to an otherwise nonductile system by support flexibility. For instance, transversely loaded fillet welds are known to have limited ductility. If a plate is fillet welded near the center of a column or beam web and subjected to a load transverse to the web, the flexibility of the web under transverse load will tend to mitigate the low ductility of the fillet weld and will allow redistribution to occur. This same effect can be achieved with transversely loaded fillet welds to rigid supports by using a fillet weld larger than that required for the given loads. The larger fillet weld allows the given applied loads to redistribute within the length of the weld without local fracture. The term “admissible force field” perhaps needs some further explanation. Bracing connections are inherently statically indeterminate. Therefore, there will be many possible force distributions within the connection. All of those force distributions that satisfy equilibrium are said to be “admissible” or “statically admissible.” There are theoretically an infinite number of possible admissible force fields for any statically indeterminate structure. There will also be an infinite number of internal force fields that do not satisfy equilibrium; these are said to be “inadmissible.” If such a force field is used, the lower bound theorem is not valid and any design obtained with this inadmissible force field cannot be said to be safe; i.e., the failure load may be less than the applied load. When an admissible force field is used, the calculated failure load will be less than, or at most equal to, the load at which failure occurs; therefore, a safe design is achieved. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 1 2 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Chapter 2 Common Bracing Systems Figure 2-1 shows some common concentric bracing configurations. The sketches of the member are meant to show the member orientation (all elements are W-shapes), which is often missing on computer generated drawings. These sketches are not meant to show work point locations. In a concentrically braced frame, the gravity axes of all members at any one joint (such as Detail A of Figure 2-1) meet at a common point, called the work point (W.P.). Figure 2-2 shows the connection at Detail A of Figure 2-1(a). As with trusses, all joints in concentrically braced frames are assumed to be pinned. Nonconcentrically braced frames are similar to concentrically braced frames; however, the work point is not located at the common gravity axis point. Figure 2-3 illustrates a nonconcentric bracing arrangement. The work point location results in a couple at the joint. This couple must be considered in the design of the system’s connections and main members in order to ensure that the internal force system is admissible. In many cases, this couple will be small and will not affect member size. Eccentrically braced frames have a different appearance than concentrically braced frames. The braces intersect the beams at points quite distinct from the usual joint working point, as shown in Figure 2-4. Eccentrically braced frames can be used to provide better access through the braced bay for doors, windows or equipment access. They are also used in seismic design because they can provide significant inelastic deformation capacity primarily through shear or flexural yielding in the links. 2.1 ANALYSIS CONSIDERATIONS Braced frames can be analyzed as simple trusses with all joints pinned. In most cases, secondary forces (also called distortional or rotational forces) due to joint rigidity can be ignored. The AASHTO Bridge Code (AASHTO, 2012) Section 6.14.2.3, for example, states that these forces can be ignored if member lengths are greater than 10 times their cross-sectional dimension in the plane of distortion. In regions of high seismicity, braced frames can be designed as pinned, but corner connections (those involving a column, beam and brace) may have to explicitly include consideration of distortional forces in the design of the connections. In concentrically braced frames, all members are assumed to be subjected only to axial forces due to lateral loads. This greatly simplifies the structural analysis because, in many cases, the frame will be statically determinate. In nonconcentrically braced frames, it is common practice to analyze the frame as concentric and then, when the work point location is moved, to superimpose the resulting member moments with the originally calculated axial member forces. Eccentrically braced frames are usually analyzed with the brace work point displaced from the beam-column work point when the braces do not intersect the beam at a common point (Figure 2-4). The beam-column work point is usually considered to be concentric as shown in Figure 2-3, but if nonconcentric beam-column joints are used as shown in Figure 2-4, the small resulting member moment can be superimposed on the original eccentric analysis results. 2.2 CHEVRON BRACED FRAMES (CENTER TYPE) Typical chevron bracing configurations are shown in Figure 2-5. These arrangements are used extensively in commercial buildings because the story height is typically half of the bay width. The chevron arrangement will, in this case, provide a brace slope close to a 12 on 12 bevel. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 3 Fig. 2-1. Various vertical bracing arrangements. 4 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 CL column CL brace W.P. CL beam Fig. 2-2. Detail A—typical concentric gusset connection. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 5 CL column CL brace Brace work point (gusset corner is a typical location) CL beam Gravity axis (concentric) work point Fig. 2-3. Nonconcentric brace work point. 6 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Link (typical) Fig. 2-4. Typical eccentrically braced frame. "Inverted V" "V" Fig. 2-5. Chevron braced frames. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 7 8 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Chapter 3 Brace-to-Gusset Connection Arrangements Every bracing connection at the beam/column/brace (corner) consists of at least four connections, and sometimes five, as shown in Figure 3-1. These connections are labeled as follows: (A) Brace to gusset (B) Gusset to beam (C) Gusset to column (D) Beam to column orientation are typically connected to the gussets by WTs or double angles back-to-back on the near and far side of the gusset. Alternatively, single angles on each side of the brace could be employed. If the brace is subjected to compression as well as tension, plates should not be used in place of the WTs or angles. Figure 3-3 shows a wide-flange brace, flange to view in elevation. This brace is connected to the gusset with four angles, two on the near side and two on the far side of the gusset. (E) Collector, or drag, beam to column 3.2 The collector beam-to-column connection occurs when there is a drag or transfer force at the column, as would clearly exist in the bracing arrangement shown in Figures 2-1(c) and 2-1(f), but can also exist in any of the configurations shown in Figure 2-1. Regardless of whether the gusset is a corner or center type, the brace-to-gusset connection is designed in the same way and should constitute the first phase of the connection design. Once the size and location of the brace-to-gusset connection is known, the size of the gusset and the remaining connections can be determined. When larger wide-flange sections are used as braces with the web to view in elevation, they can be attached to the gusset at both the flanges and at the web as shown in Figure 3-4. Plates can be used to attach the web, and “claw” angles can be used to attach the flanges. The outstanding angle legs provide for stability. Since about 80% of the force will be transferred through the flanges when a W14 column section is used, the web plates can often be eliminated. This is especially true for tension/compression braces, because the buckling strength of the brace will usually be less than 80% of the tensile strength. A very compact flange connection can be made to wide-flange braces, web to view in elevation, as shown in Figure 3-5. This arrangement resembles a truss tension-chord splice. 3.1 SMALL WIDE-FLANGE BRACES Figures 3-1 and 3-2 show wide-flange braces, with the web to view in elevation. Small wide-flange braces with this LARGE WIDE-FLANGE BRACES (C) Gusset to column P (A) Brace to gusset W.P. (B) Gusset to beam A (E) Collector beam to column (D) Beam to column CL Fig. 3-1. Concentric (corner gusset) bracing connection. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 9 W-shape W.P. W W pe -s a sh - WT, both sides or two angles e ac br CL CL beam ha pe LC b ra ce Fig. 3-2. Typical chevron brace connection configuration (center gusset). CL column CL brace Wide flange 4 Angles: 2 NS, 2 FS W.P. CL beam Fig. 3-3. Wide-flange brace (flange to view) with four angles connecting to gusset. 10 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 When wide-flange braces are flange to view in elevation, the flanges can be very effectively connected to the gusset as shown in Figure 3-6. The plates that connect to the gusset are “bird mouthed” and welded to the gusset. brace. These types of bracing arrangements are rarely used and are generally expensive compared to other bracing options. 3.5 3.3 ANGLE AND WT-BRACES Figure 3-7 shows a double-angle brace. The angles will be stitched together as shown. This is an effective brace for tension and compression, but is often not desirable for use in external locations or in industrial applications because it is difficult to maintain (i.e., keep painted). The double angles should be galvanized if this configuration is used in external applications. Single-angle braces and WT-braces (Figure 3-8) are excellent for maintenance purposes and are probably the most economical bracing members for tension-only bracing. Because of member eccentricity, they are not desirable for compression bracing. 3.4 CHANNEL BRACES Channel braces can be either back-to-back with stitches (Figure 3-9) or toe-to-toe with stitches (Figure 3-10). The back-to-back arrangement has little out-of-plane buckling strength. The toe-to-toe arrangement is similar to an HSS FLAT BAR BRACES Flat bars can be used as tension-only braces. As shown in Figure 3-11, the bars can be lapped onto the gusset and field bolted. When an available flat bar size can be used, there is very little fabrication cost involved, but these are very slender for erection and may be subject to objectionable vibration due to building movement and equipment frequency input. 3.6 HSS BRACES An HSS brace connection is shown in Figure 3-12. HSS sections are very commonly used as braces. As shown in Figure 3-12, the HSS is slotted and field welded to the gusset. The slot must be made long enough to allow for erection. HSS sections are more expensive than wide-flange sections, but they are a more efficient section for buckling, and may be a lighter alternative to a wide-flange section of equivalent strength. They are extensively used for seismic design because the tensile and compressive strengths of the members are closer than they are for wide-flange sections. W-shape girder CL 2 Claw angles W-shape brace W.P. C L Brace 2 Web plates Gusset plate Fig. 3-4. Wide-flange brace (web to view) with “claw” angles and web plate. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 11 CL column C L brace Plates A: splice plates (4 per flange) Plates B: welded both sides of gusset W.P. CL beam Fig. 3-5. Compact splice-type connection. Fig. 3-6. Wide-flange brace connection (flange to view). 12 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 W-shape girder W.P. CL CL brace Gusset plate Double angle brace Use two parallel columns of bolts as angle leg allows Stitch plate (thickness to match gusset thickness) Fig. 3-7. Double-angle brace connection. Fig. 3-8. WT- and single-angle brace. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 13 CL brace CL column Stitch plate typ. 2 Channels W.P. CL beam Fig. 3-9. Channel bracing back-to-back. CL column CL brace 2 Channels Stitch plate typ. W.P. CL beam Fig. 3-10. Channel bracing toe-to-toe. 14 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 CL brace CL column Flat bar brace Gusset plate W.P. CL beam Fig. 3-11. Flat bar brace, tension only. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 15 CL brace CL column HSS brace top & bottom Adjust for slot gap Erection bolt 2 ASTM A36 angles W.P. Gusset plate Cope top & bottom flange CL beam W-shape girder W-shape column Continuous except @ copes CL brace Note: Beam copes can be eliminated by using discontinuous connecting angles to column. Disparity between web thickness and gusset thickness may require fills. Fig. 3-12. HSS bracing connection. 16 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Chapter 4 Distribution of Forces Generally, two types of bracing connections are considered in this Design Guide—corner connections and central (or chevron) connections. Of the two types, corner connections are by far the more interesting (i.e., complex) because the distribution of internal forces within the connection cannot be determined by statics alone. Corner connections are statically indeterminate, and there has long been controversy about the correct force distribution that should be assumed for design. Central connections, on the other hand, are statically determinate, making the force distribution in them unique. The discussion earlier in this Design Guide on the lower bound theorem (LBT) is directed primarily to corner connections. There is no unique solution to the distribution of forces within these connections, but any solution that involves an admissible internal force distribution and satisfies all of the limit states of the configuration (with appropriate steps taken to account for those of limited ductility) is an acceptable solution, in accordance with the LBT. All structural analysis is based on three fundamental sets of equations: (of the capacity) by the LBT also comes closer than any other to satisfying the compatibility equations. This is another reason to use the LBT as the basis for method choice. 4.1 OVERVIEW OF COMMON METHODS There are four methods in published literature that are in common use for corner connections. There are many other methods in use by various designers that have not been published, but may be satisfactory. However, only the four common published methods will be considered in this Design Guide. Because they are statically determinate, central connections allow only one of these methods, although some variations are possible, as will be shown. 4.1.1 Corner connections The four published methods for the design of corner connections are: 1. The KISS (keep it simple, stupid) method (Thornton, 1991; Astaneh-Asl, 1998) 2. The parallel force method (Ricker, 1989; Thornton, 1991; Astaneh-Asl, 1998) 1. Equilibrium 2. Constitutive (including limit states) 3. Compatibility (deformation and displacement) The LBT satisfies two of these sets of equations (1 and 2). The upper bound theorem (UBT) also satisfies two of these sets of equations (1 and 3) (Baker et al., 1956). The true solution satisfies all three sets of equations. The LBT states that among all possible admissible force fields, the one that is closest to the true solution is the one that produces the maximum capacity. The UBT states that among all possible admissible force fields, the one that gives the least capacity is closest to the true solution. The LBT converges to the true solution from below, and the UBT converges to the true solution from above. It is reasonable to use the fact that, based on the LBT, the method chosen to design a corner connection will be best (closest to the true solution) if it produces the maximum capacity for a given connection. This will be the basis for the following review of methods presented in this Design Guide. Based on the preceding discussion it is also reasonable to suppose that, among the infinite number of possible admissible force fields, the one providing the greatest lower bound 3. The truss analogy method (Astaneh-Asl, 1989) 4. The uniform force method (Thornton, 1991, 1995; Astaneh-Asl, 1998) KISS Method The KISS method (Figure 4-1) was formalized by Thornton (1991) but had been used for many years prior and is still in use. The method is indeed simple: the horizontal component of the brace force is assumed to be carried through the gussetto-beam connection, and the vertical component of the brace force is assumed to be carried through the gusset-to-column connection. It is worth noting that the gusset edge forces (the gusset-beam and gusset-column forces) are not a function of the gusset size. However, Astaneh-Asl (1998) noted that when using this method, many users omit the gusset edge couples. While this may be common practice, unless these couples are included, neither the gusset nor the beam and column will be in equilibrium. Thus, this practice will invalidate the method, because it does not provide an admissible internal force field and the LBT is not applicable. Thornton (1991) has shown that the KISS method does not provide a AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 17 greatest lower bound solution, and thus, is less efficient than other available design methods. Therefore, the KISS method is acceptable in its full form (with couples considered in the design) but does not provide the most economical designs. Parallel Force Method The parallel force method (PFM) (Figure 4-2a and Figure 4-2b), was first proposed by Ricker (1989) and was published by Thornton (1991). In the PFM, eccentricities are calculated from the brace centerline to the centroids of the gusset plate edges along the beam and column faces. The gusset-to-beam connection is designed for the force Pb, and the gusset-to-column connection is designed for the force Pc. This method results in efficient designs when the gusset is connected to a column flange, but when the gusset is connected to a column web, a large force directed normal to the center of the web will require excessive stiffening, and if the stiffening is not provided the design fails. Therefore, the method is not practical for all design applications. Figure 4-2 illustrates that this method requires a moment, M, to be Fig. 4-1. KISS method admissible force field. Fig. 4-2a. Parallel force method admissible force field. 18 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 applied to either the gusset edges, or to the beam-to-column connection. This couple is not required for gusset equilibrium, but it is required for the equilibrium of the beam and column. It can be shown that when M = 0, the PFM reduces to the uniform force method (UFM). Truss Analogy Method The truss analogy method (Figure 4-3) was proposed by Astaneh-Asl (1989) as a simplified method to determine gusset edge forces. Unfortunately, this method suffers from limitations similar to the parallel force method. When connecting the gusset to a column web, the large force normal to the web must be accommodated by a series of stiffeners, which would not be required by the KISS method or the uniform force method. Another problem with this method is that it delivers most of the brace horizontal component to the column, and most of the brace vertical component to the beam. This load path is contrary to what is typically expected of this type of connection and normally requires the beam-to-column connection to be similar to a full strength beam-to-column moment and shear connection. Uniform Force Method The uniform force method (Figure 4-4a and Figure 4-4b) was developed by Thornton (1991) based on research performed cos sin e1 = (ec + α) cos θ − eb sin θ e2 = (eb + β)sin θ − eb cos θ Pb = e2 P e1 + e2 M= e2 P (α cos θ − eb sin θ) e1 + e2 Pc = e1 e1 + e2 Fig. 4-2b. Parallel force method admissible force field. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 19 C L ec C P L − α Centroid of brace to gusset connection Truss member (typical) Rc − β Centroid of gusset to column connection W.P. eb Rb C L Centroid of gusset to beam connection Fig. 4-3. Truss analogy method (Astaneh-Asl, 1989). Fig. 4-4a. Uniform force method admissible force field at gusset plate interface. 20 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 by Richard (1986). The method is universally applicable and has been shown to produce safe and economical designs by accurately predicting the failure loads of six full-scale physical tests (Thornton, 1991, 1995). Since this method first appeared in the AISC Manual of Steel Construction (AISC, 1992), it has been widely used in the industry. Appendix A gives a derivation and generalization of the uniform force method (UFM). The derivation involves the three control points shown in Figure 4-4a. These are labeled A, B and C. The gusset-to-beam resultant force, Rb, and the gusset-to-column resultant force, R c , are forced to pass through control points B and C respectively, and they intersect the brace line of action at control point A. The constraint, shown in Figure 4-4b, namely α − β tan θ = eb tan θ − ec causes the coincident set of connection forces P, Rb and Rc to exist (i.e., no couples on the connection interfaces). These control points will be explicitly used in the examples. When all of these four methods are applied to the same completely designed connection, the uniform force method will always produce a design capacity greater than or equal to that produced by any of the other three, thus in keeping with the LBT. 4.1.2 Central or Chevron Connections Chevron connections are illustrated in Figures 4-5, 4-6 and 4-7 (Thornton, 1987). The force distribution shown in the figures is essentially the only one possible, but some practitioners choose to assume that there is a cut at Section b-b and analyze the connection as two separate connections. This is an acceptable approach, but changes the assumed behavior α α e α β α β Fig. 4-4b. Uniform force method admissible force field at beam-to-column interface. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 21 4.1.3 of the connection. These changes will mean that the beam must take all of the shear due to the vertical components of the braces through its web, and will more frequently require web doubler plates. Also, since the gusset is cut at Section b-b, the tension brace cannot be used to stabilize the plate at the compression brace. This Design Guide will not use the vertical cut method. The method developed in Figures 4-5, 4-6 and 4-7 is based on Section a-a, the gusset-to-beam interface. This is the “control” interface, and all the resulting interface force distributions are determined by equilibrium. It is possible to use Section b-b, extended to include the beam web, as the control interface. This is not a very common approach and will not be pursued here. Comparison of Designs—Uniform Force Method vs. Parallel Force Method Figure 4-8 shows a typical corner bracing connection to a column web. The arrangement is similar to that of Figure 2-1(a), except the beam and gusset are connected to the column web. There is no transfer force. Figure 4-9 shows the UFM admissible force field, and Figure 4-10 shows the UFM design. The PFM admissible force field is shown in Figure 4-11 and the resulting design is shown in Figure 4-12. It can be seen by a comparison of Figures 4-10 and 4-12 that the PFM gives a very expensive and cumbersome design compared to the UFM. It is not practical or desirable to use the PFM when the connection is to a column web. A similar W.P. e b a a h Plate thickness = t H1 P1 b L 2 V1 L1 H2 Δ L 2 L2 L Sign convention P1, P2 + for tension, - for compression If P1 is +, V1 and H1 are + also If P1 is -, V1 and H1 are - also Same for P2 (V2 , H 2 ) 1 ( L 2 − L1 ) 2 M1 = H1e + V1Δ Δ= Note Δ is negative if L 2 < L1 M2 = H2 e − V2 Δ 1 1 1 M1ʹ = V1 L − H1h − M1 8 4 2 1 1 1 M2ʹ = V2 L − H2 h − M2 8 4 2 Fig. 4-5. Chevron brace gusset forces. 22 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 V2 P2 L 2 L 2 W.P. N M V a a H2 H1 P1 V2 V1 P2 Forces on Section a-a: Axial: N = V1 + V2 Shear: V = H1 − H2 Moment: M = M1 − M2 Fig. 4-6. Forces on Section a-a (positive directions shown). L 2 Forces on Section b-b: 1 Axial: N' = ( H1 + H 2 ) 2 1 2 Shear: V' = (V1 − V2 ) − ( M ) L 2 Moment: M' = M'1 + M'2 W.P. b V' M' N' H2 h 2 H1 P1 h 2 V1 P2 V2 b Fig. 4-7. Forces on Section b-b (positive directions shown). AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 23 conclusion would be reached if the problem of Figure 4-8 were designed using the truss analogy method. 4.2 THE UNIFORM FORCE METHOD The essence of the UFM is the selection of a connection geometry that will not produce moments on the connection interfaces (gusset-to-beam, gusset-to-column, and beam-tocolumn). In the absence of moment, these connections are designed for shear and normal forces only, hence the origin of the name UFM. This Design Guide will use the UFM exclusively for examples on the design of corner connections. Of the four methods discussed above, only the UFM and the KISS method are universally applicable. The other two methods will be used in the design examples only to illustrate the designs achieved by them for comparison with UFM designs for the same loads and geometry. The KISS method, while universally applicable, does not generally result in the most economical design, and is not recommended for general use. 4.2.1 The Uniform Force Method—General Case The UFM formulation shown in Figures 4-4a and 4-4b is the general case. This formulation achieves an admissible internal force field that provides a coincident force field on the gusset, the beam and the column. The dimensions α and β in Figures 4-4a and 4-4b describe the ideal locations of the centroids of the gusset-to-beam and gusset-to-column connections, respectively. These dimensions must satisfy the constraint of Equation 4-1: α − β tan θ = eb tan θ − ec (4-1) where eb = one half of the depth of the beam, in. ec = one half of the depth of the column, in. Note that for a column web support, ec ≈ 0. α = distance from the face of the column flange or web to the ideal centroid of the gusset-to-beam connection, in. β = distance from the face of the beam flange to the ideal centroid of the gusset-to-column connection, in. θ = angle between the brace axis and vertical This constraint can always be satisfied for new connections, but in checking existing structures (or for convenience) the connection centroids may not satisfy Equation 4‑1. In general, this Design Guide will use the terms α and β to describe the actual locations of the connection centroids. The dimensions α and α, and β and β may be the same, but often are different. When the ideal and actual connection centroids are not the same, couples will exist on the gusset edges. Appendix A contains a generalization of the UFM that dispenses with the constraint of Equation 4-1, then there is no need to distinguish between α and α or β and β. However, the examples in this Design Guide, except for the examples Fig. 4-8. Bracing connection to demonstrate the application of the parallel force method to a column web. 24 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 given in Appendix A, will only illustrate the application of the traditional UFM. There are several ways to approach the design: If the brace force, P, is tension, this couple is clockwise on the gusset when α > α and counterclockwise when α < α. If the gusset-to-beam connection is the more flexible of the two, using Equation 4-1, set α = α and calculate β. If β ≠ β there is a couple on the gusset-to-column connection equal to: 1. Lay out the connection geometry to satisfy Equation 4-1. This is the true UFM and will not induce any moment in the connection edges. 2. If the ideal geometry cannot be accommodated, decide which of the two gusset edge connections is stiffer and assign the moment to the stiffer of the two connections. If the gusset-to-column connection is the more flexible of the two, using Equation 4-1, set β = β and calculate α. If α ≠ α, there is a couple on the gusset-to-beam connection equal to: Mb = Vb ( α − α ) Mc = Hc ( β − β ) (4-3) If the brace force, P, is tension, this couple is counterclockwise on the gusset when β > β and clockwise when β < β. The direction of the couple assumes that the gusset is in the first quadrant, where the work point is the origin. (4-2) CL 300 kips 212 kips 212 kips 125 kips 125 kips 1 1 10" 212 kips 17" 7" 87 kips W.P. 87 kips 212 kips 212 kips 17" W.P. 7" 87 kips W.P . 87 kips 212 kips Fig. 4-9. Uniform force method admissible force field. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 25 Column W14x90 2" 33" 300 kips 212 kips x x 20" 5@3"=15" 212 kips 4 4 1 W.P. Beam W14×43 Web only x x PL2×8×34 (A36) Bolts: d" dia. A325-N Holes: std ," dia. Beam/Col: A992 Column: gage 52" Plate: A36 212 kips Fig. 4-10. Design by uniform force method. 26 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 3@3"=9" 22" 3" 212 kips 1 7" PL s" 3. When a decision as to relative stiffness of the two gusset interfaces cannot be made, use the method presented in the AISC Manual and distribute the moment based on minimized eccentricities α − α and β − β, by minimizing the objective function, ξ, as follows: The values of α and β that minimize ξ are: ⎛ α⎞ K ʹ tan θ + K ⎜ ⎟ ⎝ β⎠ α = D K ʹ − K tan θ β = D 2 ⎛α−α⎞ ⎛β−β⎞ ξ=⎜ ⎟ − λ ( α − β tan θ − K ) ⎟ +⎜ ⎝ α ⎠ ⎝ β ⎠ (4-4) where 2 ⎛ α⎞ K ʹ = α ⎜tan θ + ⎟ β⎠ ⎝ ⎛α⎞ D = tan 2 θ + ⎜ ⎟ ⎝β⎠ 2 670 kip-in. CL 300 kips 212 kips 212 kips 79 kips 1 1 79 kips 79 kips 10" 79 kips 134 kips 7" 17" 134 kips 17" 134 kips 134 kips W.P. 17" 134 kips 212 kips W.P. 7" 2 λ is a Lagrange multiplier and K = eb tan θ − ec 79 kips 79 kips W.P . 1,340 kip-in. 134 kips 670 kip-in. 212 kips Fig. 4-11. Parallel force method admissible force field. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 27 Bolts: d" dia. A325-N Holes: std ," dia. Beam/Col: A992 Column: gage 52" Plates: A36 Fig. 4-12. Design by parallel force method. 28 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 4.2.2 Nonconcentric Brace Force—Special Case 1 from the eccentricity, e, with the work point at the gusset corner. The forces in Figure 4-14 are given by: The case where the work point is located at the gusset corner, instead of at the intersection of the beam and column centerlines, is referred to as Special Case 1 in Figure 13-3 of the AISC Steel Construction Manual (AISC, 2011). The force distribution shown in Manual Figure 13-3 is admissible, but it does not agree with analytical results obtained by Richard (1986). A better approach is shown in Figure 4-13. Here, the work point can be located at the gusset corner or at any other reasonable location, defined by the coordinates x and y. The corner gusset location is defined by the coordinates x = 0 and y = 0. Figure 4-14 shows the force distribution that results Hʹ = Vʹ = (1 − η) M β + eb M − H ʹβ α M = Pe (4-5) (4-6) (4-7) CL column ec α Specified line of action x e ac L C br P e Gravity line of action Uniform force method control point (typical) Rc ˥ β y Rb eb Specified work point CL beam Gravity axis work point e = ( eb − y ) sin θ − ( ec − x ) cos θ Fig. 4-13. Nonconcentric uniform force method. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 29 where η is the fraction of the moment that is distributed to the beam. η can be estimated at ultimate load by: η= Z beam Z beam + ∑ Z column (4-8) or at service loads by: η= ( L) ( I L) I beam beam ( L) +∑ I (4-9) Note that when the connection is to a column web, η = 1 and all of the moment, M, is in the beam. In this case, the column stiffness or strength is not mobilized because of web distortion. This is the distribution found by Gross (1990) in essentially full scale physical tests. The forces H′ and V′ are superimposed on the UFM forces on the same interface, and the connection is designed for the resultant forces. The beam and column must be checked for ⎛ 1 − η⎞ ηM and ⎜ ⎟ M, respectively. ⎝ 2 ⎠ column (1-η)M 2 ec α P e V' H' H' β V' Specified W.P. H' V' eb Gravity axis W.P. eb + β V' V' H' V' H' H' (1-η)M 2 Fig. 4-14. Extra forces due to nonconcentric work point. 30 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ηM 4.2.3 Reduced Vertical Brace Shear Force in Beam-toColumn Connection—Special Case 2 carry much, if any, of the brace vertical component, because they were not designed for this load. To resolve this problem, some or all of the brace vertical component that the UFM places on the gusset-to-beam interface, and hence on the beam-to-column connection, can be removed by introducing a parameter ΔVb. ΔVb acts at the gusset-to-beam connection centroid, located at a distance α from the beam end, in a direction opposite to Vb as shown in Figure 4-15. Then, the total vertical force on the gusset-to-column interface is Vc + ΔVc , and on the gusset-to-beam interface the force is Vb − ΔVb. In Figure 4-15, the general case of α ≠ α and β ≠ β In many cases, large braces are connected to similarly sized columns, but the beams are essentially struts. This often happens at the outside walls of buildings and elsewhere where gravity loads are small. This is often due to the use of computer analysis to design the members. Computer models for a concentrically braced frame usually have all members meeting at a node. How these members are to be connected to each other is not considered until the connections are designed. Small beams or struts generally cannot CL column P CL brace ΔV b Hc β β Vc Hb Control points eb ΔV b W.P. CL column Vb CL beam α Mb = V b( α - α)+ ΔV b α α (a) CL Vc Hc ΔV b β α CL brace ΔV b Mb R eb W.P. Hc Vb R CL beam Hc Vb ΔV b (b) (c) Fig. 4-15. Special Case 2. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 31 is assumed. The introduction of ΔVb in Figure 4-15 produces a moment equal to ΔVb α on the gusset-to-beam interface; therefore, the total moment there is: Mb = Vb ( α − α ) + ΔVb α (4-10) which is clockwise on the gusset when the brace force, P, is tension and α > α. When ΔVb = Vb, it is referred to as Special Case 2 in the AISC Manual, i.e., no vertical force on the beam-to-column interface due to the brace force, and Equation 4-10 becomes: Mb = Vb α = H b eb β (4-11) Note that the force component Hc still acts as an axial force on the beam-to-column interface. 4.2.4 No Gusset-to-Column Connection— Special Case 3 This case occurs when the brace bevel is shallow, i.e., approximately θ > 60°. Figure 4-16 shows the general case and the admissible force field. Setting β = 0, α = ebtanθ − ec. If α ≠ α, a moment Mb = V(α − α) exists on the beam-togusset interface, which is counterclockwise when α > α, the usual case. Note also that there is a moment on the beam-tocolumn interface, Mbc = Vec , in addition to the vertical brace component V. In this case, it is impossible to have uniform θ Fig. 4-16. Special Case 3. 32 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 4.2.5 forces on all interfaces unless the connection is to a column web, where ec = 0 and thus Mbc = 0. If the brace bevel is steep (i.e., θ < 30°), a similar formulation for the gusset connected only to the column can be developed. Note that the force distribution of Figure 4-16 can be determined less formally by statics alone. α − β(cos γ tan θ − sin γ ) = eb (tan θ − tan γ ) − Nonorthogonal Corner Connections Figure 4-17 shows a generalization of the UFM to nonorthogonal corner connections. These occur when the beam or column is sloping, and also in trusses with a sloping top chord. The angle γ in Figure 4-17 is positive when the angle between the beam and column is less than 90°, and negative otherwise. A design example of this type of connection is presented later in this Design Guide. ec cos γ α + eb tan γ P r Vb = eb P r Hb = Vc = β cos γ P r e ⎞ P ⎛ H c = ⎜ β sin γ + c ⎟ cos γ ⎠ r ⎝ T1 θ γ Q = H c − P cos θ tan γ ec 2 ec ⎞ ⎛ 2 r = ⎜α + eb tan γ + β sin γ + ⎟ + ( eb + β cos γ ) cos γ ⎝ ⎠ P β T1 Control point Hc Rc Hb Vc α W.P. Rb V b eb T3 Control point Vc Hc Control point Vb T2 Vb Q Hb Vb Q T3 T2 Fig. 4-17. Nonorthogonal uniform force method. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 33 4.2.6 P = brace force, kips Ib = moment of inertia of the beam, in.4 Ic = moment of inertia of the column, in.4 A = brace area, in.2 D = subscript denoting distortion b = length of beam to inflection point, in. (assumed at beam midpoint) c = length of column to inflection point, in. (assumed at column midlength) Effect of Frame Distortion As was mentioned earlier, braced frames are usually analyzed as pin-connected members but, obviously, the connections are not pins. This gives rise to additional forces in the connections known as distortional forces. An admissible distribution of distortional forces is presented in Figure 4-18. If an elastic analysis is used, an estimate of the moment in the beam can be determined from Tamboli (2010, pp. 2–72): ⎛ P ⎞ ⎛ Ib Ic MD = 6 ⎜⎜ ⎟⎟ ⎜ ⎝ Abc⎠ ⎜ I b + 2 I c ⎝ b c ⎞ ⎟ ⎟ ⎠ ⎛ b2 + c 2 ⎞ ⎜⎜ ⎟⎟ ⎝ bc ⎠ (4-12) where This formula is limited to bracing arrangements such as those given in Figure 2-1, except those of Figure 2-1(c) at column line D and Figure 2-1(e). Similar formulas can be derived for these two cases and others that occur in practice. Alternatively, a computer-based frame analysis can be used to produce MD for design. CL column CL column ec 12 MD α FD HD VD P VD HD β VD HD W.P. FD HD W.P. HD eb CL beam VD VD HD VD CL beam MD 12 MD Fig. 4-18. Admissible distribution of distortional forces. 34 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 If MD, previously calculated, exceeds the elastic flexural M strength of the beam, or D exceeds the elastic flexural 2 strength of the column, then Equation 4-12 is no longer valid. The plastic moments of the beam or column could be used as an upper bound on MD: { MD = min ϕM p( beam ) , ∑ ϕM p( column ) } (4-13) In either case, the forces of Figure 4-18 are calculated as: HD = MD β + eb (4-14) VD = β HD α (4-15) FD = H D2 + VD2 (4-16) Note that when P is tension, FD is compression, and therefore it is possible for the gusset plate to buckle when the Fig. 4-19. Connection to minimize distortional forces. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 35 brace is in tension. This is called “gusset pinching” and has been observed in physical testing (Lopez et al., 2004). Note also that the distortional forces can be controlled by controlling MD. If an actual physical pin were introduced into the beam as shown in Figure 4-19, the distortional forces would be theoretically reduced to zero. Short of using a pin as shown in Figure 4-19, a shear connection (Figure 4-20) or a reduced beam section (Figure 4-21) can be introduced at this point to control the distortional forces. An idea similar to this has been investigated in recent research at Lehigh (Fahnestock, 2005). A well-thought-out detail to decouple the moment frame, which produces the distortional forces, from the braced frame is given by Walters et al. (2004). Another way to consider distortional forces is to design the connection ignoring them, then determine the maximum distortional forces that can be developed by the existing design. The connection is then checked using the combined original and distortional forces. This method will be demonstrated in this Design Guide, where the distortional forces are controlled via a defined hinge in the beam. Figures 4-18 through 4-21 all illustrate connections to column flanges. When the connection is made to a column web, the web itself distorts and prevents the development of significant distortional forces in the gusset plate. Therefore, HD = 0, VD = 0 and MD = 0. This was shown in physical testing reported by Gross (1990). Typically, distortional forces have been ignored in bracing connection design, just as they have been ignored in truss CL column CL brace W.P. CL beam End plates or angles Fig. 4-20. Shear splice to control distortional forces. 36 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 design, with no known negative consequences for nonseismic and low-seismic (R = 3) applications. However, because of the high drift ratios of about 2 to 2.5%, which produce very high distortional forces, these forces cannot be ignored in high-seismic design (R > 3). When comparing the uniform force method to nonseismic physical results (Gross, 1990), which included distortional forces (frame action), Thornton (1995) showed that the uniform force method produced conservative results. Because of this, and the uncertainties involved in their determination, distortional forces are generally not included in the examples presented in this Design Guide, except for the high-seismic design example of Section 6.1. 4.3 BRACING CONNECTIONS TO COLUMN BASE PLATES A bracing connection to a column base is shown to the column strong axis in Figure 4-22, along with an admissible force field. As with the UFM, the admissible force field of Figure 4-22 produces uniform forces on the gusset edges, as shown. The work point is shown at a distance, e, above the top of the base plate because this position is favored by many designers to keep some or all of the brace above the surface of the finished floor. The ideal location of the work point is at the top surface of the base plate, because this location eliminates bending and shear forces on the column cross section. Fig. 4-21. Reduced beam section (RBS) to control distortional forces. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 37 In the position shown at e above the base plate, the column will be subjected to shear force, Hc, and moment, Hc e, in addition to any other column loads. This force and moment (Hc and Hc e) are probably negligible, but the designer should be aware of them. The work point is often specified to be at the bottom of the base plate, at e = −tbp . This point is very convenient for layout geometry because the base plate thickness is not required to determine bevels (brace inclinations). If e is large, it may be desirable to attach the gusset to the column only. Figure 4-22 shows the gusset connected to a column flange. When the connection is to a column web, ec = 0, and Figure 4-23 results. It is recommended that the work point be located at the top of the base plate (e = 0) whenever possible. It can be seen that if e is not equal to zero, there will be a force, Hc , acting normal to the column web. This force can be split between the optional stiffener of Figure 4-23 with 2Hc added to Hb at the base plate. However, the column web is capable of carrying some or all of Hc , as can be determined by a yield line analysis. Figure 4-24 shows an appropriate yield line pattern. From Abolitz and Warner (1965), the moment that can be carried by the column web, with the gusset pivoting about point A of Figure 4-24, is: Mn = k m p L(4-17) ec CL column V P H Hc θ L V W.P. β e Hc tbp Hb H V Lb Clip Hb = Hc = H (β - e) - Vec β Cut to avoid extension plate if possible Extension plate if required He + Vec β Fig. 4-22. Gusset-to-base plate geometry and admissible force field, strong-axis case. 38 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 where L = depth of gusset plate, in. k = 4 + 2 2 + 6 (L h) + (h L ) m p = (4) Fy t w2 , kips ( for the column ) h = clear distance between web to flange fillets, in. The expression for k is the average of the k values for the web pinned at the flanges and fixed at the flanges. The average is used because at the base plate, the flanges (being welded to the base plate) cannot rotate. At some point above the base plate, the flanges can rotate. The average of these two values is a reasonable approximate value to use in lieu of a more detailed analysis. For most cases, L will be larger than h and k will be larger than 16. Using k = 16: Mn = 16(4)Fy tw2L(4-18) = 4Fy tw2L For an applied moment of Hcβ for ASD, the web will be Fig. 4-23. Gusset-to-base plate geometry and admissible force field, weak-axis case. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 39 Fig. 4-24. Yield line for column web at base plate. 40 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 adequate to carry Hc if: Hc β ≤ or Hc ≤ Mn Ω = 4 Fy t w2 L Ω 2 1 ⎡ 4 Fy t w L ⎤ ⎢ ⎥ Ω⎢ β ⎥ ⎣ ⎦ (4-19) For LRFD, the web will be adequate to carry Hc if: Hcβ ≤ ϕMn = ϕ(4Fy tw2L) or ⎛ 4 Fy t w2 L ⎞ ⎟ Hc ≤ ϕ ⎜ ⎜ β ⎟ ⎝ ⎠ (4-20) If Hc exceeds the ASD strength for ASD formulations or the LRFD required strength for LRFD formulations, the optional stiffener of Figure 4-23 should be used. Note that while the same symbol, Hc , is used for both ASD and LRFD formulations, they are not the same number due to the different load combinations used for ASD versus LRFD Because of web flexibility, the fillet weld of the gusset plate to the base plate at point A of Figure 4-24 will be subjected to a rotation that could cause a weld fracture. To avoid this, the fillet weld can be made large enough to force the gusset to yield before the weld fractures. From the AISC Manual discussion of single-plate connections, the fillet weld between the plate and the support is sized to be s of the plate thickness. This provision can be extended to the gusset plate to column web connection. Alternatively, the optional top stiffener can be used as shown in Figure 4-23. Note that the above analysis is for wide-flange columns. For HSS columns, a k-value for fixed flanges would be appropriate. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 41 42 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Chapter 5 Design Examples The design examples of this section are worked in a generally complete manner and the design approach for each connection type is presented in the example. Design aid tables that are currently printed in the AISC Steel Construction Manual (AISC, 2011), hereafter referred to as the AISC Manual, have been used where possible. Examples 5.1 through 5.4 address a corner bracing connection with the gusset plate connected to the column flange (strong-axis bracing connection). Examples 5.5 through 5.8 address a corner bracing connection with the gusset plate connected to the column web (weak-axis bracing connection). For each configuration, four different uniform force method (UFM) procedures are exemplified: general UFM, Special Case 1, Special Case 2, and Special Case 3. Example 5.9 demonstrates the design of a chevron bracing connection. Nonorthogonal bracing connections, truss connections, and brace-to-column base plate connections are addressed in Examples 5.10, 5.11 and 5.12, respectively. Example 5.1—Corner Connection-to-Column Flange: General Uniform Force Method Given: Design the corner bracing connection shown in Figure 5-1 given the listed members, geometry and loads. The bay width is 25 ft. The connection designed in this problem is shown in Figure 5-1, in completed form (note that the final bolt type changes in the course of the design example). Fig. 5-1. Strong axis bracing connection—general uniform force method. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 43 The required strengths of the brace connection were chosen to force many limit states to be critical. The required strength of the brace-to-gusset connection is: LRFD ASD Pu = 840 kips Pa = 560 kips The transfer force, as shown in the elevation in Figure 5-1, is: LRFD ASD Aub = 100 kips Aab = 66.7 kips The beam shear end reaction is: LRFD ASD Vu = 50.0 kips Va = 33.3 kips Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A572 Grade 50 Fy = 50 ksi Fu = 65 ksi ASTM A36 Fy = 36 ksi Fu = 58 ksi From AISC Manual Tables 1-1, 1-7 and 1-15, the geometric properties are as follows: Beam W21×83 d = 21.4 in. tw = 0.515 in. bf = 8.36 in. tf = 0.835 in. kdes = 1.34 in. Column W14×90 d = 14.0 in. tw = 0.440 in. bf = 14.5 in. tf = 0.710 in. Ix = 999 in.4 k1 = d in. Ix = 1,830 in.4 Brace 2L8×6×1 LLBB Ag = 26.2 in.2 x = 1.65 in. (single angle) Brace-to-Gusset Connection The brace-to-gusset connection should be designed first so that a minimum required size of the gusset plate can be determined. In order to facilitate the connection design, a sketch of the connection should be drawn to scale (Figure 5-2). From this sketch, the important dimensions can be checked graphically (either manually or on a computer) at the same time as they are being calculated analytically. Determine required number of bolts The preliminary design uses d-in.-diameter ASTM A325-X bolts. The calculations begin with the assumption that A325-X bolts will work. However, calculations later in this example will show that the beam-to-column connection requires d-in.-diameter ASTM A490-X bolts. It is not advisable to use different grade bolts of the same diameter, so ASTM A490-X bolts are used for all d-in.-diameter bolts as shown in Figure 5-1. For now, proceed with ASTM A325-X bolts. 44 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 From the AISC Specification for Structural Steel Buildings (AISC, 2010c), hereafter referred to as the AISC Specification, Section J3.6, the available shear strength (in double shear) and available tensile strength are determined using Equation J3-1 and Table J3.2, as follows: LRFD ASD ϕrnv = ϕFn A b = 2 ( 0.75 )( 68 ksi ) ⎡⎢ π ( d in. ) ⎣ = 61.3 kips 2 ϕrnt = ϕFn A b 2 = 0.75 ( 90 ksi ) ⎡⎢ π ( d in. ) 4 ⎤⎥ ⎣ ⎦ = 40.6 kips 4 ⎤⎥ ⎦ rnv Fn Ab = Ω Ω 2 ( 68 ksi ) ⎡⎢ π ( d in. ) ⎣ = 2.00 = 40.9 kips rnt Fn A b = Ω Ω ( 90 ksi ) ⎡⎢⎣ π ( d in.)2 = 2.00 = 27.1 kips 2 4 ⎤⎥ ⎦ 4 ⎤⎥ ⎦ Alternatively, the available shear and tensile strengths can be determined directly from AISC Manual Tables 7-1 and 7-2. The minimum number of d-in.-diameter ASTM A325-X bolts in double shear required to develop the required strength is: Fig. 5-2. Geometry for Example 5.1 calculations. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 45 LRFD Nb = ASD Pu ϕPn Nb = 840 kips 61.3 kips/bolt = 13.7 bolts Pa ( Pn Ω) 560 kips 40.9 kips/bolt = 13.7 bolts = = Use two rows of seven bolts with 3-in. spacing, 3-in. pitch, and 12-in. edge distance as shown in Figure 5-1. Check tensile yielding on the brace gross section From AISC Specification Section D2(a), use Equation D2-1 to determine the available tensile yielding strength of the doubleangle brace: LRFD ASD ϕPn = ϕFy Ag ( = 0.90 ( 36 ksi ) 26.2 in.2 ) = 849 kips > 840 kips o.k. Fy Ag Pn = Ω Ω = ( 36 ksi ) ( 26.2 in.2 ) 1.67 = 565 kips > 560 kips o.k. Alternatively, from AISC Manual Table 5-8, the available tensile yielding strength of the double-angle brace is: LRFD ASD ϕPn = 849 kips > 840 kips o.k. Pn = 565 kips > 560 kips o.k. Ω Check tensile rupture on the brace net section The net area of the double-angle brace is determined in accordance with AISC Specification Section B4.3, with the bolt hole diameter, dh = , in., from AISC Specification Table J3.3: An = Ag − 4t ( dh + z in.) = 26.2 in.2 − 4 (1.00 in. ) (, in. + z in.) = 22.2 in.2 Because the outstanding legs of the double angle are not connected to the brace, an effective net area of the double angle needs to be determined. From AISC Specification Section D3 and Table D3.1, Case 2, the effective net area is: Ae = AnU(Spec. Eq. D3-1) 46 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 where x l l = 6 ( 3.00 in. ) U = 1− = 18.0 in. 1.65 in. 18.0 in. = 0.908 U = 1− Ae = (22.2 in.2)(0.908) = 20.2 in.2 From AISC Specification Section D2(b), the available tensile rupture strength of the double-angle brace is: LRFD ϕPn = ϕFu Ae ASD ( = 0.75 ( 58 ksi ) 20.2 in.2 ) = 879 kips > 840 kips o.k. Pn FA = u e Ω Ω = ( 58 ksi ) ( 20.2 in.2 ) 2.00 = 586 kips > 560 kips o.k. Alternatively, because Ae > 0.75Ag, AISC Manual Table 5-8 could be used conservatively to determine the available tensile rupture strength. The calculated values provide a more precise solution, however. Check block shear rupture on the brace The block shear rupture failure path is assumed as shown in Figure 5-2a. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = 2(1.00 in.)[6(3.00 in.) + 1.50 in.] = 39.0 in.2 0.60Fy Agv = 0.60(36ksi)(39.0 in.2) = 842 kips Shear rupture component: Anv = 39.0 in.2 − 6.5(2)(1.00 in.)(1.00 in.) = 26.0 in.2 0.60Fu Anv = 0.60(58 ksi)(26.0 in.2) = 905 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = 2(1.00 in.)[(3.00 in. + 2.00 in.) − 1.5(d in. + z in. + z in.)] = 7.00 in.2 UbsFu Ant = 1(58 ksi)(7.00 in.2) = 406 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 47 The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 905 kips + 406 kips = 1,310 kips 0.60Fy Agv + UbsFu Ant = 842 kips + 406 kips = 1,250 kips Because 1,310 kips > 1,250 kips, use Rn = 1,250 kips. LRFD ϕRn = 0.75(1,250 kips) = 938 kips > 840 kips o.k. ASD Rn 1, 250 kips = Ω 2.00 = 625 kips > 560 kips o.k. Check block shear rupture on the gusset plate Assume that the gusset plate is 1 in. thick and verify the assumption later. The block shear rupture failure path is assumed as shown in Figure 5-2b. Shear yielding component: Agv = 2(1.00 in.)(19.5 in.) = 39.0 in.2 0.60Fy Agv = 0.60(50 ksi)(39.0 in.2) = 1,170 kips Block shear failure path on double angle Fig. 5-2a. Block shear rupture failure path on double-angle brace. 48 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Shear rupture component: Anv = 39.0 in.2 − 6.5(1.00 in.)(1.00 in.)(2) = 26.0 in.2 0.60Fu Anv = 0.60(65 ksi)(26.0 in.2) = 1,010 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = (1.00 in.)(3.00 in.) − 2(0.5)(1.00 in.)(1.00 in.) = 2.00 in.2 UbsFu Ant = 1(65 ksi)(2.00 in.2) = 130 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 1,010 kips + 130 kips = 1,140 kips 0.60Fy Agv + UbsFu Ant = 1,170 kips + 130 kips = 1,300 kips Therefore, Rn = 1,140 kips. From AISC Specification Section J4.3, the available block shear rupture strength is: LRFD ASD ϕRn = 0.75(1,140 kips) Rn 1,140 kips = Ω 2.00 = 570 kips > 560 kips o.k. = 855 kips > 840 kips o.k. Block shear failure path on gusset plate Fig. 5-2b. Block shear rupture failure path on gusset plate. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 49 Check bolt bearing on the gusset plate Bearing strength at bolt holes on the gusset plate will control over bearing strength at bolt holes on the brace because the brace has two angles for every bolt hole; therefore, only the gusset plate will be checked here. Assume the gusset plate is 1 in. thick. Standard size holes are used in the brace. From AISC Specification Table J3.3, for a d-in.-diameter bolt, dh = , in. According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 3.00 in. − 1.0dh = 3.00 in. − 1.0(, in.) = 2.06 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(2.06 in.)(1.00 in.)(65 ksi) = 121 kips 1.2lctFu/Ω = 1.2(2.06 in.)(1.00 in.)(65 ksi)/ 2.00 = 80.3 kips ϕ2.4dtFu = 0.75(2.4)(d in.)(1.00 in.)(65 ksi) = 102 kips 2.4dtFu/Ω = 2.4 (d in.)(1.00 in.)(65 ksi)/ 2.00 = 68.3 kips Therefore, ϕrn = 102 kips. Therefore, rn/Ω = 68.3 kips. Because the available bolt shear strength determined previously (61.3 kips for LRFD and 40.9 kips for ASD) is less than the bearing strength, the limit state of bolt shear controls the strength of the inner bolts. For the end bolts: lc = 1.50 in. − 0.5dh = 1.50 in. − 0.5(, in.) = 1.03 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(1.03 in.)(1.00 in.)(65 ksi) = 60.3 kips 1.2lctFu/Ω = 1.2(1.03 in.)(1.00 in.)(65 ksi)/ 2.00 = 40.2 kips ϕ2.4dtFu = 0.75(2.4)(d in.)(1.00 in.)(65 ksi) = 102 kips 2.4dtFu/Ω = 2.4 (d in.)(1.00 in.)(65 ksi)/ 2.00 = 68.3 kips Therefore, ϕrn = 60.3 kips. Therefore, rn/Ω = 40.2 kips. For the end bolts, the available bearing strength governs over the available bolt shear strength determined previously. To determine the available strength of the bolt group, sum the individual effective strengths for each bolt. The total available strength of the bolt group is: LRFD ASD ϕRn = (2 bolts)(60.3 kips) + (12 bolts)(61.3 kips) Rn = (2 bolts)(40.2 kips) + (12 bolts)(40.9 kips) Ω = 571 kips > 560 kips o.k. = 856 kips > 840 kips o.k. Therefore, the 1-in.-thick gusset plate is o.k. Additional checks are required as follows. 50 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check the gusset plate for tensile yielding on the Whitmore section From AISC Manual Part 9, the width of the Whitmore section is: lw = 3.00 in. + 2(18.0 in.) tan 30° = 23.8 in. Approximately 4.70 in. of this length runs into the beam web, as shown in Figure 5-2, which is thinner than the gusset. AISC Manual Part 9 states that the Whitmore section may spread across the joint between connected elements, which in this case includes the beam web. Thus, the effective area of the Whitmore section is: Aw = (23.8 in. − 4.70 in.)(1.00 in.) + (4.70 in.)(0.515 in.) = 21.5 in.2 From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate is: LRFD ASD ϕRn = ϕFy Aw = 0.90(50 ksi)(21.5 in. ) 2 = 968 kips > 840 kips o.k. Rn Fy A w = Ω Ω = ( 50 ksi ) ( 21.5 in.2 ) 1.67 = 644 kips > 560 kips o.k. Check the gusset plate for compression buckling on the Whitmore section The available compressive strength of the gusset plate based on the limit state of flexural buckling is determined from AISC Specification Section J4.4, using an effective length factor, K, of 0.50 as established by full scale tests on bracing connections (Gross, 1990). Note that this K value requires the gusset to be supported on both edges. Alternatively, the effective length factor for gusset buckling could be determined according to Dowswell (2006). In this case, because KL/r is found to be less than 25 assuming K = 0.50, the same conclusion, that buckling does not govern, will be reached using either method. r= = tg 12 1.00 in. 12 = 0.289 in. From Figure 5-2, the gusset plate unbraced length along the axis of the brace has been determined graphically to be 9.76 in. KL 0.50 ( 9.76 in. ) = r 0.289 in. = 16.9 Because KL < 25, AISC Specification Equation J4-6 is applicable, and the available compressive strength is: r LRFD ASD ϕPn = ϕFyAg = 0.90(50 ksi)(20.9 in. ) 2 = 941 kips > 840 kips o.k. Pn Fy Ag = Ω Ω = ( 50 ksi ) ( 20.9 in.2) 1.67 = 626 kips > 560 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 51 This completes the brace-to-gusset connection calculations and the gusset plate width and depth size can be determined as shown in Figure 5-1. Squaring off the gusset on top shows that seven rows of bolts will fit. Connection Interface Forces The forces at the gusset-to-beam and gusset-to-column interfaces are determined using the general case of the UFM as discussed in AISC Manual Part 13 and this Design Guide: db 2 21.4 in. = 2 = 10.7 in. dc ec = 2 14.0 in. = 2 = 7.00 in. eb = From Figure 5-2: tan θ = θ 12 118 = 47.2° Choose β = 12 in. Then, the constraint α − βtanθ = ebtanθ − ec from AISC Manual Part 13, with β = β, yields: α = (10.7 in. + 12.0 in.)(1.08) − 7.00 in. = 17.5 in. This can also be done graphically as shown in Figure 5-2, with β = β = 12 in. Start at point a and construct a line through b to intersect the brace line at c. From c, construct a line through point d until the top flange of the beam is intersected at point e. The distance from point e to the face of the column flange is α. The points b, c and d, are known as the “control points” of the uniform force method. Arrange the horizontal edge of the gusset so that α = α and from Figure 5-1: α= lh + w in. + tp 2 where tp is the yet to be determined end-plate thickness, w in. is the gusset corner clip thickness, and lh is the horizontal gusset length. Assume tp = 1 in. and solving for lh: lh = 2(17.5 in.) − 2(1.00 in.) − w in. = 32.3 in. So, the tentative gusset plate size is 1 in. × 242 in. × 2 ft 84 in. The top edge of the gusset could be shaped as shown by the broken line to reduce the excessive (but acceptable) edge distance. Proceeding with α = α = 17.5 in. and β = β = 12 in., there will be no couples on any connection interface. This is the general case of the UFM. 52 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Using the previously determined variables and guidelines from AISC Manual Part 13: r = = ( α + ec ) 2 + ( β + eb ) (Manual Eq. 13-6) 2 (17.5 in. + 7.00 in.) 2 + (12.0 in. + 10.7 in.) = 33.4 in. 2 The required shear force at the gusset-to-column connection and required normal force at the gusset-to-beam connection are determined as follows: Vc = β P r (Manual Eq. 13-2) Vb = eb P r (Manual Eq. 13-4) LRFD β Pu r ⎛ 12.0 in. ⎞ =⎜ ⎟ ( 840 kips ) ⎝ 33.4 in. ⎠ ASD β Pa r ⎛ 12.0 in. ⎞ =⎜ ⎟ ( 560 kips ) ⎝ 33.4 in. ⎠ Vuc = Vac = = 302 kips e Vub = b Pu r ⎛ 10.7 in. ⎞ =⎜ ⎟ ( 840 kips ) ⎝ 33.4 in. ⎠ = 201 kips e Vab = b Pa r ⎛ 560 kips ⎞ = (10.7 in. ) ⎜ ⎟ ⎝ 33.4 in. ⎠ = 179 kips = 269 kips Verify that the sum of the vertical gusset forces equals the vertical component of the brace force: LRFD ASD Σ(Vuc + Vub) = 302 kips + 269 kips = 571 kips Σ(Vac + Vab) = 201 kips + 179 kips = 380 kips Pu(cos θ) Pa(cos θ) = (840 kips)(cos 47.2°) = 571 kips o.k. = (560 kips)(cos 47.2°) = 380 kips o.k. The required normal force at the gusset-to-column connection and required shear force at the gusset-to-beam connection are determined as follows: Hc = ec P r (Manual Eq. 13-3) Hb = α P r (Manual Eq. 13-5) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 53 LRFD ASD ec Pu r ⎛ 7.00 in. ⎞ =⎜ ⎟ ( 840 kips ) ⎝ 33.4 in. ⎠ ec Pa r ⎛ 7.00 in. ⎞ =⎜ ⎟ ( 560 kips ) ⎝ 33.4 in. ⎠ Huc = H ac = = 176 kips = 117 kips α Pu r ⎛ 17.5 in. ⎞ =⎜ ⎟ ( 840 kips ) ⎝ 33.4 in. ⎠ α Pa r ⎛ 17.5 in. ⎞ =⎜ ⎟ ( 560 kips ) ⎝ 33.4 in. ⎠ Hub = H ab = = 440 kips = 293 kips The total horizontal force at the brace-to-gusset connection is: LRFD ASD Σ(Huc + Hub) = 176 kips + 440 kips = 616 kips Σ(Hac + Hab) = 117 kips + 293 kips = 410 kips heck that the sum of the horizontal gusset forces equals C the brace horizontal component heck that the sum of the horizontal gusset forces equals C the brace horizontal component Σ(Huc + Hub) = (840 kips)(sin 47.2°) = 616 kips o.k. Σ(Hac + Hab) = (560 kips)(sin 47.2°) = 411 kips o.k. Figures 5-3a and 5-3b show the free body diagram (admissible force field) determined by the UFM. Gusset-to-Beam Connection Using the same symbols that were used in determining the UFM forces, the required strengths and weld length at the gusset-tobeam interface are: LRFD ASD Required shear strength, Hub = 440 kips Required shear strength, Hab = 293 kips Required normal strength, Vub = 269 kips Required normal strength, Vab = 179 kips Length of weld, l = 324 in. − w in. = 31.5 in. Length of weld, l = 324 in. − w in. = 31.5 in. Check gusset plate for shear yielding and tensile yielding along the beam flange The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1, as follows: 54 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Fig. 5-3a. Admissible force field for Example 5.1—LRFD. Fig. 5-3b. Admissible force field for Example 5.1—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 55 LRFD ASD ϕRn = ϕ0.60Fy Agv = 1.00(0.60)(50 ksi)(1.00 in.)(31.5 in.) = 945 kips > 440 kips o.k. ϕRn = ϕFyAg = 0.90(50 ksi)(1.00 in.)(31.5 in.) = 1,420 kips > 269 kips Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )(1.00 in. )( 31.5 in. ) = 1.50 = 630 kipps > 293 kips o.k. Rn Fy Ag = Ω Ω 50 ( ksi )(1.00 in.)( 31.5 in.) = 1.67 = 943 kips > 179 kips o.k. o.k. Consider force interaction for gusset plate It is usually suggested that the von Mises yield criterion be used for checking interaction; however, in its underlying theory the von Mises criterion requires three stresses at a point and only two stresses (normal and shear) are available. A better choice of interaction check is the equation derived from plasticity theory (Neal, 1977) and suggested by Astaneh-Asl (1998) as: LRFD 2 ASD 4 2 ⎛ Mub ⎞ ⎛ Vub ⎞ ⎛ Hub ⎞ ⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ≤1 ⎝ ϕM n ⎠ ⎝ ϕN n ⎠ ⎝ ϕVn ⎠ From AISC Specification Equation F2-1 From AISC Specification Equation F2-1 ϕM n = ϕM p M n Mp = Ω Ω Fy Z x = Ω = 0.90 Fy Z x ⎛ tg l 2 ⎞ ⎟ = 0.90 Fy ⎜ ⎜⎝ 4 ⎟⎠ ⎡ (1.00 in. )( 31.5 in. ) = 0.90 ( 50 ksi ) ⎢ 4 ⎢ ⎣ = 11, 200 kip-in. 2 Incorporating the previously determined values ⎛ 0 kip-in. ⎞ ⎛ 269 kips ⎞ ⎜ ⎟ +⎜ ⎟ ⎝ 11, 200 kip-in. ⎠ ⎝ 1, 420 kips⎠ 4 ⎡ M ab ⎤ ⎡ Vab ⎤ ⎡ H ab ⎤ ⎢ ⎥+⎢ ⎥ +⎢ ⎥ ≤1 ⎢⎣ ( M n Ω ) ⎥⎦ ⎢⎣ ( N n Ω ) ⎥⎦ ⎢⎣ (Vn Ω ) ⎦ ⎤ ⎥ ⎥ ⎦ 2 ⎛ Fy ⎞ ⎛ t g l ⎞ ⎟ = ⎜⎜ ⎟⎜ ⎝ 1.67 ⎠ ⎜⎝ 4 ⎟⎠ 2 ⎛ 50 ksi ⎞ ⎡⎢ (1.00 in. )( 31.5 in. ) ⎤⎥ =⎜ ⎟ 4 ⎥ ⎝ 1.67 ⎠ ⎢⎣ ⎦ = 7, 430 kip-in. 2 4 ⎛ 440 kips ⎞ +⎜ ⎟ = 0.0829 < 1.0 o.k. ⎝ 945 kips ⎠ Incorporating the previously determined values ⎛ 0 kip-in. ⎞ ⎛ 179 kips ⎞ ⎜ ⎟ +⎜ ⎟ ⎝ 7, 430 kip-in. ⎠ ⎝ 943 kips ⎠ 2 4 ⎛ 293 kips ⎞ +⎜ ⎟ = 0.0828 < 1.0 ⎝ 630 kips ⎠ o.k. Therefore, the 1-in.-thick gusset plate is adequate. Note that interaction of the forces at the gusset-to-beam interface is generally not a concern in gusset design, as can be seen from this result. The AISC Specification and Manual generally do not require that this interaction be checked. 56 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Design weld at gusset-to-beam flange connection The forces involved are: Vub = 269 kips and Vab = 179 kips Hub = 440 kips and Hab = 293 kips The UFM postulates uniform forces on each connection interface. For this to be possible, sufficient ductility must be present in the system. As discussed in Section 1.2, most connection limit states have some ductility, or force redistribution can be accomplished either by support or element flexibility or by increasing the strength of nonductile elements to allow redistribution to occur without premature local fracture. To allow for redistribution of stress in welded gusset connections, the weld is designed for a stress of fpeak or 1.25favg, as defined in the following. The 1.25 factor is the weld ductility factor discussed in the AISC Manual Part 13 (Hewitt and Thornton, 2004), which is actually an overstrength factor, and is intended to allow redistribution to occur before fracture of nonductile elements. The required shear force and normal force per linear inch of weld on the gusset-to-beam flange interface are: LRFD ASD Vub l 269 kips = 31.5 in. = 8.54 kip/in. Hub = l 440 kips = 31.5 in. = 14.0 kip/in. Vab l 179 kips = 31.5 in. = 5.68 kip/in. H = ab l 293 kips = 31.5 in. = 9.30 kip/in. fua = faa = fuv fav Use a vector sum (square root of the sum of the squares) to combine the shear, axial and bending stresses on the gusset-to-beam interface. The peak and average stresses are: f peak = favg = ( fa + fb )2 + fv2 1⎡ 2 ⎣⎢ ( fa − fb )2 + fv2 + ( fa + fb )2 + fv2 ⎤⎥ ⎦ Because fb, the bending stress, is zero in this case, the average stress on the gusset-to-beam flange interface is: LRFD fu avg = fu peak = ASD fa avg = fa peak (8.54 kip/in. )2 + (14.0 kip/in.)2 = 16.4 kip/in. The resultant load angle is: ⎛ 8.54 kip/in. ⎞ θ = tan −1 ⎜ ⎟ ⎝ 14.0 kip/in. ⎠ = 31.4° = ( 5.68 kip/in. )2 + ( 9.30 kip/in.)2 = 10.9 kip/in. The resultant load angle is: ⎛ 5.68 kip/in. ⎞ θ = tan −1 ⎜ ⎟ ⎝ 9.30 kip/in. ⎠ = 31.4° AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 57 According to AISC Manual Part 13, because the gusset is directly welded to the beam, the weld is designed for the larger of the peak stress and 1.25 times the average stress: LRFD ASD ( fu weld = max 1.25 fu avg , fu peak = 1.25 (16.4 kip/in. ) ) ( fa weld = max 1.25 fa avg , fa peak = 1.25 (10.9 kip/in. ) = 20.5 kip/in. ) = 13.6 kip/in. The strength of fillet welds defined in AISC Specification Section J2.4 can be simplified, as explained in AISC Manual Part 8, to AISC Manual Equations 8-2a and 8-2b. Also, incorporating the increased strength due to the load angle from AISC Specification Equation J2-5, the required weld size is: LRFD D= = ASD fu weld ( 2 (1.392 kip/in. ) 1.0 + 0.50sin θ 1.5 ) 20.5 kip/in. ( D= 2 (1.392 kip/in. ) 1.0 + 0.50sin 31.4° 1.5 = ) = 6.20 sixteenths fa weld ( 2 ( 0.928 kip/in. ) 1.0 + 0.50sin1.5 θ ) 13.6 kip/in. ( 2 ( 0.928 kip/in. ) 1.0 + 0.50sin1.5 31.4° ) = 6.17 sixteenths From AISC Specification Table J2.4, the minimum fillet weld required is c in., which does not control in this case. Therefore, use a two-sided v-in. fillet weld to connect the gusset plate to the beam. Check beam web local yielding The normal force is applied at 16.8 in. away from the beam end (assuming a w-in.-thick end plate), which is less than the beam depth of 21.4 in. Therefore, use AISC Specification Equation J10-3 as follows: LRFD ϕRn = ϕFywtw(2.5k + lb) = 1.00(50 ksi)(0.515 in.)[2.5(1.34 in.) + 31.5 in.] = 897 kips > 269 kips o.k. ASD Rn Ω = = ( Fyw tw 2.5k + l b ) Ω 50 ksi ( )( 0.515 in.) ⎣⎡2.5 (1.34 in.) + 31.5 in.⎤⎦ 1.50 = 598 kips > 179 kips o.k. Alternatively, AISC Manual Table 9-4 may be used to determine the available strength of the beam due to web local yielding. When the compressive force is applied at a distance less than the beam depth, use AISC Manual Equation 9-45 as follows: LRFD ϕRn = ϕR1 + lb(ϕR2) = 86.3 kips + (31.5 in.)(25.8 kip/in.) = 899 kips > 269 kips o.k. ASD Rn R1 ⎛R ⎞ = + lb ⎜ 2 ⎟ Ω Ω ⎝ Ω⎠ = 57.5 kips + ( 31.5 in. )(17.2 kip/in. ) = 599 kips > 179 kips The differences in the calculated strength versus the table values are due to rounding. 58 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 o.k. Check equivalent normal force, Ne There is no moment on the gusset-to-beam connection, so there is no additional normal force due to the moment. If there were a moment on the gusset-to-beam connection, the following calculation could be used to determine the equivalent normal force: LRFD Nue = Vub + ASD 2 Mub ⎛l⎞ ⎜2⎟ ⎝ ⎠ = 269 kips + N ae = Vab + 2 ( 0 kip-in. ) 2 M ab ⎛l⎞ ⎜2⎟ ⎝ ⎠ = 179 kips + ⎛ 31.5 in. ⎞ ⎜ 2 ⎟ ⎝ ⎠ = 269 kips 2 ( 0 kip-in. ) ⎛ 31.5 in. ⎞ ⎜ 2 ⎟ ⎝ ⎠ = 179 kips Check beam web local crippling The normal force is applied at 16.8 in. away from the beam end (assuming a w-in.-thick end plate), which is greater than d/ 2. Therefore, AISC Specification Equation J10-4 is applicable: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝ d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜ tf ⎟ ⎝ ⎠ ASD 1.5 ⎤ ⎥ ⎥ ⎦ EFyw t f tw 0.80t w2 Rn = Ω 1.5 ⎡ ⎛ 31.5 in. ⎞ ⎛ 0.515 in. ⎞ ⎤ 2 ϕRn = 0.75 ( 0.80 )( 0.515 in. ) ⎢1 + 3 ⎜ ⎟ ⎥ ⎟⎜ ⎢⎣ ⎝ 21.4 in.⎠ ⎝ 0.835 in. ⎠ ⎥⎦ × ( 29, 000 ksi )( 50 ksi )( 0.835 in. ) 0.515 in. Rn = Ω = 766 kips > 269 kips o.k. 1.5 ⎡ ⎛ tw ⎞ ⎤ EFyw t f l ⎛ ⎞ b ⎢1 + 3 ⎥ ⎜ d ⎟ ⎜⎜ t ⎟⎟ ⎥ ⎢ tw ⎝ ⎠⎝ f ⎠ ⎣ ⎦ Ω 1.5 ⎡ ⎡ ⎞ ⎤⎤ ⎛ ⎞⎛ ⎢( 0.80 )( 0.515 in. )2 ⎢1 + 3 ⎜ 31.5 in. ⎟ ⎜ 0.515 in. ⎟ ⎥ ⎥ ⎢ ⎢⎣ ⎝ 21.4 in. ⎠ ⎝ 0.835 inn. ⎠ ⎥⎦ ⎥ ⎢ ⎥ ⎢ ⎥ 29, 000 ksi )( 50 ksi )( 0.835 in. ) ( ⎢ ⎥ × 0.515 in. ⎢⎣ ⎥⎦ 2.00 = 511 kips > 179 kips o.k. Alternatively, AISC Manual Table 9-4 and Equation 9-49 (for x ≥ d/ 2) may be used to determine the available strength of the beam for web local crippling: LRFD ASD ϕRn = 2 ⎡⎣( ϕR3 ) + lb ( ϕR4 ) ⎤⎦ = 2 ⎡⎣122 kips + ( 31.5 in. )( 8.28 kip/in. ) ⎤⎦ = 766 kips > 269 kips o.k. Rn = 2 ⎡⎣( R3 Ω ) + lb ( R4 Ω ) ⎤⎦ Ω = 2 ⎡⎣81.3 kips+ ( 31.5 in. )( 5.52 kip/in. ) ⎤⎦ = 510 kips > 179 kips o.k. The differences in the calculated strength versus the table values are due to rounding. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 59 Gusset-to-Column Connection The required strengths at the gusset-to-column interface are: LRFD ASD Required normal strength, Huc = 176 kips Required axial strength, H ac = 117 kips Required shear strength,Vuc = 302 kips Required shear strength, Vac = 201 kips Use an end-plate connection between the gusset plate and the column flange and the beam and the column flange, where the end plate is continuous. Design bolts at gusset-to-column connection Use d-in.-diameter ASTM A325-X bolts in standard holes for preliminary analysis. If this bolt type proves to be insufficient, use d-in.-diameter ASTM A490-X bolts with standard holes. Note that if A490 d-in.