APPENDIX I. GEORGIA TECH PARAMETRIC STUDY FINAL REPORT

FINITE ELEMENT SIMULATION AND ASSESSMENT OF THE STRENGTH BEHAVIOR OF RIVETED AND BOLTED GUSSET-PLATE CONNECTIONS IN

STEEL TRUSS BRIDGES

FINAL REPORT

Prepared for Federal Highway Administration and NCHRP Transportation Research Board of the National Academies

Donald W. White, Georgia Institute of Technology, Atlanta, GA Roberto T. Leon, Virginia Polytechnic Institute and State University, Blacksburg, VA

Yoon Duk Kim, Georgia Institute of Technology, Atlanta, GA Yavuz Mentes, MMI Engineering, Houston, TX

Mohammad Towhidur Rahman Bhuiyan, Georgia Southern University, Statesboro, GA

March 2013

 

ACKNOWLEDGMENT OF SPONSORSHIP

This work was sponsored by one or more of the following as noted:

�� American Association of State Highway and Transportation Officials, in cooperation with the Federal Highway Administration, and was conducted in the National Cooperative Highway Research Program,

�� Federal Transit Administration and was conducted in the Transit Cooperative Research Program,

�� American Association of State Highway and Transportation Officials, in cooperation with the Federal Motor Carriers Safety Administration, and was conducted in the Commercial Truck and Bus Safety Synthesis Program,

�� Federal Aviation Administration and was conducted in the Airports Cooperative Research Program,

which is administered by the Transportation Research Board of the National Academies.

DISCLAIMER This is an uncorrected draft as submitted by the research agency. The opinions and

conclusions expressed or implied in the report are those of the research agency. They are not necessarily those of the Transportation Research Board, the National Academies, or the program sponsors.

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TABLE OF CONTENTS

1. INTRODUCTION ...................................................................................................................... 3 

1.1 BACKGROUND ........................................................................................................................ 3 1.2 PROBLEM STATEMENT AND LITERATURE REVIEW ................................................................. 5 1.3 OBJECTIVE AND SCOPE ........................................................................................................ 17 1.4 REPORT ORGANIZATION ...................................................................................................... 17 

2. PARAMETRIC STUDY DESIGN ........................................................................................... 19 

2.1 OVERVIEW ....................................................................................................................... 19 2.2 EXPERMENTAL TEST GEOMETRIES AND VARIATIONS ON EXPERIMENTAL

TESTS ....................................................................................................................................... 19 2.3 PARAMETRIC STUDY TESTS ........................................................................................ 23 

2.3.1 Joint Configurations ..................................................................................................... 23 2.3.2 Other Joint Parametric Considerations ........................................................................ 33 2.3.3 Joint Design ................................................................................................................. 35 2.3.4 Member Design ............................................................................................................ 48 

3. TEST SIMULATION PROCEDURES .................................................................................... 51 

3.1 FINITE ELEMENT MODELS ........................................................................................... 51 3.2 GEOMETRIC IMPERFECTIONS ..................................................................................... 53 3.3 MATERIAL PROPERTIES ................................................................................................ 55 3.4 FASTENER STRENGTHS ................................................................................................ 56 3.5 MODELING OF TENSION RUPTURE OR SHEAR RUPTURE RESISTANCES .......... 57 

4. TEST SIMULATION RESULTS ............................................................................................. 59 

4.1 DEFINITION OF FAILURE AND DETERMINATION OF FAILURE MODES ............. 59 4.2 OVERVIEW ....................................................................................................................... 61 4.3 WARREN WITH VERTICAL EXPERIMENTAL TEST CONFIGURATIONS, UNCHAMFERED MEMBERS ................................................................................................ 76 

4.3.1 E1-U-307SS-WV ......................................................................................................... 76 4.3.2 E2-U-307LS-WV ......................................................................................................... 79 4.3.3 E3-U-307SL-WV ......................................................................................................... 82 4.3.4 E4-U-490SS-WV ......................................................................................................... 86 4.3.5 E5-U-490LS-WV ......................................................................................................... 89 

4.4 WARREN WITHOUT VERTICAL VARIATIONS ON EXPERIMENTAL TEST

CONFIGURATIONS, UNCHAMFERED MEMBERS ........................................................... 92 4.4.1 E1-U-307SS-W ............................................................................................................ 92 4.4.2 E2-U-307LS-W ............................................................................................................ 95 4.4.3 E3-U-307SL-W ............................................................................................................ 97 4.4.4 E4-U-490SS-W .......................................................................................................... 100 4.4.5 E5-U-490LS-W .......................................................................................................... 102 

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4.5 WARREN WITH VERTICAL PARAMETRIC TEST CONFIGURATIONS ................. 105 4.5.1 P1-C-WV-M .............................................................................................................. 105 4.5.2 P2-C-TCS-WV-M ...................................................................................................... 112 4.5.3 P3-C-WV-P ................................................................................................................ 116 4.5.4 P4-C-WV-P ................................................................................................................ 121 4.5.5 P5-WV-NP ................................................................................................................. 127 4.5.6 P6-WV-NP ................................................................................................................. 141 4.5.7 P7-WV-INF ................................................................................................................ 150 4.5.8 P8-WV-INF ................................................................................................................ 158 

4.6 PRATT PARAMETRIC TEST CONFIGURATIONS ..................................................... 171 4.6.1 P9-C-P-NP ................................................................................................................. 171 4.6.2 P10-C-P-NP ............................................................................................................... 175 

4.7 WARREN WITHOUT VERTICAL PARAMETRIC TEST CONFIGURATIONS ........ 179 4.7.1 P11-C-W-M ............................................................................................................... 179 4.7.2 P12-C-W-P ................................................................................................................. 183 4.7.3 P13-W-NP .................................................................................................................. 187 4.7.4 P14-W-INF ................................................................................................................ 195 

4.8 CORNER JOINT PARAMETRIC TEST CONFIGURATIONS ...................................... 204 4.8.1 P15-C-CJ .................................................................................................................... 204 4.8.2 P16-C-CJ .................................................................................................................... 210 

4.9 PARAMETRIC TEST CONFIGURATIONS WITH A POSITIVE ANGLE BETWEEN

THE CHORD MEMBERS...................................................................................................... 216 4.9.1 P17-C-POS ................................................................................................................. 216 4.9.2 P18-C-POS ................................................................................................................. 221 

4.10 PARAMETRIC TEST CONFIGURATIONS WITH A NEGATIVE ANGLE BETWEEN

THE CHORD MEMBERS...................................................................................................... 225 4.10.1 P19-NEG .................................................................................................................. 225 4.10.2 P20-NEG .................................................................................................................. 232 

4.11 PARAMETRIC TEST CONFIGURATIONS WITH SHINGLE PLATES .................... 240 4.11.1 P3-C-SP(0.4:0.2)-WV-P .......................................................................................... 240 4.11.2 P3-C-SP(0.5:0.25)-WV-P ........................................................................................ 245 4.11.3 P3-C-SP(0.3:0.3)-WV-P .......................................................................................... 246 4.11.4 P5-C-SP(0.3:0.2)-WV-NP ....................................................................................... 249 4.11.5 P5-C-SP(0.3:0.3)-WV-NP ....................................................................................... 254 4.11.6 P12-C-SP(0.5:0.5)-W-P ........................................................................................... 257 

4.12 TEST CONFIGURATIONS WITH EDGE STIFFENERS ............................................. 261 4.12.1 E4-U-490SS(3/8)-SES-WV ..................................................................................... 261 4.12.2 E4-U-490SS(3/8)-EES-WV ..................................................................................... 264 4.12.3 E5-U-490LS(3/8)-SES-WV ..................................................................................... 267 4.12.4 E5-U-490LS(3/8)-EES-WV ..................................................................................... 269 4.12.5 P5-U-EES-WV-NP .................................................................................................. 273 

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4.12.6 P14-C(0.5)-EES-W-INF .......................................................................................... 278 4.13 PARAMETRIC TEST CONFIGURATIONS WITH CORRODED GUSSET PLATES 282 

4.13.1 P8-C-C1-WV-INF and P8-C-C2-WV-INF .............................................................. 284 4.13.2 P8-C-COS-WV-INF ................................................................................................ 288 4.13.3 P14-U-C1-W-INF and P14-U-C2-W-INF ............................................................... 290 4.13.4 P14-U-COS-W-INF ................................................................................................. 293 

4.14 PARAMETRIC CONFIGURATIONS WITH CORRODED GUSSET PLATES

REINFORCED BY SHINGLE PLATES ................................................................................ 295 4.14.1 P8-C-C1-SP(0.5:0.25)-WV-INF .............................................................................. 295 4.14.2 P8-C-COS-SP(0.5:0.25)-WV-INF ........................................................................... 298 4.14.3 P14-U-C1-SP(0.5:0.25)-W-INF ............................................................................... 300 4.14.4 P14-U-COS-SP(0.5:0.25)-W-INF ............................................................................ 303 

5. DISCUSSION OF RESULTS AND DEVELOPMENT OF DESIGN RECOMMENDATIONS..................................................................................................................................................... 307 

5.1 OVERVIEW ..................................................................................................................... 307 5.2 CHORD SPLICE ECCENTRIC COMPRESSION ........................................................... 314 

5.2.1 Method 1 .................................................................................................................... 314 5.2.2 Method 2 .................................................................................................................... 321 

5.3 CHORD SPLICE ECCENTRIC TENSION ...................................................................... 326 5.3.1 Method 1 .................................................................................................................... 326 5.3.2 Method 2 .................................................................................................................... 330 5.3.3 Importance of considering splice eccentricity ........................................................... 333 

5.4 CONCENTRIC TENSION OR COMPRESSION ............................................................ 335 5.5 FULL SHEAR PLANE YIELDING ................................................................................. 337 

5.5.1 Method 1 .................................................................................................................... 337 5.5.2 Theoretical Plastic Strength Interaction on Full Shear Plane, Elastic-Plastic Material Idealization .......................................................................................................................... 340 5.5.3 Method 2 .................................................................................................................... 343 

5.6 DIAGONAL BUCKLING, FULL WHITMORE SECTION ............................................ 344 5.6.1 Method 1 .................................................................................................................... 344 5.6.2 Method 2 .................................................................................................................... 361 

5.7 DIAGONAL BUCKLING, TRUNCATED WHITMORE SECTION .............................. 366 5.7.1 Method 1 .................................................................................................................... 366 5.7.2 Method 2 .................................................................................................................... 371 

5.8 INFLUENCE OF SHINGLE PLATES ............................................................................. 382 5.8.1 Method 1 .................................................................................................................... 383 5.8.2 Method 2 .................................................................................................................... 387 