-diameter bolts are used for this part of the connection, they should be used everywhere in this connection and on this job. It is not good practice to use different grades of the same size bolts on a specific job. For preliminary design, assume two rows, seven bolts per row, in the gusset-to-column portion of the connection. The available shear and tensile strengths per bolt, from AISC Manual Tables 7-1 and 7-2, and the required shear and tensile strengths per bolt are: LRFD ϕrnv = 30.7 kips ϕrnt = 40.6 kips Required shear strength per bolt 302 kips 14 bolts = 21.6 kips/bolt < 30.7 kips o.k. ruv = Required tensile strength per bolt rut = 176 kips = 12.6 kips < 40.6 kips o.k. 14 ASD rnv = 20.4 kips Ω rnt = 27.1 kips Ω Required shear strength per bolt 201 kips 14 bolts = 14.4 kips/bolt < 20.4 kips o.k. rav = Required tensile strength per bolt rat = 117 kips = 8.36 kips < 27.1 kips o.k. 14 The tensile strength requires an additional check due to the combination of tension and shear. From AISC Specification Section J3.7, the available tensile strength of the bolts subject to combined tension and shear is: LRFD ASD Interaction of shear and tension on bolts is given by Interaction of shearon and tension on bolts Interaction of shear and tension on bolts is given by Interaction of shear and tension bolts is given by is given by AISC Specification J3-3aEquation J3-3a AISC Specification J3-3b AISC Equation Specification AISC SEquation pecification Equation J3-3b Fntʹ = 1.3Fnt − Fnt frv ≤ Fnt ϕFnv Note that the AISC Specification Commentary Section J3 has an alternative elliptical formula, Equation C-J3-8a, that can be used. Fntʹ = 1.3Fnt − ΩFnt frv ≤ Fnt Fnv Note that the AISC Specification Commentary Section J3 has an alternative elliptical formula, Equation C-J3-8b, that can be used. 60 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD From AISC Specification Table J3.2 Fnt = 90 ksi Fnv = 68 ksi Fntʹ = 1.3 ( 90 ksi ) − ASD From AISC Specification Table J3.2 ⎛ 21.6 kips ⎞ 90 ksi ⎜ ⎟ 0.75 ( 68 ksi ) ⎜⎝ 0.601 in.2 ⎟⎠ Fnt = 90 ksi Fnv = 68 ksi 2.00 ( 90 ksi ) ⎛ 14.4 kips ⎞ ⎜⎜ 2⎟ ⎟ 68 ksi ⎝ 0.601 in. ⎠ = 53.6 ksi < 90 ksi o.k. Fntʹ = 1.3 ( 90 ksi ) − = 53.6 ksi < 90 ksi o.k. he reduced available tensile strength per bolt due to comT bined forces is: ϕrnt = ϕFntʹ Ab ( = 0.75 ( 53.6 kips ) 0.601 in. 2 he reduced available tensile strength per bolt due to comT bined forces is: rnt ) Ω = 24.2 kips > 12.6 kips o.k. = = Fntʹ Ab Ω ( 53.6 kips ) ( 0.601 in.2) 2.00 = 16.1 kips > 8.36 kips o.k. A 1-in.-end-plate thickness, tp, has been assumed. The gage of the bolts is taken as 52 in., which is the standard gage for a W14×90 and will work with a 1-in.-thick gusset plate and assuming 2-in. fillet welds. From Figure 5-4a, the clear distance from the center of the bolt hole to the toe of the fillet is 1w in. From AISC Manual Table 7-15, the entering clearance for aligned d-in.diameter ASTM A325 and A490 bolts is d in. < 1w in. The next step is to determine the fillet weld size of the end plate-to-gusset plate connection to ensure that the 52-in. gage will work. Design gusset-to-end plate weld The resultant load, R, and its angle with the vertical, θ, are: LRFD Ru = (176 kips ) 2 ASD + ( 302 kips ) 2 Ra = = 350 kips (117 kips )2 + ( 201 kips )2 = 233 kips ⎛ 176 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 302 kips ⎠ = 30.2° ⎛ 117 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 201 kips ⎠ = 30.2° 2" 1" " 1w"" Gusset plate End plate 5 2" Fig. 5-4a. Washer clearance for end-plate bolts. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 61 The weld size is determined from AISC Manual Equation 8-2, including the increase in strength allowed by AISC Specification Section J2.4 when the angle of loading is not along the weld longitudinal axis. In the following, only the weld length tributary to the group of bolts is used; therefore, l = 7(3.0 in.) = 21.0 in. LRFD Rearranging AISC Manual Equation 8-2a, and incorporating AISC Specification Equation J2-5, the weld size required is: 350 kips Dreq ’d = 2(1.392 kip/in.)(21.0 in.)(1.0 + 0.50 sin1.5 30.22°) = 5.08 ASD Rearranging AISC Manual Equation 8-2b, and incorporating AISC Specification Equation J2-5, the weld size required is: 233 kips Dreq ’d = 2(0.928 kip/in.)(21.0 in.)(1.0 + 0.50 sin1.5 30.22°) = 5.07 Therefore, use a two-sided a-in. fillet weld at the gusset-to-end plate connection. Note that this exceeds the minimum size fillet weld of c in. required by AISC Specification Table J2.4. Note that in the equation for Dreq’d, no ductility factor has been used. This is because the flexibility of the end plate will allow nonuniform gusset edge forces to be redistributed. If the edge force is high at one point, the flexible end plate will deform and shed load to less highly loaded locations. Since the required fillet weld of a-in. is less than the assumed 2-in. fillet weld in Figure 5-4a, the assumed 52-in. gage can be used. Check gusset plate tensile and shear yielding at the gusset-to-end-plate interface The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1 as follows: LRFD ASD ϕVn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 50 ksi )(1.00 in. )( 23.8 in. ) = 714 kips > 302 kips o.k. ϕNn = ϕFy Ag = 0.90 ( 50 ksi )(1.00 in. )( 23.8 in. ) = 1, 070 kips > 176 kips o.k. Vn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )(1.00 in. )( 23.8 in. ) = 1.50 = 476 kipps > 201 kips o.k. N n Fy Ag = Ω Ω ( 50 ksi )(1.00 in.)( 23.8 in.) = 1.67 = 713 kips > 117 kips o.k. Check prying action on bolts at the end plate Using AISC Manual Part 9, determine the effect of prying action on the end-plate connection. The end plate is initially assumed to be 1 in. thick × 10 in. wide. The final end plate thickness may be different. From AISC Manual Part 9 and Figure 5-4a: 52 in. − 1.00 in. 2 = 2.25 in. b = bʹ = b − db 2 = 2.25 in. − = 1.81 in. (Manual Eq. 9-21) d in. 2 62 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 10.0 in. − 52 in. 2 = 2.25 in. a = d ⎛ aʹ = ⎜ a + b 2 ⎝ db ⎞ ⎞ ⎛ ⎟ ≤ ⎜ 1.25b + ⎟ 2 ⎠ ⎠ ⎝ d in.⎞ ⎡ d in. ⎤ ⎛ = ⎜ 2.25 in. + ⎟ ≤ ⎢1.25 ( 2.25 in. ) + 2 ⎥ 2 ⎠ ⎣ ⎝ ⎦ o.k. = 2.69 in. < 3.25 in. (Manual Eq. 9-27) From AISC Specification Table J3.3 for a d-in.-diameter bolt, the standard hole dimension is , in. bʹ aʹ 1.81 in. = 2.69 in. = 0.673 ρ = (Manual Eq. 9-26) p = pitch of bolts = 3 in. δ = 1− dʹ p (Manual Eq. 9-24) , in. 3.00 in. = 0.688 = 1− From AISC Manual Part 9, the available tensile strength, including prying action effects, is: LRFD From AISC Manual Equation 9-30a, with B = ϕrnt = 24.2 kips previously determined: tc = = 4 Bbʹ ϕpFu From AISC Manual Equation 9-30b, with B = rnt/Ω = 16.1 kips previously determined: tc = 4 ( 24.2 kips )(1.81 in. ) 0.90 ( 3.00 in. )( 65 ksi ) = 0.999 in. αʹ = ASD = Ω4 Bbʹ pFu 1.67 ( 4 )(16.1 kips )(1.81 in. ) ( 3.00 in.)( 65 ksi ) = 0.999 in. ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.999 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ 1.00 in. ⎠ ⎥⎦ ⎣ = 0.00 Because 0 ≤ αʹ ≤ 1, determine Q from AISC Manual Equation 9-33 = 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δαʹ ) ⎝ tc ⎠ 2 αʹ = ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.999 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ 1.00 in. ⎠ ⎥⎦ ⎣ = 0.00 Because 0 ≤ αʹ ≤ 1, determine Q from AISC Manual Equation 9-33 = 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δαʹ ) ⎝ tc ⎠ 2 ⎛ 1.00 in. ⎞AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS ⎛ 1.00 in. ⎞ ⎡⎣1 + 0.688 ( 0.00 ) ⎤⎦ =⎜ =⎜ )⎤⎦ DESIGN / 63 ⎟ ⎟ ⎡⎣1 + 0.688 ( 0.00AND ⎝ 0.999 in. ⎠ ⎝ 0.999 in. ⎠ = 1.00 = 1.00 αʹ = ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.999 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ 1.00 in. ⎠ ⎥⎦ ⎣ = 0.00 LRFD αʹ ≤ 1, determine Q BecauseBecause 0 ≤ α′ ≤ 01, ≤determine Q from AISC Manual Equationfrom 9-33:AISC Manual Equation 9-33 = 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δαʹ ) ⎝ tc ⎠ 2 ⎛ 1.00 in. ⎞ =⎜ ⎟ ⎡⎣1 + 0.688 ( 0.00 ) ⎤⎦ ⎝ 0.999 in. ⎠ = 1.00 From AISC Manual Equation 9-31: αʹ = ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.999 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ 1.00 in. ⎠ ⎥⎦ ⎣ = 0.00 ASD ʹ Because 0 α 1 , determine Q ≤ ≤ Because 0 ≤ α′ ≤ 1, determine Q from AISC Manual Equationfrom 9-33: AISC Manual Equation 9-33 = 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δαʹ ) ⎝ tc ⎠ 2 ⎛ 1.00 in. ⎞ =⎜ ⎟ ⎡⎣1 + 0.688 ( 0.00 ) ⎤⎦ ⎝ 0.999 in. ⎠ = 1.00 From AISC Manual Equation 9-31: Tavail = BQ Tavail = BQ = ( 24.2 kips )(1.00 ) = (16.1 kips )(1.00 ) = 24.2 kips > 12.6 kips o.k. = 16.1 kips > 8.36 kips o.k. Physically, note that tc is the plate thickness required to develop the bolt strength, B, with no prying action, i.e., q = 0. It is, therefore, the maximum effective plate thickness. A thicker plate would not increase the connection’s strength. From this we can see that the 1-in.-thick end plate is unnecessarily thick. A thinner plate will allow some prying force to develop, resulting in double curvature in the plate. Try a s-in.-thick end plate. From AISC Manual Part 9, the available bolt tensile strength, including the effects of prying action, is: LRFD ⎡⎛ tc ⎞ ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ASD 2 αʹ = αʹ = ⎡⎛ tc ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.999 in. ⎞2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ = 1.35 ⎡⎛ 0.999 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ = 1.35 Because α′ > 1, determine Q from AISC Manual Equation 9-34 Because α′ > 1, determine Q from AISC Manual Equation 9-34 = 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎛ s in. ⎞ =⎜ ⎟ (1 + 0.688 ) ⎝ 0.999 in. ⎠ = 0.661 Tavail = BQ = ( 24.2 kips )( 0.661) = 16.0 kips > 12.6 kips o.k. = ⎛ t⎞ Q =⎜ ⎟ ⎝ tc ⎠ 2 (1 + δ ) 2 ⎛ s in. ⎞ =⎜ ⎟ (1 + 0.688 ) ⎝ 0.999 in. ⎠ = 0.661 Tavail = BQ = (16.1 kips )( 0.661) = 10.6 kips > 8.36 kips o.k. Use a s-in.-thick end plate. Now that an end-plate thickness is known, check the end plate for the remaining limit states. Note that the beam-to-column portion of the end-plate connection may require a thicker end plate. 64 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check bolt bearing at bolt holes on end plate Check the bearing strength at the top bolts, assuming the optional cut on the gusset plate occurs and the end plate edge distance correspondingly decreases to 1w in., as the available bearing strength will have the lowest value at that location. This edge distance is reflected in Figure 5-4b. The clear distance, based on this edge distance is: lc = 1w in. − 0.5(, in.) = 1.28 in. Assuming that deformation at the bolt hole is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) The available strength is determined as follows: LRFD ASD ϕ1.2lctFu = 0.75(1.2)(1.28 in.)(s in.)(65 ksi) = 46.8 kips/bolt 1.2lctFu/Ω = 1.2(1.28 in.)(s in.)(65 ksi)/ 2.00 = 31.2 kips/bolt ϕ2.4dtFu = 0.75(2.4)(d in.)(s in.)(65 ksi) = 64.0 kips/bolt 2.4dtFu/Ω = 2.4(d in.)(s in.)(65 ksi)/ 2.00 = 42.7 kips/bolt Therefore, ϕrn = 46.8 kips/bolt Therefore, rn/Ω = 31.2 kips/bolt Because 46.8 kips/bolt > 30.7 kips/bolt, LRFD (31.2 kips/bolt > 20.4 kips/bolt, ASD), bolt shear controls over bearing strength at bolt holes, and this check will not govern. Check block shear rupture of the end plate The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = (18.0 in. + 1w in.)(s in.) = 12.3 in.2 0.60Fy Agv = 0.60(50 ksi)(12.3 in.2) = 369 kips Shear rupture component: Anv = 12.3 in.2 − 6.5(, in. + z in.)(s in.) = 8.24 in.2 0.60Fu Anv = 0.60(65 ksi)(8.24 in.2) = 321 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = [2.25 in. − 0.5(, in. + z in.)](s in.) = 1.09 in.2 UbsFu Ant = 1(65 ksi)(1.09 in.2) = 70.9 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 65 The available strength for the limit state of block shear rupture is determined as follows: 0.60Fu Anv + UbsFu Ant = 321 kips + 70.9 kips = 392 kips 0.60Fy Agv + UbsFu Ant = 369 kips + 70.9 kips = 440 kips Therefore, Rn = 392 kips. LRFD ASD ϕRn = 0.75 ( 392 kips )( 2 ) = 588 kips > 302 kips Rn ( 392 kips )( 2 ) = Ω 2.00 = 392 kips > 201 kips o.k. o.k. Check prying action on column flange From AISC Manual Part 9: 52 in. − 0.440 in. 2 = 2.53 in. b = bʹ = b − db 2 = 2.53 in. − (Manual Eq. 9-21) d in. 2 = 2.09 in. 14.5 in. − 5.50 in. a = 2 = 4.50 in. 10" 3'-11d"" 5 @ 3" 3" 3" 6 @ 3" Optional cut Bolts: d" dia. A490-X Holes: std. ," dia. PLw"× 10"× 3'-11d" Fig. 5-4b. End-plate geometry. 66 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The fulcrum point for prying is controlled by the end plate with a = 2.25 in., so use this value for a. d ⎞ d ⎞ ⎛ ⎛ a ʹ = ⎜ a + b ⎟ ≤ ⎜1.25b + b ⎟ 2⎠ ⎝ 2⎠ ⎝ d in. ⎡ d in.⎤ = 2.25 in. + ≤ ⎢1.25 ( 2.53 in. ) + 2 2 ⎥⎦ ⎣ = 2.69 in. < 3.60 in. (Manual Eq. 9-27) bʹ aʹ 2.09 in. = 2.69 in. = 0.777 ρ= (Manual Eq. 9-26) p = pitch of bolts = 3.00 in. Using p = 3.00 in. assumes that the column flange is cut above and below the bolt group, which is conservative. δ = 1− dʹ p (Manual Eq. 9-24) , in. 3.00 in. = 0.688 = 1− From AISC Manual Part 9, the available tensile strength including prying action effects is determined as follows: LRFD From AISC Manual Equation 9-30a: tc = = 4 Bbʹ ϕpFu tc = 4 ( 24.2 kips )( 2.09 in. ) 0.90 ( 3.00 in. )( 65 ksi ) = 1.07 inn. αʹ = ASD From AISC Manual Equation 9-30b: = Ω4 Bbʹ pFu 1.67 ( 4 )(16.1 kips )( 2.09 in. ) ( 3.00 in.)( 65 ksi ) = 1.07 in. ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.07 in. ⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.688 (1 + 0.777 ) ⎢⎝ 0.710 in. ⎠ ⎥⎦ ⎣ = 1.04 αʹ = ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.07 in. ⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.688 (1 + 0.777 ) ⎢⎝ 0.710 in. ⎠ ⎥⎦ ⎣ = 1.04 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 67 LRFD Because α′ > 1, determine Q from AISC Manual Equation 9-34: 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎛ 0.710 in. ⎞ =⎜ ⎟ (1 + 0.688 ) ⎝ 1.07 in. ⎠ = 0.743 From AISC Manual Equation 9-31: Tavail = BQ = ( 24.2 kips )( 0.743 ) = 18.0 kips > 12.6 kips o.k. ASD Because α′ > 1, determine Q from AISC Manual Equation 9-34: 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎛ 0.710 in. ⎞ =⎜ ⎟ (1 + 0.688 ) ⎝ 1.07 in. ⎠ = 0.743 From AISC Manual Equation 9-31: Tavail = BQ = (16.1 kips )( 0.743 ) = 12.0 kips > 8.36 kips o.k. If this method were to fail, a more realistic model is given in Tamboli (2010). The effective tributary length of column flange for a continuous column is: peff = n p b a peff ( n − 1) p + πb + 2a n = number of rows of bolts =7 = bolt pitch = 3.00 in. =b = 2.53 in. b f − gage = 2 14.5 in. − 52 in. = 2 = 4.50 in. ( 7 − 1)( 3.00 in. ) + π ( 2.53 in.) + 2 ( 4.50 in.) = 7 = 4.99 in. Using peff in place of p in the AISC Manual prying equations and following the procedure from AISC Manual Part 9: bʹ = 2.09 in. a = 4.50 in. > a = 2.25 in. a ʹ = 2.69 in. Use the smaller of a and a. ρ = 0.777 d δ = 1− h p , in. = 1− 4.99 in. = 0.812 68 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD tc = = ASD 4 Bb′ ϕpFu tc = 4 ( 24.2 kips )( 2.09 in. ) 0.90 ( 4.99 in. )( 65 ksi ) = 0.832 in. = 4Ω Bb′ pFu 4 (1.67 )(16.1 kips )( 2.09 in. ) ( 4.99 in.)( 65 ksi ) = 0.832 in. ⎡⎛ tc ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ α′ = δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.832 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.812 (1 + 0.777 ) ⎢⎝ 0.710 in. ⎠ ⎥⎦ ⎣ = 0.259 = Because 0 ≤ α′ ≤ 1, determine Q from AISC Manual Equation 9-33 αʹ = ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 0.832 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.812 (1 + 0.777 ) ⎢⎝ 0.710 in. ⎠ ⎥⎦ ⎣ = 0.259 = Because 0 ≤ α′ ≤ 1, determine Q from AISC Manual Equation 9-33 2 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δα′) ⎝ tc ⎠ ⎛ t⎞ Q = ⎜ ⎟ (1 + δα′ ) ⎝ tc ⎠ 2 ⎛ 0.710 ⎞ =⎜ ⎟ ⎡⎣1 + ( 0.812 )( 0.259 ) ⎤⎦ ⎝ 0.832 ⎠ = 0.881 Tavail = BQ = ( 24.2 kips ) ( 0.881) 2 ⎛ 0.710 in. ⎞ =⎜ ⎟ ⎡⎣1 + ( 0.812 )( 0.259 ) ⎤⎦ ⎝ 0.832 in. ⎠ = 0.881 Tavail = BQ = (16.1 kips )( 0.881) = 21.3 kips > 12.6 kips o.k. = 14.2 kips > 8.36 kips o.k. By this method the calculated available strength has increased by: LRFD ( 21.3 kips − 18.0 kips ) 18.0 kips ASD × 100% = 18.3% (14.2 kips − 12.0 kips ) 12.0 kips × 100% = 18.3% Thus, this method is considerably less conservative than the “cut column” model. Check bearing on column flange Because tf = 0.710 in. > tp = s in., bolt bearing on the column flange will not control the design, by inspection. Beam-to-Column Connection This section addresses the design of the portion of the end plate that connects the beam to the column. The required shear strength is equal to Vub (LRFD) or Vab (ASD), based on the brace force, plus the beam reaction from the Given section of 50 kips (LRFD) or 33.3 kips (ASD). The Vub and Vab force is reversible as the brace force goes from tension to compression, but the gravity beam shear always remains in the same direction. Therefore, Vub (or Vab) and the reaction should always be added even if shown in opposite directions as in Figure 5-3. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 69 The axial force in the beam shown in Figure 5-3 is affected by the transfer force, Aub, of 100 kips (LRFD) or Aab of 66.7 kips (ASD). When the braces are in tension, the transfer force is compression as shown in Figure 5-1, and will increase the required brace tensile strength of the next lower brace to: LRFD ASD 716 kips = 976 kips sin 47.2° 477 kips = 650 kips sin 47.2° Figure 5-3 shows that the axial force between the beam and column from equilibrium would be Aub + Huc for LRFD and Aab + Hac for ASD, but it has been shown that frame action, as discussed in Section 4.2.6, will reduce the force Huc for LRFD or Hac for ASD. Section 4.2.6 provides the following formula to estimate the moment in the beam due to the effect of the admissible distortional forces from Tamboli (2010): ⎛ P ⎞ ⎛ Ib Ic MD = 6 ⎜ ⎟⎜ ⎝ Abc ⎠ ⎜ I b + 2 I c ⎜ c ⎝ b ⎞ ⎛ b2 + c 2 ⎞ ⎟ ⎜⎜ ⎟⎟ ⎟⎟ ⎝ bc ⎠ ⎠ (4-12) This formula is valid for bracing arrangements such as those shown in Figure 2-1, i.e., those involving one beam and two columns. Other arrangements would involve a different formula. For cantilever situations, gravity forces rather than lateral forces will probably dominate and Aub + Huc for LRFD or Aab + Hac for ASD should be used. As given, the required strength of the brace is: LRFD ASD Pu = 840 kips Pa = 560 kips The values of b and c to be used in the Tamboli equation are: b = = lb 2 ( 25.0 ft )(12.0 in. 1.00 ft ) 2 = 150 in. c = = lc 2 ⎛ 118 in.⎞ ⎛ 12.0 in. ⎞ ⎟⎜ ⎟ ⎝ 12.0 in. ⎠ ⎝ 1.00 ft ⎠ 2 ( 25.0 ft ) ⎜ = 139 in. Ib b = 1, 830 in.4 150 in. = 12.2 in.3 ( 4 2 I c 2 999 in. = 139 in. c ) = 14.4 in.3 70 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The required flexural strength of the beam is: LRFD ASD ⎡ ⎤ 840 kips ⎥ M uD = 6 ⎢ ⎢ 26.2 in.2 (150 in. )(139 in. ) ⎥ ⎣ ⎦ ( ) ⎡ (1, 830 in. ) ( 999 in. ) ⎤ ⎡ 150 in. ) ⎥ ⎢( ×⎢ 4 4 ⎢ 12.2 in.3 + 14.4 in.3 ⎣ ⎛ ⎞ 560 kips ⎟ M aD = 6 ⎜ ⎜ 26.2 in.2 (150 in. )(139 in. ) ⎟ ⎝ ⎠ 2 + (139 in. ) ⎤ ⎥ ⎥ ⎢ (150 in. )(139 in. ) ⎥ ⎦ ⎦ ⎣ = 4 2 = 1, 270 kip-in. Using previously determined variables, β and eb: HuD = ( ) ⎡(1, 830 in. ) ( 999 in. ) ⎤ ⎡ 150 in. ) ⎥ ⎢( ×⎢ MuD β + eb ⎢ 12.2 in.3 + 14.4 in.3 ⎣ 2 + (139 in. ) ⎤ ⎥ ⎥ ⎢ (150 in. )(139 in. ) ⎥ ⎦ ⎦⎣ 2 = 848 kip-in. Using previously determined variables, β and eb: HaD = 1, 270 kip-in. 12.0 in. + 10.7 in. 4 = M aD β + eb 848 kip-in. 12.0 in. + 10.7 in. = 37.4 kips = 55.9 kips If this value of HD is used, the axial force between the beam and column will be: LRFD Hu = 176 kips − 55.9 kips + 100 kips = 220 kips ASD Ha = 117 kips − 37.4 kips + 66.7 kips = 146 kips A recent study (Fortney and Thornton, 2014) has shown that the average ratio of (Hc − Hd + A)/(Hc + A) is about 70%, but the standard deviation of about 0.33 was too large to make a recommendation for design. This study also shows that using max (Hc, A) as the axial force may not be justified. This Design Guide will use the justifiable axial force of Hc − Hd + A for the following calculations: LRFD Required Shear Strength Vu = Vub + Reaction = 269 kips + 50 kips = 319 kips ASD Required Shear Strength Va = Vab + Reaction = 179 kips + 33.3 kips = 212 kips Required Axial Strength Tu = 176 kips − 55.9 kips + 100 kips = 220 kips Required Axial Strength Ta = 117 kips − 37.3 kips + 66.7 kips = 146 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 71 Design bolts at beam-to-column connection For the connection geometry, see Figure 5-1. Try (12) d-in.-diameter ASTM A325-X bolts in the end-plate connection of the beam to the column. As determined previously, the available shear strength of a d-in.-diameter ASTM A325-X bolt is 30.7 kips for LRFD and 20.4 kips for ASD. The required shear and tensile bolt strengths are: LRFD ASD 319 kips 12 bolts = 26.6 kips/bolt < 30.7 kips o.k. 212 kips 12 bolts = 17.7 kips/bolt < 20.4 kips o.k. ruv = rav = 220 kips 12 bolts = 18.3 kips/bolt 146 kips 12 bolts = 12.2 kips/bolt rut = rat = The available tensile strength considers the effects of combined tension and shear. From AISC Specification Section J3.7, the available tensile strength of the bolts subject to combined tension and shear is: LRFD Interaction of shear and tension on bolts is given by AISC Specification Equation J3-3a: Fntʹ = 1.3Fnt − Fnt frv ≤ Fnt ϕFnv ASD Interaction of shear and tension on bolts is given by AISC Specification Equation J3-3b: Fntʹ = 1.3Fnt − From AISC Specification Table J3.2: From AISC Specification Table J3.2: Fnt = 90 ksi Fnt = 90 ksi Fnv = 68 ksi Fnv = 68 ksi Fntʹ = 1.3 ( 90 ksi ) − 90 ksi ⎛ 26.6 kips ⎜ 0.75 ( 68 ksi ) ⎜⎝ 0.601 in.2 ⎞ ⎟⎟ ⎠ = 38.9 ksi < 90 ksi o.k. The reduced available tensile force per bolt due to combined forces is: ϕrnt = ϕFntʹ Ab ( ΩFnt frv ≤ Fnt Fnv = 0.75 ( 38.9 kips ) 0.601 in. 2 = 17.5 kips < 18.3 kips n.g. ) 2.00 ( 90 ksi ) ⎛ 17.7 kips ⎞ ⎜⎜ ⎟ 2⎟ 68 ksi ⎝ 0.601 in. ⎠ = 39.0 ksi < 90 ksi o.k. Fnt′ = 1.3 ( 90 ksi ) − The reduced available tensile force per bolt due to combined forces is: rnt Fntʹ Ab = Ω Ω = ( 38.9 kips ) ( 0.601 in.2 ) 2.00 = 11.7 kips < 12.2 kips n.g. 72 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Try d-in.-diameter ASTM A490-X bolts. LRFD From AISC Specification Table J3.2: ASD From AISC Specification Table J3.2: Fnt = 113 ksi Fnt = 113 ksi Fnv = 84 ksi Fnv = 84 ksi Fntʹ = 1.3 (113 ksi ) − 2.00 (113 ksi ) ⎛ 17.7 kips ⎜⎜ 2 84 ksi ⎝ 0.601 in. = 67.7 ksi < 113 ksi o.k. 113 ksi ⎛ 26.66 kips ⎞ ⎜ ⎟ 0.75 ( 84 ksi ) ⎜⎝ 0.601 in.2 ⎟⎠ Fntʹ = 1.3 (113 ksi ) − = 67.5 ksi < 113 ksi o.k. The reduced available tensile force per bolt due to combined forces is: ϕrnt = ϕFntʹ Ab ( = 0.75 ( 67.5 kips ) 0.601 in. 2 ⎞ ⎟⎟ ⎠ The reduced available tensile force per bolt due to combined forces is: rnt Fntʹ Ab = Ω Ω ) = 30.4 kips > 18.3 kips o.k. = ( 67.7 kips ) ( 0.601 in.2 ) 2.00 = 20.3 kips > 12.2 kips o.k. Because A490-X bolts are required for the beam-to-column connection, they will be used everywhere in the connection. Using A325 and A490 bolts of the same diameter in any one connection, or even within the same project, is not recommended. Design beam web-to-end plate weld Using V and T, which were previously determined, the resultant load, R, and its angle with the vertical, θ, are: LRFD ASD Ru = Vu2 + Tu2 = Ra = Va2 + Ta2 ( 220 kips )2 + ( 319 kips )2 = = 388 kips (146 kips )2 + ( 212 kips )2 = 257 kips ⎛ 220 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 319 kips ⎠ = 34.6° ⎛ 146 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 212 kips ⎠ = 34.6° The weld size is determined from AISC Manual Equation 8-2, including the increase in strength allowed by AISC Specification Section J2.4 when the angle of loading is not along the weld longitudinal axis. In the following, only the weld length tributary to the group of bolts is used; therefore, l = 6(3.00 in.) = 18.0 in. LRFD Rearranging AISC Manual Equation 8-2a, and incorporating AISC Specification Equation J2-5, the weld size required is: Dreq ’d = 388 kips ( 2 (1.392 kip/in. )(18.0 in. ) 1.0 + 0.50 sin = 6.38 1.5 34.6° ASD Rearranging AISC Manual Equation 8-2b, and incorporating AISC Specification Equation J2-5, the weld size required is: ) Dreq ’d = 257 kips ( 2 ( 0.928 kip/in. )(18.0 in. ) 1.0 + 0.50 sin1.5 34.6° ) = 6.34 Therefore, use a two-sided v-in. fillet weld to connect the beam web to the end plate. Note that this exceeds the minimum size fillet weld of 4 in. required by AISC Specification Table J2.4. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 73 Check the 52-in. gage with v-in. fillet welds From AISC Manual Table 7-15, the required clearance for d-in.-diameter bolts with circular washers is: C3, req ’d = d in. 52 in. 0.515 in. − − v in. 2 2 = 2z in. > d in. o.k. Clearance = Therefore, the 52-in. gage is acceptable with v-in. fillet welds. Check prying action on bolts and end plate Using AISC Manual Part 9, determine the effect of prying action on the end-plate connection. The end plate is assumed to be w in. thick × 10 in. wide: 52 in. − 0.515 in. 2 = 2.49 in. db bʹ = b − 2 d in. = 2.49 in. − 2 = 2.05 in. b = (Manual Eq. 9-21) 10.0 in. − 52 in. 2 = 2.25 in. < 1.25b o.k. a = db ⎞ ⎛ db ⎞ ⎛ a ʹ = ⎜ a + ⎟ ≤ ⎜1.25b + ⎟ 2⎠ ⎝ 2 ⎠ ⎝ d in.⎞ ⎡ d in. ⎤ ⎛ = ⎜ 2.25 + ≤ 1.25 ( 2.49 ) + 2 ⎟⎠ ⎢⎣ 2 ⎥⎦ ⎝ = 2.69 in. < 3.55 in. (Manual Eq. 9-27) From AISC Specification Table J3.3 for a d-in.-diameter bolt, the standard hole dimension is , in. bʹ aʹ 2.05 in. = 2.69 in. = 0.762 p = pitch of bolts ρ = = 3.00 in. dʹ δ = 1− p , in. = 1− 3.00 in. = 0.688 74 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 (Manual Eq. 9-26) From the AISC Manual Part 9, the available tensile strength including prying action effects is: LRFD From AISC Manual Equation 9-30a: tc = = 4 Bbʹ ϕpFu tc = 4 ( 30.4 kips )( 2.05 in. ) 0.90 ( 3.00 in. )( 65 ksi ) = 1.19 in n. = = Ω4 Bb′ pFu 1.67 ( 4 )( 20.3 kips )( 2.05 in. ) ( 3.00 in.)( 65 ksi ) = 1.19 in. From AISC Manual Equation 9-35: αʹ = ASD From AISC Manual Equation 9-30b: ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.19 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.762 ) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ = 2.17 Because α′ > 1, α′ = 1.00 Determine Q from AISC Manual Equation 9-33: 2 ⎛ t ⎞2 Q = ⎜⎛ t ⎟⎞ (1 + δαʹ ) Q = ⎜⎝ tc ⎟⎠ (1 + δαʹ ) ⎝ tc ⎠ 2 s in. in. ⎞⎞ 2 ⎛⎛ s = = ⎜⎜ + (( 00..688 688 )( 00 )) ⎦⎤⎤⎦ ⎟ ⎡⎡⎣11 + )(11..00 19 in. in. ⎟⎠⎠ ⎣ ⎝⎝ 11..19 = 0.466 From AISC Manual Equation 9-31: From AISC Manual Equation 9-31: Tavail = BQ = ( 30.4 kips ) ( 0.466 ) = 14.2 kips < 18.3 kips n.g. From AISC Manual Equation 9-35: ⎡⎛ t ⎞2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥ ⎡⎣⎛ t ⎞2 ⎤⎦ 1 c ⎢⎜ ⎟ − 1⎥ α′ = 2 ⎤ δ (1 + ρ )1⎢⎝ t ⎠ ⎡⎛⎥ 1.19 in.⎞ ⎣ ⎢⎜⎦ = ⎟ − 1⎥ 0.688 (1 + 0.762 ) ⎡⎢⎝ s in. ⎠ 2 ⎤⎥ ⎦ ⎣⎛ 1.19 in.⎞ 1 ⎢⎜ = − 1⎥ ⎟ = 02.688 17 (1 + 0.762 ) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ 2.17 Because α′ >= 1, α′ = 1.00 ′ > 1, use Because α′ = 1Equation .00 Determine Q fromαAISC Manual 9-33: Determine Q from AISC Manual Equation 9-33: α′ = 2 ⎛⎛ tt ⎞⎞ 2 Q = + δα δα′′ )) Q = ⎜⎜ t ⎟⎟ ((11 + ⎝⎝ tcc ⎠⎠ 2 ⎛ s in.⎞ 1 0.688 1.00 ⎤ )( )⎦ =⎜ ⎟ ⎡⎣ + ( ⎝ 1.19 ⎠ = 00..466 466 = From AISC Manual Equation 9-31: From AISC Manual Equation 9-31: Tavail = BQ = ( 20.3 kips )( 0.466 ) = 9.46 kips < 12.2 kips n.g. The connection has failed. Since α′ > 1, it is known that the s-in.-thick end plate contributes to the failure. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 75 Try an end plate of increased thickness, tp = w in. LRFD ASD ⎡⎛ 1.19 in.⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.762 ) ⎢⎝ w in. ⎠ ⎥⎦ ⎣ = 1.25 α′ = Because α′ > 1, use α′ = 1.00 Determine Q from AISC Manual Equation 9-33: 2 ⎡⎛ 1.19 in.⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 0.762 ) ⎢⎝ w in. ⎠ ⎥⎦ ⎣ = 1.25 α′ = Because α′ > 1, use α′ = 1.00 Determine Q from AISC Manual Equation 9-33: 2 ⎛ w in. ⎞ Q =⎜ ⎟ ⎡⎣1 + ( 0.688 )(1.00 ) ⎤⎦ ⎝ 1.19 in.⎠ = 0.671 ⎛ w in. ⎞ Q =⎜ ⎟ ⎡⎣1 + ( 0.688 )(1.00 ) ⎤⎦ ⎝ 1.19 in. ⎠ = 0.671 Tavail = 0.671 ( 30.4 kips ) Tavail = 0.671 ( 20.3 kips ) = 20.4 kips > 18.3 kips o.k. = 13.6 kips > 12.2 kips o.k. Use the w-in.-thick end plate. Check prying action on column flange From the calculations for the gusset-to-column connection with d-in.-diameter bolts: b′ = 2.09 in. a′ = 2.69 in. ρ = 0.777 p = 3.00 in. δ = 0.688 LRFD tc = ASD 4 ( 30.4 kips )( 2.09 in. ) tc = 0.90 ( 3.00 in. )( 65 ksi ) = 1.20 in. 4 (1.67 )( 20.3 kips )( 2.09 in. ) ( 3.00 in.)( 65 ksi ) = 1.21 in. ⎤ ⎡⎛ 1.20 ⎞ 1 − 1⎥ ⎢⎜ ⎟ 0.688 (1 + 0.777 ) ⎢⎣⎝ 0.710 ⎠ ⎥⎦ = 1.52, use α ′ = 1. 00 2 α′ = 2 ⎛ 0.710 ⎞ Q =⎜ ⎟ ⎡⎣1 + 0.688 (1.00 ) ⎤⎦ ⎝ 1.20 ⎠ = 0.591 Tavail = 0.591 ( 30.4 kips ) = 18.0 kips < 18.3 kips n.g. ⎡⎛ 1.21 ⎞ 2 ⎤ 1 − 1⎥ ⎢ 0.688 (1 + 0.777 ) ⎢⎣⎜⎝ 0.710 ⎟⎠ ⎥⎦ = 1.56, use α ′ = 1. 00 α′= 2 ⎛ 0.710 ⎞ Q =⎜ ⎟ ⎡⎣1 + 0.688 (1.00 ) ⎤⎦ ⎝ 1.20 ⎠ = 0.591 Tavail = 0.591 ( 20.3 kips ) = 12.0 kips < 12.2 kips n.g. The check indicates that the column flange is deficient. However, the effective length of the connection need not be limited to the 18 in. assumed. An increase of less than z in. per bolt, or slightly more than 4 in. total, in effective length is all that is required to make the flange sufficient. This slight increase is okay by inspection. A yield-line analysis could be used to determine a better estimate of the effective length. 76 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check bolt bearing at end plate By the calculations for the gusset-to-column connection, bearing strength at the bolt holes in the beam-to-column connection will not control. The calculations for the bearing strength at bolt holes at this location would be similar to those presented for the gusset-to-column connection, except that the edge distance will be greater than the 14 in. used there. Check block shear rupture on end plate The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + Ubs Fu Ant(Spec. Eq. J4-5) The edge distance to the bottom of the end plate will be 21.4 in.− 18.0 in.+ 1.00 in. = 4.40 in., with a 1.00 in. bottom projection. The controlling block shear failure path cuts through each line of bolts and then to the outer edge of the end plate on each side. Shear yielding component: Agv = (15.0 in. + 4.40 in.)(w in.) = 14.6 in.2 0.6Fy Agv = 0.6(50 ksi)(14.6 in.2) = 438 kips Shear rupture component: Anv = 14.6 in.2 − 5.5(, in. + z in.)(w in.) = 10.5 in.2 0.6FuAnv = 0.6(65 ksi)(10.5 in.2) = 410 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = (w in.)[2.25 in. − 0.5(, in. + z in.)] = 1.31 in.2 UbsFuAnt = 1(65 ksi)(1.31 in.2) = 85.2 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60Fu Anv + UbsFuAnt = 410 kips + 85.2 kips = 495 kips 0.60Fy Agv + UbsFuAnt = 438 kips + 85.2 kips = 523 kips Therefore, Rn = 495 kips. LRFD ASD ϕRn = 0.75 ( 495 kips )( 2 ) = 743 kips > 319 kips o.k. Rn ( 495 kips )( 2 ) = Ω 2.00 = 495 kips > 212 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 77 Check beam shear strength The available beam shear strength is determined from AISC Specification Section J4.2, assuming shear yielding controls: LRFD ASD ϕRn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 50 ksi )( 21.4 in. )( 0.515 in. ) = 331 kipss > 319 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )( 21.4 in. )( 0.515 in. ) = 1.50 = 220 kips > 212 kips o.k. Check column shear strength The available column shear strength is: LRFD ϕRn = ϕ0.60Fy Agv = 1.00 ( 0.60 )( 50 ksi )(14.0 in. )( 0.440 in. ) = 185 kips > 176 kips o.k. ASD Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )(14.0 in. )( 0.440 in. ) = 1.50 = 123 kips > 117 kips o.k. Discussion A final comment on the end-plate thickness is in order. To calculate lh and α, tp = 1.00 in. was assumed. The final plate thickness was tp = w in. This will cause α and α to be slightly different and will cause a very small couple to exist on the gusset-to-beam interface. This couple can be ignored in the design of this connection. The final completed design is shown in Figure 5-1. Note that no weld is specified for the flanges of the W21×83 beam to the end plate other than those out to approximately the k1 dimension (the 1-in. returns). These welds are not required for the uniform force transfer assumed in the calculations for the forces shown in Figure 5-3. If the flanges are fully welded to the end plate, the bolts close to the flanges will see more load than those further away. This will affect the force distribution, which was assumed to be supplied uniformly to the column flange as well, and column flange stiffeners may be required adjacent to these more highly loaded bolts. It is important to recognize that a certain load path has been assumed by the use of the forces in Figure 5-3 and that the connection parts have been developed from that assumed load path. Thus, it is important to arrange the connection to allow the assumed load path to be realized. If the flanges are welded to the end plate, this fact should be considered in the design from the outset. It sometimes happens that a shop will weld the flanges to the end plate with AISC minimum fillet welds because it “looks better” and “more is better.” But in this case, more may not be better. Fracture of the bolts and/or weld close to the flange may occur. However, if no column stiffeners are provided adjacent to the beam flanges, the bolts near the flanges will probably not be overloaded because of column flange flexibility. The gusset “clip” of Figure 5-1 is used to separate the welds of the gusset to the beam and the gusset to the end plate. No weld passes through this clip. Example 5.2—Corner Connection-to-Column Flange: Uniform Force Method Special Case 1 Given: It is sometimes necessary that the brace line of action has a work point other than the intersection of the beam and column axes. This is referred to as a nonconcentric connection and is treated as Special Case 1 of the uniform force method. The situation is discussed in Section 4.2.2 and is shown in Figure 4-13, where the work point is at an arbitrary point located by the coordinates 78 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 x and y from the intersection of the beam and column flanges for a strong-axis connection (for a weak-axis connection the origin would be at the intersection of the beam flange and the face of the column web). Thus the origin (x = 0, y = 0) is at the gusset corner as shown in Figure 4-13, or close to this point when end plates, clip angles or shear plates are used. Consider the connection of Example 5.1, as shown in Figure 5-1, but choose the gusset line of action to pass through the (theoretical) gusset corner at (x = 0, y = 0). In this example, the admissible force distribution proposed in Section 4.2.2 will be compared with that presented in the AISC Manual Part 13 for Special Case 1, Modified Working Point Location. Solution: From AISC Manual Table 1-1: Beam W21×83 Zx = 196 in.4 Column W14×90 Zx = 157 in.4 From the geometry of Figure 5-2 and the terminology of Figure 4-13, and assuming that the brace bevel does not change (it will, but assume it does not for the sake of illustration) the eccentricity from the theoretical gusset corner to the gravity working point (the intersection of the beam and column centerlines) is: e = (eb − y)sin θ − (ec − x)cos θ = (10.7 in. − 0 in.)sin 47.2° − (7.00 in. − 0 in.)cos 47.2° = 3.09 in. Then from Section 4.2.2, Equations 4-5 through 4-8: LRFD η = = ASD Z beam Z beam + 2 Z column 196 in.3 ( 196 in.3 + 2 157 in.3 η = ) = 0.384 M = Pe = Z beam Z beam + 2 Z column 196 in.3 ( 196 in.3 + 2 157 in.3 = 0.384 M = Pe = ( 840 kips )( 3.09 in. ) = ( 560 kips )( 3.09 in. ) = 2, 600 kip-in. = 1, 730 kip-in. H′ = = (1 − η) M H′ = β + eb (1 − 0.384 ) ( 2, 600 kip-in.) 12.0 in. + 10.7 in. = 70.6 kips V′ = = M − H′β α 2, 600 kip-in. − ( 70.6 kips )(12.0 in. ) = 100 kips ) 17.5 in. = (1 − η) M β + eb (1 − 0.384 ) (1, 730 kip-in.) 12.0 in. + 10.7 in. = 46.9 kips V′ = = M − H ′β α 1, 730 kip-in. − ( 46.9 kips )(12.0 in. ) 17.5 in. = 66.7 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 79 These quantities are in the directions shown in Figure 4-14 when the brace force, P, is tension and e is positive. They are superimposed on the UFM forces, which are calculated as for a concentric case (with the correct bevel). The result of this superposition is shown in Figures 5-5a and 5-5b by showing the forces separately, and the summarized forces (summing the forces together) are shown in Figures 5-5c and 5-5d. It can be seen that the interface forces increase in some places and decrease in others. The connection must be designed with these forces. The brace-to-gusset connection will not change. The couple, M = 2,600 kip-in. (LRFD) and M = 1,730 kip-in. (ASD) is applied to the connection by the offset brace work line, but the brace itself is still an axial-force only member. This is why M is not shown on Figure 5-5a and 5-5b or Figure 5-5c and 5-5d, but its results on the beam and column are shown. The beam must be checked for an extra couple of 998 kip-in. (LRFD) and 668 kip-in. (ASD); the column must be checked for an extra couple of 801 kip-in. (LRFD) and 534 kip-in. (ASD), respectively. Comparison with AISC Manual Part 13, Special Case 1 method As noted in the problem statement, this treatment of Special Case 1, nonconcentric connections, does not follow the AISC Manual procedure provided. The Manual method is an admissible method, but Richard (1986) has reported that the gusset interface forces were not greatly affected by moving the work point from a concentric to a nonconcentric location. The Manual method would have only shear on the gusset edges, but the moments in the beam and column would be determined from the free body diagram given in Figure 5-6 as follows: Fig. 5-5a. Admissible force fields for nonconcentric Example 5.2—LRFD. 80 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Fig. 5-5b. Admissible force fields for nonconcentric Example 5.2—ASD. Fig. 5-5c. Admissible forces for Example 5.2 summarized—LRFD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 81 LRFD ASD In the beam: In the beam: Mub = ( 616 kips )( 21.4 in. 2 ) M ab = ( 410 kips )( 21.4 in. 2) = 6 590 kip-in. In the column: = 4 390 kip-in. In the column: Muc = ( 571 kips )(14.0 in. 2 ) 2 = 2, 000 kip-in. M ac = ( 380 kips )(14.0 in. 2 ) 2 = 1, 330 kip-in. This admissible force distribution is shown in Figure 5-6. This distribution, while admissible, is possible only if the gusset-tocolumn and gusset-to-beam connections can carry only shear. If very thin end plates or shear tabs with slots perpendicular to the shear loads were used in both locations, it is possible to achieve this distribution, but the authors have never seen such connections. Also, if these types of connections were achieved, the beam and column would be subjected to significant moments. Using AISC Manual Table 3-2, determine the available flexural strength for the beam and column. Assume the beam is fully braced and the column unbraced length is less than Lp = 15.1 ft: Fig. 5-5d. Admissible forces for Example 5.2 summarized—ASD. 82 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Fig. 5-6a. Admissible forces—Special Case 1 approach of the AISC Manual—LRFD. Fig. 5-6b. Admissible forces—Special Case 1 approach of the AISC Manual—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 83 LRFD ASD For the beam: Mp = 489 kip-ft Ωb For the beam: ϕbMp = 735 kip-ft As calculated previously: Mub = 6,590 kip-in. 12 in./ft As calculated previously: = 549 kip-ft Mub = 4,400 kip-in. 12 in./ft = 367 kip-ft Mub is 75% of the beam’s flexural strength, ϕbMp. Muc = 2,000 kip-in. 12 in./ft Mab is 75% of the beam’s flexural strength Mp/Ωb. = 167 kip-ft M ac = 1,330 kip-in. 12 in./ft = 111 kip-ft For the column: ϕbMp = 574 kip-ft Muc is 29% of the column’s flexural strength, ϕbMp. hese moments are too large to ignore. Note that the difT ference between the beam and column moments, Mub − Muc = 6,590 kip-in.− 2(2,000 kip-in.) = 2,590 kip-in. is equal to the actual applied moment of 2,600 kip-in. (with round off). For the column: Mp = 382 kip-ft Ωb Mac is 29% of the column’s flexural strength, Mp/ Ωb. hese moments are too large to ignore. Note that the difT ference between the beam and column moments, Mab − Mac = 4,390 kip-in.− 2(1,330 kip-in.) = 1,730 kip-in. is equal to the actual applied moment of 1,730 kip-in. Therefore, the authors recommend that the distribution discussed in Section 4 and illustrated in the first part of this example be used instead of Special Case 1 presented in the AISC Manual, for most applications. Example 5.3—Corner Connection-to-Column Flange: Uniform Force Method Special Case 2 Given: Verify the connection in Figure 5-1, using a W21×44 beam and the uniform force method Special Case 2. Special Case 2 minimizes the shear in the beam-to-column connection as discussed in Section 4.2.3. The other given properties are the same as those given in Example 5.1. Solution: From AISC Manual Table 1-1, the geometric properties are: Beam W21×44 d = 20.7 in. tw = 0.350 in. tf = 0.450 in. k1 = m in. kdes = 0.950 in. From Figure 5-3 and the AISC Manual Table 3-2, the required and available shear strengths of the beam adjacent to the end plate are: 84 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Vu = 269 kips + 50 kips = 319 kips ϕVn = 217 kips < 319 kips n.g. he beam web fails in shear and a web doubler plate will T be necessary to support a required shear strength of: Vu = 319 kips − 217 kips = 102 kips Va = 179 kips + 33.3 kips = 212 kips Vn = 145 kips < 212 kips n.g. Ω he beam web fails in shear and a web doubler plate will T be necessary to support a required shear strength of: Va = 212 kips − 145 kips = 67.0 kips Rather than using a doubler plate, it is possible to carry less shear in the beam-to-column connection and more in the gusset-tocolumn connection. This redistribution of shear is called Special Case 2 of the uniform force method. This is discussed in the AISC Manual Part 13 and in Section 4.2.3 of this Design Guide. For the configuration shown in Figure 5-3, set ΔVub = 102 kips (LRFD) and ΔVab = 68 kips (ASD). (The value of 67 kips calculated previously is increased to 68 kips due to rounding considerations.) This means that the gusset-to-column connection must support a required shear strength equal to the vertical interface force, Vuc (LRFD) or Vac (ASD), calculated in Example 5.1 in addition to the extra shear force, ΔVub (LRFD) or ΔVab (ASD). LRFD ASD Vu = 102 kips + 302 kips = 404 kips Va = 68 kips + 201 kips = 269 kips in addition to a required tensile strength of: Huc = 176 kips (see Figure 5-3a) in addition to a required tensile strength of: Hac = 117 kips (see Figure 5-3b) Preliminary Gusset-to-Column Connection Check Check bolts at gusset-to-column connection With the increased required shear strength of 404 kips (LRFD) and 269 kips (ASD) and required tensile strength of 176 kips (LRFD) and 117 kips (ASD), previous calculations for the gusset-to-column connection with 14 bolts can be reviewed to see if more bolts are required. Bolts are d-in.-diameter ASTM A490-X in single shear. The available shear and tensile strengths per bolt, from AISC Manual Tables 7-1 and 7-2, and the required shear and tensile strengths per bolt are: LRFD ϕrnv = 37.9 kips ϕrnt = 51.0 kips Required shear strength per bolt: 404 kips 14 = 28.9 kips < 37.9 kips o.k. ruv = Required tensile strength per bolt: 176 kips 14 = 12.6 kips < 51.0 kips o.k. rut = ASD rnv = 25.2 kips Ω rnt = 34.0 kips Ω Required shear strength per bolt: 269 kips 14 = 19.2 kips < 25.2 kips o.k. rav = Required tensile strength per bolt: 117 kips 14 = 8.36 kips < 34.0 kips o.k. rat = AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 85 From AISC Specification Section J3.7, the available tensile strength of the bolts subject to combined tension and shear is: LRFD Interaction of shear and tension on bolts is given by AISC Specification Equation Fnt J3-3a: Fnt′ = 1.3Fnt − frv ≤ Fnt ϕFnv Fnt Fnt′ = 1.3Fnt − frv ≤ Fnt ϕFnv From AISC Specification Table J3..2: FromFAISC Specification Table J3.2: nt = 113 ksi From AISC Specification Table J3..2: Fnv = = 113 84 ksi ksi F nt 113 ksi ⎛ 28.9 kips ⎞ F ′ = Fnv = 184 .3 ksi (113 ksi ) − ⎟ ⎜ nt 0.75 ( 84 ksi ) ⎛⎜⎝ 0.601 in.2 ⎞⎟⎠ 113 ksi 28.9 kips .3.6(113 Fnt′ = ) − ksi o.k. ⎜⎜ ⎟ = 160 < 113 ksi ksi 0.75 ( 84 ksi ) ⎝ 0.601 in.2 ⎟⎠ = 60.6 ksi < 113 ksi o.k. The reduced available tensile force per bolt due to The reduced available tensile force per bolt due to comcombined forces is s: reduced binedThe forces is: available tensile force per bolt due to combined forces iss: ϕrnt = ϕFnt′ Ab ( ) .3 (kips 12.6) (ki s o.k. in. ) .75 60.6>kips 0.p601 = 027 0F .75 .6 kips ) 0.601 in.2 ( 60 ′A ϕrnt = =ϕ nt b 2 = 27.3 kips > 12.6 kips o.k. ASD Interaction of shear and tension on bolts is given by AISC Specification Equation J3-3b: Fnt′ = 1.3Fnt − ΩFnt frv ≤ Fnt Fnv From AISC Specification Table J3.2: Fnt = 113 ksi Fnv = 84 ksi Fnt′ = 1.3 (113 ksi ) − ( 2.00 )(113 ksi ) ⎛ 19.2 kips ⎞ 84 ksi ⎜⎜ 2⎟ ⎟ ⎝ 0.601 in. ⎠ = 60.9 ksi < 113 ksi o.k. The reduced available tensile force per bolt due to combined forces is: rnt Fnt′ Ab = Ω Ω = ( 60.9 kips) ( 0.601 in.2 ) 2.00 = 18.3 kips > 8.36 kips It would appear from these preliminary calculations that 14 bolts will carry the force. This means that the geometry of Figure 5-1 can be used, except that the eb = 2(20.7 in.) = 10.4 in. must be used rather than 10.7 in., which corresponds to the W21×83 beam. Because this change is small, the UFM geometry and force distribution need not be changed. The force distribution for this case is derived from Figure 5-7 and shown in Figure 5-8. Interface Forces The moment between the gusset and beam is: LRFD Mub = ΔVub α ASD M ab = ΔVab α = (102 kips )(17.5 in. ) = ( 68.0 kips )(17.5 in. ) = 1, 790 kip-in. = 1,190 kip-in. Table 5-1a compares the interface forces for the general UFM from Figure 5-3 to those of Special Case 2 of the UFM from Figure 5-8. From Table 5-1, it can be seen that the portion of the connection that likely requires adjustment from the Figure 5-1 (general case) design is the gusset-to-column connection. This will be checked first, and then the gusset-to-beam connection will be checked. The brace-to-gusset connection and beam-to-column connection used in the general case will be satisfactory for Special Case 2. 86 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Fig. 5-7a. Special Case 2 superposition of forces—LRFD. Fig. 5-7b. Special Case 2 superposition of forces—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 87 Fig. 5-8a. Admissible force fields—Special Case 2—LRFD. Fig. 5-8b. Admissible force fields—Special Case 2—ASD. 88 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Table 5-1a. Comparison of Interface Forces (LRFD) General Case Interface Special Case 2 Shear Normal Moment Shear Normal Moment kips kips kip-in. kips kips kip-in. Brace-to-Gusset 0 840 0 0 840 0 Gusset-to-Beam 440 269 0 440 167 1790 Gusset-to-Column 302 176 0 404 176 0 Beam-to-Column 319 176 0 217 176 0 Note: The distortional force HD is not shown in the figures, but it is included in the table. Table 5-1b. Comparison of Interface Forces (ASD) General Case Interface Special Case 2 Shear Normal Moment Shear Normal Moment kips kips kip-in. kips kips kip-in. Brace-to-Gusset 0 560 0 0 560 0 Gusset-to-Beam 293 179 0 293 111 1190 Gusset-to-Column 201 117 0 269 117 0 Beam-to-Column 212 117 0 144 117 0 Note: The distortional force HD is not shown in the figures, but it is included in the table. Gusset-to-Column Connection (continued) The required shear and axial strengths are (Figure 5-8): LRFD Required shear strength, Vu = 404 kips Required normal strength, Pu = 176 kips ASD Required shear strength, Va = 269 kips Required normal strength, Pa = 117 kips Design gusset-to-end plate weld The resultant load, R, and its angle with the vertical, θ, are: LRFD Ru = ( 404 kips )2 + (176 kips )2 = 441 kips ⎛ 176 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 404 kips ⎠ = 23.5° ASD Ra = ( 269 kips )2 + (117 kips )2 = 293 kips ⎛ 117 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 269 kips ⎠ = 23.5° The weld size is determined from AISC Manual Equation 8-2, including the increase in strength allowed by AISC Specification Section J2.4 when the angle of loading is not along the weld longitudinal axis. In the following, only the weld length tributary to the group of bolts is used, therefore use l = 7(3.00 in.) = 21.0 in.: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 89 LRFD Rearranging AISC Manual Equation 8-2a, and incorporating AISC Specification Equation J2-5, the weld size required is: Dreq’d = 441 kips ( 2 (1.392 kip/in. )( 21.0 in. ) 1.0 + 0.50 sin 1.5 23.5° = 6.70 ASD Rearranging AISC Manual Equation 8-2b, and incorporating AISC Specification Equation J2-5, the weld size required is: ) Dreq’d = 293 kips ( 2 ( 0.928 kip/in. )( 21.0 in. ) 1.0 + 0.50 sin1.5 23.5° ) = 6.68 Therefore, use a two-sided v-in. fillet weld at the gusset-to-end plate connection. Note that this exceeds the minimum size fillet weld of c-in. required by AISC Specification Table J2.4. From previous calculations, this will be adequate for the 52-in. gage and 1-in.-thick gusset plate. As noted before, a ductility factor is not required here because of the end plate flexibility. Check bolts at gusset-to-column connection Use d-in.-diameter ASTM A490-X bolts with standard holes. Preliminary calculations have established that (14) A490-X d-in.diameter bolts are satisfactory. Check prying action on bolts at the end plate Using AISC Manual Part 9, determine the effect of prying action on the end-plate connection with a w-in.-thick end plate and the geometry shown in Figure 5-4b. From Example 5.1, the parameters are: b = 2.25 in. a = 2.25 in. < 1.25b o.k. b′ = 1.81 in. a′ = 2.69 in. ρ = 0.673 δ = 0.688 p = 3.00 in. From the AISC Manual Part 9, the available tensile strength, including prying action effects, is: LRFD From AISC Manual Equation 9-30a, with B = ϕrnt = 27.3 kips, previously determined: tc = = 4 Bb′ ϕpFu 4 ( 27.3 kips )(1.81 in. ) 0.90 ( 3.00 in. )( 65 ksi ) = 1.06 inn. ASD From AISC Manual Equation 9-30b, with B = rnt/ Ω = 18.3 kips, previously determined: tc = = Ω4 Bb′ pFu 1.67 ( 4 )(18.3 kips )(1.81 in. ) ( 3.00 in.)( 65 ksi ) = 1.07 in. 90 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.06 in.⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ w in. ⎠ ⎥⎦ ⎣ = 0.867 α′ = Because 0 ≤ α′ ≤ 1, determine Q from AISC Manual Equation 9-33: 2 ⎛ t ⎞2 Q = ⎛⎜ t ⎞⎟ (1 + δα ′) Q = ⎜ t ⎟ (1 + δα ′) ⎝ tc ⎠ ⎝ c⎠ 2 ⎛ w in. ⎞ 2 = ⎛⎜ w in. ⎞⎟ [1 + 0.688(0.867)] = ⎜⎝ 1.06 ⎟⎠ [1 + 0.688(0.867)] ⎝ 1.06 ⎠ = 0.799 = 0.799 From AISC Manual Equation 9-31: From AISC Manual 9-31: Manual Equation From AISCEquation 9-31: Tavail = BQ Tavail = BQ = ( 27.3 kips )( 0.800 ) = ( 27.3 kips )( 0.800 ) = 21.8 kips > 12.6 kips o.k. = 21.8 kips > 12.6 kips o.k. ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.07 in. ⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.688 (1 + 0.673 ) ⎢⎝ w in. ⎠ ⎥⎦ ⎣ = 0.900 α′ = Because 0 ≤ α′ ≤ 1, determine Q from AISC Manual Equation 9-33: 2 ⎛⎛ tt ⎞⎞ 2 Q Q = = ⎜⎜ t ⎟⎟ ((1 1+ + δα δα′′ )) ⎝⎝ t cc ⎠⎠ 2 2 ⎛⎛ w w in. in. ⎞⎞⎟ [1 + 0.688(0.900)] = ⎜ = ⎜⎝ 1.07 ⎟⎠ [1 + 0.688(0.900)] ⎝ 1.07 ⎠ = = 00..796 796 F AISC Manual 9-31: rom AISC F Manual 9-31: Manual Equation From rom AISCEquation Equation 9-31: T BQ = avai T l = BQ avail = = ((18 18..5 5 kips kips )( 796 )) )( 00..796 = = 14 14..7 7 kips kips > >8 8..36 36 kips kips o.k. o.k. Check bolt bearing on end plate From calculations to determine the available bearing strength for a s-in.-thick end plate in Example 5.1, the available bearing strength per bolt is: LRFD ϕrn = 46.8 kips > 28.9 kips o.k. ASD rn = 31.2 kips > 19.2 kips Ω o.k. Therefore, the thicker w-in.-thick end plate ultimately chosen will have adequate bearing strength. Check block shear rupture of the end plate From the similar calculations to determine block shear rupture strength for a s-in.-thick end plate in Example 5.1, the available strength is: LRFD ϕRn = 588 kips > 404 kips ASD o.k. Rn = 392 kips > 269 kips Ω o.k. Therefore, the thicker w-in.-thick end plate ultimately chosen will have adequate bearing strength. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 91 Check prying action on column flange From the calculations for Example 5.1: b = 2.53 in. a = 2.25 in. < 1.25b o.k. b′ = 2.09 in. a′ = 2.69 in. ρ = 0.777 p = 3.00 in. δ = 0.688 From AISC Manual Part 9, the available tensile strength including prying action effects is determined as follows: LRFD tc = 4 ( 27.3 kips )( 2.09 in. ) 0.90 ( 3.00 in. )( 65 ksi ) ASD tc = = 1.14 in. ( 3.00 in. )( 65 ksi ) = 1.15 in. ⎡⎛ 1.14 ⎞ 2 ⎤ 1 − 1⎥ ⎢ 0.688 (1 + 0.777 ) ⎢⎣⎜⎝ 0.710 ⎟⎠ ⎥⎦ = 1.29 Because α′ > 1, determine Q from AISC Manual Because α′ > 1, determine Q from AISC Manual Equation 9-34: Equation 9-34: α′ = 2 2 0.710 ⎛ 0.⎛710 ⎞ ⎞ Q 0.688 Q = ⎜ = ⎜ ⎟ (⎟1 +(10.+688 ) ) 1 .1 1 ⎝ 1⎝.114 ⎠4 ⎠ 0.655 = 0.=655 T = ( 27 Tavail = kips )( 00..655 27..33 kips 655) avail 1.67 ( 4 )(18.5 kips )( 2.09 in. ) = = 17 > 12 12..66 kips o.k. kipsss > kips o.k. 17..99 kips ⎡⎛ 1.15 ⎞ 2 ⎤ 1 − 1⎥ ⎢ 0.688 (1 + 0.777 ) ⎢⎣⎜⎝ 0.710 ⎟⎠ ⎥⎦ = 1.33 α′ = Because α′ > 1, determine Q from AISC Manual Equation 9-34: 2 ⎛ 0.710 ⎞ Q=⎜ ⎟ (1 + 0.688 ) ⎝ 1.115 ⎠ = 0.643 Tavail = (18.5 kips )( 0.643 ) = 11.9 kipss > 8.36 kips o.k. Gusset-to-Beam Connection From Figure 5-8, the required strengths at the gusset-to-beam connection are: LRFD ASD Required shear strength, Hub = 440 kips Required normal strength, Vub = 167 kips Required flexural strength, Mub = 1,790 kip-in. Required shear strength, Hab = 293 kips Required normal strength, Vab = 111 kips Required flexural strength, Mab = 1,190 kip-in. Check gusset plate for tensile yielding and shear yielding along the beam flange As shown in Figure 5-1, the gusset-to-beam interface length is the length of the gusset less the clip, or 31.5 in. In this example, to allow combining the required normal strength and flexural strength, stresses will be used to check the tensile and shear yielding limit states. Tensile yielding is checked using AISC Specification Section J4.1, and shear yielding is checked using Section J4.2: 92 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD fua = = Vub Ag faa = 167 kips (1.00 in.)( 31.5 in.) = = 5.30 ksi fub fab = 1, 790 kip-in. (1.00 in.)( 31.5 in.) 4 2 = 7.22 ksi 111 kips (1.00 in.)( 31.5 in.) = M ab Z (1,190 kip-in. ) (1.00 in.)( 31.5 in.)2 4 = 4.80 ksi Total normal stress = fun Tensile yielding strength: Tensile yielding strength: fun = fua + fub = 5.30 ksi + 7.22 ksi = 12.55 ksi < 0.90 ( 50 ksi ) = 45.0 ksi o.k. Shear yielding strength: fuv = Vab Ag = 3.52 ksi M = ub Z = ASD 440 kips (1.00 in.)( 31.5 in.) = 14.0 ksi < 1.00 ( 0.60 )( 50 ksi ) = 30.0 ksi o.k. Total normal stress = fan Tensile yielding strength: Tensile yielding strength: fan = faa + fab = 3.52 ksi + 4.80 ksi 50 ksi = 29.9 ksi o.k. = 8.32 ksi < 1.67 Shear yielding strength: fav = 293 kips (1.00 in.)( 31.5 in. ) 0.60 ( 50 ksi ) = 9.30 ksi < 1.50 = 20.00 ksi o.k. Design weld at gusset-to-beam flange connection As determined previously, the stresses on this interface are: LRFD ASD fua = 5.30 ksi fub = 7.22 ksi fun = fua + fub = 12.5 ksi fuv = 14.0 ksi faa = 3.52 ksi fab = 4.80 ksi fan = faa + fab = 8.32 ksi fav = 9.30 ksi AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 93 Following the method of Hewitt and Thornton (2004) and AISC Manual Part 13, where it is explained that if the gusset plate is directly welded to the beam, the weld is designed for the larger of the peak stress and 1.25 times the average stress: LRFD fu peak = = ASD ( fua + fub )2 + fuv2 fa peak = ( 5.30 ksi + 7.22 ksi )2 + (14.0 ksi )2 = = 18.8 ksi fu avg = 1⎡ 2 ⎢⎣ ( faa + fab )2 + fav2 ( 3.52 ksi + 4.80 ksi )2 + ( 9.30 ksi )2 = 12.5 ksi ( fua − fub )2 + fuv2 + ( fua + fub )2 + fuv2 ⎤⎥ ⎦ 2 2 ⎡ ⎤ 1 ⎢ ( 5.30 ksi − 7.22 ksi ) + (14.0 ksi ) ⎥ = ⎢ 2 2⎥ 2 + ( 5.30 ksi + 7.22 ksi ) + (14.0 ksi ) ⎥⎦ ⎢⎣ = 16.5 ksi ( fu weld = max 1.25 fu avg , fu peak ) fa avg = 1⎡ 2 ⎢⎣ ( faa − fab )2 + fav2 + ( faa + fab )2 + fav2 ⎤⎥ ⎦ 2 2 ⎡ ⎤ 1 ⎢ ( 3.52 ksi − 4.80 ksi ) + ( 9.30 ksi ) ⎥ = ⎢ 2 2⎥ 2 + ( 3.52 ksi + 4.80 ksi ) + ( 9.30 ksi ) ⎥⎦ ⎢⎣ = 10.9 ksi ( fa weld = max 1.25 fa avg , fa peak ) = max ⎡⎣1.25 (16.5 ksi ) , 18.8 ksi ⎤⎦ = max ⎡⎣1.25 (10.9 ksi ) , 12.5 ksi ⎤⎦ = 20.6 ksi = 13.6 ksi To account for the directional strength increase of the fillet weld, the total normal stress and the shear stress can be used to determine the load angle. Then, from AISC Manual Part 8, Equations 8-2a and 8-2b: LRFD ASD ⎛f ⎞ θ = tan −1 ⎜ un ⎟ ⎝ fuv ⎠ ⎛ 12.5 ksi ⎞ = tan −1 ⎜ ⎟ ⎝ 14.0 ksi ⎠ ⎛f ⎞ θ = tan −1 ⎜ an ⎟ ⎝ fav ⎠ ⎛ 8.32 ksi ⎞ = tan −1 ⎜ ⎟ ⎝ 9.30 ksi ⎠ = 41.8° = 41.8° The required number of sixteenths of weld is: D= = fu weld t ( 2 (1.392 kip/in. ) 1 + 0.50sin1.5 θ ) ( 20.6 ksi ) (1.00 in.) 2 (1.392 kip/in. ) (1 + 0.50sin1.5 41.8° ) = 5.82 sixteenths The required number of sixteenths of weld is: D= = fa weld t ( 2 ( 0.928 kip/in. ) 1 + 0.50sin1.5 θ ) (13.6 ksi ) (1.00 in.) 2 ( 0.928 kip/in. ) (1 + 0.50sin1.5 41.8° ) = 5.76 sixteenths From AISC Specification Table J2.4, the minimum fillet weld required is c in. which does not control in this case. Use a twosided a-in. fillet weld to connect the gusset plate to the beam. 94 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check beam web local yielding The total equivalent normal force (including both the force and the moment resolved into a force couple) at the gusset-to-beam interface is: LRFD ASD 2 Mub ⎛L⎞ ⎜2⎟ ⎝ ⎠ N u equiv = Vub + = 167 kips + N a equiv = Vab + ( 4 )(1, 790 kip-in.) 2 Mab ⎛L⎞ ⎜2⎟ ⎝ ⎠ = 111 kips + 31.5 in. = 394 kips ( 4 )(1,190 kip-in.) 31.5 in. = 262 kips See Appendix B, Figure B-1 for an explanation of Nequiv . From AISC Specification Section J10.2, because the resultant force is applied at a distance, α = 17.5 in. (determined in Example 5.1) from the member end that is less than or equal to the depth of the W21×44 beam, the available strength is determined from Equation J10-3, with k = kdes. Alternatively, AISC Manual Table 9-4 could be used. LRFD ASD ϕRn = ϕFyw t w ( 2.5kdes + lb ) Rn Fy tw ( 2.5k des + lb ) = Ω Ω ( 50 ksi )( 0.350 in.) ⎡⎣2.5 ( 0.950 in.) + 31.5 inn.⎤⎦ = 1.50 = 395 kips > 262 kips o.k. = 1.00 ( 50 ksi )( 0.350 in. ) [ 2.5 ( 0.950 in. ) + 31.5 in. ] = 593 kips > 394 kips o.k. Check beam web local crippling From AISC Specification Section J10.3, because the resultant force is applied at a distance greater than d/ 2 from the member end, use Equation J10-4: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝ d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜ tf ⎟ ⎝ ⎠ 1.5 ⎤ ϕRn = ( 0.75 )( 0.80 )( 0.350 in. ) × = 415 kips > 394 kips ⎥ ⎥ ⎦ 2 ASD EFyw t f 0.80t w2 tw ⎡ ⎛31.5 in.⎞ ⎢1 + 3 ⎜ ⎟ ⎢ ⎝20.7 in.⎠ ⎣ 1.5 ⎛ 0.350 in.⎞ ⎜ ⎟ ⎝ 0.450 in.⎠ ⎤ ⎥ ⎥ ⎦ Rn = Ω ⎡ ⎛ 31.5 in. ⎞ ⎛ 0.350 in.⎞ ⎟⎜ ⎟ ⎝ 20.7 in. ⎠ ⎝ 0.450 in.⎠ ( 0.80 )( 0.350 in.)2 ⎢1 + 3 ⎜ ⎢⎣ ( 29, 000 ksi )( 50 ksi )( 0.450 in. ) 0.350 in. 1.5 ⎡ ⎛ ⎞ ⎤ EFyw tf ⎢1 + 3 ⎛ lb ⎞ ⎜ tw ⎟ ⎥ ⎜ d⎟ ⎜t ⎟ ⎥ ⎢ tw ⎝ ⎠⎝ f⎠ ⎣ ⎦ Ω Rn = Ω × 1.5 ⎤ ⎥ ⎥⎦ ( 29, 000 ksi )( 50 ksi )( 0.450 in.) 0.350 in. 2.00 = 276 kips > 262 kips o.k. Beam-to-Column Connection The required shear strength of this connection has been reduced from Vu = 319 kips to Vu = 217 kips (LRFD) or from Va = 212 kips to Va = 144 kips (ASD) (see Table 5-1), but the beam also has been made lighter and must be checked for strength. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 95 The required strengths for this connection are: LRFD ASD Vu = 217 kips Va = 144 kips Au = max ( Aub , H uc ) Aa = max ( Aab , H ac ) = max (100 kips, 176 kips ) = max ( 66.7 kips, 117 kips ) = 176 kips = 117 kips For a discussion on this, see Section 4.2.6 and the beam-to-column connection analysis for Example 5.1. Check shear yielding on beam web From AISC Specification Section J4.2: LRFD ASD ϕVn = ϕ0.60 Fy Agv Vn Ω = 1.00 ( 0.60 )( 50 ksi )( 0.350 in. )( 20.7 in. ) = 217 kipss = 217 kips o.k. = = 0.60 Fy Agv Ω 0.60 ( 50 ksi )( 0.350 in. )( 20.7 in. ) 1.50 = 145 kips > 144 kips o.k. Design beam web-to-end plate weld The resultant load, R, and its angle with the vertical, θ, are: LRFD R= ASD ( 217 kips )2 + (176 kips )2 R= = 279 kips (144 kips )2 + (117 kips )2 = 186 kips ⎛ 176 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 217 kips ⎠ = 39.0° ⎛ 117 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 144 kips ⎠ = 39.1° The number of sixteenths of weld required is determined from AISC Manual Equation 8-2, including the increase in strength allowed by AISC Specification Section J2.4 when the angle of loading is not along the weld longitudinal axis. In the following, only the weld length tributary to the group of bolts is used; therefore, use l = 6(3.00 in.) = 18.0 in: LRFD Dreq’d = ASD 279 kips ( 2 (1.392 kip/in. )(18.0 in. ) 1.0 + 0.50 sin = 4.46 sixteenths 1.5 ) 39.0° Dreq’d = 186 kips ( 2 ( 0.928 kip/in. )(18.0 in. ) 1.0 + 0.50 sin1.5 39.1° ) = 4.45 sixteenths Therefore, use a two-sided c-in. fillet weld to connect the beam web to the end plate. Note that this exceeds the minimum size fillet weld of 4 in. required by AISC Specification Table J2.4. Check W21×44 web weld to deliver load to bolts In the preceding weld calculation, only the weld adjacent to the bolts, 18.0 in., is used to size the weld. The actual effective weld length to the k1 distance inside the flange is: 96 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 leff = d − 2tf + 2(k1 − tw/ 2) = 20.7 in. − 2(0.450 in.) + 2(m in. − 0.350 in./ 2) = 21.1 in. Since 21.1 in. > 20.7 in., the weld and web are fully effective. Check tensile yielding in the web adjacent to the weld For the 18-in. length of weld assumed to be tributary to the load on the bolts, the available tensile yielding strength of the web from AISC Specification Equation J4-1 is: LRFD ASD ϕRn = ϕFy Ag = 0.90 ( 50 ksi )( 0.350 in. )(18.0 in. ) = 284 ksi > 176 kips o.k. Rn Fy Ag = Ω Ω ( 50 ksi )( 0.350 in.)(18.0 in.) = 1.67 = 189 kips > 117 kips o.k. Summary Figure 5-9 shows the final design of this connection. The final connection is similar to that of Example 5.1 for the general UFM shown in Figure 5-1. As a result of the decision to carry less shear in the beam-to-column connection and more in the Fig. 5-9. Special Case 2—strong axis. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 97 gusset-to-column connection as discussed in the beginning of the example, the gusset-to-end-plate weld has increased from a in. to v in., the gusset-to-beam-flange weld has changed to a in., and the beam-web to end-plate weld has decreased from v in. to c in. Example 5.4—Corner Connection-to-Column Flange with Gusset Connected to Beam Only: Uniform Force Method Special Case 3 Given: Special Case 3 of the uniform force method was discussed in Section 4.2.4, where it was noted that a moment Mbc = Vec will exist on the beam-to-column connection, in addition to the vertical brace component, V (and the beam shear, R, if any). A design example of the connection shown in Figure 5-10 will illustrate this case. The bracing arrangement is as shown in Figure 2-1(a). The loading, member sizes, material strengths, bolt size and type, hole size, and connection element sizes are as given in Figure 5-10. Use 70-ksi electrodes. α α Fig. 5-10. Special Case 3—strong axis. 98 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The required strengths are: LRFD ASD Brace required strength, Pu = ± 100 kips Beam required shear strength, Vu = 30.0 kips Brace required strength, Pa = ± 66.7 kips Beam required shear strength, Va = 20.0 kips Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A36 Fy = 36 ksi Fu = 58 ksi ASTM A992 Fy = 50 ksi Fu = 65 ksi From AISC Manual Tables 1-1, 1-7 and 1-15, the geometric properties are as follows: Beam W18×35 tw = 0.300 in. d = 17.7 in. bf = 6.00 in. tf = 0.425 in. tw = 0.360 in. d = 12.2 in. bf = 10.0 in. tf = 0.640 in. Brace 2L4×4×a in. Ag = 5.72 in.2 x = 1.13 in. (single angle) kdes = 0.827 in. Column W12×58 From AISC Specification Table J3.3, the hole dimensions for d-in.-diameter bolts are as follows: Brace, gusset plate and column Standard: , in. diameter Angles Short slots: , in. × 18 in. (short-slotted holes transverse to the line of force) From AISC Manual Tables 7-1 and 7-2, the available shear and tensile strengths of a d-in.-diameter ASTM A325-N bolt are: LRFD ASD ϕrnv = 48.7 kips double shear at gusset-to-brace connection rnv = 32.5 kips double shear at gusset-to-brace connection Ω rnv = 16.2 kips single shear at beam-to-column connection Ω rnt = 27.1 kips Ω ϕrnv = 24.3 kips single shear at beam-to-column connection ϕrnt = 40.6 kips Brace-to-Gusset Connection Check tension yielding on the brace The available tensile yielding strength is determined from AISC Specification Equation J4-1. Alternatively, AISC Manual Table 5-8 may be used. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 99 LRFD ASD ϕt Rn = ϕt Fy Ag = 0.90 ( 36 ksi ) 5.72 in.2 ( ) = 185 kips > 100 kips o.k. Rn Fy Ag = Ωt Ωt = ( 36 ksi ) ( 5.72 in.2 ) 1.67 = 123 kips > 66.7 kips o.k. Check tensile rupture on the brace The net area of the double-angle brace is determined in accordance with AISC Specification Section B4.3: An = Ag − 2t(dh + z in.) = 5.72 in.2 − 2(a in.)(, in. + z in.) = 4.97 in.2 Because the outstanding legs of the double angle are not connected to the gusset plate, an effective net area of the double angle needs to be determined. From AISC Specification Section D3 and Table D3.1, Case 2, the effective net area is: Ae = AnU(Spec. Eq. D3-1) where x l 1.13 in. = 1− (with 6 in. connection length) 6.00 in. = 0.812 U = 1− Then: Ae = (4.97 in.2)0.812 = 4.04 in.2 From AISC Specification Section J4.1, the available strength for tensile rupture on the net section is: LRFD ϕt Rn = ϕt Fu Ae ASD ( = 0.75 ( 58 ksi ) 4.04 in.2 = 176 kips > 100 kips ) o.k. Rn Fu Ae = Ωt Ωt = ( 58 ksi ) ( 4.04 in.2 ) 2.00 = 117 kips > 66.7 kips o.k. Check bolt shear strength From AISC Manual Table 7-1, the available shear strength of the three d-in.-diameter ASTM A325-N bolts in double shear in the brace-to-gusset connection is: 100 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ϕRn = nϕrnv = 3 ( 48.7 kips ) = 146 kips > 100 kips o.k. ⎛ rnv ⎞ ϕRn = n ⎜ ⎟ ⎝Ω⎠ = 3 ( 32.5 kips ) = 97.5 kips > 66.7 kips o.k. Check bolt bearing on the gusset plate and on the brace Try a 2-in.-thick gusset plate with standard bolt holes. This thickness will be increased later, as shown in Figure 5-10. For the brace, short-slotted holes transverse to the line of force are used and the hole width is the same as for a standard hole. According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength on the gusset plate or brace. Bearing strength is limited by bearing deformation of the hole or by tearout of the material. From AISC Specification Section J3.10, when deformation at the bolt hole is a design consideration, the available strength for the limit state of bearing at bolt holes on the gusset is determined as follows: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) where, for the end bolt: lc = 12 in. − 0.5dh = 12 in. − 0.5(, in.) = 1.03 in. For the end bolt, using AISC Specification Equation J3-6a: LRFD ϕ1.2lctFu = 0.75(1.2)(1.03 in.)(2 in.)(58 ksi) = 26.9 kips/bolt ϕ2.4dtFu = 0.75(2.4)(d in.)(2 in.)(58 ksi) = 45.7 kips/bolt Therefore, ϕRn = 26.9 kips/bolt. ASD 1.2lc tFu (1.2 )(1.03 in.) ( 2 in. ) (58 ksi ) = Ω 2.00 = 17.9 kipss/bolt 2.4 dtFu ( 2.4 )( d in.) ( 2 in. ) (58 ksi ) = Ω 2.00 = 30.5 kips/bolt Therefore Rn Ω = 17.9 kips/bolt. ϕrnv = 48.7 kips/bolt > 26.9 kips/bolt for LRFD and rnv/ Ω = 32.5 kips/bolt > 17.9 kips/bolt = 17.9 kips/bolt for ASD, indicates that bearing at bolt holes on the end bolt is less than the shear strength of the bolt in double shear and bearing strength controls the design. The edge distance can be increased, the gusset made thicker, or the end bolt can be used with the reduced strength. The latter approach will be attempted in this example. Bearing strength at bolt holes on the gusset for the two inner bolts is calculated as follows: lc = 3.00 in. − 1.0dh = 3.00 in. − 1.0(, in.) = 2.06 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 101 LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 2.06 in. ) ( 2 in. ) ( 58 ksi ) 1. 2 ( 2.06 in. )( 2 in. ) ( 58 ksi ) 1.2lc tFu = Ω 2.00 = 35.8 kips > 32.5 kipss = 53.8 kips > 48.7 kipss Therefore, available bolt shear strength controls the strength of the inner bolt. For the end bolt on the brace: lc = 1.03 in. ϕrn = 0.75(1.2)(1.03 in.)(a in.)(2)(58 ksi) = 40.3 kips controls < 0.75(2.4)(d in.)(a in.)(2)(58 ksi) = 68.5 kips he bearing strength of the end bolt on the brace controls T over the interior bolt strength on the gusset plate. he effective strength of the bolts, including bolt shear T and bearing at bolt holes on both the brace and the gusset is: ϕRn = 26.9 kips + 48.7 kips + 40.3 kips = 116 kips > 100 kips o.k. herefore, available bolt shear strength controls the T strength of the inner bolt. For the end bolt on the brace: lc = 1.03 in. rn 1.2 (1. 03 in. ) ( a in.) ( 2 ) ( 58 ksi ) = Ω 2.00 = 26.9 kips controls < 2.4 ( d in. ) ( a in.) ( 2 ) ( 58 ksi ) 2.00 = 45.7 kips The bearing strength of the end bolt on the brace controls over the interior bolt strength on the gusset plate. he effective strength of the connection including bolt T shear and bearing at bolt holes on both the brace and the gusset is: Rn = 17.9 kips + 32..5 kips + 26.9 kips Ω = 77.3 kips > 66.7 kips o.k. Note that bearing on both the brace and gusset should be considered in these checks. Check block shear rupture on the gusset plate The block shear rupture failure path on the gusset is assumed to run down the line of bolts and then perpendicular to the free edge of the gusset plate. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = [12 in. + 2(3.00 in.)](2 in.) = 3.75 in.2 0.60Fy Agv = 0.60(36 ksi)(3.75 in.2) = 81.0 kips Shear rupture component: Anv = 3.75 in.2 − [2.50(, in. + z in.)(2 in.)] = 2.50 in.2 0.60Fu Anv = 0.60(58 ksi)(2.50 in.2) = 87.0 kips 102 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = [5.00 in. − 0.5(1.00 in.)](2 in.) = 2.25 in.2 UbsFu Ant = 1(58 ksi)(2.25 in.2) = 131 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 87.0 kips + 131 kips = 218 kips 0.60Fy Agv + UbsFuAnt = 81.0 kips + 131 kips = 212 kips Therefore, because 218 kips > 212 kips, Rn = 212 kips. LRFD ASD ϕRn = 0.75 ( 212 kips ) o.k. = 159 kips > 100 kips Rn 212 kips = Ω 2.00 = 106 kips > 66.7 kips o.k. Check the gusset plate for tensile yielding on the Whitmore section From AISC Manual Part 9, the width of the Whitmore section, using a 2-in.-thick gusset plate, is: lw = 2(6.00 in.)tan 30° = 6.93 in. Therefore, by geometry, the Whitmore section stays within the gusset plate. The effective area is: Aw = (2 in.)(6.93 in.) = 3.47 in.2 From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate is: LRFD ASD ϕRn = ϕFy Aw ( = 0.90 ( 36 ksi ) 3.47 in.2 = 112 kips > 100 kips ) o.k. Rn Fy Aw = Ω Ω = ( 36 ksi ) ( 3.47 in.2 ) 1.67 = 74.8 kips > 66.7 kips o.k. Check the gusset plate for compression buckling on the Whitmore section The available compressive strength of the gusset plate based on the limit state of flexural buckling is determined from AISC Specification Section J4.4. Use an effective length factor of K = 1.2 due to the possibility of sidesway buckling (see AISC Specification Commentary Table C-A-7.1). From Figure 5-10, the unbraced length of the gusset plate is shown as 62 in. KL r = 1.2 ( 62 in. ) 2 in. = 54.0 12 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 103 Because KL/ r > 25, use AISC Manual Table 4-22 to determine the available critical stress for the 36-ksi gusset plate. Then, the available compressive strength can be determined from AISC Specification Sections E1 and E3: LRFD ASD Fcr = 18.5 ksi Ωc ϕc Fcr = 27.8 ksi ϕc Pn = ϕc Fcr Aw ( = (27.8 ksi ) 3.47 in.2 ) = 96.5 kips < 100 kips n.g. Pn Fcr Aw = Ωc Ωc ( = (18.5 ksi ) 3.47 in.2 ) = 64.2 kips < 66.7 kips n.g. Because the gusset fails in buckling, a thicker gusset could be used, or the material could be changed to ASTM A572 Grade 50. The thicker gusset will be used here. Try a s-in.-thick gusset plate: KL 1.2 ( 62 in.) = s in. 12 r = 43.2 Use AISC Manual Table 4-22 to find the available critical stress, and determine the available compressive strength from AISC Specification Sections E1 and E3: LRFD ϕc Fcr = 29.4 ksi ϕc Pn = ϕc Fcr Ag = ( 29.4 ksi ) ( s in.) (6.93 in.) = 127 kips > 100 kips o.k. ASD Fcr = 19.6 ksi Ωc Pn Fcr = Ag Ωc Ωc = (19.6 ksi ) ( s in.) (6.93 in.) = 84.9 kipss > 66.7 kips o.k. Therefore, use a gusset plate thickness of s in. Check block shear rupture on the brace The available strength for the limit state of block shear rupture on the brace is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = 2(7.50 in.)(a in.) = 5.63 in.2 0.60Fy Agv = 0.60(36 ksi)(5.63 in.2) = 122 kips Shear rupture component: Anv = 5.63 in.2 − 2.5(2)(a in.)(1.00 in.) = 3.76 in.2 0.60Fu Anv = 0.60(58 ksi)(3.76 in.2) = 131 kips 104 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Tension rupture component: Ubs = 1 from AISC Specifications Section J4.3 because the bolts are uniformly loaded Ant = [1.50 in. − 0.5(1.00 in.)](a in.)(2) = 0.750 in.2 ( ) UbsFu Ant = 1(58 ksi) 0.750 in.2 = 43.5 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 131 kips + 43.5 kips = 175 kips 0.60Fy Agv + UbsFu Ant = 122 kips + 43.5 kips = 166 kips Therefore, because 175 kips > 166 kips, Rn = 166 kips. LRFD ASD ϕRn = 0.75 (166 kips ) = 125 kips > 100 kips o.k. Rn 166 kips = Ω 2.00 = 83.0 kips > 66.7 kips o.k. Connection Interface Forces The forces at the gusset-to-beam and gusset-to-column interfaces are determined using Special Case 3 of the UFM as discussed in Section 4.2.4 and AISC Manual Part 13. From Figure 5-10 and the given beam and column geometry: db 2 17.7 in. = 2 = 8.85 in. eb = ec = dc 2 12.2 in. = 2 = 6.10 in. tan θ = θ 12 7 ⎛ 12 ⎞ = tan −1 ⎜ ⎟ ⎝7⎠ = 59.7° Because there is no connection of the gusset plate to the column, β = 0, and α = eb tan θ − ec (Manual Eq. 13-1) ⎛ 12 ⎞ = ( 8.85 in. ) ⎜ ⎟ − 6.10 in. ⎝ 7⎠ = 9.07 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 105 From Figure 5-10, the horizontal length of the gusset, lg, is 22.0 in., therefore: α= lg + 2 in. 2 22.0 in. = + 2 in. 2 = 11.5 in. From Figure 4-16 and Section 4.2.4, the forces on the gusset-to-beam interface are: LRFD ASD Hub = H = Pu sin θ = (100 kips) sin 59.7° = 86.3 kips Hab = H = Pa sin θ = (66.7 kips) sin 59.7° = 57.6 kips Vub = V = Pu cos θ = (100 kips) cos 59.7° = 50.5 kips Vab = V = Pa cos θ = (66.7 kips) cos 59.7° = 33.7 kips Mub = Vub(α − α) = (50.5 kips)(11.5 in. − 9.07 in.) = 123 kip-in. Mab = Vab(α − α) = (33.7 kips)(11.5 in. − 9.07 in.) = 81.9 kip-in. The forces on the beam-to-column interface are: LRFD ASD Mbc = Vubec = (50.5 kips)(6.10 in.) = 308 kip-in. Mbc = Vabec = (33.7 kips)(6.10 in.) = 206 kip-in. Vbc = Vub + R = 50.5 kips + 30 kips = 80.5 kips Vbc = Vab + R = 33.7 kips + 20 kips = 53.7 kips Gusset-to-Beam Connection Check gusset plate for shear yielding and tensile yielding along the beam flange In this example, to allow combining the required normal strength and flexural strength, stresses will be used to check the tensile and shear yielding limit states. The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1: 106 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Vub LRFD fua = V ub A fua = Vub fua = A 50.5 kips = A 50.5 kips in..)5(kips 22.0 in.) = ( s 50 = ( s in.) ( 22.0 in.) = 3( s .67in. ksi ) ( 22.0 in.) = 3M.67 ksi ub = 3M.67 ksi fub = ub Z fub = Mub fub = Z 123 kip-in.. = Z 123 kip-in.. 2 in.. ) 4 = ( s in. ) ( 22.0 123 kip-in. = ( s in.) ( 22.0 in.)22 4 = 1(.s 63in. ksi ) ( 22.0 in.) 4 = 1 63 ksi . Total = 1normal .63 ksi stress = fun Total normal stress = fun Total normal stress = fun Check tensile yielding: Check tensile yielding: Check tensile fun = fua + fubyielding: Check tensile yielding: fun == 3fua fub + 1.63 ksi + ksi . 67 fun = fua + fub = 1.63 = 335...67 30 ksi .90ksi ksi + ( 36 ksi ) = 32.4 ksi o.k. = 67 +<10.63 ksi ksi = 5.30 ksi < 0.90 ( 36 ksi ) = 32.4 ksi o.k. = 5.30 ksi < 0.90 ( 36 ksi ) = 32.4 ksi o.k. Check shear yielding: Check shear yielding: CheckHshear yielding: ub Check fuv = Hshear yielding: ub A fuv = Hub fuv = A 86.3 kips = A 86.3 kips in..)3(22.0 = ( s 86 kips in.) = ( s in.) (22.0 in.) .28in. k. = 6( s ksi < 1.00 in.()0.60 )( 36 ksi ) = 21.6 ksi o.k ) (22.0 k. = 6.28 ksi < 1.00 ( 0.60 )( 36 ksi ) = 21.6 ksi o.k k. = 6.28 ksi < 1.00 ( 0.60 )( 36 ksi ) = 21.6 ksi o.k Design weld at gusset-to-beam flange connection Vab ASD = Vab = A VA ab 33.7 kips = = A 33.7 kips = ( s in.) ( 22.0 in.) in..)7( kips 22.0 in.) ( s 33 = 2.45 ksi s in. ) ( 22.0 in.) = 2( .45 ksi M ab f ab = 2M.45 ab ksi f ab = Z MZab (81.9 kip-inn. ) f ab = = Z ( 81.9 kip-inn. )2 = ( s in.) ( 22.0 in.)2 4 81).9( 22.0 kip-in n.)) 4 in. ( s (in. = = 1.08 ksi ) ( 22.0 in.)2 4 ksi = 1(.s 08in. Total normal stress = fan .08 ksistress = fan Total= 1normal Total normal stress = fan Check tensile yielding: Check tensile yielding: Che ck tensile yielding: fan = faa + fab Che c k tensile fan = faa + fabyielding: = 2.45 ksi + 1.08 ksi fan = 2f.aa45+ksi fab+ 1.08 ksi 36 ksi = 32.53 < 13.608ksi 45 ksi + ksi= 21.6 ksi o.k. = 3.53 ksi < 1.67 = 21.6 ksi o.k. 316.67 ksi = 3.53 ksi < = 21.6 ksi o.k. . 1 67 Check shearshear yielding: Check yielding: Check shear yielding: H ab fav = Hshear Check yielding: ab fav = A HAab 57.6 kips fav == 57.6 kips A = ( s in.) ( 22.0 in.) kips in.) 57.)6( 22.0 ( s in. = 0.60 ( 36 ksi ) in.()36 ksi ) = 14.4 ksi ) ( 22.0 .19in. = 4( s ksi < 0.60 1.50 = 4.19 ksi < = 14.4 ksi 0.601(.36 50 ksi ) = 4.19 ksi < = 14.4 ksi 1.50 f aa f aa f aa o.k. o.k. o.k. Following the method of Hewitt and Thornton (2004) and AISC Manual Part 13, the weld is designed for the larger of the peak stress and 1.25 times the average stress: LRFD f peak = ASD ( 5.30 ksi )2 + ( 6.28 ksi )2 f peak = 8.22 ksi fave ( 3.53 ksi )2 + ( 4.19 ksi )2 = 5.48 ksi 2 2 ⎛ 1 ⎜ ( 3.67 ksi − 1.63 ksi ) + ( 6.28 ksi ) 2 2 2 ⎜⎜ + ( 5.30 ksi ) + ( 6.28 ksi ) ⎝ = 7.441 ksi = 1.25 fave = 1.25 ( 7.41 ksi ) = 9.26 ksi = ⎞ ⎟ ⎟ ⎟ ⎠ fave 2 2 ⎛ 1 ⎜ ( 2.45 ksi − 1.08 ksi ) + ( 4.19 ksi ) 2 2 2 ⎜⎜ ⎝ + ( 3.53 ksi ) + ( 4.19 ksi ) = 4.94 ksi = ⎞ ⎟ ⎟ ⎟ ⎠ 1.25 fave = 1.25 ( 4.94 ksi ) = 6.18 ksi AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 107 The direction of the load angle is calculated and then AISC Manual Equations 8-2a and 8-2b are used to determine the number of sixteenths of fillet weld required: LRFD θ = tan −1 ASD fun fuv θ = tan −1 ⎛ 5.30 ksi ⎞ = tan −1 ⎜ ⎟ ⎝ 6.28 ksi ⎠ = 40.2° Dreq’d = = fan fav ⎛ 3.53 ksi ⎞ = tan −1 ⎜ ⎟ ⎝ 4.19 ksi ⎠ = 40.1° fu weld t ( 2 (1.392 kip/in. ) 1.0 + 0.50 sin 1.5 θ ) Dreq’d = ( 9.26 ksi )( s in.) 2 (1.392 kip/in. ) (1.0 + 0.50 sin1.5 40.2° ) = = 1.65 sixteenths fa weld t ( 2 ( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 θ ) ( 6.18 ksi ) ( s in.) 2 ( 0.928 kip/in. ) (1.0 + 0.50 sin1.5 40.1° ) = 1.65 sixteenths From AISC Specification Table J2.4, the minimum fillet weld required is x in. for the 0.425-in. beam flange. Therefore, use a two-sided x-in. fillet weld. Check beam web local yielding The total equivalent normal force at the gusset-to-beam interface is (see Appendix B, Figure B-1): LRFD Nu equiv = Vub + ASD 2 Mub ⎛L⎞ ⎜2⎟ ⎝ ⎠ = 50.5 kips + N a equiv = Vab + 4 (123 kip-in. ) 2 M ab ⎛L⎞ ⎜2⎟ ⎝ ⎠ = 33.7 kips + 22.0 in. 4 ( 81.9 kip-in. ) 22.0 in. = 48.6 kips = 72.9 kips From AISC Specification Equation J10-3, because the resultant force is applied at a distance, α = 11.5 in., from the member end that is less than or equal to the depth of the W18×35 beam, the available strength for web local yielding is: LRFD ϕRn = ϕFyw tw ( 2.5kdes + lb ) = 1.00 ( 50 ksi )( 0.300 in. ) ⎡⎣2.5 ( 0.827 in. ) + 22.0 in.⎤⎦ = 361 kips > 72.9 kips o.k. ASD Rn Fyw t w ( 2.5kdes + lb ) = Ω Ω = ( 50 ksi )( 0.300 in. ) [2.5 ( 0.827 in. ) + 22.0 in.] 1.50 = 241 kips > 48.6 kips o.k. 108 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check beam web local crippling Using AISC Specification Equation J10-4 because the force to be resisted is applied a distance from the member end, α = 11.5 in., that is greater than or equal to d/ 2, the available strength for web local crippling is: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝ d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜ tf ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ ASD EFyw tf tw ⎡ ⎛ 22.0 in. ⎞ 2 = 0.75 ( 0.80 )( 0.300 in. ) ⎢1 + 3 ⎜ ⎟ ⎢⎣ ⎝ 17.7 in. ⎠ × 1.5 ⎡ ⎛ ⎞ ⎤ EFyw t f ⎢1 + 3 ⎛ lb ⎞ ⎜ tw ⎟ ⎥ ⎜ d⎟ ⎜t ⎟ ⎥ ⎢ tw ⎝ ⎠⎝ f⎠ Rn ⎣ ⎦ = Ω Ω 1.5 ⎡ ⎛ 22.0 in.⎞ ⎛ 0.300 in. ⎞ ⎤ ( 0.80 )( 0.300 in.)2 ⎢1 + 3 ⎜ ⎟⎜ ⎟ ⎥ ⎢⎣ ⎝ 17.7 in. ⎠ ⎝ 0.425 in.. ⎠ ⎥⎦ 0.80t w2 ⎛ 0.300 in.⎞ ⎜ ⎟ ⎝ 0.425 in..⎠ 1.5 ⎤ ⎥ ⎥⎦ ( 29, 000 ksi )( 50 ksi )( 0.425 in.) 0.300 in. = 249 kips > 72.9 kips o.k. × ( 29, 000 ksi )( 50 ksi )( 0.425 in.) 0.300 in. 2.000 = 166 kips > 48.6 kips o.k. Beam-to-Column Connection LRFD ASD Required shear strength, Vu = 80.5 kips Required flexural strength, Mu = 308 kip-in. Required shear strength, Va = 53.7 kips Required flexural strength, Ma = 206 kip-in. For convenience of calculation, the moment, Mu or Ma, can be converted into an equivalent axial force, Nu or Na (see Appendix B): N= 4M L where L is the length of the angles. Design bolts at beam-to-column connection The available tensile and shear strength are determined from AISC Manual Tables 7-1 and 7-2 for ASTM A325-N bolts. The equivalent axial force is: LRFD Nu = ASD 4 ( 308 kip-in. ) Na = 14.5 in. = 85.0 kips The required tensile strength per bolt is: 85.0 kips 10 bolts = 8.50 kips < ϕrnt = 40.6 kips 14.5 in. = 56.8 kips The required tensile strength per bolt is: rut = 56.8 kips 10 bolts = 5.68 kips < rnt Ω = 27.1 kips rat = o.k. The required shear strength per bolt is: 80.5 kips 10 bolts = 8.05 kips < ϕrnv = 24.3 kips 4 ( 206 kip-in. ) o.k. The required shear strength per bolt is: ruv = 53.7 kips 10 bolts = 5.37 kips < rnv Ω = 16.2 kips o.k. rav = o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 109 The available tensile strength considers the effects of combined tension and shear. From AISC Specification Section J3.7 and Table J3.2, the available tensile strengths are: LRFD Fnt′ = 1.3Fnt − ASD Fnt frv ≤ Fnt ϕFnv Fnt′ = 1.3Fnt − Fnt = 90 ksi Fnt = 90 ksi Fnv = 54 ksi Fnv = 54 ksi Fnt′ = 1.3 ( 90 ksi ) − 90 ksi ⎛ 8.05 kips ⎞ ⎜ ⎟ 0.75 ( 54 ksi ) ⎜⎝ 0.601 in.2 ⎟⎠ = 87.2 ksi < 90 ksi o.k. he reduced available tensile strength per bolt due to T combined forces is: ϕrnt = ϕFnt′ Ab ( = 0.75 ( 87.2 kips ) 0.601 in.2 = 39.3 kips > 8.50 kips ) o.k. ΩFnt frv ≤ Fnt Fnv 2.00 ( 90 ksi ) ⎛ 5.37 kips ⎞ ⎜⎜ 2⎟ ⎟ 54 ksi ⎝ 0.601 in. ⎠ = 87.2 ksi < 90 ksi o.k. Fnt′ = 1.3 ( 90 ksi ) − he reduced available tensile strength per bolt due to T combined forces is: rnt Fnt′ Ab = Ω Ω = (87.2 kips ) ( 0.601 in.2 ) 2.00 = 26.2 kips > 5.68 kips o.k. Check prying action on bolts and angles Following the notation of the AISC Manual Part 9: tw g − 2 2 0.300 in. 52 in. = 4 in. + − 2 2 = 1.40 in. a = leg width + b = g − tw − t 2 52 in. − 0.300 in. − s in. = 2 = 2.29 in. db ⎞ db ⎛ ≤ ⎜1.25b + ⎟ 2 ⎝ 2⎠ d in. = 1.40 in. + 2 = 1.84 in. a′ = a + db d in. = 1.25 ( 2.29 in. ) + 2 2 = 3.30 in. 1.84 in. < 3.30 in. o.k. 1.25b + 110 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 (Manual Eq. 9-27) b′ = b − db 2 (Manual Eq. 9-21) = 2.29 in. − d in. 2 = 1.85 in. b′ a′ 1.85 in. = 1.84 in. = 1.01 ρ = (Manual Eq. 9-26) Note that end distances of 14 in. are used on the angles, so p is the average pitch of the bolts: 14.5 in. 5 = 2.90 in. p= From AISC Specification Table J3.3, for a d-in.-diameter bolt, the standard hole dimension is , in. d′ p , in. = 1− 2.90 in. = 0.677 δ = 1− (Manual Eq. 9-24) Note that tc is the angle thickness required to eliminate prying action. It is, therefore, an upper bound on angle thickness. Any angle thicker than tc would provide no additional strength. LRFD From AISC Manual Equation 9-30a: tc = = 4 Bb′ ϕpFu tc = 4 ( 39.3 kips )(1.85 in. ) 0.90 ( 2.90 in. )( 58 ksi ) = 1.39 inn. = 4ΩBb′ pFu 4 (1.67 )( 26.2 kips )(1.85 in. ) ( 2.90 in.) ( 58 ksi ) = 1.39 inn. From AISC Specification Equation 9-35: ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.39 in.⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.677 (1 + 1.01) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ = 2.90 α′ = ASD From AISC Manual Equation 9-30b: From AISC Specification Equation 9-35: ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎮ δ (1 + ρ ) ⎢⎝ t ⎠ ⎮ ⎣ ⎦ ⎡⎛ 1.39 in. ⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎮ 0.677 (1 + 1.01) ⎢⎝ s in. ⎠ ⎮ ⎦ ⎣ = 2.90 α′ = AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 111 LRFD ASD Because α′ > 1, the angles, not the bolts, will govern the available strength of the connection. Use AISC Manual Equation 9-34: Because α′ > 1, the angles, not the bolts, will govern the available strength of the connection. Use AISC Manual Equation 9-34: 2 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 2 ⎛ s in. ⎞ =⎜ ⎟ (1 + 0.677 ) ⎝1.39 in.⎠ = 0.339 Tavail = BQ ⎛ s in. ⎞ =⎜ ⎟ (1 + 0.677 ) ⎝ 1.39 in. ⎠ = 0.339 Tavail = BQ = ( 39.3 kips )( 0.339 ) = 13.3 kips > 8.50 kips = ( 26.2 kips )( 0.339 ) o.k. = 8.88 kips > 5.68 kips o.k . Check prying action on column flange The fulcrum point for the column is governed by that of the angles. Follow the notation of Part 9 of the AISC Manual. The variables a and a′ are as calculated previously for the angle. g − tw 2 52 in. − 0.360 in. = 2 = 2.57 in. b = a ′ ≤ 1. 25b + db 2 = 1.25 ( 2.57 in. ) + = 3.65 in. 1.84 in. < 3.65 in. b′ = b − d in. 2 o.k. db 2 = 2.57 in. − = 2.13 in. (Manual Eq. 9-21) d in. 2 b′ a′ 2.13 in. = 1.84 in. = 1.16 ρ = p = 3.00 in. 112 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 (Manual Eq. 9-26) d′ p , in. = 1− 3.00 in. = 0.688 δ = 1− From AISC Manual Part 9, the available tensile strength including prying action effects, Tavail, is determined as follows: LRFD From AISC Manual Equation 9-30a: tc = = ASD From AISC Manual Equation 9-30b: 4 Bb′ ϕpFu tc = 4 ( 39.3 kips )( 2.13 in. ) = 0.90 ( 3.00 in. )( 65 ksi ) = 1.38 inn. 1.67 ( 4 )( 26.2 kips )( 2.13 in. ) ( 3.00 in.)( 65 ksi ) = 1.38 in. From AISC Manual Equation 9-35: α′ = Ω4 Bb′ pFu From AISC Manual Equation 9-35: ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ α′ = ⎡⎛ 1.38 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 1.16 ) ⎢⎝ 0.640 inn.⎠ ⎥⎦ ⎣ = 2.46 = Because α′ > 1, use Equation 9-34. ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.38 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.688 (1 + 1.16 ) ⎢⎝ 0.640 inn.⎠ ⎥⎦ ⎣ = 2.46 = Because α′ > 1, use Equation 9-34. 2 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 2 ⎛ 0.640 in.⎞ =⎜ ⎟ (1 + 0.688 ) ⎝ 1.38 in. ⎠ = 0.363 ⎛ 0.640 in. ⎞ =⎜ ⎟ (1 + 0.688 ) ⎝ 1.38 in. ⎠ = 0.363 From AISC Manual Equation 9-31: From AISC Manual Equation 9-31: Tavail = BQ Tavail = BQ = ( 39.3 kips )( 0.363 ) = 14.3 kips > 8.50 kips = ( 26.2 kips )( 0.363 ) o.k. = 9.51 kips > 5.68 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 113 Design angle-to-beam web welds The resultant force on the welds and resultant load angle are determined as follows: LRFD Required shear strength: Vu = 50.5 kips + 30 kips = 80.5 kips Required shear strength: Va = 33.7 kips + 20 kips = 53.7 kips Requuired equivalent axial strength: Nu equiv = 85.0 kips R= ASD (80.5 kips )2 + (85.0 kips )2 = 117 kips Reqquired equivalent axial strength: Na equiv = 56.8 kips R= ( 53.7 kips )2 + ( 56.8 kips )2 = 78.2 kips ⎛ 85.0 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 80.5 kips ⎠ = 46.6° ⎛ 56.8 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 53.7 kips ⎠ = 46.6° Refer to AISC Manual Table 8-8 with θ = 45°, kl = 3.5 in.: 3.50 in. 14.5 in. = 0.241 al = 4 − xl k = x = k2 (1 + 2k ) ( 0.241)2 = ⎡⎣1 + 2 ( 0.241) ⎤⎦ = 0.0392 xl = ( 0.0392 )(14.5 in. ) = 0.568 in. al = 4 in. − 0.568 in. = 3.43 in. 3.43 in. a = 14.5 in. = 0.237 By interpolation, C = 2.98. C1 is taken as 1.00 for E70 electrodes from AISC Manual Table 8-3. LRFD Dreq’d = = Pu ϕCC1l (117 kips ) 0.75 ( 2.98 )(1.00 )( 2 )(14.5 in. ) = 1.81 sixteenths ASD Dreq’d = = ΩPa CC1l 2.00 ( 78.2 kips ) 2.98 (1.00 )( 2 )(14.5 in. ) = 1.81 siixteenths 114 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Therefore, use a x-in. fillet weld (minimum size according to AISC Specification Table J2.4). Since there is no simple way to check the beam web under the “C” shaped weld, use the conservative AISC Manual Part 9 equation for the minimum beam web thickness: t min = 6.19 D Fu (Manual Eq. 9-3) For the present problem, D = 1.81. t min = 6.19 (1.81) 65 ksi = 0.172 in. < 0.300 in. o.k. Note: Further discussion of the weld design presented here is given in Appendix B. Check shear yielding on double angles From AISC Specification Section J4.2, the available shear yielding strength of the double angles is determined from Equation J4-3: LRFD ASD ϕRn = ϕ0.60 Fy Agv Rn Ω = 1.00 ( 0.60 )( 36 ksi )( 2 )(14.5 in. )(s in.) = 392 kips > 80.5 kips o.k. = = 0.60 Fy Agv Ω ( 0.60 )( 36 ksi )( 2 )(14.5 in.)( s in.) 1.50 = 261 kips > 53.7 kips o.k. Check shear rupture on double angles From AISC Specification Section J4.2, the available shear rupture strength of the double angles is determined through the line of bolts using Equation J4-4: LRFD ASD ϕRn = ϕ0.60 Fu Anv = 0.75 ( 0.60 )( 58 ksi )( 2 ) ⎡⎣14.5 in. − 5 (1.00 in. ) ⎤⎦ ( s in.) = 310 kips > 80.5 kips o.k. Rn 0.60 Fu Anv = Ω Ω 0.60 ( 58 ksi )( 2 ) ⎡⎣14.5 in. − 5 ( , in. + z in.)⎤⎦ ( s in.) = 2.00 = 207 kips > 53.7 kips o.k. Check block shear rupture on the double angle at the column flange Block shear is checked relative to the shear force assuming the controlling failure path would run through the bolt lines, and then perpendicular to the angle edge, as shown in Figure 5-10a. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = 2(134 in.)(s in.) = 16.6 in.2 ( ) 0.60Fy Agv = 0.60(36 ksi) 16.6 in.2 = 359 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 115 Shear rupture component: Anv = 16.6 in.2 − 4.5(2)(, in. + z in.)(s in.) = 11.0 in.2 ( ) 0.60Fu Anv = 0.60(58 ksi) 11.0 in.2 = 383 kips Tension rupture component: Ubs = 1 from AISC Specifications Section J4.3 because the bolts are uniformly loaded ⎡ 4 in. + 4 in. + 0.300 in. − 52 in. ⎤ Ant = 2 (s in.) ⎢ − 0.5 (, in. + z in.)⎥ 2 ⎦ ⎣ = 1.13 in.2 ( ) UbsFu Ant = 1(58 ksi) 1.13 in.2 = 65.5 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 383 kips + 65.5 kips = 449 kips 0.60Fy Agv + UbsFu Ant = 359 kips + 65.5 kips = 425 kips Because 449 kips > 425 kips, use Rn = 425 kips: LRFD ASD ϕRn = 0.75 ( 425 kips ) = 319 kips > 80.5 kips o.k. Rn 425 kips = Ω 2.00 = 213 kips > 53.7 kips o.k. Fig. 5-10a. Block shear rupture failure path on double angle. 116 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check shear yielding on beam web The available shear yielding strength of the beam web is determined using AISC Specification Equation J4-3 of Section J4.2: LRFD ASD ϕRn = ϕ0.60Fy Agv Rn Ω = 1.00(0.60)(50 ksi)(17.7 in.)(0.300 in.) = 159 kips > 80.5 kips o.k. = = 0.60 Fy Agv Ω 0.60 ( 50 ksi )(17.7 in. )( 0.300 in. ) 1.50 = 106 kips > 53.7 kips o.k. Check block shear rupture on beam web due to axial force Block shear rupture is checked relative to the axial force on the beam with the failure path running along the welds of the double angle to the beam web. By inspection, the shear yielding component will control; therefore, AISC Specification Equation J4-5 reduces to: Rn = 0.60Fy Agv + UbsFu Ant(from Spec. Eq. J4-5) where Agv = 2tLe = 2(0.300 in.)(3.50 in.) = 2.10 in.2 Ant = tL = (0.300 in.)(14.5 in.) = 4.35 in.2 The available block shear rupture strength of the beam web is: LRFD ( ) ASD ϕRn = ϕ 0.60 Fy Agv + U bs Fu Ant ( ) ( ) = 0.75 ⎡0.60 ( 50 ksi ) 2.10 in.2 + 1 ( 65 ksi ) 4.35 in.2 ⎤ ⎣ ⎦ = 259 kips > 85.0 kips o.k. Rn 0.60 Fy Agv + Fu Ant = Ω Ω = ( ) ( 0.60 ( 50 ksi ) 2.10 in.2 + ( 65 ksi ) 4.35 in.2 ) 2.00 = 173 kips > 56.8 kips o.k. Check bolt bearing on the column flange Because tf = 0.640 in. is greater than the s-in. angle thickness, bolt bearing on the column flange will not control the design. Check bolt bearing on the double angles Check bearing strength on the double angle at the top bolts, as the available bearing strength will have the lowest value at that location. Assuming that deformation at the bolt holes is a design consideration, use AISC Specification Equation J3-6a to determine the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) where lc is based on the angle edge distance of 14 in. lc = 14 in. − 0.5(, in.) = 0.781 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 117 The available bearing strength per bolt is determined as follows: LRFD ASD ϕ1.2lctFu = 0.75(1.2)(0.781 in.)(s in.)(58 ksi) = 25.5 kips/bolt 1.2lctFu/Ω = 1.2(0.781 in.)(s in.)(58 ksi)/ 2.00 = 17.0 kips/bolt ϕ2.4dtFu = 0.75(2.4)(d in.)(s in.)(58 ksi) = 57.1 kips/bolt 2.4dtFu/Ω = 2.4(d in.)(s in.)(58 ksi)/ 2.00 = 38.1 kips/bolt Therefore, ϕrn = 25.5 kips/bolt. Therefore, rn/Ω = 17.0 kips/bolt. Because 25.5 kips/bolt > 24.3 kips/bolt for LRFD and 17.0 kips/bolt >16.2 kips/bolt for ASD, bolt shear strength controls over the bearing strength for the top bolts (as well as the inner bolts, which would have a higher bearing strength due to the larger clear distance between the bolts). Therefore, the limit state of bearing does not apply. Example 5.5—Corner Connection-to-Column Web: General Uniform Force Method Given: Verify the corner bracing connection design shown in Figure 5-11 using the general uniform force method. The loading, member sizes, material strengths, bolt size and type, hole size, and connection element sizes are as given in Figure 5-11. The unbraced length of the brace is 24 ft. Use 70-ksi electrodes. The required strengths are: LRFD ASD Brace: ±Pu = 270 kips Beam: Vu = 60 kips Transfer force = ±37.5 kips Brace: ±Pa = 180 kips Beam: Va = 40 kips Transfer force = ±25 kips Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A36 Fy = 36 ksi Fu = 58 ksi From AISC Manual Tables 1-1 and 1-8, the member geometric properties are as follows: Beam W18×97 Ag = 28.5 in.2 tw = 0.535 in. d = 18.6 in. bf = 11.1 in. tf = 0.870 in. d = 16.0 in. bf = 15.9 in. tf = 1.72 in. h/ tw = 10.7 tw = 0.360 in. d = 12.2 in. bf = 10.0 in. tf = 0.640 in. kdes = 1.27 in. Column W14×233 tw = 1.07 in. Brace W12×58 Ag = 17.0 in.2 Brace-to-Gusset Connecting Element WT5×16.5 Ag = 4.85 in.2 tf = 0.435 in. y = 0.869 in. bf = 7.96 in. 118 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Fig. 5-11. General case—weak axis. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 119 From AISC Specification Table J3.3, the hole dimensions for 1-in.-diameter bolts are as follows: Brace and column Standard: dh = 1z in. diameter Angles Short slots: 1z in. × 1c in. From AISC Manual Tables 7-1 and 7-2, the available shear and tensile strengths of the bolts are: LRFD ASD ϕrnv = 31.8 kips (single shear at beam-to-column connection) rnv = 21.2 kips (single shear at beam-to-column connection) Ω rnv = 42.4 kips (double shear at brace-to-gusset connection) Ω rnt = 35.3 kips Ω ϕrnv = 63.6 kips (double shear at brace-to-gusset connection) ϕrnt = 53.0 kips The brace-to-gusset connection should be designed first to determine an approximate gusset size and then the remaining parts of the connection can be designed. The distribution of forces by the UFM is simpler in this case than it was for the connection to the column’s strong axis, as will be seen. Brace-to-Gusset Connection Determine the number of bolts required The number of bolts required is: LRFD n= ASD Pu ϕrnv n= 270 kips 63.6 kips/bolt = 4.25 bolts Pa rn Ω 180 kips 42.4 kips/bolt = 4.25 bolts = = This calculation indicates that two lines of three bolts will be sufficient to carry the brace force; however, Figure 5-11 shows five bolts per line. The reason for this difference will become apparent as the calculations proceed. This example will continue with the two lines of five bolts, as shown in Figure 5-11. Check tensile yielding on the brace From AISC Specification Section J4.1(a), Equation J4-1, the available tensile yielding strength is: LRFD ϕRn = ϕFy Ag ( = 0.90(50 ksi) 17.0 in. ASD ) 2 = 765 kips > 270 kips o.k. Rn Fy Ag = Ω Ω = ( 50 ksi ) (17.0 in.2 ) 1.67 = 509 kips > 180 kips o.k. 120 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check tensile rupture on the brace The net area of the brace is determined according to AISC Specification Section B4.3, with the bolt hole diameter, dh = 1z in. An = Ag − 2t(dh + z in.) = 17.0 in.2 − 2(0.360 in.)(1z in. + z in.) = 16.2 in.2 Because the flanges of the brace are not connected, an effective net area of the brace needs to be determined. From AISC Specification Section D3, the effective net area is: Ae = UAn where U = 0.70 from Table D3.1, Case 7 Alternatively, Table D3.1, Case 2 may also be used. From AISC Specification Commentary Figure C-D3.1, use x = 1.93 in. for the W12×58: x l 1.93 in. = 1− 4 ( 42 in. ) U = 1− = 0.893 The larger value of U may be used; therefore, the effective net area is: Ae = UAn ( ) = 0.893 16.2 in.2 2 = 14.5 in. From AISC Specification Section D2(b), the available strength for tensile rupture on the net section is: LRFD ϕtRn = ϕtFuAe ( ASD ) = 0.75(65 ksi) 14.5 in.2 = 707 kips > 270 kips o.k. Rn Fu Ae = Ωt Ωt = ( 65 ksi ) (14.5 in.2 ) 2.00 = 471 kips > 180 kips o.k. The designs of braces that are subjected to reversible loads are usually controlled by compression. The available compressive strength of a W12×58 is ϕcPn = 292 kips (LRFD) or Pn/ Ωc = 194 kips (ASD) at a length KL = 24 ft, per AISC Manual Table 4-1. The choice of a W12×58 brace to support a required strength of 270 kips (LRFD) and 180 kips (ASD) is appropriate. Check block shear rupture on the brace The block shear rupture failure path on the brace is assumed to run down both bolt lines and then across a perpendicular line connecting the two bolt lines. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 121 Shear yielding component: Agv = [4(42 in.) + 12 in.](0.360 in.)(2) = 14.0 in.2 0.60Fy Agv = 0.60(50 ksi)(14.0 in.2) = 420 kips Shear rupture component: Anv = 14.0 in.2 − 4.5(1z in. + z in.)(0.360 in.)(2) = 10.4 in.2 0.60Fu Anv = 0.60(65 ksi)(10.4 in.2) = 406 kips Tension rupture component: Ubs = 1 from AISC Specifications Section J4.3 because the bolts are uniformly loaded Ant = [5.00 in. − (0.5 + 0.5) (1z in. + z in.)](0.360 in.) = 1.40 in.2 UbsFu Ant = 1(65 ksi)(1.40 in.2) = 91.0 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 406 kips + 91.0 kips = 497 kips 0.60Fy Agv + UbsFu Ant = 420 kips + 91.0 kips = 511 kips Therefore, Rn = 497 kips. LRFD ϕRn = 0.75(497 kips) = 373 kips > 270 kips o.k. ASD Rn 497 kips = Ω 2.00 = 249 kips > 180 kips o.k. Note that the 42-in. bolt spacing in the brace is required to provide adequate available strength due to block shear rupture. A spacing of 3 in. would provide a reduced available strength of ϕRn = 246 kips for LRFD and Rn/ Ω = 164 kips for ASD. Check bolt bearing on the brace According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use Equation J3-6a: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 42 in. − (0.5 + 0.5)dh = 42 in. − 1(1z in.) = 3.44 in. 122 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available bearing strength per bolt is determined as follows: LRFD ASD ϕ1.2lctFu = 0.75(1.2)(3.44 in.)(0.360 in.)(65 ksi) = 72.4 kips/bolt 1.2lctFu/Ω = 1.2(3.44 in.)(0.360 in.)(65 ksi)/ 2.00 = 48.3 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(0.360 in.)(65 ksi) = 42.1 kips/bolt 2.4dtFu/Ω = (2.4)(1.00 in.)(0.360 in.)(65 ksi)/ 2.00 = 28.1 kips/bolt Therefore, ϕRn = 42.1 kips/bolt. Therefore, Rn/Ω = 28.1 kips/bolt. Because these strengths are less than the available bolt shear strength determined previously (63.6 kips for LRFD and 42.4 kips for ASD), the limit state of bolt bearing controls the strength of the inner bolts. For the edge bolts: lc = 12 in. − 0.5dh = 12 in. − 0.5(1z in.) = 0.969 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(0.969 in.)(0.360 in.)(65 ksi) = 20.4 kips 1.2lctFu/ Ω = 1.2(0.969 in.)(0.360 in.)(65 ksi)/ 2.00 = 13.6 kips ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(0.360 in.)(65 ksi) = 42.1 kips 2.4dtFu/ Ω = 2.4(1.00 in.)(0.360 in.)(65 ksi)/ 2.00 = 28.1 kips Therefore, ϕRn = 20.4 kips. Therefore, Rn/ Ω = 13.6 kips. Because these strengths are less than the available bolt shear strength determined previously (63.6 kips for LRFD and 42.4 kips for ASD), the limit state of bolt bearing controls the strength of the edge bolts. To determine the available strength of the bolt group, sum the individual effective strengths for each bolt. The total available strength of the bolt group is: LRFD ϕRn = (2 bolts)(20.4 kips) + (8 bolts)(42.1 kips) = 378 kips > 270 kips o.k. ASD Rn = (2 bolts)(13.6 kips) + (8 bolts)(28.1 kips) Ω = 252 kips > 180 kips o.k. Note that this is significantly less than the bolt shear strength of ϕRn = 636 kips for LRFD and Rn/Ω = 424 kips for ASD. Check the gusset plate for tensile yielding on the Whitmore section From AISC Manual Part 9, the width of the Whitmore section is: lw = 5 in. + 2(12 in.)tan 30° = 18.9 in. The Whitmore section enters the beam web, but tw = 0.535 in. > tg = 0.500 in., so the entire section is effective. For simplicity, use a thickness of 2 in. for the entire Whitmore area: Aw = (2 in.)(18.9 in.) = 9.45 in.2 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 123 From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate is: LRFD ASD Rn Fy Ag = Ω Ω ϕRn = ϕFy Aw = 0.90(36 ksi)(9.45 in.2) = 306 kips > 270 kips o.k. = ( 36 ksi ) ( 9.45 in.2 ) 1.67 = 204 kips > 180 kips o.k. From this calculation, it can be seen that Whitmore yield requires the two lines of 5 bolts. A thicker gusset plate could also be used. Check the gusset plate for compression buckling on the Whitmore section From Dowswell (2006), this is a compact corner gusset and buckling is not an applicable limit state. However, if buckling is checked, the available compressive strength based on the limit state of flexural buckling is determined from AISC Specification Section J4.4. Use the common AISC Manual value for the effective length factor of K = 0.5 (Gross, 1990) and the unbraced length of the gusset plate shown in Figure 5-11. KL 0.5 ( 82 in.) = r 2 in. 12 = 29.4 Because KL/ r > 25, use AISC Manual Table 4-22 to determine the available critical stress for the 36-ksi gusset plate. Then, the available strength can be determined from AISC Specification Sections E1 and E3: LRFD ASD ϕcFcr = 31.0 ksi ϕPn = ϕcFcr Aw Fcr Ωc = 20.6 ksi ( ) = (31.0 ksi) 9.45 in.2 = 293 ksi > 270 kips o.k. Pn Fcr A w = Ω Ωc ( = ( 20.6 ksi ) 9.45 in.2 = 195 kips > 180 kips ) o.k. Check block shear rupture on the gusset plate As demonstrated previously for the brace, using AISC Specification Section J4.3, determine the available strength due to block shear rupture on the 2-in.-thick gusset plate. Shear yielding component: Agv = 2(2 in.)[4(3.00 in.) + 12 in.] = 13.5 in.2 0.60Fy Agv = 0.60(36 ksi)(13.5 in.2) = 292 kips Shear rupture component: Anv = 13.5 in.2 − 4.5(1z in. + z in.)(2 in.)(2) = 8.44 in.2 0.60Fu Anv = 0.60(58 ksi)(8.44 in.2) = 294 kips 124 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Tension rupture component: Ubs = 1 from AISC Specifications Section J4.3 because the bolts are uniformly loaded Ant = (2 in.)[5.00 in. − (2 + 2)(1z in. + z in.)] = 1.94 in.2 UbsFu Ant = 1(58 ksi)(1.94 in.2) = 113 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 294 kips + 113 kips = 407 kips 0.60Fy Agv + UbsFu Ant = 292 kips + 113 kips = 405 kips Therefore, Rn = 405 kips. LRFD ϕRn = 0.75(405 kips) = 304 kips > 270 kips o.k. ASD Rn 405 kips = Ω 2.00 = 203 kips > 180 kips o.k. Check bolt bearing on the gusset plate According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use Equation J3-6a: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, assuming standard holes, the clear distance is: lc = 3.00 in. − (0.5 + 0.5)dh = 3.00 in. − 1(1z in.) = 1.94 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(1.94 in.)(2 in.)(58 ksi) = 50.6 kips 1.2lctFu/Ω = 1.2(1.94 in.)(2 in.)(58 ksi)/ 2.00 = 33.8 kips ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(2 in.)(58 ksi) = 52.2 kips 2.4dtFu/Ω = 2.4(1.00 in.)(2 in.)(58 ksi)/ 2.00 = 34.8 kips Therefore, ϕRn = 50.6 kips. Therefore, Rn/Ω = 33.8 kips. Because these strengths are less than the available bolt shear strength determined previously (63.6 kips for LRFD and 42.4 kips for ASD), the limit state of bolt bearing controls the strength of the inner bolts. For the edge bolts: lc = 12 in. − 0.5dh = 12 in. − 0.5(1z in.) = 0.969 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 125 LRFD ASD ϕ1.2lctFu = 0.75(1.2)(0.969 in.)(2 in.)(58 ksi) = 25.3 kips 1.2lctFu/Ω = 1.2(0.969 in.)(2 in.)(58 ksi)/ 2.00 = 16.9 kips ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(2 in.)(58 ksi) = 52.2 kips 2.4dtFu/Ω = 2.4(1.00 in.)(2 in.)(58 ksi)/ 2.00 = 34.8 kips Therefore, ϕRn = 25.3 kips. Therefore, Rn/Ω = 16.9 kips. Because these strengths are less than the available bolt shear strength determined previously (63.6 kips for LRFD and 42.4 kips for ASD), the limit state of bolt bearing controls the strength of the edge bolts. To determine the available strength of the bolt group, sum the individual effective strengths for each bolt. The total available strength of the bolt group is: LRFD ASD ϕRn = (2 bolts)(25.3 kips) + (8 bolts)(50.6 kips) = 455 kips > 270 kips o.k. Rn = (2 bolts)(16.9 kips) + (8 bolts)(33.8 kips) Ω = 304 kips > 180 kips o.k. Checks on double-tee connecting elements For these members, the critical side of the connection will be the gusset side, because the bolt spacing is 3 in. rather than 42 in. Therefore, the double-tee to the gusset plate connection will be checked in the following. Check tensile yielding on the double-tee connecting elements The gross area of the 2-WT5×16.5 is: Ag = 2(4.85 in.2) = 9.70 in.2 From AISC Specification Section J4.1(a), the available tensile yielding strength is: LRFD ASD ϕPn = ϕFyAg ( ) = 0.90(50 ksi) 9.70 in.2 = 437 kips > 270 kips o.k. Pn Fy Ag = Ω Ω = ( 50 ksi 9.70 in.2 ) 1.67 = 290 kips > 180 kips o.k. Check tensile rupture on the double-tee connecting elements The net area of the double-tee brace is determined in accordance with AISC Specification Section B4.3, with the bolt hole diameter, dh = 1z in., from AISC Specification Table J3.3: An = Ag − 2(2)t(dh + z in.) = 9.70 in.2 − 4(0.435 in.)(1z in. + z in.) = 7.74 in.2 Because the outstanding webs of the tees are not connected, an effective net area of the double tee needs to be determined. From AISC Specification Section D3, the effective net area is: Ae = AnU(Spec. Eq. D3-1) 126 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 where x from AISC Specification Table D3.1, Case 2 l 0.869 in. = 1− 12.0 in. = 0.928 Ae = AnU U = 1− ( ) = 7.74 in.2 (0.928) = 7.18 in.2 From AISC Specification Section D2(b), the available tensile rupture strength of the double-tee connecting element is: LRFD ASD ϕPn = ϕFu Ae ( ) = 0.75(65 ksi) 7.18 in.2 = 350 kips > 270 kips o.k. Pn Fu Ae = Ω Ω = ( 65 ksi ) ( 7.18 in.2 ) 2.00 = 233 kips > 180 kips o.k. Check block shear rupture on the double-tee connecting element The block shear failure path through the double-tee section is assumed to follow along the bolt lines on each side of the web and then perpendicular to the bolt line to the edge of the tee flange. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 and determined as previously discussed. Shear yielding component: Agv = 2(0.435 in.)[4(3.00 in.) + 12 in.] = 11.7 in.2 0.60Fy Agv = 0.60(50 ksi)(11.7 in.2) = 351 kips Shear rupture component: Anv = 11.7 in.2 − 4.5(2)(1z in. + z in.)(0.435 in.) = 7.30 in.2 0.60Fu Anv = 0.60(65 ksi)(7.30 in.2) = 285 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = 2(0.435 in.)[7.96 in. − 5.00 in. − 1(1z in. + z in.)] = 1.60 in.2 UbsFu Ant = 1(65 ksi)(1.60 in.2) = 104 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 285 kips + 104 kips = 389 kips 0.60Fy Agv + UbsFu Ant = 351 kips + 104 kips = 455 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 127 Therefore, Rn = 389 kips. LRFD ϕRn = 0.75(2)(389 kips) = 584 kips > 270 kips o.k. ASD Rn ⎛ 389 kips ⎞ = 2⎜ ⎟ Ω ⎝ 2.00 ⎠ = 389 kips > 180 kips o.k. Bolt bearing and tearout will not control for the WTs because the combined thickness of the double-tee flanges is greater than the thickness of the gusset plate, i.e., 2(0.435 in.) > 0.500 in. This completes the design calculations for the brace-to-gusset connection. These calculations will be the same for weak and strong-axis connections and for Special Cases 1, 2 and 3 in both the weak and strong axes. However, the calculations that follow will change depending on the case being considered. Connection Interface Forces The forces at the gusset-to-beam and gusset-to-column interfaces are determined using the general case of the UFM as discussed in AISC Manual Part 13. The geometric properties of the connection configuration given in Figure 5-11 are: ec = 0 (neglecting the column web thickness) db 2 18.6 in. = 2 = 9.30 in. eb = ⎛ 12 ⎞ θ = tan −1 ⎜ ⎟ ⎝ 9⎠ = 53.1° 312 in. + 2 in. 2 = 16.2 in. α= β = 4 in. + 5 in. = 9.00 in. In the preceding calculation, ec is assumed to equal zero. However, ec is actually half the column web thickness, or 1.07 in./ 2 = 0.535 in. If ec = 0.535 in. were used in the calculations, it would result in a force of: LRFD Huc = ( 270 kips )( 0.535 in. ) cos 53.1° ( 9.00 in. + 9.30 in. ) = 4.74 kips ASD H ac = (180 kips )( 0.535 in. ) cos 53.1° ( 9.00 in. + 9.30 in. ) = 3.16 kips Considering the other forces on the connection, this force is negligible and will not change the resulting design in any significant way. Therefore, for ease of calculation, ec is taken as zero, which results in Huc = 0 and Hac = 0. Because ec = 0, α can be taken as α, and β is computed from: α − β tan θ = eb tan θ − ec 128 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 (Manual Eq. 13-1) β= α − ( eb tan θ ) + ec tan θ 16.2 in. − ⎡( 9.30 in. ) tan 53.1°⎤ + 0 ⎣ ⎦ tan 53.1° = 2.86 in. = This value of β is used in all the subsequent equations. The fact that β ≠ β is unimportant, because Huc is 0 and therefore, there is no moment [Mc = Hc(β − β)]. r= = ( α + ec )2 + (β + eb )2 (Manual Eq. 13-6) (16.2 in. + 0 )2 + ( 2.86 in. + 9.30 in. )2 = 20.3 in. LRFD ASD Pu 270 kips = r 20.3 in. = 13.3 kip/in. Pa 180 kips = r 20.3 in. = 8.87 kip/in. From AISC Manual Equation 13-5: From AISC Manual Equation 13-5: Pu r = (16.2 in. )(13.3 kip/in. ) Pa r = (16.2 in. )( 8.87 kip/in. ) Hub = α H ab = α = 215 kips = 144 kips From AISC Manual Equation 13-3: From AISC Manual Equation 13-3: Pu r = ( 0 in. )(13.3 kip/in. ) Pa r = ( 0 in. )( 8.87 kip/in. ) = 0 kips = 0 kips From AISC Manual Equation 13-4: From AISC Manual Equation 13-4: Huc = ec H ac = ec Pa r = ( 9.30 in. )( 8.87 kip/in. ) Pu r = ( 9.30 in. )(13.3 kip/in. ) Vab = eb Vub = eb = 82.5 kips = 124 kips From AISC Manual Equation 13-2: From AISC Manual Equation 13-2: Pa r = ( 2.86 in. )( 8.87 kip/in. ) Pu r = ( 2.86 in. )(13.3 kip/in. ) Vuc = β Vac = β = 38.0 kips ∑ V = 124 kips + 38.3 kips = 162 kips = 25.4 kips ∑ V = 82.5 kips + 25.4 kips = 108 kips ( 270 kips ) cos 53.1° = 162 kips o.k. (180 kips ) cos 53.1° = 108 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 129 These admissible forces are shown in Figure 5-12. The figure shows that the line of action of the gusset-to-beam interface passes through the control point at the centerline of the beam and face of the column web. As discussed previously, this point and the work point differ only by the column web half thickness, which is small compared to the other dimensions and is therefore neglected in the calculations. Because the location of the control point is known and point A is at the center of the gusset-to-beam connection, the interface forces can be calculated as: LRFD ASD Hub = Hu = 215 kips Hab = Ha = 144 kips eb Hu α ⎛ 9.30 in. ⎞ =⎜ ⎟ ( 215 kips ) ⎝ 16.2 in. ⎠ = 123 kips eb Ha α ⎛ 9.30 in. ⎞ =⎜ ⎟ (144 kips ) ⎝ 16.2 in. ⎠ = 82.7 kips Vub = Vab = These values are similar to that achieved by the UFM, Vub = 124 kips for LRFD and Vab = 82.5 kips for ASD (the difference is due to rounding). If more significant figures were carried, the results would be exactly the same. Once Vub (Vab) and Hub (Hab) are known, all the other forces of Figure 5-12 can be determined by a free body diagram. Therefore, for weak-axis connections, either the simplified method given in the foregoing or the formal UFM can be used. α α Fig. 5-12a. Admissible force fields for Figure 5-11 connection—LRFD. 130 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Gusset-to-Beam Connection Check the gusset plate for shear yielding along the beam flange The available shear yielding strength of the gusset plate is determined from AISC Specification Section J4.2, Equation J4-3, as follows: LRFD ASD Vn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( 2 in.) ( 312 in. ) = 1.50 = 227 kips > 144 kips o.k. ϕVn = ϕ0.60Fy Agv = 1.00(0.60)(36 ksi)(2 in.)(312 in.) = 340 kips > 215 kips o.k. Check the gusset plate for tensile yielding along the beam flange The available tensile yielding strength of the gusset plate is determined from AISC Specification Section J4.1, Equation J4-1, as follows: α α Fig. 5-12b. Admissible force fields for Figure 5-11 connection—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 131 LRFD ASD = 0.90(36 ksi)(2 in.)(312 in.) N n Fy Ag = Ω Ω ( 36 ksi ) ( 2 in.) ( 312 in.) = 1.67 = 340 kips > 82.5 kips o.k. ϕNn = ϕFy Ag = 510 kips > 124 kips o.k. Note that as shown in a previous example, the interaction of the forces at the gusset-to-beam interface is assumed to have no impact on the design. Design weld at gusset-to-beam flange connection According to AISC Manual Part 13, because the gusset is directly welded to the beam, the weld size determination includes a ductility factor of 1.25. Therefore, the weld is designed for 1.25 times the resultant force. The strength determination of fillet welds defined in AISC Specification Section J2.4 can be simplified as explained in AISC Manual Part 8 to AISC Manual Equations 8-2a and 8-2b. Also, incorporating the increased strength due to the load angle from AISC Specification Equation J2-5, the required weld size is determined as follows: LRFD ASD Resultant load: Resultant load: ( 215 kips )2 + (124 kips )2 Ru = = 248 kips = 166 kips Resultant load angle: Resultant load angle: ⎛ 124 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 215 kips ⎠ ⎛ 82.5 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 144 kips ⎠ = 30.0° Dreq = = (144 kips )2 + (82.5 kips )2 Ra = = 29.8° 1.25 Ru ( 2l (1.392 kip/in. ) 1.0 + 0.50sin θ 1.5 1.25 ( 248 kips ) ( ) 2 (312 in.)(1.392 kip/in. ) 1.0 + 0.50sin1.5 30° = 3.00 sixteenths Dreq = ) = 1.25R a ( 2l ( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 θ 1.25 (166 kips ) ( ) 2 ( 312 in.)( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 29.8° ) = 3.02 sixteenths From AISC Specification Table J2.4, the AISC minimum fillet weld size is x in., but many fabricators use 4 in. minimum; therefore, use 4-in. fillet welds. Check beam web local yielding The normal force is applied at a distance α = 16.2 in. from the beam end, which is less than the depth of the W18×97 beam. Therefore, use AISC Specification Equation J10-3 to determine the available strength due to beam web local yielding: 132 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ϕRn = ϕFy tw(2.5k + lb) = 1.00(50 ksi)(0.535 in.)[2.5(1.27 in.) + 312 in.] = 928 kips > 124 kips o.k. Rn Fy t w ( 2.5k + lb ) = Ω Ω ( 50 ksi )( 0.535 in.) ⎡⎣2.5 (1.27 in.) + 312 in.⎤⎦ = 1.50 = 618 kips > 82.5 kips o.k. Check beam web local crippling The normal force is applied at a distance, α = 16.2 in., from the beam end, which is greater than d/ 2, where d is the depth of the W18×97 beam. Therefore, determine the beam web local crippling strength from AISC Specification Equation J10-4: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝ d⎠ ⎣ ASD 1.5 ⎤ ⎡ Rn 1 ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFyw t f 2 ⎢ = 0.80t w 1 + 3 ⎜ ⎟ ⎜ ⎟ ⎢ tw Ω Ω ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ 1.5 ⎤ ⎛ tw ⎞ EFyw t f ⎜ ⎟ ⎥ ⎜t f ⎟ ⎥ tw ⎝ ⎠ ⎦ 1.5 2⎡ ⎛ 312 in.⎞ ⎛ 0.535 in.⎞ ⎤ = 0.75 ( 0.80 )( 0.535 in. ) ⎢1 + 3 ⎜ ⎥ ⎟ ⎜ ⎟ ⎝ 18.6 in.⎠ ⎝ 0.870 in.⎠ ⎥⎦ ⎢⎣ × ( 29, 000 ksi )( 50 ksi ) ( 0.870 in. ) 0.535 in. = 910 kips > 124 kips o.k. = ⎡ 1 312 in.⎞ ⎛ 0.535 in.⎞ ( 0.80 )( 0.535 in. )2 ⎢1 + 3 ⎛⎜ ⎟⎜ ⎟ 2.00 ⎝ 18.6 in.⎠ ⎝ 0.870 in.⎠ ⎢⎣ × 1.5 ⎤ ⎥ ⎥⎦ ( 29, 000 ksi )( 50 ksi ) ( 0.870 in.) 0.535 in. = 607 kips > 82.5 kips o.k. Gusset-to-Column Connection This connection must support a required shear strength of 38.3 kips (LRFD) or 25.5 kips (ASD). Check bolt shear strength The required shear strength per bolt (six bolts total) is: LRFD ASD 38.3 kips = 6.38 kips/bolt 6 25.5 kips = 4.25 kips/bolt 6 From AISC Manual Table 7-1,the available shear strength of 1-in.-diameter ASTM A325-N bolts in single shear is: LRFD ϕrnv = 31.8 kips/bolt > 6.38 kips/bolt o.k. ASD rnv = 21.2 kips /bolt > 4.25 kips/bolt Ω o.k. Check bolt bearing on the angles Check the bearing strength at the top bolts on the angles. The clear distance, based on the 14-in. edge distance is: lc = 14 in. − 0.5(1z in.) = 0.719 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 133 Assuming that deformation at the bolt hole is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength. Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) The available bearing strength is determined as follows: LRFD ϕ1.2lctFu = 0.75(1.2)(0.719 in.)(c in.)(58 ksi) = 11.7 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(c in.)(58 ksi) = 32.6 kips/bolt Therefore, ϕrn = 11.7 kips/bolt. ASD 1.2lc tFu 1.2 ( 0.719 in. ) ( c in.) ( 58 ksi) = Ω 2.00 = 7.82 kips 2.4 dtFu 2.4 (1.00 in. ) ( c in.) ( 58 ksi) = Ω 2.00 = 21.8 kips Therefore, rn Ω = 7.82 kips/bolt. For the top bolts, because the available strength due to bolt bearing on the angles is less than the bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD), the limit state of bolt bearing controls the strength of the top bolts. For the inner bolts in the angles, the clear distance is: lc = 5.00 in. − 1(1z in.) = 3.94 in. The available bearing strength is determined as follows: LRFD ϕ1.2lctFu = 0.75(1.2)(3.94 in.)(c in.)(58 ksi) = 64.3 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(c in.)(58 ksi) = 32.6 kips/bolt Therefore, ϕrn = 32.6 kips/bolt. ASD 1.2lc tFu 1.2 ( 3.94 in. ) ( c in.) ( 58 ksi) = Ω 2.00 = 42.8 kips/boolt 2.4 (1.00 in. ) ( c in.) ( 58 ksi) 2.4 dtFu = Ω 2.00 = 21.8 kips/bolt Therefore, rn Ω = 21.8 kips/bolt. For the inner bolts, because the available shear strength of the bolts is less than the bolt bearing strength, the limit state of bolt shear controls the strength of the inner bolts. Check bolt bearing on the column web Because tw/ 2 = 1.07 in./ 2 = 0.535 in. > angle thickness of c in., bolt bearing on the column web will not control the design, by inspection. Note that in this check, one half of the column web thickness is used. The other one half is reserved for connections, if any, on the other side of the column web. If there are no connections on the other side, the entire web thickness of 1.07 in. may be used here. This is the simplest way to deal with this problem, and is correct for connections of the same size and load on both sides. It is conservative for any other case. Considering the limit states of bolt shear and bolt bearing on the angles, the available strength of the six bolts is: LRFD ASD ϕRn = (2 bolts)(11.7 kips/bolt) + (4 bolts)(31.8 kips/bolt) Rn = ( 2 bolts )( 7.82 kips/bolt ) + ( 4 bolts )( 21.2 kips/bolt ) Ω = 100 kips > 25.5 kips o.k. = 151 kips > 38.3 kips o.k. 134 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check shear yielding on the angles The available shear yielding strength from AISC Specification Section J4.2(a) is: LRFD ϕVn = ϕ0.60Fy Agv = ϕ0.60Fy lata(2) = 1.00(0.60)(36 ksi)(122 in.)(c in.)(2) = 169 kips > 38.3 kips o.k. ASD Vn 0.60 Fy Agv = Ω Ω 0.60 Fy la t a ( 2 ) = Ω 0.60 ( 36 ksi )(122 in.)( c in.) ( 2 ) = 1.50 = 113 kips > 25.5 kips o.k. Check shear rupture on the angles The net area is determined from AISC Specification Section B4.3: Anv = [122 in. − 3(1z in. + z in.)](c in.)(2) = 5.70 in.2 The available shear rupture strength from AISC Specification Section J4.2(b) is: LRFD ϕVn = ϕ0.60Fu Anv = 0.75(0.60)(58 ksi)(5.70 in.2) = 149 kips > 38.3 kips o.k. ASD Rgv Ω = = 0.60 Fu Anv Ω ( 0.60 ( 58 ksi ) 5.70 in.2 2.00 = 99.2 kips > 25.5 kips ) o.k. Check block shear rupture on the angles The controlling block shear failure path is assumed to start at the top of the angles and follows the vertical bolt lines and then turns perpendicular to the edge of each angle. For the tension area calculated in the following, the net area is based on the length of the short-slotted hole dimension given previously. For the shear area, the net area is based on the width of the short-slotted hole, which is the same as for a standard hole. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = (122 in. − 14 in.)(c in.)(2) = 7.03 in.2 0.60Fy Agv = 0.60(36 ksi)(7.03 in.2) = 152 kips Shear rupture component: Anv = 7.03 in.2 − 2.5(1z in. + z in.)(c in.)(2) = 5.27 in.2 0.60Fu Anv = 0.60(58 ksi)(5.27 in.2) = 183 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 135 Tension rupture component: U bs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded 4.00 in. + 4.00 in. + 2 in. − 52 in. Le = − 0.50 ( 1c in. + z in.) 2 = 0.813 in. Ant = tLe ( 2 ) = ( c in.)( 0.813 in. )( 2 ) = 0.508 in.2 UbsFu Ant = 1(58 ksi)(0.508 in.2) = 29.5 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 183 kips + 29.5 kips = 213 kips 0.60Fy Agv + UbsFu Ant = 152 kips + 29.5 kips = 182 kips Therefore, Rn = 182 kips. LRFD ASD Rn Fu Ant + 0.6 Fy Agv = Ω Ω 182 kips = 2.00 = 91.0 kips > 25.5 kips ϕRn = 0.75(182 kips) = 137 kips > 38.3 kips o.k. o.k. Check block shear rupture on the gusset plate It is possible for the gusset to fail in block shear rupture under the angles at the weld lines. This will be an L-shaped tearout. By inspection, the shear yielding component will control over the shear rupture component; therefore, AISC Specification Equation J4-5 reduces to: Rn = 0.60Fy Agv + UbsFu Ant(from Spec. Eq. J4-5) where Agv = tg(Lev + Langle) = (2 in.)(2.50 in. + 122 in.) = 7.50 in.2 Ant = tgLeh = (2 in.)(3.50 in.) = 1.75 in.2 Ubs = 1 The available block shear rupture strength of the gusset plate at the angle-to-gusset plate weld is: LRFD ϕRn = ϕ(0.60Fy Agv + UbsFu Ant) = 0.75[0.60(36 ksi)(7.50 in.2) + 1(58 ksi)(1.75 in.2)] = 198 kips > 38.3 kips o.k. ASD Rn 0.60 Fy Agv + U bs Fu Ant = Ω Ω = ( ) ( 0.60 ( 36 ksi ) 7.50 in.2 + 1 ( 58 ksi ) 1.75 in.2 = 132 kips > 25.5 kips 2.00 o.k. 136 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ) Design gusset plate-to-angles weld From the AISC Manual Table 8-8: Angle = 0° kl = 3.50 in. l = 122 in. 3.50 in. k = 122 in. = 0.280 The distance to the center of gravity from the vertical weld line can be determined as follows: x = = k2 1 + 2k ( 0.280 )2 1 + 2 ( 0.280 ) = 0.050 xl = 0.050 ( 122 in.) = 0.625 inn. al = 4.00 in. − xl = 4.00 in. − 0.625 in. = 3.38 in. al a = l 3.38 in. = 122 in. = 0.270 By interpolation: C = 2.80 from Table 8-8 and C1 = 1.0 for E70XX electrodes The number of sixteenths of fillet weld required is: LRFD Dreq = = Pu ϕCC1l ASD Dreq = 38.3 kips 0.75 ( 2.80 )(1.0 ) ( 122 in.) ( 2 ) = 0.730 siixteenths = Pa Ω CC1l ( 25.5 kips )( 2.00 ) 2.80 (1.0 ) ( 122 in.) ( 2 ) = 0.729 sixteenths Use a minimum fillet weld of x in. according to AISC Specification Table J2.4 for a thickness of c in. for the thinner part joined (the angle). Using the formula tmin = 6.19D/Fu (AISC Manual Equation 9-3) is not required in this case, because the gusset had to be checked for the block shear rupture limit state. Generally, the tmin formula should only be used when it is not possible to estimate the stresses in the gusset under the weld. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 137 Beam-to-Column Connection The maximum forces on this interface, as shown in Figure 5-12, are: LRFD Required shear strength ASD Required shear strength Vu = 124 kips + 60.0 kips Va = 82.5 kips + 40.0 kips = 184 kips Required axial strength = 123 kips Required axial strength Aub = 37.5 kips transfer force Aab = 25.0 kips transfer force Except for the 60-kip (LRFD) and 40-kip (ASD) gravity beam shear, these loads can reverse; therefore, the worst case envelope of shear and axial forces is considered. Design bolts at beam-to-column connection Using the available strengths from AISC Manual Tables 7-1 and 7-2, check the shear and tension per bolt: LRFD Vu = Vu Nb Va = 184 kips 10 bolts = 18.4 kips/bolt < 31.8 kips o.k. = Tu = ASD Aub Nb 123 kips 10 bolts = 12.3 kips/bolt < 21.2 kips o.k. = Ta = 37.5 kips 10 bolts = 3.75 kips/bolt < 53.0 kips o.k. = Va Nb Aab Nb 25.0 kips 10 bolts = 2.50 kips/bolt < 35.3 kips o.k. = The tensile strength requires an additional check due to the combination of tension and shear. From AISC Specification Section J3.7 and AISC Specification Table J3.2, the available tensile strength of ASTM A325-X bolts subject to tension and shear is: LRFD Fnt′ = 1.3Fnt − Fnt frv ≤ Fnt ϕFnv Fnt = 90 ksi Fnv = 54 ksi Fnt′ = 1.3 ( 90 ksi ) − ASD Fnt′ = 1.3Fnt − ΩFnt frv ≤ Fnt Fnv Fnt = 90 ksi Fnv = 54 ksi 90 ksi ⎛ 18.4 kips ⎞ ⎜ ⎟ 0.75 ( 54 ksi ) ⎜⎝ 0.785 in.2 ⎟⎠ = 64.9 ksi < 90 ksi o.k. 2.00 ( 90 ksi ) ⎛ 12.3 kips ⎞ ⎜⎜ 2⎟ ⎟ 54 ksi ⎝ 0.785 in. ⎠ = 64.8 ksi < 90 ksi o.k. Fnt′ = 1.3 ( 90 ksi ) − 138 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD The reduced available tensile force per bolt due to combined forces is: ϕrnt = ϕF′nt Ab ASD The reduced available tensile force per bolt due to combined forces is: ϕrnt = = 0.75(64.9 kips)(0.785 in.2) = 38.2 kips > 3.75 kips o.k. = Fnt′ Ab Ω ( 65.2 ksi ) ( 0.785 in.2 ) 2.00 = 25.6 kips > 2.50 kips o.k. Check prying action on the angles From AISC Manual Part 9: gage − t w − t 2 52 in. − 0.535 in. − 2 in. = 2 = 2.23 in. b = b′ = b − db 2 = 2.23 in. − (Manual Eq. 9-21) 1.00 in. 2 = 1.73 in. a= = 2 ( angle leg ) + t w − gage 2 2 ( 4.00 in. ) + 0.535 in. − 52 in. 2 = 1.52 in. d ⎞ d ⎞ ⎛ ⎛ a′ = ⎜ a + b ⎟ ≤ ⎜1.25b + b ⎟ 2 2⎠ ⎝ ⎠ ⎝ 1.00 in.⎞ ⎡ 1.00 in. ⎤ ⎛ = ⎜ 1.52 in. + ≤ ⎢1.25 ( 2.23 in. ) + ⎟ 2 ⎠ ⎣ 2 ⎥⎦ ⎝ = 2.02 in. < 3.29 in. b′ a′ 1.73 in. = 2.02 in. = 0.856 ρ= (Manual Eq. 9-27) (Manual Eq. 9-26) 142 in. 5 = 2.90 in. ≤ 2b = 2(2.23 in.) = 4.46 in. p= AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 139 Using the width dimension along the length of the fitting given in AISC Specification Table J3.3 for a 1-in.-diameter bolt: d ′ = 1z in. d′ δ = 1− p 1z in. = 1− 2.90 in. = 0.634 From AISC Manual Part 9, the available tensile strength including prying action effects is determined as follows: LRFD From AISC Manual Equation 9-30a: tc = = 4 Bb′ ϕpFu ASD From AISC Manual Equation 9-30b: tc = 4 ( 38.2 kips )(1.73 in. ) 0.90 ( 2.90 in. )( 58 ksi ) = 1.32 in. = 4ΩBb ′ pFu 4 (1.67 )( 25.6 kips )(1.73 in. ) ( 2.90 in. ) ( 58 ksi ) = 1.33 in. ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.32 in.⎞ 2 ⎤ 1 − 1⎥ = ⎢ 0.634 (1 + 0.856 ) ⎢⎣⎜⎝ 2 in. ⎟⎠ ⎥⎦ = 5.07 α′ = Because α′ > 1, determine Q from AISC Manual Equation 9-34: 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.33 in. ⎞ 2 ⎤ 1 − 1⎥ = ⎢ 0.634 (1 + 0.856 ) ⎢⎣⎜⎝ 2 in. ⎟⎠ ⎥⎦ = 5.16 α′ = Because α′ > 1, determine Q from AISC Manual Equation 9-34: 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎛ 2 in. ⎞ =⎜ ⎟ (1 + 0.634 ) ⎝ 1.32 in. ⎠ = 0.234 ⎛ 2 in. ⎞ =⎜ ⎟ (1 + 0.634 ) ⎝ 1.33 in.⎠ = 0.231 From AISC Manual Equation 9-31: From AISC Manual Equation 9-31: Tavail = BQ = ( 38.2 kips )( 0.234 ) = 8.94 kips > 3.75 kips o.k. Tavail = BQ = ( 25.6 kips )( 0.231) = 5.91 kips > 2.50 kips o.k. The transfer force, Aub = 37.5 kips for LRFD and Aab = 25.0 kips for ASD, usually comes from the other side of the column, but it could be induced into the bracing from a column shear. If this is the case, the web may “oil can,” and the above prying analysis will be incorrect and unconservative. This occurs only when there is no matching connection on the opposite side of the column web. If the connection on the opposite side is also designed for the transfer force, there is no “oil-canning.” Figure 5-13 shows this “oil-canning” effect as a yield line problem where the transverse force required to produce the yield line deformation is (Tamboli, 2010): 140 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ⎡ 2h ⎤ l Pn = 8m p ⎢ + ⎥ ⎢⎣ h − g 2 ( h − g ) ⎥⎦ where mp = g h l tw Fy t w2 4 = gage of bolts, in. = clear distance between flange fillets, in. = depth of bolt pattern, in. = column web thickness, in. thus: mp = ( 50 ksi )(1.07 in.)2 4 = 14.3 kip-in./in. u= h(h − g) 2 Fig. 5-13. “Oil can” yield line pattern for transversely loaded column web. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 141 Therefore: h ⎛h⎞ = tw ⎜ ⎟ ⎝ tw ⎠ = (1.07 in. )(10.7 ) = 11.4 in. l = 12.0 in. g = 52 in. h − g = 11.4 in. − 52 in. = 5.90 in. Assuming ϕ = 0.90 and Ω = 1.67, the available tensile strength of the web is: LRFD ASD ⎧⎪ ⎛ kip-in. ⎞ ⎡ 2 (11.4 in. ) 12.0 in. ⎤ ⎫⎪ ⎢ ⎥⎬ ϕPn = 0.90 ⎨8 ⎜14.3 + 2 ( 5.90 in. ) ⎥ ⎪ in. ⎟⎠ ⎢ 5.90 in. ⎣ ⎦⎭ ⎩⎪ ⎝ = 307 kips Pn 1 ⎧⎪ ⎛ kip-in. ⎞ ⎡ 2 (11.4 in. ) 12.0 in.. ⎤ ⎫⎪ ⎢ ⎥⎬ = + ⎨8 ⎜14.3 2 ( 5.90 in. ) ⎥ ⎪ Ω 1.67 ⎪ ⎝ in. ⎟⎠ ⎢ 5.90 in. ⎣ ⎦⎭ ⎩ = 204 kips The column web itself can carry a transverse load of 307 kips (LRFD) and 204 kips (ASD), but the issue of concern here is stiffness. The authors suggest that unless the oil-can strength of the column web is at least 10 times the specified transfer force the web will not be stiff enough to cause the prying force, q, to develop at the angle tips, i.e., the “a” distance. In the case where the oil-can strength of the column web is less than 10 times the specified transfer force, it is assumed that the web deforms, and q cannot exist because the angle tips separate from the column web. In this case, the prying action analysis must be replaced by a nonprying analysis of the angles, which considers only bending in the angles without the additional prying force. This analysis is provided in AISC Manual Part 9, Equation 9-20, specifically the formula for tmin, where: LRFD t min = ASD 4Tb′ ϕpFu Because t min = 4ΩTb ′ pFu Because ϕPn = 307 kips < 10 ( 37.5 kips ) = 375 kips the alternate analysis gives: t min = 4 ( 3.75 kips )(1.73 in. ) 0.90 ( 2.90 in. )( 58 ksi ) = 0.414 in.. < 2 in. o.k. Pn = 204 kips Ω < 10 ( 25.0 kips ) = 250 kips the alternate analysis gives: t min = 4 (1.67 )( 2.50 kips )(1.73 in. ) ( 2.90 in. )( 58 ksi ) = 0.414 in. < 2 in. o.k. The 2L4×4×½ have sufficient thickness. If instead of knowing the required angle thickness, it is desired to know what load the 2L4×4×½ angles can carry, the parameter α′ can be set equal to zero, which is equivalent to having the angle legs in single curvature bending due to the prying force, q, being equal to zero. 142 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD tc = ASD 4 Bb′ ϕpFu tc = = 1.32 in., as before ⎛ t⎞ Q=⎜ ⎟ ⎝ tc ⎠ 4ΩBb′ pFu = 1.33 in., as before 2 ⎛ t⎞ Q =⎜ ⎟ ⎝ tc ⎠ ⎛ 2 in. ⎞ =⎜ ⎟ ⎝1.32 in.⎠ = 0.143 2 ⎛ 2 in. ⎞ =⎜ ⎟ ⎝ 1.33 in.⎠ = 0.141 Tavail = BQ 2 Tavail = BQ = ( 38.2 kips )( 0.143 ) = 5.46 kips > 3.75 kips 2 = ( 25.6 kips )( 0.141) o.k. = 3.61 kips > 2.50 kips o.k. Check bolt bearing on the angles According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 3.00 in. − (0.5 + 0.5)dh = 3.00 in. − 1(1z in.) = 1.94 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(1.94 in.)(2 in.)(58 ksi) = 50.6 kips 1.2lctFu /Ω = 1.2(1.94 in.)(2 in.)(58 ksi)/ 2.00 = 33.8 kips ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(2 in.)(58 ksi) = 52.2 kips 2.4dtFu /Ω = 2.4(1.00 in.)(2 in.)(58 ksi)/ 2.00 = 34.8 kips Therefore, ϕRn = 50.6 kips. Therefore, Rn /Ω = 33.8 kips. Because the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD) is less than the bearing strength, the limit state of bolt shear controls the strength of the inner bolts. For the end bolts: lc = 14 in. − 0.5dh = 14 in. − 0.5(1z in.) = 0.719 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 143 LRFD ASD ϕ1.2lctFu = 0.75(1.2)(0.719 in.)(2 in.)(58 ksi) = 18.8 kips 1.2lctFu / Ω = 1.2(0.719 in.)(2 in.)(58 ksi)/ 2.00 = 12.5 kips ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(2 in.)(58 ksi) = 52.2 kips 2.4dtFu / Ω = (2.4)(1.00 in.)(2 in.)(58 ksi)/ 2.00 = 34.8 kips Therefore, ϕRn = 18.8 kips. Therefore, Rn / Ω = 12.5 kips. This controls over the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD). The total available bolt strength at the angles, including the limit states of bolt shear and bolt bearing, is: LRFD ASD ϕRn = 18.8 kips (2) + 31.8 kips (8) Rn = 12.5 kips ( 2 ) + 21.2 kips ( 8 ) Ω = 195 kips > 123 kips o.k. = 292 kips > 184 kips o.k. Check bolt bearing on the column web According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) Tearout, the limit state captured by the first part of Equation J3-6a, is not applicable to the column because there is no edge through which the bolts can tear out. LRFD ϕrn = ϕ2.4 dtFu ⎛ 1.07 in.⎞ = 0.75 ( 2.4 )(1.00 in. ) ⎜ ⎟ ( 65 ksi ) ⎝ 2 ⎠ = 62.6 kips ASD rn 2.4 dtFu = Ω Ω ⎛ 1.07 in.⎞ 2.4 (1.00 in. ) ⎜ ( 65 ksi ) ⎝ 2 ⎟⎠ = 2.00 = 41.7 kips Dividing the web by 2 in the above calculation assumes that the same connection and load exist on both sides of the web. It is conservative for other cases. Because the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD) is less than the bearing strength, the limit state of bolt shear controls the strength of the inner bolts. Thus, the total available bolt strength on the column web, including the limit states of bolt shear and bolt bearing, is: LRFD ASD ϕRn = (31.8 kips)(10 bolts) Rn = (21.2 kips)(10 bolts) Ω = 212 kips > 123 kips o.k. = 318 kips > 184 kips o.k. 144 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Design beam web-to-angle weld The resultant load and load angle are: LRFD Pu = (184 kips )2 + ( 37.5 kips )2 = 188 kips ⎛ 37.5 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 184 kips ⎠ = 11.5° ASD Pa = (123 kips )2 + ( 25.0 kips )2 = 126 kips ⎛ 25.0 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 123 kips ⎠ = 11.5° AISC Manual Table 8-8 provides coefficients, C, for angle values, θ, equal to 0° and 15°. It could be unconservative to interpolate between the two tables; therefore, the Manual discussion recommends that the value from the table with the next lower angle be used, which is 0° in this case. Also, the required force, Pu or Pa, will equal the vertical load at the beam-to-column interface. From AISC Manual Table 8-8 and the geometry in Figure 5-11: Angle = 0° kl = 3.50 in. l = 142 in. 3.50 in. k = 142 in. = 0.241 The distance to the center of gravity from the vertical weld line can be determined as follows: x = = k2 1 + 2k ( 0.241)2 1 + 2 ( 0.241) = 0.0392 al = 4.00 in. − xl = 4.00 in. − 0.0392 (142 in.) = 3.43 3.43 142 in. = 0.237 a = By interpolation: C = 2.73 from AISC Manual Table 8-8 and C1 = 1.0 for E70XX electrodes AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 145 The number of sixteenths of fillet weld required is: LRFD Dreq = = Pu ϕCC1l ASD Dreq = 184 kips 0.75 ( 2.73 )(1.0 1.0 )(142 in.)( 2 ) = = 3.10 sixteenths Pa Ω CC1l (123 kips ) ( 2.00 ) 2.73 (1.0 )(142 in.)( 2 ) = 3.11 sixteenths Use a 4-in. fillet weld, which is also the minimum fillet weld size required according to AISC Specification Table J2.4. Check shear rupture strength of the beam web under the weld This is a situation where it is not possible by simple analysis to determine the stresses in the beam web under the fillet welds. Therefore, use the following equation to determine the minimum web thickness that will match its available shear rupture strength to that of the weld. Because there is a connection on both sides of the beam web, use AISC Manual Equation 9-3: LRFD t min = = 6.19 D Fu 6.19 ( 3.10 sixteenths ) 65 ksi = 0.295 in. < 0.535 in n. o.k. ASD t min = = 6.19 D Fu 6.19 ( 3.11 sixteenths ) 65 ksi = 0.296 in. < 0.535 in n. o.k. Note that the calculated required weld size is used rather than the rounded up nominal size of 4 in. in this calculation. Check block shear rupture on the beam web If the tensile transfer force is large enough, it is possible for block shear to control. The beam web can tear out in block shear at the weld lines only due to the axial (not the shear) force in the beam. Therefore, the shear and tension areas are relative to the axial force. This will be a C-shaped failure path. By inspection, the shear yielding component will control over the shear rupture component; therefore, AISC Specification Equation J4-5 reduces to: Rn = 0.60Fy Agv + UbsFu Ant(from Spec. Eq. J4-5) where Agv = twl(2) = (0.535 in.)(3.50 in.)(2) = 3.75 in.2 Agt = twl = (0.535 in.)(142 in.) = 7.76 in.2 Ubs = 1 146 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available block shear rupture strength of the beam web at the double angle-to-beam web weld is: LRFD ASD ϕRn = ϕ(0.60Fy Agv + UbsFu Ant) Rn 0.60 Fy Agv + U bs Fu Ant = Ω Ω = 0.75[0.60(50 ksi)(3.75 in.2) + 1(65 kips)(7.76 in.2)] = = 463 kips > 37.5 kips o.k. ( 65 ksi ) ( 7.76 in.2 ) + 0.60 ( 50 ksi ) ( 3.75 in.2 ) 2.00 = 308 kips > 25.0 kips o.k. Block shear on the outstanding angle legs should also be checked as was done for the gusset-to-column connection. It will not control. Check shear yielding on the double angle Agv = (142 in.)(2 in.)(2) = 14.5 in.2 From AISC Specification Equation J4-3, the available shear yielding strength of the double angle is: LRFD ASD ϕRn = ϕ0.60 Fy Agv ( = 1.00 ( 0.60 )( 36 ksi ) 14.5 in.2 ) Rn 0.6 Fy Agv = Ω Ω = = 313 kips > 184 kips o.k. ( 0.60 ( 36 ksi ) 14.5 in.2 ) 1.50 = 209 kips > 123 kips o.k. Check shear rupture on the double angle Anv = Ag − 2(5t)(dh + z in.) = 142 in.2 − 2(5)(2 in.)(1z in. + z in.) = 8.88 in.2 From AISC Specification Equation J4-4: LRFD ϕRn = ϕ0.60 Fu Anv ASD Rn ( = 0.75 ( 0.60 )( 58 ksi ) 8.88 in.2 = 232 kips > 184 kips o.k. ) Ω = = 0.60 Fu Anv Ω ( 0.60 ( 58 ksi ) 8.88 in.2 ) 2.00 = 155 kips > 123 kips o.k. Example 5.6—Corner Connection-to-Column Web: Uniform Force Method Special Case 1 Given: When the work point is moved from the gravity axis location to a point near the gusset corner, a substantial reduction in gusset size can be achieved. This can be seen by comparing Figures 5-11 and 5-14. In addition to a reduction in gusset size, moving the work point to the corner of the gusset can also result in less demand on the connections and therefore a more economical AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 147 structure. However, the beam and the column, if the column is oriented flange-to-view in elevation, must be designed for the resulting eccentricities. If the additional moments cause an increase in member size, then this increase in weight must be weighed against the potential decrease in labor to determine the most economical design. In Figure 5-14, the work point is located at the center of the column web at the elevation of the beam top flange. This point is very close to the gusset corner and is used in geometry layout because it is known before the connection is designed. The actual position of the gusset corner is not known at this time because the setback may change during fabrication. The brace bevel is calculated from this work point. This example will illustrate the design of the connection shown in Figure 5-14. The theory was discussed in Section 4.2.2. The details for this connection are the same as those given for Example 5.5. Fig. 5-14. Special Case 1—weak axis. 148 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Solution: See the Example 5.5 Solution for material and geometric properties, and hole dimensions. Brace-to-Gusset Connection Note that checks of the following limit states were included in Example 5.5 and will not be repeated here: bolt shear checks; tensile yielding, tensile rupture, block shear rupture, and bolt bearing on the brace; tensile yielding, block shear rupture, and bolt bearing on the gusset plate; and tensile yielding, tensile rupture, and block shear rupture on the double-tee connecting elements. The limit state for compression buckling on the gusset plate will be affected due to the different gusset plate size and is checked in the following. Check the gusset plate for compression buckling on the Whitmore section From AISC Manual Part 9, the width of the Whitmore section is: lw = 5 in. + 2(12 in.)(tan 30°) = 18.9 in. Use K = 0.5 (Gross, 1990) and the length of buckling, L, is given in Figure 5-14. KL 0.5 (12 w in. ) 12 = r 2 in. = 44.2 From AISC Specification Section J4.4 and using values from AISC Manual Table 4-22, the available strength is: LRFD ϕcFcr = 29.3 ksi ASD Fcr = 19.5 ksi Ωc ϕRn = ϕcFcr Ag = (29.3 ksi)(2 in.)(18.9 in.) = 277 kips > 270 kips o.k. Fcr Ag Rn = Ω Ωc = (19.5 ksi ) ( 2 in.) ( 18.9 in.) = 184 kips > 180 kips o.k. Connection Interface Forces With the brace-to-gusset connection sized, the gusset minimum size can be determined as shown in Figure 5-14. The preliminary gusset size is 192 in. vertical by 23 in. horizontal. Thus, using terminology from AISC Manual Part 13 and dimensions given in Figure 5-14: α=α 23.0 in. = + 2 in. 2 = 12.0 in. where α is measured from the face of the column web. 12 tan θ = 9 ⎛ 12 ⎞ θ = tan −1 ⎜ ⎟ ⎝ 9⎠ = 53.1° AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 149 18.6 in. 2 = 9.30 in. =0 = eb ec The theoretical value of β such that the connection interfaces are free of moments is: β= α − eb tan θ + ec tan θ (from Manual Eq. 13-1) ⎛12 ⎞ 12.0 in. − (9.30 in.) ⎜ ⎟ + 0 in. ⎝9⎠ = ⎛12 ⎞ ⎜9⎟ ⎝ ⎠ = − 0.300 in. The value of β here is used in subsequent UFM calculations but is otherwise irrelevant because Huc is 0 and therefore there is no moment, [Mc = Hc (β − β)]. Note that the work point location will generate shear only on the gusset edges. The calculations shown here are provided for reference to show the general method. In determining α and β, the gravity brace centerline (that is, the centerline of the brace if it framed into the beam and column centerlines as in Example 5.5) shown in Figure 5-14 is used. The bevel used is the actual bevel. The actual bevel and the gravity bevel will be the same if the connection at the other end of the brace is to a column web and the beam size at the next floor of W18×97 is also the same, which is assumed to be the case here. If they are not the same, the actual bevel should be used in all of the calculations. The fictitious gravity brace centerline of Figure 5-14 is parallel to the actual (true) brace centerline and therefore has the same bevel. The required axial and shear forces at the gusset-to-beam and gusset-to-column interface are determined as follows: r= = ( α + ec )2 + (β + eb )2 (Manual Eq. 13-6) (12.0 in. + 0 in.)2 + ( −0.300 in. + 9.30 in.)2 = 15.0 in. LRFD ASD Pu 270 kips = r 15.0 in. = 18.0 kip/in. Pa r From AISC Manual Equation 13-5: 180 kips 15.0 in. = 12.0 kip/in. = From AISC Manual Equation 13-5: Pu r = (12.0 in. )(18.0 kip/in. ) Pa r = (12.0 in. )(12.0 kip/in. ) = 144 kips H ab = α Hub = α = 216 kips From AISC Manual Equation 13-3: Pu Huc = ec r = ( 0 in. )(18 kip/in. ) 150 / VERTICAL BRACING = 0 kipsCONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ΣHu = 216 kips + 0 kips = 216 kips ( 270 kips ) sin53.1° = 216 kips o.k. Pu r = (12.0 in. )(18.0 kip/in. ) Hub = α = 216 kips LRFD From AISC Manual Equation 13-3: From AISC Equation 13-3: Manual Pu Huc = ec r = ( 0 in. )(18 kip/in. ) ASD From AISC Manual Equation 13-3: Pa r = ( 0 in. )(12 kip/in. ) H ac = ec = 0 kips ΣHu = 216 kips + 0 kips = 0 kips ΣHa = 144 kips + 0 kips = 216 kips ( 270 kips ) sin53.1° = 216 kips = 144 kips o.k. From AISC Manual Equation 13-4: (180 kips ) sin53.1° = 144 kips o.k. From AISC Manual Equation 13-4: Pu r = ( 9. 30 in. )(18. 0 kip/in. ) Pa r = ( 9.30 in. )(12.0 kip/in. ) Vub = eb Vab = eb = 167 kips = 112 kips From AISC Manual Equation 13-2: From AISC Manual Equation 13-2: Pu r = ( −0.300 in.)(18.0 kip/in. ) Pa r = (−0.300 in. )(12.0 kip/in. ) Vuc = β Vac = β = − 5.40 kips = −3.60 kips ΣVa = 167 kips − 5.40 kips ΣVa = 112 kips − 3.60 kips = 162 kips ( 270 kips ) cos 53.1° = 162 kips = 108 kips o.k. (180 kips ) cos 53.1° = 108 kips o.k. These forces are shown in Figures 5-15a(a) and 5-15b(a) (UFM admissible forces for gravity axis work point). Superimposed on the above forces are those caused by the nongravity axis work point. From Figure 4-13, e = (eb − y)sinθ − (ec − x)cosθ = (9.30 in. − 0 in.)sin 53.1° − (0 in. − 0 in.)cos 53.1° = 7.44 in. The required flexural strength of the beam is: LRFD ASD Mu = Pue = (270 kips)(7.44 in.) = 2,010 kip-in. Ma = Pae = (180 kips)(7.44 in.) = 1,340 kip-in. From Section 4.2.2, the quantity η, which is the proportion of the required flexural strength, Mu for LRFD and Ma for ASD, that goes to the beam, is 1.0. This is because the connection to a column web does not mobilize the column stiffness (I/L)col or strength (Zcol ). The web simply deforms and all of the moment goes to the beam. This distribution was confirmed by Gross (1990) using full scale tests. With no flange engagement, η = 1.0, and from Section 4.2.2: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 151 (a) UFM admissible forces for gravity axis work point (b) Extra admissible forces due to nongravity work point location (c) Superimposed forces from (a) and (b) including beam shear and transfer force Fig. 5-15a. Admissible forces for Special Case 1—weak axis—LRFD. 152 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 (a) UFM admissible forces for gravity axis work point (b) Extra admissible forces due to nongravity work point location (c) Superimposed forces from (a) and (b) including beam shear and transfer force Fig. 5-15b. Admissible forces for Special Case 1—weak axis—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 153 LRFD ASD From Equation 4-5: From Equation 4-5: Hu′ = (1 − η) Mu β + eb H a′ = =0 From Equation 4-6: Vu′ = = (1 − η) M a β + eb =0 From Equation 4-6: Mu − Hu′ β α 2, 010 kip-in. − ( 0 kips )( −0.300 in. ) Va′ = = 12.0 in. = 168 kips M a − H a′ β α 1, 340 kip-in. − ( 0 kips )( −0.300 in. ) 12.0 in. = 112 kips These forces are shown in Figure 5-15a(b) and Figure 5-15b(b) (extra admissible forces due to nongravity work point location). The superimposed forces are also shown in Figure 5-15a(c) and Figure 5-15b(c). As mentioned earlier in this example, the fact that β − β ≠ 0 is irrelevant because it is not used in the calculations. The value of β, which is not used, is 10 in. for this example. Note that the superimposed force distribution of Figure 5-15a(c) and Figure 5-15b(c) is exactly what is obtained from the AISC Manual Part 13 presentation of the UFM, Special Case 1. The connection shown in Figure 5-14 is designed for the superimposed interface forces shown in Figure 5-15a(c) and Figure 5-15b(c). The 2,010 kip-in. (LRFD) or 1,340 kip-in. (ASD) moment must be considered in the design of the W18×97 beam. Gusset-to-Beam Connection Referring to Figure 5-15a(c) and 5-15b(c) the forces on the interface are: LRFD ASD Required shear strength, Vu = 216 kips Required axial strength, Nu = 0 kips Required flexural strength, Mu = 0 kip-in. Required shear strength, Va = 144 kips Required axial strength, Na = 0 kips Required flexural strength, Ma = 0 kip-in. Check the gusset plate for shear yielding along the beam flange Use a gusset plate thickness of 2 in. as shown in Figure 5-14. The available shear yielding strength of the gusset plate is determined from AISC Specification Section J4.2, Equation J4-3, as follows: LRFD ASD = 1.00(0.60)(36 ksi)(2 in.)