5.9 INFLUENCE OF EDGE STIFFENERS ........................................................................... 390 5.10 HANDLING OF CORROSION EFFECTS .................................................................... 391 

6. SUMMARY AND CONCLUSIONS ..................................................................................... 397 

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7. REFERENCES ....................................................................................................................... 401 

 

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LIST OF FIGURES

Figure 1.  Geometry of the effective column assumed in the “Thornton Method.” .................. 7 Figure 2.  Offset distance c used by Dowswell (2006), along with Lmid, as a predictor of when

stability effects may be neglected in the calculation of the gusset plate compressive resistance. .................................................................................................................. 8 

Figure 3.  Gusset plate configurations evaluated by Dowswell (2006 and 2012a). .................. 9 Figure 4.  Elevation rendering of the critical U10 joint of the I-35W Bridge. ........................ 13 Figure 5.  Buckled gusset plates in the I-90 Bridge over Grand River in Lake County, Ohio

(courtesy of FHWA). ............................................................................................... 13 Figure 6.  Constant thickness contours on the critical gusset plate in the I-90 Bridge over

Grand River in Lake County, Ohio (ODOT, 2008). ............................................... 14 Figure 7.  Photo of one of truss joints in the DeSoto Bridge, which carried Minnesota

Highway 23 over the Mississippi River (MnDOT, 2013). ...................................... 15 Figure 8.  Photo of gusset plates in the Interstate 40 Bridge over the French Broad River in

Jefferson County, TN (bridgehunter.com, 2013). ................................................... 15 Figure 9.  E1-U-307SS-WV test geometry. ............................................................................. 20 Figure 10.  E2-U-307LS-WV test geometry. ............................................................................. 20 Figure 11.  E3-U-307SL-WV test geometry. ............................................................................. 21 Figure 12.  E4-U-490SS-WV test geometry. ............................................................................. 21 Figure 13.  E5-U-490LS-WV test geometry. ............................................................................. 22 Figure 14.  Truss subassembly configurations for gusset plate parametric studies. .................. 25 Figure 15.  Warren with vertical configurations. ....................................................................... 27 Figure 16.  Pratt configurations. ................................................................................................ 28 Figure 17.  Warren without vertical configurations. .................................................................. 28 Figure 18.  Other configurations. ............................................................................................... 29 Figure 19.  Typical gusset plate joint design for initial set of tests (P5-C-WV-NP). ................ 36 Figure 20.  Nonlinear shear-force shear-displacement curves for A307 and A490 bolts in single

shear used in modeling the experimental tests and variations on these tests. ......... 37 Figure 21.  Nonlinear shear-force shear-displacement curves for hot driven rivets in single

shear used in the parametric study test simulations. ............................................... 38 Figure 22.  Free-body diagram of end of left-hand chord member in joint P5-C-WV-NP, based

on design assumption of equal average stress in the chord flanges and webs associated with the web and flange forces transferred to the joint. ......................... 40 

Figure 23.  Free-body diagram of an individual gusset plate in joint P5-C-WV-NP. ............... 41 Figure 24.  Free-body diagram of end of right-hand chord member in joint P5-C-WV-NP. .... 42 Figure 25.  Free-body diagram of left-hand chord member in joint P5-C-WV-NP, based on the

assumption that the fasteners in the flange and web splice plates are loaded to their shear capacity. ......................................................................................................... 43 

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Figure 26.  Alternate free-body diagram of an individual gusset plate in joint P5-C-WV-NP, based on the assumption that the fasteners in the flange and web splice plates are loaded to their shear capacity. ................................................................................. 43 

Figure 27.  Alternate free-body diagram of right-hand chord member in joint P5-C-WV-NP, based on the assumption that the fasteners in the flange and web splice plates are loaded to their shear capacity. ................................................................................. 44 

Figure 28.  Second alternate free-body diagram of right-hand chord member in joint P5-C-WV-NP, based on developing the shear capacity of 28 fasteners to the flange splice plates and 18 fasteners to the web splice plates on the right-hand side of the joint. 45 

Figure 29.  Second alternate free-body diagram of an individual gusset plate in joint P5-C-WV-NP, based on developing the shear capacity of 28 fasteners to the flange splice plates and 18 fasteners to the web splice plates on the right-hand side of the joint. 45 

Figure 30.  Second alternate free-body diagram of left-hand chord member in joint P5-C-WV-NP, based on developing the shear capacity of 28 fasteners to the flange splice plates and 18 fasteners to the web splice plates on the right-hand side of the joint. 46 

Figure 31.  Typical test simulation model (shown for P5-C-WV-NP). ..................................... 51 Figure 32.  Length from work point within which the members are modeled using shell finite

elements. .................................................................................................................. 52 Figure 33.  Typical loading and boundary conditions (shown for P5-C-WV-NP). ................... 53 Figure 34.  Typical geometric imperfection shapes on gusset plate joints (shown for P5-C-WV-

NP). ......................................................................................................................... 55 Figure 35.  True stress-strain curves for Grade 50 and Grade 100 steel. ................................... 56 Figure 36.  Nonlinear shear-force shear-displacement curve for A307 and A490 bolts in single

shear. ....................................................................................................................... 57 Figure 37.  Nonlinear shear-force shear-displacement curve for hot driven rivets in single

shear. ....................................................................................................................... 58 Figure 38.  Gusset plate geometry and design forces for E1-U-307SS-WV. ............................ 77 Figure 39.  In-plane vs. out-of-plane displacements, E1-U-307SS(3/8)-WV. .......................... 78 Figure 40.  von Mises stress response contours at the limit load (ALF=0.94), E1-U-

307SS(3/8)-WV ....................................................................................................... 78 Figure 41.  Equivalent plastic strain response contours at the limit load (ALF=0.94), E1-U-

307SS(3/8)-WV ....................................................................................................... 79 Figure 42.  Gusset plate geometry and design forces for E2-U-307LS-WV. ............................ 80 Figure 43.  In-plane vs. out-of-plane displacements, E2-U-307LS(3/8)-WV. .......................... 81 Figure 44.  von Mises stress response contours at the limit load (ALF=1.10), E2-U-

307LS(3/8)-WV. ..................................................................................................... 81 Figure 45.  Equivalent plastic strain response contours at the limit load (ALF=1.10, E2-U-

307LS(3/8)-WV. ..................................................................................................... 82 Figure 46.  Gusset plate geometry and design forces for E3-U-307SL-WV. ............................ 83 Figure 47.  In-plane vs. out-of-plane displacements, E3-U-307SL(3/8)-WV. .......................... 84 

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Figure 48.  von Mises stress response contours at the limit load (ALF=1.06), E3-U-307SL(3/8)-WV. ..................................................................................................... 84 

Figure 49.  Equivalent plastic strain response contours at the limit load (ALF=1.06), E3-U-307SL(3/8)-WV. ..................................................................................................... 85 

Figure 50.  Equivalent plastic strain response contours at the 4% PEEQ Limit (ALF=1.02), E3-U-307SL(3/8)-WV .................................................................................................. 85 

Figure 51.  Shear stress response contours at the 4% PEEQ Limit (ALF=1.02), E3-U-307SL(3/8)-WV. ..................................................................................................... 86 

Figure 52.  Gusset plate geometry and design forces for E4-U-490SS-WV. ............................ 87 Figure 53.  In-plane vs. out-of-plane displacements, E4-U-490SS(3/8)-WV. .......................... 87 Figure 54.  von Mises stress response contours at the limit load (ALF=1.04), E4-U-

490SS(3/8)-WV. ...................................................................................................... 88 Figure 55.  Equivalent plastic strain response contours at the limit load (ALF=1.04), E4-U-

490SS(3/8)-WV ....................................................................................................... 88 Figure 56.  Gusset plate geometry and design forces for E5-U-490LS-WV. ............................ 90 Figure 57.  In-plane vs. out-of-plane displacements, E5-U-490LS(3/8)-WV ........................... 91 Figure 58.  von Mises stress response contours at the limit load (ALF=0.97), E5-U-

490LS(3/8)-WV. ..................................................................................................... 91 Figure 59.  Equivalent plastic strain response contours at the limit load (ALF=0.97), E5-U-

490LS(3/8)-WV. ..................................................................................................... 92 Figure 60.  Design forces for E1-U-307SS-W. .......................................................................... 93 Figure 61.  Load-displacement plot, E1-U-307SS(3/8)-W. ....................................................... 93 Figure 62.  von Mises stress response contours at the limit load (ALF=0.83), E1-U-

307SS(3/8)-W (DSF = 5). ....................................................................................... 94 Figure 63.  Equivalent plastic strain response contours at the limit load (ALF=0.83), E1-U-

307SS(3/8)-W (DSF = 5). ....................................................................................... 94 Figure 64.  Design forces for E2-U-307LS-W. ......................................................................... 95 Figure 65.  Load-displacement plot, E2-U-307LS(3/8)-W. ...................................................... 96 Figure 66.  von Mises stress response contours at the limit load (ALF=0.95), E2-U-

307LS(3/8)-W (DSF = 5). ....................................................................................... 96 Figure 67.  Equivalent plastic strain response contours at the limit load (ALF=0.95), E2-U-

307LS(3/8)-W (DSF = 5). ....................................................................................... 97 Figure 68.  Design forces for E3-U-307SL-W. ......................................................................... 97 Figure 69.  Load-displacement plot, E3-U-307SL(3/8)-W. ...................................................... 98 Figure 70.  von Mises stress response contours at the limit load (ALF=1.0), E3-U-307SL(3/8)-

W (DSF = 5). ........................................................................................................... 99 Figure 71.  Equivalent plastic strain response contours at the limit load (ALF=1.0), E3-U-

307SL(3/8)-W (DSF = 5). ....................................................................................... 99 Figure 72.  Design forces for E4-U-490SS-W. ........................................................................ 100 Figure 73.  Load-displacement plot, E4-U-490SS(3/8)-W. ..................................................... 101 

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Figure 74.  von Mises stress response contours at the limit load (ALF=0.98), E4-U-490SS(3/8)-W (DSF = 5). ..................................................................................... 101 

Figure 75.  Equivalent plastic strain response contours at the limit load (ALF=0.98), E4-U-490SS(3/8)-W (DSF = 5). ..................................................................................... 102 

Figure 76.  Design forces for E5-U-490LS-W. ....................................................................... 102 Figure 77.  Load-displacement plot, E5-U-490LS(3/8)-W. .................................................... 103 Figure 78.  von Mises stress response contours at the limit load (ALF=1.23), E5-U-

490LS(3/8)-W (DSF = 5). ..................................................................................... 104 Figure 79.  Equivalent plastic strain response contours at the limit load (ALF=1.23), E5-U-

490LS(3/8)-W (DSF = 5). ..................................................................................... 104 Figure 80.  Applied loads and boundary conditions for P1-C-CCS-WV-M. .......................... 105 Figure 81.  Gusset plate geometry and design forces for P1-C-CCS-WV-M. ........................ 106 Figure 82.  Load-displacement plot for P1-C-CCS(0.4)-WV-M. ............................................ 108 Figure 83.  von Mises stress contours for P1-C-CCS(0.4)-WV-M at the PEEQ > 4 % strength

condition occurring at an ALF of 1.37 (DSF = 5). ................................................ 108 Figure 84.  Equivalent plastic strain contours for P1-C-CCS(0.4)-WV-M at the PEEQ > 4 %

strength condition occurring at an ALF of 1.37 (DSF = 5). .................................. 109 Figure 85.  Load-displacement curves for P1-C-MTB(0.35)-WV-M and P1-C-MTB(0.4)-WV-

M. .......................................................................................................................... 110 Figure 86.  von Mises stress contours for P1-C-MTB(0.35)-WV-M at the limit load occurring

at an ALF of 1.96 (DSF = 5). ................................................................................ 111 Figure 87.  Equivalent plastic strain contours for P1-C-MTB(0.35)-WV-M at the limit load

occurring at an ALF of 1.96 (DSF = 5). ................................................................ 111 Figure 88.  Applied loads and boundary conditions of P2-C-TCS-WV-M. ............................ 112 Figure 89.  Gusset plate geometry and design forces for P2-C-TCS-WV-M. ......................... 113 Figure 90.  Load-displacement plot for P2-C-TCS(0.4)-WV-M. ............................................ 114 Figure 91.  von Mises stress contours for P2-C-TCS(0.4)-WV-M at the PEEQ > 4 % strength

condition occurring at an ALF of 1.33 (DSF = 5). ................................................ 115 Figure 92.  Equivalent plastic strain contours for P2-C-TCS(0.4)-WV-M at the PEEQ > 4 %

strength condition occurring at an ALF of 1.33 (DSF = 5). .................................. 115 Figure 93.  Applied loads and boundary conditions for P3-C-WV-P. ..................................... 116 Figure 94.  Gusset plate geometry and design forces for P3-C-WV-P. ................................... 117 Figure 95.  Load-displacement plot for P3-C(0.5)-WV-P. ...................................................... 119 Figure 96.  von Mises stress contours for P3-C(0.5)-WV-P at the PEEQ > 4 % strength

condition occurring at an ALF of 0.98 (DSF = 10). .............................................. 119 Figure 97.  Equivalent plastic strain contours for P3-C(0.5)-WV-P at the PEEQ > 4 % strength

condition occurring at an ALF of 0.98 (DSF = 10). .............................................. 120 Figure 98.  von Mises, normal, and shear stresses in the gusset plate along the vertical plane at

the right-hand edge of the vertical member in P3-C(0.5)-WV-P, generated at the PEEQ > 4 % strength condition occurring at an ALF of 0.98. ............................. 121 

Figure 99.  Applied loads and boundary conditions of P4-C-WV-P. ...................................... 122 

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Figure 100.  Gusset plate geometry and design forces for P4-C-WV-P. ................................... 123 Figure 101.  Load-displacement plot for P4-C(0.8)-WV-P. ...................................................... 125 Figure 102.  von Mises stress contours for P4-C(0.8)-WV-P at the PEEQ > 4 % strength

condition occurring at an ALF of 0.97 (DSF = 10). .............................................. 125 Figure 103.  Equivalent plastic strain contours for P4-C(0.8)-WV-P at the PEEQ > 4 % strength

condition occurring at an ALF of 0.97 (DSF = 10). .............................................. 126 Figure 104.  von Mises, normal, and shear stresses along the vertical plane of P4-C-WV-P at the

PEEQ > 4 % strength condition occurring at an ALF of 0.97. ............................. 126 Figure 105.  Applied loads and boundary conditions of P5-C-WV-NP. ................................... 127 Figure 106.  Gusset plate geometry and design forces for P5-C-WV-NP. ................................ 128 Figure 107.  Load-displacement plot for P5-C(0.40)-WV-NP. ................................................. 130 Figure 108.  von Mises stress contours for P5-C(0.40)-WV-NP at the limit load occurring at an

ALF of 0.94 (DSF = 5). ......................................................................................... 130 Figure 109.  Equivalent plastic strain contours for P5-C(0.40)-WV-NP at the limit load

occurring at an ALF of 0.94 (DSF = 5). ................................................................ 131 Figure 110.  von Mises, normal, and shear stresses along the vertical plane of P5-C(0.40)-WV-

NP at the limit load occurring at an ALF of 0.94. ................................................. 131 Figure 111.  von Mises, normal, and shear stresses along the horizontal plane of P5-C(0.40)-

WV-NP at the limit load occurring at an ALF of 0.94. ......................................... 132 Figure 112.  Gusset plate geometry and design forces for P5-U-WV-NP. ................................ 133 Figure 113.  Load-displacement plot for P5-U(0.4)-WV-NP. ................................................... 135 Figure 114.  von Mises stress contours for P5-U(0.4)-WV-NP at the limit load occurring at an

ALF of 0.78 (DSF = 5). ......................................................................................... 136 Figure 115.  Equivalent plastic strain contours for P5-U(0.4)-WV-NP at the limit load occurring

at an ALF of 0.78 (DSF = 5). ................................................................................ 136 Figure 116.  von Mises stress contours for P5-U(0.4)-WV-NP at a post-peak ALF of 0.55 (DSF

= 5). ....................................................................................................................... 137 Figure 117.  Equivalent plastic strain contours for P5-U(0.40)-WV-NP at a post-peak ALF of

0.55 (DSF = 5). ...................................................................................................... 137 Figure 118.  Load-displacement plot of P5-C(0.4)-WV-NP, P5-C-HS(0.4)-WV-NP, and P5-C-

HS(0.2)-WV-NP. ................................................................................................... 139 Figure 119.  von Mises stress contours of P5-C-HS(0.4)-WV-NP at the limit load occurring at

an ALF of 1.69 (DSF = 5). .................................................................................... 139 Figure 120.  Equivalent plastic strain contours of P5-C-HS(0.4)-WV-NP at the limit load

occurring at an ALF of 1.69 (DSF = 5). ................................................................ 140 Figure 121.  von Mises stress contours of P5-C-HS(0.2)-WV-NP at the limit load occurring at

an ALF of 0.64 (DSF = 5). .................................................................................... 140 Figure 122.  Equivalent plastic strain contours of P5-C-HS(0.2)-WV-NP at the limit load

occurring at an ALF of 0.64 (DSF = 5). ................................................................ 141 Figure 123.  Applied loads and boundary conditions of P6-C-WV-NP. ................................... 142 Figure 124.  Gusset plate geometry and design forces for P6-C-WV-NP. ................................ 142 

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Figure 125.  Load-displacement plot for P6-C(0.5)-WV-NP. ................................................... 144 Figure 126.  von Mises stress contours for P6-C(0.5)-WV-NP at the limit load occurring at an

ALF of 0.98 (DSF = 10). ....................................................................................... 144 Figure 127.  Equivalent plastic strain contours for P6-C(0.5)-WV-NP at the limit load occurring

at an ALF of 0.98 (DSF = 10). .............................................................................. 145 Figure 128.  Gusset plate geometry and design forces for P6U-WV-NP. ................................. 146 Figure 129.  Load-displacement plot for P6-U(0.5)-WV-NP. ................................................... 148 Figure 130.  von Mises stress contours for P6-U(0.5)-WV-NP at the limit load occurring at an

ALF of 0.80 (DSF = 5). ......................................................................................... 148 Figure 131.  Equivalent plastic strain contours for P6-U(0.5)-WV-NP at the limit load occurring

at an ALF of 0.80 (DSF = 5). ................................................................................ 149 Figure 132.  von Mises stress contours for P6-U(0.5)-WV-NP at a post-peak ALF of 0.59 (DSF

= 5). ....................................................................................................................... 149 Figure 133.  Equivalent plastic strain contours for P6-U(0.5)-WV-NP at a post-peak ALF of

0.59 (DSF = 5). ...................................................................................................... 150 Figure 134.  Applied loads and boundary conditions of P7-C-WV-INF. .................................. 151 Figure 135.  Gusset plate geometry and design forces for P7-C-WV-INF. .............................. 151 Figure 136.  Load-displacement plot for P7-C(0.7)-WV-INF. .................................................. 153 Figure 137.  von Mises stress contours for P7-C(0.7)-WV-INF at the PEEQ > 4 % strength limit

occurring at an ALF of 1.28 (DSF = 5). ................................................................ 154 Figure 138.  Equivalent plastic strain contours for P7-C(0.7)-WV-INF at the PEEQ > 4 %

strength limit occurring at an ALF of 1.28 (DSF = 5). ......................................... 154 Figure 139.  Load-displacement plot for P7-C(0.7)-WV-INF, P7-C-HS(0.7)-WV-INF, and P7-

C-HS(0.35)-WV-INF. ........................................................................................... 156 Figure 140.  von Mises stress contours for P7-C-HS(0.7)-WV-INF at the PEEQ > 4 % strength

limit occurring at an ALF of 2.28 (DSF = 5). ....................................................... 156 Figure 141.  Equivalent plastic strain contours for P7-C-HS(0.7)-WV-INF at the PEEQ > 4 %

strength limit occurring at an ALF of 2.28 (DSF = 5). ......................................... 157 Figure 142.  von Mises stress contours for P7-C-HS(0.35)-WV-INF at limit load occurring at an

ALF of 0.99 (DSF = 5). ......................................................................................... 157 Figure 143.  Equivalent plastic strain contours for P7-C-HS(0.35)-WV-INF at the limit load

occurring at an ALF of 0.99 (DSF = 5). ................................................................ 158 Figure 144.  Applied loads and boundary conditions of P8-C-WV-INF. .................................. 159 Figure 145.  Gusset plate geometry and design forces for P8-C-WV-INF. .............................. 160 Figure 146.  Load-displacement plot for P8-C(0.5)-WV-INF. .................................................. 161 Figure 147.  von Mises stress contours for P8-C(0.5)-WV-INF at the PEEQ > 4 % strength limit

occurring at an ALF of 0.97 (DSF = 10). .............................................................. 162 Figure 148.  Equivalent plastic strain contours for P8-C(0.5)-WV-INF at the PEEQ > 4 %

strength limit occurring at an ALF of 0.97 (DSF = 10). ....................................... 162 Figure 149.  Gusset plate geometry and design forces for P8-U-WV-INF. .............................. 163 Figure 150.  Load-displacement plot for P8-U(0.5)-WV-INF. ................................................. 165 