(23.0 in.) Vn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi ) ( 2 in.) ( 23.0 in.) = 1.50 = 166 kips > 144 kips o.k. ϕVn = ϕ0.60Fy Agv = 248 kips > 216 kips o.k. 154 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Design weld at gusset-to-beam flange connection The strength determination of fillet welds defined in AISC Specification Section J2.4 can be simplified as explained in AISC Manual Part 8 to AISC Manual Equations 8-2a and 8-2b. The required number of sixteenths of fillet weld on each side of the gusset plate is: LRFD Dreq = = ASD Ru 2l (1.392 kip/in.) Dreq = 216 kips 2 ( 23.0 in.)(1.392 kip/in.) = = 3.37 sixteenths Ra 2l ( 0.928 kip/in. ) 144 kips 2 ( 23.0 in. )( 0.928 kip/in. ) = 3.37 sixteenths The weld ductility factor is not used for welds that resist shear forces only. From AISC Specification Table J2.4, the minimum fillet weld size is x in. Use a 4-in. two-sided fillet weld. Gusset-to-Column Connection The forces at the gusset plate-to-column flange interface are: LRFD Required shear strength, Vu = 162 kips Required axial strength, Nu = 0 kips Required flexural strength, Mu = 0 kip-in. ASD Required shear strength, Va = 108 kips Required axial strength, Na = 0 kips Required flexural strength, Ma = 0 kip-in. Check the gusset plate for shear yielding along the column web The available shear yielding strength of the gusset plate is determined from AISC Specification Section J4.2, Equation J4-3, as follows: LRFD ASD = 1.00(0.60)(36 ksi)(2 in.)(192 in.) Vn 0.60 Fy Ag = Ω Ω 0.60 ( 36 ksi ) ( 2 in.) ( 192 in. ) = 1.50 = 140 kips > 108 kips o.k. ϕVn = ϕ0.60Fy Agv = 211 kips > 162 kips o.k. Design gusset plate-to-angles weld Assume 4-in. angles will be selected. From AISC Manual Table 8-8: Angle = 0° kl = 3.50 in. l = 142 in. 3.50 in. k = 142 in. = 0.241 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 155 The distance to the center of gravity from the vertical weld line can be determined as follows: x = k2 1 + 2k ( 0.241)2 = 1 + 2 ( 0.241) = 0.0392 al = 4.00 in. − xl = 4.00 in. − 0.0392 ( 142 in.) = 3.43 al a = l 3.43 = 142 in. = 0.237 From Table 8-8, by interpolation: C = 2.73 and C1 = 1.0 for E70XX electrodes The required number of sixteenths of fillet weld size is: LRFD Dreq = = Pu ϕCC1 l ASD Dreq = 162 kips 0.75 ( 2.73 )(1.0 ) ( 142 in.) ( 2 ) = = 2.73 sixteenths Pa Ω CC1l (108 kips )( 2.00 ) 2.73 (1.0 ) ( 142 in.) ( 2 ) = 2.73 sixteenths Use a x-in. fillet weld. The minimum weld size required by AISC Specification Table J2.4 will also have to be checked once the double angle size is selected. Try 2L4×4×c to connect the gusset plate to the column web. Check shear yielding on the angles at the column web connection Agv = (142 in.)(c in.)(2) = 9.06 in.2 The available shear yielding strength is determined from AISC Specification Section J4.2(a), Equation J4-3: LRFD ϕVn = ϕ0.60Fy Agv = 1.00(0.60)(36 ksi)(9.06 in.2) = 196 kips > 162 kips o.k. ASD Vn 0.60 Fy Agv = Ω Ω = ( 0.60 ( 36 ksi ) 9.06 in.2 1.50 = 130 kips > 1088 kips 156 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ) o.k. Check shear rupture on the angles at the column web connection The net area is determined from AISC Specification Section B4.3: Anv = [(142 in.)(c in.) − 5(c in.)(1z in. + z in.)](2) = 5.55 in.2 From AISC Specification Section J4.2(b), Equation J4-4: LRFD ASD Vn 0.60 Fu Anv = Ω Ω ϕVn = ϕ0.60Fu Anv = 0.75(0.60)(58 ksi)(5.55 in.2) = 145 kips < 162 kips = n.g. ( 0.60 ( 58 ksi ) 5.55 in.2 2.00 = 96.6 kips < 108 kips ) n.g. Try 2L4×4×a connection angles. The updated shear rupture strength can be calculated by using a ratio of the new angle thickness to the previous thickness: LRFD ⎛ a in. ⎞ ϕRn = (149 kips ) ⎜ ⎟ ⎝ c in.⎠ = 179 kips >162 kips ASD o.k. Rn ⎛ a in. ⎞ = ( 96.6 kips ) ⎜ ⎟ Ω ⎝ c in.⎠ = 116 kips > 108 kips o.k. Use 2L4×4×a. Check block shear rupture on the angles The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) The edge distance for the net tension area calculation, given the 52-in. column bolt gage, is: 4.00 in. + 4.00 in. + 2 in. − 52 in. 2 = 1.50 in. leh = Shear yielding component: Agv = (13.25 in.)(a in.)(2) = 9.94 in.2 0.60Fy Agv = 0.60(36 ksi)(9.94 in.2) = 215 kips Shear rupture component: Anv = 9.94 in.2 − 4.5(1z in. + z in.)(a in.)(2) = 6.14 in.2 0.60Fu Anv = 0.60(58 ksi)(6.14 in.2) = 214 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 157 Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = [(1.50 in.)(a in.) − 0.5(1c in. + z in.)(a in.)](2) = 0.609 in.2 UbsFu Ant = 1(58 ksi)(0.609 in.2) = 35.3 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60Fu Anv + UbsFu Ant = 214 kips + 35.3 kips = 249 kips 0.60Fy Agv + UbsFu Ant = 215 kips + 35.3 kips = 250 kips Therefore, Rn = 249 kips. LRFD ϕRn = 0.75(249 kips) = 187 kips > 162 kips o.k. ASD Rn 249 kips = Ω 2.00 = 125 kips > 108 kips o.k. Check bolt shear strength From AISC Manual Table 7-1, for 1-in.-diameter ASTM A325-N bolts in single shear, the available shear strength is: LRFD ϕRn = (31.8 kips/bolt)(10 bolts) = 318 kips > 162 kips o.k. ASD Rn = ( 21.2 kips/bolt )(10 bolts ) Ω = 212 kips > 108 kips o.k. Check bolt bearing on the angles According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 3.00 in. − (0.5 + 0.5)dh = 3.00 in. − 1(1z in.) = 1.94 in. The available bearing strength per bolt is determined as follows for the inner bolts: LRFD ASD ϕ1.2lctFu = 0.75(1.2)(1.94 in.)(a in.)(58 ksi) = 38.0 kips/bolt 1.2lctFu/Ω = 1.2(1.94 in.)(a in.)(58 ksi)/ 2.00 = 25.3 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(a in.)(58 ksi) = 39.2 kips/bolt 2.4dtFu/Ω = 2.4(1.00 in.)(a in.)(58 ksi)/ 2.00 = 26.1 kips/bolt Therefore, ϕRn = 38.0 kips/bolt. Therefore, Rn/Ω = 25.3 kips/bolt. 158 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Because these strengths are greater than the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD), the limit state of bolt shear controls the strength of the inner bolts. For the edge bolts, the clear distance is: lc = 14 in. − 0.5dh = 14 in. − 0.5(1z in.) = 0.719 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(0.719 in.)(a in.)(58 ksi) = 14.1 kips/bolt 1.2lctFu/Ω = 1.2(0.719 in.)(a in.)(58 ksi)/ 2.00 = 9.38 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(a in.)(58 ksi) = 39.2 kips/bolt 2.4dtFu/Ω = 2.4(1.00 in.)(a in.)(58 ksi)/ 2.00 = 26.1 kips/bolt Therefore, ϕRn = 14.1 kips. Therefore, Rn/Ω = 9.38 kips. Because these strengths are less than the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD), the limit state of bolt bearing controls the strength of the edge bolts. To determine the available strength of the bolt group, sum the individual effective strengths for each bolt. The total available strength of the bolt group is: LRFD ϕRn = (14.1 kips/bolt)(2 bolts) + (31.8 kips/bolt)(8 bolts) = 283 kips > 162 kips o.k. ASD Rn = ( 9.38 kips/bolt )( 2 bolts ) + ( 21.2 kips/bolt )(8 bolts ) Ω = 188 kips > 108 kips o.k. The limit state of bolt bearing on the column web, with tw = 1.07 in., is ok by inspection. The 2L4×4×a angles are satisfactory. Beam-to-Column Connection The forces on this interface, as shown in Figure 5-15, are: LRFD Required shear strength, Vu = 60.0 kips Required axial strength, Nu = 37.5 kips Required flexural strength, Mu = 0 kip-in. ASD Required shear strength, Va = 40.0 kips Required axial strength, Na = 25.0 kips Required flexural strength, Ma = 0 kip-in. Check shear yielding on gross area of the beam From AISC Manual Table 3-6: LRFD ϕvVn = 299 kips > 60.0 kips o.k. ASD Rn = 199 kips > 40.0 kips Ωv o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 159 Design angle-to-beam web weld The resultant load and angle of the resultant load are: LRFD Ru = ASD ( 37.5 kips )2 + ( 60.0 kips )2 = 70.8 kips ⎛ 37.5 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 60.0 kips ⎠ Ra = ( 25.0 kips )2 + ( 40.0 kips )2 = 47.2 kips ⎛ 25.0 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 40.0 kips ⎠ = 32.0° from vertical = 32.0° from vertical From AISC Manual Table 8-8 and Example 5.5: l = 142 in. kl = 3.50 in. k = 0.241 in. x = 0.0392 in. al = 3.43 in. a = 0.237 in. The AISC Manual discussion says to use the table for the next lower angle; therefore, from Table 8-8 for θ = 30°, by interpolation: C = 2.86 and C1 = 1.0 for E70XX electrodes The required number of sixteenths of fillet weld is: LRFD Dreq = = ASD Pu ϕCC1l Dreq = 70.8 kips 0.75 ( 2.86 )(1.00 ) ( 142 in.) ( 2 ) = 1.14 sixxteenths = Pa Ω CC1l ( 47.2 kips )( 2.00 ) 2.86 ( 142 in.) ( 2 welds ) = 1.14 sixteenths Use a x-in. fillet weld, which is the minimum permitted in AISC Specification Table J2.4 based on the thickness of the thinner part joined (assume 4 in. < tmin < 2 in.). Try 2L4×4×c to join the beam to the column. Check shear yielding on the gross area of the angles Agv = (142 in.)(c in.)(2) = 9.06 in.2 From AISC Specification Equation J4-3: LRFD ASD ϕVn = 1.00(0.60)(36 ksi)(9.06 in. ) 2 = 196 kips > 60.0 kips o.k. ( ) 2 Vn 0.60 ( 36 ksi ) 9.06 in. = Ω 1.50 = 130 kips > 40.0 kips o.k. 160 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check shear rupture on the net section of the angles at the column web connection Anv = 2[(142 in.)(c in.) − 5(c in.)(1z in. + z in.)] = 5.55 in.2 From AISC Specification Equation J4-4: LRFD ASD ϕVn = 0.75(0.60)(58 ksi)(5.55 in. ) 2 = 145 kips > 60.0 kips o.k. ( ) 2 Vn 0.60 ( 58 ksi ) 5.55 in. = 2.00 Ω = 96.6 kips > 40.0 kips o.k. Check block shear rupture on the angles at the column web connection Check block shear relative to the required shear strength of the connection. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) The edge distance for the net tension area calculation, given the 52-in. column bolt gage, is: 4.00 in. + 4.00 in. + 2 in. − 52 in. 2 = 1.50 in. leh = Shear yielding component: Agv = (13.25 in.)(c in.)(2) = 8.28 in.2 0.60Fy Agv = 0.60(36 ksi)(8.28 in.2) = 179 kips Shear rupture component: Anv = 8.28 in.2 − 4.5(1z in. + z in.)(c in.)(2) = 5.12 in.2 0.60Fu Anv = 0.60(58 ksi)(5.12 in.2) = 178 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = [1.50 in. − 0.5(1c in. + z in.)](c in.)(2) = 0.508 in.2 UbsFu Ant = 1(58 ksi)(0.508 in.2) = 29.5 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60Fu Anv + UbsFu Ant = 178 kips + 29.5 kips = 208 kips 0.60Fy Agv + UbsFu Ant = 179 kips + 29.5 kips = 209 kips Therefore, Rn = 208 kips. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 161 LRFD ASD ϕRn = 0.75(208 kips) R n 208 kips = Ω 2.00 = 104 kips > 40.0 kips = 156 kips > 60.0 kips o.k. o.k. Design bolts at beam-to-column connection Using the available strengths from AISC Manual Tables 7-1 and 7-2, check the shear strength and tensile strength of the connection: LRFD ASD Rnv = ( 21.2 kips/bolt )(10 bolts ) Ω = 212 kips > 40.0 kips o.k. Rnt = ( 35.3 kips/bolt )(10 bolts ) Ω = 353 kips > 25 kips o.k. ϕRnv = (31.8 kips/bolt)(10 bolts) = 318 kips > 60.0 kips o.k. ϕRnt = (53.0 kips/bolt)(10 bolts) = 530 kips > 37.5 kips o.k. The tensile strength requires an additional check due to the combination of tension and shear. From AISC Specification Section J3.7 and AISC Specification Table J3.2, the available tensile strength of the bolts subject to tension and shear is: LRFD Fnt′ = 1.3Fnt − ASD Fnt frv ≤ Fnt ϕFnv Fnt′ = 1.3Fnt − Fnt = 90 ksi Fnv = 54 ksi Fnt = 90 ksi Fnv = 54 ksi ⎛ ⎞ 90 ksi ⎜ 60.0 kips ⎟ 0.75 ( 54 ksi ) ⎜ 10 0.785 in.2 ⎟ ⎝ ⎠ = 100 ksi > 90 ksi Therefore, = 90 there is no reduction herefore, T F′ F=nt′ 90 ksiksi andand there is no reduction in the Fnt′ = 1.3 ( 90 ksi ) − nt in the available tensile strength. available tensile strength. ΩFnt frv ≤ Fnt Fnv ( ) 2.00 ( 90 ksi ) ⎛⎜ 40.0 kips ⎞⎟ ⎜ 10 0.785 in.2 ⎟ 54 ksi ⎝ ⎠ = 100 ksi > 90 ksi Therefore, = 90 there is no reduction in the herefore, T F′ F=nt′ 90 ksiksi andand there is no reduction in the Fnt′ = 1.3 ( 90 ksi ) − ( ) nt available tensile strength. available tensile strength. Check bolt bearing on angles According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 3.00 in. − (0.5 + 0.5)dh = 3.00 in. − 1(1z in.) = 1.94 in. 162 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available bearing strength per bolt is determined as follows for the inner bolts: LRFD ASD ϕ1.2lctFu = 0.75(1.2)(1.94 in.)(c in.)(58 ksi) = 31.6 kips/bolt 1.2lctFu/Ω = 1.2(1.94 in.)(c in.)(58 ksi)/ 2.00 = 21.1 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(c in.)(58 ksi) = 32.6 kips/bolt 2.4dtFu/Ω = 2.4(1.00 in.)(c in.)(58 ksi)/ 2.00 = 21.8 kips/bolt Therefore, ϕRn = 31.6 kips/bolt. Therefore, Rn/Ω = 21.1 kips/bolt. Because these strengths are less than the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD), the limit state of bolt bearing controls the strength of the inner bolts. For the edge bolts: lc = 14 in. − 0.5dh = 14 in. − 0.5(1z in.) = 0.719 in. LRFD ASD ϕ1.2lctFu = 0.75(1.2)(0.719 in.)(c in.)(58 ksi) = 11.7 kips/bolt 1.2lctFu/Ω = 1.2(0.719 in.)(c in.)(58 ksi)/ 2.00 = 7.8 2 kips/bolt ϕ2.4dtFu = 0.75(2.4)(1.00 in.)(c in.)(58 ksi) = 32.6 kips/bolt 2.4dtFu/Ω = (2.4)(1.00 in.)(c in.)(58 ksi)/ 2.00 = 21.8 kips/bolt Therefore, ϕRn = 11.7 kips. Therefore, Rn/ Ω = 7.82 kips. Because these strengths are less than the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD), the limit state of bolt bearing controls the strength of the edge bolts. Thus, the strength of the bolt group is: LRFD ASD ϕRn = (11.7 kips/bolt)(2 bolts) + (31.6 kips/bolt)(8 bolts) = 276 kips > 60 kips o.k. Rn = ( 7.82 kips/bolt ) (2 bolts) + ( 21.1 kips/bolt ) (8 bolts) Ω = 184 kips > 40 kips o.k. Bearing on the column web is o.k. by inspection because tw = 1.07 in. is much greater than the angle thickness = c in. Check prying action on angles The example given in Example 5.5 had the same axial transfer force of 37.5 kips (LRFD) or 25.0 kips (ASD) and, because of column “oil-canning,” a 2-in.-thick angle was required. The 2-in.-thick angles will be required in this example also; therefore, use 2L4×4×½. Example 5.7—Corner Connection-to-Column Web: Uniform Force Method Special Case 2 Given: Special Case 2 was discussed for strong-axis connections in Example 5.3. This case is used when the beam web is too light to carry the combined beam-to-column shear load of Vb + R. Some of the vertical force that is typically assigned to the gusset-tobeam connection, Vb , can instead be carried by the gusset-to-column connection, thereby reducing the beam-to-column shear force. The theory is discussed in Section 4.2.3 where the quantity ΔVb is introduced. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 163 Using the uniform force method Special Case 2, consider again the design of the connection given in Example 5.5 with a W18×35 beam in place of the W18×97 beam shown in Figure 5-11, and a required shear strength, Ru = 90.0 kips (LRFD) or Ra = 60.0 kips (ASD). The brace-to-gusset connection will not change from the Example 5.5 design; therefore, the gusset plate size and thickness determined in Example 5.5 is a good starting point. Solution: From AISC Manual Table 1-1, the beam geometric properties are: Beam W18×35 d = 17.7 in. tw = 0.300 in. tf = 0.425 in. kdes = 0.827 in. From Figure 5-11, the connection geometry is: =0 ec 17.7 in. 2 = 8.85 in. 12 tan θ = 9 = 1.33 θ = 53.1° eb = α =α 312 in. = + 2 in. 2 = 16.3 in. 17 w in. 2 = 8.88 in. α − eb tan θ − ec = tan θ 16.3 in. − ( 8.85 in.) (1.33) − 0 = 1.33 = 3.41 in. β = β (from Manual Eq. 13-1) This value of β is used in all the subsequent equations. The fact that β ≠ β is unimportant, because β – β is not used. r= = ( α + ec )2 + (β + eb )2 (Manual Eq. 13-6) (16.3 in. + 0 )2 + ( 3.41 in. + 8.85 in. )2 = 20.4 in. 164 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Pa 180 kips = r 20.4 in. = 8.82 kip/in. Pu 270 kips = r 20.4 in. = 13.2 kip/in. From AISC Manual Equation 13-5: From AISC Manual Equation 13-5: Pa H ab = α r = (16.3 in. )( 8.82 kip/in. ) Pu r = (16.3 in. )(13.2 kip/in. ) Hub = α = 144 kips = 215 kips From AISC Manual Equation 13-3: Pa H ac = ec r = ( 0 in. )( 8.82 kip/in. ) From AISC Manual Equation 13-3: Pu r = ( 0 in. )(13.2 kip/in. ) Huc = ec = 0 kips = 0 kips ΣH a = 144 kips ΣHu = 215 kips ( 270 kips ) sin 53.1° = 216 kips o.k. From AISC Manual Equation 13-4: o.k. From AISC Manual Equation 13-4: Pa Vab = eb r = ( 8.85 in. )( 8.82 kip/in. ) Pu r = ( 8.85 in. )(13.2 kip/in. ) Vub = eb = 78.1 kips = 117 kips From AISC Manual Equation 13-2: From AISC Manual Equation 13-2: Pa Vac = β r = ( 3.41 in. ) ( 8.82 kip/in. ) Pu r = ( 3.41 in. ) (13.2 kip/in. ) Vuc = β = 30.1 kips = 45.0 kips ΣVa = 30.1 kips + 78.1 kips ΣVu = 45.0 kips + 117 kips = 108 kips = 162 kips ( 270 kips ) cos 53.1° = 162 kips (180 kips ) sin 53.1° = 144 kips o.k. (180 kips ) cos 53.1° = 108 kips o.k. Figures 5-16a and 5-16b show the admissible force fields for this case. The total required shear strength of the beam-to-column connection is: LRFD Vu = Vub + Ru = 117 kips + 90.0 kips = 207 kips ASD Va = Vab + Ra = 78.1 kips + 60.0 kips = 138 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 165 From AISC Manual Table 3-6, the available shear strength of the W18×35 is: LRFD ϕ vVn = 159 kips < 207 kips ASD n.g. Vn Ω v = 106 kips < 138 kips n.g. This means that a web doubler plate is required. However, the required shear strength of the beam can be reduced by moving a portion of the Vub (LRFD) or Vab (ASD) force to the gusset-to column interface, as follows: LRFD Vub − ΔVub + Ru = 159 kips Solving for ΔVub , ΔVub = 117 kips + 90.0 kips − 159 kips = 48.0 kips ASD Vab − ΔVab + Ra = 106 kips Solving for ΔVab , ΔVab = 78.1 kips + 60.0 kips − 106 kips = 32.1 kips Fig. 5-16a. UFM General Case: Admissible force fields for Example 5.7 connection—LRFD. 166 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 From Section 4.2.3, Equation 4-10: LRFD ( ) Mub = Vub α − α + ΔVub α ASD ( ) M ab = Vab α − α + ΔVab α = (117 kips )( 0 ) + ( 48.0 kips )(16.3 in. ) = ( 78.1 kips )( 0 ) + ( 32.1 kips ) (16.3 in. ) = 782 kip-in. = 523 kip-in. Figure 5-17 shows the Special Case 2 interface forces. Brace-to-Gusset Connection The calculations are the same as for the general case in Example 5.5. CL Fig. 5-16b. UFM General Case: Admissible force fields for Example 5.7 connection—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 167 Gusset-to-Beam Connection From Figures 5-16 and Figures 5-17 and AISC Manual Part 13 for Special Case 2, the required strengths at the gusset-to-beam connection are: LRFD ASD Required shear strength, Hu = 215 kips Required normal strength, Vu = Vub − ΔVub Required shear strength, Ha = 144 kips Required normal strength, Va = Vab − ΔVab = 117 kips – 48.0 kips = 69.0 kips Required flexural strength, Mu = 782 kip-in. = 78 1 kips − 32.1 kips = 46.0 kips Required flexural strength, Ma = 523 kip-in. Fig. 5-17a. Admissible forces Special Case 2—LRFD. 168 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check the gusset plate for tensile yielding and shear yielding along the beam flange LRFD ASD Check shear yielding fuv = = Check shear yielding Hu Ag fav = 215 kips (2 in.)( 312 in.) = = 13.7 ksi < 1.00 ( 0.60 ) ( 36 ksi ) = 21.6 ksi o.k. Ha Ag 144 kips (2 in.)( 312 in.) = 9.14 ksi < 0.60 ( 36 ksi ) 1.50 = 14.4 ksi o.k. Fig. 5-17b. Admissible forces Special Case 2—ASD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 169 LRFD fua = = fub ASD Vu Ag faa = 69.0 kips ( 2 in. ) ( 312 in.) = = 4.38 ksi Mu = Z 782 kip-in.) ( 4 ) ( = (2 in.)( 312 in.)2 Va Ag 46.0 kips ( 2 in.) ( 312 in.) = 2.92 ksi Ma fab = Z ( 523 kip-in.)( 4 ) = ( 2 in.) ( 312 in.)2 = 6.30 ksi = 4.22 ksi Total normal stress = fun Total normal stress = fan Check tensile yielding Check tensile yielding fun = fua + fub fan = faa + fab = 4.38 ksi + 6.30 ksi = 10.7 ksi < 0.90 ( 36 ksi ) = 32.4 ksi o.k. = 2.92 ksi + 4.22 ksi 36 ksi = 7.14 ksi < = 21.6 ksi 1.67 o.k. Design weld at gusset-to-beam flange connection LRFD Ru = ( 215 kips )2 + ( 69.0 kips )2 = 226 kips ⎛ 69.0 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 215 kips ⎠ = 17.8° ASD Ra = (144 kips )2 + ( 46.0 kips )2 = 151 kips ⎛ 46.0 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 144 kips ⎠ = 17.7° Measured relative to the horizontal Measured relative to the horizontal The effective eccentricity is: LRFD al = Mu Hu 782 kip-in. 215 kips = 3.64 in. = ASD al = Ma Ha 523 kip-in. 144 kips = 3.63 in. = From AISC Manual Table 8-4: 170 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Angle = 15° k =0 al a C C1 = 3.63 in. = 3.63 in. 312 in. = 0.115 = 3.76 = 1.0 for E70XX electrodes LRFD Dreq = = ASD Ru (1.25 ) Dreq = ϕ CC1l ( 226 kips )(1.25) = 0.75 ( 3.76 )(1.0 )( 312 in.) = 3.18 Ra Ω (1.25 ) CC1l (151 kips )( 2.00 )(1.25) 3.76 (1.0 )( 312 in.) = 3.19 Use a 4-in. double-sided fillet weld. The 1.25 factor in the preceding calculation is the weld ductility factor which is explained in AISC Manual Part 13. An alternative method to size the weld is to introduce an equivalent normal force: The maximum stress is fn = fa + fb = 10.4 ksi (LRFD) or 7.14 ksi (ASD). This maximum stress occurs over half of the 312-in. gusset edge; the other half of the gusset edge has a stress of fa – fb. It is convenient when dealing with web yielding and crippling to work with an equivalent normal force that produces the same maximum stress. The maximum equivalent normal force is (see Appendix B, Figure B-1): LRFD N e = Vu + = Vu + ASD M u (2 ) N e = Va + ( lg / 2 ) 4 Mu lg = 69.0 kips + = Va + 4 ( 782 kip-in. ) 312 in. = 168 kips M a (2 ) ( lg / 2 ) 4 Ma lg = 46.0 kips + 4 ( 523 kip-in. ) 312 in. = 112 kips Check that the normal stress is the same as previously calculated: LRFD fun = = Ne t g lg ASD fan = 168 kips ( 2 in.) ( 312 in.) = 10.7 ksi o.k. N u max = N e = Ne t g lg 112 kips ( 2 in.) ( 312 in.) = 7.11 ksi o.k. Na max = Ne = 168 kips = 112 kips Except for small differences due to rounding, fn is the same as previously computed. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 171 The required weld size is determined using the directional strength increase and the simplified equation given in AISC Manual Part 8 as Equation 8-2: LRFD Ru ASD = N u2 max + Hu2 = Ra (168 kips )2 + ( 215 kips )2 = = 273 kips θ Dreq (112 kips )2 + (144 kips )2 = 182 kips ⎛ 168 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 215 kips ⎠ = 38.0° 1.25 Ru = 2l (1.392 kip/in. ) 1 + 0.50 sin1.5 θ ( = = N a2 max + H a2 1.25 ( 273 kips ) θ Dreq ) ( 2 ( 312 in. ) (1.392 kip/in. ) 1 + 0.50 sin = 3.13 sixteenths 1.5 38.0° ) ⎛ 112 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 144 kips ⎠ = 37.9° 1.25 Ru = 2l ( 0.928 kip/in. ) 1 + 0.50 sin1.5 θ ( = 1.25 (182 kips ) ) ( 2 ( 312 in. ) ( 0.928 kip/in. ) 1 + 0.50 sin1.5 37.9° ) = 3.14 sixteenths This is essentially the same value as obtained from the AISC Manual Table 8-4. Note that the alternative method as used here is a conservative variation on the approach presented by Hewitt and Thornton (2004) and discussed in AISC Manual Part 13, in that the ductility factor is applied to the peak rather than to the average stress. If the more exact procedure of Hewitt and Thornton were used (see Appendix B): LRFD 2 ASD Ru max = N u max + Hu = Nu min 2 Ra max = N a max 2 + H a 2 (168 kips )2 + ( 215 kips )2 = 273 kips 4 Mu = Vu − l = 69.0 kips − = N a min 4 ( 782 kip-in.) Ru min = Nu min + Hu = Ravg = 182 kips 4 Ma = Va − l = 46.0 kips − 312 in. = 30.3 kips 2 (112 kips )2 + (144 kips )2 4 ( 523 kip-in. ) 312 in. = 20.4 kips 2 Ra min = N a min 2 + H a 2 ( 30.3 kips )2 + ( 215 kips )2 = 217 kips Ru max + Ru min = 2 273 kips + 217 kips = 2 = 245 kips = ( 20.4 kips )2 + (144 kips )2 = 145 kips Ra max + Ra min 2 182 kips + 145 kips = 2 = 164 kips R avg = The weld size is based on max (Ra max, 1.25Ravg): 172 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Ru = max ⎡⎣ 273 kips, 1.25 ( 245 kips ) ⎤⎦ Ra = max ⎡⎣182 kips, 1.25 (164 kips ) ⎤⎦ = 306 kips Dreq = = = 205 kips Ru ( 2l (1.392 kip/in. ) 1 + 0.50 sin 1.5 θ Dreq = ) 306 kips ( 2 ( 312 in. ) (1.392 kip/in. ) 1 + 0.50 sin 1.5 38.0° = ) = 2.81 sixteenths Ra ( 2l ( 0.928 kip/in. ) 1 + 0.50 sin1.5 θ ) 205 kips ( 2 ( 312 in. ) ( 0.928 kip/in. ) 1 + 0.50 sin1.5 37.9° ) = 2.83 sixteenths If D = 2.81 (LRFD) or 2.83 (ASD) sixteenths is taken as the most accurate value, the AISC Manual Table 8-4 method is con3.19 − 2.83 servative by approximately (100% ) = 12.7% (ASD) and the simplified Hewitt-Thornton method is conservative by 2.83 3.14 − 2.83 (100% ) = 11.0% (ASD). In any case, a 4-in. fillet weld is used here because it is the minimum size used by many fabrica2.83 tors. (Note that AISC Specification Table J2.4 requires a minimum x-in. fillet weld based on the thinner part joined.) These comparisons are intended to demonstrate that there are several ways to perform these calculations that yield similar results. Beam Checks Check beam web local yielding The normal force is assumed to act at the center of the gusset-to-beam interface, or 312 in. / 2 = 15.8 in. from the end of the beam. Because this distance is less than or equal to the depth of the beam (17.7 in.), use AISC Specification Equation J10-3: LRFD ASD ϕRn = ϕFy t w ( 2.5k + lb ) Rn Fy t w ( 2.5k + lb ) = Ω Ω ( 50 ksi )( 0.300 in.) ⎡⎣2.5 ( 0.827 in.) + 312 in.⎤⎦ = 1.50 = 336 kips > N e = 112 kips o.k. = 1.00 ( 50 ksi )( 0.300 in. ) ⎡⎣2.5 ( 0.827 in.. ) + 312 in.⎤⎦ = 504 kips > N e = 168 kips o.k. Check beam web local crippling The normal force is applied at 15.8 in. from the beam end, which is greater than d/ 2, where d is the depth of the beam. Therefore, determine the beam web local crippling strength from AISC Specification Equation J10-4: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎛ ⎞ ⎢1 + 3 ⎛ lb ⎞ ⎜ t w ⎟ ⎜ ⎟ ⎢ ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎣ 1.5 ⎤ ⎥ ⎥ ⎦ ASD EFyw t f tw ≥ Ne 2⎡ ⎛ 312 in.⎞ = 0.75 ( 0.80 )( 0.300 in. ) ⎢1 + 3 ⎜ ⎟ ⎝ 17.7 in. ⎠ ⎢⎣ × ( 29, 000 ksi )( 50 ksi ) ( 0.425 in.) 0.300 in. = 322 kips > 168 kips o.k. 1.5 ⎤ ⎡ Rn 1 ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFyw t f 2 ⎢ = 0.80t w 1 + 3 ⎜ ⎟ ⎜ ⎟ ≥ Ne ⎢ tw Ω Ω ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ ⎛ 0.300 in. ⎞ ⎜ 0.425 in. ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥⎦ = ⎡ 1 312 in.⎞ (0.80) ( 0.300 in. )2 ⎢1 + 3 ⎛⎜ ⎟ 2.00 ⎝ 17.7 in.⎠ ⎢⎣ × ⎛ 0.300 in.⎞ ⎜ 0.425 in. ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥⎦ ( 29, 000 ksi )( 50 ksi ) ( 0.425 in. ) = 215 kips > 112 kips 0.300 in. o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 173 Gusset-to-Column Connection From Figure 5-17, the required shear strengths are 93.0 kips (LRFD) and 62.2 kips (ASD). The connection angle sizes are as given in Figure 5-11. Check bolt shear strength From AISC Manual Table 7-1 for a 1-in.-diameter ASTM A325-N bolt in single shear, the required shear strength is: LRFD ASD ϕrn = 31.8 kips rn = 21.2 kips Ω The required shear strength per bolt for the six-bolt double-angle connection is: LRFD 93.0 kips 6 = 15.5 kips < 31.8 kips ASD ru = 62.2 kips 6 = 10.4 kips < 21.2 kips ra = o.k. o.k. Check bolt bearing on the angles According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the spacing between bolts is large (5.0 in. as shown in Figure 5-11); therefore, the first part of the equation (tearout) will not control by inspection. LRFD ϕrn = ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )( c in. )( 58 ksi ) = 32.6 kips ASD rn 2.4 dtFu = Ω Ω 2.4 (1.00 in. ) ( c in.) ( 58 ksi) = 2.00 = 21.8 kips This does not control over the available bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD). For the top bolts, the clear distance, based on the 14-in. edge distance is: lc = 14 in. − 0.5dh = 14 in. − 0.5(1z in.) = 0.719 in. 174 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 0.719 in. )( c in.) ( 58 ksi ) = 11.7 kips/bolt ϕ2.4dtFu = 0.75 ( 2.4 )(1.00 in. ) ( c in.) ( 58 ksi ) = 32.6 kips/bolt Therefore, ϕrn = 11.7 kips/bolt. 1.2lc tFu 1.2 ( 0.719 in. ) ( c in.) ( 58 ksi ) = Ω 2.00 = 7.82 kips/bolt 2.4 (1.00 in. ) ( c in.) ( 58 ksi ) 2.4dtFu = Ω 2.00 = 21.8 kips/bolt Therefore, rn/ Ω = 7.82 kips/bolt. For the top bolts, because the available strength due to bolt bearing on the angles is less than the bolt shear strength determined previously (31.8 kips for LRFD and 21.2 kips for ASD), the limit state of bolt bearing controls the strength of the top bolts. Check bolt bearing on the column web There are no end bolts on the column web because the column is continuous. Similar to the inner bolts bearing on the angles, the first part of AISC Specification Equation J3-6a (tearout) does not control. Therefore: LRFD ASD ϕrn = ϕ2.4 dtFu ⎛ 1.07 in. ⎞ = 0.75 ( 2.4 )(1.00 in. ) ⎜ ⎟ ( 65 ksi ) ⎝ 2 ⎠ = 62.6 kips > 31.8 kips bolt shear controls rn 2.4 dtFu = Ω Ω ⎛ 1.07 in.⎞ 2.4 (1.00 in. ) ⎜ ⎟ ( 65 ksi ) ⎝ 2 ⎠ = 2.00 = 411.7 kips > 21.2 kips bolt shear controls Note that in this check, one half of the column web thickness is used. The other one half is reserved for connections, if any, on the other side of the column web. If there are no connections on the other side, the entire web thickness of 1.07 in. may be used here. This is the simplest way to deal with this problem, and is correct for connections of the same size and load on both sides. It is conservative for any other case. The available bearing strength of the bolts on the angles and the column web is: LRFD ASD ϕRn = 2 (11.7 kips ) + 4 ( 31.8 kips ) Rn = 2 ( 7.82 kips ) + 4 ( 21.2 kips ) Ω = 100 kips > 62.2 kips o.k. = 151 kips > 93.0 kips o.k. Check shear yielding on the angles The available shear yielding strength from AISC Specification Section J4.2(a) is: LRFD ASD ϕRn = ϕ0.60 Fy Agv = ϕ0.60 Fy la t a ( 2 ) = 1.00 ( 0.60 )( 36 ksi ) ( 122 in.) ( c in.) ( 2 ) = 169 kips > 93.0 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 Fy l a ta ( 2) = Ω 0.60 ( 36 ksi )(122 in.) ( c in.)( 2 ) = 1.50 = 113 kips > 62.2 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 175 Check shear rupture on the angles The net area is determined from AISC Specification Section J4.3: Anv = [122 in. − 3(1z + z in.)](c in.)(2) = 5.70 in.2 The available shear rupture strength from AISC Specification Section J4.2(b) is: LRFD ϕRn = ϕ0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 58 ksi ) 5.70 in. = 149 kips > 93.0 kips 2 ) o.k. Rn 0.60 Fu Anv = Ω Ω = ( 0.60 ( 58 ksi ) 5.70 in.2 2.00 = 99.2 kips > 62.2 kips ) o.k. Check block shear rupture on the angles The controlling block shear failure path is assumed to start at the top of the angles and follows the vertical bolt lines and then turns perpendicular to the edge of each angle. For the tension area calculated in the following, the net area is based on the length of the short-slotted hole dimension given in AISC Specification Table J3.3 and the requirements for net area determination in AISC Specification Section B4.3b. For the shear area, the net area is based on the width of the short-slotted hole, which is the same as for a standard hole. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = (122 in. − 14 in.)(c in.)(2) = 7.03 in.2 0.60Fy Agv = 0.60(36 ksi)(7.03 in.2) = 152 kips Shear rupture component: Anv = 7.03 in.2 − 2.5(1z in. + z in.)(c in.)(2) = 5.27 in.2 0.60Fu Anv = 0.60(58 ksi)(5.27 in.2) = 183 kips Tension rupture component: U bs = 1 from AISC Specification Section J4.3 because the boltts are uniformly loaded 4.00 in. + 4.00 in. + 2 in. − 52 in. − 0.50 (1c in. + z in.) 2 = 0.813 in. Le = Ant = tLe ( 2 ) = ( c in.) ( 0.813 in. )( 2 ) = 0.508 in.2 UbsFu Ant = 1(58 ksi)(0.508 in.2) = 29.5 kips 176 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 183 kips + 29.5 kips = 213 kips 0.60Fy Agv + UbsFu Ant = 152 kips + 29.5 kips = 182 kips Therefore, Rn = 182 kips. LRFD ASD ϕRn = 0.75 (182 kips ) = 137 kips > 93.0 kips Rn 182 kips = Ω 2.00 = 91.0 kips > 62.2 kips o.k. o.k. Check block shear rupture on the gusset plate It is possible for the gusset to fail in block shear rupture under the angles at the weld lines. This will be an L-shaped tearout. By inspection, the shear yielding component will control over the shear rupture component; therefore, AISC Specification Equation J4-5 reduces to: Rn = 0.60Fy Agv + UbsFu Ant(from Spec. Eq. J4-5) where Agv = tg(Lev + Langle) = (2 in.)(22 in. + 122 in.) = 7.50 in.2 Ant = tg Leh = (2 in.)(3.50 in.) = 1.75 in.2 Ubs = 1 The available block shear rupture strength of the gusset plate at the angle-to-gusset plate weld is: LRFD ( ϕRn = ϕ 0.60 Fy Agv + U bs Fu Ant ( ) ASD ) = 0.75 ⎡0.60 ( 36 ksi ) 7.50 in.2 + 1 ( 58 ksi )(1.75 in. ) ⎤ ⎣ ⎦ = 197 kips > 93.0 kips o.k. Rn 0.60 Fy Agv + U bs Fu Ant = Ω Ω = ( ) ( 0.60 ( 36 ksi ) 7.50 in.2 + 1 ( 58 ksi ) 1.75 in.2 = 132 kips > 62.2 kips ) 2.00 o.k. Check shear yielding on the gusset plate On a vertical section of the gusset plate, the available shear yielding strength is, from AISC Specification Equation J4-3: LRFD ASD ϕRn = ϕ0.60 Fy Agv = ϕ0.60 Fy lv t g = 1.00 ( 0.60 )( 36 ksi )(17w in. ) ( 2 in.) = 192 kips > 93.0 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 Fy lv t g = 1.50 0.60 ( 36 ksi ) (17w in. ) ( 2 in.) = 1.50 = 128 kips > 62.2 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 177 Design gusset plate-to-angles weld Use AISC Manual Table 10-2, Case I, conservatively using L = 112 in. for the 122-in. connection in this example. The angle leg used in this example is slightly larger than that assumed in Table 10-2, which is also conservative. A x-in. fillet weld (Weld A) will have an available strength of: LRFD ASD ϕRn = 142 kips > 93.0 kips o.k. Rn = 95.0 kips > 62.2 kips o.k. Ω Use a x-in. fillet weld, which is also the minimum required fillet weld size according to AISC Specification Table J2.4. Beam-to-Column Connection From Figure 5-17, the beam-to-column interface forces are: LRFD ASD Required shear strength: Required shear strength: Vu = 90.0 kips + 69.0 kips Va = 60.0 kips + 46.0 kips = 159 kips Required axial strength: = 106 kips Required axial strength: Aub = 37.5 kips transfer force Aab = 25.0 kips transfer force The required shear strength is less than that used in Example 5.5 (Figure 5-11) and the calculations will not be repeated here. The final design is the same as that of Figure 5-11, with the beam size changed to a W18×35. Example 5.8—Corner Connection-to-Column Web with Gusset Connected to Beam Only: Uniform Force Method Special Case 3 This case is discussed in Section 4.2.4, shown in Figure 4-16, and addressed in an example (for the case where the beam is connected to the column flange or strong axis) in Example 5.4. A similar example will be considered here in the weak-axis configuration, that is, where the beam connects to the column web. The connection is shown in Figure 5-18. The optional flat bar (FB) configuration will be used, with a thickness of s in. The column is a W12×58 flange to view, the beam is a W18×35, and the brace is 2L4×4×a. All properties and loads are the same as those given for Example 5.4, and as shown in Figure 5-10. The required strengths are: LRFD Brace required strength, Pu = ± 100 kips Beam required shear strength, Vu = 30.0 kips ASD Brace required strength, Pa = ± 66.7 kips Beam required shear strength, Va = 20.0 kips Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A36 Fy = 36 ksi Fu = 58 ksi ASTM A992 Fy = 50 ksi Fu = 65 ksi 178 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 From AISC Manual Tables 1-1, 1-7 and 1-15, the geometric properties are as follows: Beam W18×35 tw = 0.300 in. d = 17.7 in. bf = 6.00 in. tf = 0.425 in. tw = 0.360 in. d = 12.2 in. bf = 10.0 in. tf = 0.640 in. Brace 2L4×4×a in. Ag = 5.72 in.2 x = 1.13 in. kdes = 0.827 in. Column W12×58 Fig. 5-18. Special Case 3—weak axis. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 179 From AISC Specification Table J3.3, the hole dimensions for d-in.-diameter bolts are as follows: Brace, column and flat bar Standard: dh = , in. diameter Angles Short slots: , in. × 8 in. Connection Interface Forces The forces at the gusset-to-beam and gusset-to-column interfaces are determined using Special Case 3 of the UFM as discussed in Section 4.2.4 and AISC Manual Part 13. From Figure 5-18 and the given beam and column geometry: eb ec db 2 17.7 in. = 2 = 8.85 in. = 0 in. = tan θ = θ 12 7 ⎛ 12 ⎞ = tan −1 ⎜ ⎟ ⎝ 7⎠ = 59.7° Because there is no connection of the gusset plate to the column, β = 0, and α = eb tan θ − ec (Manual Eq. 13-1) ⎛12 ⎞ = ( 8.85 in. ) ⎜ ⎟ − 0 in. ⎝7⎠ = 15.2 in. The forces can be calculated from Figure 4-16 or by inspection of Figure 5-18. Because ec = 0, Mbc = 0. With the flat bar (FB) straddling the brace line of action, α = α; therefore, Mb = 0. The forces on the gusset-to-beam connection are: LRFD Hub = H Vub ASD H ab = H = Pu sin θ = Pa sin θ = (100 kips ) sin 59.7° = ( 66.7 kips ) sin 59.7° = 86.3 kips =V = 57.6 kips =V Vab = Pu cos θ = Pa cos θ = (100 kiips ) cos 59.7° = ( 66.7 kips ) cos 59.7° = 50.5 kips = 33.7 kips 180 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The forces on the beam-to-column interface are: LRFD ASD Vbc = Vub + R Vbc = Vab + R = 50.5 kips + 30 kips = 80.5 kips = 33.7 kips + 20 kips = 53.7 kips Brace-to-Gusset Connection The contact length between the flat bar (FB) and the beam flange is: 8.00 in. cos θ 8.00 in. = cos 59.7° = 15.9 in. lg = The calculations for the design of this connection are the same as those shown in Example 5.4, except as follows. Check the flat bar for tensile yielding on the Whitmore section As explained in AISC Manual Part 9, the gross area may be limited by the Whitmore section, where the width of the Whitmore section is: lw = 2(6.00 in.)(tan 30°) = 6.93 in. If lw were greater than 8 in., a Whitmore width of 8 in. would be used because 8 in. is the full width of the flat bar. Aw = (6.93 in.)(s in.) = 4.33 in.2 From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate is: LRFD ASD ϕRn = ϕFy Aw ( = 0.90 ( 36 ksi ) 4.33 in.2 = 140 kips > 100 kips ) o.k. Rn Fy Aw = Ω Ω = ( 36 ksi ) ( 4.33 in.2 ) 1.67 = 93.3 kips > 66.7 kips o.k. Check the flat bar for compression buckling on the Whitmore section The available compressive strength of the flat bar based on the limit state of flexural buckling is determined from AISC Specification Section J4.4. Use an effective length factor of K = 1.2 due to the possibility of sidesway buckling (see AISC Specification Commentary Table C-A-7.1). From Figure 5-18, the unbraced length of the flat bar is shown as 62 in. KL 1.2 ( 62 in.) = r ( s in.) 12 = 43.2 Because KL/r > 25, use AISC Manual Table 4-22 to determine the available critical stress for the 36-ksi flat bar. Then, the available compressive strength can be determined from AISC Specification Sections E1 and E3, with Ag = Aw: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 181 LRFD ASD ϕc Fcr = 29.4 ksi ϕc Pn = ϕc Fcr Aw Fcr = 19.6 ksi Ωc ( = ( 29.4 ksi ) 4.33 in.2 ) = 127 kips > 100 kips o.k. Pn Fcr Aw = Ωc Ωc ( = (19.6 ksi ) 4.33 in.2 ) = 84.9 kips > 66.7 kips o.k. Check tension rupture on the net section of the flat bar Because Section a-a of Figure 5-18 passes through a flat bar cross section, net tension is a more likely failure mode (limit state). The net area on the section marked a-a through the flat bar on Figure 5-18 is determined in accordance with AISC Specification Section B4.3, with the bolt hole diameter, dh = , in., from AISC Specification Table J3.3: An = [8.00 in. − 1(, in. + z in.)](s in.) = 4.38 in.2 The available tensile rupture strength of the flat bar according to AISC Specification Equation J4-2, with Ae = An (U = 1), is: LRFD ϕRn = ϕFu Ae ASD ( = 0.75 ( 58 ksi ) 4.38 in.2 = 191 kips > 100 kips ) o.k. Rn Fu Ae = Ω Ω = ( 58 ksi ) ( 4.38 in.2 ) 2.00 = 127 kips > 66.7 kips o.k. Check block shear rupture on the flat bar The available strength for the limit state of block shear rupture on the flat bar is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = (7.50 in.)(s in.) = 4.69 in.2 0.60Fy Agv = 0.60(36 ksi)(4.69 in.2) = 101 kips Shear rupture component: Anv = 4.69 in.2 − 2.5(, in. + z in.)(s in.) = 3.13 in.2 0.60Fu Anv = 0.60(58 ksi)(3.13 in.2) = 109 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = [4.00 in. − 0.5(, in. + z in.)](s in.) = 2.19 in.2 182 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 UbsFu Ant = 1(58 ksi)(2.19 in.2) = 127 kips The available strength for the limit state of block shear rupture is: 0.60Fu Anv + UbsFu Ant = 109 kips + 127 kips = 236 kips 0.60Fy Agv + UbsFu Ant = 101 kips + 127 kips = 228 kips Therefore, Rn = 228 kips. LRFD ASD ϕRn = 0.75 ( 228 kips ) = 171 kips > 100 kips o.k. Rn Fu Ant + 0.6 Fy Agv = Ω Ω 228 kips = 2.00 = 114 kips > 66.7 kips o.k. Gusset-to-Beam Connection Check gusset plate for shear yielding and tensile yielding along the beam flange The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1 as follows: LRFD ASD ϕVn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi ) ( s in.) ( 15.9 in.) = 215 kipss > 86.3 kips o.k. Nn Fy Ag = Ω Ω ( 36 ksi ) ( s in.) (15.9 in.) = 1.67 = 214 kips > 33.7 kips o.k. ϕN n = ϕFy Ag = 0.90 ( 36 ksi ) ( s in.) (15.9 in. ) = 322 kips > 50.5 kips Vn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi.) ( s in.) (15.9 in.) = 1.50 = 143 kips > 57.6 kips o.k. o.k. Note that as shown previously, the interaction of the forces at the gusset-to-beam interface is assumed to have no impact on the design. Design weld at gusset-to-beam flange connection The resultant load and load angle are: LRFD Ru = (86.3 kips ) 2 ASD + ( 50.5 kips ) = 100 kips ⎛ 50.5 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 86.3 kips ⎠ = 30.3° 2 Ra = ( 57.6 kips )2 + ( 33.7 kips )2 = 66.7 kips ⎛ 33.7 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 57.6 kips ⎠ = 30.3° AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 183 From AISC Manual Table 8-4: Angle = 30° a =0 k =0 = 4.37 = 1.0 for E70XX electrodes C C1 Use AISC Manual Table 8-4 equation for Dmin to determine the number of sixteenths of weld required: LRFD Dreq = = ASD Ru ϕCC1l Dreq = 100 kips 0.75 ( 4.37 )(1.0 )(15.9 in. ) = = 1.92 sixteenths Ra Ω CC1l ( 66.7 kips )( 2.00 ) ( 4.37 )(1.0 )(15.9 in. ) = 1.92 sixteenths Note that the ductility factor need not be used here because the stress is uniform and statically determinate. An Appendix B approach could also be used. Alternatively, the AISC Manual Equation 8-2 with the directional strength increase may be used. Use a double-sided x-in. fillet weld based on the minimum weld size required by AISC Specification Table J2.4. Check beam web local yielding From AISC Specification Equation J10-3, because the resultant force is applied at a distance, α = 15.2 in., from the member end that is less than or equal to the depth of the W18×35 beam, the available strength for web local yielding is: LRFD ϕRn = ϕFy t w ( lb + 2.5kdes ) = 1.00 ( 50 ksi )( 0.300 in. ) ⎡⎣15.9 inn. + 2.5 ( 0.827 in. ) ⎤⎦ = 270 kips > 50.5 kips o.k. ASD Rn Fy t w = [lb + 2.5kdes ] Ω Ω ( 50 ksi )( 0.300 in.) ⎡ = ⎣15.9 in. + 2.5 ( 0.827 in. ) ⎤⎦ 1.50 = 180 kips > 33.7 kips o.k. Check beam web local crippling Using AISC Specification Equation J10-4 because the force to be resisted is applied a distance from the member end, α = 15.2 in., that is greater than or equal to d/ 2, the available strength for web local crippling is: LRFD 1.5 ⎤ ⎡ ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFyw t f ⎢ ϕRn = 1+ 3⎜ ⎟ ⎜ ⎟ ⎢ tw ⎝ d ⎠ ⎜⎝ tf ⎟⎠ ⎥ ⎣ ⎦ 1.5 2⎡ ⎛ 15.9 in.⎞ ⎛ 0.300 in.⎞ ⎤ = 0.75 ( 0.80 )( 0.300 in. ) ⎢1 + 3 ⎜ ⎟⎜ ⎟ ⎥ ⎝ 17.7 in.⎠ ⎝ 0.425 in. ⎠ ⎥⎦ ⎢⎣ ϕ0.80t w2 × ( 29, 000 ksi )( 50 ksi ) ( 0.425 in.) 0.300 in. = 201 kips > 50.5 kips o.k. ASD 1.5 ⎤ ⎡ Rn 0.80t w2 ⎢ ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFyw t f = 1+ 3⎜ ⎟ ⎜ ⎟ Ω Ω ⎢ tw ⎝ d ⎠ ⎜⎝ tf ⎟⎠ ⎥ ⎣ ⎦ = ( 0.80 )( 0.300 in. )2 ⎡ 2.00 × ⎛ 15.9 in. ⎞ ⎛ 0.300 in. ⎞ ⎢1 + 3 ⎜ ⎟⎜ ⎟ ⎝ 17.7 in. ⎠ ⎝ 0.425 in. ⎠ ⎢⎣ ( 29, 000 ksi )( 50 ksi ) ( 0.425 in.) 0.300 in. = 134 kips > 33.7 kips o.k. 184 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 1.5 ⎤ ⎥ ⎥⎦ Beam-to-Column Connection The required shear strength is: LRFD ASD Vu = 30.0 kips + 50.5 kips Va = 20.0 kips + 33.7 kips = 80.5 kips = 53.7 kips Use a double-angle connection as shown in Figure 5-18. Design bolts at beam-to-column connection Use d-in.-diameter ASTM A325-N bolts with horizontal short slots in the angles. From AISC Manual Table 7-1, the available shear strength is: LRFD ϕrnv = 24.3 kips/bolt ASD rnv = 16.2 kips/bolt Ω The number of bolts required is: LRFD n= 80.5 kips 24.3 kips/bolt = 3.31 bolts ASD n= 53.7 kips 16.2 kips/bolt = 3.31 bolts The AISC Manual recommends a minimum depth connection for a W18 to be 3 rows of bolts, or 6 bolts total. Thus, try a minimum depth connection as shown in Figure 5-18 with six bolts arranged in three rows of 2 on the column, as indicated. Try angles 2L4×4×a×8½. Check bolt bearing on the connection angles According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength. Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 3.00 in. − (0.5 + 0.5)dh = 3.00 in. − 1(, in.) = 2.06 in. For the inner bolts, the bearing strength per bolt is: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 2.06 in. )( a in.) ( 58 ksi ) 1.2lc tFu Ω = 1.2 ( 2.06 in. )( a in.) (58 ksi ) 200 = 40.3 kips/bolt ϕ2.4dtFu = 0.75 ( 2.4 ) ( d in.) ( a in.) ( 58 ksi ) = 26.9 kips/bolt 2.4dtFu Ω = 2.4 ( d in. ) ( a in.) ( 58 ksi ) 2.00 = 34.3 kips/bolt = 22.8 kips/bolt Therefore, ϕrn = 34.3 kips/bolt. Therefore, rn / Ω = 22.8 kips/bolt. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 185 Because the available bolt shear strength determined previously (24.3 kips for LRFD and 16.2 kips for ASD) is less than the bearing strength, the limit state of bolt shear controls the strength of the inner bolts. For the end bolts, based on a 14-in. edge distance, the clear distance is: lc = 14 in. − 0.5dh = 14 in. − 0.5(, in.) = 0.781 in. For the end bolts, the available bearing strength per bolt is: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 0.781 in. )( a in.) ( 58 ksi ) 1.2lc tFu Ω = 1.2 ( 0.781 in. )( a in.)( 58 ksi ) 2.00 = 15.3 kips/bolt = 10.2 kips/bolt ϕ2.4dtFu = 0.75 ( 2.4 )( d in.) ( a in.) ( 58 ksi ) 2.4dtFu Ω = 2.4 ( d in. ) (a in.) ( 58 ksi ) 2.00 = 34.3 kips/bolt = 22.8 kips/bolt Therefore, rn / Ω = 10.2 kips/bolt. Therefore, ϕrn = 15.3 kips/bolt. Bolt bearing on the angles controls over bolt shear for the end bolts. Check bolt bearing on the column web Tearout, the limit state captured by the first part of Equation J3-6a, is not applicable to the column because there is no edge through which the bolts can tear out. LRFD ASD ϕrn = ϕ2.4 dtFu ⎛ 0.360 in. ⎞ = 0.75 ( 2.4 )( d in.) ⎜ ⎟ ( 65 ksi ) 2 ⎝ ⎠ = 18.4 kips/bolt rn 2.4 dtFu = Ω Ω ⎛ 0.360 in.⎞ 2.4 ( d in.) ⎜ ⎟⎠ ( 65 ksi ) ⎝ 2 = 2.00 = 12.3 kips/bolt Bolt bearing on the column web controls over bolt shear. Dividing the web by 2 in the above calculation assumes that the same connection and load exist on both sides of the web. It is conservative for other cases. Thus, the total available bolt strength, including the limit states of bolt shear and bolt bearing on the connection angles and the column web, is: LRFD ASD ϕRn = 2 (15.3 kips ) + 4 (18.4 kips ) Rn = 2 (10.2 kips ) + 4 (12.3 kips ) Ω = 69.6 kips > 53.7 kips o.k. = 104 kips > 80.5 kips o.k. Check shear yielding on the angles The gross area is: Agv = 2(82 in.)(a in.) = 6.38 in.2 186 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available shear yielding strength, from AISC Specification Section J4.2(a), Equation J4-3, is: LRFD ASD ϕRn = ϕ 0.60 Fy Agv ( = 1.00 ( 0.60 )( 36 ksi ) 6.38 in.2 = 138 kips > 80.5 kips ) o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( 6.38 in. ) = 1.50 = 91.9 kips > 53.7 kips Check shear rupture on the angles The net area is determined from AISC Specification Section B4.3: Anv = Ag − 2(3t)(dh + z in.) = 6.38 in.2 − 2(3)(a in.)(, in. + z in.) = 4.13 in.2 From AISC Specification Section J4.2(b), Equation J4-4, the available shear rupture strength is: LRFD ϕRn = ϕ 0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 58 ksi ) 4.13 in.2 = 108 kips > 80.5 kips o.k. ) Rn 0.60 Fu Anv = Ω Ω = ( 0.60 ( 58 ksi ) 4.13 in.2 2.00 = 71.9 kips > 53.7 kips ) o.k. Check block shear rupture on the angles The controlling block shear failure path is assumed to start at the top of the angles and follows the vertical bolt lines and then runs perpendicular to the outer edge of each angle, as shown in Figure 5-18a. For the tension area calculated in the following, the net area is based on the length of the short-slotted hole dimension given in AISC Specification Table J3.3. For the shear area, the net area is based on the width of the short-slotted hole, which is the same as for a standard hole. Fig. 5-18a. Block shear rupture failure path on double angles. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 187 The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Agv = (7.25 in.)(a in.)(2) = 5.44 in.2 Anv = 5.44 in.2 − 2.5(, in. + z in.)(a in.)(2) = 3.57 in.2 Ant = [(4.00 in. + 4.00 in. + 0.300 in. − 52 in.)/ 2 − 0.5(18 in. + z in.)](a in.)(2) = 0.605 in.2 Ubs is 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded. Shear yielding component: 0.60Fy Agv = 0.60(36 ksi)(5.44 in.2) = 118 kips Shear rupture component: 0.60Fu Anv = 0.60(58 ksi)(3.57 in.2) = 124 kips Tension rupture component: UbsFu Ant = 1(58 ksi)(0.605 in.2) = 35.1 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60Fu Anv + UbsFu Ant = 124 kips + 35.1 kips = 159 kips 0.60Fy Agv + UbsFu Ant = 118 kips + 35.1 kips = 153 kips Therefore, Rn = 153 kips. LRFD ASD ϕRn = 0.75 (153 kips ) = 115 kips > 80.5 kips o.k. Rn 153 kips = Ω 2.00 = 76.5 kips > 53.7 kips o.k. Design angle-to-beam web weld From AISC Manual Table 10-2, with L = 82 in. for Weld A, choose a x-in. fillet weld. Note that Table 10-2 is conservative for this example because it assumes an angle size of L4×3½, with the shorter leg attached to the beam. This example has an angle leg of 4 in. attached to the beam. The available weld strength of the double-angle connection is: LRFD ASD ϕRn = 110 kips > 80.5 kips Rn = 73.5 kips > 53.7 kips Ω Note: This is the case when the minimum web thickness is 0.286 in. tw = 0.300 in. > 0.286 in. o.k. 188 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The minimum web thickness of Table 10-2 is calculated from: t min = = 6.19 D Fu 6.19 ( 3 ) 65 ksi = 0.286 in. The completed design is shown in Figure 5-18. Example 5.9—Chevron Brace Connection Given: The general arrangement of a chevron braced frame is shown in Figures 2-1(e), 2-5 and 3-2, and the theory is given in Figures 4-5 through 4-7. There are two types of chevron braced frames: V and inverted-V. Design the chevron heavy bracing connection for the W27×114 beam and the (2) HSS8×8×½ braces shown in Figure 5-19. Figure 5-19 is typical of chevron bracing of the inverted-V type. The required axial strength of the bracing connections is: LRFD ASD Pu = ± 289 kips Pa = ± 193 kips Fig. 5-19. Typical chevron brace connection. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 189 Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A500 Grade B Fy = 46 ksi Fu = 58 ksi ASTM A572 Grade 50 Fy = 50 ksi Fu = 65 ksi From AISC Manual Table 1-1 and Table 1-12, the member geometric properties are as follows: Beam: W27×114 tw = 0.570 in. tf = 0.930 in. Braces: HSS8×8×½ Ag = 13.5 in.2 r = 3.04 in. d = 27.3 in. bf = 10.1 in. kdes = 1.53 in. tdes = 0.465 in. The geometry of this connection is shown in Figure 5-19. The work point is at the concentric location at the beam gravity axis, e = 13.65 in. from the outside face of the bottom flange. The brace bevels are equal and the brace loads are equal at ±289 kips (LRFD) or ±193 kips (ASD). Thus the gusset will be symmetrical and Δ = 0, using the notation defined in Figure 4-5. Because the brace forces are reversible, it is possible that both are tension or both are compression, but the most common case is for one to be tension and the other to be compression, as will be assumed here. Figure 5-20 shows the free body diagram (admissible force fields) for this case. Using the general notation given in Figure 4-5: 1 ( L2 − L1 ) = 0 2 L1 = 32 in. Δ = L2 = 32 in. Fig. 5-20. Admissible force fields. 190 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 L = 64 in. e = 13.65 in. h = 18 in. In the following calculations, the subscript “1” denotes Brace 1, on the left side of Figure 5-19, and the subscript “2” denotes Brace 2 on the right side. Axial force, shear force and moment acting on Section b-b have a “prime” symbol to distinguish them from the axial force, shear force and moment acting on Section a-a. The horizontal and vertical components of the brace forces are: LRFD ASD Pu1 = −289 kips Pa1 = −193 kips Hu1 = ( −289 kips ) cos 45° H a1 = ( −193 kips ) cos 45° = − 204 kips = − 136 kips Vu1 = ( −289 kips ) sin 45° Va1 = ( −193 kips ) sin45° = −204 kips Pu 2 = 289 kips = −136 kips Pa 2 = 193 kips Hu 2 = ( 289 kips ) cos 45° H a 2 = (193 kips ) cos 45° = 204 kips Vu 2 = ( 289 kips ) sin 45° = 136 kips Va 2 = (193 kips ) sin45° = 204 kips = 136 kips Using the equations presented in Figure 4-5, the moments are: LRFD Mu1 = Hu1e + Vu1Δ ASD M a1 = H a1e + Va1Δ = ( −204 kips )(13.65 in. ) + ( −204 kips )( 0 ) = − 2, 780 kip-in. = (−136 kips) (13.65 in.) + ( −136 kips)( 0 ) = − 1,860 kip-in. M a2 = Ha2e − Va2 Δ Mu 2 = Hu 2 e − Vu 2 Δ = ( 204 kips )(13.65 in. ) − ( 204 k ips) ( 0 ) = 2, 780 kip-in. = (136 kips) (13.65 in.) − (136 kips)( 0) = 1,860 kip-in. ′ = 8Va1L − 4 Ha1 h − 2Ma1 Ma1 ′ = 8 Vu1 L − 4 Hu1 h − 2 Mu1 Mu1 = 8 ( −204 kips )( 64.0 in. ) − 4 ( −204 kips )(18.0 in. ) −2 (− 2,780 kips ) = 8( −136 kips)( 64.0 in.) − 4 ( − 136 kips) (18.0 in. ) − 2 ( −1,860 kips) = 454 kip-in. ′ = 8Va2L − 4 Ha2 h − 2Ma2 Ma2 = 676 kip-in. = 8 (136 kips)( 64.0 in. ) ′ = 8 Vu2 L − 4 Hu2 h − 2Mu2 Mu2 = 8 ( 204 kips)( 64.0 in. ) − 4(136 kips) (18.0 in.) − 2(1,860 kips) = − 454 kip-in. − 4 ( 204 kips )(18.0 in. ) −2 ( 2,780 kips ) = −676 kip-in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 191 Note that the signs on the various forces and equations are important. Calculating the forces on Section a-a—the gusset-to-beam interface (using equations based on Figure 4-6): LRFD ASD Axial Axial N u = Vu1 + Vu 2 Na = V a1 + V a2 = −204 kips + 204 kips = 0 kips = −136 kips + 136 kips = 0 kips Shear Vu = Hu1 − Hu 2 Shear Va = Ha1 − Ha2 = −204 kips − 204 kips = − 408 kips = −136 kips − 136 kips = −272 kips Moment Mu = Mu1 − Mu 2 Moment M a = M a1 − M a 2 = −2, 780 kip-in. − 2, 780 kip-in. = −5, 560 kip-in. = −1, 860 kip-in. − 1, 860 kip-in. = −3, 720 kip-in. The negative signs on the shear and moment indicate that these forces act opposite to the positive directions assumed in Figure 4-6. An important location in the gusset is Section b-b shown in Figure 4-7. The forces on this section are: LRFD ASD Axial Axial N u′ = 2 (Hu1 + Hu2 ) Na′ = 2 (Ha1 + Ha2 ) = 2 ( −204 kips + 204 kips) = 2 ( −136 kips + 136 kips) = 0 kips = 0 kips Shear Shear V u′ = 2 ( V u1 − V u2 ) − 2 Mu L = 2 (−204 kips − 204 kips) − V a′ = 2 ( Va1 − V a2 ) − 2 ( −5, 560 kip-in. ) 64.0 in. 2 Ma L = 2 ( −136 kips − 136 kips) − = − 30.3 kips = −119.8 kips Moment Mu′ = Mu′1 + Mu′ 2 Moment Ma′ = Ma′1 + Ma′ 2 = 676 kip-in. + ( −676 kip-in. ) = 454 kip-in. + ( −454 kip-in. ) =0 =0 2 ( −3,720 kip-in. ) 64.0 in. These forces are shown in Figure 5-20 acting on the left half of the gusset. Vu′ is acting opposite to the direction shown, as indicated by the minus sign. With all interface (Section a-a) and internal (Section b-b) forces known, the connection can now be designed. 192 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Brace-to-Gusset Connection This part of the connection should be designed first because it will give a minimum required size of the gusset plate. Check tension yielding on the brace The available tensile yielding strength is determined from AISC Specification Section J4.1(a), Equation J4-1: LRFD ASD ϕRn = ϕFy Ag = 0.90 ( 46 ksi ) 13.5 in.2 ( ) = 559 kips > 289 kips o.k. Rn Fy Ag = Ω Ω = ( 46 ksi ) (13.5 in.2 ) 1.67 = 372 kips > 193 kips o.k. Check tensile rupture on the brace From AISC Specification Equation J4-4, determine the minimum weld length required to provide adequate available shear rupture strength of the brace material. Choose a connection length between the brace and the gusset plate based on the limit state of shear rupture in the brace wall: LRFD Pu = 0.60 ⎡⎣ϕ Fu tl ( 4 ) ⎤⎦ 289 kips = 0.60 ( 0.75 )( 58 ksi )( 0.465 in. ) l ( 4 ) Therefore, l = 5.95 in. ASD ⎡F ⎤ Pa = 0.60 ⎢ a tl ( 4 ) ⎥ ⎣Ω ⎦ ⎛ 58 ksi ⎞ 193 kips = 0.60 ⎜ ⎟ ( 0.465 in. ) l ( 4 ) ⎝ 2.00 ⎠ Therefore, l = 5.96 in. If a c-in. fillet weld is assumed with four lines of welds, AISC Specification Equation J2-4 and Table J2.5 give: LRFD ⎛ ϕPn = ϕ0.60 FEXX ⎜ ⎝ Pu l ≥ ⎛ ϕ0.60 FEXX ⎜ ⎝ 1 ⎞ ⎟ w ( 4l ) ≥ Pu 2⎠ 1 ⎞ ⎟ w (4) 2⎠ 289 kips ≥ ⎛ 1 ⎞ 0.75 ( 0.60 )( 70 ksi ) ⎜ ⎟ ( c in.) ( 4 ) ⎝ 2⎠ ≥ 10.4 in. ASD ⎛ 1 ⎞ 0.60 FEXX ⎜ ⎟ w ( 4l ) ⎝ 2⎠ ≥ Pa Ω Ω Pa ≥ ⎛ 1 ⎞ 0.60 FEXX ⎜ ⎟ w (4) ⎝ 2⎠ Pn = Ω l ≥ 2.00 (193 kips ) ⎛ 1 ⎞ 0.60 ( 70 ksi ) ⎜ ⎟ ( c in.) ( 4 ) ⎝ 2⎠ ≥ 10.4 in. Note that in this application, the actual weld size used would be c in. because of the slot gap; that is, with 4-in. fillet welds called out on the drawing, the fabricator would increase the weld size to c in. in order to compensate for the gap between the brace and the gusset (see AWS D1.1 clause 5.22.1). This is a single pass fillet whereas the specified c-in. fillet weld will be a multi-pass a-in. fillet weld. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 193 From the AISC Manual Part 8, Equations 8-2a and 8-2b, this calculation could more simply be written: LRFD ϕPn ASD = 1.392 Dl Pn = 0.928 Dl Ω 193 kips = 0.928 ( 5 )( 4l ) 289 kips = 1.392 ( 5 )( 4l ) = 10.4 in. l = 10.4 in. l Use a 12-in.-long c-in. fillet weld. With l = 12.0 in., determine the shear lag factor, U. The symbol, dslot, used in the following is the width of the slot in the HSS brace, which is taken as the gusset thickness plus z in. on either side. A w-in.-thick gusset plate is assumed and later checked. Ag = 13.5 in.2 An = Ag − 2tdslot (dslot = width of slot) = 13.5 in.2 − 2(0.465 in.)(w in. + z in. + z in.) = 12.7 in.2 From AISC Specification Table D3.1, Case 6 with a single concentric gusset plate: x = = B2 + 2 BH 4(B + H ) ( B = H = 8.00 in. ) (8.00 in.)2 + 2 (8.00 in. )(8.00 in. ) 4 ( 8.00 in. + 8.00 in. ) = 3.00 in. ⎛x⎞ U = 1− ⎜ ⎟ ⎝l⎠ ⎛ 3.00 in. ⎞ = 1− ⎜ ⎟ ⎝ 12.0 in. ⎠ = 0.750 From AISC Specification Equation D3-1: Ae = AnU(Spec. Eq. D3-1) = (12.7 in.2)(0.750) = 9.53 in.2 From AISC Specification Section J4.1(b), the available tensile rupture strength of the brace is: LRFD ϕRn = ϕFu A e ASD ( = 0.75 ( 58.0 ksi ) 9.53 in. = 415 kips > 289 kips 2 ) o.k. Rn Fu Ae = Ω 2.00 = ( 58.0 ksi ) ( 9.53 in.2) 2.00 = 276 kips > 193 kips 194 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 o.k. Check block shear rupture on the gusset plate Assume a gusset plate thickness of w in. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant (Spec. Eq. J4-5) Shear yielding component: Agv = Anv = 2tgl = (2)(w in.)(12.0 in.) = 18.0 in.2 0.60Fy Agv = 0.60(50 ksi)(18.0 in.2) = 540 kips Shear rupture component: 0.60Fu Anv = 0.60(65 ksi)(18.0 in.2) = 702 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Agt = Ant = tgB = (w in.)(8.00 in.) = 6.00 in.2 UbsFu Ant = 1(65 ksi)(6.00 in.2) = 390 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60Fu Anv + UbsFu Ant = 702 kips + 390 kips = 1,090 kips 0.60Fy Agv + UbsFu Ant = 540 kips + 390 kips = 930 kips Therefore, Rn = 930 kips. LRFD ASD ϕRn = 0.75 ( 930 kips ) = 698 kips > 289 kips o.k.. Rn 930 kips = Ω 2.00 = 465 kips > 193 kips o.k. If the braces had different loads, block shear would need to be checked for each brace. Check the gusset plate for tensile yielding on the Whitmore section From AISC Manual Part 9, the width of the Whitmore section is: lw = B + 2l tan 30° = 8.00 in. + 2(12.0 in.) tan 30° = 21.9 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 195 It can be seen from Figure 5-20 that about 2.00 in. of the Whitmore section is in the beam web. The web thickness, tw = 0.570 in. < tg = w in.; therefore, the portion of the Whitmore area in the web is accounted for. Aw = (21.9 in. − 2.00 in.)(w in.) + (2.00 in.)(0.570 in.) = 16.1 in.2 From AISC Specification Section J4.1(a): LRFD ASD ϕRn = 0.90 Fy Aw = 0.90 ( 50 ksi ) 16.1 in.2 ( ) = 725 kips > 289 kips o..k. Rn Fy Aw = Ω 1.67 = ( 50 ksi ) (16.1 in.2 ) 1.67 = 482 kips > 193 kips o.k. Check the gusset plate for buckling on the Whitmore section The available compressive strength of the gusset plate based on the limit state of flexural buckling is determined from AISC Specification Section J4.4, using an effective length factor, K, of 0.65 according to Dowswell (2006, 2009, 2012). As shown in Figure 5-20, the unbraced length of the gusset plate is 8.00 in. KL 0.65 ( 8.00 in. ) = r ⎛ w in. ⎞ ⎜ ⎟ ⎝ 12 ⎠ = 24.0 KL < 25.0, according to AISC Specification Section J4.4, the strength in compression is the same as that for tension r yielding, which was previously checked. Because Gusset-to-Beam Connection (Section a-a) The required strengths are: LRFD Required shear strength, Vu = 408 kips Required axial strength, Nu = 0 kips Required flexural strength, Mu = 5,560 kip-in. ASD Required shear strength, Va = 272 kips Required axial strength, Na = 0 kips Required flexural strength, Ma = 3,720 kip-in. Check the gusset plate for tensile yielding and shear yielding along the beam flange In this example, to allow combining the required normal strength and flexural strength, stresses will be used to check shear yielding and tensile yielding limit states on the gusset plate. Shear yielding is checked using AISC Specification Section J4.2 and tensile yielding is checked using Section J4.1: 196 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Shear yielding stress: Va fav = Ag Shear yielding stress: fuv = = Vu Ag 408 kips w in. ( )( 64.0 in.) = = 8.50 ksi < 1.00 ( 0.60 )( 50 ksi ) = 30 ksi 0 kips ( w in.)( 64.0 in.) = = 0 ksi fub = = = 5.67 ksi < 0.60 ( 50 ksi ) 1.50 = 20 ksi o.k. Tensile yielding stress: Na faa = Ag Tensile yielding stress: Nu fua = Ag = o.k. 272 kips in. w ( )( 64.0 in.) 0 kips w in. ( )( 64.0 in.) = 0 ksii Mu Z fab = 5, 560 kip-in. ( w in.)( 64.0 in.) 2 = 4 Ma Z 3, 720 kip-in. ( w in. )( 64.0 in.)2 = 7.24 ksi = 4.84 ksi The total normal stress = fun The total normal stress = fan fun = fua + fub 4 fan = faa + fab = 0 ksi + 7.24 ksi = 7.224 ksi < 0.90 ( 50 ksi ) = 45.0 ksi = 0 ksi + 4.84 ksi o.k. = 4.84 ksi < 50 ksi = 29.9 ksi 1.67 o.k. Design weld at gusset-to-beam flange connection The 408-kip (LRFD) or 272-kip (ASD) shear force is to be delivered to the center of the beam. This is a function of the required flexural strength, 5,560 kip-in (LRFD) or 3,720 kip-in. (ASD). The effective eccentricity of the shear force is: LRFD e= Mu Vu 5, 560 kip-in. 408 kips = 13.6 in. = ASD e= Ma Va 3, 720 kip-in. 272 kips = 13.7 in. = Therefore, from AISC Manual Table 8-4 for Angle = 0, k = 0, a = 13.6 in./ 64.0 in. = 0.213, by interpolation C = 3.46. With C1 = 1.0, the number of sixteenths of weld required with a ductility factor of 1.25 (as discussed in AISC Manual Part 13) is determined from AISC Manual Equation 8-13: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 197 LRFD Dreq’d = = ASD Pu ϕCC1l Dreq’d = ( 408 kips )(1.25) ( 0.75)( 3.46 )(1.0 )( 64.0 in.) = = 3.07 ΩPa CC1l 2.00 ( 272 kips )(1.25 ) ( 3.46 )(1.0 )( 64.0 in.) = 3.07 Use a 4-in. double-sided fillet weld. An alternative method to determine the required weld size that does not require the use of an AISC Manual table follows. See Appendix B for a discussion of the method. This method is similar to the method of Hewitt and Thornton (2004) except that it allows the use of the increased strength of fillet welds loaded at an angle relative to the weld longitudinal axis. The required shear strength is Vu = 408 kips (LRFD) or Va = 272 kips (ASD). The normal force and the moment can be combined to generate a maximum equivalent normal force (see Appendix B, Figure B-1): LRFD ASD N max = N u + 2 Mu L/2 N max = N a + 2 Ma L/2 = Nu + 4 Mu L = Na + 4 Ma L = 0 kips + 4 ( −5, 560 kip-in. ) = 0 kips + 64 in. = 348 kipss and a minimum equivalent normal force: N min = N u − N min = N a − 4 ( −5, 560 kip-in. ) 64 in. R peak = Va 2 + N max 2 ( 408 kips )2 + ( 3448 kips )2 = The average weld resultant force: ⎛ N max + N min ⎞ = Vu + ⎜ ⎟ 2 ⎝ ⎠ ⎛ N max + N min ⎞ Ravg = Va2 + ⎜ ⎟ 2 ⎝ ⎠ 2 2 = 536 kips 2 ( 272 kips )2 + ( 2333 kips )2 = 358 kips The average weld resultant force: = 536 kips ( 408 kips ) 64.0 in. The peak weld resultant force: R peak = Vu 2 + N max 2 = 4 ( −3, 720 kip-in. ) = 233 kips The peak weld resultant force: Ravg 4 Ma L = 0 kips − = 348 kips = 64.0 in. = 233 kiips and a minimum equivalent normal force: 4 Mu L = 0 kips − 4 ( −3, 720 kip-in. ) ⎛ 348 kips + 348 kips ⎞ +⎜ ⎟ 2 ⎠ ⎝ 2 = 2 233 kips + 233 kips ⎞ ⎟ 2 ⎝ ⎠ ( 272 kips )2 + ⎛⎜ = 358 kips 198 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 2 Because 1.25Ravg > Rpeak, use 1.25Ravg to size the weld. From AISC Manual Equation 8-2, including the directional strength increase of the fillet weld provided in AISC Specification Equation J2-4: LRFD ASD ⎛ N max ⎞ θ = tan −1 ⎜ ⎟ ⎝ Vu ⎠ ⎛ 348 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 408 kips ⎠ ⎛ N max ⎞ θ = tan −1 ⎜ ⎟ ⎝ Va ⎠ ⎛ 233 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 272 kips ⎠ = 40.5° = 40.6° LRFD Dreq’d = = ( ASD 1.25Ravg 1.5 1.392 1.0 + 0.50 sin ( ) θ (2L ) 1.25 ( 536 kips ) 1.5 1.392 1.0 + 0.50 sin ) 40.5° ⎡⎣2 ( 64.0 in. ) ⎤⎦ Dreq’d = = ( 1.25 Ravg ) 0.928 1.0 + 0.50 sin1.5 θ ( 2 L ) ( 1.25 ( 358 kips ) ) 0.928 1.0 + 0.50 sin1.5 40.6° ⎡⎣2 ( 64.0 in. ) ⎤⎦ = 2.98 = 2.98 % difference betweeen methods % difference betweeen methods ⎡ ( 2.98 − 3.07 ) ⎤ ⎢ ⎥ × 100% = 2.93% 3.07 ⎣ ⎦ ⎡ ( 2.98 − 3.07 ) ⎤ ⎢ ⎥ × 100% = 2.93% 3.07 ⎣ ⎦ Note that the directional strength increase of 1.0 + 0.50 sin1.5 40.5° = 1.26 indicates a 26% increase over the strength of a longitudinally loaded fillet weld. This alternative method produces a result that is 2 to 3% smaller than the result of using the AISC Manual Table 8-4 method, which is essentially the same answer. The required weld size is 4 in., which is the AISC minimum weld size from AISC Specification Table J2-4. A discussion on the alternative method is given in Appendix B. Check gusset internal strength (Section b-b) As mentioned previously, the prime symbol after the force quantity indicates that it acts on Section b-b through the gusset plate. The control section for this chevron gusset is the horizontal Section a-a. Section b-b, which is an internal gusset section, plus the beam cross section above the gusset Section b-b, must be able to transfer the brace vertical components, V1 and V2, across this section. The forces H′, N′ and M′ are a direct result of the forces H, N and M assumed on the control Section a-a. See Figures 4-5, 4-6 and 4-7. The required strengths are: LRFD ASD Required shear strength, Vu′ = −30.3 kips Required shear strength, Va′ = −19.8 kips Required axial strength, N u′ = 0 kips Required axial strength, N a′ = 0 kips Required flexural strength, Mu′ = 0 kip-in. Required flexural strength, M a′ = 0 kip-in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 199 Check the gusset plate for shear yielding at Section b-b From AISC Specification Section J4.2(a), the available shear yielding strength of the gusset plate is: LRFD ASD ϕRn = ϕ0.60 Fy Agv Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )( w in. )(18.0 in. ) = 1.50 = 270 kips > 19.8 kips o.k. = 1.00 ( 0.60 )( 50 ksi )( w in. )(18.0 in. ) = 405 kips > 30.3 kips o.k. Because N′and M′ are zero, there are no further checks. If N′ and M′ were not zero, it is possible for a compressive stress to exist on the gusset free edge at Section b-b. In this case, the gusset should be checked for buckling under this stress. The procedure given in the AISC Manual on pages 9-8 and 9-9 for buckling of double-coped beams can be used. This Manual procedure is developed into a gusset free-edge buckling method in Appendix C, Section C.4, where it is referred to as the admissible force maintenance method (AFMM). Check gusset stresses on Section b-b for a hypothetical load case As an example, consider that both braces of Figure 5-19 are in compression simultaneously. Then P1 = P2, and from Figure 4-5: LRFD ASD Mu1 = ( −204 kips )(13.65 in. ) M a1 = ( −136 kips )(13.65 in. ) = −2, 780 kip-in. = −1, 860 kip-in. Mu 2 = ( −204 kips ) (13.65 in. ) M a 2 = ( −136 kips ) (13.65 in. ) = −2, 780 kip-in. Mu′1 = 1 8 = −1, 860 kip-in. ( −204 kips )( 64.0 in.) − ( −204 kips ) (18.0 in.) − 21 ( −2, 780 kip-in. ) 1 4 = 676 kip-in. ′ Mu 2 = 81 ( −204 kips )( 64.0 in. ) − 14 ( −204 kips )(18.0 in. ) − 21 ( −2, 780 kip-in. ) = 676 kip-in. M a′ 1 = ( −136 kips )( 64.0 in.) − 14 ( −136 kips )(18.0 in. ) − 21 ( −1, 860 kip-in. ) 1 8 = 454 kip-in. M a′ 2 = 81 ( −136 kips )( 64.0 in. ) − 14 ( −136 kips )(18.0 in. ) − 21 ( −1, 860 kip-in. ) = 454 kip-in. From Figure 4-6, the forces on Section a-a are: LRFD Nu = − 204 kips + ( −204 kips ) = − 408 kips Vu = − 204 kips − ( −204 kips ) = 0 kipss Mu = −2, 780 kip-in. − ( −2, 780 kip-in. ) = 0 kip-in. ASD Na = −136 kips + ( −136 kips ) = −272 kips Va = −136 kips − ( −136 kips ) = 0 kipss Ma = −1, 860 kip-in. − ( −1, 860 kip-in. ) = 0 kip-in. 200 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 And those on Section b-b from Figure 4-7 are: LRFD ASD 1 ⎡ −204 kips + ( −204 kips ) ⎤⎦ 2⎣ = −204 kips 2 1 V ′ = ⎡⎣ −204 kips − ( −204 kipss ) ⎤⎦ − (0) 64 in. 2 = 0 kips M ′ = 676 kip-in. + 676 kip-in. = 1, 350 kip-in. 1 ⎡ −136 kips + ( −136 kips ) ⎤⎦ 2⎣ = −136 kips 2 1 V ′ = ⎡⎣ −136 kips − ( −136 kipss ) ⎤⎦ − (0) 64 in. 2 = 0 kips M ′ = 454 kip-in. + 454 kip-in. = 908 kip-in. N′ = N′ = The entire Section b-b is under a uniform compression of N′ and an additional compressive force due to M′. The worst case equivalent normal force (compression) can be calculated (see Appendix B, Figure B-1) as: LRFD ASD ⎛1, 350 kip -in. ⎞ N ue = 204 kips + ⎜ ⎟ (2) ⎝ 9.00 in. ⎠ = 504 kips ⎛ 908 kip- in. ⎞ N ae = 136 kips + ⎜ ⎟ (2) ⎝ 9.00 in. ⎠ = 338 kips Note that this equivalent normal force will produce the correct gusset maximum normal stress but is conservative for buckling because the compressive portion of the stress due to M′ acts over the upper half of gusset Section b-b, because M′ is greater than zero. The dimensions a and b, scaled or calculated from Figure 5-19, are a = 52 in. and b = 18 in. From Appendix C, Section C.4, λ and Q can be determined as follows: (b λ= t w ) Fy 5 475 + = (C-7) 1,120 ( a b )2 (18.0 in. w in. ) 5 475 + 50 ksi 1,120 ( 52.0 in. 18.0 in. )2 = 1.38 Since 0.70 < λ ≤ 1.41: Q = 1.34 − 0.486λ = 1.34 − 0.486 (1.38 ) = 0.670 LRFD ASD ϕFcr = 0.90 ( 0.670 )( 50 ksi ) Fcr ( 0.670 )( 50 ksi ) = Ω 1.67 = 20.1 ksi = 30.2 ksi AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 201 The actual (worst case) compressive stress is: LRFD fua = ASD 504 kips ( w in.)(18.0 in.) = 37.3 ksi > 30.2 ksi faa = n.g. 338 kips ( w in.)(18.0 in.) = 25.0 ksi > 20.1 ksi n.g. Because gusset buckling does control, an edge stiffener could be added, or a thicker plate could be used. The latter is usually the more economical solution. A d-in.-thick plate will give: λ = (18.0 in. 5 475 + d in. ) 50 ksi 1,120 ( 52.0 in. 18.0 in. )2 = 1.18 Q = 1.34 − 0.486 (1.18 ) = 0.767 LRFD ASD Fcr ( 0.767 )( 50 ksi ) = Ω 1.67 = 23.0 ksi ϕFcr = 0.90 ( 0.767 )( 50 ksi ) = 34.5 ksi fua = 504 kips ( d in.) (18.0 in.) = 32.0 ksi < 34.5 ksi o.k. faa = 338 kips ( d in. ) (18.0 in.) = 21.5 ksi < 23.0 ksi o.k. A less conservative method for calculating the maximum compressive stress is based on the fact that the moment M′ is a free vector and does not need to be applied at the half depth of Section b-b of the gusset. A discussion of this matter can be found in Tamboli (2010). If this method is used, it will be found that fu = 29.9 ksi for LRFD and fa = 19.9 ksi for ASD, and the compression zone occurs at the free edge when both braces are in tension. To calculate λ, b = 13.6 in. and use a w-in.-thick gusset plate: λ= (13.6 in. w in. ) 5 475 + 50 ksi 1,120 ( 52.0 in. 13.6 in. )2 = 1.09 Q = 1.34 − 0.486 (1.09 ) = 0.810 LRFD ASD ϕFcr = 0.90 ( 0.810 )( 50 ksi ) = 36.5 ksi > 29.9 ksi o.k. Fcr ( 0.810 )( 50 ksi ) = Ω 1.67 = 24.3 ksi > 19.9 ksi 202 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 o.k. Thus, the more complex and less conservative method verifies that the w-in.