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Figure 151.  von Mises stress contours of P8-U(0.5)-WV-INF at the limit load occurring at an ALF of 0.94 (DSF = 5). ......................................................................................... 165 

Figure 152.  Equivalent plastic strain contours of P8-U(0.5)-WV-INF at the limit load occurring at an ALF of 0.94 (DSF = 5). ................................................................................ 166 

Figure 153.  von Mises stress contours of P8-U(0.5)-WV-INF at a post-peak ALF of 0.70 (DSF = 5). ....................................................................................................................... 166 

Figure 154.  Equivalent plastic strain contours of P8-U(0.5)-WV-INF at a post-peak ALF of 0.70 (DSF = 5). ...................................................................................................... 167 

Figure 155.  Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-HS(0.5)-WV-INF, and P8-C-HS(0.25)-WV-INF. ...................................................................................... 168 

Figure 156.  von Mises stress contours for P8-C-HS(0.5)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 1.79 (DSF = 5). ....................................................... 169 

Figure 157.  Equivalent plastic strain contours for P8-C-HS(0.5)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 1.79 (DSF = 5). ......................................... 169 

Figure 158.  von Mises stress contours for P8-C-HS(0.25)-WV-INF at the limit load occurring at an ALF of 0.77 (DSF = 5). ................................................................................ 170 

Figure 159.  Equivalent plastic strain contours for P8-C-HS(0.25)-WV-INF at the limit load occurring at an ALF of 0.77 (DSF = 5). ................................................................ 170 

Figure 160.  Applied loads and boundary conditions of P9-C-P-NP. ....................................... 171 Figure 161.  Gusset plate geometry and design forces for P9-C-P-NP. .................................... 172 Figure 162.  Load-displacement plot for P9-C(0.2)-P-NP. ....................................................... 173 Figure 163.  von Mises stress contours for P9-C(0.2)-P-NP at the limit load occurring at an ALF

of 0.96 (DSF = 5). ................................................................................................. 174 Figure 164.  Equivalent plastic strain contours for P9-C(0.2)-P-NP at the limit load occurring at

an ALF of 0.96 (DSF = 5). .................................................................................... 174 Figure 165.  Applied loads and boundary conditions for P10-C-P-NP. .................................... 175 Figure 166.  Gusset plate geometry and design forces for P10-C-P-NP. .................................. 176 Figure 167.  Load-displacement plot for P10-C(0.2)-P-NP. ..................................................... 177 Figure 168.  von Mises stress contours for P10-C(0.2)-P-NP at the PEEQ > 4 % strength limit

occurring at an ALF of 1.73 (DSF = 2). ................................................................ 178 Figure 169.  Equivalent plastic strain contours for P10-C(0.2)-P-NP at the PEEQ > 4 % strength

limit occurring at an ALF of 1.73 (DSF = 2). ....................................................... 178 Figure 170.  Applied loads and boundary conditions of P11-C-W-M. ..................................... 179 Figure 171.  Gusset plate geometry and design forces for P11-C-W-M. .................................. 180 Figure 172.  Load-displacement plot for P11-C(0.45)-W-M. ................................................... 181 Figure 173.  von Mises stress contours for P11-C(0.45)-W-M at the PEEQ > 4 % strength limit

occurring at an ALF of 1.41 (DSF = 5). ................................................................ 182 Figure 174.  Equivalent plastic strain contours for P11-C(0.45)-W-M at the PEEQ > 4 %

strength limit occurring at an ALF of 1.41 (DSF = 5). ......................................... 182 Figure 175.  Applied loads and boundary conditions of P12-C-W-P. ....................................... 183 Figure 176.  Gusset plate geometry and design forces for P12-C-W-P. .................................... 184 

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Figure 177.  Load-displacement plot for P12-C(1)-W-P. .......................................................... 185 Figure 178.  von Mises stress contours for P12-C(1)-W-P at the limit load occurring at an ALF

of 1.05 (DSF = 5). ................................................................................................. 186 Figure 179.  Equivalent plastic strain contours for P12-C(1)-W-P at the limit load occurring at

an ALF of 1.05 (DSF = 5). .................................................................................... 186 Figure 180.  Applied loads and boundary conditions of P13-C-W-NP. .................................... 187 Figure 181.  Gusset plate geometry and design forces for P13-C-W-NP. ................................. 188 Figure 182.  Load-displacement plot for P13-C(0.4)-W-NP. .................................................... 190 Figure 183.  von Mises stress contours for P13-C(0.4)-W-NP at the limit load occurring at an

ALF of 0.99 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load level in this problem). ............................................................................................ 190 

Figure 184.  Equivalent plastic strain contours for P13-C(0.4)-W-NP at the limit load occurring at an ALF of 0.99 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load level in this problem). .................................................................................... 191 

Figure 185.  von Mises, normal, and shear stresses in the gusset plate along the vertical plane at the left-hand edge of the vertical member of P13-C(0.4)-W-NP at the limit load occurring at an ALF of 0.99. ................................................................................. 191 

Figure 186.  Gusset plate geometry and design forces for P13-U-W-NP. ................................ 192 Figure 187.  Load-displacement plot for P13-U(0.4)-W-NP. .................................................... 194 Figure 188.  von Mises stress contours for P13-U(0.4)-W-NP at the limit load occurring at an

ALF of 0.85 (DSF = 10). ....................................................................................... 194 Figure 189.  Equivalent plastic strain contours for P13-U(0.4)-W-NP at the limit load occurring

at an ALF of 0.85 (DSF = 10). .............................................................................. 195 Figure 190.  Applied loads and boundary conditions of P14-C-W-INF. ................................... 196 Figure 191.  Gusset plate geometry and design forces for P14-C-W-INF. ............................... 197 Figure 192.  Load-displacement plot for P14-C(0.5)-W-INF. ................................................... 199 Figure 193.  von Mises stress contours for P14-C(0.5)-W-INF at the limit load occurring at an

ALF = 1.22 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load level in this problem). .................................................................................................... 199 

Figure 194.  Equivalent plastic strain contours for P14-C(0.5)-W-INF at the limit load occurring at an ALF of 1.22 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load level in this problem). .................................................................................... 200 

Figure 195.  von Mises, normal, and shear stresses along the horizontal plane of P14-C(0.5)-W-INF at the limit load occurring at an ALF of 1.22 (the PEEQ > 4 % strength limit also occurs at this load level in this problem). ...................................................... 200 

Figure 196.  Gusset plate geometry and design forces for P14-U-WV-INF. ............................ 201 Figure 197.  Load-displacement plot for P14-U(0.5)-W-INF. .................................................. 203 Figure 198.  von Mises stress contours for P14-U(0.5)-W-INF at the limit load occurring at an

ALF of 1.14 (DSF = 10). ....................................................................................... 203 Figure 199.  Equivalent plastic strain contours for P14-U(0.5)-W-INF at the limit load occurring

at an ALF of 1.14 (DSF = 10). .............................................................................. 204 

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Figure 200.  Applied loads and boundary conditions of P15-C-CJ. .......................................... 205 Figure 201.  Gusset plate geometry and design forces for P15-C-CJ. ....................................... 206 Figure 202.  Load-displacement plot for P15-C(0.5)-CJ. .......................................................... 208 Figure 203.  von Mises stress contours for P15-C(0.5)-CJ at the PEEQ > 4 % strength limit

occurring at an ALF of 1.15 (DSF = 2). ................................................................ 208 Figure 204.  Equivalent plastic strain contours for P15-C(0.5)-CJ at the PEEQ > 4 % strength

limit occurring at an ALF of 1.15 (DSF = 2). ....................................................... 209 Figure 205.  von Mises, normal, and shear stresses in the gusset plate along the vertical plane at

the right-hand edge of the vertical member in P15-C(0.5)-CJ at the PEEQ > 4 % strength condition occurring at an ALF of 1.15. ................................................... 209 

Figure 206.  von Mises, normal, and shear stresses in the gusset plate along the plane at the bottom of the horizontal chord in P15-C(0.5)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.15. ................................................................................. 210 

Figure 207.  Applied loads and boundary conditions of P16-C-CJ. .......................................... 211 Figure 208.  Gusset plate geometry and design forces for P16-C-CJ. ....................................... 212 Figure 209.  Load-displacement plot for P16-C(0.85)-CJ. ........................................................ 214 Figure 210.  von Mises stress contours for P16-C(0.85)-CJ at the PEEQ > 4 % strength limit

occurring at an ALF of 1.16 (DSF = 2). ................................................................ 214 Figure 211.  Equivalent plastic strain contours P16-C(0.85)-CJ at the PEEQ > 4 % strength limit

occurring at an ALF of 1.16 (DSF = 2). ................................................................ 215 Figure 212.  von Mises, normal, and shear stresses in the gusset plate along the vertical plane on

the right-hand side of the vertical member in P16-C(0.85)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.16. ........................................................... 215 

Figure 213.  von Mises, normal, and shear stresses in the gusset plate along the plane at the bottom of the horizontal chord in P16-C(0.85)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.16. ................................................................................. 216 

Figure 214.  Applied loads and boundary conditions of P17-C-POS. ....................................... 217 Figure 215.  Gusset plate geometry and design forces for P17-C-POS. .................................... 218 Figure 216.  Load-displacement plot for P17-C(0.6)-POS. ....................................................... 219 Figure 217.  von Mises stress contours for P17-C(0.6)-POS at the PEEQ > 4 % strength limit

occurring at an ALF of 1.47 (DSF = 5). ................................................................ 220 Figure 218.  Equivalent plastic strain contours for P17-C(0.6)-POS at the PEEQ > 4 % strength

limit occurring at an ALF of 1.47 (DSF = 5). ....................................................... 220 Figure 219.  Applied loads and boundary conditions of P18-C-POS. ....................................... 221 Figure 220.  Gusset plate geometry and design forces for P18-C-POS. .................................... 222 Figure 221.  Load-displacement plot for P18-C(0.6)-POS. ....................................................... 223 Figure 222.  von Mises stress contours for P18-C(0.6)-POS at the PEEQ > 4 % strength limit

occurring at an ALF of 1.30 (DSF = 10). .............................................................. 224 Figure 223.  Equivalent plastic strain contours for P18-C(0.6)-POS at the PEEQ > 4 % strength

limit occurring at an ALF of 1.30 (DSF = 10). ..................................................... 224 Figure 224.  Applied loads and boundary conditions of P19-C-CCS-NEG. ............................. 225 