-thick gusset plate is satisfactory. The design checks will continue with the w-in.-thick gusset plate. Check the gusset plate for buckling on the Whitmore section Taking a conservative estimate of the buckling length as the vertical distance from the ends of the braces at their centerlines to the beam flange and K = 1.2: ⎛ 8.00 in. ⎞ 1.2 ⎜ ⎟ KL ⎝ 2 ⎠ = r ⎛ w in. ⎞ ⎜ ⎟ ⎝ 12 ⎠ = 31.4 As previously determined, lw = 21.9 in. and Aw = 16.1 in.2 From AISC Specification Section J4.4, because KL/r > 25, AISC Specification Chapter E applies. From AISC Manual Table 4-22, interpolating for KL/r = 31.4, and using AISC Specification Section E3, the available compressive strength is determined as follows: LRFD ASD ϕc Fcr = 41.9 ksi ϕc Pn = ϕcFcr Aw ( = ( 41.9 ksi ) 16.1 in.2 ) = 675 kips > 289 kips o.k. Fcr = 27.9 ksi Ωc Pn Fcr Aw = Ωc Ωc ( = ( 27.9 ksi ) 16.1 in.2 ) = 449 kips > 193 kips o.k. Check the gusset plate for sidesway buckling For this alternate load case, with both braces in compression, consider gusset sidesway buckling under the compressive force of 408 kips (LRFD) or 272 kips (ASD) on Section a-a. From AISC Specification Section J4.4, because KL/ r > 25, AISC Specification Chapter E applies. From AISC Manual Table 4-22 and using AISC Specification Section E3, the available compressive strength is determined as follows: LRFD ASD ϕc Fcr = 41.9 ksi Fcr = 27.9 ksi Ωc ϕRn = ϕFcr tL = ( 41.9 ksi )( w in. )( 64 in. ) = 2, 010 kips > 408 kips o.k. Fcr tL Ω = ( 27.9 ksi )( w in. )( 64 in. ) Rn = = 1, 340 kips > 272 kips o.k.. In this buckling calculation it is assumed that the beam web is supported laterally by in-fill beams that frame orthogonally to the W27×114 and are of a similar depth. If the infill beams are shallow or nonexistent, a beam bottom flange bracing strut may be necessary. Returning now to the original problem, check the beam for web local yielding and local crippling. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 203 Check beam web local yielding The force used is the maximum normal force, Nmax. The normal force is applied near the midspan of the beam, which is at a distance from the end of the beam greater than the beam depth. Therefore, use AISC Specification Equation J10-2. From AISC Specification Equation J10-2: LRFD ASD N max = 348 kips Nmax = 233 kips ϕRn = ϕ Fy tw ( 5k + lb ) Rn Ω = 1.00 ( 50 ksi )( 0.570 in. ) ⎡⎣5 (1.53 in. ) + 64.0 in.⎤⎦ = Fy tw ( 5k + lb ) Ω ( 50 ksi )( 0.570 in.) ⎡⎣5 (1.53 in. ) + 64.0 in.⎤⎦ 1.50 = 1, 360 kips > 233 kips o.k. = 2, 040 kips > 348 kips o.k. = Check beam web shear yielding The maximum required shear strength is Vu = 408 kips (LRFD) and Va = 272 kips (ASD). From AISC Specification Equation J4-3: LRFD ASD ϕRn = ϕ 0.60 Fy Agv Rn 0.6 Fy Agv = Ω Ω 0.6 Fy tw l = Ω 0.60 ( 50 ksi )( 0.570 in. )( 64.0 in. ) = 1.50 = 730 kips > 272 kips o.k. = ϕ0.60 Fy tw l = 1.00 ( 0.60 )( 50 ksi )( 0.570 in. ) ( 64.0 inn. ) = 1, 090 kips > 408 kips o.k. The gusset length, l, is used here, but any length up to about half the beam length can be used. Check beam web local crippling The normal force is applied near the midspan of the beam, which is greater than a distance of d/ 2 from the beam end. Therefore, AISC Specification Equation J10-4 is applicable: LRFD 1.5 ⎤ ⎡ ⎛ lb ⎞ ⎛ tw ⎞ ⎥ EFyw tf ⎢ ϕRn = 1+3⎜ ⎟ ⎜ ⎟ ⎢ tw ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ 1.5 ⎡ ⎛ 64.0 in. ⎞ ⎛ 0.570 in. ⎞ ⎤ 2 = ( 0.75 )( 0.80 )( 0.570 in. ) ⎢1 + 3 ⎜ ⎟ ⎥ ⎟⎜ ⎢⎣ ⎝ 27.3 in. ⎠ ⎝ 0.930 in. ⎠ ⎥⎦ ϕ0.80t w2 × ( 29, 000 ksi )( 50 ksi )( 0.930 in.) 0.570 in. = 1, 310 kips > 348 kips o.k. ASD ⎡ Rn 0.80t w2 ⎢ ⎛ lb ⎞ = 1+ 3 ⎜ ⎟ ⎢ Ω Ω ⎝d⎠ ⎣ = ⎛ tw ⎞ ⎜ ⎟ ⎜tf ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ EFyw tf tw ( 0.80 )( 0.570 in.)2 ⎡⎢ 2.00 × ⎛ 64.0 in.⎞ ⎛ 0.570 in.⎞ 1+ 3 ⎜ ⎟ ⎟⎜ ⎢⎣ ⎝ 27.3 in.⎠ ⎝ 0.930 in..⎠ ( 29, 000 ksi )( 50 ksi )( 0.930 in. ) 0.570 in. = 874 kips > 233 kips o.k. 204 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 1.5 ⎤ ⎥ ⎥⎦ Check transverse section web yielding The transverse shear induced in the beam at the centerline of the gusset (gusset Section b-b) is: LRFD ASD Vu = 204 kips − 30.3 kips Va = 136 kips − 19.8 kips = 174 kips = 116 kips From AISC Specification Section G2.1, the available shear yielding strength is: From AISC Specification Section G2.1, the available shear yielding strength is: ϕVn = ϕ0.6 Fy AwCv Vn 0.6 Fy tw dCv = Ω Ω 0 . 6 ( )( 50 ksi )( 0.570 in.)( 27.3 in.)(1.0 ) = 1.50 = 311 kips > 116 kips o.k. = 1.00 ( 0.6 )( 50 ksi )( 0.570 in. )( 27.3 in. )(1.0 ) = 467 kips > 174 kips o.k. This completes the calculations supporting the connection design shown in Figure 5-19. Example 5.10—Nonorthogonal Bracing Connection Given: Building columns must sometimes be designed for changing floor arrangements from floor to floor. Building facades are sometimes arranged with a slope, resulting in nonorthogonal beam-to-column connections and bracing connections. This topic is discussed in Section 4.2.5. The theory is shown in Figure 4-17, and Figure 5-21 shows the connection designed in this example. The required strengths of the brace (and bracing connection) and the beam are: LRFD ASD Brace required strength, Pu = ± 525 kips Brace required strength, Pa = ± 350 kips Beam required shear strength, Vu = 30.0 kips Beam required shear strength, Va = 20.0 kips Use 1-in.-diameter ASTM A490-X bolts in the beam-to-column and gusset-to-column connections. Solution: From AISC Manual Tables 1-1, 1-7 and 1-12, the geometric properties are as follows: Beam W18×50 tw = 0.355 in. d = 18.0 in. bf = 7.50 in. tf = 0.570 in. kdes = 0.972 in. d = 17.5 in. bf = 16.4 in. tf = 2.47 in. h/tw = 7.41 in. Column W14×342 tw = 1.54 in. Connection Angles: 2L5×5×½ and 2L8×6×s (6 in. leg trimmed to 5 in.) Brace HSS10×10×½ Ag = 17.2 in.2 tdes = 0.465 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 205 From AISC Manual Table 2-4 and 2-5, the material properties are as follows: ASTM A36 Fy = 36 ksi Fu = 58 ksi ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A500 Grade B Fy = 46 ksi Fu = 58 ksi Fig. 5-21. Nonorthogonal bracing connection. 206 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 From AISC Manual Tables 7-1 and 7-2 the available bolt strength for the 1-in.-diameter ASTM A490-X bolts is: LRFD ASD ϕrnv = 49.5 kips single shear rnv = 33.0 kips single shear Ω rnt = 44.4 kips tension Ω ϕrnt = 66.6 kips tension Brace-to-Gusset Connection This part of the connection should be designed first to arrive at a preliminary gusset plate size. Check tension yielding on the brace The available tensile yielding strength is determined from AISC Specification Section D2, Equation D2-1. Alternatively, AISC Manual Table 5-5 may be used. LRFD ASD ϕ t Pn = ϕt Fy Ag Fy Ag = 0.90 ( 46 ksi ) 17.2 in. ( ) = 712 kips > 525 kips o.k. 2 Pn = Ωt Ωt = ( 46 ksi ) (17.2 in.2 ) 1.67 = 474 kips > 350 kips o.k. Check tensile rupture on the brace For preliminary calculations, assume a s-in.-thick gusset plate. Assume that the slot in the brace is 8 in. wider than the gusset plate. The effective net area is: Ae = AnU(Spec. Eq. D3-1) where An = Ag − 2 t g + 8 in. tdes ( ) = 17.2 in.2 − 2 ( s in. + 8 in. )( 0.465 in. ) = 16.5 in.2 The shear lag factor, U, is determined from AISC Specification Table D3.1, Case 6, with a single concentric gusset plate, assuming l ≥ H: x= = B 2 + 2 BH 4(B + H ) ( B = H = 10.0 in.) (10.0 in.)2 + 2 (10.0 in.)(10.0 in.) 4 (10.0 in. + 10.0 in. ) = 3.75 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 207 To proceed, the length, l, of the HSS-to-gusset weld is needed, which is based on the brace-to-gusset lap length. This is determined by checking the limit state of shear rupture in the brace wall. From AISC Specification Section J4.2: Rn = 0.60Fu Anv(Spec. Eq. J4-4) In this calculation, the net cross-sectional area of the brace is taken as the four walls of the brace, Anv = 4ltdes. Solving for the lap length or weld length, l: LRFD l≥ ≥ ASD Pu ϕ0.60 Fu t des 4 l≥ 525 kips 0.75 ( 0.60 )( 58 ksi )( 0.465 in. )( 4 ) ≥ 10.8 in. ≥ ΩPa 0.60 Fu t des 4 2.00 ( 350 kips ) 0.60 ( 58 ksi ) ( 0.465 in. )( 4 ) ≥ 10.8 in. Therefore, the minimum lap length is 10.8 in. Determine the length in the following based on assumed weld size. Design the weld between the brace and the gusset The strength of fillet welds defined in AISC Specification Section J2 can be simplified, as explained in Part 8 of the AISC Manual, to AISC Manual Equations 8-2a and 8-2b: LRFD ASD Rn = 0.928 Dl Ω ϕRn = 1.392 Dl Try c-in. fillet welds for the four lines of weld, which can be made in a single pass, and solve for l: LRFD ASD 4 ( 0.928 ) Dl ≥ Pa 4 (1.392 ) Dl ≥ Pu l≥ 525 kips 4 (1.392 )( 5 sixteenths ) l≥ 350 kips 4 ( 0.928 )( 5 sixteenths ) ≥ 18.9 in. ≥ 18.9 in. Use 19.0-in.-long, c-in. fillet welds for the brace-to-gusset connection. The minimum required weld size according to AISC Specification Table J2.4 for 12-in.-thick material is x in. Check that there are no limitations on the fillet welds according to AISC Specification Section J2.2b. For end-loaded fillet welds with a length up to 100 times the weld size, it is permitted to take the effective length equal to the actual length: ( c in. )(100 ) = 31.3 in. > 19.0 in. o.k. Returning to the tensile rupture check on the brace, the equation in AISC Specification Table D3.1, Case 6, with a single concentric gusset plate can be used because the length of weld, l, exceeds the wall of the HSS, H: ⎛x⎞ U = 1− ⎜ ⎟ ⎝l⎠ ⎛ 3.75 in. ⎞ = 1− ⎜ ⎟ ⎝ 19.0 in. ⎠ = 0.803 208 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Therefore, the effective net area is: Ae = AnU (Spec. Eq. D3-1) ( = 16.5 in.2 = 13.2 in.2 ) ( 0.8003) From AISC Specification Section D2(b), Equation D2-2, the available strength for the limit state of tensile rupture is: LRFD ϕt Rn = ϕt Fu Ae ( ASD = 0.75 ( 58 ksi ) 13.2 in.2 ) = 574 kips > 525 kips o.k. Rn Fu Ae = Ωt Ωt = ( 58 ksi ) (13.2 in.2 ) 2.00 = 383 kips > 350 kips o.k. The gusset size can now be approximated as shown in Figure 5-21 as 24 in. high and 27 in. wide. This is a preliminary size. The 27 in. width is shown as the initial size in Figure 5-21. Check the gusset plate for tensile yielding on the Whitmore section The gusset has been assumed to be s in. thick. From AISC Manual Part 9, the width of the Whitmore section is: lw = 10.0 in. + 2 (19.0 in. ) tan 30° = 31.9 in. This section penetrates the beam web by 6 in. and passes out into “space” past the column web by 2 in. To keep the Whitmore section symmetric, subtract 2 in. off of each side so that lw = 31.9 in. − 4.00 in. = 27.9 in. The beam web is thinner than the gusset, so the effective Whitmore area is: ( ) Aw = ( lw − 4.00 in. ) t g + ( 4.00 in. )( t w ) = ( 27.9 in. − 4.00 in. )( s in. ) + ( 4.00 in. )( 0.355 in. ) = 16.4 in.2 From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate on the Whitmore section is: LRFD ASD ϕRn = ϕFy Aw = 0.90 ( 36 ksi ) 16.4 in.2 ( ) = 531 kips > 525 kips o.k. Rn Fy Aw = Ω Ω = ( 36 ksi ) (16.4 in.2 ) 1.67 = 354 kips > 350 kips o.k. Check the gusset plate for compression buckling on the Whitmore section From Dowswell (2006), this is a compact gusset and buckling is not a limit state. However, if buckling is checked, the available compressive strength based on the limit state of flexural buckling is determined from AISC Specification Section J4.4. Whitmore buckling is based on the gusset free length, lb = 8 in., in Figure 5-21. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 209 Check block shear rupture on the gusset plate The block shear rupture failure path is assumed to follow the overlap of the brace on the gusset plate. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = (19.0 in. )( 2 )( s in. ) = 23.8 in.2 ( ) ( ) 0.60 Fy Agv = 0.60 ( 36 ksi ) 23.8 in.2 = 514 kips Shear rupture component: Anv = Agv = 23.8 in.2 0.60 Fu Anv = 0.60 ( 58 ksi ) 23.8 in.2 = 828 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded Ant = (10.0 in. )( s in. ) = 6.25 in.2 ( Ubs Fu Ant = 1 ( 58 ksi ) 6.25 in.2 ) = 363 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60 Fu Anv + U bs Fu Ant = 828 kips + 363 kips = 1,190 kips 0.60 Fy Agv + U bs Fu Ant = 514 kips + 363 kips = 877 kips Therefore, Rn = 877 kips. LRFD ASD ϕRn = 0.75 ( 877 kips ) = 658 kips > 525 kips o.k. Rn 877 kips = Ω 2.00 = 439 kips > 350 kips o.k. Interface Forces Following the nonorthogonal UFM as presented in general in Figure 4-17, with the connection to a column web: 210 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ec = 0 eb = 9.00 in. ⎛ 2 ⎞ γ = tan −1 ⎜ ⎟ ⎝ 12 ⎠ = 9.46° ⎛ 10 w ⎞ θ = tan −1 ⎜ ⎟ ⎝ 12 ⎠ = 41.9° 18.0 in. = 13.5 in. (the distance to the centroid of the gusset-to-column connection), thhe distance to the 2 centroid of the gusset-to-beam connectiion is: Taking β = β = 42 in. + ec + β ( cos γ tan θ − sin γ ) cos γ 0 = ( 9.00 in. ) ( tan 41.9° − tan 9.46° ) − + (13.5 in. ) [ cos 9.46° tan 41.9° − sin 9.46°] cos 9.46° = 16.3 in. α = eb ( tan θ − tan γ ) − To get α = α, set: 16.3 in. = lg 2 lg = 31.6 in. + 1 2 and solve for lg Set the horizontal length of the gusset, lg, equal to 31s in., as shown in Figure 5-21. From Figure 4-17: 2 ec ⎞ ⎛ 2 r = ⎜α + eb tan γ + β sin γ + ⎟ + ( eb + β cos γ ) cos γ ⎝ ⎠ = 2 ⎡16.3 in. + ( 9.00 in. ) tan 9.46° + (13.5 in. ) sin 9.46° + 0⎤ + ⎡ 9.00 in. + (13.5 in. ) cos 9.46°⎤ ⎣ ⎦ ⎣ ⎦ 2 = 30.0 in. LRFD Pu 525 kips = r 30.0 in. = 17.5 kip/in. Pu r = ( 9.00 in. ) (17.5 kip/in. ) Vub = eb = 158 kips Pu Vuc = β cos γ r = (13.5 in. ) cos 9.46 (17.5 kip/in. ) = 233 kips ASD Pa 350 kips = r 30.0 in. = 11.7 kip/in. Pa r = ( 9.00 in. ) (11.7 kip/in. ) Vab = eb = 105 kips Pa r = (13.5 in. ) cos 9.46° (11.7 kip/in. ) Vac = β cos γ = 156 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 211 LRFD ASD Σ (Vub + Vuc ) = 158 kips + 233 kips Σ (Vab + Vac ) = 105 kips + 156 kips = 391 kips = 261 kips = ( 525 kips ) cos 41.9° = ( 350 kips ) cos 41.9° = 391 kips o.k. Pu Hub = ( α + eb tan γ ) r = ⎡⎣16.3 in. + ( 9.00 in. ) tan 9.46 ⎤⎦ (17.5 kip/in. ) = 312 kips ec ⎞ Pu ⎛ Huc = ⎜ β sin γ + ⎟ cos γ ⎠ r ⎝ = ⎡⎣(13.5 in. ) sin 9.46° + 0 ⎤⎦ (17.5 kip/inn. ) = 38.7 kips ∑ ( Huc + Hub ) = 38.7 kips + 312 kips = ⎡⎣(13.5 in. ) sin 9.46 + 0 ⎤⎦ (11.7 kip/in. ) = 25.9 kips ∑ ( H ac + H ab ) = 208 kips + 25.9 kips = 234 kips = 351 kips = ( 525 kips )( sin 41.9°) = 351 kips Qu = Huc − Pu cos θ tan γ = 261 kips o.k. Pa H ab = ( α + eb tan γ ) r = ⎡⎣16.3 in. + ( 9.00 in. ) tan 9.46° ⎤⎦ (11.7 kip/in. ) = 208 kips ec ⎞ Pa ⎛ H ac = ⎜ β sin γ + ⎟ cos γ ⎠ r ⎝ o.k. = ( 350 kips ) sin 41.9° = 234 kips Qa = H ac − Pa cos θ tan γ o.k. = 38.7 kips − ( 525 kips ) cos 41.9° tan 9.46° = 25.9 kips − ( 350 kips ) cos 41.9° tan 9.46° = −26.6 kips = −17.6 kips The admissible force field is shown in Figure 5-22. The negative sign on Qu and Qa indicates that it is opposite in direction to that shown in Figure 5-22. Figure 5-22 and Figure 4-17 are referred to as positive diagrams; all forces are shown in positive directions. The angle γ is positive when the angle between the beam and column is less than 90°, and negative when the angle is greater than 90°. Gusset-to-Column Connection The forces Huc , Hac , Vuc and Vac need to be converted into normal and tangential forces, oriented relative to the column’s longitudinal axis, for connection design. For the normal force: LRFD N uc = Huc cos γ − Vuc sin γ ASD N ac = H ac cos γ − Vac sin γ = ( 38.7 kips ) cos 9.46° − ( 233 kips ) sin 9.46° = ( 25.9 kips ) cos 9.46° − (156 kips ) sin 9.46° = −0.0538 kips ≈ 0 kips = −0.0466 kips ≈ 0 kips The small normal force is due to round off, and is taken as zero. The normal force is in tension if positive and compression if negative. For the tangential force: LRFD Tuc = Vuc cos γ + Huc sin γ ASD Tac = Vac cos γ + H ac sin γ = ( 233 kips ) cos 9.46° + ( 38.7 kips ) sin 9.46° = (156 kips ) cos 9.46° + ( 25.9 kips ) sin 9.46° = 236 kips = 158 kips 212 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The tangential force is positive when acting down on the gusset. Since the tangential force is the shear, its sign is irrelevant. The connection required strengths are: LRFD ASD Required axial strength, Nu = 0 kips Required axial strength, N a = 0 kips Required shear strength, Vu = 236 kips Required shear strength, Va = 158 kips Design bolts at gusset-to-column web connection From the gusset geometry, a maximum of seven rows of bolts will fit. At the column web double-angle connection, from AISC Manual Table 7-1, the available shear strength of 1-in.-diameter ASTM A490-X bolts in single shear is: Fig. 5-22a. Admissible force fields for nonorthogonal connection in Example 5.10—LRFD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 213 LRFD ASD ϕRnv = ( 49.5 kips/bolt )( 7 bolts )( 2 ) Rnv = ( 33.0 kips/bolt )( 7 bolts )( 2 ) Ω = 462 kips > 158 kips o.k. = 693 kips > 236 kips o.k. ince the available strength is almost three times what is S required, try four bolts with 6 in. spacing: ϕRnv = ( 49.5 kips/bolt )( 4 bolts )( 2 ) = 396 kips > 236 kips o.k. ince the available strength is almost three times what is S required, try four bolts with 6 in. spacing: Rnv = ( 33.0 kips/bolt )( 4 bolts )( 2 ) Ω = 264 kips > 158 kips o.k. Fig. 5-22b. Admissible force fields for nonorthogonal connection in Example 5.10—ASD. 214 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Figure 5-21 shows four rows of bolts at 6-in. spacing. Continue with this arrangement. At the gusset side of the double-angle connection, the bolts have an eccentricity of 3 in. As stipulated in the discussion of doubleangle connections in AISC Manual Part 10, for a single column of bolts, eccentricity is generally ignored up to about 3 in. From AISC Manual Table 7-1, the available shear strength of the four 1-in.-diameter ASTM A490-X bolts in double shear is: LRFD ASD ϕRnv = ( 98.9 kips/bolt )( 4 ) = 396 kips > 236 kips Rnv = ( 65.9 kips/bolt )( 4 ) Ω = 264 kips > 158 kips o.k. o.k. Use four bolts at 6-in. spacing. Check shear yielding on double angles Start with the angle size given in Figure 5-21, 2L5×5×½. From AISC Specification Section J4.2(a), the available shear yielding strength of the double angles is determined as follows: Agv = (2 in. )( 21.0 in. )( 2 ) (multiply by two for two angles) = 21.00 in.2 LRFD ASD ϕRgv = ϕ0.60 Fy Agv Rgv ( = 1.00 ( 0.60 )( 36 ksi ) 21.0 in.2 = 454 kips > 236 kips ) o.k. Ω = = 0.60 Fy Agv Ω ( 0.60 ( 36 ksi ) 21.0 in.2 1.50 = 302 kips > 158 kips ) o.k. Check shear rupture on double angles From AISC Specification Section J4.2(b), the available shear rupture strength of the double angles is determined as follows, with the standard hole size for a 1-in.-diameter bolt equal to 1z in. from AISC Specification Table J3.3: Anv = 21.0 in.2 − 4 (1z in. + z in. )(2 in.)( 2 ) = 16.5 in.2 LRFD ϕRnv = ϕ0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 58 ksi ) 16.5 in.2 = 431 kips > 236 kips o.k. ) Rnv 0.60 Fu Anv = Ω Ω = ( 0.60 ( 58 ksi ) 16.5 in.2 2.00 = 287 kips > 158 kips ) o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 215 Check block shear rupture on double angles The block shear rupture failure path is assumed to begin at the top of each angle and follow the bolt line, and then turn perpendicular to the outside free edge of each angle. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = (12 in. + 18.0 in. )(2 in. )( 2) = 19.5 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 19.5 in.2 ) = 421 kips Shear rupture component: Anv = 19.5 in.2 − ⎡⎣3.5 (1z in. + z in. )(2 in. ) ⎤⎦ ( 2 ) = 15.6 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 15.6 in.2 ) = 543 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded Le , the horizontal edge distance to the center of the bolt hole is 2.00 in. on the gusset side and 2.06 in. on the column side. Use Le = 2.00 in. Ant = ⎡⎣2.00 in. − 0.5 (1z in. + z in. ) ⎦⎤ (2 in. )( 2 ) = 1.44 in.2 ( U bs Fu Ant = 1 ( 58 ksi ) 1.44 in.2 ) = 83.5 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 543 kips + 83.5 kips = 627 kips 0.60 Fy Agv + U bs Fu Ant = 421 kips + 83.5 kips = 505 kips Therefore, Rn = 505 kips. LRFD ASD ϕRn = 0.75 ( 505 kips ) = 379 kips > 236 kips o.k. Rn 505 kips = Ω 2.00 = 253 kips > 158 kips 216 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 o.k. Check block shear rupture on the gusset plate The block shear rupture failure path on the gusset plate at the gusset-to-column connection is assumed to run from the top of the gusset plate through the bolt line and then on a perpendicular path to the closest free edge of the gusset. From Figure 5-21, the gusset top (measured parallel to the column) edge distance is 2 in. Shear yielding component: Agv = ( 2.00 in. + 18.0 in. )( s in. ) = 12.5 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 12.5 in.2 ) = 270 kips Shear rupture component: ( ) Anv = 12.5 in.2 − ( 3.5 ) (1z in. + z in. )( s in. ) 2 = 10.0 in. ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 10.0 in.2 ) = 348 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded Ant = ⎡⎣2.50 in. − 0.5 (1z in. + z in. ) ⎤⎦ ( s in. ) = 1.21 in.2 ( U bs Fu Ant = 1 ( 58 ksi ) 1.21 in.2 ) = 70.2 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 348 kips + 70.2 kips = 418 kips 0.60 Fy Agv + U bs Fu Ant = 270 kips + 70.2 kips = 340 kips Therefore, Rn = 340 kips. LRFD ASD ϕRn = 0.75 ( 340 kips ) = 255 kips > 236 kips o.k. Rn 340 kips = Ω 2.00 = 170 kips > 158 kips o.k. Check bolt bearing on the double angles and the gusset plate Using the 2-in.-thick double angles shown in Figure 5-21 and dh = 1z in., check bolt bearing on the angle material and the gusset plate at the angle-to-gusset connection. According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength on the double angles or on the gusset plate. Bearing strength is limited by bearing AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 217 deformation of the hole or by tearout of the material. From AISC Specification Section J3.10, when deformation at the bolt hole is a design consideration, the available strength for the limit state of bearing at bolt holes is determined as follows: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) Determine the effective strength of the top bolt in the double angle. The edge distance is: lc = 12 in. − 0.5dh = 12 in. − 0.5 ( 11 1z in. ) = 0.969 in. For the top bolt, using AISC Specification Equation J3-6a: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 0.969 in. )(2 in. )( 58 ksi )( 2 ) 1.2lc tFu (1.2 )( 0.969 in.) (2 in.) (58 ksi) (2) = Ω 2.00 = 33.7 kips/bolt = 50.6 kips/bolt ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )(2 in. )( 58 ksi )( 2 ) = 104 kips/bolt Therefore, ϕrn = 50.6 kips/bolt < 98.9 kips/bolt (bolt shear) 2.4 dtFu ( 2.4 )(1.00 in. ) (2 in.) (58 ksi) (2) = Ω 2.00 = 69.6 kips/bolt Therefore rn/ Ω = 33.7 kips/bolt < 65.9 kips/bolt (bolt shear) Bearing at the top bolt hole in the double angles controls over bolt shear strength at the angle-to-gusset connection. Determine the effective strength of the top bolt in the gusset plate. The edge distance is: lc = 2.00 in. − 0.5dh = 2.00 in. − 0.5 (1z in. ) = 1.47 in. For the top bolt, using AISC Specification Equation J3-6a: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.47 in. )( s in. )( 58 ksi ) 1.2lc tFu (1.2 )(1.47 in.)( s in.)( 58 ksi ) = Ω 2.00 = 32.0 kips/bolt = 48.0 kips/bolt ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )( s in. )( 58 ksi ) = 65.3 kips/bolt Therefore, ϕrn = 48.0 kips/bolt < 98.9 kips/bolt (bolt shear) 2.4 dtFu ( 2.4 )(1.00 in. )( s in. )( 58 ksi ) = Ω 2.00 = 43.5 kips/bolt Therefore rn/ Ω = 32.0 kips/bolt < 65.9 kips/bolt (bolt shear) The bearing strength on the gusset plate controls the strength of the top bolt. Determine the bearing strength at bolt holes on the gusset plate for the inner bolts. The edge distance is: lc = 6.00 in. − d h = 6.00 in. − 1z in. = 4.94 in. 218 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 For the inner bolts using AISC Specification Equation J3-6a: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 4.94 in. )( s in. )( 58 ksi ) 1.2lc tFu (1.2 )( 4.94 in. )( s in. )( 58 ksi ) = Ω 2.00 = 107 kips/bolt = 161 kips/bolt ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )( s in. )( 58 ksi ) = 65.3 kips/bolt Therefore, ϕrn = 65.3 kips/bolt < 98.9 kips/bolt (bolt shear) 2.4 dtFu ( 2.4 )(1.00 in. )( s in. )( 58 ksi ) = Ω 2.00 = 43.5 kips/bolt Therefore rn/ Ω = 43.5 kips/bolt < 65.9 kips/bolt (bolt shear) By inspection, the bolt bearing strength on the angles need not be checked because the strength of the gusset will control. The bolt spacing (6 in.) is the same for both, and the thickness of the two angles is greater than the thickness of the gusset plate. Considering the limit states of bolt shear, gusset bearing and tearout, and angle bearing and tearout, the available strength is: LRFD ASD ϕRn = 48.3 kips + 3 ( 65.3 kips ) Rn = 32.0 kips + 3 ( 43.5 kips ) Ω = 163 kips > 158 kips o.k. = 244 kips > 236 kips o.k. Check bolt bearing on the column web The column web needs to be checked for bolt bearing. The right side of AISC Specification Equation J3-6a will control. With tw = 1.54 in.: LRFD ASD ϕRn = ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )(1.54 in. )( 65 ksi ) = 180 kips > 236 = 29.5 kips 8 bolts o.k. Rn 2.4 dtFu = Ω Ω 2.4 (1.00 in. )(1.54 in. )( 65 ksi ) = 2.00 158 kiips = 120 kips > = 19.8 kips 8 bolts o.k. Gusset-to-Beam Connection From Figure 5-22, the required strengths at the gusset-to-beam interface are: LRFD ASD Required shear strength, Hu = 312 kips Required shear strength, H a = 208 kips Required axial strength, Vu = 158 kips Required axial strength, Va = 105 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 219 Check gusset plate for shear yielding and tension yielding along the beam flange The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1 as follows: LRFD ASD ϕRn = ϕ0.60 Fy Agv Rn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( s in. )( 31s in. ) = 1.50 = 285 kips > 208 kips o.k. Rn Fy Ag = Ω Ω ( 36 ksi )( s in.)( 31s in.) = 1.67 = 426 kips > 105 kips o.k. = 1.00 ( 0.60 )( 36 ksi )( s in. )( 31s in. ) ϕRn = 427 kips > 312 kips o.k. = ϕFy Ag = 0.90 ( 36 ksi )( s in. )( 31s in. ) = 640 kips > 158 kips o.k. Design weld at gusset-to-beam flange connection As discussed previously in Example 5.1, and in AISC Manual Part 13, to allow for redistribution of force in welded gusset connections, the weld is designed for an increased strength of 1.25R. The strength of the fillet weld can be determined from AISC Manual Equation 8-2, incorporating the increased strength permitted from AISC Specification Equation J2-5, as follows: LRFD ASD The resultant load is: The resultant load is: Ru = ( 312 kips )2 + (158 kips )2 = 350 kips The resultant load angle is: = 233 kips The resultant load angle is: ⎛ 158 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 312 kips ⎠ = 26.8° The number of sixteenths of fillet weld required is: 1.25 Ru Dreq′d = 2 ( l )(1.392 kip/in. ) 1.0 + 0.50 sin1.5 θ ( = 1.25 ( 350 kips ) ( ⎛ 105 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 208 kips ⎠ = 26.8° The number of sixteenths of fillet weld required is: 1.25 Ra Dreq′dd = 2 ( l )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 θ ) ( 2 ( 31s in. )(1.392 kip/in. ) 1.0 + 0.50 sin = 4.32 sixteenths ( 208 kips )2 + (105 kips )2 Ra = 1.5 26.8° ) = 1.25 ( 233 kips ) ( ) 2 ( 31s in. )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 26.8° ) = 4.31 sixteenths Use a double-sided c-in. fillet weld. This also satisfies the 4-in. minimum fillet weld required by AISC Specification Table J2.4. 220 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check beam web local yielding The normal force is applied at 31s in./ 2 = 15.8 in. from the beam end, which is less than the beam depth of 18 in. Therefore, use AISC Specification Equation J10-3, with k = kdes: LRFD ASD ϕRn = ϕFy t w ( 2.5kdes + lb ) = 1.00 ( 50 ksi )( 0.355 in. ) ⎡⎣2.5 ( 0.972 in. ) + 31s in.⎤⎦ = 604 kips > 158 kips o.k. Rn Fy t w ( 2.5kdes + lb ) = Ω Ω ( 50 ksi )( 0.355 in.) ⎡⎣2.5 ( 0.972 in.) + 31s in.⎤⎦ = 1.50 = 403 kips > 105 kips o.k. Check beam web shear yielding and shear buckling This check is used to guard against excessive horizontal web shear in the beam. This can be a concern when a thick gusset is welded to a beam with a thin web. The effective length of web is not well defined, but one half of the beam length is a reasonable maximum. The beam length for this problem is 10 ft, therefore: ⎛ 12 in.⎞ leff = 2 (10 ft ) ⎜ ⎟ ⎝ 1 ft ⎠ = 60.0 in. The available shear strength is determined from AISC Specification Section G2.1, where Cv = 1.0: LRFD ASD ϕv Rn = ϕ v 0.6 Fy AwCv = 1.00 ( 0.6 )( 50 ksi )( 0.355 in. )( 60.0 in. )(1.0 ) = 639 kips > 312 kips o.k. Rn 0.6 Fy AwCv = Ωv Ωv = 0.6 ( 50 ksi )( 0.355 in. )( 60.0 in. )(1.0 ) 1.50 = 426 kips > 208 kips o.k. Check beam web local crippling The normal force is applied at 15.8 in. from the beam end, which is greater than d/ 2. Therefore, AISC Specification Equation J10-4 is applicable: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜t f ⎟ ⎝ ⎠ = 0.75 ( 0.80 )( 0.355 in. ) × 2 1.5 ⎤ ⎥ ⎥ ⎦ ASD EFyw t f tw 1.5 ⎡ ⎛ 31s in. ⎞ ⎛ 0.355 in.⎞ ⎤ ⎢1 + 3 ⎜ ⎟ ⎥ ⎟⎜ ⎢⎣ ⎝ 18.0 in. ⎠ ⎝ 0.570 in.⎠ ⎥⎦ ( 29, 000 ksi )( 50 ksi )( 0.570 in.) 0.355 in. = 414 kips > 158 kips o.k. 1.5 ⎤ ⎡ Rn 0.80t w2 ⎢ ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFy tf 1 + 3⎜ ⎟ ⎜ ⎟ = Ω Ω ⎢ tw ⎝ d ⎠ ⎜⎝ tf ⎟⎠ ⎥ ⎣ ⎦ 2 1.5 0.80 ( 0.355 in. ) ⎡ ⎛ 31s in.⎞ ⎛ 0.355 in. ⎞ ⎤ ⎢1 + 3 ⎜ = ⎟ ⎥ ⎟⎜ 2.00 ⎢⎣ ⎝ 18.0 in. ⎠ ⎝ 0.570 in. ⎠ ⎥⎦ × ( 29, 000 ksi )( 50 ksi )( 0.570 in.) = 276 kips > 105 kips 0.355 in. o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 221 Beam-to-Column Connection The forces required are the tangential and normal forces. The beam shear reaction, Ru = 30 kips (LRFD) or Ra = 20 kips (ASD) does not reverse, but Vub (LRFD) or Vab (ASD), and Qu (LRFD) or Qa (ASD) do when Pu (LRFD) or Pa (ASD) goes from tension to compression. Figure 5-23 shows this load case on the beam end. In Figure 5-22, the force Qu = −26.6 kips and Qa = −17.6 kips, are negative by the sign convention of Figure 4-17. This force is shown in Figure 5-23(a) and 5-23(b) in its actual direction. Tbc and Nbc are also shown in their directions in Figures 5-23(a) and 5-23(b) to avoid the confusion of having negative quantities (i.e., quantities acting in directions opposite to the directions shown). LRFD ASD From Figure 5-23(a) : From Figure 5-23(a): Tubc = (158 kips + 30 kips ) cos 9.46° − ( 26.6 kips ) sin 9.46° Tabc = (105 kips + 20 kips ) cos 9.46° − (17.6 kips ) sin 9.46° N ubc = (158 kips + 30 kips ) sin 9.46° + ( 26.6 kips ) cos 9.46° N abc = (105 kips + 20 kips ) sin 9.46° + (17.6 kips ) cos 9.46° = 181 kips ( shear ) = 57.1 kips ( tension ) From Figure 5-23(b): = 120 kips ( shear ) = 37.9 kips ( tension ) From Figure 5-23(b): Tubcc = (158 kips − 30 kips ) cos 9.46° − ( 26.6 kips ) sin 9.46° = 122 kips Tabc = (105 kips − 20 kips ) cos 9.46° − (17.6 kips ) sin 9.46° = 80.9 kips N ubc = (158 kips − 30 kips ) sin 9.46° + ( 26.6 kips ) cos 9.46 ° = 47.2 kips ( compression ) N abc = (105 kips − 20 kips ) sin 9.46° + (17.6 kips ) cos 9.46° = 31.3 kips ( compression ) From the preceding analysis, the critical load case is when the brace is in tension. The beam-to-column interface required strengths are: LRFD ASD Required shear, Tubc = 181 kips Required shear, Tabc = 120 kips Required axial, N ubc = 57.1 kips ( tension ) Required axial, N abc = 37.9 kips ( tension ) Figure 5-23. Beam end forces. 222 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Determine bolt shear strength at beam web The load angle is: LRFD ASD ⎛ 57.1 kips ⎞ angle = tan −1 ⎜ ⎟ ⎝ 181 kips ⎠ ⎛ 37.9 kips ⎞ angle = tan −1 ⎜ ⎟ ⎝ 120 kips ⎠ = 17.5° = 17.5° With ex = 4.50 in., n = 4, s = 3.00 in., Angle = 15.0°, use AISC Manual Table 7-7 to determine C = 4.53. From AISC Manual Table 7-1, the available bolt shear strength for 1-in.-diameter bolts in double shear is: LRFD ASD rn = 65.9 kips/bolt Ω ϕrn = 98.9 kips/bolt From AISC Manual Chapter 7, Equation 7-19: LRFD The resultant load is: ASD The resultant load is: (181 kips )2 + ( 57.1 kips )2 Ru = = 190 kips ϕrnv = = ϕrnv C n ( 98.9 kips/bolt )( 4.53 bolts ) 8 bolts = 56.0 kips/boolt Ra = (120 kips )2 + ( 37.9 kips )2 = 126 kips rnv ⎛ rnv ⎞ C =⎜ ⎟ Ω ⎝ Ω⎠ n ( 65.9 kips/bolt )( 4.53 bolts ) = 8 bolts = 37.3 kipps/bolt Check bolt bearing on the double angles and the beam web Initially assume there is one vertical line of four bolts in the beam web. According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) Assuming conservatively that the horizontal edge distance is 2.50 in., for a bolt hole size of 1z in., the distance to the nearest edge for all of the bolts in the beam web is: l c = 2.50 in. − 0.5 (1z in. ) = 1.97 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 223 LRFD The required shear strength per bolt is: 190 kips 4 = 47.5 kips/bolt ruv = The available bearing strengths of the beam web are: ϕrn = ϕ1.2lc tFu = 0.75 (1.2 )(1.97 in. )( 0.355 in. )( 65 ksi ) = 40.9 kips//bolt ϕrn = ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )( 0.355 in. )( 65 ksi ) = 41.5 kips/bbolt Therefore, ϕrn = 40.9 kips/bolt < 47.5 kips/bolt n.g. ASD The required shear strength per bolt is: 126 kips 4 = 31.5 kips rav = The available bearing strengths of the beam web are: rn 1.2lc tFu = Ω Ω 1.2 (1.97 in. )( 0.355 in. )( 65 ksi ) = 2.00 = 27.3 kips rn 2.4 dtFu = Ω Ω 2.4 (1.00 in. )( 0.355 in. )( 65 ksi ) = 2.00 = 27.7 kips Therefore rn/ Ω = 27.3 kips/bolt < 31.5 kips/bolt n.g. Because the available bearing strength for one bolt is less than the required shear strength per bolt, a four-bolt connection fails. Try two vertical lines of four bolts spaced 3 in. apart at the beam web connection as shown in Figure 5-21. For the four bolts nearest to the beam web edge, the available bearing strength is as determined previously. For the four additional bolts furthest from the edge, the bearing strength is determined as follows, where lc = 3.00 in.−1z in. = 1.94 in.: LRFD The required shear strength per bolt is: 190 kips 8 = 23.8 kips/bolt ruv = The available tearout strength of the beam web is: ϕrn = ϕ1.2lc tFu = 0.75 (1.2 )(1.94 in. )( 0.355 in. )( 65 ksi ) = 40.3 kips//bolt The available bearing strength of the beam web is: ϕrn = ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )( 0.355 in. )( 65 ksi ) = 41.5 kips/bbolt Therefore, the tearout strength controls and ϕrn = 40.3 kips/bolt. ASD The required shear strength per bolt is: 126 kips 8 = 15.8 kips/bolt rav = The available tearout strength of the beam web is: rn 1.2lc tFu = Ω Ω 1.2 (1.94 in. )( 0.355 in. )( 65 ksi ) = 2.00 = 26.9 kips The available bearing strength of the beam web is: rn 2.4 dtFu = Ω Ω 2.4 (1.00 in. )( 0.355 in. )( 65 ksi ) = 2.00 = 27.7 kips/bollt Therefore, the tearout strength controls and rn/ Ω = 26.9 kips/bolt. 224 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check bolt bearing on double angle—beam side Try 2L8×6×½ with the 6-in. leg trimmed to 5 in. Check bearing on the double angle at the beam web side. The clear distance to the top edge of the angle, based on a 1z-in.-diameter hole, is: lc = 12 in. − 0.5 (1z in. ) = 0.969 in. For the top and bottom bolts, using AISC Specification Equation J3-6a: LRFD The available tearout strength of the double angle at the beam web is: ϕrn = ϕ1.2lc tFu = 0.75 (1.2 )( 0.969 in. )(2 in. )( 2 angles )( 58 ksi ) = 50.6 kips/bolt he available bearing strength of the double angle at the T beam web is: ϕrn = ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )(2 in. )( 2 )( 58 ksi ) = 104 kips/bolt > 56.0 kips/bolt herefore, bolt shear controls the strength of the top and T bottom bolts compared to bearing or tearout on the angles. ASD The available tearout strength of the double angle at the beam web is: rn 1.2lc tFu = Ω Ω 1.2 ( 0.969 in. )(2 in. )( 2 )( 58 ksi ) = 2.00 = 33.7 kips/bolt he available bearing strength of the double angle at the T beam web is: 2.4 dtFu rn = Ω Ω 2.4 (1.00 in. )(2 in. )( 2 )( 58 ksi ) = 2.00 = 69.6 kips/bolt > 37.3 kips/bolt herefore, bolt shear controls the strength of the top and T bottom bolts compared to bearing or tearout on the angles. For the inner bolts bearing on the double angle at the beam web connection, where lc = 3.00 in.−1z in. = 1.94 in.: LRFD The available bearing strength of the double angle at the beam web is: ϕrn = ϕ1.2lc tFu = 0.75 (1.2 )(1.94 in. )(2 in. )( 2 angles )( 58 ksi ) = 101 kips/bolt The available bearing strength is 104 kips/bolt as determined previously. ASD The available bearing strength of the double angle at the beam web is: rn 1.2lc tFu = Ω Ω 1.2 (1.94 in. )(2 in. )( 2 angles )( 58 ksi ) = 2.00 = 67.5 kips/bolt The available bearing strength is 69.6 kips/bolt as determined previously. Bolt shear strength controls the strength of the bolts bearing on the double angle. Edge tearout controls the bolts bearing on the beam web; therefore, ϕrn = 40.3 kips/bolt and rn/ Ω = 26.9 kips/bolt. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 225 The effective strength of the bolt group at the double angle-to-beam web connection is based on the controlling available strength based on bolt shear, and bearing and tearout of the double angle and beam web: LRFD ASD ϕRn = 4.53 ( 40.3 kips ) = 183 kips < 190 kips Rn = 4 ( 26.9 kips ) Ω = 108 kips < 126 kips n.g. n.g. Based on this simplified bearing check, the beam web has insufficient bearing strength. However, a more rigorous approach which considers the actual magnitude and direction of the load on each bolt will prove the web to be sufficient. The more rigorous approach, using the same instantaneous center of rotation method used to generate the C-value of 4.53, provides ϕRn = 190 kips, which is exactly equal to Ru = 190 kips and Rn/ Ω = 126 kips, which is exactly equal to Ra = 126 kips. Check bolt shear strength at column web From AISC Manual Table 7-1, the available bolt shear strength for 1-in.-diameter bolts in single shear is: LRFD ASD rn = 33.0 kips/bolt Ω ϕrn = 49.5 kips/bolt The required bolt shear at the double angle-to-column web connection is: LRFD n the angle legs attached to the column web side, the O maximum shear per bolt is: Vu 181 kips = n 8 = 22.6 kips/bolt < 49.5 kips/bolt ASD n the leg attached to the column web side, the maximum O shear per bolt is: Va 120 kips = n 8 = 15.0 kips/bolt < 33.0 kips/bolt o.k. o.k. Check bolt bearing on double angle and column web With a bolt spacing of 3.00 in. (lc = 3.00 in.−1z in. = 1.94 in.), the available bearing and tearout strengths of the column web are: LRFD The available tearout strength of the column web is: ASD The available tearout strength of the column web is: ϕrn = ϕ1.2lc tFu rn 1.2lc tFu = Ω Ω 1.2 (1.94 in. )(1.54 in. )( 65 ksi ) = 2.00 = 117 kips/boltt > 15.0 kips/bolt o.k. = 0.75 (1.2 )(1.94 in. )(1.54 in. )( 65 ksi ) = 175 kips/boolt > 22.6 kips/bolt o.k. The available bearing strength of the column web is: ϕrn = ϕ 2.4 dtFu = 0.75 (2.4)(1.00 in.)(1.54 in.)(65 ksi) = 180 kips/bbolt > 22.6 kips/bolt The available bearing strength of the column web is: o.k. rn 2.4dtFu = Ω Ω 2.4 (1.00 in. )(1.54 in. )( 65 ksi ) = 2.00 = 120 kips/bolt > 15.0 kips/bolt o.k. Bolt shear controls the strength of the bolt group in the column web. 226 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 For bearing of the outer bolts on the double angle at the column web connection, with 12-in. edge distance, from the previous calculation, the available bearing strength per bolt is: LRFD ASD ϕrn = 50.6 kips > 22.6 kips/bolt o.k. he available bolt shear of 49.5 kips/bolt controls the T strength of the outer bolts. Rn = 33.7 kips/bolt > 15.0 kips/bolt Ω o.k. he available bolt shear of 33.0 kips/bolt controls the T strength of the outer bolts. For the inner bolts in the double angle at the column web connection, where lc = 3.00 in.−1z in. = 1.94 in.: LRFD The available tearout strength of the double angle is: ϕrn = ϕ1.2lc tFu = 0.75 (1.2 )(1.94 in. )(2 in. )( 58 ksi ) = 50.6 kips/bolt > 22.6 kips/bolt o.k. The available bearing strength of the double angle is: ϕrn = ϕ 2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )(2 in. )( 58 ksi ) = 52.2 kips/bolt > 22.6 kips/bolt o.k. he available bolt shear of 49.5 kips/bolt controls the T strength of the inner bolts. ASD The available tearout strength of the double angle is: 1.2lc tFu rn = Ω Ω 1.2 (1.94 in. )(2 in. )( 58 ksi ) = 2.00 = 33.8 kips/bolt > 15.0 kips/bolt o.k. The available bearing strength of the double angle is: 2.4 dtFu rn = Ω Ω 2.4 (1.00 in. )(2 in. )( 58 ksi ) = 2.00 = 34.8 kips/bolt > 15.0 kips/bolt o.k. he available bolt shear of 33.0 kips/bolt controls the T strength of the inner bolts. The effective strength of the bolt group at the double angle-to-column web connection is based on the controlling available strength due to bolt shear and bearing and tearout of the double angle and column web. Bolt shear controls. LRFD ASD ϕR n = 8 ( 49.5 kips ) = 396 kips > 181 kips o.k. Rn = 8 ( 33.0 kips ) Ω = 264 kips > 120 kips o.k. Check gross shear on double angle—column side The gross area of the double angle is: Agv = (12.0 in. )(2 in. )( 2 ) = 12.0 in.2 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 227 From AISC Specification Equation J4-3, the available shear yielding strength of the double angle is: LRFD ASD ϕRn = ϕ 0.60 Fy Agv ( = 1.00 ( 0.60 )( 36 ksi ) 12.0 in.2 = 259 kips > 181 kips ) o.k. Rn 0.60 Fy Agv = Ω Ω = ( 0.60 ( 36 ksi ) 12.0 in.2 ) 1.50 = 173 kips > 120 kips o.k. Check net shear on double angle Based on the required hole size for a 1-in.-diameter bolt in standard holes from AISC Specification Table J3.3 and the z in. increase required by AISC Specification Section B4.3b, the net area is: Anv = 12.0 in.2 − 4 (1z in.+ z in.)(2 in. )( 2 ) = 7.50 in.2 From AISC Specification Equation J4-4, the available shear rupture strength of the double angle is: LRFD ASD ϕRn = ϕ0.60 Fu Anv ( = 0.75 ( 0.60 )( 58 ksi ) 7.50 in. 2 ) = 196 kips > 181 kips o.k. Rn 0.60 Fu Anv = Ω Ω = ( 0.60 ( 58 ksi ) 7.50 in.2 ) 2.00 = 131 kips > 120 kips o.k. Check block shear rupture on angles—beam side The controlling block shear rupture failure path is assumed to be the path shown in Figure 5-23c. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = 2 (10.5 in. )(2 in. ) = 10.5 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 10.5 in.2 ) = 227 kips Shear rupture component: Anv = 10.5 in.2 − 3.5 (1z in.+ z in.)(2 in. )( 2 ) = 6.56 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 6.56 in.2 ) = 228 kips 228 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Tension rupture component: Ant = ⎡⎣5.00 in. − 1.5 (1z in. + z in.)⎤⎦ (2 in. )( 2 ) = 3.31 in.2 In this case, because of the double column of bolts, Ubs = 0.5. ( U bs Fu Ant = 0.5 ( 58 ksi ) 3.31 in.2 ) = 96.0 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 228 kips + 96.0 kips = 324 kips 0.60 Fy Agv + U bs Fu Ant = 227 kips + 96.0 kips = 323 kips Therefore, Rn = 323 kips. Fig. 5-23c. Block shear rupture failure path on double angle in beam web. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 229 LRFD ϕRn = 0.75 ( 323 kips ) = 242 kips > 181 kips o.k. ASD Rn 323 kips = Ω 2.00 = 162 kips > 120 kips o.k. Check block shear rupture on angles—column side The controlling block shear failure path is similar to that shown in Figure 5-10a. Shear yielding component (same as for beam side): Agv = 10.5 in.2 0.60 Fy Agv = 227 kips Shear rupture component (same as for beam side): Anv = 6.56 in.2 0.60 Fu Anv = 228 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Agt = (10.0 in. + 0.355 in. − 62 in. )(2 in. ) = 1.93 in.2 Ant = 1.93 in.2 − 1 (1z in.+z in. )(2 in. ) = 1.37 in.2 ( U bs Fu Ant = 1 ( 58 ksi ) 1.37 in.2 ) = 79.5 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 228 kips + 79.5 kips = 308 kips 0.60 Fy Agv + U bs Fu Ant = 227 kips + 79.5 kips = 307 kips Therefore, Rn = 307 kips. LRFD ϕRn = 0.75 ( 307 kips ) = 230 kips > 181 kips o.k. ASD Rn 307 kips = Ω 2.00 = 154 kips > 120 kips o.k. 230 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check block shear rupture on beam web due to tensile force The controlling block shear failure path forms a rectangle bounded by the beam end, the top and bottom row of bolts, and the inner vertical line of bolts. Shear yielding component: Agv = 2 ( 5.50 in. )( 0.355 in. ) = 3.91 in.2 ( 0.60 Fy Agv = 0.60 ( 50 ksi ) 3.91 in.2 ) = 117 kips Shear rupture component: Anv = 3.91 in.2 − 1.5 (1z in. + z in. )( 0.355 in.)( 2 ) = 2.71 in.2 ( 0.60 Fu Anv = 0.60 ( 65 ksi ) 2.71 in.2 ) = 106 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = ⎡⎣9.00 in. − 3 (1z in. + z in. )⎤⎦ ( 0.355 in. ) = 2.00 in.2 ( U bs Fu Ant = 1 ( 65 ksi ) 2.00 in.2 ) = 130 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 106 kips + 130 kips = 236 kips 0.60 Fy Agv + U bs Fu Ant = 117 kips + 130 kips = 247 kips Therefore, Rn = 236 kips. LRFD ϕRn = 0.75 ( 236 kips ) = 177 kips > 57.1 kips o.k. ASD Rn 236 kips = Ω 2.00 = 118 kips > 37.9 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 231 Check interaction of shear and tension on the bolts in the column web The interaction of shear and tension on a bolt is checked using AISC Specification Section J3.7: LRFD ASD The required shear per bolt is: ruv = The required shear per bolt is: Tubc Nb rav = 181 kips 8 bolts = 22..6 kips/bolt 120 kips 8 bolts = 15..0 kips/bolt = = and the required tension per bolt is: and the required tension per bolt is: rut = N ubc Nb rat = 57.1 kips 8 bolts = 7.14 kips/bolt frv = N abc Nb 37.9 kips 8 bolts = 4.74 kips/bolt = Fnt′ = 1.3Fnt − Tabc Nb = Fnt frv ≤ Fnt ϕFnv Fnt′ = 1.3Fnt − ruv Ab frv = ΩFnt frv ≤ Fnt Fnv rav Ab Fnt = 113 ksi from AISC Speecification Table J3.2 Fnt = 113 ksi from AISC Speecification Table J3.2 Fnv = 84 ksi from AISC Specification Table J3.2 Fnv = 84 ksi from AISC Specification Table J3.2 Fnt′ = 1.3 (113 ksi ) − (113 ksi ) ⎛ 22.6 kips/bolt ⎞ ⎜ ⎟ ( 0.75)(84 ksi ) ⎜⎝ 0.785 in.2 ⎟⎠ = 95.3 ksi < 113 ksi o.k. The reduced available tensile force per bolt due to combined forces is: ϕrnt = ϕFnt′ Ab ( = 0.75 ( 95.3 kips ) 0.785 in. 2 ) = 56.1 kips/bolt > 7.14 kips/bolt o.k. 2.00 (113 ksi ) ⎛15.0 kips/bolt ⎞ ⎜ ⎟ (84 ksi ) ⎜⎝ 0.785 in.2 ⎟⎠ = 95.5 ksi < 113 ksi o.k. Fnt′ = 1.3 (113 ksi ) − The reduced available tensile force per bolt due to combined forces is: rnt Ω = = Fnt′ Ab Ω ( 95.5 kips ) ( 0.785 in.2 ) 2.00 = 37.5 kips/bolt > 4.74 kips/bolt o.k. Check prying action on bolts and double angle From AISC Manual Figure 9-4b: b = ( g − twb ) − t angle 2 2 62 in. − 0.355 in. 2 in. = − 2 2 = 2.82 in. 232 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 a = = 2 ( angle leg width ) + t wb − g 2 2 ( 5.00 in. ) + 0.355 in. − 62 in. 2 = 1.93 in. db ⎛ d ⎞ ≤ ⎜1.25b + b ⎟ 2 ⎝ 2 ⎠ 1.25b = 1.25 ( 2.82 in. ) a′ = a + = 3.53 in. 1.93 in. < 3.53 in. o.k. Use a = 1.93 in. d a′ = a + b 2 1.00 in. = 1.93 in. + 2 = 2.43 in. d b′ = b − b 2 1.00 in. = 2.82 in. − 2 = 2.32 in. ′ b ρ = a′ 2.32 in. = 2.43 in. = 0.955 p = pitch of bolts = 3.00 in. ′ d δ = 1− p = 1− (Manual Eq. 9-27) (Manual Eq. 9-21) (Manual Eq. 9-26) (Manual Eq. 9-24) (1z in.) 3.00 in. = 0.646 From AISC Manual Part 9: LRFD From AISC Manual Equation 9-30a: tc = = 4 Bb′ ϕpFu 4 ( 56.1 kips )( 2.32 in. ) 0.90 ( 3.00 in. )( 58 ksi ) = 1.82 in n. ASD From AISC Manual Equation 9-30b: tc = = 4ΩBb′ pFu 4 (1.67 )( 37.5 kips )( 2.32 in. ) ( 3.00 in.)( 58 ksi ) = 1.83 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 233 Note that tc is the angle thickness required to eliminate prying action, i.e., it is the angle thickness required to develop the strength of the bolt with no prying force. It is, therefore, an upper bound on angle thickness. Any angle thicker than tc would provide no additional strength. The optimal ratio of the moment in the angle leg at the bolt line to that at the center of the other leg is α′. From AISC Specification Equation 9-35: LRFD α′ = ASD ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ α′ = ⎡⎛ 1.82 in. ⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.646 (1 + 0.955 ) ⎢⎝ 2 in. ⎠ ⎥⎦ ⎣ = 9.70 in. Because α′ > 1 the angle leg will control and the prying action effect is determined from AISC Manuaal Equation 9-34: ⎛ 2 in. ⎞ =⎜ ⎟ ⎝ 1.82 in. ⎠ = 0.124 2 ⎡⎛ 1.83 in.⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.646 (1 + 0.955 ) ⎢⎝ 2 in. ⎠ ⎥⎦ ⎣ = 9.81 in. Because α′ > 1 the angle leg will control and the prying action effect is determined from AISC Manual Equation 9-34: 2 2 ⎛ t⎞ Q = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ ⎛ t⎞ = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ Q ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ (1 + 0.6646 ) Tavail = BQ 2 ⎛ 2 in. ⎞ =⎜ ⎟ (1 + 0.646 ) ⎝ 1.83 in. ⎠ = 0.123 Tavail = BQ = ( 56.1 kips )( 0.124 ) = ( 37.5 kips )( 0.123 ) = 6.96 kips < 7.14 kips n.g. = 4.61 kips < 4.74 kips n.g. The 2L8×6×½ angles are inadequate. Try 2L8×6×s angles. From AISC Manual Figure 9-4b: b = ( g − twb ) − t angle 2 2 62 in. − 0.355 in. s in. = − 2 2 = 2.76 in. a = 1.93 in. as determined previously db ⎛ db ⎞ ≤ ⎜ 1.25b + ⎟ 2 ⎝ 2 ⎠ 1.25b = 1.25 ( 2.76 in. ) a′ = a + = 3.45 in. 1.93 in. < 3.45 in. (Manual Eq. 9-27) Use a = 1.93 in. a′ = a + db 2 = 1.93 in. + 1.00 in. 2 = 2.43 in. 234 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 b′ = b − db 2 = 2.76 in. − 1.00 in. 2 = 2.26 in. b′ ρ = a′ 2.26 in. = 2.43 in. = 0.930 p = 3.00 in. δ = 0.646 as determined previously LRFD From AISC Manual Equation 9-30a: tc = = 4 Bb′ ϕ pFu ASD From AISC Manual Equation 9-30b: tc = 4 ( 56.1 kips )( 2.26 in. ) = 0.90 ( 3.00 in. )( 58 ksi ) = 1.80 inn. 4 (1.67 )( 37.5 kips )( 2.26 in. ) ( 3.00 in.)( 58 ksi ) = 1.80 in. ⎡⎛ t c ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ α′ = δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ α′ = Because α′ > 1, the angle leg will control and the prying action effect is determined from AIS SC Manual Equation 9-34: 2 ⎛ t⎞ = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎛ s in. ⎞ =⎜ ⎟ (1 + 0.646 ) ⎝ 1.80 in. ⎠ = 0.198 Tavail = BQ = ( 56.1 kips )( 0.198 ) = 11.1 kips > 7.14 kips o.k. ⎡⎛ t ⎞ 2 ⎤ 1 ⎢⎜ c ⎟ − 1⎥ δ (1 + ρ ) ⎢⎝ t ⎠ ⎥⎦ ⎣ ⎡⎛ 1.80 in. ⎞ 2 ⎤ 1 ⎢⎜ ⎟ − 1⎥ 0.646 (1 + 0.930 ) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ = 5.85 in. ⎡⎛ 1.80 in. ⎞ 2 ⎤ 1 ⎢⎜ = ⎟ − 1⎥ 0.646 (1 + 0.930 ) ⎢⎝ s in. ⎠ ⎥⎦ ⎣ = 5.85 in. Q 4ΩBb′ pFu = Because α′ > 1, the angle leg will control and the prying action effect is determined from AIS SC Manual Equation 9-34: 2 Q ⎛ t⎞ = ⎜ ⎟ (1 + δ ) ⎝ tc ⎠ 2 ⎛ s in. ⎞ =⎜ ⎟ (1 + 0.646 ) ⎝ 1.80 in. ⎠ = 0.198 Tavail = BQ = ( 37.5 kips )( 0.198 ) = 7.43 kips > 4.74 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 235 Note that the prying action design did not require the actual calculation of the prying force q. If this force is needed for analysis as is sometimes the case with fatigue design, it can be calculated as follows. Calculate α from AISC Manual Equation 9-29: LRFD α= 2 ⎤ 1 ⎡ T ⎛ tc ⎞ ⎢ ⎜ ⎟ − 1⎥ δ ⎢B⎝ t ⎠ ⎥⎦ ⎣ 2 ⎤ 1 ⎡⎛ 7.14 kips ⎞ ⎛1.80 in. ⎞ ⎢ = ⎟ − 1⎥ ⎜ ⎟⎜ 0.646 ⎢⎝ 56.1 kips ⎠ ⎝ s in. ⎠ ⎥⎦ ⎣ = 0.0861 ASD α= 2 ⎤ 1 ⎡ T ⎛ tc ⎞ ⎢ ⎜ ⎟ − 1⎥ δ ⎢B⎝ t ⎠ ⎣ ⎦⎥ 2 ⎤ 1 ⎡⎛ 4.74 kips ⎞ ⎛1.80 in. ⎞ ⎢⎜ = ⎟ − 1⎥ ⎟⎜ 0.646 ⎢⎝ 37.5 kips ⎠ ⎝ s in. ⎠ ⎥⎦ ⎣ = 0.0749 Note that α ≠ α′. α is the actual ratio of bolt line to leg line moment, while α′ is the optimal value (see Thornton, 1985). Since α is an actual value, it is limited to the range 0 ≤ α ≤ 1. From AISC Manual Equation 9-28: LRFD 2 ⎡ ⎛ t⎞ ⎤ q = B ⎢δαρ ⎜ ⎟ ⎥ ⎢ ⎝ tc ⎠ ⎥⎦ ⎣ 2 ⎡ ⎛ s in. ⎞ ⎤ = ( 56.1 kips ) ⎢( 0.646 )( 0.0861)( 0.930 ) ⎜ ⎟ ⎥ ⎝ 1.80 in. ⎠ ⎥⎦ ⎢⎣ = 0.350 kips ASD 2 ⎡ ⎛ t⎞ ⎤ q = B ⎢δαρ ⎜ ⎟ ⎥ ⎢ ⎝ tc ⎠ ⎥⎦ ⎣ 2 ⎡ ⎛ s in. ⎞ ⎤ = ( 37.5 kips ) ⎢( 0.646 )( 0.0749 )( 0.930 ) ⎜ ⎥ ⎟ ⎝ 1.80 in. ⎠ ⎥⎦ ⎢⎣ = 0.203 kips The actual maximum force per bolt is T + q T + q = 7.14 kips + 0.350 kips = 7.49 kips The actual maximum force per bolt is T + q T + q = 4.74 kips + 0.203 kips = 4.94 kips < B = 56.11 kips o.k. < B = 37.5 kips o.k. This same conclusion was reached before with Tavail > T, without calculating q. Check yielding in the column web When the normal force, Nbc , is large and the column web is thin, web yielding can be a controlling limit state. Also, the prying action analysis of the beam-to-column connection angles assumes that the column web is sufficiently stiff to allow double curvature bending in the angle legs to develop, or the axial force of 57.1 kips (LRFD) or 37.9 kips (ASD), is picked up by matching connection angles on the other side of the web. Assume for the present example that the column web must carry the normal force of 57.1 kips (LRFD) or 37.9 kips (ASD) (i.e., there is no opposing matching connection). A yield line for this case is assumed as given in Figure 5-13, and the theory is presented in Example 5.5. The column web transverse strength is given by: ⎡ 2h ⎤ l Pn = 8m p ⎢ + ⎥ ⎢⎣ h − g 2 ( h − g ) ⎥⎦ where the terms are defined in Example 5.5. For the present case: mp = = Fy t w2 4 ( 50 ksi )(1.54 in.)2 4 = 29.6 kip-in./in. 236 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ⎛ h⎞ h = ⎜ ⎟ tw ⎝ tw ⎠ = 7.41 (1.54 in. ) l g = 11.4 in. = 9 in. = 62 in. ⎡ Pn = 8 ( 29.6 kip-in./in. ) ⎢ ⎢⎣ = 728 kips 2 (11.4 in. ) (11.4 in. − 62 in.) LRFD ϕPn = 0.90 ( 728 kips ) = 655 kips > 57.1 kips o.k. + ⎤ 9.00 in. ⎥ 2 (11.4 in. − 62 in. ) ⎥ ⎦ ASD Pn 728 kips = Ω 1.67 = 436 kips > 37.9 kips o.k. Prying Action Validation In order to allow the double curvature prying action to exist, the authors recommend that the strength of the column web in yielding be at least 10 times the required strength. The normal double curvature prying action formulation assumes that the support is rigid, i.e., infinitely stiff. This is the reason for the factor of 10 on strength. LRFD ϕPn 0.90 ( 728 kips ) = 10 10 = 65.5 kips > 57.1 kips o.k. ASD Pn 728 kips = Ω (10 ) 1.67 (10 ) = 43.6 kips > 37.9 kips o.k. Check Beam Web Vertical Shear Adjacent to the beam-to-column connection, the beam web will have a required shear strength and available shear strength from AISC Manual Table 3-6, as shown in the following: LRFD Vu = 158 kips + 30 kips = 188 kips ϕvVn = 192 kips > 188 kips o.k. ASD Va = 105 kips + 20 kips = 125 kips Vn = 128 kips > 125 kips o.k. Ωv If this check failed, a web doubler plate could be used or some of the required shear strength could be transferred to the gussetto-column connection via a Special Case 2 approach to the nonorthogonal problem. This would be similar to the ΔVab analysis of Section 4.2.3. The completed design is shown in Figure 5-21. The calculations for this problem show some of the adjustments that must be made as the design proceeds. Example 5.11—Truss Connection Given: The versatility of the uniform force method will be demonstrated in a truss-to-truss connection as shown in Figure 5-24. Truss connections are similar to bracing connections, as bracing connections are part of a vertical truss that provides lateral support to the building. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 237 Figure 5-24 shows a secondary truss, Truss B, connected to a primary truss, Truss D. The overall objective of this connection is to deliver the vertical component of the secondary truss diagonal, 902 kips for LRFD and 599 kips for ASD, to the centerline of the primary truss. The given properties of the connection are as shown in Figure 5-24. Use 1-in.-diameter ASTM A490-SC bolts, surface Class B, in standard holes in the members and oversized holes in the plates. Threads are included in the shear plane. Use 70-ksi electrodes. LRFD ASD Brace force, Pu = ± 1,100 kips Brace force, Pa = ± 730 kips Truss B, top chord, beam shear, Vu = 61.5 kips Truss B, top chord, beam shear, Va = 41 kips Fig. 5-24. Connection of Truss B to Truss D completed design. 238 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A572 Grade 50 Fy = 50 ksi Fu = 65 ksi From AISC Manual Table 1-1, the geometric properties are as follows: Beam (Truss D top chord) W36×231 d = 36.5 in. tw = 0.760 in. bf = 16.5 in. tf = 1.26 in. bf = 12.0 in. tf = 0.790 in. k = 1.54 in. bf = 16.0 in. tf = 1.89 in. k1 = 1 in. tw = 0.485 in. bf = 14.6 in. tf = 0.780 in. Beam (Truss B top chord) W36×135 d = 35.6 in. tw = 0.600 in. h/ tw = 54.1 Column (Truss D vertical web member) W14×257 d = 16.4 in. tw = 1.18 in. Brace (Truss B web diagonal) W14×99 Ag = 29.1 in.2 d = 14.2 in. Bolt Shear Strength From AISC Manual Tables 7-2 and 7-3, the available tensile and shear strengths for 1-in.-diameter ASTM A490 bolts in slipcritical connections, Class B, are: LRFD ϕrnt = 66.6 kips/bolt For standard holes: ϕrnv = ( 21.7 kips )(1.67 ) = 36.2 kips/bolt ( single shear ) ϕrnv = ( 43.4 kips )(1.67 ) = 72.5 kips/bolt ( double shear ) For oversized holes: ϕrnv = (18.4 kips )(1.67 ) = 30.7 kips/bolt ( single shear ) ϕrnv = ( 36.9 kips )(1.67 ) = 61.6 kips/bolt ( double shear ) ASD rnt = 44.4 kips/bolt Ω For standard holes: rnv = (14.5 kips )(1.67 ) Ω = 24.2 kips/bolt ( single shear ) rnv = ( 28.9 kips )(1.67 ) Ω = 48.3 kips/bolt ( double shear ) For oversized holes: rnv = (12.3 kips )(1.67 ) Ω = 20.5 kips/bolt ( single shear ) rnv = ( 24.7 kips )(1.67 ) Ω = 41.2 kips/bolt ( double shear ) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 239 AISC Specification Section J3.8 requires that slip-critical connections be designed for the limit states of bearing-type connections. From AISC Manual Table 7-1, the available bolt shear strength with threads included in the shear planes is: LRFD ASD ϕrnv = 40.0 kips/bolt > 30.7 kips/bolt o.k. rnv = 26.7 kips/bolt > 20.5 kips/bolt Ω Brace-to-Gusset Connection Check tension yielding on the brace From AISC Specification Section D2(a): LRFD ASD ϕt Pn = ϕt Fy Ag ( = 0.90 ( 50 ksi ) 29.1 in.2 ) = 1, 310 kips > 1,100 kips o.k. Pn Fy Ag = Ωt Ωt = ( 50 ksi ) ( 29.1 in.2 ) 1.67 = 871 kips > 730 kips o.k. Check tensile rupture on the brace The net area of the brace is determined from AISC Specification Section B4.3b: An = Ag − 4 ( dh + z in. ) t f 2 = 29.1 in. ( dh = bolt hole diameter ) − 4 (1z in. + z in. )( 0.780 in. ) = 25.6 in.2 From AISC Specification Section D3 and Table D3.1, Case 7: U = 0.90 Table D3.1, Case 7, says that the shear lag factor can also be calculated from Table D3.1, Case 2, and the larger value is permitted to be used. The tee shape that corresponds to a W14×99 is a WT7×49.5. From Table 1-8, use x = y = 1.14 in., to calculate U as follows: x l 1.14 in. = 1− 24.0 in. = 0.953 U = 1− Using U = 0.953, the effective net area is: Ae = AnU ( = 25.6 in.2 = 24.4 in.2 (Spec. Eq. D3-1) ) ( 0.953) 240 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 From AISC Specification Section D2(b), the available tensile rupture strength of the brace is: LRFD ϕRn = ϕFu Ae ASD ( = 0.75 ( 65 ksi ) 24.4 in.2 ) = 1,190 kips > 1,100 kips o.k. Rn Fu Ae = Ω Ω = ( 65 ksi ) ( 24.4 in.2 ) 2.00 = 793 kips > 730 kips o.k. Check bolt shear The bolts in the brace flanges are in single shear. Using the controlling available bolt shear strength based on the oversized holes in the connection plates, the available bolt shear strength is: LRFD ASD ϕRnv = nϕrnv = ( 36 )( 30.7 kips ) = 1,110 kips > 1,100 kips o.k. Rnv ⎛ rnv ⎞ = n⎜ ⎟ Ω ⎝Ω⎠ = ( 36 )( 20.5 kips ) = 738 kips > 730 kips o.k. Check block shear rupture on the brace Block shear rupture on the brace flanges will control over block shear rupture of the connection plates because the flanges are thinner. The controlling block shear rupture failure path is assumed to run through both lines of bolts and then on a perpendicular line to the edge of each beam flange. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Shear yielding component: Agv = ⎡⎣12 in. + 8 ( 3.00 . in. ) ⎤⎦ ( 0.780 . in. )( 4 ) = 79.6 in.2 ( 0.60 Fy Agv = 0.60 ( 50 ksi ) 79.6 in.2 ) = 2, 390 kips Shear rupture component: Anv = 79.6 in.2 − 8.5 (1z in. + z in. )( 0.780 in. )( 4 ) = 49.8 in.2 ( 0.60Fu Anv = 0.60 ( 65 ksi ) 49.8 in.2 ) = 1, 940 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 241 Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the bolts are uniformly loaded Ant = ⎡⎣14.6 in. − 52 in. − 1 (1z in.+z in. ) ⎤⎦ ( 0.780 in. )( 2 ) = 12.4 in.2 ( U bs Fu Ant = 1 ( 65 ksi ) 12.4 in.2 ) = 806 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 1, 940 kips + 806 kips = 2, 750 kips 0.60 Fy Agv + U bs Fu Ant = 2, 390 kips + 806 kips = 3, 200 kips Therefore, Rn = 2,750 kips. LRFD ϕRn = 0.75 ( 2, 750 kips ) = 2, 060 kips > 1,100 kips o.k. ASD Rn 2, 750 kips = Ω 2.00 = 1, 380 kips > 730 kips o.k. Check bolt bearing on the brace The available bearing and tearout strengths of the brace are determined as follows: LRFD ϕrn = ϕ1.2lc tFu lc = 12 in. − 0.5 (1z in. ) = 0.969 in. ϕrn = 0.75 (1.2 ) ( 0.969 in. )( 0.780 in. )( 65 ksi ) = 44.2 kips/bolt ϕrn = ϕ2.4dtFu = 0.75 ( 2.4 )(1.00 in. )( 0.780 in. )( 65 ksi ) = 91.3 kips/bolt ASD rn 1.2lc tFu = Ω Ω lc = 12 in. − 0.5 (1z in. ) = 0.969 in. 1.2 ( 0.969 in. ) ( 0.780 in. )( 65 ksi ) rn = Ω 2.00 = 29.5 kips/bolt rn 2.4 dtFu = Ω Ω 2.4 (1.00 in. )( 0.780 in. )( 65 ksi ) = 2.00 = 60.8 kips/bolt The bolt shear strength of 30.7 kips/bolt for LRFD and 20.5 kips/bolt for ASD controls the strength of the bolt group compared to bearing or tearout on the brace. Bolt shear for the bolt group was checked previously. 242 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check bolt bearing on the connection plate Because tp = 1.00 in. > tf = 0.780 in., the plate is satisfactory for bearing. For tearout: lc = 12 in. – 0.5(14 in.) = 0.875 in. LRFD ASD ϕrn = 0.75 (1.2 )( 0.875 in. )(1.00 in. )( 65 ksi ) rn 1.2 ( 0.875 in. )(1.00 in. )( 65 ksi ) = Ω 2.00 = 34.1 kips/bollt = 51.2 kips/bbolt The bolt shear strength of 30.7 kips/bolt for LRFD and 20.5 kips/bolt for ASD controls the strength of the bolt group compared to bearing or tearout on the plate. Bolt shear for the bolt group was checked previously. Check tensile rupture of Plates A at gusset plate slot The available tensile rupture strength of Plates A is determined from AISC Specification Section D2(b) assuming an 8-in. gap at the slot, where Ae = AnU. The net area is: An = (14.0 in. − 18 in. )(1.00 in. ) = 12.9 in.2 U = 1.0 from AISC Specification Table D3.1 LRFD ϕRnt = ϕFu AnU ( 2 ) ( ASD ) = 0.75 ( 65 ksi ) 12.9 in.2 (1.0 )( 2 ) = 1, 260 kips > 1,100 kips o.k. Rnt Fu AnU = (2) Ω Ω ⎡ 65 ksi 12.9 in.2 (1.0 ) ⎤ ⎥ (2) =⎢ ⎢ ⎥ 2.00 ⎣ ⎦ = 839 kips > 730 kips o.k. ( ) Check tensile yielding of Plates A The available tensile yielding strength of Plates A is determined from AISC Specification D3(a) as follows. The gross area is: Agt = (14.0 in. )(1.00 in. ) = 14.0 in.2 LRFD ϕRn = ϕFy Agt ( 2 ) ( = 0.90 ( 50 ksi ) 14.0 in.2 ASD ) (2) = 1, 260 kips > 1,100 kips o.k. Fy Agt Rn = (2) Ω Ω ⎡ ( 50 ksi ) 14.0 in.2 =⎢ ⎢ 1.67 ⎣ = 838 kips > 730 kips ( ) ⎤⎥ ( 2 ) ⎥ ⎦ o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 243 Check tensile rupture of Plates A at bolts The available tensile rupture strength of Plates A is determined from AISC Specification D3(b) with an oversized hole diameter of 14 in. in the plates, and Ae = AnU: An = ⎡⎣14.0 in. − 2 (14 in. + z in. ) ⎤⎦ (1.00 in. ) = 11.4 in.2 U = 1.0 from AISC Specification Table D3.1 LRFD ϕRn = ϕFu Ae ( 2 ) ( ASD ) = 0.75 ( 65 ksi ) 11.4 in.2 (1.0 )( 2 ) Rn Fu Ae = (2) Ω Ω = 1,110 kips > 1,100 kips o.k. = ( 65 ksi ) (11.4 in.2 ) (1.0 ) (2) 2.00 = 741 kips > 730 kips o.k. Check weld of Plates A to gusset plate The welds to the gusset are loaded eccentrically by 3.22 in. (See Figure 5-25). The load is applied at the center of the plate above and below the slot and is transferred at the face of the slot. Try a weld length of l = 25 in. Then al = 3.22, a = 0.129 and from AISC Manual Table 8-4, Angle = 0°, k = 0, C = 3.69. Fig. 5-25. Weld of plate to gusset. 244 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD Dreq′d = ASD Pu ϕCC1l = Dreq′d = 275 kips = ( 0.75)( 3.69 )(1.0 )( 25.0 in.) = 3.97 ΩPa CC1l 2.00 (183 kips ) ( 3.69 )(1.0 )( 25.0 in.) = 3.97 Use 4-in. fillet welds. Based on strength alone, 4-in. fillet welds are indicated; however, AISC Specification Table J2.4 requires a minimum fillet weld size of c in. for material thicknesses over w in. Therefore, use c-in. fillet welds to connect the plate to the gusset. Using AISC Manual Equation 9-3, the minimum base metal thickness required for the weld size determined from Table 8-4 previously is: t min = = 6.19 D Fu 6.19 ( 3.97 ) 65 ksi = 0.380 in. Because the gusset plate and the connection plates are both 1 in. thick, this criterion is satisfied. Note that the formula for tmin is valid for both LRFD and ASD designs. Check the Whitmore section in the gusset plate The width of the Whitmore section, as defined in AISC Manual Part 9, is: lw = 14.2 in. + 1.00 in. + 1.00 in. + 2 ( 25.0 in. ) tan 30° = 45.1 in. This section enters the column, which has a web thickness, tw = 1.18 in. > gusset thickness, tg. Therefore, the entire Whitmore section is effective, and: Aw = tg lw = (1.00 in. )( 45.1 in. ) = 45.1 in.2 The tensile yielding strength of the gusset plate is determined from AISC Specification Section D2(a), with Ag = Aw: LRFD ASD ϕRn = ϕFy Aw ( = 0.90 ( 50 ksi ) 45.1 in.2 ) = 2, 030 kips > 1,100 kips o.k. Rn Fy Aw = Ω Ω = ( 50 ksi ) ( 45.1 in.2 ) 1.67 = 1, 350 kips > 730 kips o.k. Because the brace is always in tension, there is no need to check gusset buckling. The 1-in.-thick gusset plate is adequate. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 245 Force Distributions Figure 5-26 shows the uniform force method control points determined as described in AISC Manual Part 9 or Section 4.1.1 of this Design Guide. ec = 8.2 in. tan θ = 8a in. 12.0 in. eb = 0.698 = 17.8 in. θ = 34.9° CL W14×257, W36 ×231 ec = 8.2" α = 19.2" CL W36×135 chord eb = 17.8" A β = 21.5" B 12 θ 8a CL W14×99 Fig. 5-26. Control points for the uniform force method. 246 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 From Figure 5-24: 38w in. + distance from top end of gusset-to-truss vertical weld to bottom of secondary truss chord flange 2 38w in. = + 2 in. + w in. + d in. 2 = 21.5 in. Set β = β = With β = β = 21.5, calculate α required to eliminate moments at the gusset interfaces: α = ( β + eb ) tan θ − ec (from AISC Manual Eq. 13-1) ⎛ 8a in. ⎞ = ( 21.5 in. + 17.8 in. ) ⎜ ⎟ − 8.20 in. ⎝ 12.0 in. ⎠ = 19.2 in. The gusset-to-secondary chord connection has been arranged so that α = 19.0 in., thus there is no moment on this interface: r= = ( α + ec )2 + (β + eb )2 (AISC Manual Eq. 13-6) (19.2 in. + 8.20 in.)2 + ( 21.5 in. + 17.8 in.)2 = 47.9 in. Where 2 in. is the clearance between the cap plate and the gusset plate, w in. is the cap plate thickness, and d in. is the difference between the Truss B top chord and the Truss D top chord. LRFD Pu 1,100 kips = r 47.9 in. = 23.0 kip/in. From AISC Manual Equation 13-5: Pu r = (19.2 in. )( 23.0 kip/in. ) Hub = α = 442 kips From AISC Manual Equation 13-3 Pu r = ( 8.20 in. )( 23.0 kip/in. ) Huc = ec = 189 kips ∑ ( Huc + Hub ) = 442 kips + 189 kips = 631 kips = (1,100 kips ) sin 34.9° = 629 kips o.k. ASD Pa 730 kips = r 47.9 in. = 15.2 kip/in. From AISC Manual Equation 13-5: Pa r = (19.2 in. )(15.2 kip/in. ) H ab = α = 292 kips From AISC Manual Equation 13-3: Pa r = ( 8.20 in. )(15.2 kip/in. ) H ac = ec = 125 kips ∑ ( H ac + H ab ) = 292 kips + 125 kips = 417 kips = ( 730 kips ) sin 34.9° = 418 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 247 LRFD From AISC Manual Equation 13-4: Pu r = (17.88 in. )( 23.0 kip/in. ) Vub = eb ASD From AISC Manual Equation 13-4: Pa r = (17.8 in. )(15.2 kip/in. ) Vab = eb = 409 kips From AISC Manual Equation 13-2: Pu Vuc = β r = ( 21.5 in. )( 23.0 kip/in. ) = 271 kips From AISC Manual Equation 13-2: Pa r = ( 21.5 in. )(15.2 kip/in. ) Vac = β = 495 kips ∑ (Vuc + Vub ) = 495 kips + 409 kips = 327 kips ∑ (Vac + Vab ) = 327 kipps + 271 kips = 904 kips = 598 kips = (1,100 kips ) cos 34.9° = ( 730 kips ) cos 34.9° = 902 kips o.k. = 599 kips o.k. This distribution of forces is shown in Figure 5-27. Gusset-to-Column Connection (Primary Truss Vertical) Design weld LRFD The resultant load is: Ru = ( 495 kips )2 + (189 kips )2 = 530 kips ⎛ 189 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 495 kips ⎠ = 20.9° From AISC Manual Table 8-4 with Angle = 15°, l = 38.75 in., a = k = 0, C = 3.96 : Pu Dreq′d = ( ductility factor ) ϕCC1l = 530 kips (1.25) 0.75 ( 3.96 )(1.0 ) ( 38w in. ) = 5.76 sixteenths ASD The resultant load is: Ra = ( 327 kips )2 + (125 kips )2 = 350 kips ⎛ 125 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 327 kips ⎠ = 20.9° From AISC Manual Table 8-4 with Angle = 15°, l = 38.75 in., a = k = 0, C = 3.96 : Pa Ω Dreq′d = ( ductility factor ) CC1l = ( 350 kips )( 2.00 ) (1.25) 3.96 (1.0 )( 38w in. ) = 5.70 sixteenths 248 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Use a a-in. fillet weld. From AISC Specification Table J2.4, the minimum fillet weld required for a 1-in.-thick plate is c in. The use of the ductility factor in this manner has been explained in previous examples. Check gusset plate for shear yielding and tensile yielding along the column flange The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1 as follows: Fig. 5-27a. Admissible force fields for truss-to-truss example—LRFD. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 249 LRFD ASD ϕRn = ϕ 0.60 Fy Agv = 1.00 ( 0.60 )( 50 ksi )(1.00 in. )( 38w in. ) = 1,160 kips > 495 kips o.k. ϕRn = ϕFy Ag = 0.90 ( 50 ksi )(1.00 in. )( 38w in. ) = 1,740 kips > 189 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )(1.00 in. )( 38w in. ) = 1.50 = 775 kips > 327 kips o.k. Fy Ag Rn = Ω Ω ( 50 ksi )(1.00 in.)( 38w in.) = 1.67 = 1,160 kips > 125 kips o.k. Fig. 5-27b. Admissible force fields for truss-to-truss example—ASD. 250 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Gusset-to-Beam Connection (Secondary Truss Chord) Design tab plate-to-chord weld LRFD The resultant load on the weld is: Ru = ASD The resultant load on the weld is: ( 409 kips )2 + ( 442 kips )2 Ra = = 602 kips = 398 kips ⎛ 409 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 442 kips ⎠ = 42.8° From AISC Manual Table 8-4 with Angle = 30°, a = k = 0, C = 4.37: Dreqq′d = = ( 271 kips )2 + ( 292 kips )2 ⎛ 271 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 292 kips ⎠ = 42.9° From AISC Manual Table 8-4 with Angle = 30°, a = k = 0, C = 4.37: Pu ϕCC1l Dreqq′d = 602 kips 0.75 ( 4.37 )(1.0 )( 33.0 in. ) = = 5.57 sixteenths Pa Ω CC1l ( 398 kips )( 2.00 ) ( 4.37 )(1.0 )( 33.0 in. ) = 5.52 sixteenths Use a a-in. fillet weld. Note that a ductility factor is not used here because the slip in the bolted connection on the other side of the tab plate and bolt shear ductility will allow redistribution to occur. Check web local yielding on secondary truss chord due to normal force The normal force is applied at a distance from the member end that is less than the depth of the member, d = 35.6 in.; therefore, use AISC Specification Equation J10-3: LRFD ϕRn = ϕ Fyw t w ( 2.5k + lb ) = 1.00 ( 50 ksi )( 0.600 in. ) ⎡⎣2.5 (1.54 in. ) + 33.0 in.⎤⎦ = 1,110 kips > 409 kips o.k. ASD Rn Fyw t w ( 2.5k + lb ) = Ω Ω ( 50 ksi )( 0.600 in.) ⎡⎣2.5 (1.54 in.) + 33.0 in.⎤⎦ = 1.50 = 737 kips > 271 kips o.k. Check beam web local crippling on secondary truss chord due to normal force The normal force of 409 kips (LRFD) or 271 kips (ASD) is always tension, so this limit state does not exist. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 251 Check bolts connecting tab plate to gusset plate As determined previously, the required strength at the gusset-to-beam interface is: LRFD ASD Ru = 602 kips Ra = 398 kips The angle of the resultant is θ = 42.9° as determined previously. The load on the bolt group is eccentric, i.e., ex = 3.5 in. From AISC Manual Table 7-7 for Angle = 30°, n = 11, ex = 3.5 in., s = 3 in., C = 19.6, and the available bolt shear strength values determined previously for oversized holes in single shear, the available strength of the bolt group is: LRFD ϕRnv = Cϕrnv = 19.6 ( 30.7 kips/bolt ) = 602 kips ≥ 602 kips o.k. ASD Rnv = C ϕrnv Ω = 19.6 ( 20.5 kips/bolt ) = 402 kips > 398 kips o.k. Check bolt bearing on the tab plate and the gusset plate Because of the 3.5-in. eccentricity, the 22 bolts are equivalent to 19.6 directly loaded bolts (C = 19.6 as determined previously). Therefore, the maximum force per bolt is: LRFD Ru C 602 kips = 19.6 bolts = 30.7 kips/bolt Pmax = ASD Ra C 398 kips = 19.6 bolts = 20.3 kips/bolt Pmax = According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength on the tab plate or on the gusset plate. Bearing strength is limited by bearing deformation of the hole or by tearout of the material. From AISC Specification Section J3.10, when deformation at the bolt hole is a design consideration, the available strength for the limit state of bearing at bolt holes is determined as follows: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) Bearing and tearout will not control as long as they provide more than 30.7 kips per bolt for LRFD and 20.3 kips per bolt for ASD. Because the tab and the gusset have the same thickness, the following checks apply to both. The clear distance to the edge is determined as follows, with a bolt hole diameter of 14 in. and assuming the edge distance is 12 in.: lc = 12 in. − 0.5 (14 in. ) = 0.875 in. 252 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 For the outer bolt, using AISC Specification Equation J3-6a: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 0.875 in. )(1.00 in. )( 65 ksi ) 1.2lc tFu 1.2 ( 0.875 in. )(1.00 in. )( 65 ksi ) = Ω 2.00 = 34.1 kips = 51.2 kips ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )(1.00 in. )( 65 ksi ) = 117 kips Therefore, ϕrn = 51.2 kips/bolt > 30.7 kips/bolt (bolt shear) 2.4 dtFu 2.4 (1.00 in. )(1.00 in. )( 65 ksi ) = Ω 2.00 = 78.0 kips Therefore, rn/ Ω = 34.1 kips/bolt > 20.3 kips/bolt (bolt shear) Bolt shear controls the strength of the connection and is o.k. as previously determined. Check tab plate for shear yielding and tensile yielding The available shear yielding strength of the tab plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1 as follows: LRFD ASD ϕRn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 50 ksi )(1.00 in. )( 33.0 in. ) = 990 kips > 442 kips o.k. ϕRn = ϕFy Ag = 0.90 ( 50 ksi )(1.00 in. )( 33.0 in. ) = 1, 490 kips > 409 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )(1.00 in. )( 33.0 in. ) = 1.50 = 660 kipps > 292 kips o.k. Rn Fy Ag = Ω Ω ( 50 ksi )(1.00 in.)