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Figure 225.  Gusset plate geometry and design forces for P19-C-CCS-NEG. .......................... 226 Figure 226.  Load-displacement plot for P19-C-CCS(0.6)-NEG. ............................................. 228 Figure 227.  von Mises stress contours for P19-C-CCS(0.6)-NEG at the PEEQ > 4 % strength

limit occurring at an ALF of 0.89 (DSF = 10). ..................................................... 228 Figure 228.  Equivalent plastic strain contours for P19-C-CCS(0.6)-NEG at the PEEQ > 4 %

strength limit occurring at an ALF of 0.89 (DSF = 10). ....................................... 229 Figure 229.  Load-displacement plot for P19-C-MTB(0.6)-NEG. ............................................ 230 Figure 230.  von Mises stress contours for P19-C-MTB(0.6)-NEG at the limit load occurring at

an ALF of 1.01 (DSF = 10). .................................................................................. 230 Figure 231.  Equivalent plastic strain contours for P19-C-MTB(0.6)-NEG at the limit load

occurring at an ALF of 1.01 (DSF = 10). .............................................................. 231 Figure 232.  von Mises stress contours for P19-C-MTB(0.6)-NEG at a post-peak ALF of 0.93

(DSF = 10). ............................................................................................................ 231 Figure 233.  Equivalent plastic strain contours for P19-C-MTB(0.6)-NEG at a post-peak ALF of

0.93 (DSF = 10). .................................................................................................... 232 Figure 234.  Applied loads and boundary conditions of P20-C-CCS-NEG. ............................. 233 Figure 235.  Gusset plate geometry and design forces for P20-C-CCS-NEG. .......................... 234 Figure 236.  Load-displacement plot for P20-C-CCS(0.6)-NEG. ............................................. 235 Figure 237.  von Mises stress contours for P20-C-CCS(0.6)-NEG at the PEEQ > 4 % strength

limit occurring at an ALF of 1.03 (DSF = 10). ..................................................... 236 Figure 238.  Equivalent plastic strain contours for P20-C-CCS(0.6)-NEG at the PEEQ > 4 %

strength limit occurring at an ALF of 1.03 (DSF = 10). ....................................... 236 Figure 239.  Load-displacement plot for P20-C-MTB(0.6)-NEG. ............................................ 238 Figure 240.  von Mises stress contours for P20-C-MTB(0.6)-NEG at the limit load occurring at

an ALF of 1.22 (DSF = 10). .................................................................................. 238 Figure 241.  Equivalent plastic strain contours for P20-C-MTB(0.6)-NEG at the limit load

occurring at an ALF of 1.22 (DSF = 10). .............................................................. 239 Figure 242.  von Mises stress contours for P20-C-MTB(0.6)-NEG at a post-peak ALF of 1.01

(DSF = 10). ............................................................................................................ 239 Figure 243.  Equivalent plastic strain contours for P20-C-MTB(0.6)-NEG at a post-peak ALF of

1.01 (DSF = 10). .................................................................................................... 240 Figure 244.  Gusset and shingle plate geometries for P3-C-SP(0.4:0.2)-WV-NP (units = inches).

............................................................................................................................... 241 Figure 245.  Load-displacement plot for P3-C-SP(0.4:0.2)-WV-NP. ....................................... 243 Figure 246.  Equivalent plastic strain contours for P3-C-SP(0.4:0.2)-WV-NP at the PEEQ > 4 %

strength limit occurring at an ALF of 0.90 (DSF=10). ......................................... 243 Figure 247.  Equivalent plastic strain contours for P3-C-SP(0.4:0.2)-WV-NP at the PEEQ > 4 %

strength limit occurring at an ALF of 0.90 (DSF=10), shingle plate removed to show the contours on the gusset plate. .................................................................. 244 

Figure 248.  Distribution of vertical member force between main gusset and shingle plate for P3-C-SP(0.4:0.2)-WV-NP . ................................................................................... 244 

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Figure 249.  Distribution of right-hand diagonal force between main gusset and shingle plate for P3-C-SP(0.4:0.2)-WV-NP. .................................................................................... 245 

Figure 250.  Load-displacement plot for P3-C-SP(0.5:0.25)-WV-P. ........................................ 246 Figure 251.  Load-displacement plot for P3-C-SP(0.3:0.3)-WV-P. .......................................... 247 Figure 252.  Distribution of the vertical member force between main gusset and shingle plate for

P3-C-SP(0.3:0.3)-WV-P. ....................................................................................... 248 Figure 253.  Distribution of the right-hand diagonal force between main gusset and shingle plate

for P3-C-SP(0.3:0.3)-WV-P. ................................................................................. 248 Figure 254.  Gusset and shingle plate geometry for P5-C-SP(0.3:0.2)-WV-NP (units = inches).

............................................................................................................................... 249 Figure 255.  Load-displacement plot for P5-C-SP(0.3:0.2)-WV-NP. ....................................... 251 Figure 256.  Equivalent plastic strain contours for P5-C-SP(0.3:0.2)-WV-NP at the limit load

occurring at an ALF of 0.95 (DSF=5). .................................................................. 252 Figure 257.  Equivalent plastic strain contours for P5-C-SP(0.3:0.2)-WV-NP at the limit load

occurring at an ALF of 0.95 (DSF=5), shingle plate removed to show the contours on the gusset plate. ................................................................................................ 252 

Figure 258.  Distribution of compression diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.2)-WV-NP ............................................................................... 253 

Figure 259.  Distribution of tension diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.2)-WV-NP ..................................................................................... 253 

Figure 260.  Load-displacement plot for P5-C-SP(0.3:0.2)-WV-NP and P5-C-SP(0.3:0.3)-WV-NP. ......................................................................................................................... 255 

Figure 261.  von Mises contours for P5-C-SP(0.3:0.3)-WV-NP at the limit load occurring at an ALF of 1.14 (DSF=5). ........................................................................................... 255 

Figure 262.  von Mises contours for P5-C-SP(0.3:0.3)-WV-NP at the limit load occurring at an ALF of 1.14 (DSF=5), shingle plate removed to show the contours on the gusset plate. ...................................................................................................................... 256 

Figure 263.  Distribution of compression diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.3)-WV-NP. .............................................................................. 256 

Figure 264.  Distribution of tension diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.3)-WV-NP. .................................................................................... 257 

Figure 265.  Gusset plate and shingle plate geometry for P12-C-SP(0.5:0.5)-W-P (units = inches). .................................................................................................................. 258 

Figure 266.  Load-displacement curves for P12-C(1.0)-W-P and P12-C-SP(0.5:0.5)-W-P. .... 259 Figure 267.  Equivalent plastic strain contours for P12-C-SP(0.5:0.5)-W-P at the limit load

occurring at an ALF of 0.71 (DSF=5). .................................................................. 260 Figure 268.  Equivalent plastic strain contours for P12-C-SP(0.5:0.5)-W-P at the limit load

occurring at an ALF of 0.71 (DSF=5) ), shingle plate removed to show the contours on the gusset plate. ................................................................................................ 260 

Figure 269.  Configuration of E4-U-490SS(3/8)-SES-WV with short angle stiffeners attached to the inside of the gusset plates. ............................................................................... 262 

xvi

Figure 270.  Reference member forces of E4-U-490SS(3/8)-SES-WV. ................................... 262 Figure 271.  Load-displacement plot for E4-U-490SS(3/8)-SES-WV. ..................................... 263 Figure 272.  von Mises stress contours for E4-U-490SS(3/8)-SES-WV at the limit load

occurring at an ALF of 0.72 (DSF = 5). ................................................................ 263 Figure 273.  Equivalent plastic strain contours for E4-U-490SS(3/8)-SES-WV at the limit load

occurring at an ALF of 0.72 (DSF = 5). ................................................................ 264 Figure 274.  Load-displacement plot for E4-U-490SS(3/8)-EES-WV. ..................................... 265 Figure 275.  von Mises stress contours for E4-U-490SS(3/8)-EES-WV at the PEEQ > 4 %

strength limit occurring at an ALF of 0.84 (DSF = 5). ......................................... 266 Figure 276.  Equivalent plastic strain contours for E4-U-490SS(3/8)-EES-WV at the PEEQ > 4

% strength limit occurring at an ALF of 0.84 (DSF = 5). ..................................... 266 Figure 277.  Load-displacement plot for E5-U-490LS(3/8)-SES-WV. ..................................... 268 Figure 278.  von Mises stress contours for E5-U-490LS(3/8)-SES-WV at the limit load

occurring at an ALF of 0.60 (DSF = 5). ................................................................ 268 Figure 279.  Equivalent plastic strain contours for E5-U-490LS(3/8)-SES-WV at the limit load

occurring at an ALF of 0.60 (DSF = 5). ................................................................ 269 Figure 280.  Long stiffening angles applied externally on the horizontal and vertical edges of

E5-U-490LS(3/8)-EES-WV. ................................................................................. 270 Figure 281.  Load-displacement plot for E5-U-490LS(3/8)-EES-WV. .................................... 271 Figure 282.  von Mises stress contours for E5-U-490LS(3/8)-EES-WV at limit load and the

PEEQ > 4 % strength limit occurring at an ALF of 0.81 (DSF = 5). .................... 272 Figure 283.  Equivalent plastic strain contours for E5-U-490LS(3/8)-EES-WV at the limit load

and the PEEQ > 4 % strength limit occurring at an ALF of 0.81 (DSF = 5). ....... 272 Figure 284.  Edge stiffener geometry for P5-U-EES-WV-NP (units = inches). ....................... 273 Figure 285.  Load-displacement plot for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ edge

stiffener. ................................................................................................................. 274 Figure 286.  von Mises stress contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ edge

stiffener at the limit load occurring at an ALF of 0.97 (DSF = 5). ....................... 275 Figure 287.  Equivalent plastic strain contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½

edge stiffener at the limit load occurring at an ALF of 0.97 (DSF = 5). ............... 275 Figure 288.  von Mises stress contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ stiffener

at a post-peak ALF of 0.73 (DSF = 5). .................................................................. 276 Figure 289.  Equivalent plastic strain contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½

edge stiffener at a post-peak ALF of 0.73 (DSF = 5). ........................................... 276 Figure 290.  Percentage increase in joint capacity vs. relative stiffness of edge stiffener to gusset

plates (Istiffener/Ig) for P5-U-EES-WV-NP. ............................................................. 277 Figure 291.  Calculation of Istiffener and Ig. .................................................................................. 278 Figure 292.  Edge stiffener geometry for P14-C-EES-W-INF (units = inches). ....................... 279 Figure 293.  Load-displacement curves for P14-C(0.5)-W-INF and P14-C-EES(0.5)-W-INF. 280 

xvii

Figure 294.  von Mises stress contours for P14-C-EES(0.5)-W-INF with L3½ x 3½ x ⅜ stiffeners at the PEEQ > 4 % strength limit occurring at an ALF of 1.32 (DSF = 5). ............................................................................................................................... 281 

Figure 295.  Equivalent plastic strain contours for P14-C-EES(0.5)-W-INF with L3½ x 3½ x ⅜ stiffeners at the PEEQ > 4 % strength limit occurring at an ALF of 1.32 (DSF = 5). ............................................................................................................................... 281 

Figure 296.  Percentage increase in joint capacity vs. relative stiffness of edge stiffeners and gusset plate (Istiffener/Ig) for P14-C-EES(0.5)-W-INF. ............................................ 282 

Figure 297.  Corrosion of a gusset plate shown with the percentage loss in the gusset plate thickness highlighted at several locations (courtesy of Mn/DOT). ....................... 283 

Figure 298.  Corrosion on gusset plate just above the chord including holes just below the end of the diagonal (courtesy of Illinois DOT). ............................................................... 284 

Figure 299.  Corroded gusset plate geometry for P8-C-C1-WV-INF (units = inches). ............ 285 Figure 300.  Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-

C2-WV-INF. ......................................................................................................... 286 Figure 301.  Equivalent plastic strain contours for P8-C-C1-WV-INF at the PEEQ > 4 % limit

occurring at an ALF of 0.61 (DSF = 1). ................................................................ 287 Figure 302.  Equivalent plastic strain contours for P8-C-C2-WV-INF at the PEEQ > 4 % limit

occurring an ALF of 0.76 (DSF = 1). .................................................................... 287 Figure 303.  Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-

COS-WV-INF. ...................................................................................................... 288 Figure 304.  Equivalent plastic strain contours for P8-C-COS-WV-INF for the corroded gusset

plate at the PEEQ > 4 % strength limit occurring at an ALF of 0.83 (DSF = 1). . 289 Figure 305.  Equivalent plastic strain contours for P8-C-COS-WV-INF for the gusset plate

without corrosion at the PEEQ > 4 % strength limit occurring at an ALF of 0.83 (DSF = 1). .............................................................................................................. 289 

Figure 306.  Corroded gusset plate geometry for P14-U-C1-W-INF (units = inches). ............. 290 Figure 307.  Corroded gusset plate geometry for P14-U-C2-W-INF (units = inches). ............. 291 Figure 308.  Load-displacement plots for P14-U(0.5)-W-INF, P14-U-C1-W-INF and P14-U-C2-

W-INF. .................................................................................................................. 292 Figure 309.  Equivalent plastic strain contours for P14-U-C1-W-INF at the limit load occurring

at an ALF of 0.94 (DSF = 1). ................................................................................ 292 Figure 310.  Equivalent plastic strain contours for P14-U-C2-W-INF at the limit load occurring

at an ALF of 0.95 (DSF = 1). ................................................................................ 293 Figure 311.  Load-displacement curves for P14-U(0.5)-W-INF, P14-U-C1-W-INF, and P14-U-

COS-W-INF. ......................................................................................................... 294 Figure 312.  Equivalent plastic strain contours for P14-U-COS-W-INF at the limit load

occurring at an ALF of 1.07. ................................................................................. 294 Figure 313.  Gusset and shingle plate geometry for P8-C-C1-SP(0.5:0.25)-WV-INF (units =

inches). .................................................................................................................. 295 

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Figure 314.  Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-C1-SP(0.5:0.25)-WV-INF. .................................................................................... 296 

Figure 315.  Equivalent plastic strain contours for P8-C-C1-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.93 (DSF = 1). ............................................... 297 

Figure 316.  Equivalent plastic strain contours for P8-C-C1-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.93 (DSF = 1), shingle plate removed to show the contours on the gusset plate. ............................................................................ 297 

Figure 317.  Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, P8-C-COS-WV-INF, and P8-C-COS-SP(0.5:0.25)-WV-INF. ................................................ 299 

Figure 318.  Equivalent plastic strain contours for P8-C-COS-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.95 (DSF = 1). .................................... 299 

Figure 319.  Equivalent plastic strain contours for P8-C-COS-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.95 (DSF = 1) ), splice plate removed to show the contours on the gusset plate. .................................................................. 300 

Figure 320.  P14-U-C1-SP(0.5:0.25)-W-INF shingle plate geometry. ...................................... 301 Figure 321.  Load-displacement curves for P14-U(0.5)-W-INF, P14-U-C1-W-INF, and P14-U-

C1-SP(0.5:0.25)-W-INF. ....................................................................................... 302 Figure 322.  Equivalent plastic strain contours for P14-U-C1-SP(0.5:0.25)-W-INF at the limit

load occurring at an ALF of 1.22 (DSF = 1). ........................................................ 302 Figure 323.  Equivalent plastic strain contours for P14-U-C1-SP(0.5:0.25)-W-INF at the limit

load occurring at an ALF of 1.22 (DSF = 1) ), shingle plate removed to show the contours on the gusset plate. ................................................................................. 303 

Figure 324.  Load-displacement curves for P14-U(0.5)-W-INF, P14-U-COS-W-INF, and P14-U-COS-SP(0.5:0.25)-W-INF. ............................................................................... 304 

Figure 325.  Equivalent plastic strain contours for P14-U-COS-SP(0.5:0.25)-W-INF at the limit load occurring at an ALF of 1.20 (DSF = 1). ........................................................ 305 

Figure 326.  Equivalent plastic strain contours for P14-U-COS-SP(0.5:0.25)-W-INF at the limit load occurring at an ALF of 1.20 (DSF = 1), splice plate removed to show the contours on the gusset plate. ................................................................................. 305 

Figure 327.  Method 1 professional factor for cases governed by chord-splice eccentric compression or chord-splice eccentric tension ...................................................... 308 

Figure 328.  Method 2 professional factor for cases governed by chord-splice eccentric compression or chord-splice eccentric tension ...................................................... 308 

Figure 329.  Method 1 professional factor for cases with no chamfer governed by combined diagonal buckling (DB) or Partial Shear Plane Yielding (PSPY) ......................... 310 

Figure 330.  Method 2 professional factor for cases with no chamfer governed by diagonal buckling (DB) ........................................................................................................ 310 

Figure 331.  Method 2 professional factor for cases with no chamfer governed by full shear plane yielding (FSPY) ........................................................................................... 311 

Figure 332.  Method 2 professional factor for cases with no chamfer governed by diagonal buckling with a truncated Whitmore section (DB-TWS) ...................................... 312 

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Figure 333.  Method 1, chamfered cases governed by diagonal buckling, diagonal yielding, or partial shear plane yielding (DB, TY or PSPY) .................................................... 312 

Figure 334.  Method 2 professional factor for chamfered cases governed by diagonal buckling or diagonal yielding with a truncated Whitmore section (DB-TWS or DY-TWS) ... 313 

Figure 335.  Method 2 professional factor for chamfered cases governed by full shear plane yielding (FSPY) ..................................................................................................... 313 

Figure 336.  Method 1 base stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P19-C-CCS(0.6)-NEG. ................................... 315 

Figure 337.  Method 1 elastic stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P19-C-CCS(0.6)-NEG. ................................... 316 

Figure 338.  Method 1 base stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P4-C(0.8)-WV-P. ............................................ 319 

Figure 339.  Method 1 elastic stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P4-C(0.8)-WV-P. ............................................ 320 

Figure 340.  Method 2 plastic stress distribution in gusset plate and splice plates for evaluation of chord splice eccentric compression. ................................................................. 322 

Figure 341.  Method 2 plastic stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P19-C-CCS(0.6)-NEG. ................................... 324 

Figure 342.  Method 2 plastic stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P4-C-(0.8)-WV-P. ........................................... 325 

Figure 343.  Method 1 base stress distribution in gusset plate for evaluation of chord splice eccentric tension, shown on P2-C-TCS-WV-M. ................................................... 327 

Figure 344.  Method 1 elastic stress distribution in gusset plate for evaluation of chord splice eccentric tension, shown on P2-C-TCS-WV-M. ................................................... 328 

Figure 345.  Method 2 plastic stress distribution in gusset plate and splice plates for evaluation of chord splice eccentric tension, shown for a case in which all the plates are governed by tension yielding. ............................................................................... 331 

Figure 346.  Method 2 plastic stress distribution in gusset plate for evaluation of chord splice eccentric tension, shown on P2-C-TCS-WV-M. ................................................... 332 

Figure 347.  Professional factors for Whitmore section calculation of chord splice eccentric compression and chord-splice eccentric tension resistance. ................................. 333 

Figure 348.  Whitmore section model of chord splice neglecting gusset eccentricity, shown on P2-C-TCS(0.4)-WV-M. ........................................................................................ 334 

Figure 349.  Stress distribution for concentric compression check, shown on P12-C(1.0)-W-P. ............................................................................................................................... 336 

Figure 350.  Full shear plane and joint free-body diagram for P15-C-CJ. ................................ 338 Figure 351.  Full shear plane and joint free-body diagram for E3-U-307SL-WV. ................... 339 Figure 352.  Method 1 gusset plate diagonal buckling model with full Whitmore section, shown

on E1-U-307SS-WV. ............................................................................................ 346 Figure 353.  Method 1 partial shear planes on E1-U-307SS-WV. ............................................ 347 

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Figure 354.  Method 1 gusset plate diagonal buckling model with full Whitmore section, shown on E1-U-307SS-W. ............................................................................................... 350 

Figure 355.  Method 1 partial shear planes on E1-U-307SS-W. ............................................... 352 Figure 356.  Method 1 gusset plate diagonal buckling model with full Whitmore section, shown

on E3-U-307SL-WV. ............................................................................................ 354 Figure 357.  Method 1 gusset plate diagonal buckling model with full Whitmore section, shown

on E3-U-307SL-W. ............................................................................................... 356 Figure 358.  Method 1 gusset plate diagonal buckling model with full Whitmore section, shown

on P6-U-WV-NP. .................................................................................................. 357 Figure 359.  Method 1 partial shear planes on P6-U-WV-NP. .................................................. 357 Figure 360.  Method 1 gusset plate diagonal tension yielding model with full Whitmore section,

shown on P18-C(0.6)-POS .................................................................................... 360 Figure 361.  Method 2 gusset plate diagonal buckling model with full Whitmore section, shown

on E1-U-307SS-WV. ............................................................................................ 363 Figure 362.  Method 2 gusset plate diagonal buckling model with full Whitmore section, shown

on E1-U-307SS-W. ............................................................................................... 365 Figure 363.  Method 1 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P7-C-WV-INF. ..................................................................................... 367 Figure 364.  Method 1 partial shear planes on P7-C-WV-INF. ................................................. 368 Figure 365.  Method 1 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P5-C-HS-WV-NP. ................................................................................ 369 Figure 366.  Method 1 partial shear planes on P5-C-HS-WV-NP. ........................................... 370 Figure 367.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P7-C-WV-INF. ..................................................................................... 372 Figure 368.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P7-C-WV-INF – illustration of assumed state of stress at the bolt lines truncating the Whitmore Section. .......................................................................... 372 