( 33.0 in.) = 1.67 = 988 kips > 271 kips o.k. Check tab plate for shear rupture and tensile rupture The available shear rupture strength of the tab plate is determined from AISC Specification Equation J4-4, and the available tensile rupture strength is determined from AISC Specification Equation J4-2. The net area, with an oversized hole diameter of 14 in., is: Anv = ⎡⎣33.0 in. − 11 (14 in. + z in. ) ⎤⎦ (1.00 in. ) = 18.6 in.2 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 253 LRFD ϕRn = ϕ0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 65 ksi ) 18.6 in.2 ) Rn 0.60 Fu Anv = Ω Ω = 544 kips > 442 kips o.k. ϕRn = ϕFu Ae = 0.75 ( 65 ksi ) 18.6 in.2 ( ) = 907 kips > 409 kips o.k. = ( 0.60 ( 65 ksi ) 18.6 in.2 ) 2.00 = 363 kips > 292 kips o.k. Rn Fu Ae = Ω Ω = ( 65 ksi ) (18.6 in.2 ) 2.00 = 605 kips > 271 kips o.k. Check combined shear, normal and flexural strength on the tab plate The forces on the critical section are: LRFD ASD Vu = 442 kips Va = 292 kips N u = 409 kips N a = 271 kips Mu = ( 442 kips ) ( 3.50 in.) M a = ( 292 kips ) ( 3.50 in. ) = 1,550 kip-in. = 1,020 kip-in. An interaction equation that includes shear, normal and flexural stresses is the fourth order interaction equation proposed by Neal (1977) and derived based on plastic theory: LRFD 2 ASD 4 2 4 ΩM a ⎛ ΩN a ⎞ ⎛ ΩVa ⎞ +⎜ ⎟ +⎜ ⎟ ≤ 1.0 M n ⎝ N n ⎠ ⎝ Vn ⎠ ⎛ Vu ⎞ Mu ⎛ N u ⎞ +⎜ ⎟ +⎜ ⎟ ≤ 1.0 ϕM n ⎝ ϕN n ⎠ ⎝ ϕVn ⎠ where Mn , Nn and Vn , as well as ϕ and Ω, vary depending on whether yield or rupture is being considered. The plate gross plastic section modulus is: Z g = 4 (1.00 in. )( 33.0 in.) 2 = 272 in.3 The plate net plastic section modulus is: ( Z n = 4t ( s − dh′ ) n2 s + dh′ ) (Manual Eq. 15-4) ( ) = 4 (1.00 in. ) ⎡⎣3.00 in. − (14 in. + 1z in. )⎤⎦ ⎡ 112 ( 3.00 in. ) + (14 in. + 1z in. ) ⎤ ⎣ ⎦ = 154 in.3 254 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available flexural yield strength of the tab plate is determined from AISC Manual Equation 15-2, and the available flexural rupture strength is determined from AISC Manual Equation 15-3 as follows. LRFD ASD ϕM n = 0.90 Fy Z g ( = 0.90 ( 50 ksi ) 272 in.3 M n Fy Z g = Ω Ω ) = 12, 200 kip-in. > 1,550 kiip-in. o.k. ϕM n = 0.75Fu Z n ( = 0.75 ( 65 ksi ) 154 in.3 ) = 7, 510 kip-in. > 1,550 kip-in. o.k. = ( 50 ksi ) ( 272 in.3 ) 1.67 = 8,140 kip-in. > 1,020 kip-in. o.k. M n Fu Z n = Ω Ω = ( 65 ksi ) (154 in.3) 2.00 = 5, 010 kip-in. > 1,020 kip-in. o.k. From Neal (1977), for yielding: LRFD 2 ASD 4 1, 550 kip-in. ⎛ 409 kips ⎞ ⎛ 442 kips⎞ +⎜ ⎟ +⎜ ⎟ = 0.242 < 1.0 o.k. 12, 200 kip-in. ⎝1, 490 kips⎠ ⎝ 990 kips⎠ 2 4 1, 020 kip-in. ⎛ 271 kips ⎞ ⎛ 292 kips ⎞ +⎜ ⎟ +⎜ ⎟ = 0.239 < 1.0 o.k. 8,140 kip-in. ⎝ 988 kips ⎠ ⎝ 660 kips ⎠ For rupture: LRFD 2 ASD 4 2 4 1, 550 kip-in. ⎛ 409 kips ⎞ ⎛ 442 kips ⎞ 1, 020 kip-in. ⎛ 271kips ⎞ ⎛ 292 kips ⎞ +⎜ +⎜ ⎟ +⎜ ⎟ = 0.845 < 1.0 o.k. ⎟ +⎜ ⎟ = 0.823 < 1.0 o.k. 7, 510 kip-in. ⎝ 907 kips ⎠ ⎝ 544 kips ⎠ 5, 010 kip-in. ⎝ 605 kips ⎠ ⎝ 363 kips ⎠ Because the tab plate and the gusset plate are both 1 in. thick, the above checks on the tab plate suffice for the gusset plate. Truss B-to-Truss D Connection (Secondary Truss Chord to Primary Truss Chord) Figure 5-27 gives the forces, and Figure 5-28 gives greater definition as to the force distribution in the shaped tab plate denoted Tab Plate C in Figure 5-24. Check bolt shear strength The bolts must support the following required strengths: LRFD ASD Pu = 189 kips Pa = 125 kips Vu = 409 kips + 62.0 kips Va = 271 kips + 41.0 kips = 471 kips = 312 kips Note that Pu = 189 kips (LRFD) and Pa =125 kips (ASD) is not a transfer force, but rather the “Hc” required to maintain equilibrium. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 255 The secondary chord reaction of 62 kips for LRFD and 41 kips for ASD must be delivered to the centerline of the primary truss member. Thus, a moment of (62.0 kips) (11Xa in.) = 705 kip-in. for LRFD and (41.0 kips)(11Xa in.) = 466 kip-in. for ASD must be carried by the bolts. Also, because the control point A is 3.5 in. from the centroid of the bolt group, an additional moment is required to deliver the Vb = 409 kips (LRFD) and Vb = 271 kips (ASD) force from the control point to the bolt group centroid; thus (409 kips)(3.5 in.) = 1,430 kip-in. (LRFD) and (271 kips)(3.5 in.) = 949 kip-in. (ASD). The total moment at the Truss B to Truss D interface is: LRFD ASD Mu = 705 kip-in. + 1,430 kip-in. Ma = 466 kip-in. + 949 kip-in. = 2,140 kip-in. The resultant load is: R = = 1,420 kip-in. The resultant load is: (189 kips )2 + ( 471 kips )2 = 508 kips R = (125 kips )2 + ( 312 kips )2 = 336 kips Fig. 5-28. Tab Plate C admissible force field—transfer of shear force only. 256 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD The resultant load angle is: ASD The resultant load angle is: ⎛ 189 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 471 kips ⎠ = 21.9° ⎛ 125 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 312 kips ⎠ = 21.8° The effective eccentricity is: 2,140 kip-in. ex = 471 kips = 4.54 in. From AISC Manual Table 7-7 with Angle = 15°, n = 10, e x = 4.54 in., and s = 3 in.: The effective eccentricity is: 1, 420 kip-in. ex = 312 kips = 4.55 in. From AISC Manual Table 7-7 with Angle = 15°, n = 10, e x = 4.55 in., and s = 3 in.: C = 17.0 ϕRn = Cϕrn = 17.0 ( 30.7 kips/bolt ) = 522 kips > 508 kips o.k. C = 17.0 Rnv Crnv = Ω Ω = 17.0 ( 20.5 kips/bolt ) = 349 kips > 336 kips o.k. Check bolt bearing on the Truss B top chord web and Tab Plate C A 1-in.-thick plate is used for the initial calculations. Figure 5-24 shows a 14-in.-thick plate that is required to satisfy the interaction checks. The maximum force per bolt is: LRFD 508 kips 17.0 = 29.9 kips/bolt < 30.7 kips/bolt o.k. ru = ASD 336 kips 17.0 = 19.8 kips/bolt < 20.5 kips/bolt o.k. ra = Note that the eccentricity degrades the strength of the bolts so that the 20 bolts can only support a force equivalent to that which could be supported by 17 directly loaded bolts. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the Truss B top chord web, assume a 1.75-in. end distance and 1z-in.-diameter holes. The clear distance perpendicular to the outer line of bolts to the beam end is: lc = 1.75 in. − 0.5 (1z in. ) = 1.22 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 257 The available tearout and bearing strengths of the Truss B top chord web are: LRFD ϕ1.2lc tFu = 0.75 (1.2 )(1.22 in. )( 0.600 in. )( 65 ksi ) = 42.8 kips/boltt ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )( 0.600 in. )( 65 ksi ) = 70.2 kips/bolt Therefore, ϕrn = 42.8 kips/bolt. The bolt shear strength of 30.7 kips governs over bearing or tearout strength. ASD 1.2 Lc tFu 1.2 (1.22 in. )( 0.600 in. )( 65 ksi ) = Ω 2.00 = 28.6 kips/bolt 2.4 (1.00 in. )( 0.600 in. )( 65 ksi ) 2.4 dtFu = Ω 2.00 = 46.8 kips/bolt Theerefore, rn Ω = 28.6 kips/bolt. The bolt shear strength of 20.5 kips governs over bearing or tearout strength. Note that because of eccentric loading, the directions of the bolt forces are unknown. The tearout strength calculations for both the top chord web and the tab plate are conservative and assume the worst case, where the bolt force is perpendicular to the edge. Note also that all bolts in the outer bolt line are assumed to have the tearout capacity of the worst bolt. If the direction and magnitude of the force on each bolt is determined, it is acceptable to use a less conservative approach. Assuming a 1.5-in. end distance and 14-in.-diameter holes, the clear distance from the outer bolt line to the Tab Plate C edge is: lc = 1.50 in. − 0.5 (14 in. ) = 0.875 in. LRFD ϕ1.2lc tFu = 0.75 (1.2 )( 0.875 in. )(1.00 in. )( 65 ksi ) = 51.2 kips ϕ2.4 dtFu = 0.75 ( 2.4 )(1.00 in. )(1.00 in. )( 65 ksi ) = 117 kips Thereforee, ϕrn = 51.2 kips/bolt The bolt shear strength of 30.7 kips governs over bearing or tearout strength. ASD 1.2 Lc tFu 1.2 ( 0.875 in. )(1.00 in. )( 65 ksi ) = Ω 2.00 = 34.1 kips 2.4 (1.00 in. )(1.00 in. )( 65 ksi ) 2.4 dtFu = 2.00 Ω = 78.0 kips Therefore, rn Ω = 34.1 kips/bolt The bolt shear strength of 20.5 kips governs over bearing or tearout strength. Note that spacing should also be checked for tearout between bolts, but this seldom controls. For example: lc = 3.00 in. − 1 (14 in. ) = 1.75 in. LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.75 in. )(1.00 in. )( 65 ksi ) 1.2lc tFu 1.2 (1.75 in. )(1.00 in. )( 65 ksi ) = Ω 2.00 = 68.3 kips = 102 kips Again, bolt shear strength of 30.7 kips for LRFD and 20.5 kips for ASD controls over tearout strength; therefore, the effective strength of the bolt group is controlled by bolt shear strength, which was previously checked. 258 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check gross and net sections of Tab Plate C at Section a-a The forces at Section a-a are: LRFD ASD Vu (shear ) = 4 71 kips Va (shear ) = 312 kips N u (normal ) = 189 kips N a (normal ) = 125 kips The moment at the centroid of the bolt group is 2,130 kip-in. (LRFD) and 1,420 kip-in. (ASD). The portion of this moment due to the beam shear [62 kips (LRFD) and 41 kips (ASD)] at the edge of Tab Plate C attached to the web of the Truss D top chord (W36×231) is zero. At the first line of bolts, the moment, due to the offset control point A, is reduced from the moment at the bolt centroid because of its closer proximity to control point A. Therefore: LRFD ASD Mu = ( 62.0 kips ) (11.375 in. − 1.50 in. ) + ( 409 kips )( 2.00 in. ) M a = ( 41.0 kips ) (11.375 in. − 1.50 in. ) + ( 271 kips )( 2.00 in. ) = 1, 430 kiip-in. = 947 kip--in. An interaction equation that includes shear, normal and flexural stresses is the fourth order interaction equation proposed by Neal (1977) and derived based on plastic theory: LRFD 2 ASD 4 2 ⎛ Vu ⎞ Mu ⎛ N u ⎞ +⎜ ⎟ +⎜ ⎟ ≤ 1.0 ϕM n ⎝ ϕN n ⎠ ⎝ ϕVn ⎠ 4 ⎛ ΩVa ⎞ Ω Ma ⎛ Ω Na⎞ +⎜ ⎟ +⎜ ⎟ ≤ 1.0 Mn ⎝ Nn ⎠ ⎝ Vn ⎠ where Mn, Nn and Vn , as well as ϕ and Ω, vary depending on whether yield or rupture is being considered. The gross and net areas are: Ag = (1.00 in. )( 30.0 in. ) = 30.0 in.2 An = (1.00 in. ) ⎡⎣30.0 in. − 10 ( 14 in. + 1z in.) ⎤⎦ = 16.9 in.2 The plate gross plastic section modulus is: Z g = 4 (1.00 in. )( 30.0 in. ) 2 = 225 in.3 The plate net plastic section modulus is: ( Z n = 4t ( s − dh′ ) n2 s + dh′ ) (Manual Eq. 15-4) ( ) = 4 (1.00 in. ) ⎡⎣3.00 in. − (14 in. + 1z in. ) ⎤⎦ ⎡ 102 ( 3.00 in. ) + (14 in. + 1z in. ) ⎤ ⎣ ⎦ = 127 in.3 The available shear yielding strength of Tab Plate C is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1 as follows: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 259 LRFD ASD ϕRn = ϕ 0.60 Fy Agv ( = 1.00 ( 0.60 )( 50 ksi ) 30.0 in.2 = 900 kips > 468 kips ) o.k. = ( ) = 1, 350 kips > 188 kips ( 0.60 ( 50 ksi ) 30.0 in.2 1.50 = 600 kips > 312 kips ϕRn = ϕFy Ag = 0.90 ( 50 ksi ) 30.0 in.2 Rn 0.60 Fy Agv = Ω Ω ) o.k. Rn Fy Ag = Ω Ω o.k. = ( 50 ksi ) ( 30.0 in.2 ) 1.67 = 898 kips > 125 kips o.k. The available shear rupture strength of Tab Plate C is determined from AISC Specification Equation J4-4, and the available tensile rupture strength is determined from AISC Specification Equation J4-2. LRFD ASD ϕRn = ϕ 0.60Fu Anv ( = 0.75 ( 0.60 )( 65 ksi ) 16.9 in. 2 ) = 494 kips > 468 kips o.k. ϕRn = ϕFu Ae = 0.75 ( 65 ksi ) 16.9 in.2 ( ) = 824 kips > 188 kips o.k. Rn 0.60 Fu Anv = Ω Ω = ( 0.60 ( 65 ksi ) 16.9 in.2 ) 2.00 = 330 kips > 312 kips o.k. Rn Fu Ae = Ω Ω = ( 65 ksi ) (16.9 in.2 ) 2.00 = 549 kips > 125 kips o.k. The available flexural yield strength of Tab Plate C is determined from AISC Manual Equation 15-2, and the available flexural rupture strength is determined from AISC Manual Equation 15-3 as follows: LRFD ( ASD ϕM n = 0.90 ( 50 ksi ) 225 in. 3 ) = 10,100 kip-in. > 1,430 kip-in. o.k. ( ϕM n = 0.75 ( 65 ksi ) 127 in.3 ) = 6,190 kip-in. > 1,430 kip-in. o.k. ( ) ( ) 3 M n ( 50 ksi ) 225 in. = 1.67 Ω = 6, 740 kip-in. > 947 kip-in. o.k. 3 M n ( 65 ksi ) 127 in. = Ω 2.00 = 4,130 kip-in. > 947 kip-in. o.k. 260 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 For yielding: LRFD 2 ASD 4 2 4 1, 430 kip-in. ⎛ 189 kips ⎞ ⎛ 471 kips⎞ 947 kip-in. ⎛ 125 kips ⎞ ⎛ 312 kips ⎞ +⎜ +⎜ ⎟+⎜ ⎟ = 0.236 < 1.0 o.k. ⎟ +⎜ ⎟ = 0.233 < 1.0 o.k. 10,100 kip-in. ⎝1, 350 kips⎠ ⎝ 900 kips⎠ 6, 740 kip-in. ⎝ 898 kips ⎠ ⎝ 600 kips ⎠ For rupture: LRFD 2 ASD 4 1, 430 kip-in. ⎛ 189 kips ⎞ ⎛ 471 kips ⎞ +⎜ ⎟ +⎜ ⎟ = 1.11 > 1.0 n.g. 6,190 kip-in. ⎝ 824 kips ⎠ ⎝ 494 kips ⎠ 2 4 947 kip-in. ⎛ 125 kips ⎞ ⎛ 312 kips ⎞ +⎜ ⎟ +⎜ ⎟ = 1.08 > 1.0 n.g. 4,130 kip-in. ⎝ 549 kips ⎠ ⎝ 330 kips ⎠ Plate C must be increased to 18 in., which will give an interaction value of 0.762 < 1.0. Most fabricators will use a 14-in.-thick plate. Figure 5-24 shows Plate C as a 14 in. plate. Check Truss B top chord web shear The W36×135 web has h/ tw = 54.1, which exceeds the AISC Specification Section G2.1(a) limit of: h E = 2.24 Fy tw = 53.9 Therefore, from AISC Specification Section G2.1(b), with kv = 5.0: 5 ( 29, 000 ksi ) h = 54.1 < 1.10 tw 50 ksi = 59.2 Therefore, Cv =1.0, ϕ = 0.90 and Ω = 1.67. The available shear strength is determined from AISC Specification Equation G2-1: LRFD ϕVn = ϕ0.6 Fy AwCv = 0.90 ( 0.6 )( 50 ksi )( 35.6 in. )( 0.600 in. )(1.0 ) = 577 kips > 471 kips o.k. ASD 0.6 Fy AwCv Vn = Ω Ω 0.6 ( 50 ksi )( 35.6 in. )( 0.600 in.)(1.0 ) = 1.67 = 384 kiips > 312 kips o.k. Design welds of Tab Plate C to Truss D top chord Figure 5-28 shows that the resultant of the brace induced forces, 189 kips and 409 kips (LRFD) and 125 kips and 271 kips (ASD), acts through the control point B at the center of the W14×257 and W36×231 at the elevation of the bottom flange of the W36×135 secondary truss chord. This means that the vertical component of 409 kips (LRFD) and 271 kips (ASD) can be carried by the Tab Plate C weld to the W36×231 web, and the horizontal component of 189 kips (LRFD) and 125 kips (ASD) can be carried by the Tab Plate C weld to the inside of the W36×231 bottom flange. This would be a correct solution (an admissible force field) if the load were to remain in the W36×231 primary truss chord. In this case, however, the load needs to be delivered to the primary truss vertical W14×257 member. It is best in situations like this to consider where the load needs to be delivered, AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 261 and then determine a force field that will accomplish this. With this approach, consider Figure 5-29. The forces acting on Tab Plate C need to be delivered to the chord/truss vertical interface. In order to do this a Stiffener D is introduced. The forces on Tab Plate C and the Stiffener D are shown in Figure 5-30. Point P for each plate is sufficiently close to the control point B shown in Figure 5-29 to assume that they are the same. For the forces shown in Figure 5-30 to be admissible (i.e., satisfy equilibrium) a couple, M, must be introduced. Taking moments about point P for either plate: LRFD ASD Mu = ( 94.5 kips )(17.0 in. ) − ( 205 kips + 31.0 kips )( 4 b in. ) M a = ( 62.5 kips )(17 in. ) − (136 kips + 20.5 kips )( 4 b in. ) = 530 kip-in. = 348 kip-in. This moment can be applied as shown in Figure 5-30, or it can be converted into horizontal forces at the top and bottom of the plates as H = 530 kip-in./ 34 in. = 15.6 kips (LRFD) or H = 348 kip-in./ 34 in. = 10.2 kips (ASD). The force field shown in Figure 5-30 will be used here. CL Tab Plate C Stiffener D 189 kips (LRFD) 125 kips (ASD) 705 kip-in. (LRFD) 466 kip-in. (ASD) 62 kips (LRFD) 42 kips (ASD) Control point B 3 409 kips (LRFD) 271 kips (ASD) Cap plate 4x18"x18" A A 189 kips (LRFD) 125 kips (ASD) 471 kips (LRFD) 312 kips (ASD) Fig. 5-29. Delivery point of Tab Plate C forces. 262 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Fig. 5-30a. Admissible force fields for Tab Plate C and Stiffener D to deliver load to chord and vertical interface—LRFD Fig. 5-30b. Admissible force fields for Tab Plate C and Stiffener D to deliver load to chord and vertical interface—ASD AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 263 Design the weld of Tab Plate C to the W36×231 (primary truss chord) web. Following Appendix B and the loading given in Figure 5-30: LRFD ASD Nu = 94.5 kips Na = 62.5 kips Vu = 205 kips + 31.0 kips Va = 136 kips + 20.5 kips = 236 kips Mu = 530 kip-in. = 157 kips Ma = 348 kip-in. = 34.0 in. − 2 (14 in. ) l l = 31.5 in. From Appendix B, Equation B-1, the equivalent normal force is: 4M Nue = Nu + l 4 ( 530 kip-in. ) = 94.5 kips + 31.5 in. = 162 kips = 31.5 in. From Appendix B, Equation B-1, the equivalent normal force is: 4M Nae = N a + l 4 ( 348 kip-in. ) = 62.5 kips + 31.5 in. = 107 kips ( 235 kips )2 + (162 kips )2 Ru = Ra = ⎛ 162 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 235 kips ⎠ θ = 34.6° = Ru ( 2l (1.392 kip/in. ) 1.0 + 0.50 sin1.5 θ Dreq = ) 284 kips ( 2 ( 31.5 in. )(1.392 kip/in. ) 1.0 + 0.50 sin = 2.67 sixteenths ⎛ 107 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 157 kips ⎠ = 34.3° The required weld size is determined using the directional strenggth increase from AISC Specification Section J2.4, and the simplified equation given in AISC Manual Part 8 as Equation 8-2 : The required weld size is determined using the directional streng gth increase from AISC Specification Section J2.4, and the simplified equation given in AISC Manual Part 8 as Equation 8-2 : Dreq = (157 kips )2 + (107 kips )2 = 190 kips = 285 kips θ = 34.0 in. − 2 (14 in. ) 1.5 34.6° ) = Ra ( 2l ( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 θ ) 190 kips ( 2 ( 31.5 in. )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 34.3° ) = 2.68 sixteenths Although a x-in. fillet weld is indicated, use the minimum c-in. fillet weld required for this connection according to AISC Specification Table J2.4. The stresses at the weld in the plate could be checked as was done in previous examples, or by using the conservative formula, tmin = 6.19D/Fu . Taking the latter approach, check the rupture strength of the tab plate at the weld using AISC Manual Equation 9-3: LRFD t min = 6.19 D Fu 6.19(2.67 sixteenths) 65 ksi = 0.254 in. < 1.00 in. o.k. = ASD t min = 6.19 D Fu 6.19(2.68 sixteenths) 65 ksi = 0.255 in. < 1.00 in. o.k. = 264 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Design the weld of Tab Plate C to the W36×231 (primary truss chord) flange. The resultant load and load angle on the weld is: LRFD R= ( 94.5 kips )2 + ( 205 kips + 31.0 kips )2 = 254 kips ASD R= ( 62.5 kips )2 + (136 kips + 20.5 kips )2 = 169 kips ⎛ 236 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 94.5 kips ⎠ = 68.2° ⎛ 157 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 62.5 kips ⎠ = 68.3° Length of weld, l = 7d in.−14 in. = 6.625 in. From AISC Manual Table 8-4 with Angle = 60°: a =k = 0 C = 5.21 C1 = 1.0 LRFD Dmin = 254 kips 0.75 ( 5.21)(1.0 )( 6.625 in. ) ASD Dmin = = 9.81 sixteenths 2.00 (169 kips ) 5.21 (1.0 )( 6.625 in. ) = 9.79 sixteenths Use s-in. fillet welds. This weld could be reduced if the Tab Plate C were “finished to bear” on the inside flange of the W36, as the normal force can then be ignored. This is not the usual practice because of mill tolerances and shop operations. If the plate were finished to bear, the fillet weld size required could be calculated from AISC Manual Equation 8-2: LRFD D= 94.5 kips (1.392 kip/in.)( 6.625 in.)( 2 ) = 5.12 sixteenths ASD D= 62.5 kips ( 0.928 kip/in.)( 6.625 in.)( 2 ) = 5.08 sixteenths Therefore, a a-in. fillet weld would be required in this case. Check the rupture strength of the plate at the weld using AISC Manual Equation 9-3, and assume the plate is not finished to bear: LRFD t min = = 6.19 Dmin Fu 6.19 ( 9.81 sixteenths ) 65 ksi = 0.934 in. < 1.00 in. o.k. ASD t min = = 6.19 Dmin Fu 6.19 ( 9.79 sixteenths ) 65 ksi = 0.932 in. < 1.00 in. o.k. Therefore, by inspection, the plate is adequate for either size of weld. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 265 Design weld of Stiffener D to the W36×231 web and flange The loads are the same as for Tab Plate C, so likewise the welds will be the same. Also, use 1-in.-thick plate for this stiffener. Figure 5-24 shows c-in. fillet welds connecting Tab Plate C and Stiffener D to the inside of the W36×231 top flange. These welds are not required for strength, but are used to control distortion during fabrication, to provide stability to the plates, and for aesthetics. Connection of Primary Truss D Top Chord (W36×231) to Primary Truss D Vertical Web Member (W14×257) This connection is shown in Figure 5-29. The forces are 471 kips (LRFD) and 312 kips (ASD) axial compression and 189 kips (LRFD) and 125 kips (ASD) shear. The cap plate to W14 connection will be “finished to bear.” The axial compressive force will be carried in bearing to the W14×257 web. All of the force goes to the web because Tab Plate C and Stiffener D are aligned with the W14 web. Check the bearing strength of the W14×257 web Apb is the projected bearing area, which is the W14×257 web area in this case. Therefore: Apb = (16.4 in.)(1.18 in.) = 19.4 in.2 From AISC Specification Section J7: LRFD ASD ϕRn = ϕ1.8 Fy Apb ( = 0.75 (1.8 )( 50 ksi ) 19.4 in.2 ) = 1, 310 kips > 471 kips o.k. Rn 1.8 Fy Apb = Ω Ω = ( 1.8 ( 50 ksi ) 19.4 in.2 ) 2.00 = 873 kips > 312 kips o.k. Design weld of cap plate to W14×257 The weld to the W14 web, plus weld to the inside and outside of the W14 flanges up to the k1 distance, will be effective in resisting the shear load of 189 kips (LRFD) and 125 kips (ASD). Therefore, the effective length of weld is: tw ⎞ ⎛ leff = 2 d − 2t f + 4 k1 + 4 ⎜ k1 − ⎟ 2⎠ ⎝ ( ) 1.18 in. ⎞ ⎛ = 2 ⎡⎣16.4 in. − 2 (1.89 in. ) ⎤⎦ + 4 (1m in. ) + 4 ⎜ 1m in. − 2 ⎟⎠ ⎝ = 37.4 in. LRFD Dmin = 189 kips (1.392 kip/in.)( 37.4 in. ) = 3.63 sixteenths ASD Dmin = 125 kips ( 0.928 kip/in.)( 37.4 in. ) = 3.60 sixteenths Use the minimum fillet weld required by AISC Manual Table J2.4 of c-in. as shown in Figure 5-24. 266 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check bolt shear strength The bolts carry the shear load into the cap plate from the W36×231 flange. Using the bolt shear value for ASTM A490 1-in.diameter bolts in slip-critical connections, Class B, in oversized holes, single shear, the number of bolts required is: LRFD ASD 189 kips 30.7 kips/bolt = 6.16 bolts 125 kips 20.5 kips/bolt = 6.10 bolts Nb = Nb = Use eight bolts. Check bolt bearing on the cap plate Try a w-in.-thick cap plate. Check the bearing strength using AISC Specification Equation J3-6a, where the right side of the equation controls by inspection: LRFD ASD ϕrn = ϕ2.4 dtFu rn 2.4 dtFu = Ω Ω 2.4 (1.00 in. )( w in. )( 65 ksi ) = 2.00 = 58.5 kips/bolt = 0.75 ( 2.4 )(1.00 in. )( w in. )( 65 ksi ) = 87.8 kips/bolt Bolt shear strength controls the effective strength of the bolt group, which was checked previously. Figure 5-31 shows the bolt pattern. This completes the design of this truss-to-truss connection. At every step of the way, admissible force fields were determined. The final design is, by the lower bound theorem of limit analysis, a safe design. 3" 4" CL Primary truss (truss B) 4" 3" CL Secondary truss (truss D) PLwx18x18 4" 4" Fig 5-31. Cap plate bolt pattern. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 267 Example 5.12—Brace-to-Column Base Plate Connection Example 5.12.1—Strong-Axis Case Given: The theory for this case is given in Section 4.3 and Figure 4-22. Figure 5-32 shows a completed connection design, the calculations for which follow. The brace force given in Figure 5-32 is 180 kips (LRFD) and 120 kips (ASD). Use d-in.-diameter ASTM A325-N bolts in standard holes and 70-ksi electrodes. Fig. 5-32. Brace-to-column base plate, strong axis. 268 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A36 Fy = 36 ksi Fu = 58 ksi From AISC Manual Table 1-1, the member properties are as follows: Column W21×83 d = 21.4 in. tw = 0.515 in. bf = 8.36 in. tf = 0.835 in. k = 1.34 in. tw = 0.390 in. bf = 12.0 in. tf = 0.605 in. Brace W12×65 Ag = 19.1 in.2 d = 12.1 in. Angles L4×4×2 Ag = 3.75 in.2 t = 2 in. x = 1.18 in. From AISC Manual Table 7-1 the available shear strength of the ASTM A325-N d-in.-diameter bolts in standard holes are: LRFD ASD ϕrnv = 24.3 kips/bolt single shear ϕrnv = 48.7 kips/bolt doublee shear rnv = 16.2 kips/bolt single shear Ω rnv = 32.5 kips/bolt double shear Ω Brace-to–Gusset Connection As noted in the previous example, this part of the connection should be designed first in order to obtain a gusset plate size. Check tension yielding on the brace From AISC Specification Section J4.1, the available tension yielding strength is: LRFD ASD ϕRn = ϕFy Ag ( = 0.90 ( 50 ksi ) 19.1 in. 2 Rn Fy Ag = Ω Ω ) = 860 kips > 180 kips o.k. = ( 50 ksi ) (19.1 in.2 ) 1.67 = 572 kips > 120 kips o.k. It may seem that the brace is oversized, but remember that the brace force can be compression, and this will control the brace size. Check tensile rupture on the brace From AISC Specification Section J4.1, the available tensile rupture strength is determined as follows. The effective net area is determined from AISC Specification Section D3, using the hole diameter as given in AISC Specification Table J3.3 of , in.: An = 19.1 in.2 − 2 ( , in. + z in. )( 0.390 in. ) = 18.3 in.2 Ae = AnU (Spec. Eq. D3-1) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 269 The value of U is obtained from AISC Specification Table D3.1, Case 2, by considering that the W12×65 can be assumed to be two channels back-to-back with: 0.390 in. 2 = 0.195 in. 12.0 in. bf = 2 = 6.00 in. t f = 0.605 in. tw = Then determine the distance to the center of gravity for the channel section, x: x= ∑ Ai xi A where A = ( 6.00 in. )( 0.605 in. )( 2 ) + ( 0.195 in. ) ⎡⎣12.1 in. − 2 ( 0.605 in. ) ⎤⎦ = 7.26 in.2 + 2.12 in.2 = 9.38 in.2 A1 = area of fflanges = 7.26 in.2 x1 = 6.00 in. 2 = 3.00 in. A2 = area of web = 2.12 in.2 x2 = 0.195 in. 2 = 0.0975 in. ( 7.26 in. ) (3.00 in.) + ( 2.12 in. ) ( 0.0975 in.) = 2 x 2 9.38 in.2 = 2.34 in. x U = 1− l 2.34 in. = 1− 6.00 in . = 0.610 Ae = AnU ( = 18.3 in.2 ) ( 0.610 ) = 11.2 in.2 270 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available tensile rupture strength is: LRFD ϕRn = ϕFu Ae ASD ( = 0.75 ( 65 ksi ) 11.2 in.2 ) = 546 kips > 180 kips o.k. Rn Fu Ae = Ω Ω = ( 65 ksi ) (11.2 in.2 ) 2.00 = 364 kips > 120 kips o.k. Check bolt bearing on the brace According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance is: lc = 3.00 in. − 1.0 dh = 3.00 in. − 1.0 ( , in. ) = 2.06 in. The available tearout and bearing strengths are: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 2.06 in. )( 0.390 in. )( 65 ksi ) 1.2lc tFu Ω = 1.2 ( 2.06 in. )( 0.390 in. )( 65 ksi ) 2.00 = 47.0 kips/boltt ϕ2.4 dtFu = 0 .75 ( 2.4 )( d in. )( 0.390 in. )( 65 ksi ) = 39.9 kips/bolt Therefore, ϕrn = 39.9 kips/bolt. = 31.3 kips/bolt 2.4 dtFu 2.4 ( d in. )( 0.390 in. )( 65 ksi ) = Ω 2.00 = 26.6 kips/bollt Therefore, rn / Ω = 26.6 kips/bolt. Because the available bolt shear strength for double shear determined previously (48.7 kips for LRFD and 32.5 kips for ASD) is greater than the bearing strength, the limit state of bearing controls the strength of the inner bolts. For the outer bolts: lc = 12 in. − 0.5 ( , in. ) = 1.03 in. The available tearout and bearing strengths are: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.03 in. )( 0.390 in. )( 65 ksi ) 1.2lc tFu Ω = 1.2 (1.03 in. )( 0.390 in. )( 65 ksi ) 2.00 = 23.5 kips/boltt = 15.7 kips/bolt AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 271 LRFD ASD ϕ2.4 dtFu = 0.75 ( 2.4 )( d in. )( 0.390 in. )( 65 ksi ) = 39.9 kips/bolt Therefore, ϕrn = 23.5 kips/bolt. 2.4 dtFu 2.4 ( d in. )( 0.390 in. )( 65 ksi ) = Ω 2.00 = 26.6 kips/bolt Therefore, rn / Ω = 15.7 kips/bolt. Because the available bolt shear strength for double shear determined previously (48.7 kips for LRFD and 32.5 kips for ASD) is greater than the tearout strength, the limit state of tearout controls the strength of the outer bolts. The effective strength of the bolt group based on bearing, tearout and shear strength is: LRFD ASD ϕRn = 2 ( 23.5 kips/bolt ) + 4 ( 39.9 kips/bolt ) Rn = 2 (15.7 kips ) + 4 ( 26.6 kips ) Ω = 138 kips > 120 kips o.k. = 207 kips > 180 kips o.k. . Bearing and tearout on the connection angles will be checked subsequently. Check block shear rupture on the brace The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Assume the controlling failure path forms a rectangle bounded by the two bolt lines on the brace web. Shear yielding component: Agv = 2 ( 7.50 in. )( 0.390 in. ) = 5.85 in.2 ( 0.60 Fy Agv = 0.60 ( 50 ksi ) 5.85 in.2 ) = 176 kips Shear rupture component: Anv = 5.85 in.2 − 2.5 ( , in. + z in. )( 0.390 in. )( 2 ) = 3.90 in.2 ( 0.60Fu Anv = 0.60 ( 65 ksi ) 3.90 in.2 ) = 152 kips Tension rupture component: Ant = ⎡⎣5.00 in. − ( 0.5 + 0.5 ) ( , in. + z in. ) ⎤⎦ ( 0.390 in. ) = 1.56 in.2 Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded ( Ubs Fu Ant = 1 ( 65 ksi ) 1.56 in.2 ) = 101 kips 272 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available strength for the limit state of block shear rupture is determined as follows: 0.60 Fu Anv + U bs Fu Ant = 152 kips + 101 kips = 253 kips 0.60 Fy Agv + U bs Fu Ant = 176 kips + 101 kips = 277 kips Therefore, Rn = 253 kips. LRFD ASD ϕRn = 0.75 ( 253 kips ) = 190 kips > 180 kips o.k. Rn 253 kips = Ω 2.00 = 127 kips > 120 kips o.k. Check tensile yielding on the connection angles The gross area of each angle is 3.75 in.2 From AISC Specification Section J4.1, the available tensile yielding strength is: LRFD ASD ϕRn = ϕFy Ag ( = 0.90 ( 36 ksi ) 3.75 in.2 ) (4) = 486 kips > 180 kips o.k. Rn Fy Ag = Ω Ω = ( 36 ksi 3.75 in.2 )(4) 1.67 = 323 kips > 120 kips o.k. Check tensile rupture on connection angles From AISC Specification Section J4.1, the available tensile rupture strength is determined as follows. The effective net area is determined from AISC Specification Section D3, using the hole diameter as given in AISC Specification Table J3.3 of , in. The net area of one angle is: Ant = 3.75 in.2 − (2 in. ) ( , in. + z in. ) = 3.25 in.2 From AISC Specification Table D3.1, Case 2: x U = 1− l 1.18 in. = 1− 6.00 in. = 0.803 From AISC Specification Table D3.1, Case 8: U = 0.600 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 273 The AISC Specification allows the larger value determined from Case 2 and Case 8 to be used for angles; therefore, in this example, use the larger value of U = 0.803: Ae = UAnt ( = 0.803 3.25 in.2 ) 2 = 2.61 in. The available tensile rupture strength is: LRFD ϕRnt = ϕFu Ae ( = 0.75 ( 58 ksi ) 2.61 in.2 ASD ) (4) = 454 kips > 180 kips o.k. Rnt Fu Ae = Ω Ω ( 58 ksi ) ( 2.61 in.2 ) (4) 2.00 = 303 kips > 120 kips o.k. = Check bolt bearing on the connection angles For the inner bolts, using AISC Specification Equation J3-6a, and lc = 2.06 as determined previously for the brace, the available tearout and bearing strengths are: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 2.06 in. )(2 in. )( 58 ksi ) 1.2lc tFu = 1.2 ( 2.06 in. )(2 in. )( 58 ksi ) 2.00 Ω = 35.8 kips/bolt = 53.8 kips/bolt ϕ2.4 dtFu = 0.75 ( 2.4 )( d in. )(2 in. )( 58 ksi ) = 45.7 kips/bollt Therefore, ϕrn = 45.7 kips/bolt. 2.4 dtFu 2.4 ( d in. )(2 in. )( 58 ksi ) = Ω 2.00 = 30.5 kips/bolt Therefore, rn / Ω = 30.5 kips/bolt. Because the available bolt shear strength for single shear determined previously (24.3 kips for LRFD and 16.2 kips for ASD) is less than the bearing strength, the limit state of bolt shear controls the strength of the inner bolts. For the outer bolts, using lc = 1.03 in. determined previously: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.03 in. )(2 in. )( 58 ksi ) 1.2 lc tFu 1.2 (1.03 in. )(2 in. )( 58 ksi ) = Ω 2.00 = 17.9 kips/bolt = 26.9 kips/bolt ϕ2.4 dtFu = 0.75 ( 2.4 )( d in. )(2 in. )( 58 ksi ) = 45.7 kips/bollt Therefore, ϕrn = 26.9 kips/bolt. 2.4 dtFu 2.4 ( d in. )(2 in. )( 58 ksi ) = Ω 2.00 = 30.5 kips/bolt Therefore, rn / Ω = 17.9 kips/bolt. 274 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Because the available bolt shear strength for single shear determined previously (24.3 kips for LRFD and 16.2 kips for ASD) is less than the bearing strength, the limit state of bolt shear also controls the strength of the outer bolts. The determination of the effective strength of the bolt group based on bolt shear follows. Check bolt shear strength The available bolt shear strength at the brace-to-gusset connection is: LRFD ASD ϕRn = ( 48.7 kips/bolt )( 6 bolts ) Rn = ( 32.5 kips/bolt )( 6 bolts ) Ω = 195 kips > 120 kips o.k. = 292 kips > 180 kips o.k. Check block shear rupture on the connection angles Determine the available strength for the limit state of block shear rupture from AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Assume the controlling failure path follows each bolt line and then runs perpendicular to the edge of each angle. Shear yielding component: Agv = 4 ( 7.50 in. )(2 in. ) = 15.0 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 15.0 in.2 ) = 324 kips Shear rupture component: Anv = 4 ⎡⎣3.75 in.2 − 2.5 ( , in. + z in. )(2 in. ) ⎤⎦ = 10.0 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 10.0 in.2 ) = 348 kips Tension rupture component: { } Ant = 4 ⎡⎣( 8.00 in. − 5.00 in.) 2 ⎤⎦ (2 in.) − 0.5 ( , in. + z in.) (2 in.) = 2.00 in.2 Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded ( U bs Fu Ant = 1 ( 58 ksi ) 2.00 in.2 ) = 116 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60 Fu Anv + U bs Fu Ant = 348 kips + 116 kips = 464 kips 0.60 Fy Agv + U bs Fu Ant = 324 kips + 116 kips = 440 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 275 Therefore, Rn = 440 kips. LRFD ASD ϕRn = 0.75 ( 440 kips ) = 330 kips > 180 kips o.k. Rn 440 kips = Ω 2.00 = 220 kips > 120 kips o.k. Check bolt bearing on gusset plate Try a s-in.-thick gusset plate. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) For the inner bolts, the clear distance between bolts is: lc = 3.00 in. − , in. = 2.06 in. The available tearout and bearing strengths are: LRFD ASD ϕ1.2 lc tFu = 0.75 (1.2 )( 2.06 in. )( s in. )( 58 ksi ) 1.2lc tFu 1.2 ( 2.06 in. )( s in. )( 58 ksi ) = Ω 2.00 = 44.8 kips/bolt = 67.2 kips/bolt ϕ2.4 dtFu = 0.75 ( 2.4 )( d in. )( s in. )( 58 ksi ) = 57.1 kips/bolt Therefore, ϕrn = 57.1 kips/bolt. 2.4 dtFu 2.4 ( d in. )( s in. )( 58 ksi ) = Ω 2.00 = 38.1 kips/bolt Therefore, rn / Ω = 38.1 kips/bolt. The available bolt shear strength of 48.7 kips (LRFD) and 32.5 kips (ASD) controls over bearing strength. For the outer bolts, the clear end distance is: lc = 12 in. − 0.5 ( , in. ) = 1.03 in. The available tearout and bearing strengths are: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.03 in. )( s in. )( 58 ksi ) 1.2lc tFu 1.2 (1.03 in. )( s in. )( 58 ksi ) = Ω 2.00 = 22.4 kips = 33.6 kips/bolt 276 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ϕ2.4 dtFu = 0.75 ( 2.4 )( d in. )( s in. )( 58 ksi ) 2.4 dtFu 2.4 ( d in. )( s in. )( 58 ksi ) = Ω 2.00 = 38.1 kips/bolt = 57.1 kips/bolt Therefore, ϕrn = 33.6 kips/bolt. Therefore, rn / Ω = 22.4 kips/bolt. Note that the tearout strength controls over bolt shear strength for the outer bolts. The effective strength of the bolt group in the gusset plate is: LRFD ASD ϕRn = (33.6 kips/bolt ) ( 2 bolts ) + ( 48.7 kips/bolt ) ( 4 bolts ) Rn = (22.4 kips/bolt ) ( 2 bolts ) + ( 32.5 kips/bolt ) ( 4 bolts ) Ω = 175 kipps > 120 kips o.k. = 262 kipss > 180 kips o.k. Check the gusset plate for tension yielding on the Whitmore section From AISC Manual Part 9, the width of the Whitmore section is: lw = 5.00 in. + 2 ( 6.00 in. ) tan 30° = 11.9 in. The Whitmore section remains in the gusset plate. From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate is: LRFD ϕRn = ϕFy Aw = ϕFy lw tg = 0.90 ( 36 ksi )(11.9 in. )( s in. ) = 241 kips > 180 kips o.k. ASD Rn Fy Aw = Ω Ω Fy lw t g = Ω ( 36 ksi )(11.9 in.)( s in. ) = 1.67 = 160 kips > 120 kips o.k. Check the gusset plate for compression buckling on the Whitmore section From Dowswell (2006), this is an extended gusset, for which K = 0.60 is recommended. With a laterally unbraced length of 8.32 in., as given in Figure 5-32, the effective slenderness ratio is: KL 0.60 ( 8.32 in. ) = r s in. 12 = 27.7 AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 277 KL > 25, according to AISC Specification Section J4.4, Chapter E applies. From AISC Manual Table 4-22, r ϕcFcr = 31.1 ksi and Fcr / Ωc = 20.7 ksi. Because From AISC Specification Section E3: LRFD ASD ϕc Rn = ϕc Fcr Aw Rn Fcr Aw = Ωc Ωc = ϕc Fcr lw tg = ( 31.1 ksi )(11.9 in. )( s in. ) = 231 kips > 180 kips o.k. = Fcr lw tg Ωc = ( 20.7 ksi )(11.9 in. )( s in. ) = 154 kips > 120 kips o.k. Check block shear rupture on the gusset plate Determine the available strength for the limit state of block shear rupture from AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Assume the controlling failure path forms a rectangle bounded by the two bolt lines on the gusset plate. Shear yielding component: Agv = ( 7.50 in. )( s in. )( 2 ) = 9.38 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 9.38 in.2 ) = 203 kips Shear rupture component: Anv = 9.38 in.2 − 2.5 ( , in. + z in. )( s in. )( 2 ) = 6.26 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 6.26 in.2 ) = 218 kips Tension rupture component: Ant = ( 5.00 in. )( s in. ) − 1.0 ( , in. + z in. )( s in. ) = 2.50 in.2 Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded ( U bs Fu Ant = 1 ( 58 ksi ) 2.50 in.2 ) = 145 kips 278 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available strength for the limit state of block shear rupture is determined as follows: 0.60 Fu Anv + Ubs Fu Ant = 218 kips + 145 kips = 363 kips 0.60 Fy Agv + Ubs Fu Ant = 203 kips + 145 kips = 348 kips Therefore, Rn = 348 kips. LRFD ASD ϕRn = 0.75 ( 348 kips ) = 261 kips > 180 kips Rn 348 kips = Ω 2.00 = 174 kips > 120 kips o.k. o.k. Distribution of Forces The force distribution is obtained from the formulas of Figure 4-22 and shown in Figure 5-32: 36.0 in. − w in. + w in. 2 = 18.375 in. e = 6 in. 21.4 in. ec = 2 = 10.7 in. β = LRFD Hu = 109 kips Vu = 143 kips Hub = = Ha = 72.6 kips Va = 95.5 kips Hu ( β − e ) − Vu ec H ab = β (109 kips ) (18.375 in. − 6.00 in. ) − (143 kips )(10.7 in. ) 18.375 in. = −9.86 kips Huc = = ASD = H a ( β − e ) − Va ec β ( 72.6 kips ) (18.375 in. − 6.00 in. ) − ( 95.5 kips )(10.7 in.) 18.375 in. = − 6.72 kips Hu e + Vu ec β H ac = (109 kips )( 6.00 in. ) + (143 kips )(10.7 in. ) 18.375 in. = 119 kips = H a e + Va ec β ( 72.6 kips )( 6.00 in. ) + ( 95.5 kips ) (10.7 in.) 18.375 in. = 79.3 kips Gusset-to-Column Connection Check shear and tensile yielding on the gusset plate along the column flange The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 279 The weld length is: l = 36.0 in. − w in. = 35.3 in. LRFD ASD ϕRn = ϕ0.60Fy Agv Rn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( s in. )( 35.3 in. ) = 1.50 = 318 kips > 95.5 kips o.k. = 1.00 ( 0.60 )( 36 ksi )( s in. )( 35.3 in. ) = 477 kips > 143 kips o.k. ϕRn = Fy Ag = 0.90 ( 36 ksi )( s in. )( 35.3 in. ) = 715 kips > 119 kips Rn Fy Ag = Ω Ω ( 36 ksi )( s in.)( 35.3 in.) = 1.67 = 476 kips > 79.3 kips o.k. o.k. Design weld for combined forces at gusset-to-column flange connection As discussed previously in Example 5.1, and in AISC Manual Part 13, to allow for redistribution of force in welded gusset connections, the weld is designed for an increased strength of 1.25R. The strength of the fillet weld can be determined from AISC Manual Equation 8-2, incorporating the increased strength permitted from AISC Specification Equation J2-5, as follows: LRFD ASD The resultant load is: Ru = The resultant load is: (119 kips )2 + (143 kips )2 ( 79.3 kips )2 + ( 95.5 kips )2 Ra = = 186 kips The resultant load angle is: = 124 kips The resultant load angle is: ⎛ 119 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 143 kips ⎠ = 39.8° The number of sixteenths of fillet weld required is: ⎛ 79.3 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 95.5 kips ⎠ = 399.7° Dreq′d = = 1.25 Ru ( 2 ( l )(1.392 kip/in. ) 1.0 + 0.50 sin1.5 θ 1.25 (186 kips ) ( ) ( 2 ( 35.3 in. )(1.392 kip/in. ) 1.0 + 0.50 sin = 1.88 sixteenths The number of sixteenths of fillet weld required is: 1.25 R a Dreq′d = 2 ( l )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 θ 1.5 39.8° ) = 1.25 (124 kips ) ( ) ) 2 ( 35.3 in. )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 39.7° = 1.88 sixteenths Use the AISC Specification Table J2.4 minimum fillet weld size of 4 in. Note that the 1.25 ductility factor is conservatively applied to the resultant force, which is based on both the peak stress and the average stress in this case. See Hewitt and Thornton (2004). 280 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check local yielding on the column web The normal force is applied at a distance, β = 18.375 in., from the base plate edge, which is less than the column depth. Therefore, use AISC Specification Equation J10-3: LRFD ASD ϕRn = ϕFy t w ( 2.5k + lb ) ⎫⎪ ⎧⎪50 ksi ( 0.515 in. ) = 1.00 ⎨ ⎬ ⎩⎪× ⎡⎣2.5 (1.34 in. ) + ( 36.0 in. − w in. ) ⎤⎦ ⎭⎪ = 994 kips > 119 kips o.k. Rn Fy t w ( 2.5k + lb ) = Ω Ω ( 50 ksi )( 0.515 in.) ⎡⎣2.5 (1.34 in.) + ( 36.0 in. − w in. )⎤⎦ = 1.50 = 663 kips > 79.3 kips o.k. Check local crippling on the column web The normal force is applied at β = 18.375 in. from the base plate edge, which is greater than a distance d/ 2, where d is the column depth. Therefore, AISC Specification Equation J10-4 is applicable: LRFD ASD 1.5 ⎤ ⎧ ⎫ ⎡ ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFy tf ⎪ ⎪ 2⎢ ϕRn = ϕ ⎨0.80t w 1 + 3 ⎜ ⎟ ⎜ ⎟ ⎬ ⎢ tw ⎪ ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎪ ⎣ ⎦ ⎩ ⎭ ⎧ ⎫ 2 ⎪ ⎪ ⎪0.80 ( 0.515 in. ) ⎪ ⎪ ⎡ 1.5 ⎤ ⎪ ⎛ ( 36.0 in. − w in. ) ⎞ ⎛ 0.515 in. ⎞ ⎪ ⎪ = 0.75 ⎨× ⎢1 + 3 ⎜⎜ ⎟⎟ ⎜ ⎟ ⎥⎬ 0 835 21 . 4 in. in. . ⎝ ⎠ ⎥ ⎢ ⎝ ⎠ ⎪ ⎣ ⎦⎪ ⎪ ⎪ ( 29, 000 ksi )( 50 ksi )( 0.835 in.) ⎪ ⎪ ⎪× ⎪ 0.515 in. ⎩ ⎭ = 829 kips > 119 kips o.k. ⎧ ⎡ Rn 1 ⎪ ⎛ lb ⎞ = ⎨0.80t w2 ⎢1 + 3 ⎜ ⎟ ⎢ Ω Ω⎪ ⎝d⎠ ⎣ ⎩ ⎛ tw ⎞ ⎜ ⎟ ⎜ tf ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ EFy tf ⎫⎪ ⎬ tw ⎪ ⎭ ⎧ ⎫ 2 ⎪ ⎪ ⎪0.80 ( 0.515 in. ) ⎪ ⎪ ⎡ 1.5 ⎤ ⎪ ⎛ ( 36.0 in. − w in. ) ⎞ ⎛ 0.515 in. ⎞ 1 ⎪ ⎪ = ⎟⎟ ⎜ ⎨× ⎢1 + 3 ⎜⎜ ⎟ ⎥⎬ . 2.00 ⎪ ⎢⎣ . 4 in. 0 835 in. 21 ⎝ ⎠ ⎥ ⎝ ⎠ ⎦⎪ ⎪ ⎪ ( 29, 000 ksi )( 50 ksi )( 0.835 in.) ⎪ ⎪ ⎪× ⎪ 0.515 in. ⎩ ⎭ = 553 kips > 79.3 kips o.k. Gusset-to-Base Plate Connection Check shear yielding on gusset plate From AISC Specification Section J4.2, the available shear yielding strength is: LRFD ASD ϕRn = ϕ 0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi )( s in. )( 3b in. ) = 48.1 kips > 9.86 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( s in. )( 3b in. ) = 1.50 = 32.1 kips > 6.72 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 281 Design weld at gusset-to-base plate connection The strength of the fillet weld can be determined from AISC Manual Equation 8-2: LRFD Dreq′d = = ASD Ru 2 ( l )(1.392 kip/in. ) Dreq’d = 9.86 kips 2 ( 3b in. ) (1.392 kip/inn. ) = = 0.994 sixteenths Ra 2 ( l )( 0.928 kip/in. ) 6.72 kips 2 ( 3b in. ) ( 0.928 kip/inn. ) = 1.02 sixteenths Use the AISC Specification Table J2.4 minimum fillet weld of 4 in. Note that an extension plate similar to that which is shown in Figure 4-22 could be used if the gusset stress and gusset weld to the base plate were excessive. Alternatively, the base plate could be extended to accommodate the gusset. Column-to-Base Plate Connection AISC Manual Figure 14-2 shows a typical column-to-base plate weld for axial compression. The welds of AISC Manual Figure 14-2 are for handling, shipping and erection. The column compressive axial load is carried by bearing for which no weld is required. Figure 5-33 shows a more commonly used weld arrangement. The purpose of both these arrangements is to provide enough weld for handling, shipping and erection without turning the column in the shop. When axial tension and shear are present, as in this example, welds specific for these loads are required. For gussets to the column strong axis at the base plate, the most effective force distribution is to take the horizontal shear to the web-to-base plate weld, and the axial tension to the flange-to-base plate welds. This force distribution assumes that the anchor bolts are placed close to the flanges to avoid excessive base plate bending due to the axial tension (which is also called uplift). Base plate W Column W L W L W = weld size L= length of weld None for columns less than 10 in. deep 3 in. for columns 10 in. through 16 in. deep 6 in. for columns 18 in. deep and over Fig. 5-33. Weld pattern for column-to-base plate for axial column compression. 282 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Base plate-to-column web connection Check shear yielding on the column web From AISC Specification Section J4.2, the available shear yielding strength is: LRFD ASD ϕRn = ϕ 0.60Fy Agv Rn 0.60 Fy Agv = Ω Ω 0.60 ( 50 ksi )( 0.515 in. )( 21.4 in. ) = 1.50 = 220 kiips > 79.3 kips o.k. = 1.00 ( 0.60 )( 50 ksi )( 0.515 in. )( 21.4 in. ) = 331 kips > 119 kips o.k. Design column web-to-base plate welds Use the minimum 4-in. fillet weld required by AISC Specification Table J2.3 and determine the required length from AISC Manual Equation 8-2: LRFD ASD Huc ≤ 2 (1.392 kip/in. ) Dl l≥ H ac ≤ 2 ( 0.928 kip/in. ) Dl 119 kips = 10.7 in. 2 (1.392 kip/in. )( 4 sixteenths ) l≥ 79.3 kips = 10.7 in. 2 ( 0.928 kip/in. )( 4 sixteenths ) Use 11-in.-long 4-in. fillet welds on each side of the web. Check that the column web shear rupture strength along the weld is adequate: t min = = 6.19 D Fu (Manual Eq. 9-3) 6.19 ( 4 sixteenths ) 65 ksi = 0.381 in. < 0.515 in. o.k. Base plate-to-column flange connection 143 kips 95.5 kips = 71.5 kips (LRFD) and = 47.8 kips (ASD) tension. Using the minimum c-in. fillet 2 2 weld required by AISC Specification Table J2.4, the length required can be determined from AISC Manual Equation 8-2. The directional strength increase factor from AISC Specification Section J2.4 is also incorporated due to the 90° angle of loading: Each flange carries LRFD l= = ASD Vu 2 ( 2 (1.392 kip/in. ) D 1.0 + 0.50 sin 1.5 71.5 kips ( θ l= ) 2 (1.392 kip/in. )( 5 sixteenths ) 1.0 + 0.50 sin = 3.42 in. 1.5 90° ) = Va 2 ( 2 ( 0.928 kip/in. ) D 1.0 + 0.50 sin1.5 θ 47.8 kips ( ) 2 ( 0.928 kip/in. )( 5 sixteenths ) 1.0 + 0.50 sin1.5 90° ) = 3.43 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 283 Use a 4-in.-long c-in. fillet weld on the outside of each flange, and (2) 2-in. long welds on the inside of each flange. Check that the column-flange shear rupture strength along the weld is adequate: t min = = 6.19 D Fu (Manual Eq. 9-3) 6.19 ( 5 sixteenths ) 65 ksi = 0.476 in. < 0.835 in. o.k. Example 5.12.2—Weak-Axis Case Given: The theory for this case is given in Section 4.3 and Figure 4-23. Figure 5-34 shows a completed sketch of the connection designed in this example. The brace is the same as that shown in Figure 5-32, as is the brace load. Therefore, with the exception of gusset plate buckling, the brace, connection angle and gusset plate limit state checks shown in Example 5.12.1 apply here also and will not be repeated. The given forces and section properties are also the same as those given in the previous example, but the bevel changes to 108 on 12. Solution: Check compression buckling of gusset plate on Whitmore section As given in Figure 5-34, the buckling length is 14 in. The K factor is 0.60 from Dowswell (2006). Therefore, the effective slenderness ratio is: KL 0.60 (14.0 in. ) = r s in. 12 = 46.6 KL > 25, AISC Specification Chapter E r applies, as indicated in AISC Specification Section J4.4. From AISC Manual Table 4-22: As determined in Example 5.12.1, the Whitmore section width, lw, is 11.9 in. Because ϕcFcr = 28.9 ksi and Fcr / Ωc = 19.2 ksi From AISC Specification Section E3: LRFD ϕc Rn = ϕc Fcr Aw = ϕc Fcr lw t g = ( 28.9 ksi )(11.9 in. )( s in. ) = 215 kips > 180 kips o.k. ASD Rn Fcr Aw = Ωc Ωc = Fcr lw t g Ωc = (19.2 ksi )(11.9 in. )( s in. ) = 143 kips > 120 kips o.k. Distribution of Forces From Figure 4-23, and the specific case shown in Figure 5-34: 284 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 25 in. − w in. + w in. 2 = 12.875 in. e = 6 in. ec = 0 in. β = LRFD ASD Hu = 116 kips Vu = 138 kips Hub = = Ha = 77.4 kips Va = 91.7 kips Hu ( β − e ) Hab = β (116 kips ) (12.875 in. − 6.00 in. ) 12.875 in. = 61.9 kips Huc = = Hu e β = 12.875 in. = 54.1 kips β ( 77.4 kips ) (12.875 in. − 6.00 in. ) 12.875 in. = 41.3 kips Hac = (116 kips )( 6.00 in. ) Ha ( β − e ) = Ha e β ( 77.4 kips )( 6.00 in. ) 12.875 in. = 36.1 kips Fig. 5-34. Brace-to-column base plate, weak axis. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 285 Gusset-to-Column Connection Check shear and tension yielding on the gusset plate The available shear yielding strength of the gusset plate is determined from AISC Specification Equation J4-3, and the available tensile yielding strength is determined from AISC Specification Equation J4-1: The weld length is, l = 25.0 in. − w in. = 24.3 in. LRFD ASD ϕRn = ϕ 0.60 Fy Agv Rn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( s in. )( 24.3 in. ) = 1.50 = 219 kips > 91.7 kips o.k. = 1.00 ( 0.60 )( 36 ksi )( s in. )( 24.3 in. ) = 328 kips > 138 kips o.k. ϕRn = ϕFy Ag = 0.90 ( 36 ksi )( s in. )( 24.3 in. ) = 492 kips > 54.1 kips Rn Fy Ag = Ω Ω ( 36 ksi )( s in.)( 24.3 in.) = 1.67 = 327 kips > 36 1 kips o.k. o.k. Design weld for combined forces at gusset-to-column web connection The strength of the fillet weld can be determined from AISC Manual Equation 8-2, incorporating the directional strength increase permitted from AISC Specification Equation J2-5, as follows: LRFD ASD The resultant load is: Ru = ( 54.1 kips ) 2 The resultant load is: + (138 kips ) 2 Ra = = 148 kips ( 36.1 kips )2 + ( 91.7 kips )2 = 98.5 kips The resultant load angle is: The resultant load angle is: ⎛ 54.1 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 138 kips ⎠ = 21.4° The number of sixteenths of fillet weld required is: Ru Dreq’d = 2 ( l )(1.392 kip/in. ) 1.0 + 0.50 sin1.5 θ ⎛ 36.1 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 91.7 kips ⎠ = 21.5° The number of sixteenths of fillet weld required is: ( = Dreq’d = ) 148 kips ( 2 ( 24.3 in. )(1.392 kip/in. ) 1.0 + 0.50 sin = 1.97 sixteenths 1.5 21.4° ) = Ra ( 2 ( l )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 θ ) 98.5 kips ( 2 ( 24.3 in. )( 0.928 kip/in. ) 1.0 + 0.50 sin1.5 21.5° ) = 1.97 sixteenths Use the minimum fillet weld size of ¼ in. stipulated in AISC Specification Table J2.4. Note that the ductility factor of 1.25 is omitted here. The flexibility of the web will allow redistribution to occur and maintain the assumed uniform force distribution. 286 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check column web strength to carry Hc The theory for this limit state is discussed in Section 4.3 and Figure 4-24. From Section 4.3: LRFD Huc ≤ ASD 4ϕFy t w2 L H ac ≤ β 4 ( 0.90 )( 50 ksi )( 0.515 in. ) ( 25.0 in. ) 2 = 4 Fy t w2 L βΩ 4 ( 50 ksi )( 0.515 in. ) ( 25.0 in. ) 2 = 12 d in. = 92.7 kips (12 d in. ) (1.67 ) = 61.7 kips Because Huc = 54.1 kips < 92.7 kips for LRFD and Hac = 36.1 kips < 61.7 kips for ASD, the optional stiffener of Figure 4-23 is not required. Gusset-to-Base Plate Connection Check shear yielding on the gusset plate From AISC Specification Section J4.2, the available shear yielding strength is: LRFD ASD ϕRn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi )( s in.)( 5w in.) = 77.6 kips > 61.9 kips o.k. Rn 0.60 Fy Agv = Ω Ω 0.60 ( 36 ksi )( s in. )( 5w in. ) = 1.50 = 51.8 kips > 41.3 kips o.k. Design weld Design weld at gusset-to-base plate connection The strength of the fillet weld can be determined from AISC Manual Equation 8-2: LRFD Dreq’d = = Ru 2 ( l )(1.392 kip/in. ) 61.9 kips 2 ( 5w in. ) (1.392 kip/in. ) = 3.87 sixteenths Check weld for ductility using the recommendations in Sectiion 4.3: Dmin = s ( s in. )(16 ) = 6.25 sixteenths ASD Dreq’d = = Ra 2 ( l )( 0.928 kip/in. ) 41.3 kips 2 ( 5w in. ) ( 0.928 kip/in. ) = 3.87 sixteenths Check weld for ductility using the recommendation in Sectioon 4.3: Dmin = s ( s in. )(16 ) = 6.25 sixteenths AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 287 Use a v-in. fillet weld. As noted in Section 4.3, this larger weld can be avoided by the use of a top stiffener as shown in Figure 4-23. Column-to-Base Plate Connection The force Hub = 61.9 kips (LRFD) and Hab = 41.3 kips (ASD) is already in the base plate and needs no further consideration. The gusset-to-column shear force Vu = 138 kips (LRFD) and Va = 91.7 kips (ASD) is an axial tension force between the column and base plate and needs to exit the column near the anchor rods. Because the anchor rods are usually near the flanges, one half of this shear force will be distributed to each flange. The force Huc (LRFD) and Hac (ASD) also needs to exit the column through the column flanges. Thus, the forces per flange are: LRFD ASD Vu 2 138 kips = 2 = 69.0 kips Va 2 91.7 kips = 2 = 45.9 kips Axial = Axial = Huc 2 54.1 kips = 2 = 27.1 kips Hac 2 36.1 kips = 2 = 18.1 kips Shear = Shear = The resultant force per flange is: Ru = The resultant force per flange is: ( 69.0 kips )2 + ( 27.1 kips )2 Ra = = 74.1 kips ( 45.9 kips )2 + (18.1 kips )2 = 49.3 kips The resultant load angle is: The resultant load angle is: ⎛ 69.0 ⎞ θ = tan −1 ⎜ ⎟ ⎝ 27.1 ⎠ ⎛ 45.9 ⎞ θ = tan −1 ⎜ ⎟ ⎝ 18.1 ⎠ = 68.6° = 68.5° If a c-in. fillet weld is assumed on each of the flanges, the length required is determined from AISC Manual Equation 8-2, LRFD l= = ASD Ru ( 2 (1.392 kip/in. ) D 1.0 + 0.50 sin 1.5 θ l= ) 74.1 kips ( 2 (1.392 kip/in. )( 5 sixteenths ) 1.0 + 0.50 sin = 3.67 in. 1.5 68.6° ) = Ra ( 2 ( 0.928 kip/in. ) D 1.0 + 0.50 sin1.5 θ 49.3 kips ) ( 2 ( 0.928 kip/in. ) ( 5 sixteenths ) 1.0 + 0.50 sin1.5 68.5° = 3.67 in. Use a 4-in.-long c-in. fillet weld on the outside of the flanges and two 2-in.-long welds each on the inside of the flanges. 288 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ) Check that the column flange shear rupture strength along the weld is adequate: t min = = 6.19 D Fu (Manual Eq. 9-3) 6.19 ( 5 sixteenths ) 65 ksi = 0.476 in. < tf = 0.835 in. o.k. Check that the web can deliver 2V to each flange by checking the available shear yielding strength of the column web along the gusset plate: LRFD ϕRn = ϕ0.60 Fy t w L ≥ ASD V 2 = 1.00 ( 0.60 )( 50 ksi )( 0.515 in. ) ( 25.0 in. ) ≥ = 386 kips > 69.0 kips o.k. 138 kips 2 Rn 0.60 Fy t w L V = ≥ Ω Ω 2 0.60 ( 50 ksi )( 0.515 in. )( 25.0 in. ) 91.7 kips = ≥ 1.50 2 = 258 kips > 45.9 kips o.k. Design weld of column web-to-base plate The column web delivers no load to the base plate in this case. A nominal AISC minimum weld of the length given in Figure 5-33 can be used to facilitate shipping, handling and erection. Concluding Remarks: The designs shown in Figures 5-32 and 5-34 could occur at the same point. In this case, a question arises as to whether or not the column shears and tensions (uplifts) need to be considered simultaneously. In most cases this is not necessary because the lateral loads do not act simultaneously in two orthogonal directions. Some building codes, however, require that the full lateral load in one direction be combined with 70% of the lateral load in the other (orthogonal) direction. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 289 290 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Chapter 6 Design of Bracing Connections for Seismic Resistance As discussed in Section 4.2.6, Effect of Frame Distortion, for high-seismic applications where story drifts of 2 to 2.5% must be accommodated, frame distortion cannot be ignored. These story drifts of 2 to 2.5% are on the order of ten times the drifts that are expected for wind and low seismic (R ≤ 3) design. These large deformations occur because the actual maximum considered earthquake (MCE) forces are reduced to about one ninth of the forces the MCE could produce. This is done by first using 2/3 of the MCE forces and then dividing the forces resulting from the reduced earthquake by an R factor on the order of 6, so the MCE load reduction factor is 6(3)/2 = 9 (ASCE, 2010). The rationale for this reduction factor is twofold: (1) the forces are of short duration and reversing so the response to them does not necessarily achieve the maximum values; and (2) when the structure deforms into the plastic response range, its fundamental natural frequency is reduced, and its period of vibration is increased. With an increased period of vibration, the maximum effect of the seismic base input is reduced. The softening of the structure tends to isolate it from the ground motion. The price paid for this MCE force reduction is the high drift ratios, and the requirement for ductile response that allows large distortions without fracture and resulting building collapse. If R = 1 is used, or even 2/3, the drift would be no greater (and probably less because of the short duration) than traditional wind design. Some designers of hospitals and nuclear power plants use these lower R factors. The AISC Seismic Provisions for Structural Steel Buildings (AISC, 2010b), hereafter referred to as the AISC Seismic Provisions, provide three ways to design special concentrically braced frame (SCBF) connections involving a beam, a column and a brace. The requirements of these SCBF connections are given by AISC Seismic Provisions Section F2.6b. The first option, given in AISC Seismic Provisions Section F2.6b(a), is to provide a simple connection. This is one that satisfies AISC Specification Section B3.6a, except using a required rotation of 0.025 rad. The simple shear connections of AISC Manual Parts 9 and 10 are deemed to satisfy this requirement. Basically, the moment that can exist in the beam-column-brace connection, because of continuity of the column and the rigid connection that results between the beam and the column because of the gusset plate connection for the brace, needs to be released. This can be done by means of a beam moment release as shown in Figure 4-20 with a beam splice or Figure 4-19 with a pin. An option given by Thornton and Muir (2009) is to move the workpoint and use long slots to accommodate the 0.025 rad rotation. The second and third options, given in AISC Seismic Provisions Section F2.6b(b), are to provide a connection that can resist a moment equal to the lesser of 1.1 times the expected beam flexural strength (1.1Ry Mp)beam for LRFD, or the expected column flexural strength Σ(1.1Ry Mp)column for LRFD. There are two ways to accomplish this (providing options two and three): 1. Provide a prescriptive fully restrained beam-to-column moment connection designed for the lesser of the beam or column expected flexural strength. One way to achieve this connection is to satisfy the prescriptive option for beam-to-column connections in ordinary moment frames (OMF) given in AISC Seismic Provisions Section E1.6b(c). 2. Provide a connection in which the beam, column and gusset plate participate in carrying the required flexural strength prescribed. This approach requires distributing the rotational or distortional forces that result from the required flexural strength, MD = min[(1.1Ry Mp)beam, Σ(1.1RyMp)column]. MD will be referred to as the distortional moment. Figure 4-18 shows an admissible distortional force distribution that can be added algebraically to the uniform force method (UFM) distribution. The AISC Seismic Design Manual (AISC, 2012) gives examples of the use of these three methods. 6.1 COMPARISON BETWEEN HIGH-SEISMIC DUCTILE DESIGN AND ORDINARY LOWSEISMIC DESIGN In the following examples, two connections are designed. The connections are at the same plan location, in the same building, and at the same geographic location. This location allows for a structure not specifically detailed for seismic resistance with R = 3, but the designer has the option to use a lower seismic force level when the seismic detailing requirements of a special concentrically braced frame (SCBF), with R = 6, are satisfied. In Example 6.1a, the SCBF option is designed. In Example 6.1b, the R = 3 version is designed and a comparison between the two is discussed. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 291 Example 6.1a—High-Seismic Design in Accordance with the AISC Specification and the AISC Seismic Provisions Given: The frame is designed as a special concentrically braced frame (SCBF) with R = 6. Design the HSS heavy bracing connection, shown in its final state in Figure 6-1, between the W16×100 beam and the W14×283 column. Use 1-in.-diameter ASTM A490-X bolts in standard holes and 70-ksi electrodes. The shape materials are given in Figure 6-1. The brace length is 24 ft (work pointto-work point). The moment release option will be used. The required axial strengths (tension or compression) of the brace from seismic and wind loads are as follows: LRFD ASD Pu = 300 kips (wind) Pa = 200 kips (wind) Pu = 145 kips (seismic) Pa = 96.7 kips (seismic) The shear reaction on the beam due to factored gravity loads is: LRFD ASD Vu = 70.0 kips Va = 46.7 kips Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A500, Grade B Fy = 46 ksi Fu = 58 ksi ASTM A572, Grade 50 Fy = 50 ksi Fu = 65 ksi From AISC Manual Table 1-1 and Table 1-12 the geometric properties are as follows: Beam W16×100 Ag = 29.4 in.2 d = 17.0 in. tw = 0.585 in. bf = 10.4 in. tw = 1.29 in. tf = 2.07 in. kdes = 2.67 in. t = 0.581 in. b / t = 10.8 in. r = 2.99 in. tf = 0.985 in. kdes = 1.39 in. Column W14× 283 d = 16.7 in. Brace HSS8×8×s Ag = 16.4 in.2 Design HSS Brace The AISC Seismic Provisions are applicable for the design of a special concentrically braced frame (SCBF). Because the required strength of the bracing connection is based on the expected strength of the brace, it is important not to oversize the brace. Try an HSS8×8×12. At a length of 24 ft, AISC Manual Table 4-4 gives an available compressive strength of: LRFD ASD ϕc Pn = 306 kips (compression) Pn = 203 kips (compression) Ωc > 200 kips o.k. > 300 kips o.k. 292 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 According to AISC Seismic Provisions Section F2.5a, braces shall satisfy the requirements of Section D1.1 for highly ductile members. From AISC Seismic Design Manual Table 1-5b, it can be seen that an HSS8×8×2 does not satisfy the width-tothickness requirements for an SCBF brace. From AISC Seismic Provisions Table D1.1, for highly ductile HSS: E b ≤ 0.55 t Fy ≤ 0.55 29, 000 ksi = 13.8 46 ksi Try an HSS8×8×s. For this shape, b/t = 10.8 < 13.8, therefore the highly ductile width-to-thickness requirement for an SCBF brace is satisfied. According to AISC Seismic Provisions Section F2.5, columns in SCBF must satisfy the requirements for highly ductile members and beams must satisfy the requirements for moderately ductile members. To meet the width-to-thickness limits, a similar check to the above could be performed. Alternatively, AISC Seismic Design Manual Table 1-3 demonstrates that the W14×283 column and the W16×100 beam satisfy the width-to-thickness requirements for SCBF columns and beams, respectively. Fig. 6-1. High-seismic design with hinge (moment release). AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 293 Brace-to-Gusset Connection Determine the required tensile strength of the connection Assuming that AISC Seismic Provisions Section F2.6c(1)(a) controls, with Ry given in Table A3.1, the required tensile strength of the bracing connection is: LRFD ASD Pu = Ry Fy Ag ( = 1.4 ( 46 ksi ) 16.4 in.2 ) = 1, 060 kips Pa = = Ry Fy Ag 1.5 ( 1.4 ( 46 ksi ) 16.4 in.2 ) 1.5 = 704 kips Note that the required strength of the brace and the required strength of the bracing connection are significantly different in an SCBF, whereas in an R = 3 design they are the same. Furthermore, the SCBF highly ductile width-to-thickness requirement requires a heavier section (HSS8×8×s) than is required for strength (HSS8×8×2), thereby driving up the required strength of the connection even more. Determine the required compressive strength of the connection AISC Seismic Provisions Section F2.5b(1) requires that the slenderness ratio of SCBF braces be less than 200. The calculation uses the work point-to-work point length of the brace, with K = 1.0. The actual buckling length will be less. KL 1.0 ( 24.0 ft )(12 in./ft ) = r 2.99 in. = 96.3 < 200 o.k. From AISC Seismic Provisions Section F2.3, the expected brace buckling strength must be based on a length not more than the brace end-to-end length. With the pull-off dimension given in Figure 6-1 assumed to exist at each end of the brace, the length of the brace, lbrace = (24 ft)(12 in./ft) − 2(43 in.) = 202 in. = 16 ft 10 in. The geometry of Figure 6-2 can be used to determine the pull-off dimension. The “pull-off” dimension is the terminology used by fabricators and detailers to determine the actual cut length of the brace. It is based on the point where the brace ends. With a K value of 1.0: KL 1.0 ( 202 in. ) = r 2.99 in. = 67.6 According to AISC Seismic Provisions Section F2.3, the expected brace strength in compression is permitted to be taken as the lesser of Ry Fy Ag or 1.14Fcre Ag, where Fcre is determined from AISC Specification Chapter E using Ry Fy in lieu of Fy. From AISC Specification Section E3: Fe = = π 2E ⎛ KL ⎞ ⎜ r ⎟ ⎝ ⎠ (Spec. Eq. E3-4) 2 π 2 ( 29, 000 ksi ) ( 67.6 )2 = 62.6 ksi 294 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 E 29,000 ksi = 4.71 Ry Fy 1.4 ( 46 ksi ) 4.71 = 99.9 Because Fcre KL = 67.6 < 99.9: r Ry Fy ⎤ ⎡ ⎢ = Ry Fy 0.658 Fe ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ 1.4 ( 46 ksi ) ⎤ ⎡ = 1.4 ( 46 ksi ) ⎢0.658 62.6 ksi ⎥ ⎢ ⎥ ⎣ ⎦ = 41.9 ksi L = [ eb tan θ + (a sin θ)(tan θ)] + (eb + a sin θ) 2 2 2 Fig. 6-2. Geometry to locate critical Point A of the bending zone. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 295 The expected brace strength in compression is, according to AISC Seismic Provisions Section F2.3, the lesser of RyFy Ag and 1.14Fcre Ag for LRFD or RyFy Ag/1.15 and 1.14Fcre Ag/1.5 for ASD: LRFD ASD Pu = 1.14 Fcre Ag ( = 1.14 ( 41.9 ksi ) 16.4 in.2 ) = 783 kips < 1,060 kips determined previously Therefore, use Pu = 783 kips. Pa = = 1.14 Fcre Ag 1.5 ( 1.14 ( 41.9 ksi ) 16.4 in.2 ) 1.5 = 522 kips < 704 kips determined previously Therefore, use Pa = 522 kips. The expected post-buckling strength does not govern in this example, which is equal to 0.3 times the expected brace strength in compression. Summary of required strengths of the connection for tension and compression in the brace LRFD ASD Pu = 1, 060 kips (tension) Pa = 704 kips (tension) Pu = 783 kips (compression) Pa = 522 kips (compression) Check shear yielding strength in the brace wall Choose a lap length between the brace and the gusset based on the limit state of shear yielding in the brace wall. Shear yielding is given by AISC Specification Equation J4-3: LRFD ϕRn = ϕ0.60 Fy tl ≤ Pu = 1.00 ( 0.60 )( 46 ksi )( 0.581 in. ) l ( 4 walls ) ≤ 1, 060 kips l ≥ 16.5 in. Try l = 19.0 in. ASD Rn 0.60 Fy tl = ≤ Pu Ω Ω ( 0.60 )( 46 ksi )( 0.581 in.) l ( 4 walls ) = ≤ 704 kips 1.50 l ≥ 16.5 in. Try l = 19.0 in. Design brace-to-gusset weld There are four lines of fillet weld between the brace and the gusset plate. From AISC Manual Part 8, Equation 8-2, where l = weld length: LRFD ϕRn = 1.392 Dl D 1, 060 kips = 4 (1.392 kip/in. )(19.0 in. ) = 10.0 sixteeenths ASD Rn = 0.928 Dl Ω D = 704 kips 4 ( 0.928 kip/in. )(19.0 in. ) = 9.98 sixteentths Try a s-in. fillet weld. 296 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 AISC Specification Section J2.2b states that, for end-loaded fillet welds, a reduction factor must be applied to the length when the length exceeds 100 times the weld size: ( s in.)100 = 62.5 in. > 19 in. o.k. Therefore, no reduction on the length is required. Figure 6-2 shows a “bending zone” of width equal to two times the gusset thickness. This is discussed in the AISC Seismic Provisions Commentary Section F2.6c(3). Figure 6-2 gives the geometry required to locate the critical Point A, which locates the bending zone. For steep braces, this point will be at the column and a similar geometry derivation can be developed. From Figures 6-1 and 6-2 and the member geometry, where d is the dimension of the gusset perpendicular to the brace at the beginning of the brace-to-gusset lap, the following parameters can be determined: ec = 8.35 in. eb = 8.50 in. d = 8.00 in. + 1.00 in. + 1.00 in. = 10.0 in. θ = 56.3° tan θ = 1.50 l = 19.0 in. Check block shear rupture on the gusset plate The block shear rupture failure path is assumed to follow the overlap of the brace on the gusset plate. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) Assume a 1-in.-thick gusset plate. Shear yielding component: Agv = (19.0 in. )( 2 )(1.00 in. ) = 38.0 in.2 ( ) ( ) 0.60 Fy Agv = 0.60 ( 50 ksi ) 38.0 in.2 = 1,140 kips Shear rupture component: Anv = Agv = 38.0 in.2 0.60 Fu Anv = 0.60 ( 65 ksi ) 38.0 in.2 = 1, 480 kips Tension rupture component: Ubs = 1 from AISC Specification Section J4.3 because the connection is uniformly loaded Ant = Agt = ( 8.00 in. )(1.00 in. ) = 8.00 in.2 ( U bs Fu Ant = 1 ( 65 ksi ) 8.00 in.2 ) = 520 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 297 The available block shear rupture strength of the gusset plate is: 0.60Fu Anv + U bs Fu Ant = 1, 480 kips + 520 kips = 2, 000 kips 0.60Fy Agv + Ubs Fu Ant = 1,140 kips + 520 kips = 1, 660 kips Therefore, Rn = 1,660 kips. LRFD ϕRn = 0.75 (1, 660 kips ) = 1, 250 kips > 1, 060 kips o.k. ASD Rn 1, 660 kips = Ω 2.00 = 830 kips > 704 kips o.k. Check tensile yielding on the Whitmore section l w = 8.00 in. + 2 (19.0 in. ) tan 30° = 29.9 in. LRFD ϕRn = 0.90 Fy lw t = 0.90 ( 50 ksi )( 29.9 in. )(1.00 in. ) = 1, 350 kips > 1, 060 kips o.k. ASD Rn Fy lw t = Ω Ω ( 50 ksi )( 29.9 in.)(1.00 in.) = 1.67 = 895 kips > 704 kips o.k. Check flexural buckling in compression on the Whitmore section From Dowswell (2006), the critical gusset thickness is tβ. If the actual gusset thickness is greater than tβ, the gusset will yield before it buckles and the limit state of flexural buckling does not apply. From Figure 6-2: l ′ = l + 2t = 19.0 in. + 2 (1.00 in. ) = 21.0 in. d + l ′ tan 30° 2 10.0 in. = + ( 21.0 in. ) tan 30° 2 = 17.1 in. lb = a tan θ + 2t a= = (17.1 in. )(1.50 ) + 2 (1.00 in. ) = 27.7 in. d c = lb cos θ − sin θ 2 298 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Use d = 8.00 in. here (the actual width of the brace) to locate the corner of the HSS connection at the Whitmore section: ⎛ 8.00 in.⎞ c = ( 27.7 in. ) cos 56.3° − ⎜ ⎟ sin 56.3° ⎝ 2 ⎠ = 12.0 in. From Dowswell (2006): tβ = 1.5 = 1.5 Fy c3 Es lb ( 50 ksi )(12.0 in.)3 ( 29, 000 ksi )( 27.7 in.) = 0.492 in. < 1.00 in. o.k. Because t = 1.00 in. > 0.492 in., the gusset is “compact” and will yield before it buckles according to Dowswell (2006); however, the gusset flexural buckling will be checked here using AISC Specification Section J4.4. The gusset satisfies Dowswell’s compact criterion. Dowswell refers to this type of gusset as an “extended” gusset and recommends K = 0.60. KL 0.60 ( 27.7 in. ) = r (1.00 in.) 12 = 57.6 KL > 25, AISC Specification Chapter E applies. Interpolating for KL/r = 57.6 in AISC Manual Table 4-22, for Fy = Because 50 ksi: r LRFD ϕc Fcr = 35.3 ksi From AISC Specification Section E3: ϕc Rn = ϕc Fcr Aw = ϕc Fcr lw tg = ( 35.3 ksi)( 29.9 in. )(1.00 in. ) = 1, 060 kips > 783 kips o.k. ASD Fcr = 23.5 ksi Ωc From AISC Specification Section E3: Rn Fcr Aw = Ωc Ωc = Fcr lw t g Ωc = ( 23.5 ksi )( 29.9 in. )(1.00 in. ) = 703 kips > 522 kips o.k. Use a 1-in.