Figure 369.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section, shown on E3-U-307SL-WV. ................................................................................. 374 

Figure 370.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section, shown on E3-U-307SL-W. .................................................................................... 376 

Figure 371.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section, shown on P5-C-HS-WV-NP. ................................................................................ 378 

Figure 372.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section, shown on P6-U-WV-NP. ....................................................................................... 380 

Figure 373.  Method 2 gusset plate diagonal tension yielding model with truncated Whitmore section, shown on P18-C(0.6)-POS. ...................................................................... 381 

Figure 374.  Method 1 shingle plate diagonal buckling model with full Whitmore section, shown on P5-C-SP-WV-NP. ............................................................................................ 383 

Figure 375.  Method 1 splice plate partial shear plane for P5-C-SP-WV-NP. .......................... 384 Figure 376.  Full shear plane on gusset and shingle plates for P5-C-SP-WV-NP. .................... 386 

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Figure 377.  Method 2 shingle plate diagonal buckling model with truncated Whitmore section, shown on P5-C-SP-WV-NP. ................................................................................. 388 

Figure 378.  Method 2 gusset plate diagonal buckling model with truncated Whitmore section, shown on P14-U-C1-WV-INF. ............................................................................. 392 

Figure 379.  Distribution of HS20 truck loads to truss panel points for load cases causing maximum shear effects in the representative continuous- or cantilever span truss bridge. .................................................................................................................... 408 

Figure 380.  Distribution of HS20 truck loads to truss panel points for load cases causing maximum moment effects in the representative continuous- or cantilever span truss bridge. .................................................................................................................... 408 

Figure 381.  Factored loads causing the maximum panel shear force at the mid-span of the representative continuous- or cantilever-span truss bridge. .................................. 409 

Figure 382.  Factored loads causing the maximum mid-span or pier-section moment for the representative continuous- or cantilever-span truss bridge. .................................. 409 

Figure 383.  Factored loads causing the maximum shear force in the right-hand end truss panel for the representative continuous- or cantilever-span truss bridge. ...................... 410 

Figure 384.  Factored loads causing the maximum shear force in the third truss panel from the right-hand support for the representative continuous- or cantilever-span truss bridge. .................................................................................................................... 410 

Figure 385.  Distribution of HS20 truck loads to truss panel points for load cases causing maximum shear effects in the representative simple-span truss bridge. ............... 413 

Figure 386.  Distribution of HS20 truck loads to truss panel points for load cases causing maximum moment effects in the representative simple-span truss bridge. .......... 414 

Figure 387.  Factored loads causing the maximum panel shear force second panel from the support of the representative simple-span truss bridge. ........................................ 414 

Figure 388.  Factored loads causing the maximum moment at the second panel point from the support of the representative simple-span truss bridge. ........................................ 415 

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LIST OF TABLES

Table 1 Relevant forces and dimensions for the 20 parametric test configurations selected for this research. ................................................................................................................... 31 

Table 2 Force ratios and location information for the 20 parametric test configurations selected for this research. ...................................................................................................... 32 

Table 3 Member designs for all configurations other than Pratt joints. ........................................ 48 Table 4 Member designs for Pratt joints. ...................................................................................... 49 Table 5 Test simulations for Warren with vertical experimental configurations, unchamfered

members .................................................................................................................. 64 Table 6 Test simulations for Warren without vertical variations on experimental configurations,

unchamfered members ............................................................................................ 65 Table 7 Test simulations for Warren with vertical parametric configurations ............................. 66 Table 8 Test simulations for Pratt parametric configurations ....................................................... 69 Table 9 Test simulations for Warren without vertical parametric configurations ........................ 70 Table 10 Test simulations for other base parametric configurations ............................................ 71 Table 11 Test simulations for parametric configurations with shingle plates .............................. 72 Table 12 Test simulations for parametric configurations with edge stiffeners ............................. 73 Table 13 Test simulations for configurations with corroded gusset plates ................................... 74 Table 14 Test simulations for parametric configurations with corroded gusset plates reinforced

by shingle plates ...................................................................................................... 75 Table 15 Key required gusset plate thicknesses from the FHWA Guide and from a

supplementary pseudo-plastic section analysis of the compression splice for P1-C-CCS-WV-M. ......................................................................................................... 107 

Table 16 Key required gusset plate thicknesses from the FHWA Guide and from a supplementary pseudo-plastic section analysis of the tension splice for P2-C-TCS-WV-M. .................................................................................................................. 114 

Table 17 Key required gusset plate thicknesses from the FHWA Guide for P3-C-WV-P. ........ 118 Table 18 Key required gusset plate thicknesses from the FHWA Guide for P4-C-WV-P. ........ 124 Table 19 Key required gusset plate thicknesses from the FHWA Guide for P5-C-WV-NP. ..... 129 Table 20 Key required gusset plate thicknesses from the FHWA Guide for P5-U-WV-NP. ..... 135 Table 21 Key required gusset plate thicknesses from the FHWA Guide for P6-C-WV-NP. ..... 143 Table 22 Key required gusset plate thicknesses from the FHWA Guide for P6-U-WV-NP. ..... 147 Table 23 Key required gusset plate thicknesses from the FHWA Guide for P7-C-WV-INF. .... 152 Table 24 Key required gusset plate thicknesses from the FHWA Guide for P8-C-WV-INF. .... 161 Table 25 Key required gusset plate thicknesses from the FHWA Guide for P8-U-WV-INF. ... 164 Table 26 Key required gusset plate thicknesses from the FHWA Guide for P9-C-P-NP. ......... 173 Table 27 Key required gusset plate thicknesses from the FHWA Guide for P10-C-P-NP. ....... 177 Table 28 Key required gusset plate thicknesses from the FHWA Guide for P11-C-W-M. ....... 181 Table 29 Key required gusset plate thicknesses from the FHWA Guide for P12-C-W-P. ......... 185 Table 30 Key required gusset plate thicknesses from the FHWA Guide for P13-C-W-NP. ...... 189 

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Table 31 Key required gusset plate thicknesses from the FHWA Guide for P13-U-W-NP. ...... 193 Table 32 Key required gusset plate thicknesses from the FHWA Guide for P14-C-W-INF. .... 198 Table 33 Key required gusset plate thicknesses from the FHWA Guide for P14-U-W-INF and

P14-C-W-INF. ....................................................................................................... 202 Table 34 Key required gusset plate thicknesses from the FHWA Guide for P15-C-CJ. ............ 207 Table 35 Key required gusset plate thicknesses from the FHWA Guide for P16-C-CJ. ............ 213 Table 36 Key required gusset plate thicknesses from the FHWA Guide for P17-C-POS. ......... 219 Table 37 Key required gusset plate thicknesses from the FHWA Guide for P18-C-POS. ......... 223 Table 38 Key required gusset plate thicknesses from the FHWA Guide for P19-C-CCS-NEG. 227 Table 39 Key required gusset plate thicknesses from the FHWA Guide for P20-C-CCS-NEG. 235 Table 40 Key required gusset plate and shingle plate thicknesses from the FHWA Guide for P3-

C-SP(0.4:0.2)-WV-NP . ........................................................................................ 242 Table 41 Key required gusset plate and shingle plate thicknesses from the FHWA Guide for P5-

C-SP(0.3:0.2)-WV-NP. ......................................................................................... 250 Table 42 Key required gusset plate and shingle plate thicknesses for P12-C-SP(0.5:0.5)-W-P. 259 Table 43 Summary assessment of the professional factor from chord-splice eccentric tension and

chord-splice eccentric compression prediction equations, 28 parametric tests ..... 307 Table 44 Summary assessment of professional factors for Method 1 combined diagonal buckling

or partial shear plane yielding, Method 2 diagonal bucking, Method 1 or Method 2 diagonal tension yielding, and Method 1 or Method 2 full shear plane yielding. . 309 

Table 47 Internal forces in the representative simple-span truss bridge. .................................... 415 

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ACKNOWLEDGEMENTS

The research presented in this report was performed under the direction of the Federal Highway Administration in collaboration with the National Cooperative Highway Research Program (NCHRP) Project 12-84, “Guidelines for the Load and Resistance Factor Design and Rating of Riveted and Bolted Gusset-Plate Connections for Steel Bridges.” The authors are grateful for the input and advice provided by the panel overseeing the broader NCHRP project. Special thanks are extended to Drs. Justin M. Ocel and Robert S. Zobel of FHWA for extensive discussions, feedback, and collaboration during these studies, as well as for their skillful management of the NCHRP project and execution of the project experimental tests. Drs. William Wright and Joseph Hartmann, who were the first and second principle investigators on the NCHRP project, are thanked for setting the project on a successful path. The opinions, findings and conclusions expressed in this report are those of the authors and do not necessarily reflect the views of the above individuals and organizations.

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xxvii

ABSTRACT

This report presents the development, execution and interpretation of refined parametric finite element test simulation studies of steel truss bridge gusset plates conducted in support of NCHRP Project 12-84, “Guidelines for the Load and Resistance Factor Design and Rating of Riveted and Bolted Gusset-Plate Connections for Steel Bridges.” The research focuses primarily on the buckling and shear failure modes of steel truss bridge gusset plates, including the effects of a wide range of joint geometries and loadings, fastener types, material strengths, combined action of the gusset plates as chord splice elements along with the force transfer from the truss web members, combined action of gusset plates and shingle plates, edge stiffening, and section loss due to corrosion. Two groups of equations are developed and applied, in coordination with the broader NCHRP 12-84 research, to estimate the resistances determined from the simulation studies. The equations termed Method 1 are an early version of the equations ultimately recommended by NCHRP 12-84. The equations termed Method 2 require some additional calculation, but provide improved estimates of the mean resistances from the simulation studies and smaller dispersion of the ratio of the test simulation resistances to the predicted strengths.