-thick gusset plate. Check tensile rupture on the HSS brace The available tensile rupture strength of the HSS effective net area must be larger than the required tensile strength of the bracing connection. Also, in accordance with AISC Seismic Provisions Section F2.5b(3), the brace effective net area cannot be less than the brace gross area. Assume that the slot in the brace is z in. wider than the gusset plate on each side of the gusset plate, and determine the effective net area of the brace from AISC Specification Section D3: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 299 Ag = 16.4 in.2 A n = Ag − 2tdslot ( dslot = width of slot ) = 16.4 in. − 2 ( 0.581 in. ) (1.00 in. + z in. + z in. ) 2 = 15.1 in.2 From AISC Specification Table D3.1, Case 6: x = = B2 + 2 BH 4(B + H ) ( B = H = 8.00 in. ) (8.00 in.)2 + 2 (8.00 in.)(8.00 in.) 4 ( 8.00 in. + 8.00 in. ) = 3.00 in. ⎛x⎞ U = 1− ⎜ ⎟ ⎝l⎠ ⎛ 3.00 in. ⎞ = 1− ⎜ ⎟ ⎝ 19.0 in. ⎠ = 0.842 Ae = AnU ( = 15.1 in.2 (Spec. Eq. D3-1) ) ( 0.842 ) = 12.7 in.2 Because 12.7 in.2 < 16.4 in.2, reinforcement is required. AISC Seismic Provisions Section F2.5b(3)(i) requires the specified minimum yield strength of the reinforcement to be at least that of the brace. Therefore, ASTM A572 Grade 50 must be used for the reinforcing plates. Design reinforcing plates Try 4 in. × 14 in. ASTM A572 Grade 50 plates, as shown in Figures 6-3 and 6-4: Ar = (4.00 in.)(14 in.) = 5.00 in.2 Determine the available strength of the HSS effective area with the reinforcing plates, which must be greater than the gross area. Using AISC Specification Table D3.1, Case 6, and expanding it to include the reinforcing plates, the shear lag factor is determined as follows: x = 2 BHt + B2 t + 2 Ar B 4 ( Bt + Ht + Ar ) 2 = ( ) 2 ( 8.00 in. )( 8.00 in. )( 0.581 in. ) + ( 8.00 in. ) ( 0.581 in. ) + 2 5.00 in.2 ( 8.00 in. ) 4 ⎡⎣( 8.00 in. ) ( 0.581 in. ) + ( 8.00 in. )( 0.581 in. ) + 5.00 in. ⎦ 2⎤ = 3.35 in. x l 3.35 in. = 1− 19.0 in. = 0.824 U = 1− 300 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 An = 15.1 in.2 + 2 ⎡⎣( 4.00 in. )(14 in. ) ⎤⎦ = 25.1 in.2 Ae = AnU ( (Spec. Eq. D3-1) 2 = 25.1 in. ) ( 0.824 ) = 20.7 in.2 > 16.4 in.2 o.k. Fig. 6-3. Reinforcing plates positioning and dimensioning to transfer loads into and out of plates. Gusset plate H Ar Ar B Fig. 6-4. Section A-A: HSS section with reinforcing plates of area Ar . AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 301 4 in. × 14 in. ASTM A572 Grade 50 reinforcing plates will be sufficient. Attachment of reinforcing plates to HSS brace The AISC Seismic Provisions Section F2.5b(3)(ii) requires that the attachment of the reinforcing plates to the HSS brace develop the expected reinforcing tensile strength on each side of the reduced section. The expected tensile strength of each 4 in. × 14 in. reinforcing plate is determined as follows, with Ry taken from AISC Seismic Provisions Table A3.1: LRFD ASD ( Pu = 1.1 ( 50 ksi ) 5.00 in.2 ) ( 1.1 ( 50 ksi ) 5.00 in.2 1.5 = 183 kips Pa = = 275 kips ) Determine what “a” length is required to transfer the force into the plates (refer to Figure 6-3). Applying AISC Specification Equation J2-10b to account for the fillet welds oriented longitudinally and transversely to the direction of the applied load, and AISC Manual Equation 8-2, for each reinforcing plate: LRFD ASD Pu = ⎡⎣2 a ( 0.85 )(1.392 kip/in. ) D ⎤⎦ + ⎡⎣( 4.00 in. )(1.5 ) (1.392 kip//in. ) D ⎤⎦ Pa = ⎡⎣2 a ( 0.85 )( 0.928 kip/in. ) D ⎤⎦ + ⎡⎣( 4.00 in. )(1.5 ) ( 0.928 kip/in. ) D ⎤⎦ The 0.85 factor in the equation above is the strength factor for longitudinally loaded fillet welds when the strength factor of 1.50 is used for the transversely loaded fillets, as given in AISC Specification Equation J2-10b. Try a weld length, a, of 12 in. and solve for D: LRFD D= ASD 275 kips 2 (12 )( 0.85 )(1.392 kip/in. ) + 4 (1.5 )(1.392 kip/in. ) D= = 7.48 sixteenths 183 kips 2 (12 )( 0.85 )( 0.928 kip/in. ) + 4 (1.5 )( 0.928 kip/in. ) = 7.47 sixteenths Use a 2 -in. fillet weld. Use a 2 -in. fillet weld. The final design for the reinforcing plates is shown in Figure 6-1. The x dimension is the erection clearance. Force Distribution The uniform force method (UFM) is used to determine the forces at the gusset-to-beam and gusset-to-column interfaces. As determined previously, the required parameters are: ec = 8.35 in. eb = 8.50 in. θ = 56.3° tan θ = 1.50 l = 19.0 in. sin 56.3° = 0.832 α − β tan θ = eb tan θ − ec (Manual Eq. 13-1) = ( 8.50 in. )(1.50 ) − 8.35 in. = 4.40 in. 302 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The “pull-off” dimension is L + 2t. From the geometry shown in Figure 6-2: 2 2 L = ⎡⎢( eb tan θ + a sin θ tan θ ) + ( eb + a sin θ ) ⎤⎥ ⎣ ⎦ 0.5 d + l ′ tan 30° 2 10..0 in. + ⎡⎣19.0 in. + 2 (1.00 in. ) ⎤⎦ tan 30° = 2 = 17.1 in. a= { 2 L = ⎡⎣( 8.50 in. ) (1.50 ) + (17.1 in. )( 0.832 )(1.50 )⎤⎦ + ⎣⎡8.50 in. + (17.1 in. )( 0.832 )⎤⎦ } 2 0.5 = 41.0 in. Pull-off = 41.0 in. + 2.00 in. = 43.0 in. Figure 6-1 shows the geometry required to guarantee that the hinge (2t) position clears the beam flange. For a steep brace, i.e., θ on the order of 30°, a similar formulation can be derived to guarantee that the hinge (2t) position clears the column. It should be noted that the 30° spread angle shown is used to accommodate the Whitmore section. The 30° Whitmore section defines the maximum effective width of the load spread from the beginning to the end of the brace-to-gusset connection. When the entire Whitmore section is accommodated, the entire section is effective in resisting tensile and compressive forces in the gusset; thus, accommodating the entire Whitmore section will result in the minimum gusset thickness. If a smaller gusset is required, the spread angle can be taken as any angle between 0° and 30°. Such an assumption will result in smaller, thicker gussets. While smaller gussets may be more aesthetically appealing, or may cause fewer interferences with the building services and façade, they may result in difficult-to-accomplish and expensive gusset-to-beam and gusset-to-column connections. Completejoint-penetration welds of the gusset-to-beam and gusset-to-column connections may be required, one of which will usually be a field weld. Also, column and beam web doublers may be required. Figure 6-1 shows the typical case for sizing of the gusset, which has been widely used, but it should be recognized that variations, where 0° ≤ ϕ ≤ 30°, are possible in Figures 6-1 and 6-2. From the geometry shown in Figure 6-2, the distance from the face of the column flange to Point A, Adist , is used to find the horizontal length of the gusset plate. The horizontal length of the gusset plate (measured at the gusset-to-beam interface) is: Adist = eb tan θ − ec + a cos θ = ( 8.50 in. ) tan 56.3° − 8.35 in. + 17.1 in. cos 56.3° = 35.2 in. Use 354 in. Since both the gusset-to-column and gusset-to-beam connections are welded, choose the shorter connection length to be free of moment because it will be the more flexible of the two. A more formal way to distribute the moment between the two gusset plate edges is given in the AISC Manual Part 13 and Section 4.2.1 of this Design Guide. In this example, the gusset-to-column is the shorter length. Therefore, choose β = β and apply any moment that results from the UFM to the more rigid gusset-to-beam interface. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 303 Determine forces on connection interfaces Using the notation and equations in AISC Manual Part 13: β = β = 14.5 in. α − β tan θ = eb tan θ − ec (Manual Eq. 13-1) α = eb tan θ − ec + β tan θ = ( 8.50 in. ) tan 56.3° − 8.35 in. + (14.5 in. ) tan 56.3° = 26.1 in. 354 in. − 1.00 in. + 1.00 in. α= 2 = 18.1 in. Since α ≠ α , there will be a moment on the gusset-to-beam interface. From AISC Manual Part 13, determine r: r = = ( α + ec )2 + ( β + eb )2 (Manual Eq. 13-6) ( 26.1 in. + 8.35 in.)2 + (14.5 in. + 8.50 in.)2 = 41.4 in. LRFD Pu 1, 060 kips = r 41.4 in. = 25.6 kip/in. From AISC Manual Equation 13-2: Pu r = (14.5 in.) ( 25.6 kip/in.) Vuc = β = 371 kips From AISC Manual Equation 13-4: Vub = eb Pu r ASD Pa 704 kips = r 41.4 in. = 17.0 kip/in. From AISC Manual Equation 13-2: Vac = β Pa r = (14.5 in.) (17.0 kip/in.) = 247 kips From AISC Manual Equation 13-4: Pa r = (8.50 in.)(17.0 kip/in. ) Vab = eb = (8.50 in.) ( 25.6 kip/in.) = 218 kips ∑ (Vuc + Vub ) = 371 kips + 218 kips = 145 kips ∑ (Vac + Vab ) = 247 kips + 145 kips = 589 kips = 392 kips ≈ (1,060 kips ) cos 56.3° ≈ ( 704 kips ) cos 56.3° = 588 kips o.k. = 391 kips o.k. 304 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD From AISC Manual Equation 13-3: Huc = ec ASD From AISC Manual Equation 13-3: Pu r Hac = ec Pa r = (8.35 in.) ( 25.6 kip/in.) = (8.35 in.) (17.0 kip/in.) = 214 kips = 142 kips From AISC Manual Equation 13-5: From AISC Manual Equation 13-5: Hub = α Pu r H ab = α Pa r = ( 26.1 in.)(17.0 kip/in.) = ( 26.1 in.) ( 25.6 kip/in.) = 444 kips = 668 kips ∑ ( Huc + Hub ) = 214 kips + 668 kips ∑ ( H ac + H ab ) = 142 kips + 444 kips = 586 kips = 882 kips ≈ ( 704 kips ) sin 56.3° ≈ (1,060 kips ) sin 56.3° ( = 882 kips o.k. ) = ( 218 kips )( 26.1 in. − 18.1 in.) Mub = Vub α − α = 1,740 kip-in. ( = 586 kips o.k. ) = (145 kips )( 26.1 in. − 18.1 in.) M ab = Vab α − α = 1,160 kip-in. The admissible force field of Figure 6-5 assumes the bracing configuration of Figure 6-6, called Configuration A. For this configuration there is no significant transfer force or drag force entering the braced bay. Figure 6-7 shows a possible Configuration B. In this case, there is a large transfer or drag force at the joint. The admissible force field for Configuration B is shown in Figure 6-8. As noted in Figure 6-8, the beam-to-column axial force increases from Configuration A to Configuration B. This example will proceed with Configuration B because the beam splice functions as a hinge, which should be designed as a “simple” connection in accordance with AISC Manual Parts 9 and 10. Beam-to-Column Connection Use a directly-welded connection between the beam and column. LRFD ASD Shear: Vu = 218 kips + 70 kips = 288 kips Normal: Nu = Hub = 668 kips Shear: Va = 145 kips + 46.7 kips = 192 kips Normal: Na = Hab = 444 kips Check beam shear From AISC Manual Table 3-2, the available shear strength of the W16×100 beam is: LRFD ϕvVnx = 298 kips > 288 kips ASD o.k. Vnx = 199 kips > 192 kips Ωv o.k. The axial force will be distributed to the column by the beam flange and beam web in proportion to their areas, as follows: AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 305 A f = t f bf = ( 0.985 in. )(10.4 in. ) = 10.2 in.2 Aw = Ag − 2 Af ( = 29.4 in.2 − 2 10.2 in.2 ) 2 = 9.00 in. LRFD Nuf = 2 10.2 in. ( 668 kips ) 2 29.4 in. = 232 kips N uw = 9.00 in.2 ( 668 kips ) 2 29.4 in. = 204 kips ASD Naf = 10.2 in.2 29.4 in.2 = 154 kips Naw = 9.00 in.2 29.4 in.2 = 136 kips ( 444 kips ) ( 444 kips ) Fig. 6-5a. Admissible LRFD force field for connection shown in Figure 6-1—Configuration A of Figure 6-6. 306 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Design beam web-to-column flange weld Assume all shear is carried by the web weld. Size the weld joining the beam web to the column flange. The length of weld is taken as 134 in. From AISC Manual Equation 8-2 and the directional strength increase permitted in AISC Specification Section J2.4 for fillet welds: LRFD ⎛ 204 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 288 kips ⎠ ⎛ 136 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 192 kips ⎠ = 35.3° D= ASD = 35.3° ( ϕRn 1.5 1.392l 1.0 + 0.50 sin θ ) ( 204 kips )2 + ( 288 kips )2 = (1.392 kip/in.)( 2 )(134 in.) (1.0 + 0.50 sin1.5 35.3° ) = 7.84 sixteenths Use a 2 -in. fillet weld. D= ( Rn Ω 0.928l 1.0 + 0.50 sin1.5 θ ) (136 kips )2 + (192 kips )2 = ( 0.928 kip/in.)( 2 )(134 in.) (1.0 + 0.50 sin1.5 35.3° ) = 7.84 sixteenths Use a 2 -in. fillet weld. Fig. 6-5b. Admissible ASD force field for connection shown in Figure 6-1—Configuration A of Figure 6-6. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 307 Design beam flange-to-column flange weld The required weld length is: flange weld length = 10.4 in. + 10.4 in. − 0.585 in. = 20.2 in. From AISC Manual Equation 8-2 and the directional strength increase permitted in AISC Specification Section J2.4 for fillet welds: LRFD D= = ( ASD ϕRn 1.5 1.392l 1.0 + 0.50 sin θ D = ) 232 kips (1.392 kip/in.) ( 20.2 in .) (1.0 + 0.50 sin 1.5 = 5.50 sixteenths 90° ) = ( Rn Ω 0.928l 1.0 + 0.50 sin1.5 θ ) 154 kips ( 0.928 kip/in.) ( 20.2 in.) (1.0 + 0.50 sin1.5 90° ) = 5.48 sixteenths Use a a-in. fillet weld. Use a a-in. fillet we ld. A 1 in. × 1 in. clip is provided in the gusset plate to accommodate the top flange weld. Lateral loads 13'-3" Bracing configuration assumed for free body diagrams of Figure 6-5 20'–0" Configuration A Fig. 6-6. Bracing arrangement with little to no transfer force. 308 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check for column stiffeners (continuity plates) From AISC Manual Equation 4-2, Equation 4-4 and Table 4-1 for the W14×283 column, check web local yielding and flange local bending of the column: LRFD ASD Web local yielding: Web local yielding: ϕRn = Pwo + Pwi tf Rn Ω = Pwo + tf Pwi = 861 kips + ( 64.5 kips )( 0.985 in. ) = 574 kips + ( 43.0 kips )( 0.985 in. ) = 925 kips > 232 kips = 616 kips > 154 kips o.k. eb compression buckling does not apply because only W one flange is loaded. o.k. eb buckling does not apply because only one flange is W loaded. Lateral loads 13'-3" Bracing configuration assumed for free body diagram of Figure 6-8 20'-0" 20'-0" Configuration B Fig. 6-7. Bracing arrangement with large transfer force. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 309 LRFD ASD Flange local bending: Flange local bending: ϕRn = Pfb = 1,210 kips > 232 kips o.k. Rn = Pfb Ω = 802 kips > 154 kips o.k. Therefore, no column stiffeners (continuity plates) are required. Check column web local yielding and web local crippling adjacent to the beam web AISC Specification Equation J10-2 is applicable for web local yielding in this case (the concentrated load is applied at a distance from the column end that is greater than the column depth, d); however, note that the 5k term is not used here because that portion of the web is already used by the flange connection. Refer to AISC Manual Figure 4-1. Fig. 6-8a. Admissible LRFD force field including transfer force—Configuration B. 310 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Web local yielding: Web local yielding: ϕRn = ϕ Fyw tw lb = 1.00 ( 50 ksi )(1.29 in.)(134 in. ) = 855 kips > 204 kips o.k. Rn Fyw t w lb = Ω Ω ( 50 ksi )(1.29 in.)(134 in.) = 1.50 = 570 kips > 136 kips o.k. Because lb/d = 134 in./16.7 in. = 0.793 > 0.2, use AISC Manual Equation 9-48 with Table 9-4 to check the web local crippling strength of the W14×283 column: Fig. 6-8b. Admissible ASD force field including transfer force—Configuration B. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 311 LRFD ASD Web local crippling: Web local crippling: ϕRn = ϕR5 + lb ( ϕR6 ) = 2 ( 687 kips ) +2 (134 in. )( 89.7 kips ) = 3,750 kips > 204 kips R5 ⎛ R6 ⎞ + lb ⎜ ⎟ = 2 ( 458 kips ) + 2 (134 in. )( 59.8 kips ) Ω ⎝Ω⎠ = 2,500 kips > 136 kips o.k. o.k. Gusset-to-Beam Connection Using the same symbols used for the uniform force method, the required strengths at the gusset-to-beam connection interface as determined previously and shown in Figures 6-8a and 6-8b are: LRFD ASD Required shear strength, Hub = 668 kips Required shear strength, Hab = 444 kips Required normal strength, Vub = 218 kips Required normal strength, Vab = 145 kips Required flexural strength, Mub = 1,740 kip-in. Required flexural strength, Mab = 1,160 kip-in. Gusset-to-beam contact length = 354 in. − 1-in. clip = 34.25 in. Check gusset plate for shear yielding and tensile yielding along the beam flange As determined, the gusset-to-beam interface length is the length of the gusset less the clip, or 34.25 in. In this example, to facilitate combining the required normal strength and flexural strength, stresses will be used to check the tensile and shear yielding limit states. Tensile yielding is checked using AISC Specification Section J4.1 and shear yielding is checked using Section J4.2: LRFD fua = = fub ASD Vub Ag faa = 218 kips (1.00 in.)( 34.25 in.) = Vab Ag 145 kips (1.00 in.)( 34.25 in.) = 4.23 ksi = 6.36 ksi Mub = Z 1,740 kip-in. = (1.00 in.)( 34.25 in.)2 4 fab = = M ab Z 1,160 kip-in. (1.00 in.)( 34.25 in.)2 = 5.93 ksi Total normal stress = fun = 3.96 ksi Total normal stress = fan Tensile yielding strength: fun = fua + fub Tensile yielding strength fan = f aa + fab = 4.23 ksi + 3.96 ksi = 6.36 ksi + 5.93 ksi = 12.3 ksi < 0.90 ( 50 ksi ) = 45 ksi 4 o.k. = 8.119 ksi < 50 ksi = 29.9 ksi 1.67 312 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 o.k. LRFD ASD Shear yielding strength: 668 kips fv = 1.00 in. ( )( 34.25 in.) = 19.5 ksi < 1.00 ( 0.60 )( 50 ksi ) = 30 ksi Shear yielding strength 444 kips fv = 1.00 in. ( )( 34.25 in.) o.k. =13.0 ksi < 0.60 ( 50 ksi ) 1.50 = 20.0 ksi o.k. Consider the interaction equation derived from plasticity theory from Astaneh-Asl (1998) and Neal (1977) for gusset plates. A fourth order interaction equation is recommended. Neal states that this approximate interaction equation produces results within 5% of the theoretical lower-bound failure load for all possible combinations of shear, normal and flexural strengths. The interaction equation is: LRFD ASD 2 4 2 ⎛ Hub ⎞ ⎛ Mub ⎞ ⎛ Vub ⎞ ⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ≤1 ⎝ ϕM n ⎠ ⎝ ϕ N n ⎠ ⎝ ϕVn ⎠ 4 ⎛ H ab ⎞ ⎛ M ab ⎞ ⎛ Vab ⎞ ⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ≤1 ⎝ Mn Ω ⎠ ⎝ Nn Ω ⎠ ⎝ Vn Ω ⎠ where Mn = M p = Fy Z x (Spec. Eq. F2-1) ⎛ tg l 2 ⎞ ⎟ = Fy ⎜ ⎜ 4 ⎟ ⎝ ⎠ ⎡ (1.00 in. )( 34.25 in. )2 ⎤ ⎥ = ( 50 ksi ) ⎢ 4 ⎢ ⎥ ⎣ ⎦ = 14, 700 kip-in. N n = Fy Ag (Spec. Eq. J4-1) = Fy t g l = ( 50 ksi )(1.00 in. )( 34.25 in. ) = 1, 710 kips Vn = 0.60 Fy Ag (Spec. Eq. J4-3) = 0.60 Fy tl = 0.60 ( 50 ksi )(1.00 in. )( 34.25 in. ) = 1, 030 kips LRFD ⎡ ⎤ ⎡ ⎤ 1, 740 kip-in. 218 kips ⎢ ⎥+⎢ ⎥ ⎢⎣ 0.90 (14, 700 kip-in. ) ⎥⎦ ⎢⎣ 0.90 (1, 710 kips )⎥⎦ ⎡ ⎤ 668 kips +⎢ ⎥ ⎢⎣1.00 (1,030 kips )⎥⎦ ASD 2 ⎡ ⎤ ⎡ ⎤ 1,160 kip-in. 145 kips ⎢ ⎥+⎢ ⎥ ⎢⎣ (14, 700 kip-in. 1.67 ) ⎥⎦ ⎢⎣ (1, 710 kips 1.67 ) ⎥⎦ 2 4 4 = 0.328 < 1 o.k. ⎡ ⎤ 444 kips +⎢ ⎥ = 0.327 < 1 o.k. ⎢⎣ (1, 030 kips 1.50 ) ⎥⎦ AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 313 It can be seen from this interaction calculation that interaction is not critical. It in fact very seldom controls. The authors have checked many connections of this type and have never had a case where interaction controlled. It is checked here for the sake of illustration. Design gusset-to-beam weld The AISC Manual Part 13 recommends an increase in design force for a welded connection between a gusset and a member flange to account for uneven load distribution across the weld’s surface. The average stress should thus be multiplied by a factor of 1.25 and then compared to the peak stress, and the greater of the two should be selected as the design stress for the weld. From Hewitt and Thornton (2004): LRFD ASD Vub 2l 218 kips = 2 ( 34.25 in. ) H ab 2l 145 kips = 2 ( 34.25 in. ) fua = fub fun faa = = 3.18 kip/in. 2 Mub = 2 l 2(1, 740 kip-in.) = ( 34.25 in.)2 fab = 2.97 kip/in. = fua + fub fan = 3.18 kip/in. + 2.97 kip/in. = 6.15 kip/in. Hub 2l 668 kips = 2 ( 34.25 in. ) Va 2l 444 kips = 2 ( 34.25 in. ) fav = = 9.75 kip/in. = = 6.48 kip/in. ( fua + fub )2 + fuv2 f peak = ( 6.15 kip/in. )2 + ( 9.75 kip/in.)2 = 1⎡ 2 ⎢⎣ ( fua − fub )2 + fuv2 + ( fua + fub )2 + fuv2 ⎤⎥ favg = ⎦ 1 ⎡ ( 3.18 kip/in. − 2.97 kip/in. ) + ( 9.75 kip/in. ) ⎢ 2 ⎢ +11.5 kip/in. ⎣ = 10.6 kip/in. = fan 2 + fav 2 ( 4.10 kip/in.)2 + ( 6.48 kip/in.)2 = 7.67 kip/in. = 11.5 kip/in. favg = = 1.98 kip/in. = faa + fab = 2.12 kip/in. + 1.98 kip/in. = 4.10 kip/in. fuv = f peak = = 2.12 kip/in. 2 Ma = 2 l 2(1,160 kip-in.) = ( 34.25 in.)2 2 2 ⎤ ⎥ ⎥ ⎦ 1⎡ 2 ⎢⎣ ( faa − fab )2 + fav2 + ( faa + fab )2 + fav2 ⎤⎥ ⎦ 2 2⎤ ⎡ 1 2.12 kip/in. − 1.98 kip/in. ) + ( 6.48 kip/in. ) ⎥ = ⎢ ( 2 ⎢ +7.67 kip/in. ⎥ ⎣ ⎦ = 7.08 kip/in. 314 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD 1.25 favg = 1. 25 (10.6 kip/in. ) 1.25 favg = 1.25 ( 7.08 kip/in. ) = 13.3 kip/in. > 11.5 kip/in. Use 1.25 favg . = 8.85 kip/in. > 7.67 kip/in. Use 1.255 favg . ⎛ fun ⎞ θ = tan −1 ⎜ ⎟ ⎝ fuv ⎠ ⎛ 6.15 kip/in. ⎞ = tan −1 ⎜ ⎟ ⎝ 9.75 kip/in. ⎠ ⎛ fan ⎞ θ = tan −1 ⎜ ⎟ ⎝ fav ⎠ ⎛ 4.10 kip/in. ⎞ = tan −1 ⎜ ⎟ ⎝ 6.48 kip/in. ⎠ = 32.2° = 32.3° From AISC Manual Equation 8-2: Dreq’d = From AISC Manual Equation 8-2: 13.3 kip/in. (1.392 kip/in.) (1.0 + 0.50 sin 1.5 32.2° ) = 8.00 sixteenths Dreq’d = 8.85 kip/in. ( 0.928 kip/in.) (1.0 + 0.50 sin1.5 32.3° ) = 7.98 sixteenths Use a b-in. fillet weld on both sides of the gusset plate. Determine equivalent normal force, Ne The concentrated force for which the beam must be designed includes both the normal force shown in Figure 6-8 and an additional normal force due to the moment. For convenience, convert the moment into an equivalent normal force and add it to the normal force: LRFD N e = Vub + 2 Mub ⎛l⎞ ⎜2⎟ ⎝ ⎠ = 218 kips + ASD N e = Vab + 2 (1, 740 kip-in. ) ⎛ 34.25 in. ⎞ ⎜ ⎟ 2 ⎝ ⎠ = 421 kips 2 M ab ⎛l⎞ ⎜2⎟ ⎝ ⎠ = 145 kips + 2 (1,160 kip-in. ) ⎛ 34.25 in. ⎞ ⎜ ⎟ 2 ⎝ ⎠ = 280 kips Check beam web local yielding The concentrated force, which is the resultant force from the gusset plate, acts at a distance of 34.25 in./ 2 = 17.1 in. from the end of the beam. From AISC Specification Section J10.2, use Equation J10-2 because this distance is greater than the beam depth: LRFD ϕRn = ϕ Fyw tw ( 5kdes + lb ) = 1.00 ( 50 ksi )( 0.585 in. ) ⎡⎣5 (1.39 in. ) + 34.25 in ⎤⎦ = 1, 210 kips > 421 kips o.k. ASD Fyw t w ( 5kdes + lb ) Rn = Ω Ω ( 50 ksi )( 0.585 in.) ⎣⎡5 (1.39 in.) + 34.25 in.⎤⎦ = 1.50 = 803 kips > 280 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 315 Check beam web local crippling As determined for web local yielding, the concentrated force acts at a distance of 17.1 in. from the beam end which is greater than d/ 2. Therefore, from AISC Specification Equation J10-4: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜tf ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ ASD EFyw t f tw ⎡ ⎛ 34.25 in. ⎞ 2 = 0.75 ( 0.80 )( 0.585 in. ) ⎢1 + 3 ⎜ ⎟ ⎢⎣ ⎝ 17.0 in. ⎠ × ⎛ 0.585 in.⎞ ⎜ ⎟ ⎝ 0.985 inn.⎠ 1.5 ⎤ ⎥ ⎥⎦ ( 29, 000 ksi )( 50 ksi ) ( 0.985 in.) 1.5 ⎤ ⎡ Rn 0.80tw2 ⎢ ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFyw tf = 1+ 3⎜ ⎟ ⎜ ⎟ Ω Ω ⎢ tw ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ ( 0.80 )( 0.585 in.)2 ⎡⎢ ⎛ 34.25 in.⎞ ⎛ 0.585 in.⎞ 1.5 ⎤⎥ = 1+ 3⎜ ⎟ ⎟⎜ 2.00 ⎢⎣ ⎝ 17.0 in. ⎠ ⎝ 0.985 in. ⎠ ⎥⎦ × 0.585 in. = 1, 210 kips > 421 kips o.k. 29, 000 ksi ( 50 ksi )( 0.985 in. ) 0.585 in. = 806 kips > 280 kips o.k. Column-to-Gusset Connection The required strengths are: LRFD ASD Required normal strength: Huc = 214 kips Required normal strength: Hac = 142 kips Required shear strength: Vuc = 371 kips Required shear strength: Vac = 247 kips The length of the weld is 27 in. as shown in Figure 6-1, with the 1 in. × 1 in. clip. Because there is no moment on this connection, the average and peak stresses are the same; therefore, the weld ductility factor discussed in AISC Manual Part 13 of 1.25 will be included. From AISC Manual Equation 8-2 and the directional strength increase permitted in AISC Specification Section J2.4 for fillet welds, the fillet weld size is determined as follows: LRFD ASD ⎛ 214 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 371 kips ⎠ ⎛ 142 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 247 kips ⎠ = 30.0° = 29.9° ⎡ Ru D =1.25 ⎢ ⎢1.392l 1.0 + 0.50 sin1.5 θ ⎣ ( ) ⎤ ⎥ ⎥ ⎦ ⎡ ( 214 kips )2 + ( 371 kips )2 ⎢ =1.25 ⎢ 1.392 kip/in. 2 27 in. 1.0 + 0.50 sin1.5 30° ) ( )( ) ⎢⎣ ( = 6.05 sixteenths Usse a v-in. fillet weld. ( ⎡ Ra D =1.25 ⎢ ⎢ 0.928l 1.0 + 0.50 sin1.5 θ ⎣ ( ) ⎤ ⎥ ⎥ ⎥⎦ ) ⎤ ⎥ ⎥ ⎦ ⎡ (142 kips )2 + ( 247 kips )2 ⎢ =1.25 ⎢ 0.928 kip/in. 2 27 in. 1.0 + 0.50 sin1.5 29.9° ) ( )( ) ⎢⎣ ( = 6.04 sixteenthss Use a v-in. fillet weld. ( ) ⎤ ⎥ ⎥ ⎥⎦ Column web local yielding and web local crippling Based on the beam web check, these are assumed to be o.k. because the column is a heavier shape and the required normal strength is nearly the same as that applied to the beam. 316 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check column shear strength The column required shear strength is equal to the required normal force of 214 kips (LRFD) or 142 kips (ASD). From AISC Manual Table 3-6: LRFD ASD ϕvVn = 646 kips > 214 kips o.k. Vn = 431 kips > 142 kips o.k. Ωv Moment Release Shear Splice There are two design possibilities. 1. The splice must be capable of transporting the 70 kips (LRFD) or 46.7 kips (ASD) beam gravity shear force to the face of the column. This is necessary because this is the boundary condition usually assumed in the design of the beam. Also, this splice must be checked for rotational ductility in accordance with AISC Manual Part 10 to qualify as a simple connection in accordance with AISC Specification Section B3.6a. It might be possible to satisfy AISC Specification Section B3.6a without satisfying the checks of Part 10 of the AISC Manual if the rotational demand at the end of the beam is small. However, a good deal of judgment may be involved and in most instances satisfying the rotational ductility checks in the AISC Manual provides a cleaner solution. 2. If the structural analysis is performed with an actual moment release at the splice location, then the splice could be designed for the gravity shear without consideration of the moment required to transport the shear to the column face. This splice, being a simple shear splice, will satisfy AISC Specification Section B3.6a. The first option will be used here. Use two 2-in.-thick ASTM A572 Grade 50 splice plates. Check bolt shear strength With three vertical rows of four bolts, the distance, ex, from the face of the column to the centroid of the farthest bolt group is, from Figure 6-1: ex = 354 in. + 1 in. + 1 in. + 2.5 in. + 3 in. = 42.75 in. This value exceeds the maximum eccentricity given in AISC Manual Table 7-10 of 36 in. A computer program based on the AISC Manual inelastic theory used in Table 7-10 yields C = 1.04. The 1.04 value will be used. The bolts are 1-in.-diameter ASTM A490-X in standard 1z-in.-diameter holes. From AISC Manual Table 7-1, the available shear strength of the bolts in double shear is: LRFD ASD ϕrn = 98.9 kips/bolt rn = 65.9 kips/bolt Ω The available strength of the eccentrically loaded bolt group is: LRFD ϕRn = 1.04 ( 98.9 kips/bolt ) = 103 kips > 70.0 kips o.k. ASD Rn = 1.04 ( 65.9 kips/bolt ) Ω = 68.5 kips > 46.7 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 317 Check bolt bearing on the beam web and splice plates Assuming that deformation at the bolt hole is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) With a 2.5-in. end distance in the beam (to the top of the beam) and the bolt hole diameter of 1z in., the bolt spacing controls lc: lc = 3.00 in. –1z in. = 1.94 in. The available strength of one bolt on the beam web is determined as follows: LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.94 in. )( 0.585 in. )( 65 ksi ) 1.2lc tFu 1.2 (1.94 in. )( 0.585 in. )( 65 ksi ) = Ω 2.00 = 44.3 kips/bolt = 66.4 kips/boltt ϕ2.4 dtFu = 0 .75 ( 2.4 )(1.00 in. )( 0.585 in. )( 65 ksi ) = 68.4 kips/bollt Therefore, ϕrn = 66.4 kips/bolt < 98.9 kips/bolt. 2.4 dtFu 2.4 (1.00 in. )( 0.585 in. )( 65 ksi ) = Ω 2.00 = 45.6 kips/bolt Therefore, rn/Ω = 44.3 kips/bolt < 65.9 kips/bolt. The bearing strength controls over the bolt shear strength. The available strength of the eccentrically loaded bolt group is: LRFD ϕRn = 1.04 ( 66.4 kips/bolt ) = 69.1 kips < 70.0 kips n.g. ASD Rv =1.04 ( 44.3 kips/bolt ) Ω = 46.1 kips < 46.7 kips n.g. Note that the calculations assume the worst case for tearout, the resultant force per bolt acts in the most critical direction, and all bolts are assumed to be in the same critical condition. In reality, the force resultant per bolt will have a different direction for each bolt, and some will have a much larger edge distance than 22 in. or 3 in. If the true distribution is obtained from a computer program based on the AISC Manual inelastic bolt strength, and using the actual edge distances when determining lc for each bolt, the available strength is: LRFD ASD ϕRn = 71.1kips > 70.0 kips o.k. Rn = 47.4 kips > 46.7 kips o.k. Ω Therefore, a web doubler is not required based on the bearing strength of the beam web. The available strength per bolt on the splice plates (two 2-in.-thick splice plates) based on bearing and tearout strength is determined as follows: lc = 1.50 in. –1z in. / 2 = 0.969 in. 318 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )( 0.969 in. )( 2 )(2 in. )( 65 ksi ) 1.2lc tFu 1.2 ( 0.969 in. )( 2 )(2 in. )( 65 ksi ) = Ω 2.00 = 37.8 kips/bolt = 56.7 kips/bolt ϕrn = ϕ2.4dtFu = 0.75 ( 2.4 )(1.00 in. )( 2 )(2 in. )( 65 ksi ) = 117 kips/bolt Therefore, ϕrn = 56.7 kips/bolt < 98.9 kips/bolt (bolt shear). 2.4 dtFu rn = Ω Ω 2.4 (1.00 in. )( 2 )(2 in. )( 65 ksi ) = 2.00 = 78.0 kips/bolt Therefore, rn /Ω = 37.8 kips/bolt < 65.9 kips/bolt (bolt shear). The available strength of the eccentrically loaded bolt group is: LRFD ϕRn = 1.04 ( 56.7 kips/bolt ) = 59.0 kips < 70.0 kips n.g. ASD Rn = 1.04 ( 37.8 kips/bolt ) Ω = 39.3 kips < 46.7 kips n.g. Thicker splice plates are required. Try two s-in.-thick plates: LRFD ASD ⎡ 2 ( s in. ) ⎤ ϕrn = ( 56.7 kips/bolt ) ⎢ ⎥ ⎢⎣ 2 (2 in. ) ⎥⎦ = 70.9 kips/bolt ⎡ 2 ( s in. ) ⎤ rn = ( 37.8 kips/bolt ) ⎢ ⎥ Ω ⎢⎣ 2 (2 in. ) ⎥⎦ = 47.3 kips/bolt Tearout on the splice plates controls over bolt shear (98.9 kips for LRFD and 65.9 kips for ASD). The available strength of the eccentrically loaded bolt group is: LRFD ϕRn = 1.04 ( 70.9 kips/bolt ) = 73.7 kips > 70.0 kips o.k. ASD Rn = 1.04 ( 47.3 kips/bolt ) Ω = 49.2 kips > 46.7 kips o.k. Check splice plates for global effects The shear on the splice is 70.0 kips (LRFD) or 46.7 kips (ASD). The moment on the splice, at the critical location (the first vertical line of bolts to the right of the splice) is: LRFD ASD Mu = (70.0 kips) ( 39.75 in. ) M a = (46.7 kips) ( 39.75 in.) = 2, 780 kip-in. = 1, 860 kip-in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 319 The available shear rupture and flexural strengths of the splice plates are: LRFD From AISC Specification Section J4.2: ASD From AISC Specification Section J4.2: ϕVn = ϕ0.60Fy Ag = 1.00 ( 0.60 )( 50 ksi )( 2 )( s in.)(12 in. ) = 450 kips > 70 kips o.k. ϕVn = ϕ0.60Fu Anv = 0.75 ( 0.60 )( 65 ksi )( 2 )( s in.) × ⎡⎣12.0 in. − 4 (1z in. + z in. ) ⎤⎦ = 274 kips > 70 kips o.k. Vn 0.60 Fy Ag = Ω Ω 0.60 ( 50 ksi )( 2 )( s in. )(12 in. ) = 1.50 = 300 kips > 46.7 kips o.k. Vn 0.60 Fu Anv = Ω Ω = 0.60 ( 65 ksi )( 2 )( s in. ) × ⎡⎣12.0 in. − 4 (1z in. + z in. ) ⎤⎦ 2.00 = 183 kips > 46.7 kips o.k. From AISC Specification Section J4.5: ϕM n = ϕFy Z x 2 = 0.90 ( 50 ksi ) ⎡⎢2 ( s in. )(12 in. ) 4 ⎤⎥ ⎣ ⎦ = 2, 030 kip-in.< 2, 780 kip-in. n.g. From AISC Specification Section J4.5: M n Fy Z x = Ωb Ωb = ( 50 ksi ) ⎡⎣⎢2 ( s in.)(12 in.)2 4 ⎤⎥ ⎦ 1.67 = 1, 350 kip-in. < 1, 860 kip-in. n.g. Try two d-in.-thick splice plates: LRFD ⎡ 2 ( d in. ) ⎤ ϕM n = 2,030 kip-in. ⎢ ⎥ ⎢⎣ 2 ( s in. ) ⎥⎦ = 2,840 kip-in. > 2,780 kip-in. ASD ⎡ 2 ( d in. ) ⎤ Mn = 1,350 kip-in. ⎢ ⎥ Ω ⎢⎣ 2 ( s in. ) ⎥⎦ o.k. = 1,890 kip-in. > 1,860 kip-in. o.k. Use two d-in.-thick splice plates. Ductility checks on the splice The splice can be considered to be a double extended shear plate connection as presented in the AISC Manual Part 10. Equation 10-4 of the AISC Manual is a ductility check. When the shear plate or splice plates satisfy this equation, the plate or plates are guaranteed to yield before the bolts fracture. For a shear plate connection, the weld must satisfy a ductility requirement also; that is, the fillet weld size on each side of the plate must be at least s times the plate thickness. This weld requirement is satisfied by the two bolt groups that form the splice. Following the AISC Manual: Mmax = Fv ( AbC′ ) 0.90 where Fv = 84 ksi (from AISC Specification Table J3.2) π (1.00 in.)2 4 = 0.785 in.2 Ab = C′ = 44.7 in. (from AISC Manual Table 7-10) 320 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 (Manual Eq. 10-4) Therefore: ( 84.0 ksi 0.785 in.2 0.90 = 3, 280 kip-in. M max = t max = = ) ( 44.7 in.) 6 M max (Manual Eq. 10-3) Fy d 2 6 ( 3,280 kip-in. ) ( 50 ksi )(12.0 in.)2 = 2.73 in. > 2 ( d in. ) = 1.75 in. o.k. Note that this check applies equally to both LRFD and ASD design methods. The completed connection design is shown in Figure 6-1. Example 6.1b—Bracing Connections for Systems not Specifically Detailed for Seismic Resistance (R = 3) Given: This example has the same geometry as Example 6.1a. Its purpose is to show the difference in the resulting designs when seismic detailing requirements can be avoided. In this example, the brace force (the required strength of the connection) is defined by brace buckling and not RyFy Ag, as was required in the previous example with R > 3. When designing connections for systems not specifically detailed for seismic resistance, the brace and the connection are designed for the same force. In a special concentrically braced frame (SCBF), the connection is designed for the expected strength of the brace (RyFy Ag). Refer to Figure 6-9 for the final design. Design the HSS heavy bracing connection for the W16×100 beam and W14×283 column shown. Use 1-in.-diameter ASTM A490-X bolts in standard holes and 70-ksi electrodes. The shape materials are given in Figure 6-9. The brace length is 24 ft (work point-to-work point). For the single-plate connections, use 2-in.-thick plate for the gussetto-column connection and 2-in.-thick plate for the beam-to-column connection, initially (w-in.-thick plate is used in the final design). The required strengths of the brace from seismic and wind loads are as follows: LRFD ASD Pu = 300 kips (wind) Pa = 200 kips (wind) Pu = 290 kips (seismic) Pa = 193 kips (seismic) Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A36 Fy = 36 ksi Fu = 58 ksi ASTM A572, Grade 50 Fu = 65 ksi Fy = 50 ksi ASTM A500, Grade B Fy = 46 ksi Fu = 58 ksi AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 321 From AISC Manual Table 1-1 and Table 1-12 the geometric member properties are as follows: Beam W16 ×100 tw = 0.585 in. tf = 0.985 in. d = 17.0 in. kdes = 1.39 in. kdes = 2.67 in. d = 16.7 in. tf = 2.07 in. Column W14×283 tw = 1.29 in. Fig. 6-9. Low-seismic connection for the same location as the connection shown in Figure 6-1. 322 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Brace HSS8×8×2 Ag = 13.5 in.2 t = 0.465 in. b/ t = 14.2 r = 3.04 in. Brace-to-Gusset Connection Choose an HSS brace Try an HSS8×8×½. From AISC Manual Table 4-4, for a 24-ft-long brace, the available compressive strength is: LRFD ASD ϕc Pn = 306 kips (compression) Pn = 203 kips (compression) Ωc > 300 kips o.k. > 200 kips o.k. From AISC Manual Table 5-5, the available tensile yielding strength is: LRFD ASD ϕt Pn = 559 kips Pn = 372 kips Ωt > 300 kips o.k. > 200 kips o.k. The available tensile rupture strength of the HSS effective area must be greater than the required tensile strength or reinforcement plates will be required. Assume that the slot in the brace is z in. wider than the gusset plate on each side of the gusset plate. Assume a gusset plate thickness of 2 in. and verify. Start with a brace-to-gusset weld length of 12 in. From AISC Specification Section D3, determine the effective area: Ae = AnU An = Ag − tdh (Spec. Eq. D3-1) = 13.5 in.2 − 2 ( 0.465 in. ) (2 in. + z in. + z in. ) = 12.9 in.2 U x = 1− l x = B2 + 2 BH 4( B + H ) = (8.00 in.)2 + 2(8.00 in.)(8.00 in.) 4(8.00 in. + 8.00 in.) U = 3.00 in. 3.00 in. = 1− 12.0 in. = 0.75 ( A e = 0.75 12.9 in.2 = 9.68 in.2 (Spec. Table D3-1) ) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 323 The available tensile rupture strength of the column is determined from AISC Specification Equation D2-2: LRFD ϕt Pn = ϕtFu Ae ( = 0.75 ( 58 ksi ) 9.68 in.2 ASD Pn Fu Ae = Ω Ω ) = 421 kips > 300 kips o.k. Therefore, no reinforcement plates are needed. = ( 58 ksi ) ( 9.68 in.2 ) 2.00 = 281 kips > 200 kips o.k. Therefore, no reinforcement plates are needed. The HSS8×8×½ is acceptable. No high or moderately ductile requirements apply. Determine brace-to-gusset connection geometry Choose a lap length between the brace and the gusset plate based on the limit state of shear yielding in the brace wall. By setting AISC Specification Equation J4-3 equal to the required shear strength, the required lap length can be determined: LRFD Pu = ϕ 0.60 Fy Ag = ϕ 0.60 Fy t ( 4 ) l weld ASD Pa 300 kips = 1.00 ( 0.60 )( 46 ksi ) ( 0.465 inn. )( 4 ) lweld lweld = = = 5.84 in. 200 kips = lweld 0.60 Fy Ag Ω 0.60 Fy t ( 4 ) l weld Ω 0.60 ( 46 ksi )( 0.465 in. )( 4 ) lweld 1.50 = 5.84 in. As stated previously, note that for this example, which is not specifically detailed for seismic resistance, the connection is designed for the same force as the brace. This was not the case for the SCBF connection designed in Example 6.1a. Design weld at brace-to-gusset connection Try a weld length of 12 in. to allow a c-in. fillet weld. There are four lines of fillet weld between the brace and the gusset plate. From AISC Manual Part 8, Equation 8-2, where l = weld length: LRFD ϕRn = 1.392 Dl D 300 kips = 4 (1.392 kip/in. )(12.0 in. ) = 4.49 sixteeenths ASD Rn = 0.928 Dl Ω D = 200 kips 4 ( 0.928 kip/in. )(12.0 in. ) = 4.49 sixteennths Use a c-in. fillet weld. Check block shear rupture on the gusset plate Assume the gusset plate is 2 in. thick. The block shear rupture failure path will follow the fillet welds. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) 324 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Shear yielding component: Agv = (12.0 in. )( 2 )(2 in. ) = 12.0 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 12.0 in.2 ) = 259 kips Shear rupture component: Anv = A gv = (12.0 in. )( 2 )(2 in. ) = 12.0 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 12.0 in.2 ) = 418 kips Tension rupture component: Ant = Agt = ( 8.00 in. )(2 in. ) = 4.00 in.2 ( U bs Fu Ant = 1 ( 58 ksi ) 4.00 in.2 ) = 232 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 418 kips + 232 kips = 650 kips 0.60 Fy Agv + U bs Fu Ant = 259 kips + 232 kips = 491 kips Therefore, Rn = 491 kips. LRFD ASD ϕRn = 0.75 ( 491 kips ) Rn 491 kips = Ω 2.00 = 246 kips > 200 kips o.k. = 368 kips > 300 kips o.k. Refer to Figure 6-10: d = 8.00 in. + 1.00 in. + 1.00 in. = 10.0 in. l = 12.0 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 325 Check tensile yielding on the Whitmore section The Whitmore section is determined in accordance with AISC Manual Part 9. The width of the Whitmore section is: lw = 8.00 in. + 2 (12.0 in. ) tan 30° = 21.9 in. From AISC Specification Section J4.1(a), the available tensile yielding strength on the Whitmore section is: LRFD ASD ϕR n = ϕFy Ag = 0.90 Fy lw t Rn Fy Ag = Ω Ω Fy lw t = 1.67 Fig. 6-10. Geometry for gusset plate in Example 6.1b. 326 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Approximately 4z in. of the Whitmore width is within the beam. Because tw = 0.585 in. > tg = 2 in., conservatively use the gusset plate thickness across the entire Whitmore width to determine the available tensile yielding strength: LRFD ASD ϕRn = 0.90 ⎡⎣( 36 ksi ) ( 21.9 in. − 4.06 in. )(2 in. ) Rn 1 ⎡( 36 ksi ) ( 21.9 in. − 4.06 in. )(2 in. ) = Ω 1.67 ⎣ + ( 50 ksi)( 4.06 in.) (2 in. ) ⎤⎦ + ( 50 ksi )( 4.06 in. ) (2 in. ) ⎤⎦ = 380 kips > 300 kips o.k. = 253 kips > 200 kips o.k. Check flexural buckling in compression on the Whitmore section From Dowswell (2006), the critical gusset plate thickness is tβ. If the actual gusset plate thickness is greater than tβ, the gusset plate will yield before it buckles and the limit state of flexural buckling does not apply. tβ = 1.5 Fy c3 Es lb lb = 10a in. ( approximately ) c = 2a in. tβ = 1.5 ( 36 ksi )( 2a in.)3 ( 29, 000 ksi )(10a in.) = 0.0600 in. ≤ 2 in.; therefore, the gusset plate is compact o.k. Because the gusset plate is considered to be compact, it need not be checked for compressive strength due to flexural buckling; however, the check is completed here to be consistent with the AISC Specification requirements. With K = 0.5, the effective slenderness is: KLb 0.5 (10a in. ) = r ⎛ 0.50 in. ⎞ ⎜ ⎟ ⎝ 12 ⎠ = 35.9 From AISC Manual Table 4-22, with Fy = 36 ksi: LRFD ASD ϕc Fcr = 30.3 ksi Fcr = 20.1 ksi Ωc According to AISC Specification Section J4.4, because KL/r > 25, Chapter E applies. From AISC Specification Section E1 and E3, the available compressive strength of the gusset plate is: LRFD ASD ϕc Pn = ϕc Fcr lw t = ( 30.3 ksi )( 21.9 in. )(2 in. ) = 332 kips > 300 kips o.k. Pn Fcr lw t = Ωc Ωc = ( 20.1 ksi )( 21.9 in. )(2 in. ) = 220 kips > 200 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 327 Therefore, the 2-in.-thick gusset plate is adequate. Force Distribution using the Uniform Force Method From Figure 6-10, L = 25s in. To eliminate any moment on the gusset-to-column connection interface, assume β = β, where β is the distance from the face of the beam flange to the centroid of the gusset-to-column connection, which in this case is a bolted single plate. From Figure 6-9 and AISC Manual Part 3: α − β tan θ = eb tan θ − ec (Manual Eq. 13-1) eb = 8.50 in. ec = 8.35 in. tan θ = 12 8 = 1.50 β=β = 54 in. + 3 in. + 12 in. = 9.75 in. α = eb tan θ − ec + β tan θ (from Manual Eq. 13-1) = ( 8.50 in. )(1.50 ) − 8.35 in. + ( 9.75 in. )(1.50 ) = 19.0 in. 25.0 in. + w in. 2 = 13.3 in. α= 2 2 r = ⎡⎢( α + ec ) + ( β + eb ) ⎤⎥ ⎣ ⎦ 0.5 2 2 = ⎡⎢(19.0 in. + 8.35 in. ) + ( 9.75 in. + 8.50 in. ) ⎤⎥ ⎣ ⎦ = 32.9 in. (Manual Eq. 13-6) 0.5 The forces at the connection interface are determined from AISC Manual Part 13: LRFD ASD Pu 300 kips = r 32.9 in. = 9.12 kip/in. Pa 200 kips = r 32.9 in. = 6.08 kip/in. 328 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD From AISC Manual Equation 13-2: Pu r = ( 9.75 in. ) ( 9.12 kip/in. ) Vuc = β = 88.9 kips From AISC Manual Equation 13-4: eb Pu Vub = r = ( 8.50 in. )( 9.12 kip/in. ) ASD From AISC Manual Equation 13-2: Pa Vac = β r = ( 9.75 in. ) ( 6.08 kip/in. ) = 59.3 kips From AISC Manual Equation 13-4: eb Pa Vab = r = ( 8.50 in. )( 6.08 kip/in. ) = 51.7 kips = 77.5 kips ∑ (Vuc + Vub ) = 88.9 kips + 77.5 kips ∑ (Vac + Vab ) = 59.3 kips + 51.7 kips = 166 kips = 111 kips = ( 300 kips ) cos 56.3° = ( 200 kips ) cos 56.3° = 166 kips o.k. = 111 kips o.k. From AISC Manual Equation 13-3: ec Pu r = ( 8.35 in. ) ( 9.12 kip/in. ) Huc = = 76.2 kips From AISC Manual Equation 13-5: Pu Hub = α r = (19.0 in. )( 9.12 kip/in.) = 173 kips From AISC Manual Equation 13-3: ec Pa r = ( 8.35 in. ) ( 6.08 kip/in. ) H ac = = 50.8 kips From AISC Manual Equation 13-5: Pa Hab = α r = (19.0 in. )( 6.08 kip/in. ) = 116 kips ∑ ( Huc + Hub ) = 76.2 kips + 173 kips ∑ ( H ac + H ab ) = 50.8 kips + 116 kips = 249 kips = 167 kips ≈ ( 300 kips ) sin 56.3° ≈ ( 200 kips ) sin 56.3° = 250 kips o.k. From AISC Manuall Equation 13-17: Mub = Vub ( α − α) = ( 77.5 kips ) (19.0 in. − 13.3 in. ) = 442 kip-in. clockwise on gusset = 166 kips o.k. From AISC Manual Equation 13-17: M ab = Vab ( α − α) = ( 51.7 kips ) (19.0 in. − 13.3 in. ) = 295 kip-in. clockwise on gusset The admissible force distribution shown in Figure 6-11 assumes a bracing configuration similar to configuration A of Figure 6-6. In this case there is a small addition of lateral force to the braced bay at each floor. This can usually be neglected in the calculations. Configuration A will be used here, with the admissible force distribution shown in Figure 6-11. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 329 Gusset-to-Beam Connection The required strengths at the gusset-to-beam interface are: LRFD ASD Required shear strength, Hub = 173 kips Required shear strength, Hab = 115 kips Required normal strength, Vub = 77.5 kips Required normal strength, Vab = 51.7 kips Required flexural strength, Mub = 442 kip-in. Required flexural strength, Mab = 295 kip-in. Horizontal length of gusset-to-beam connection, from Figure 6-9, is 25 in. Fig. 6-11. Admissible force field for connection shown in Figure 6-9. 330 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check gusset plate for tension yielding and shear yielding along the beam flange In this example, to facilitate combining the required normal strength and flexural strength, stresses will be used to check the tensile and shear yielding limit states. Tensile yielding is checked using AISC Specification Section J4.1: LRFD fua = = Vub Ag ASD faa = 77.5 kips = (2 in.)( 25.0 in.) = 6.20 ksi fub = = fun Vab Ag 51.7 kips (2 in.)( 25.0 in.) = 4.14 ksi Mub Z fab = 442 kip-in. (2 in.)( 25.0 in.) 2 = 4 = 5.66 ksi = fua + fub M ab Z 295 kip-in. (2 in.)( 25.0 in.)2 4 = 3.78 ksi fan = faa + fab = 6.20 ksi + 5.66 ksi = 4.114 ksi + 3.78 ksi = 11.9 ksi < 0.90 ( 36 ksi ) = 32.4 ksi o.k. = 7.92 ksi < 36 ksi = 21.6 ksi o.k. 1.67 From AISC Specification Section J4.2, the available shear yielding strength of the gusset plate is: LRFD fuv = 173 kips 2 in. ( )( 25.0 in.) = 13.9 ksi < 1.00 ( 0.60 )( 36 ksi ) = 21.6 ksi o.k. ASD fav = 115 kips in. 2 ( )( 25.0 in.) = 9.20 ksi < 0.60 ( 36 ksi ) 1.50 = 14.4 ksi o.k. Design gusset-to-beam weld The AISC Manual recommends an increase in design force for a welded connection between a gusset plate and member flange to account for uneven stress distribution across the weld surface. The average stress should be multiplied by a factor of 1.25 and compared to the peak stress, with the greater of the two being the required stress for the weld (Hewitt and Thornton, 2004) determined as follows: LRFD fua Vub 2l 77.5 kips = 2 ( 25.0 in. ) = = 1.55 kip/in. ASD faa H ab 2l 51.7 kips = 2 ( 25.0 in. ) = = 1.03 kip/in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 331 LRFD fub = = fun ASD 2 Mub l fab 2 2 ( 442 kip--in. ) = 1.41 kip/in. = fua + fub fan = 2.96 kip/in. Hub 2l 173 kips = 2 ( 25.0 in. ) = fav = 3.46 kip/in. favg = 1⎡ 2 ⎢⎣ f peak = = 1.03 kip/in. + 0.944 kip/in. = 1.97 kip/in. Va = 2l 115 kips = 2 ( 25.0 in. ) ( fua − fub )2 + fuv2 + ( fua + fub )2 + fuv2 ⎤⎥ ⎦ favg = (1.55 kip/in. + 1.41 kip/in.)2 + ( 3.46 kip/in.)2 f peak = ( faa + fab )2 + fav2 1⎡ 2 ⎢⎣ ( faa − fab )2 + fav2 + ( faa + fab )2 + fav2 ⎤⎥ ⎦ (1.03 kip/in. + 0.944 kip/in.)2 + ( 2.30 kip/in.)2 = 3.03 kipp/in. 2 2⎤ ⎡ 1 ⎢ (1.55 kip/in. − 1.41 kip/in. ) + ( 3.46 kip/in. ) ⎥ ⎥ 2 ⎢ + ( 4.55 kip/in. ) ⎣ ⎦ = 4.01 kip/in. = 1.25 favg = 1.25 ( 4.02 kip/in. ) Use fweld = 0.944 kip/in. = faa + fab f peak = = 4.55 kip//in. favg ( 25.0 in.)2 = 2.30 kip/in. ( fua + fub )2 + fuv2 f peak = l2 2 ( 295 kip-iin. ) = ( 25.0 in.)2 = 1.55 kip/in. + 1.41 kip/in. fuv 2 Ma = = 5.03 kip/in. > 4.55 kip/in. = 5.03 kip/in. −1 ⎛ fun ⎞ θ = tan ⎜ ⎟ ⎝ fuv ⎠ ⎛ 2.96 kip/in. ⎞ = tan −1 ⎜ ⎟ ⎝ 3.46 kip/in. ⎠ = 40.5° 2 2⎤ ⎡ 1 ⎢ (1.03 kip/in. − 0.944 kip/in. ) + ( 2.30 kip/in. ) ⎥ ⎥ 2 ⎢ + ( 3.03 kip/in. ) ⎣ ⎦ = 2.67 kip/in. favg = 1.25 favg = 1.25 ( 2.67 kip/in. ) = 3.34 kip/in. > 3.03 kip/in. = 3.34 kip/in. Use fweld ⎛ fan ⎞ θ = tan −1 ⎜ ⎟ ⎝ fav ⎠ ⎛ 1.97 kip/in. ⎞ = tan −1 ⎜ ⎟ ⎝ 2.30 kip/in. ⎠ = 40.6° From AISC Manual Equation 8-2: D= = fweld (1.392 kip/in.) (1.0 + 0.50 sin1.5 θ ) From AISC Manual Equation 8-2: D= 5.03 kip/in. (1.392 kip/in.) (1.0 + 0.50 sin1.5 40.5° ) = 2.86 sixteenths = fweld ( 0.928 kip/in.) (1.0 + 0.50 sin1.5 θ ) 3.34 kip/in. ( 0.928 kip/in.) (1.0 + 0.50 sin1.5 40.6° ) = 2.85 sixteenths Use a x-in. fillet weld on both sides of the gusset plate. 332 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Determine equivalent normal force, Ne The concentrated force for which the beam must be designed includes both the normal force and the moment determined previously. For convenience, convert the moment into an equivalent normal force and add it to the normal force: LRFD N e = Vub + 2 Mub ⎛l⎞ ⎜2⎟ ⎝ ⎠ = 77.5 kips + ASD N e = Vab + 2 ( 442 kip-in. ) 2 M ab ⎛l⎞ ⎜2⎟ ⎝ ⎠ = 51.7 kips + ⎛ 25.0 in. ⎞ ⎜ 2 ⎟ ⎝ ⎠ = 148 kips 2 ( 295 kip-in. ) ⎛ 25.0 in. ⎞ ⎜ 2 ⎟ ⎝ ⎠ = 98.9 kips Check beam web local yielding The concentrated force, which is the resultant force from the gusset plate, acts at a distance of 25.0 in./ 2 = 12.5 in. from the end of the beam. From AISC Specification Section J10.2, use Equation J10-3 because the concentrated force is applied at a distance from the member end that is less than the beam depth: LRFD ASD Rn Fyw t w ( 2.5k + lb ) = Ω Ω ( 50 ksi )( 0.585 in.) ⎣⎡2.5 (1.39 in.) + 25.0 in.⎤⎦ = 1.50 = 555 kips > 98.9 kips o.k. ϕRn = ϕFyw tw ( 2.5k + lb ) = 1.00 ( 50 ksi )( 0.585 in. ) ⎡⎣ 2.5 (1.39 in. ) + 25.0 in.⎤⎦ = 833 kips > 148 kips o.k. Check beam web local crippling As calculated for web local yielding, the concentrated force acts at a distance approximately 12.5 in. from the beam end, which is greater than d/ 2. Therefore, from AISC Specification Equation J10-4: LRFD 1.5 ⎤ ⎡ EF yw tf ⎛ lb ⎞ ⎛ t w ⎞ ϕRn = ϕ0.80t w2 ⎢1 + 3 ⎜ ⎟ ⎜ ⎟ ⎥ ⎢ tw ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ ⎡ ⎛ 25.0 in.⎞ 2 = 0.75 ( 0.80 )( 0.585 in. ) ⎢1 + 3 ⎜ ⎟ ⎢⎣ ⎝ 17.0 in. ⎠ × ( 29, 000 ksi )( 50 ksi )( 0.985 in.) 0.585 in. = 969 kips > 148 kips o.k. ASD 1.5 ⎤ ⎡ EFyw tf Rn 0.80t w2 ⎢ ⎛ lb ⎞ ⎛ t w ⎞ = 1+ 3⎜ ⎟ ⎜ ⎟ ⎥ Ω Ω ⎢ tw ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ ⎛ 0.585 in. ⎞ ⎜ ⎟ ⎝ 0.985 in. ⎠ 1.5 ⎤ ⎥ ⎥⎦ 2 1.5 0.80 ( 0.585 in. ) ⎡ ⎛ 25.0 in. ⎞ ⎛ 0.585 in. ⎞ ⎤ ⎢ = 1+ 3⎜ ⎟⎜ ⎟ ⎥ 2.00 ⎢⎣ ⎝ 17.0 in. ⎠ ⎝ 0.985 in. ⎠ ⎥⎦ × 29, 000 ksi ( 50 ksi )( 0.985 in. ) 0.585 in. = 646 kips > 98.9 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 333 Shear force at toe of beam flange to web fillet LRFD ϕR n = ϕ0.6 Fy t w l req lreq = 173 kips 1.00 ( 0.6 )( 50 ksi )( 0.585 in. ) = 9.86 in. < 25.0 in. o.k. Therefore, a sufficient load path into the full section is provided. ASD Rn 0.6 Fy tw lreq = Ω Ω lreq = 115 kips (1.50 ) ( 0.6 )( 50 ksi )( 0.585 in.) = 9.83 in. < 25.0 in. o.k. Therefore, a sufficient load path into the full section is provided. This is less than the gusset length of 25 in. Even if lreq were greater than 25 in., the beam web would provide sufficient length to provide a load path into the section. A reasonable maximum length to transmit the shear force into the W16×100 would be the span divided by two, or 120 in. Gusset-to-Column Connection The required strengths are: LRFD ASD Required normal strength: Huc = 76.2 kips Required normal strength: Hac = 50.8 kips Required shear strength: Vuc = 88.9 kips Required shear strength: Vac = 59.3 kips Assume the single plate is 2-in.-thick ASTM A36 material. The connection does not satisfy all of the dimensional limitations for the conventional single-plate connection given in AISC Manual Part 10, so it will be designed as an extended configuration according to Part 10. Check bolt shear From AISC Manual Table 7-1, the available shear strength per 1-in.-diameter ASTM A490-X bolt in single shear is: LRFD ASD ϕrn = 49.5 kips rn = 33.0 kips Ω The force resultant at the gusset-to-column connection is: LRFD Puc = ( 76.2 kips )2 + (88.9 kips )2 = 117 kips ⎛ 76.2 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 88.9 kips ⎠ = 40.6° ASD Pac = ( 50.8 kips )2 + ( 59.3 kips )2 = 78.1 kips ⎛ 50.8 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 59.3 kips ⎠ = 40.6° 334 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD From AISC Manual Table 7-6 with θ = 30°, ex = 3 in., From AISC Manual Table 7-6 with θ = 30°, ex = 3.0 in., and n = 4, C = 2.79. and n = 4, C = 2.79. Puc C 117 kips = 2.79 bolts = 41.9 kips/bolt Pac C 78.1 kips = 2.79 bolts = 28.0 kips/bolt ru = ra = Therefore, use 1-in.-diameter ASTM A490-X bolts. ϕrn = 49.5 kips/bolt > ru = 41.9 kips/bolt o.k. Therefore, use 1-in.-diameter ASTM A490-X bolts. rn Ω = 33.0 kips/bolt > ra = 28.0 kips/bolt o.k. Check bolt bearing on the gusset-to-column single plate Check the available bearing strength using AISC Manual Tables 7-4 and 7-5, with 3-in. bolt spacing and 14-in.-edge distance: LRFD Based on bolt spacing: ϕrn = (101 kip/in. )(2 in. ) = 50.5 kips/bolt Based on edge distance: ϕrn = ( 37.5 kip/in. )(2 in. ) = 18.8 kips/bolt < ru = 41.9 kips/bolt n.g. ASD Based on bolt spacing: rn = ( 67.4 kip/in. )(2 in. ) Ω = 33.7 kips/bolt Based on edge distance: rn = ( 25.0 kip/in. )(2 in. ) Ω = 12.5 kips/bolt < ra = 28.0 kips/bolt n.g. The available strength of the edge bolts for the limit state of bolt bearing on the single plate is less than the required shear strength per bolt. Therefore, the full bolt strength cannot be achieved on the edge bolts. Because the theory behind AISC Manual Table 7-6 assumes that all bolts can support a force equal to or less than the required shear strength, the theory is invalid unless this is possible. One way to satisfy this requirement is to increase the shear plate thickness: LRFD t p reqd = 41.9 kips (2 in. ) 18.8 kips = 1.11 in. ASD t p reqd = 28.0 kips (2 in. ) 12.5 kips = 1.12 in. Thus, a 18-in.-thick plate is required. Another option is to use a stronger plate (Fy = 50 ksi), which would require a thickness of: ⎛ 58 ksi ⎞ tp reqd = (1.12 in.) ⎜ ⎟ ⎝ 65 ksi ⎠ = 1.00 in. or increase the edge distance. There is room to accommodate an increased edge distance in this case, so consider increasing the edge distance to 1w in. (top and bottom). AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 335 With the increased edge distance, by interpolation in AISC Manual Table 7-5: LRFD ASD ϕrn = ( 63.6 kip/in. )(2 in. ) rn = ( 42.4 kip/in. )(2 in. ) Ω = 21.2 kips/bolt < ra = 28.0 kips/bolt n.g. = 31.8 kips/bolt < ru = 41.9 kips/bolt n.g. This is still inadequate. Therefore, increase the plate thickness to ensure that the available bearing strength based on edge distance at least equals the required force per bolt. Thus, the plate thickness required is: LRFD ASD 41.9 kips (2 in. ) = 0.659 in. 31.8 kips 28.0 kips (2 in. ) = 0.660 in. 21.2 kips Use a w-in.-thick plate. With the w-in.-thick plate and 1w-in. vertical edge distances: LRFD ASD rn edge ϕrn = ( 63.6 kip/in. )( w in. ) Ω = 47.7 kips/bolt > 41.9 kips/bolt o.k. = ( 42.4 kip/in. )( w in. ) = 31.8 kips/bolt > 28.0 kips/bolt o.k. Check bolt bearing on the gusset plate The minimum edge distance provided for the gusset plate is 2.25 in., in the horizontal direction. Therefore, check bearing in the horizontal direction, which will control. According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) The clear distance from the bolt hole edge to the edge of the material is: lc = 2.25 in. – 0.5(1z in.) = 1.72 in. LRFD ϕ1.2lc tFu = 0.75 (1.2 )(1.72 in. )(2 in. )( 58 ksi ) = 44.9 kips ϕ2.4dtFu = 0.75 ( 2.4 )(1.00 in. ) (2 in.) ( 58 ksi ) = 52.2 kips Therefore, ϕrn = 44.9 kips/bolt. ASD 1.2lc tFu Ω = 1.2 ( 58 ksi )(1.72 in. )(2 in. ) 2.00 = 29.9 kips/bolt 2.4dtFu Ω = 2.4 (1.00 in. ) (2 in.) ( 58 ksi ) 2.00 = 34.8 kips Therefore, rn/Ω = 29.9 kips/bolt. Because the available bolt shear strength determined previously (41.9 kips for LRFD and 28.0 kips for ASD) is less than the bearing strength, the limit state of bolt shear controls the strength of the bolt group. 336 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check shear yielding on the single plate The gross area subject to shear is: Agv = ( w in. ) ⎡⎣9.00 in. + 2 (1.75 in. ) ⎤⎦ = 9.38 in 2 From AISC Specification Section J4.2, the available shear yielding strength is: LRFD ASD ϕRn = ϕ0.60 Fy Agv ( = 1.00 ( 0.60 )( 36 ksi ) 9.38 in.2 ) = 203 kips > 88.9 kips o.k. Rn 0.60 Fy Agv = Ω Ω = ( 0.60 ( 36 ksi ) 9.38 in.2 ) 1.50 = 135 kips > 59.3 kips o.k. Check shear rupture on the single plate The net area subject to shear is: Anv = 9.38 in.2 − 4 (1z in. + z in. )( w in. ) = 6.01 in.2 LRFD ϕRn = ϕ0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 58 ksi ) 6.01 in.2 = 157 kips > 88.9 kips o.k. ) Rn 0.60Fu Anv = Ω Ω = ( 0.60 ( 58 ksi ) 6.01 in.2 ) 2.00 = 105 kips > 59.3 kips o.k. Use 1-in.-diameter ASTM A490-X bolts for uniformity throughout the connection. Use ASTM A36 plate as the connecting element between the gusset and the column. Check block shear rupture on the single plate Because the gusset is thinner than the plate and both are ASTM A36 material, block shear rupture on the gusset plate will be checked using a top edge distance of 1w in. Check block shear rupture on the gusset plate The block shear rupture failure path is assumed to follow the line of bolts and then turn perpendicular and run to the gusset plate edge at the bottom bolt. The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 337 Assume a 1w-in. top edge distance (actual edge distance is 3.25 in.), which allows the gusset plate to be shaped if required. Shear yielding component: Agv = (10.75 in. )(2 in. ) = 5.38 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 5.38 in.2 ) = 116 kips Shear rupture component: Anv = 5.38 in.2 − 3.5 (1z in. + z in. )(2 in. ) = 3.41 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 3.41 in.2 ) = 119 kips Tension rupture component: Ubs = 1 from AISC Specifications Section J4.3 because the connection is uniformly loaded Ant = ⎡⎣2.25 in. − 0.5 (1z in. + z in. ) ⎤⎦ (2 in. ) = 0.844 in.2 ( U bs Fu Ant = 1 ( 58 ksi ) 0.844 in.2 ) = 49.0 kips The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 119 kips + 49.0 kips = 168 kips 0.60 Fy Agv + U bs Fu Ant = 116 kips + 49.0 kips = 165 kips Therefore, Rn = 165 kips. LRFD ϕRn = 0.75 (165 kips ) = 124 kips > 88.9 kips o.k. ASD Rn 165 kips = Ω 2.00 = 82.5 kips > 59.3 kips o.k. The block shear rupture strength for the axial force (normal to the gusset plate edge) can also be checked and will be satisfactory. Therefore, w-in.-thick ASTM A36 plate is sufficient for the connection plate and 2-in.-thick ASTM A36 plate is sufficient for the gusset plate. 338 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check column web local yielding The concentrated force, which is the resultant force from the gusset plate, is not near the end of the column. From AISC Specification Section J10.2, use Equation J10-2 because the concentrated force is applied farther from the end of the member than its depth. The length of bearing is assumed to be the length of the connection plate, lb = 12.5 in. LRFD ASD ϕRn = ϕ Fy t w ( 5k + lb ) = 1.00 ( 50 ksi )(1.29 in. ) ⎡⎣5 ( 2.67 in. ) + 12.5 in.⎤⎤⎦ = 1, 670 kips > 76.2 kips o.k. Rn Fy t w = ( 5k + lb ) Ω Ω ( 50 ksi )(1.29 in.) ⎡ = ⎣5 ( 2.67 in. ) + 12.5 in.⎤⎦ 1.50 = 1,110 kips > 50.8 kips o.k. Check column web local crippling From AISC Specification Equation J10-4: LRFD ϕRn = ϕ0.80t w2 ⎡ ⎢1 + 3 ⎛ lb ⎞ ⎜ ⎟ ⎢ ⎝ d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜tf ⎟ ⎝ ⎠ ASD 1.5 ⎤ ⎥ ⎥ ⎦ EFyw t f tw 1.5 ⎡ ⎛ 12.5 in.⎞ ⎛ 1.29 in. ⎞ ⎤ 2 = 0.75 ( 0.80 )(1.29 in. ) ⎢1 + 3 ⎜ ⎟ ⎥ ⎟⎜ ⎢⎣ ⎝ 16.7 in.⎠ ⎝ 2.07 in.⎠ ⎥⎦ × ( 29,000 ksi ) ( 50 ksi ) ( 2.07 in.) 1.29 in. = 3, 210 kiips > 76.2 kips o.k. 1.5 ⎤ ⎡ Rn 0.80t w2 ⎢ ⎛ lb ⎞ ⎛ t w ⎞ ⎥ EFyw tf = 1 + 3⎜ ⎟ ⎜ ⎟ Ω Ω ⎢ tw ⎝ d ⎠ ⎜⎝ t f ⎟⎠ ⎥ ⎣ ⎦ 2 1.5 0.80 (1.29 in. ) ⎡ ⎛ 12.5 in. ⎞ ⎛ 1.29 in. ⎞ ⎤ ⎢1 + 3 ⎜ = ⎟ ⎥ ⎟⎜ 2.00 ⎢⎣ ⎝ 16.7 in. ⎠ ⎝ 2.07 in. ⎠ ⎥⎦ × ( 29, 000 ksi )( 50 ksi )( 2.07 in.) 1.29 in. = 2,140 kiips > 50.8 kips o.k. Check column shear Check the column for the limit state of shear yielding (on cross section at beam flange elevation) using AISC Specification Sections G1 and G2. The area of the web is: Agv = ( 16.7 in.) (1.29 in. ) = 21.5 in.2 The available shear strength is: LRFD ASD ϕvR n = ϕ v 0.60 Fy Agv ( = 1.00 ( 0.60 )( 50 ksi ) 21.5 in.2 = 645 kips > 76.2 kips o.k. ) Rn 0.60 Fy Agv = Ωv Ωv = ( 0.60 ( 50 ksi ) 21.5 in.2 ) 1.50 = 430 kips > 50.8 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 339 Check weld rupture at column flange From AISC Manual Equation 8-2: LRFD D= = ( Rn 1.5 1.392l 1.0 + 0.50 sin θ ASD D= ) 117 kips (1.392 kip/in.)( 2 ) (12.5 in.) (1.0 + 0.50 sin 1.5 = 2.67 sixteenths 40.7° ) = ( Rn 0.928l 1.0 + 0.50 sin1.5 θ ) 78.1 kips ( 0.928 kip/in.)( 2 ) (12.5 in.) (1.0 + 0.50 sin1.5 40.7° ) = 2.66 sixteenths Beam-to-Column Connection The required strengths of the connection are: LRFD ASD Required normal strength: Hu = 76.2 kips Required normal strength: Ha = 50.8 kips Required shear strength: Vu = 77.5 kips +70.0 kips Required shear strength: Va = 51.7 kips + 46.7 kips = 148 kips = 98.4 kips Assume shear plate is 2-in. thick. Check block shear rupture on single plate Determine the block shear rupture strength using AISC Specification Equation J4-5 assuming the rupture to occur is due to the shear force. The controlling failure path is assumed to follow the bolt line nearest to the column flange to the bottom row of bolts and then run perpendicular to the plate edge connected to the beam. Shear yielding component: Agv = ( 9.00 in. + 32 in. )(2 in. ) = 6.25 in.2 ( 0.60 Fy Agv = 0.60 ( 36 ksi ) 6.25 in.2 ) = 135 kips Shear rupture component: Anv = 6.25 in.2 − 3.5 (2 in. ) (1z in. + z in. ) = 4.28 in.2 ( 0.60 Fu Anv = 0.60 ( 58 ksi ) 4.28 in.2 ) = 149 kips Tensile rupture component: Ant = ⎡⎣5.00 in. − 1.5 (1z in. + z in. ) ⎤⎦ (2 in. ) = 1.66 in.2 ( U bs Fu Ant = 0.5 ( 58 ksi ) 1.66 in.2 ) = 48.1 kips 340 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 The available strength for the limit state of block shear rupture is: 0.60 Fu Anv + U bs Fu Ant = 149 kips + 48.1 kips = 197 kips 0.60 Fy Agv + U bs Fu Ant = 135 kips + 48.1 kips = 183 kips Therefore, Rn = 183 kips. LRFD ASD Rn 183 kips = Ω 2.00 = 91.5 kips < 98.4 kips n.g. ϕRn = 0.75 (183 kips ) = 137 kips < 148 kips n.g. Try w-in.-thick ASTM A36 plate: Try w-in.-thick ASTM A36 plate: ⎛ w in. ⎞ ϕRn = (137 kips) ⎜ ⎟ ⎝ 2 in. ⎠ = 206 kips > 148 kips o.k. Rn ⎛ w in. ⎞ = ( 91.5 kips ) ⎜ ⎟ Ω ⎝ 2 in. ⎠ = 137 kips > 98.4 kips o.k. A conservative way to check block shear rupture for both shear and axial forces is to treat the resultant as a shear force. The resultant is: LRFD Ru = ASD ( 76.2 kips )2 + (148 kips )2 Ra = = 166 kips < 206 kips o.k. ( 50.8 kips )2 + ( 98.4 kips )2 = 111 kips < 137 kips o.k. Check bolt shear strength The resultant load is determined in the previous calculation. The resultant load angle is: LRFD ASD ⎛ 50.8 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 98.4 kips ⎠ = 27.3° ⎛ 76.2 kips ⎞ θ = tan −1 ⎜ ⎟ ⎝ 148 kips ⎠ = 27.2° Use AISC Manual Table 7-7 for θ = 15°, ex = 4.5, n = 4, Use AISC Manual Table 7-7 for θ = 15°, ex = 4.5, n = 4, and C = 4.53. The required strength per bolt is: and C = 4.53. The required strength per bolt is: Ru C 166 kips = 4.53 bolts = 36.6 kips/bolt Therefore, use 1-in.-diameter ASTM A490-X bolts. From AISC Manual Table 7-1, the available bolt shear strength is: Ra C 111 kips = 4.53 bolts = 24.5 kips/bolt Therefore, use 1-in.-diameter ASTM A490-X bolts. From AISC Manual Table 7-1, the available bolt shear strength is: ϕrnv = 49.5 kips/bolt > ru = 36.9 kips/bolt o.k. rnv = 33.0 kips/bolt > ra = 24.5 kips/bolt o.k. Ω ru = ra = AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 341 Check bolt bearing on the beam web Assuming that deformation at the bolt hole is a design consideration, use AISC Specification Equation J3-6a to determine the bearing and tearout strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) The available strength is determined as follows: LRFD Tearout strength based on bolt spacing: ASD Tearout strength based on bolt spacing: ϕ1.2lc tFu = 0.75 (1.2 ) ( 3.00 in. − 1z in. )( 0.585 in. )( 65 ksi ) 1.2lc tFu Ω = 1.2 ( 3.00 in. − 1z in. )( 0.585 in. )( 65 ksi ) 2.00 = 44.2 kips/bolt Tearout strength based on edge distance: = 66.3 kips/bolt Tearout strength based on edge distance: ϕ1.2lc tFu = 0.75 (1.2 ) ⎡⎣2.25 in. − 0.5 (1z in. ) ⎤⎦ × ( 0.585 in. )( 65 ksi ) = 58.8 kips/bolt Bearing strength: ϕ2.4dtFu = 0.75 ( 2.4 ) (1.00 in. )( 0.585 in. )( 65 ksi ) ⎧⎪1.2 ⎡2.25 in. − 0.5 (1z in. ) ⎤⎦ ⎪⎫ 1.2lc tFu Ω = ⎨ ⎣ ⎬ 2.00 ⎩⎪× ( 0.585 in. )( 65 ksi ) ⎭⎪ = 39.2 kips/bolt Bearing strength: 2.4 dtFu Ω = 2.4 (1.00 in. )( 0.585 in. )( 65 ksi ) 2.00 = 68.4 kips/bolt = 45.6 kips/bolt Note that since 58.8 kips/bolt, LRFD (39.2 kips/bolt, ASD) > 36.9 kips/bolt, LRFD (24.5 kips/bolt, ASD), all bolts are fully effective in bearing in carrying their shear load. If ϕrn edge were less than the bolt shear strength of 49.5 kips, LRFD (33.0 kips, ASD), the edge bolts would have a reduced strength and the reduced strength value would have to be used for all of the bolts in order to satisfy the basis of AISC Manual Table 7-8. The bolt shear controls the strength of the bolt group, thus: LRFD ASD ϕRn = ϕrnC Rn rn C = Ω Ω = ( 33.0 kips/bolt )( 4.53 bolts ) = ( 49.5 kips/bolt )( 4.53 bolts ) = 224 kips > 166 kips o.k. = 149 kips > 111 kips o.k. Check bolt bearing on the single plate Determine the available bearing and tearout strength of the single plate at the beam-to-column connection. With 3-in. bolt spacing and 1w-in. edge distance, use AISC Manual Tables 7-4 and 7-5: LRFD ASD Based on bolt spacing: Based on bolt spacing: ϕrn = (101 kip/in. )( w in. ) rn = ( 67.4 kip/in. )( w in. ) Ω = 50.6 kips/bolt = 75.8 kips/bolt Based on edge distance: ϕrn edge = ( 63.6 kip/in. )( w in. ) (by interpolation) = 47.7 kips/bolt > 36.9 kips o.k. Based on edge distance: rn edge Ω = ( 42.4 kip/in. )( w in. ) (by interpolation) = 31.8 kips/bolt > 24.5 kips o.k. 342 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check shear yielding and shear rupture on single plate The available shear yielding strength is determined from AISC Specification Section J4.2(a). The gross area subject to shear is: Agv = ( w in. ) ⎡⎣9.00 in. + 2 (1w in. ) ⎤⎦ = 9.38 in.2 LRFD ASD ϕRn = ϕ0.60 Fy Agv ( = 1.00 ( 0.60 )( 36 ksi ) 9.38 in.2 ) = 203 kips > 148 kips o.k. Rn = 0.60 Fy Agv Ω Ω ( = 0.60 ( 36 ksi ) 9.38 in.2 ) 1.50 = 135 kips > 98.4 kips o.k. The available shear rupture strength is determined from AISC Specification Section J4.2(b). The net area subject to shear is: Anv = 9.38 in.2 − 4 (1z in. + z in. )( w in. ) = 6.00 in.2 LRFD ϕRn = ϕ0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 58 ksi ) 6.00 in.2 ) = 157 kips > 148 kips o.k. Rn 0.60 Fu Anv = Ω Ω ( = 0.60 ( 58 ksi ) 6.00 in.2 ) 2.00 = 104 kips > 98.4 kips o.k. Use a w-in.-thick ASTM A36 plate. Check tension yielding and tension rupture on the single plate When the brace is in compression, the beam-to-column axial force results in tension on the single plate. The available tensile yielding strength of the plate is determined from AISC Specification Section J4.1(a): LRFD ASD = 0.90 ( 36 ksi )( w in. ) ⎡⎣9.00 in. + 2 (1w in. ) ⎤⎦ = 304 kips > 76.2 kips o.k. Rn Fy Ag = Ω Ω 36 ( ksi )( w in.) ⎡⎣9.00 in. + 2 (1w in.)⎤⎦ = 1.67 = 202 kips > 50.8 kips o.k. ϕRn = ϕFy Ag The available tensile rupture strength of the plate is determined from AISC Specification Section J4.1(b), with Ae = An: LRFD ϕRn = ϕFu Ae = 0.75 ( 58 ksi )( w in. ) ⎡⎣12.5 in. − 4 (1z in. + z in. ) ⎤⎦ = 261 kips > 76.2 kips o.k. ASD Rn Fu Ae = Ω Ω ( 58 ksi )( w in.) ⎡⎣12.5 in. − 4 (1z in. + z in.)⎤⎦ = 2.00 = 174 kips > 50.8 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 343 Therefore, use a w-in.-thick ASTM A36 plate for the beam-to-column single-plate connection. Check beam shear From AISC Manual Table 3-6, the available shear strength of the beam is: LRFD ASD ϕvV n = 298 kips > 148 kips o.k. Vn = 199 kips > 98.7 kips o.k. Ωv Check compressive strength of single plate When the brace is in tension, the beam-to-column axial force is compressive. From AISC Specification Section J4.4, the available compressive strength of the plate, based on buckling between the weld to the column and the first line of bolts, is: LRFD ASD KL 0.65 ( 3 in. ) = r ⎛ w in. ⎞ ⎜ ⎟ ⎝ 12 ⎠ = 9.00 < 25 KL 0.65 ( 3 in. ) = r ⎛ w in. ⎞ ⎜ ⎟ ⎝ 12 ⎠ = 9.00 < 25 Therefore: Therefore: Pn Fy Ag = Ω Ω Fy t p l = Ω ( 36 ksi )( w in. )(12.5 in.) = 1.67 = 202 kips > 50.8 kips o.k. ϕPn = ϕ Fy Ag = ϕFy t p l = 0.90 ( 36 ksi )( w in. )(12.5 in. ) = 304 kips > 76.2 kips o.k. Check beam web local yielding and web local crippling Check in the same manner as for the gusset-to-column connection. Design weld at column flange Determine the weld size using AISC Manual Equation 8-2 and incorporating the directional weld strength provided in AISC Specification Section J2.4: LRFD D= = ( Ru 1.5 1.392l 1.0 + 0.50 sin θ ASD D= ) 166 kips (1.392 kip/in.)( 2 ) (12.5 inn.) (1.0 + 0.50 sin = 4.13 sixteenths Therefore, use a c--in. fillet weld. 1.5 27.2° ) = ( Ra 0.928l 1.0 + 0.50 sin1.5 θ ) 111 kips ( 0.928 kip/in.)( 2 ) (12.5 inn.) (1.0 + 0.50 sin1.5 27.3° ) = 4.14 sixteenths Therefore, use a c--in. fillet weld. The completed design is shown in Figure 6-9. 344 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 DISCUSSION: COMPARISON OF HIGH- AND LOW-SEISMIC CONNECTIONS The connection designs shown in Figures 6-1 and 6-9 are subjected to exactly the same loads from wind and seismic events. In the design of the connection shown in Figure 6-1, the requirements of the AISC Seismic Provisions are invoked, in addition to those of the AISC Specification. In the design of the connection shown in Figure 6-9, only the AISC Specification requirements are used. The effect of the Seismic Provisions can clearly be seen when these two figures are compared. If the Seismic Provisions are required because of building type or usage, system choice, or because of owner preference, all parties should be aware of the cost involved. The requirements of the Seismic Provisions have been known to increase the structural steel cost by 30 to 40%. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 345 346 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 APPENDIX A Derivation and Generalization of the Uniform Force Method When properly applied, the uniform force method provides a standardized procedure to arrive at a force distribution that is consistent with the boundary conditions and will satisfy equilibrium. It is also thought to result in a force distribution that will produce the most economical connection. However, in some cases it is not possible to satisfy all of the constraints inherent in the uniform force method. In such instances, it is useful to have a generalized method that can be utilized to obtain consistent force distributions. The uniform force method produces connection interfaces free of moments by placing constraints on the connection geometry and the forces distributed to the connection interfaces. A force distribution is produced such that the line of action of the gusset-to-column force passes through the column control point, the line of action of the gusset-to-beam force passes through the beam control point, and the gussetto-column, gusset-to-beam, and the brace forces all intersect at a common point. A.1 A.1.1 Beam Control Point Three forces (see Figure A-1) are applied to the beam: H, Vb2 + H c2 , and Vb2 + H b2 , where H = horizontal component of the required axial force Hb = Hub or Hab, required shear force on the gusset-tobeam connection Hc = Huc or Hac, required axial force on the gusset-to-column connection Vb = Vub or Vab, required axial force on the gusset-to-beam connection 2 2 H is assumed to act at the centerline of the beam. Vb + H c acts at the beam-to-column connection located at (ec, 0), and Vb2 + H b2 acts at the beam-to-gusset connection located at [(ec + α), eb]. H and Vb2 + H c2 intersect at (ec, 0); therefore Vb2 + H b2 GENERAL METHOD The following is a derivation of the method along with several modifications to ease its use and expand its scope of applicability. must also pass through the coordinates (ec, 0). This means that the point (ec, 0) must be the location of the beam control ( eb − 0 ) eb Vb = = . point V (point A of Figure A-1) and H b ( ec + α − ec ) α Fig. A-1. Free body diagram of beam. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 347 A.1.2 Gusset Control Point Three forces are applied to the gusset (see Figure A-2): P, Vb2 + H b2 , and Solving for y and substituting Vc2 + H c2 . P acts through the work point at the angle θ. We know the slope of the line of action of Vb eb = . Vb2 + H b2 from the discussion above is Hb α The equations for the line of action of the three vectors are determined as follows: y= y − ( eb + β ) = y= Solving for y: x tan θ (A-1) Vc ( x − ec ) Hc Substituting mc = Vc ( x − ec ) + eb + β Hc (A-3a) Setting Equations A-1 and A-3a equal and solving for x results in: Vb ⎡ x − ( ec + α ) ⎤⎦ Hb ⎣ x = mc ( x − ec ) + eb + β tan θ V CL column P α ec H Column control point V c2 + Hc2 β V b2 + Hb2 B W.P. y (A-3) Vc into Equation A-3 results in: Hc y = mc ( x − ec ) + eb + β For Vb2 + H b2 : y − eb = (A-2) Solving for y: 1 ( x − 0) tan θ y= eb ( x − ec ) α For Vc2 + H c2 : For force P: y−0 = Vb eb = : Hb α eb θ x A Beam control point Intersection Fig. A-2. Free body diagram of gusset. 348 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 CL beam Solving for x: The slope of Vb2 + H c2 is: ⎛ mc ec − eb − β⎞ x=⎜ ⎟ tan θ ⎝ mc tan θ − 1 ⎠ (A-4) Setting Equations A-2 and A-3a equal and solving for x results in: eb ( x − ec ) = mc ( x − ec ) + eb + β α Solving for x: x= α ( mc ec − β − eb ) − eb ec mc α − eb (A-5) Setting Equations A-4 and A-5 equal and solving for mc: α ( mc ec − β − eb ) − eb ec ⎛ mc ec − eb − β ⎞ ⎜ ⎟ tan θ = m − θ mc α − eb tan 1 ⎝ c ⎠ β ec ⎧⎪ eb ⎨ ⎩⎪ β ⎡ ( eb + β ) tan θ − ec ⎤ ⎫⎪ ⎢1 − ⎥ + 1⎬ α ⎣ ⎦ ⎭⎪ = eb eb tan θ + β tan θ − ec −0 α 0 + ec eb ( eb + β ) tan θ − ec α ec (A-10) We have now derived all of the requisite geometry to ensure that there are no moments at any connection interfaces. It may be noted that the relationship between α and β has not yet been established. The relationship given in the body of this Design Guide and from AISC Manual Equation 13-1 is α = (eb + β) tan θ − ec. This relationship is not necessary to ensure that moments do not exist at the interfaces and is the result of using the point (0, eb) as a control point. A.1.3 Determination of Forces Having established the requisite geometry, the forces can now be obtained. The easiest place to start is with the equal and opposite horizontal forces applied at the column. To satisfy equilibrium, Hc must be: Solving for mc: mc = Vb = Hc (A-6) Hc = The location of the gusset control point is determined as follows. Pec cos θ eb + β (A-11) Setting Equation A-1 equal to Equation A-2: eb x = ( x − ec ) tan θ α V c2 + Hc2 Solving for x, where x = xgcp: xgcp = eb ec tan θ eb tan θ − α (A-7) Column control point Using this value in Equation A-1, solve for the y-coordinate: ygcp = eb ec eb tan θ − α B (A-8) V b2 + Hc2 A Now find the location of the column control point, point B (see Figure A-3): The y-coordinate of the intersection of Vc2 + H c2 with the centerline of the column can be found by substituting x = 0 into Equation A-3: y = eb eb tan θ + β tan θ − ec α (A-9) Beam control point V Fig. A-3. Free body diagram of column. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 349 From this, determine the vertical force applied at the beamto-column connection: ⎡ eb ( eb + β ) tan θ − ec ⎤ Vb = H c ⎢ ⎥ α ⎣ ec ⎦ ⎧⎪ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎫⎪ = P⎨ ⎣ ⎬ α ( eb + β ) ⎭⎪ ⎩⎪ (A-12) The remaining forces are simply found as: r= ∑ Fx = 0 Vc = P cos θ − Vb (A-13) and ∑ Fy = 0 H b = P sin θ − H c (A-14) It should be noted that since there is no defined relationship between α and β, the equations above will produce a force distribution free of moments for any connection geometry. This eliminates the need to include the variables α and β in this procedure, where α is the actual distance from the face of the column flange or web to the centroid of the gusset-tobeam connection, and β is the actual distance from the face of the beam flange to the centroid of the gusset-to-column connection. However, there still may be a need to redistribute the vertical load between the beam-to-column and gusset-to-column connections. This redistribution will obviously result in a moment. Because the beam-to-gusset connection will generally be welded and the gusset-to-column connection is generally bolted, the moment will be assumed to act at the stiffer beam-to-gusset interface. The moment can be calculated as: Mb = H b eb − Vb α (A-15) where Vb = Vb − ΔVb (A-16) This completes the derivation of the uniform force method in its most general form. If the relationship α = (eb + β)tan θ − ec is imposed, the more familiar equations are obtained: Vc = β P r (Manual Eq. 13-2) Hc = ec P r (Manual Eq. 13-3) Vb = eb P r (Manual Eq. 13-4) Hb = α P r (Manual Eq. 13-5) ( α + ec )2 + ( eb + β )2 (Manual Eq. 13-6) A.1.4 Accounting for the Beam Reaction The uniform force method is intended to optimize the connection design by eliminating moments at the connection interfaces. In most presentations of the uniform force method, the beam reaction is neglected. It has therefore been standard practice to deliver all of the beam reaction to the face of the column through the beam-to-column connection. The eccentricity measured from the center of the column to the beam connection is ignored, as is common practice for a beam without vertical bracing. In most instances, this treatment is acceptable. However, in some instances redistributing some of the beam reaction to the gusset-to-column connection can greatly increase economy. This is especially true when the beam reaction is applied eccentric to the column. As an example, buildings are often designed with moment frames along the strong axis of the columns and vertical bracing along the weak axis of the columns. Since moment connections will often require the use of stiffeners, it can become difficult to erect the weak-axis bracing. One solution is to use extended single-plate connections, sometimes called extended shear tabs, to connect the braces. The uniform force method is particularly well suited to analyzing this type of connection. Improved economy is obtained by assuming ec is the distance from the work-point to the center of the bolt group in the extended shear tab. This is contrary to the typical assumption that ec is equal to zero, when vertical bracing frames to a column web. By analyzing the connection in this way, the eccentricity that would normally be resisted by the bolt group can now be taken as a couple, Hc. The uniform force method can be applied to the connection, with the forces normally associated with the column now associated with the single-plate connection. As long as the forces derived from the uniform force method are used without the introduction of ΔVb, no moments will need to be considered in any of the connection interfaces. However, the single-plate connection must now be checked to transfer Hc as a horizontal shear and also an internal moment that will not appear at any of the interfaces. 350 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 This procedure will eliminate moments at the connections due to the brace force, but a moment will still have to be resisted by the bolts at the beam-to-column connection. It is impossible to eliminate this moment entirely, since the beam reaction does not have a line of action that passes through the work-point. However, it is possible to move the moment from the beam-to-column connection to the beam-togusset connection. It is usually more economical to resist this moment at the beam-to-gusset connection since the beamto-gusset connection is usually welded and is not limited by the depth of the beam. In order to accomplish this, some of the beam reaction is distributed to the gusset-to-column connection through the beam-to-gusset connection. Though a model can be established that satisfies equilibrium without redistributing a portion of the vertical beam end reaction into the gusset, the model presented here has the advantage that it will result in no moment in the single plate at a section cut at the elevation of the beam flange. In order to avoid coping of the beam, the single plate is typically necked-down local to the beam flange, and in some instances, as shown in Figure A-4, stiffeners may intersect the tab plate requiring welding to transfer the Hc force. Due to the reduced width of the single plate at these locations, additional moment can lead to larger welds and/or thicker single plates. The following resulting force distribution can be superimposed on the forces resulting from the uniform force method: Gusset-to-Column Connection: ⎛ ec ⎞ H cR = R ⎜ ⎟ ⎝ eb + β ⎠ (A-17) ⎛ β ⎞ VcR = R ⎜ ⎟ ⎝ eb + β ⎠ (A-18) Gusset-to-Beam Connection: ⎛ ec ⎞ H bR = − R ⎜ ⎟ ⎝ eb + β ⎠ (A-19) ⎛ eb ⎞ VbR = R ⎜ − 1⎟ ⎝ eb + β ⎠ (A-20) M bR = − R ( ec ) (A-21) Beam-to-Column Connection: ⎛ β ⎞ VbcR = R ⎜1 − ⎟ e b + β⎠ ⎝ (A-22) H bcR = H cR (A-23) Example A.1—Vertical Brace-to-Column Web Connection Using an Extended Single Plate Given: Verify the bracing connection design shown in Figure A-4 using the general uniform force method. The loading, member sizes, material strengths, bolt size and type, hole size, and connection element sizes are as given in Figure A-4. Use 70-ksi electrodes. The required strengths are: LRFD Brace: ±Pu = 225 kips Beam: Vu = 30 kips ASD Brace: ±Pa = 150 kips Beam: Va = 20 kips Solution: From AISC Manual Tables 2-4 and 2-5, the material properties are as follows: ASTM A992 Fy = 50 ksi Fu = 65 ksi ASTM A36 Fy = 36 ksi Fu = 58 ksi AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 351 ASTM A500 Grade B Fy = 46 ksi Fu = 58 ksi From AISC Manual Tables 1-1 and 1-8, the member geometric properties are as follows: Beam W16×26 Ag = 7.68 in.2 tw = 0.250 in. d = 15.7 in. bf = 5.50 in. tf = 0.345 in. d = 14.8 in. bf = 15.5 in. tf = 1.09 in. h / tw = 16.8 r = 3.94 in. tdes = 0.291 in. kdes = 0.747 in. Column W14×145 tw = 0.680 in. Brace HSS10×10×c Ag = 11.1 in.2 From AISC Specification Table J3.3, the hole dimensions for w-in.-diameter bolts are as follows: Brace, column and plates Standard: dh = m in. diameter Fig. A-4. Extended single-plate connection example. 352 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Brace-to-Gusset Connection Design the weld between the brace and the gusset The weld will be assumed to be 4 in. This is done to maintain a single-pass c-in. fillet weld, while accounting for the gap between the tube wall and the gusset plate. Using AISC Manual Equation 8-2, and solving for l, the required length of 4-in. weld for four lines of weld is: LRFD l= = ASD Pu 4 (1.392 kip/in. ) D l= 225 kips 4 (1.392 kip/in. ) ( 4 sixteenths ) = 10.1 in. = Pa 4 ( 0.928 kip/in. ) D 150 kips 4 ( 0.928 kip/in. ) ( 4 sixteenths ) = 10.1 in. Therefore, use an 11-in. fillet weld. Check shear rupture on the HSS wall local to the weld The available shear rupture strength is determined from AISC Specification Section J4.2, Equation J4-4: LRFD ASD ϕRn = 4 (ϕ 0.60 Fu lt des ) = 4 ⎡⎣( 0.75 )( 0.60 )( 58 ksi )(11.0 in.) ( 0.291 in. ) ⎤⎦ = 334 kips > 225 kips o.k. Rn ⎛ 0.60 Fu lt des ⎞ = 4⎜ ⎟ Ω ⎝ 2.00 ⎠ ⎡ 0.60 ( 58 ksi )(11.0 in.)( 0.291 in. ) ⎤ = 4⎢ ⎥ 2.00 ⎣ ⎦ = 223 kips >150 kips o.k. Check tension yielding on the brace The available tensile yielding strength is determined from AISC Specification Section D2, Equation D2-1. Alternatively, AISC Manual Table 5-5 may be used: LRFD ASD ϕt Pn = ϕt Fy Ag = 0.90 ( 46 ksi ) 11.1 in.2 ( ) = 460 kips > 225 kips o.k. Pn Fy Ag = Ωt Ωt = ( 46 ksi ) (11.1 in.2 ) 1.67 = 306 kips > 150 kips o.k. Check tensile rupture on the brace For preliminary calculations, assume a 2-in.-thick gusset plate. Assume that the slot in the brace is 8 in. wider than the gusset plate. The effective net area is: Ae = AnU(Spec. Eq. D3-1) AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 353 where An = Ag − 2 t g + 8 in. t des ( ) = 11.1 in. − 2 (2 in. + 8 in. )( 0.291 in. ) 2 = 10.7 in.2 The shear lag factor, U, is determined from AISC Specification Table D3.1, Case 6, with a single concentric gusset plate, assuming l ≥ H: x= = B2 + 2 BH 4(B + H ) ( B = H = 10.0 in.) (10.0 in.)2 + 2 (10.0 in.)(10.0 in.) 4 (10.0 in. + 10.0 in. ) = 3.75 in. The equation in AISC Specification Table D3.1, Case 6, with a single concentric gusset plate can be used because the length of weld, l, exceeds the wall of the HSS, H: ⎛x⎞ U = 1− ⎜ ⎟ ⎝l⎠ ⎛ 3.75 in. ⎞ = 1− ⎜ ⎟ ⎝ 11.0 in. ⎠ = 0.659 Therefore, the effective net area is: Ae = AnU ( = 10.7 in.2 (Spec. Eq. D3-1) ) ( 0.659) = 7.05 in.2 From AISC Specification Section D2(b), Equation D2-2, the available strength for the limit state of tensile rupture is: LRFD ϕRn = ϕFu Ae ( ASD = 0.75 ( 58 ksi ) 7.05 in.2 ) = 307 kips > 225 kips o.k. Rn Fu Ae = Ω Ω (58 ksi ) ( 7.05 in. ) 2 = 2.00 = 205 kips > 150 kips o.k. Proceed with a 2-in.-thick ASTM A36 gusset plate and check block shear, tension on the Whitmore section, and buckling of the gusset plate. Check block shear rupture on the gusset plate The available strength for the limit state of block shear rupture is given in AISC Specification Section J4.3 as follows: Rn = 0.60Fu Anv + UbsFu Ant ≤ 0.60Fy Agv + UbsFu Ant(Spec. Eq. J4-5) 354 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Shear yielding component: Agv = Anv = 2t g l = 2 (2 in.)(11.0 in. ) = 11.0 in.2 ( ) ( ) 0.60 Fy Agv = 0.60 ( 36 ksi ) 11.0 in.2 = 238 kips Shear rupture component: 0.60 Fu Anv = 0.60 ( 58 ksi ) 11.0 in.2 = 383 kips Tension rupture component: Ubs is 1 from AISC Specification Section J4.3 because the connection is uniformly loaded. Agt = Ant = tg B = (2 in. )(10.0 in. ) = 5.00 in.2 ( U bs Fu Ant = 1 ( 58 ksi ) 5.00 in.2 ) = 290 kips The available strength for the limit state of block shear rupture is determined as follows: 0.60 Fu Anv + U bs Fu Ant = 383 kips + 290 kips = 673 kips 0.60 Fy Agv + U bs Fu Ant = 238 kips + 290 kips = 528 kips Therefore, Rn = 528 kips. LRFD ASD ϕRn = 0.75 ( 528 kips ) = 396 kips > 225 kips o.k. Rn 528 kips = Ω 2.00 = 264 kips > 150 kips o.k. Check the gusset plate for tension yielding on the Whitmore section From AISC Manual Part 9, the width of the Whitmore Section is: lw = 10.0 in. + 2 (11.0 in. ) tan 30° = 22.7 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 355 From AISC Specification Section J4.1(a), the available tensile yielding strength of the gusset plate is: LRFD ASD ϕRn = ϕ Fy A w Rn Fy Aw = Ω Ω ( 36 ksi )( 22.7 in.)(2 in.) = 1.67 = 245 kips > 150 kips o.k. = 0.90 ( 36 ksi )( 22.7 in. )(2 in. ) = 368 kips > 225 kips o.k.. Check the gusset plate for compression buckling on the Whitmore Section Assume K = 0.65 and the unbraced length of the gusset plate is 19.6 in.: KL 0.65 (19.6 in. ) = r 2 in. 12 = 88.3 Because KL/r > 25, use AISC Manual Table 4-22 to determine the available critical stress for the 36-ksi gusset plate. Then the available strength can be determined from AISC Specification Sections E1 and E3. With Aw = (22.7 in.)(2 in.) = 11.4 in.2: LRFD ASD ϕc Fcr = 21.5 ksi Fcr Ωc = 14.3 ksi ϕPn Pn Ω = ϕc Fcr Aw ( = ( 21.5 ksi ) 11.4 in.2 ) = 245 kips > 225 kips o.k. = Fcr Aw Ωc ( = (14.3 ksi ) 11.4 in.2 ) = 163 kips > 150 kips o.k. The remaining checks will assume a gusset plate that is 2 in. thick, 2 ft 11 in. long, and 2 ft 4 in. deep. Connection Interface Forces The forces at the gusset-to-beam and gusset-to-column interfaces are determined using the general case of the UFM as discussed in AISC Manual Part 13 and discussed in this Appendix. The geometric properties of the connection configuration given in Figure A-5 are as follows: Assume ec is equal to the distance from the face of the support to the center of the bolt group from the gusset to the column: ec = 104 in. d 2 15.7 in. = 2 = 7.85 in. eb = ⎛ 12 ⎞ θ = tan −1 ⎜ ⎟ ⎝ 78 ⎠ = 59.3° 356 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Because this approach assumes that ec = 104 in. rather than 0 in., then α is calculated as: 35 in. − 1w in. 2 = 15.75 in. α= β = 74 in. + 72 in. = 14.75 in. Using a modified uniform force method approach, eccentricity from both the brace force and the beam reaction will be resisted by equal and opposite horizontal forces applied at the beam-to-column and the gusset-to-column connections. Fig. A-5. Admissible force field extended single-plate connection example of Figure A-4. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 357 Gusset-to-column connection forces LRFD ASD ⎛ ec cos θ ⎞ ⎛ ec ⎞ Huc + HucR = Pu ⎜ ⎟ + Rub ⎜ ⎟ + β e ⎝ b ⎠ ⎝ eb + β ⎠ ⎡ (10.25 in. ) cos 59.3° ⎤ = ( 225 kips ) ⎢ ⎥ ⎣ 7.85 in. + 14.75 in. ⎦ 10.25 in. ⎞ ⎛ + ( 30 kips ) ⎜ ⎟ ⎝ 7.85 in. + 14.75 in. ⎠ = 65.7 kips ⎛ ec cos θ ⎞ ⎛ ec ⎞ H ac + H acR = Pa ⎜ ⎟ + Rab ⎜ ⎟ + β e ⎝ b ⎠ ⎝ eb + β ⎠ ⎡ (10.25 in. ) cos 59.3° ⎤ = (150 kips ) ⎢ ⎥ ⎣ 7.85 in. + 14.75 in. ⎦ 10.25 in. ⎛ ⎞ + ( 20 kips ) ⎜ ⎟ ⎝ 7.85 in. + 14.75 in. ⎠ = 43.8 kips ⎧⎪ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎫⎪ ⎛ β ⎞ Vuc + VucR = Pu ⎨cos θ − ⎣ ⎬ + Rub ⎜ ⎟ e α β + ( ) b ⎝ eb + β ⎠ ⎩⎪ ⎭⎪ ⎧ = ( 225 kips ) ⎨ cos 59.3° ⎩ ⎧⎪ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎫⎪ ⎛ β ⎞ Vac + VacR = Pa ⎨cos θ − ⎣ ⎬ + Rab ⎜ ⎟ α β + e ( ) b ⎝ eb + β ⎠ ⎩⎪ ⎭⎪ − ( 7.85 in. ) ⎡⎣( 7.85 in. + 14.75 in. ) sin 59.3° − (10.25 in. ) cos 59.3°⎤⎦ ⎫⎪ ⎬ ⎪⎭ (15.75 in.) ( 7.85 in. + 14.75 in.) 14.75 in. ⎛ ⎞ + ( 30 kips ) ⎜ ⎟ 7 . 85 in. + 14 . 75 in. ⎝ ⎠ = 64.1 kips ⎧ = (150 kips ) ⎨cos 59.3° ⎩ ( 7.85 in.) ⎣⎡( 7.85 in. + 14.75 in.) sin 59.3° − (10.25 in.) cos 59.3°⎤⎦ ⎫⎪ − ⎬ (15.75 in. ) ( 7.85 in. + 14.75 in.) ⎭ ⎛ ⎞ 14.75 kips + ( 20 kips ) ⎜ ⎟ kips kips 7 . 85 + 14 . 75 ⎝ ⎠ = 42.7 kips Gusset-to-beam connection forces LRFD ASD ⎛ ⎛ ec ⎞ ec cos θ ⎞ Hub + HubR = Pu ⎜ sin θ − ⎟ − Ru ⎜ ⎟ + e β b ⎝ ⎠ ⎝ eb + β ⎠ ⎛ ⎛ ec ⎞ ec cos θ ⎞ H ab + H abR = Pa ⎜ sin θ − ⎟ − Ra ⎜ ⎟ + e β b ⎝ ⎠ ⎝ eb + β ⎠ ⎧⎪ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎫⎪ ⎛ β ⎞ Vub + VubR = Pu ⎨ ⎣ ⎬ − Ru ⎜ ⎟ e α β + (b ) ⎝ eb + β ⎠ ⎩⎪ ⎭⎪ ⎧⎪ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎫⎪ ⎛ β ⎞ Vab + VabR = Pa ⎨ ⎣ ⎬ − Ra ⎜ ⎟ e α β + (b ) ⎝ eb + β ⎠ ⎩⎪ ⎭⎪ ⎡ (10.25 in. ) cos 59.3° ⎤ = ( 225 kips ) ⎢sin 59.3° − ⎥ 7.85 in. + 14.75 in. ⎦ ⎣ 10.25 in. ⎛ ⎞ − ( 30 kips ) ⎜ ⎟ ⎝ 7.85 in. + 14.75 in. ⎠ = 128 kips = ( 225 kips ) ⎡ (10.25 in. ) cos 59.3° ⎤ = (150 kips ) ⎢sin 59.3° − ⎥ 7.85 in. + 14.75 in. ⎦ ⎣ 10.25 in. ⎛ ⎞ − ( 20 kips ) ⎜ ⎟ ⎝ 7.85 in. + 14.75 in. ⎠ = 85.2 kips = (150 kips ) ⎧⎪ ( 7.85 in. ) ⎡⎣( 7.85 in. + 14.75 in. ) sin 59.3° − (10.25 in. ) cos 59.3°⎤⎦ ⎫⎪ ⎧⎪ ( 7.85 in. ) ⎡⎣( 7.85 in. + 14.75 in. ) sin 59.3° − (10.25 in. ) cos 59.3°⎤⎦ ⎫⎪ ×⎨ ⎬ ×⎨ ⎬ (15.75 in. ) ( 7.85 in. + 14.75 in. ) (15.75 in. ) ( 7.85 in. + 14..75 in. ) ⎩⎪ ⎭⎪ ⎪⎩ ⎭⎪ in. in. 14 . 75 14 75 . ⎛ ⎞ ⎛ ⎞ − ( 30 kips ) ⎜ − ( 20 kips ) ⎜ ⎟ ⎟ ⎝ 7.85 in. + 14.75 in. ⎠ ⎝ 7.85 in. + 14.75 in. ⎠ = 50.9 kips = 33.9 kips 358 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Mub = α (Vub + VubR ) − eb ( Hub + HubR ) − ec Ru M ab = α (Vab + VabR ) − eb ( H ab + H abR ) − ec Ra = (15.75 in. ) ( 50.9 kips ) − ( 7.85 in. )(128 kips ) = (15.75 in. ) ( 33.9 kips ) − ( 7.85 in. )( 85.2 kips ) − (10.25 in. ) (30 kips) − (10.25 in. ) (20 kips) = −511 kip-in. = −340 kip-in. Determine beam-to-column forces LRFD ⎡ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎤ ⎛ β ⎞ Vub + VubcR = Pu ⎢ ⎣ ⎥ + Ru ⎜ 1 − ⎟ α ( eb + β ) ⎢⎣ ⎥⎦ ⎝ eb + β ⎠ ASD ⎧⎪ eb ⎡( eb + β ) sin θ − ec cos θ ⎤⎦ ⎫⎪ ⎛ β ⎞ Vab + VabcR = Pa ⎨ ⎣ ⎬ + Ra ⎜ 1 − ⎟ α ( eb + β ) ⎝ eb + β ⎠ ⎩⎪ ⎭⎪ = ( 225 kips ) = (150 kips ) ⎧⎪ ( 7.85 in. ) ⎡⎣( 7.85 in. + 14.75 in. ) sin 59.3° − (10.25 in. ) cos 59.3°⎤⎦ ⎫⎪ ⎧⎪ ( 7.85 in. ) ⎡⎣( 7.85 in. + 14.75 in. ) sin 59.3° − (10.25 in. ) cos 59.3°⎤⎦ ⎫⎪ ×⎨ ⎬ ×⎨ ⎬ (15.75 in.) ( 7.85 in. + 14.75 in. ) (15.75 in. ) ( 7.85 in. + 14.75 in. ) ⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭ 14.75 in. 14.75 in. ⎛ ⎞ ⎛ ⎞ + ( 30 kips ) ⎜ 1 − + ( 20 kips ) ⎜ 1 − ⎟ ⎟ 7 . 85 + 14 . 7 . 5 in. + 14 . 75 in. in. 7 85 in. ⎝ ⎠ ⎝ ⎠ = 80.9 kips = 53.9 kips As determined previously: HucR + Huc = 65.7 kips As determined previously: H acR + H ac = 43.8 kips Gusset-to-Beam Connection Check the gusset plate for shear yielding along the beam flange Determine the available shear strength of the gusset plate using AISC Specification Equation J4-3: LRFD ϕVn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi )( 35.0 in. )(2 in. ) = 378 kips > 128 kips o.k. ASD Vn 0.60 Fy Agv = Ω 1.50 0.60 ( 36 ksi )( 35.0 in. )(2 in. ) = 1.50 = 252 kips > 85.2 kips o.k. Check the gusset plate for tensile yielding along the beam flange Determine the available tensile yielding strength of the gusset plate from AISC Specification Section J4.1, Equation J4-1, as follows: LRFD ϕRn = ϕFy Ag = 0.90 ( 36 ksi )( 35.0 in. )(2 in. ) = 567 kips ASD Rn Fy Ag = Ω 1.67 ( 36 ksi )( 35.0 in.)(2 in.) = 1.67 = 377 kips AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 359 LRFD Ru = Vub + ASD 4 Mub Ra = Vab + Lg = 50.9 kips + 4 ( 511 kip-in. ) 4 M ab Lg = 33.9 kips + 35.0 in. = 109 kips < 567 kips o.k. 4 ( 340 kip-in. ) 35.0 in. = 72.8 kipss < 377 kips o.k. Design weld at gusset-to-beam flange connection The strength determination of fillet welds defined in AISC Specification Section J2.4 can be simplified as explained in AISC Manual Part 8 to AISC Manual Equations 8-2a and 8-2b. Also, incorporating the increased strength due to the load angle from AISC Specification Equation J2-5, the required weld size is determined as follows: LRFD ASD Resultant load: Resultant load: R= (109 kips )2 + (128 kips )2 R= = 168 kips = 112 kips Resultant load angle: Resultant load angle: ⎛ Ru ⎞ θ = tan −1 ⎜ ⎟ ⎝ Vu ⎠ ⎛ 109 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 128 kips ⎠ ⎛ Ra ⎞ θ = tan −1 ⎜ ⎟ ⎝ Va ⎠ ⎛ 72.8 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 85.2 kips ⎠ = 40.4° Dreq = = ( 72.8 kips )2 + (85.2 kips )2 = 40.5° Ru ( )( 2 Lg 1.392 kip/in. 1.0 + 0.5 sin1.5 θ ( ) 168 )( )( 2 35.0 in. 1.392 kip/in. 1.0 + 0.5 sin1.5 40.4° Dreq = ) = 1.37 sixteenths = Ra ( )( 2 Lg 0 .928 kip/in. 1.0 + 0.5 sin1.5 θ ( ) 112 )( )( 2 35.0 in. 0.928 kip/in. 1.0 + 0.5 sin1.5 40.5° ) = 1.37 sixteenths Therefore, use a x-in. fillet weld (AISC minimum size). Check beam web local yielding The normal force is applied at a distance α = 15.75 in. from the beam end, which is approximately equal to the depth of the W16×26 beam. Therefore, use AISC Specification Equation J10-3 to determine the available strength due to beam web local yielding: LRFD ( ϕRn = ϕFy t w 2.5k + Lg ) = 1.00 ( 50 ksi )( 0.250 in. ) ⎡⎣2.5 ( 0.747 in.. ) + 35.0 in.⎤⎦ = 461 kips > 109 kips o.k. ASD ( ) Rn Fy t w 2.5k + Lg = Ω 1.50 ( 50 ksi )( 0.250 in.) ⎡⎣2.5 ( 0.747 in.) + 35.0 in.⎤⎦ = 1.50 = 307 kips > 72.8 kips o.k. 360 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check beam web local crippling The normal force is applied at a distance α = 15.75 in. from the beam end, which is greater than d/ 2, where d is the depth of the W16×26 beam. Therefore, determine the beam web local crippling strength from AISC Specification Equation J10-4: LRFD ϕRn = ϕ0.80t w2 ⎡ L ⎢1 + 3 ⎛⎜ g ⎞⎟ ⎜ ⎟ ⎢ ⎝d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜tf ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ ASD EFy t f tw 2⎡ ⎛ 35.0 in.⎞ = 0.75 ( 0.80 )( 0.250 in. ) ⎢1 + 3 ⎜ ⎟ ⎝ 15.7 in.⎠ ⎢⎣ × 0.80t w2 ⎛ 0.250 in. ⎞ ⎜ 0.345 in. ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥⎦ ( 29, 000 ksi )( 50 ksi ) (0.345 in.) 0.250 in. = 272 kips > 109 kips o.k. Rn = Ω = ⎡ L ⎢1 + 3 ⎛⎜ g ⎞⎟ ⎢ ⎝d⎠ ⎣ ⎛ tw ⎞ ⎜ ⎟ ⎜tf ⎟ ⎝ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ EFy t f tw 2.00 1.5 1 2⎡ ⎛ 35.0 in. ⎞ ⎛ 0.250 in. ⎞ ⎤ 0.80 ( 0.250 in. ) ⎢1 + 3 ⎜ ⎟⎜ ⎟ ⎥ 2.00 ⎝ 15.7 in. ⎠ ⎝ 0.345 in. ⎠ ⎥⎦ ⎢⎣ × ( 29, 000 ksi )( 50 ksi ) (0.345 in.) 0.250 in. = 181 kips > 72.8 kips o.k. Beam-to-Column Connection Determine resultant force LRFD 2 2 Rubc = Vubc + Hubc = (80.9 kips )2 + ( 65.7 kips)2 = 104 kips ASD 2 2 Rabc = Vabc + H abc = ( 53.9 kips )2 + ( 43.8 kips )2 = 69.5 kips Design bolts at beam-to-column connection From AISC Manual Table 7-1, the available shear strength for a w-in.-diameter ASTM A325-N bolt is: LRFD ASD ϕrn = 17.9 kips/bolt rn = 11.9 kips/bolt Ω Check bolt bearing on beam web According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) Conservatively assume a 12-in. distance to the edge of the beam: lc = 12 in. − 0.5dh = 12 in. − 0.5 (m in. ) = 1.09 in. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 361 LRFD ASD ϕ1.2lc t w Fu = 0.75 (1.2 )(1.09 in )( 0.250 in.)( 65 ksi ) 1.2lc tFu 1.2 (1.09 in. )( 0.250 in.)( 65 ksi ) = Ω 2.00 = 10.6 kips = 15.9 kips ϕ2.4 db t w Fu = 0.75 ( 2.4 )( w in. )( 0.250 in. )( 65 ksi ) = 21.9 kiips Therefore, ϕrn = 15.9 kips. ϕrn = 17.9 kips/bolt > ϕrn = 15.9 kips/bolt Therefore, the limit state of tearout controls. 2.4 db t w Fu 2.4 ( w in. )( 0.250 in. )( 65 ksi ) = Ω 2.00 = 14.6 kipss Therefore, rn/ Ω = 10.6 kips. rn rn = 11.9 kips/bolt > = 10.6 kips/bolt Ω Ω Therefore, the limit state of tearout controls. Check bolt bearing on single plate Assume a 2-in.-thick tab plate: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) lc = 12 in. − 0.5dh = 12 in. − 0.5 (m in. ) = 1.09 in. LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.09 in. )(2 in. )( 58 ksi ) 1.2lc tFu 1.2 (1.09 in. )(2 in. )( 58 ksi ) = Ω 2.00 = 19.0 kips = 28.4 kips ϕ2.4 db t w Fu = 0.75 ( 2.4 )( w in. )(2 in. )( 58 ksi ) = 39.2 kips Therefore, ϕrn = 28.4 kips. ϕrn = 17.9 kips/bolt < ϕrn = 28.4 kips/bolt Therefore, the limit state of bolt shear controls. 2.4 db t w Fu 2.4 ( w in. )(2 in. )( 58 ksi ) = Ω 2.00 = 26.1 kips Therefore, rn/ Ω = 19.0 kips. rn rn = 11.9 kips/bolt < = 19.0 kips/bolt Ω Ω Therefore, the limit state of bolt shear controls. Because tearout of the beam web controls, the available strength of a bolt is 15.9 kips (LRFD) or 10.6 kips (ASD). Note that since the tearout controlled using the conservative edge distance assumption, a more rigorous analysis could be performed. However, in this case there is no advantage to performing the more rigorous analysis, so the more conservative tearout value will be used. Since ec was assumed to be located 104 in. from the support, and the bolt group is 11w in. from the face of the support, a 12-in. eccentricity must be considered. 362 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD ⎛ Hubc ⎞ θ = tan −1 ⎜ ⎟ ⎝ Vubc ⎠ ⎛ 65.7 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 80.9 kips ⎠ ⎛ H abc ⎞ θ = tan −1 ⎜ ⎟ ⎝ Vabc ⎠ ⎛ 43.8 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 53.9 kips ⎠ = 39.1° = 39.1° From AISC Manual Table 7-8, with θ = 30°, ex = 1.50 in., n = 4, the coefficient C = 6.90. Therefore, the available strength of the bolt group is: LRFD ϕRn = (15.9 kips/bolt )( 6.90 ) = 110 kips > 104 kips o.k. ASD Rn = (10.6 kips/bolt )( 6.90 ) Ω = 73.1 kips > 69.5 kips o.k. Design beam single plate-to-column weld The strength determination of fillet welds defined in AISC Specification Section J2.4 can be simplified as explained in AISC Manual Part 8 to AISC Manual Equations 8-2a and 8-2b, and the required weld size is determined as follows: Note that the column only sees the vertical shear load. Conservatively, a length equal to the length of the bolted portion of the single plate is used. LRFD Vubc 2(1.392 kip/in.)l 80.9 kips = 2 (1.392 kip/in. ) (12 in. ) Dreq = = 2.42 sixteenths ASD Vabc 2(0.928 kip/in.)l 53.9 kips = 2 ( 0.928 kip/in. ) (12 in. ) Dreq = = 2.42 sixteenths Use a x-in. fillet weld. This also satisfies the minimum required weld size from AISC Specification Table J2.4. Check shear strength of the single plate To determine the available shear strength of the single plate, use AISC Specification Section J4.2. For shear yielding of the single plate, use AISC Specification Equation J4-3: LRFD ϕRn = ϕ 0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi )(12.0 in. )(2 in. ) = 130 kips > 80.9 kips o.k. ASD Rn 0.60 Fy Agv = Ω 1.50 0.60 ( 36 ksi )(12.0 in. )(2 in. ) = 1.50 = 86.4 kips > 53.9 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 363 For shear rupture of the single plate, use AISC Specification Equation J4-4: Anv = tp ⎡⎣ L − n ( d h + z ) ⎤⎦ = ( 2 in. ) ⎡⎣12.0 in. − 4 ( 0.875 in. ) ⎤⎦ = 4.25 in.2 LRFD ASD ϕRn = ϕ0.6 Fu Anv ( = 0.75 ( 0.6 )( 58 ksi ) 4.25 in. 2 ) = 111 kips > 80.9 kips o.k. Rn 0.6 Fu Anv = Ω 2.00 = ( 0.6 ( 58 ksi ) 4.25 in.2 ) 2.00 = 74.0 kips > 53.9 kips o.k. Check tensile strength of the single plate To determine the available tensile strength of the single plate, use AISC Specification Section J4.1. For tensile yielding of the single plate, use AISC Specification Equation J4-1: LRFD ASD ϕRn = ϕ Fy Ag = 0.90 ( 36 ksi )(12.0 in.)( 2 in. ) = 194 kips > 65.7 kips o.k. Rn Fy Ag = Ω 1.67 (36 ksi) (12.0 in.) ( 2 in. ) = 1.67 = 129 kips > 43.8 kips o.k. For tensile rupture of the single plate, use AISC Specification Equation J4-2: Ae = tp ⎡⎣ L − n ( dh + z ) ⎤⎦ = (2 in. ) ⎡⎣12.0 in. − 4 ( 0.875 in. ) ⎤⎦ = 4.25 in.2 LRFD ϕR n = ϕ Fu Ae ASD ( = 0.75 ( 58 ksi ) 4.25 in. 2 ) = 185 kips > 65.7 kips o.k. Rn Fu Ae = Ω 2.00 = ( 58 ksi ) ( 4.25 in.2 ) 2.00 = 123 kips > 43.8 kips o.k. 364 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Check compression buckling of beam-to-column single plate Check the compression buckling strength of the plate due to the horizontal load. Conservatively assume K = 1.2: r= = tp 12 2 in. 12 = 0.144 in. KL 1.2 (10.25 in.) = r 0.144 in. = 85.4 As specified in AISC Specification Section J4.4, the provisions of Chapter E apply, because KL/ r >25. From AISC Manual Table 4-22, the available critical stress is: LRFD ϕFcr = 22.1 ksi he available compressive strength can then be determined T by: ϕRn = ϕFcr t p L = ( 22.1 ksi )(2 in. )(12.0 in. ) = 133 kips > 65.7 kips o.k. ASD Fcr = 14.7 ksi Ω he available compressive strength can then be determined T by: Rn Fcr t p L = Ω Ω = (14.7 ksi )(2 in. )(12.0 in. ) = 88.2 kips > 43.8 kips o.k. By inspection, block shear rupture will not control and therefore will not be checked. Gusset-to-Column Connection Determine resultant force LRFD Ruc = = Vuc2 + 2 Huc ASD Rac = ( 64.1 kips )2 + ( 65.7 kips )2 = 91.8 kips = Vac2 2 + H ac ( 42.7 kips )2 + ( 43.8 kips )2 = 61.2 kips Design bolts at gusset-to-column connection As previously determined, the shear per bolt for a w-in.-diameter A325-N bolt: LRFD ASD ϕrn = 17.9 kips/bolt rn = 11.9 kips/bolt Ω AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 365 Check bolt bearing on gusset plate and single plate According to the User Note in AISC Specification Section J3.6, the strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners. The effective strength is the lesser of the fastener shear strength and the bearing strength. Assuming that deformation at the bolt hole at service load is a design consideration, use AISC Specification Equation J3-6a for the nominal bearing strength: Rn = 1.2lctFu ≤ 2.4dtFu(Spec. Eq. J3-6a) Conservatively, assume a 12-in. edge distance: lc = 12 in. − 0.5dh = 12 in. − 0.5 (m in. ) = 1.09 in. LRFD ASD ϕ1.2lc tFu = 0.75 (1.2 )(1.09 in. )(2 in. )( 58 ksi ) 1.2lc tFu 1.2 (1.09 in. )(2 in. )( 58 ksi ) = 2.00 Ω = 19.0 kips = 28.5 kips ϕ2.4 db t w Fu = 0.75 ( 2.4 )( w in. )(2 in. )( 58 ksi ) 2.4 Fu db t w 2.4 ( 58 ksi )( w in. )(2 in. ) = Ω 2.00 = 26.1 kips = 39.2 kips Therefore, ϕrn = 28.5 kips/bolt. ϕrn = 17.9 kips/bolt < ϕrn = 28.5 kips/bolt Therefore, the limit state of bolt shear controls. Therefore, rn = 19.0 kips/bolt. Ω rn rn = 11.9 kips/bolt < = 19.0 kips/bolt Ω Ω Therefore, the limit state of bolt shear controls. The available strength of one bolt is 17.9 kips (LRFD) or 11.9 kips (ASD). Note that if tearout controlled using the conservative edge distance assumption, a more rigorous analysis could be performed. The total available bolt strength on the gusset plate and single plate is: LRFD ASD ϕRb = (17.9 kips/bolt )( 6 bolts ) Rb = (11.9 kips/bolt )( 6 bolts ) Ω = 71.4 kips > 61.2 kips o.k. = 107 kips > 91.8 kips o.k. Design the gusset plate-to-column weld The strength determination of fillet welds defined in AISC Specification Section J2.4 can be simplified as explained in AISC Manual Part 8 to AISC Manual Equations 8-2a and 8-2b, and the required weld size is determined as follows. (Note that the column only sees the vertical shear load. Conservatively, a length equal to the length of the bolted portion of the single plate is used.) 366 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD ASD Vuc 2(1.392 kip/in.)L 64.1 kips = 2 (1.392 kip/in. ) (18.0 in.. ) Vac 2(0.928 kip/in.)L 42.7 kips = 2 ( 0.928 kip/in. ) (18.0 in.. ) Dreq = Dreq = = 1.28 sixteenths = 1.28 sixteenths Therefore, use a x-in. fillet weld. This also satisfies the minimum weld size required by AISC Specification Table J2.4. Check shear strength of the single plate To determine the available shear strength of the single plate, use AISC Specification Section J4.2. For shear yielding of the single plate, use AISC Specification Equation J4-3: LRFD ASD ϕRn = ϕ 0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi )(18.0 in. )(2 in. ) = 194 kips > 64.1 kips o.k. Rn 0.60 Fy Agv = Ω 1.50 0.60 ( 36 ksi )(18.0 in. )(2 in. ) = 1.50 = 130 kips > 42.7 kips o.k. For shear rupture of the single plate, use AISC Specification Equation J4-4: Anv = tp ⎡⎣ L − n ( dh + z ) ⎤⎦ = (2 in. ) ⎡⎣18.0 in. − 6 ( 0.875 in. ) ⎤⎦ = 6.38 in.2 LRFD ϕRn = ϕ0.60 Fu Anv ASD ( = 0.75 ( 0.60 )( 58 ksi ) 6.38 in.2 ) = 167 kips > 64.1 kips o.k. Rn 0.60 Fu Anv = Ω 2.00 = ( 0.60 ( 58 ksi ) 6.38 in.2 ) 2.00 = 111 kips > 42.7 kips o.k. Check tensile strength of the single plate To determine the available tensile strength of the single plate, use AISC Specification Section J4.1. For tensile yielding of the single plate, use AISC Specification Equation J4-1: LRFD ϕRn = ϕFy Ag = 0.90 ( 36 ksi )(18.0 in. )(2 in. ) = 292 kips > 65.7 kips o.k. ASD Rn Fy Ag = Ω 1.67 ( 36 ksi ) (18.0 in. )(2 in. ) = 1.67 = 194 kips > 43.8 kips o.k. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 367 For tensile rupture of the single plate, use AISC Specification Equation J4-2: Ae = tp ⎡⎣ L − n ( dh + z ) ⎤⎦ = ( 2 in. ) ⎡⎣18.0 in. − 6 ( 0.875 in. ) ⎤⎦ = 6.38 in.2 LRFD ϕRn = ϕFu Ae ( ASD = 0.75 ( 58 ksi ) 6.38 in. 2 ) = 278 kips > 65.7 kips o.k. Rn Fu Ae = Ω 2.00 = ( 58 ksi ) ( 6.38 in.2 ) 2.00 = 185 kips > 43.8 kips o.k. By inspection, block shear rupture will not control and therefore will not be checked. Design of Stiffener Connections Check single plates to strong-axis moment connection stiffeners Since the eccentricity is resisted by equal and opposite horizontal forces at the beam-to-column and gusset-to-column connections, the horizontal force must be transferred between the connections through the single plate. In this case the single plate is interrupted by column stiffeners required for the moment connection to the strong-axis of the column. Therefore, the welds of the single plates to the stiffeners must be capable of transmitting this force. Check the single plates for shear yielding Determine the available shear strength of the single plates using AISC Specification Equation J4-3: LRFD ϕVn = ϕ0.60 Fy Agv = 1.00 ( 0.60 )( 36 ksi )( 7a in. )(2 in. ) = 79.7 kips > 65.7 kips o.k. ASD Vn 0.60 Fy Agv = Ω 1.50 0.60 ( 36 ksi )( 7a in. )(2 in. ) = 1.50 = 53.1 kips > 43.8 kips o.k. Check the single plates for tensile yielding Determine the available tensile yielding strength of the single plates from AISC Specification Section J4.1, Equation J4-1, as follows (note that Mub and Mab are the moments required to keep the plate from rotating when all the forces shown in Fig. A-5 are in place): LRFD ϕRn = ϕ Fy Ag = 0.90 ( 36 ksi )( 7a in. )(2 in. ) = 119 kips ASD Rn Fy Ag = Ω 1.67 ( 36 ksi )( 7a in. )(2 in.) = 1.67 = 53.1 kips 368 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 LRFD Ru = = ASD 4 Mub Ra = L 4 ( 40.8 kip-in. ) = ( 7a in.) = 22.1 kips < 119 kips 4 M ab L 4 ( 27.2 kip-in. ) ( 7a in.) = 14.8 kips < 79.5 kips o.k. o.k. Design weld of single plates to moment connection stiffeners LRFD ASD Resultant load: Resultant load: R= ( 24.6 kips )2 + ( 65.7 kips )2 R= (16.4 kips )2 + ( 43.8 kips )2 = 70.2 kips = 46.8 kips Resultant load angle: Resultant load angle: ⎛ Ra ⎞ θ = tan −1 ⎜ ⎟ ⎝ Va ⎠ ⎛ 16.4 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 43.8 kips ⎠ ⎛ Ru ⎞ θ = tan −1 ⎜ ⎟ ⎝ Vu ⎠ ⎛ 24.6 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 65.7 kips ⎠ = 20.5° = 20.5° D= = R ( 1.5 2l (1.392 kip/in.) 1.0 + 0.5 sin θ D= ) 70.2 kips ( 1.5 2(7a in.)(1.392 kip/in.) 1.0 + 0.5 sin = 3.10 sixteenths 20.5° ) = R ( 2l (0.928 kip/in.) 1.0 + 0.5 sin1.5 θ ) 46.8 kips ( 2 ( 7a in. ) (0.928 kip/in.) 1.0 + 0.5 sin1.5 20.5° ) = 3.10 sixteenths Therefore, use a 4-in. fillet weld of the single plates to the w-in. strong-axis moment connection stiffener, and use this same weld for the single plates-to-column connection for simplicity. If a portion of the vertical beam reaction had not been redistributed to the gusset connection, the moment would have been 160 kip-in. (LRFD) or 106 kip-in. (ASD). The effect is demonstrated in the following. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 369 Check tensile yielding on single plates LRFD ASD ϕRn = ϕ Fy Ag Rn Fy Ag = Ω 1.67 ( 36 ksi )( 7a in. )(2 in. ) = 1.67 = 79.5 kips = 0.90 ( 36 ksi )( 7a in. )(2 in. ) = 119 kips Ru = = 4 Mub Lg Ra = 4 (160 kip-in. ) 7a in. = 86.8 kips < 119 kips = o.k. 4 M ab Lg 4 (106 kip-in. ) 7a in. = 57.5 kips < 79.5 kips o.k. Design weld of single plates to moment connection stiffeners LRFD ASD Resultant load: Resultant load: Ru = ( 96.6 kips )2 + ( 65.7 kips )2 Ra = = 117 kips = 77.6 kips Resultant load angle: Resultant load angle: ⎛ Ru ⎞ θ = tan −1 ⎜ ⎟ ⎝ Vu ⎠ ⎛ 96.6 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 65.7 kips ⎠ ⎛ Ra ⎞ θ = tan −1 ⎜ ⎟ ⎝ Va ⎠ ⎛ 64.0 kips ⎞ = tan −1 ⎜ ⎟ ⎝ 43.8 kips ⎠ = 55.8° D= = = 55.6° Ru ( 1.5 2l (1.392 kip/in.) 1.0 + 0.5 sin θ D= ) 117 kips ( 1.5 2(7a in.)(1.392 kip/in.) 1.0 + 0.5 sin = 4.14 sixteenths ( 64.0 kips )2 + ( 43.8 kips )2 55.8° ) = Ra ( 2l (0.928 kip/in.) 1.0 + 0.5 sin1.5 θ ) 77.6 kips ( 2 ( 7a in. ) (0.928 kip/in.) 1.0 + 0.5 sin1.5 55.6° ) = 4.12 sixteenths In this case the 2-in.-thick plate and single pass weld are still okay even without the vertical beam end reaction distributed to minimize the moment at the critical section. 370 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 APPENDIX B Use of the Directional Strength Increase for Fillet Welds In this Design Guide the authors include the increased strength of the fillet welds due to loading transverse or oblique to the longitudinal axis of the weld. In many cases, the stresses the welds must resist are based on the resultant forces from shears, axial loads and moments. The authors base the calculations on the largest resultant normal stress in such cases, though obviously the moment will add to the axial stress on one side and subtract on the other. Below is an explanation of the consequences of this procedure. It is not always conservative, but the maximum overstress in the weld resulting from the assumption is limited to about 3% and is therefore neglected. When a normal force and a moment act on a gusset edge, as shown in Figure B-1, it is convenient for analysis to convert the moment into a statically equivalent normal force and then to add this to the original normal force to arrive at an equivalent normal force: Ne = T + 4M L (B-1) The steps necessary to produce Ne are shown in Figure B-1. If there is a shear V acting on the gusset edge, then the resultant force on the edge is: R = N e2 + V 2 r= D plus = 2 ⎛ T 4M ⎞ ⎛ V ⎞ ⎜L − 2 ⎟ +⎜ L⎟ L ⎠ ⎝ ⎠ ⎝ Dminus = Let x= T 4M + L L2 (B-6) Introducing the parameters Z and Y, let xZ = (B-3) Let Dplus equal the required weld where the axial force and moment add and Dminus equals the required weld where the axial force and moment subtract: 2 1.5 ⎧ ⎡ ⎤ ⎫⎪ ⎪ T M 4 ⎛ ⎞ ⎢ ⎥ ⎪ ⎪ ⎜⎝ L − 2 ⎟⎠ ⎢ ⎥ ⎪ ⎪ L 2 (1.392 ) ⎨1 + 0.5 ⎢ ⎥ ⎬ 2 2 ⎪⎪ ⎢ ⎛ T 4M ⎞ ⎛ V ⎞ ⎥ ⎪⎪ ⎢ ⎜L − 2 ⎟ + ⎜L⎟ ⎥ ⎪ ⎪ L ⎠ ⎝ ⎠ ⎦ ⎣ ⎝ ⎩ ⎭ (B-5a) 2 R ⎛ Ne ⎞ ⎛ V ⎞ = ⎜ ⎟ +⎜ ⎟ L ⎝ L ⎠ ⎝ L⎠ 2 1.5 ⎧ ⎡ ⎤ ⎫⎪ ⎪ T M 4 ⎛ ⎞ ⎢ ⎥ ⎪ ⎪ ⎜⎝ L + 2 ⎟⎠ ⎢ ⎥ ⎪ ⎪ L 2 (1.392 ) ⎨1 + 0.5 ⎢ ⎥ ⎬ 2 2 ⎪⎪ ⎢ ⎛ T 4 M ⎞ ⎛ V ⎞ ⎥ ⎪⎪ ⎢ ⎜L + 2 ⎟ + ⎜L⎟ ⎥ ⎪ ⎪ L ⎠ ⎝ ⎠ ⎦ ⎣ ⎝ ⎩ ⎭ (B-4a) (B-2) And the resultant force per unit length of the edge is: 2 2 ⎛ T 4M ⎞ ⎛ V ⎞ ⎜L + 2 ⎟ + ⎜L⎟ L ⎠ ⎝ ⎠ ⎝ T 4M − , where 0 < Z < 1.0 L L2 and xY = V , where Y is any positive number L AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 371 Thus, Dplus and Dminus can be rewritten as: D plus = Dminus ⎡⎢ ⎤⎥ 1+Y 2 x ⎢ ⎥ 1.5 2 (1.392 ) ⎢ ⎛ 1 ⎞ ⎥ ⎢1 + 0.5 ⎜ ⎟ ⎥ ⎜ 2 ⎟ ⎢⎣ ⎝ 1 + Y ⎠ ⎥⎦ ⎡ ⎢ x Z2 +Y2 = ⎢ 2 (1.392 ) ⎢ ⎛ Z ⎢ 1 + 0.5 ⎜ ⎜ 2 2 ⎢⎣ ⎝ Z +Y It can be shown that as Z approaches 1.0, Dminus approaches the value Dplus . Therefore Dminus is most different from Dplus when Z equals zero. With Z = 0, Dminus can be rewritten as: (B-4b) Dminus = ⎤ ⎥ ⎥ (B-5b) 1.5 ⎞ ⎥ ⎟⎟ ⎥ ⎠ ⎥⎦ x (Y ) The ratio of Dminus to Dplus can be written as: Dminus Y = D plus 1+Y 2 ⎡ ⎛ ⎢1 + 0.5 ⎜ 1 ⎜ 2 ⎢ ⎝ 1+Y ⎣ L M T Actual forces M L/2 T M L /2 Moment converted into normal force L/2 M L /2 T (B-7) 2 (1.392 ) M Ne = T + 4M L L/2 Fig. B-1. Development of equivalent axial force. 372 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 ⎞ ⎟ ⎟ ⎠ 1.5 ⎤ ⎥ ⎥ ⎦ (B-8) Taking the derivative of this ratio with respect to Y and setting it equal to 0 to find the maximum, results in Y = 2.65. Therefore, the maximum ratio between Dminus and Dplus is: Dminus 2.65 = 2 D plus 1 + ( 2.65 ) = 1.03 1.5 ⎡ ⎛ ⎞ ⎤ ⎢ 1 ⎜ ⎟ ⎥ ⎢1 + 0.5 ⎜ ⎟ ⎥ ⎜ 1 + ( 2.65 )2 ⎟ ⎥ ⎢ ⎝ ⎠ ⎦ ⎣ The maximum possible variance between Dminus and Dplus is 3.37%. Because it can be shown that the weld will never be more than 3.37% overstressed, it is reasonable to base the directional strength increase on the addition of the moment and the axial stresses. It should be noted that though this analysis was performed using LRFD, the results would be identical using ASD—only the constant 1.392 would change to 0.928. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 373 APPENDIX C Buckling of Gusset Plates C.1. BUCKLING AS A STRENGTH LIMIT STATE—THE LINE OF ACTION METHOD estimate for the buckling strength of the gusset plate. The revised K factors suggested by Dowswell are used in this Design Guide and are reproduced in Table C-1 with other pertinent information, for convenience. It should be noted that prior to the publication of the LOAM in 1984, there was no published method available for line of action buckling of gusset plates. Many practitioners simply stiffened the gusset to preclude buckling or ignored buckling (Zetlin et al., 1975). The approach to gusset buckling used in the AISC Manual and in this Design Guide recognizes that the critical buckling limit state is along the line of action of the brace. If buckling occurs here, the axial strength of the connection will be compromised. The method is called the “line of action” method (LOAM) in this Design Guide. The method, as shown in Figure C-1, was originally developed by Thornton (1984). It uses the Whitmore Section to estimate the actual stress and an equivalent column of l1 + l2 + l3 length: lb = l 1 or lb = to estimate buckling stress. 3 As originally proposed by Thornton, the column effective length factor, K, was chosen as 0.65, but research by Gross (1990) indicated that K = 0.50 is conservative and this value was adopted by the AISC Manual (AISC 1992, 1994, 2001 and 2005). The method has been called the “Thornton” Method (Cheng and Grondin, 1999) or the “pseudo column” Method (Chambers and Ernst, 2005). Dowswell (2006) has shown that when compared to the results of physical tests by many researchers, the line of action method (LOAM) with revised K factors almost always gives a conservative C.2 BUCKLING AS A HIGH CYCLE FATIGUE LIMIT STATE—GUSSET PLATE EDGE BUCKLING In the design of highway bridges, gusset free edge buckling or flexing can cause fatigue cracks. For this reason, the LRFD Bridge Design Specifications (AASHTO, 2012) includes a criterion for stiffening of gusset plate free edges in Section 6.14.2.8 that defines the maximum free edge length of the gusset plate without required stiffening as: a max = 2.06 W h se itm ct or io e n l2 Column flange (or web) Brace force P (reversible) ° 30 Beam flange (or web) 30 1l 3l ° Fig. C-1. Line of action buckling method (LOAM). 374 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 E (t ) Fy (C-1) Table C-1. Summary of Effective Length Factors Gusset Configuration a Compact Corner Buckling Length, l b Effective Length Factor, K c — d —d Noncompact Corner c 1.0 lavg Extended Corner 0.6 l1 Single Brace 0.7 l1 Chevron 0.65 l1 a See Figure C-2 for gusset configurations. b See Figure C-3 for buckling length. c See Figure C-4 for determination of compactness. d Yielding is the applicable limit state for compact corner gusset plates; therefore, the effective length factor and the buckling length are not applicable. where a = length of the gusset plate free edge, in. t = thickness of gusset plate, in. E = modulus of elasticity, psi Fy = specified minimum yield stress, psi Both of the variables, E and Fy, may be applied with any consistent set of units. This criterion has been in the bridge code since at least the 1944 Edition, and is also included in Guide Specification of Seismic Design of Steel Bridges (Caltrans, 2001). The basis for Equation C-1 is elastic buckling of a plate with the boundary conditions shown in Figure C-5. This plate is imagined embedded in a gusset plate, for instance, as shown in Figure C-6. When the plate has an aspect ratio of a/ b that is approximately equal to 3 and the plate elastic bucking stress is set equal to Fy, Equation C-1 results, as will now be demonstrated. The elastic plate bucking equation can be written as: 2 ⎛a⎞ k π2E ⎜ ⎟ a ⎝b⎠ = t 12 1 − υ2 Fcr ( ) (C-2) where k = plate buckling coefficient ≈ 0.5 for a/b = 3 (Salmon and Johnson, 1971) υ = Poisson’s ratio (= 0.3 for steel) Fcr = plate buckling stress, psi Substituting these values and a/ b = 3 into Equation C-2 results in: Fig. C-2. Gusset configurations (Dowswell, 2006). AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 375 2.02), the expression is equivalent to that presented in Equation C-1. The criterion adopted by AASHTO is generally very conservative. It assumes that the material yield stress will be achieved on the gusset plate free edge. In actual gussets, the maximum possible gusset edge stress based on an 0.5π 2 E ( 3.0 ) a = t 12 1 − 0.32 Fcr 2 ( = 2.02 ) E Fcr Setting Fcr = Fy and recognizing that k and a/ b are approximate (i.e., eliminating the precision implied by the factor 2l L w c 1l 30° 3l If t g ≥ t β , then compact 1l t β , then noncompact If tg If≥ ttgβ<, then compact < tβ, then If tg where t g =noncompact thickness of gusset where tg = thickness of gusset lavg = t β = 1 .5 l1 + l2 + l3 3 Fig. C-3. Buckling length (Dowswell, 2006). Fy c3 El1 Fig. C-4. Determination of compact/noncompact (Dowswell, 2006). Fig. C-5. Basic gusset plate stability model. 376 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 admissible force system is 0.6Fy at working loads, for both ASD and LRFD. While Equation C-1 is generally conservative, as noted here, if the plate aspect ratio is less than 3, the equation can be unconservative. The authors believe that any reasonable approach to gusset plate edge buckling should be based on the statically admissible force system assumed for all of the other limit states that govern the design of a bracing connection. A subsequent section of this Appendix will present such a method. C.3. BUCKLING AS A LOW-CYCLE FATIGUE LIMIT STATE—GUSSET PLATE FREE EDGE BUCKLING In low-cycle fatigue, with excursions into the material inelastic region, it is perceived that gusset plate free edge flexing could cause premature fracture in the gusset and its connections. To prevent this flexing, Astaneh-Asl (1998), following the work of Brown (1988), suggests treating the gusset plate free edge as a column extending 1 in., or other linear unit, from the free edge and completely separated from the remainder of the gusset. This is equivalent to using the top 1 in. of the gusset plate shown in Figure C-5, and is explicitly shown in Figure C-7. The boundary conditions for this column, as shown in Figure C-7, are the fixed-sliding type, with a theoretical K of 1.0 and a recommended value of 1.2. Note that the top 1 in. of the gusset plate shown in Figure C-5, which is pinned-pinned, also has a theoretical K value of 1.0. Thus, the gusset model of Figure C-5 can be expected to yield similar results for the simple-support-tosimple-support case and the fixed-sliding case. C L column CL brace a b W.P. CL beam Embedded plate of Figure C-5 For gussets with bend lines, a and b are taken to the bend line Fig. C-6a. Embedded plate for horizontal (free) edge gusset plate buckling. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 377 The critical elastic buckling stress for the strip column of Figure C-7 is: Fcr = π2 E ( 2 12 1 − υ ) 2 ⎛ Ka ⎞ ⎜ t ⎟ ⎝ ⎠ Setting Fcr = Fy and K = 1.2: a E = 0.792 t Fy (C-3) and rounding to eliminate the precision implied by the factor 0.792: where the term (1−υ2) is included to model the increased stiffness of a plate over a simple “stand-alone” column, which will have an inelastic curvature. Solving Equation C-3 for a/t: (C-4) Astaneh-Asl (1998) further rounds Equation C-4 based on the results of one test to: a π 2E = t 12 1 − υ2 K 2 Fcr ( 4 E ⎛a⎞ = ⎜t⎟ ⎝ ⎠ max 5 Fy ) CL column CL brace b a W.P. CL beam Embedded plate of Figure C-5 For gussets with bend lines, a and b are taken to the bend line Fig. C-6b. Embedded plate for vertical (free) edge gusset plate buckling. 378 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 E ⎛a⎞ = 0.75 ⎜t⎟ F ⎝ ⎠ max y (C-5) It can be seen from the above derivation that Equation C-5 is extremely conservative. It is very likely that a gusset plate edge will not buckle even if Equation C-5 is not satisfied. Not withstanding this observation, Equation C-5 was in wide use in high-seismic regions and for that reason was adopted for use in the 1st Edition AISC Seismic Design Manual (AISC, 2006). It has been dropped from explicit use in the 2nd Edition AISC Seismic Design Manual (AISC, 2012). The reason for this is explained in the AISC Seismic Provisions Commentary Section F2.6c (AISC, 2010b), which states: Certain references suggest limiting the free edge length of gusset plates, including SCBF brace-to-beam connection design examples in the Seismic Design Manual, (AISC, 2006), and other references (Astaneh-Asl et al., 2006; ICC, 2006). However, the committee has reviewed the testing cited and has concluded that such edge stiffeners do not offer any advantages in gusset plate behavior. There is therefore no limitation on edge dimensions in these provisions. It should be noted that the use of a gusset plate free edge buckling criterion does not eliminate the need for a line of action method (LOAM) check. The stiffening of gusset plate edges will not necessarily increase the ultimate compressive strength of a gusset plate (Rabinovitch, 1993). If gusset plate free edge stiffeners are determined by the designer to be necessary, it may be more cost effective to use a thicker gusset plate to eliminate the stiffeners. Finally, if gusset plate edge stiffeners are used, they should not be attached to the beam or column. If they are, they will completely change the force distribution assumed by the UFM (or any of the other methods discussed in this Design Guide) with an increased likelihood of premature local fractures. These local fractures could propagate and cause complete connection failure. (a) In plane of gusset (Top 1 in. of gusset from free edge) (b) Normal to plane of gusset (Basis for AISC Seismic Design Manual Free-Edge Buckling) Fig. C-7. Fixed-sliding buckling model. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 379 C.4. AN APPROACH TO GUSSET PLATE FREE EDGE BUCKLING USING STATICALLY ADMISSIBLE FORCES (THE ADMISSIBLE FORCE MAINTENANCE METHOD) The Commentary to the Seismic Provisions (AISC, 2010b) indicates that edge stiffeners do not offer any advantages in gusset plate behavior. However, unusual conditions may be encountered where the engineer may wish to determine the free-edge buckling strength of a gusset. While free-edge buckling seldom, if ever, controls the gusset ultimate compressive strength as determined by LOAM, edge buckling could compromise the statically admissible force distribution determined for the system. For instance, the connection shown in Figures 6-5a and 6-5b shows a horizontal force of 214 kips (LRFD) and 142 kips (ASD) in tension, but the brace load is reversible, so the vertical gusset plate edge will see a compressive force of 214 kips (LRFD) and 142 kips (ASD), which gives rise to a compressive stress, fa = (214 kips)/ [(1.00 in.)(28.0 in.)] = 7.64 ksi for LRFD and fa = (142 kips)/ (1.00 in.)(28.0 in.) = 5.07 ksi for ASD. For this stress to be achieved, the horizontal (top) edge of the gusset plate must have a compressive strength exceeding 7.64 ksi (LRFD) and 5.07 ksi (ASD). Similarly, the horizontal edge at the beam sees an equivalent axial force of: LRFD Tequiv = 218 kips + ASD (1, 740 kip-in.)( 4 ) Tequiv = 145 kips + 354 in. = 415 kips (1,160 kip-in.)( 4 ) 354 in. = 277 kips The vertical (inclined) edge must not buckle at a load less than 415 kips (LRFD) and 277 kips (ASD) in order to sustain the postulated admissible internal force system. The method to be used in this Design Guide to check gusset plate edges for buckling is based on the plate model of Figure C-5. The critical buckling stress for this model is given by Muir and Thornton (2004) as: Fcr = QFy where Q = 1 for λ ≤ 0.7 (yield controls) = 1.34 − 0.486λ for 0.7 ≤ λ ≤ 1.41 (inelastic buckling controls) 1.3 = 2 for λ > 1.41 (elastic buckling controls) λ and ( b / t ) Fy λ= 1,120 5 475 + ( a / b )2 (C-6) (C-7) The parameter λ is the effective slenderness ratio of the plate. The formulation of Equation C-6 reduces to AISC Specification Section E7.1(c) for single angles when a/b goes to ∞. Note that Equation C-6 is the same as that used for certain double-coped beams in the AISC Manual. It will be referred to here as the admissible force maintenance method (AFMM). Applying the AFMM theory to the design of the connection in Figure 6-1, for the vertical edge to carry 214 kips (LRFD) or 142 kips (ASD), as calculated previously fa = 7.64 ksi (LRFD) or fa = 5.07 ksi (ASD). Approximate length of horizontal free edge to the bend line is a = 16.0 in. Depth of the gusset at the column, b = 28.0 in.: a 16.0 in. = b 28.0 in. = 0.571 b 28.0 in. = t 1.00 in. = 28.0 380 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 λ= 28.0 50 ksi 5 475 + (C-7) 1,120 ( 0.571)2 = 0.633 Q = 1.00 LRFD ϕFcr = 0.90 (1.00 )( 50 ksi ) = 45.0 ksi > 7.64 ksi o.k. ASD Fcr (1.00 )( 50 ksi ) = Ω 1.67 = 29.9 ksi > 5.07 ksi o.k. For the horizontal beam-to-gusset plate edge to carry 415 kips (LRFD) and 277 kips (ASD): LRFD fu = 415 kips w in. ( ) ( 354 in.) = 15.7 ksi ASD fa = 277 kips w in. ( ) ( 354 in.) = 10.5 ksi a = 12 in. (distance from beam flange to lowest edge of HSS brace—this is the “c” dimension of Figure 6-2) b = 354 in. a 12.0 in. = b 354 in. = 0.340 b 354 in. = t 1.00 in. = 35.25 λ= 35.25 50 ksi 5 475 + 1,120 ( 0.34 )2 = 0.494 Q = 1.00 LRFD ϕFcr = 0.90 (1.00 )( 50 ksi ) = 45.0 ksi > 15.7 ksi o.k. ASD Fcr (1.00 )( 50 ksi ) = Ω 1.67 = 29.9 ksi > 10.5 ksi o.k. Therefore, the gusset is sufficiently thick to prevent edge buckling under the given admissible force distribution and that distribution can thus be maintained. The AFMM is presented here to provide a viable and reasonable method for considering gusset buckling. As the example here shows, it did not control the design. Usually the LOAM method for compression and the block shear tearout limit state for tension will control the gusset thickness. This is the case for every corner gusset example presented in this Design Guide. In fact, the authors, with their considerable experience in the design of these connections, have never found a corner connection case where the AFMM controlled the gusset thickness. For this reason, it has not been included in the design examples of corner connections presented in this Design Guide. However, for the case of gussets in chevron bracing connections, if both braces are in compression or even in tension, the long horizontal edge may require a stiffener by the AFMM. This is demonstrated in Example 5.9. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 381 C.5. APPLICATION OF THE FREE EDGE APPROACH TO EXAMPLE 6.1a This section will apply the free edge approach of AstanehAsl et al. (2006) and Equation C-6 to Example 6.1a. The length of the top sloping free edge is 16 in. to the bend line if measured to the face of the column and thus: a 16 in. = t w in. = 21.3 0.75 where the terms are as previously defined. Equation C-7 is valid for aspect ratios, a/ b, from 2 to 2. For the top edge of the gusset of Example 6.1a (Figures 6-1 and 6-5): a = 0.571 b b = 28.0 t t = 1.00 in. I c = 0.92 ( 0.571)(1.00 in. ) 29, 000 ksi = 18.1 50 ksi 4 ( 28.0 )2 − 144 = 13.3 in.4 a = 21.3 > 18.11 t To size the stiffener, Dowswell (2009) can be used. From Dowswell, the required stiffener moment of inertia is: 2 ⎛ b⎞ ⎛a⎞ I c = 0.92 ⎜ ⎟ t 4 ⎜ ⎟ − 144 ⎝b⎠ ⎝t⎠ (C-8) Try a stiffener w in. × 6.00 in.: 1 3 I c = ( w in. )( 6.00 in. ) 12 = 13.5 in.4 > 13.3 in.4 o.k. The stiffener length of 14 in. is chosen to keep it clear of the column flange. The stiffener is attached to the gusset plate with x-in. fillet welds, the minimum required size based on AISC Specification Table J2.4. 382 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 APPENDIX D Transfer Forces The AISC Code of Standard Practice for Steel Buildings and Bridges (AISC, 2010a), hereafter referred to as the AISC Code of Standard Practice, states that, when the fabricator is required to design connection details, the structural design drawings shall provide, among other things, “data concerning the loads, including shears moments, axial forces and transfer forces, that are to be resisted by the individual members and their connections, sufficient to allow the fabricator to select or complete connection details…” This statement is most remarkable for what it does not require. It does not require the fabricator to be provided with all of the loads, or even loads that satisfy equilibrium. It merely requires that sufficient information be provided. It should also be noted that the term transfer force is not defined in the glossary for either the AISC Code of Standard Practice or the AISC Specification. In fact a survey of steel design handbooks will demonstrate that the term is neither ubiquitous nor well-understood. It is therefore useful to begin any discussion of transfer forces with a definition. A possible definition for a transfer force is a force that must be transferred from one element to another through connection elements and whose magnitude and direction cannot be ascertained from the maximum member end forces. An example is useful to demonstrate the concept. Assume there is bracing as shown in Figure D-1. It would be logical to assume that there is some transfer force or transfer of lateral load from the braced frame of the left-hand side of the column to the braced frame on the right-hand side of the column. In order to determine the magnitude and direction of the transfer force, the forces in the member framing to the joint must be examined. These forces are indicated in Table D-1. The transfer force, TF, can be defined using the x direction components of the forces as: in order to transfer the force. It is also interesting to note that the load combination that produces the largest transfer force contains the maximum member end force for only one of the five members of interest. D.1. THE EFFECT OF CONNECTION CONFIGURATION ON THE TRANSFER FORCE The connection configuration can often have a significant impact on the calculation of the transfer force. For example, if the configuration shown in Figure D-1 was changed so that the members were connected using near side and far side plates as shown in Figure D-2, there is a change in the transfer force information required to design the connection. In the configuration shown in Figure D-1, the connection will deliver the vertical component of the brace load to the column. Therefore, there is no need to supply a vertical transfer force. However, it is unclear how much of the horizontal component must be transferred through the joint, and how 5 2 12 3 12 8w 12 7 6 TF = 1x + 2x + 6x or TF = 3x + 7x Performing these calculations results in the transfer forces shown in Table D-1. The maximum member end forces are presented in Table D-2. As can be seen the correct transfer forces could not be determined from the maximum member end forces alone, since there is no way to ensure that equilibrium has been satisfied. An assumption could be made that all of the members on one side of the column act in the same direction. This has been done to obtain the transfer forces shown in Table D-2. This results in excessive transfer forces—on the order of 8 times those actually required. Trying to design for these forces would be expensive and would in many cases require reinforcement of the members 1 12 12 4 Fig. D-1. Configuration used to calculate transfer forces. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 383 Table D-1. Member End Forces and Transfer Forces for Connection Shown in Figure D-1 Member ID Load Combination Axial 1 Axial 2 Axial 3 Axial 4 Axial 5 Axial 6 Axial 7 TF kips kips kips kips kips kips kips kips 1 −509 217 −198 −125 −479 120 30.2 −86.7 2 366 −386 113 −867 −426 24.9 −55.6 10.9 3 323 −436 −1160 −697 68.2 −63.4 −11.6 4 −466 267 170 −209 76.8 38 −64.1 5 −185 −219 −50.3 −1130 −1070 194 −62 −91.6 34.8 145 167 −49 6 41.6 50.3 87.6 −173 36.5 16.0 + = Tension − = Compression much stays within the left-hand bay. Either statically consistent loadings or a horizontal transfer force must be provided to ensure the connection is properly designed. In the configuration shown in Figure D-2, the situation is reversed. Since all of the horizontal load can be transferred directly through the near side and far side plates, the horizontal transfer is only a secondary concern. However, the magnitude of the vertical load that must be delivered to the column is unclear. Again, either statically consistent loadings 5 2 12 3 12 7 6 1 D.2. PRESENTATION OF TRANSFER FORCES IN DESIGN DOCUMENTS 2. Determine the load combinations that produce the maximum tension and maximum compression end forces for each member framing into a joint. Additionally, determine the load combinations that produce the maximum tension and maximum compression transfer forces. 3. For each load combination found in Step 1, give the axial forces in all members associated with that load combination so that joint equilibrium is provided for each. The maximum number of load cases will be equal to 2(n+1), where n is the number of members framing into the joint. 12 12 TF = 1y + 2y + 3y There are two possible methods that can be used to satisfy the AISC Code of Standard Practice requirement that sufficient load information be provided to complete the connection design. The first method involves providing a matrix of statically consistent load combinations that can be used to determine the required transfer force. An example of this type of presentation based on Table D-1 is shown in Figure D-3. The procedure for generating such a matrix can be described as follows: 1. Define the assumed connection configuration. 8w 12 or a vertical transfer force must be provided to ensure the connection is properly designed. Assuming the members carry only axial loads, the transfer force, TF, can be defined as: 4 Fig. D-2. Alternate connection configuration. Possible advantages of this approach are as follows: • Joint equilibrium is always provided for each load case, making the load paths very clear. 384 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 Table D-2. Maximum Member End Forces and Apparent Transfer Forces for Connection Shown in Figure D-1 Member ID Axial 1 Axial 2 Axial 3 Axial 6 Axial 7 TF kips kips kips kips kips kips Maximum Compression 509 436 198 49 63.4 717 Maximum Tension 366 267 113 194 36.5 620 • The most economical connection can be provided because all forces are given. • The generation of the required load data can be automated with a post-processor program. • The second method of providing sufficient load information to complete the connection design is the approach most commonly used. It involves providing simply the maximum tension and compression axial forces and a required transfer force. An example of this type of presentation based on Table D-1 is shown in Figures D-4(a) and (b). Possible advantages of this approach are as follows: • The volume of load data is greatly reduced. Possible disadvantages of this approach are as follows: • A large volume of data must be conveyed to the fabricator. The fabricator must perform the calculations necessary to extract the transfer force. This is a transfer of labor from the engineer of record to the fabricator. 3 2 s S8 8x 8x x8 SS xs H HS • 6 7 0 x1 x2 10 SS H 1 As the generation of the connection data will undoubtedly be automated, there is a potential that less checking of computer output will be performed. Member Axial End Forces and Transfer Forces Load Case Member ID 1 2 3 6 1 -509 217 -198 120 2 366 -386 113 3 323 -436 4 -466 267 5 -185 -219 + = Tension 87.6 -173 -50.3 7 TF 30.2 -86.7 24.9 -55.6 10.9 68.2 -63.4 -11.6 76.8 38 -64.1 -62 -91.6 194 - = Compression Fig. D-3. Presentation of load information in tabular form. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 385 • The process of generating the transfer force may encourage more checking of the computer output. • The fabricator is not burdened with additional labor that has historically been performed by the engineer of record. Possible disadvantages of this approach are as follows: • It would be more difficult to automate this approach. • The fabricator has less flexibility to design more economical, alternative connections. • Without statically consistent loadings, the fabricator has a greatly diminished understanding of the engineer of record’s intent and requirements. It should be noted that, regardless of the presentation, the transfer forces must be calculated. Also the possibility exists, though unlikely, that the three-step process described previously may not capture the most critical condition for some of the internal connection plate stresses. It is difficult to conceive of a case where this could occur using the configuration illustrated in Figure D-1, but may be possible using the configurations illustrated in Figures D-2 and D-5. D.3. ADDITIONAL CONSIDERATIONS 2 In the absence of loadings that satisfy equilibrium, the fabricator is required to make assumptions about the direction of the given loads. The most reasonable assumption when faced with a bracing connection where braces frame both above and below the beam is that while one brace is in tension, the other is in compression. This would usually be consistent with the lateral loading of a braced frame. However, this assumption is not always correct. For example, in Table D-1 it can be seen that for most of the load combinations members 1 and 2 have opposite signs, conforming to the stated assumption. However, load combination 5 puts both braces into compression; while load combination 6 puts both braces into tension. It can also be problematic to check gusset plates when the given loads do not satisfy equilibrium. As an example, Figure D-5 shows a chevron brace, which in addition to resisting lateral loads, also supports a column. Given only the maximum member forces, as shown, it is impossible with any certainty to derive the correct design forces, and the design of the gusset plate must be based on potentially erroneous assumptions. It might be concluded for instance that the column is unlikely to undergo significant uplift. The connection designer may therefore assume a force distribution as shown in Figure D-6. The free-body diagram shown satisfies equilibrium, with the force in the right-hand brace reduced to a value of 207 kips, but this is only an assumption made in an attempt to satisfy equilibrium. Although it is unlikely that the column will experience significant uplift, this does not preclude the possibility entirely. Any uplift, even with a magnitude considerably less than 500 kips, will increase the magnitude of the horizontal shear force. Another approach might be to assume that the maximum brace forces shown act simultaneously. In this case, to satisfy equilibrium, the column force must be increased to 707 kips. This is shown in Figure D-7. Though the vertical force is maximized, the horizontal shear force is nonexistent. Since, in Figure D-7 a force is increased to satisfy equilibrium, while in Figure D-6 a force is decreased, it might be assumed that Figure D-7 represents the most critical case. However, this logic does not always hold true. Still another possibility exists. The column may not experience any uplift, but may unload. This is illustrated in Figure D-8. Because the horizontal shear applied in Figure D-8 is Fig. D-4. Presentation of load information using only the maximum member forces. 386 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 equal to the vertical transverse load applied in Figure D-7, the required plate thickness and weld size will be greater for the forces shown in Figure D-8. However, this is not necessarily the critical case either, for while the force shown in Figure D-7 will cause significant compression at the bottom edge of the gusset and the potential for buckling, no such concern exists for the forces shown in Figure D-8. This illustrates the intractable situation the connection designer is faced with when provided forces that do not satisfy equilibrium. D.4. EFFECTS OF MODELING ASSUMPTIONS ON TRANSFER FORCES The analysis and design of a concentrically braced frame may at first seem to be a trivial task. Since the structure is determinate, the calculation of forces within the bracing system is not complicated. However, the means of delivering the forces to the bracing system can have a significant impact on the connection design. For example, once the lateral force is known it can be delivered to the system in various ways. In Figure D-9 the loads are delivered to the bracing system in varying ways, which will affect the design of both the connections and the members. The lateral force at structure (a) would require a transfer force at the left-hand side of the frame. The lateral force at structure (b) would require a transfer force at the right-hand side of the frame, while structure (c) with the load presumably delivered through the diaphragm directly to the braced frame would not require a transfer force at all. If during the analysis the load is assumed to be delivered at a joint (or node), then it will not show up as a member force. In such cases care must be taken to ensure that the transfer is 500 kips 500 kips 12 500 kips 12 12 12 Fig. D-5. Bracing subjected to gravity and lateral loads. 500 kips 707 kips 207 kips 500 kips 207 kips Fig. D-6. A possible free body diagram for the gusset plate shown in Figure D-5. 500 kips 500 kips Fig. D-7. A possible free body diagram for the gusset plate shown in Figure D-5. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 387 Table D-3. Member End Forces and Transfer Forces for Connection Shown in Figure D-2 Member ID Load Combination Axial 1 Axial 2 Axial 3 Axial 4 Axial 5 Axial 6 Axial 7 TF kips kips kips kips kips kips kips kips 120 1 −509 217 −198 −125 −479 30.2 354 2 366 −386 113 −867 −426 24.9 −55.6 −441 3 323 −436 −1160 −697 68.2 −63.4 −466 38 379 4 −466 267 5 −185 −219 6 41.6 50.3 87.6 −173 170 −209 −50.3 −1130 −1070 194 −34.8 145 167 −49 76.8 −62 36.5 −64.6 −21.9 + = Tension − = Compression not neglected. A nodal load column could be added to Table D-3, but this too presents problems in terms of connection design. Applying the load to an infinitesimally small point in the model may seem a perfectly reasonable assumption during analysis, but there are few practical ways to transfer the load directly to the joint without first delivering it to the other members, which may or may not be capable of resisting the additional load. Therefore, providing a theoretical load transfer directly to the node again places the connection designer in the unenviable position of making assumptions about the proper mode of load transfer. Practical problems can also occur in the case of Figure D-9(c). There is an assumed transfer of force between the diaphragm and the beam. However, the presence of bracing connections at the top flange of the beam could limit the number of shear studs that can be placed in this location. The situation is illustrated in Figure D-10. 388 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29 0 kips 707 kips 500 kips 500 kips Fig. D-8. A possible free body diagram for the gusset plate shown in Figure D-5. Fig. D-9. Design assumptions for delivering lateral force to the braced frame. Shear Studs Fig. D-10. Number of shear studs limited by brace connections. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 389 REFERENCES AASHTO (1996), Standard Specifications for Highway Bridges, 16th Ed., American Association of State Highway and Transportation Officials, Washington, DC. AASHTO (2012), LRFD Bridge Design Specifications, 6th Ed., American Association of State Highway and Transportation Officials, Washington, DC. Abolitz, A.L. and Warner, M.E. (1965), “Bending Under Seated Connections,” Engineering Journal, AISC, Vol. 2, No. 1, 1st Quarter, pp. 1–5. AISC (1992), Manual of Steel Construction—Volume II: Connections, ASD 9th Ed./LRFD 1st Ed., American Institute of Steel Construction, Chicago, IL. 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(2004), “Design for Improved Performance of Buckling Restrained Brace Frames,” Proceedings of the Structural Engineers Association of California 2004 Convention, SEAOC, pp. 507–513. Zetlin, L., Thelen, J.F. and Cuocco, D. (1975), “Field Problems of Latticed Structures,” Journal of the Structural Division, ASCE, Vol. 101, No. ST11, November, pp. 2,241–2,255. AISC DESIGN GUIDE 29 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / 391 392 / VERTICAL BRACING CONNECTIONS—ANALYSIS AND DESIGN / AISC DESIGN GUIDE 29