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1

EXECUTIVE SUMMARY

The strength behavior of steel truss bridge gusset plates involves numerous complexities. Traditionally, many of these complexities have been addressed on an ad hoc and case-by-case basis, and AASHTO Specifications have generally left considerable discretion to the engineer regarding the specific calculations needed for design. Gusset plates have traditionally received little attention in rating largely because of the expectation that their capacities are generally larger than the member capacities. The collapse of the I-35W Bridge in Minneapolis on August 1, 2007 and the ensuing investigations by the National Transportation Safety Board, the FHWA and others led to the recognition that greater attention needed to be devoted in general to the resistance of the gusset plates in the rating of steel truss bridges. NCHRP Project 12-84, “Guidelines for the Load and Resistance Factor Design and Rating of Riveted and Bolted Gusset-Plate Connections for Steel Bridges,” was commissioned in 2008 to provide further scientific data and knowledge to support the development of new AASHTO procedures for design and rating of gusset plates.

This report presents the development, execution and interpretation of a suite of refined parametric finite element test simulation studies of steel truss bridge gusset plates conducted in support of NCHRP Project 12-84. The goal of these studies is to extend the results of a more limited number of experimental studies conducted in the broader research, and to provide preliminary predictive equations and data from the simulations regarding the statistical variation of these predictions. In the parent research, these predictive models and data are further evaluated, and a final model is used in LRFD calibration tasks and recommended for incorporation into AASHTO Specifications provisions.

The research presented in this report starts with the assessment of a range of geometries and loadings representative of steel truss bridges encountered in practice. Twenty different truss subassemblies are identified to capture a reasonably comprehensive set of geometries and loadings. In addition to varying the joint geometries and loadings, the following parameters also are varied:

Fastener type, including both rivets and high-strength bolts,

Steel strength, including Grade 50 and Grade 100 material,

Non-chamfered diagonal members as diagonal members having significant chamfer aimed at making the joints more compact,

Combined action of the gusset plates as chord splice elements along with the force transfer from the truss web members,

Combined action of gusset plates and shingle plates,

Edge stiffening of the gusset plates, and

Section loss due to corrosion.

2

Refined finite element models, verified and validated against experimental results in other studies conducted within the larger project, are developed in the ABAQUS software system for the various study cases. The finite element models are focused primarily on the evaluation of buckling and shear failure modes. The FEA models are not developed at the level of detail that would be required to evaluate tension or shear rupture of the steel plates. Detailed results are reported for all of the study cases regarding the load-deflection response, the distributions of stresses, the extent of yielding within the gusset plates, and the useable structural capacity of the gusset plates.

Two groups of equations are developed and applied, in coordination with the broader NCHRP 12-84 research, to estimate the resistances determined from the simulation studies. The equations termed Method 1 are an early version of the equations ultimately recommended by NCHRP 12-84. The equations termed Method 2 require some additional calculation, but provide improved estimates of the mean resistances from the simulation studies and a smaller dispersion in the ratio of the test simulation resistances to the predicted strengths. Both methods are extensions of traditional procedures involving the use of basic beam and column idealizations of the gusset plate responses.

3

1. INTRODUCTION

1.1 BACKGROUND

The collapse of the I-35W Bridge in Minneapolis in August 2007 focused attention on the reliability and safety of truss bridges in the U.S. The preliminary failure investigation and design review indicated that the collapse was initiated by buckling of the gusset plates at the U10 joint due to inadequate thickness (Holt and Hartmann, 2008). A comprehensive investigation by the National Transportation Safety Board (NTSB) ruled out other possible initiation locations and factors and concluded that the compression failure of these gusset plates was the probable cause of the collapse (NTSB, 2008). The actual margin of safety against connection failure for this bridge was significantly smaller than that of a typical truss design. The problem of the under-sized gusset plate thickness at U10, and at other joints, was due to an error in the original design. This problem was compounded by the addition of a 2-inch thick layer of concrete to the deck in 1977 (adding over 3 million pounds of dead load to the bridge), the placement of additional barriers and a deicing system in 1998 (adding approximately 1.2 million pounds of dead load to the bridge), as well as more than 0.5 million pounds of construction live load and 0.1 million pounds of traffic live load on the bridge deck in the bridge’s critical middle span at the time of the collapse. This included the staging of construction materials and machinery in the areas above the gusset plates at U10.

In the aftermath of the I-35W failure, the Federal Highway Administration issued Technical Advisory 5140.29 emphasizing the need to check gusset plate as well as member capacities in the load rating of non-load path redundant trusses (FHWA, 2008). Feedback from many sources indicated that there was no clear consensus on the specific procedures to follow for design or rating of gusset plates. The American Association of State Highway and Transportation Officials (AASHTO) Specifications (e.g., (AASHO, 2007) ) provided broad guidance for the design of gusset plates for shear, bending and axial force effects, but left the specific requirements for these checks largely up to the discretion of the engineer. For instance, Article 6.14.2.8 of the 2007 AASHTO LRFD Specifications specified a length of the unsupported edge of a gusset plate beyond which the edge was required to be stiffened, but made no mention of requirements for checking compression in gusset plates other than the statement, “The maximum stress from combined factored flexural and axial loads shall not exceed fFy based on the gross area.”

In its final report on the I-35W bridge collapse, the NTSB (2008) issued the following six recommendations to the FHWA and AASHTO to prevent these types of failures in the future:

1) Develop and implement a bridge design quality assurance / quality control program aimed at detecting design errors and verifying that the appropriate design criteria have been applied and the appropriate calculations have been performed.

4

2) Require owners to assess the truss bridges in their inventories to identify locations where visual inspections may not detect gusset plate corrosion, and thus, where appropriate nondestructive evaluation technologies should be applied to assess the gusset plate condition.

3) Modify inspector training to address inspection techniques and conditions specific to gusset plates, emphasizing issues tied to gusset plate distortion as well as the use of nondestructive evaluation at locations where visual inspections may be inadequate to assess and quantify conditions such as section loss due to corrosion. In addition, as a minimum, address any newly developed gusset plate condition rating procedures in corresponding reference materials.

4) Revise the Manual for Bridge Evaluation (AASHTO, 2008) to include guidance for conducting load ratings on new bridges before they are placed in service, and modify the guidance and procedures in this manual to include evaluation of the capacity of gusset plates as part of the load rating calculations for non-load-path-redundant steel truss bridges.

5) Develop specifications and guidelines to ensure that construction loads and stockpiled raw materials placed on a structure during construction or maintenance projects do not overload the structural members or their connections.

6) Include gusset plates as a commonly recognized (CoRe) structural element and develop guidance for bridge owners in tracking and responding to potentially damaging conditions in gusset plates, such as corrosion and distortion.

Shortly after the release of the NTSB final report on the I-35 bridge investigation, the FHWA (2009a) issued a guidance document on gusset plate design and rating (referred to herein as the FHWA Guide) based on the best available information. Early experience showed that a number of truss bridges in service failed some of the limit state checks when analyzed using the FHWA Guide. This was expected because the original bridge designers had considerable discretion and probably did not follow or have knowledge of the specific procedures outlined in the Guide. While it was generally acknowledged by experts that the FHWA Guide represented the best available knowledge on gusset plate design, it was also acknowledged that it may be overly conservative for some limit state checks. Furthermore, the existence of the FHWA Guide prompted further requests for information on some of the calculations to be performed. Therefore, a research project was launched to develop and improve design and rating procedures for gusset plate connections in steel truss bridges. This research included laboratory as well as parametric simulation studies of gusset plate joints. The research was performed under direction of the FHWA as a part of the National Cooperative Highway Research Program (NCHRP) Project 12-84 “Guidelines for the Load and Resistance Factor Design and Rating of Riveted and Bolted Gusset Plate Connections for Steel Bridges.”

5

1.2 PROBLEM STATEMENT AND LITERATURE REVIEW

This report discusses parametric test simulation studies performed at the Georgia Institute of Technology in support of NCHRP 12-84, as well as the synthesis of results from these studies. The specific focus of this effort is to evaluate the limit states response of steel truss bridge gusset plates over a wide range of geometries and configurations such as those identified in early NCHRP 12-84 assessments of bridge plans (FHWA, 2009b) as well as other potential configura-tions encountered in practice. Particular emphasis is placed on the evaluation of various potential simplified methods of calculating gusset plate strengths in compression, as well as simplified calculation of the resistance along critical shear planes. These evaluations are conducted by scrutinizing the fundamental behavior associated with the different limit state checks.

A substantial number of research studies have been conducted on the behavior of gusset plates long before the I-35W bridge tragedy and the ensuing developments. Chambers and Ernst (2005) developed a literature review on braced frame gusset plate research that identified more than 100 references. Wyss (1926) conducted the first experimental research on gusset plates and plotted stress trajectories for specimens representing a Warren truss joint. The maximum normal stress was observed to be at end of the brace member and the stress trajectories were observed along approximately 30o lines with respect to the connected member. Similar conclusions were reached subsequently by Sandel (1950), Whitmore (1952), Irvan (1957), Hardin (1958), Davis (1967), Vasarhelyi (1971) and others. Based on his test results, Whitmore proposed the well-known effective width concept commonly known as the “Whitmore Section” as a means to approximate the maximum normal stress near the ends of diagonal members.

In addition, Perna (1941), Sandel (1950), Whitmore (1952), and others conducted tests that showed that the use of beam theory to estimate the elastic internal stresses in gusset plates supported by a truss chord generally tended to be substantially in error in predicting the location of the maximum internal stresses; however, they found that beam theory produced reasonable to conservative estimates of the maximum stress magnitudes. The tests showed that the normal stresses at the extreme edges of the plate were much smaller than the maximum stresses at these locations predicted by beam theory; however, within the interior of the plate, the measured normal stresses were much larger than predicted by beam theory.

With respect to calculation of gusset plate stresses, Dowswell (2012b) states:

“Checking gusset plates for the elastic stress distribution in hopes that the stresses will not exceed the yield stress is futile, because, if the plate has flame-cut edges, it has yielded under the residual stresses before any external loading is applied. Although designing for the elastic stress distribution in gusset plates has provided safe designs in the past, the presence of residual stresses and inaccuracies in the design model make it difficult to predict the actual stress distribution in the gusset plate.

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From a designer’s perspective, the goal is to use the simplest procedure available that provides a safe and economical design. It is standard practice [in building design] to calculate the shear capacity of gusset plates based on the plastic stress distribution. Because strength design is now being used for steel members and connections, it seems appropriate to design gusset plates using the plastic capacity in bending.”

One concern that is recognized in current AASHTO Bridge Design Specifications is that the shear strength along critical shear planes can sometimes be reduced due to stability effects. In cases where a plastic interaction between moment, normal force and shear on a given plane may be a legitimate strength check, Astaneh (1998) summarized the previous research and recommended the following equation:

2 4

1p y p

M P V

M P V

where M, P, and V are the moment, normal force and shear force on the plane, and Mp, Py, and Vp are the plastic moment, axial yield load, and plastic shear capacity of the plane.

Thornton (1984) espoused a similar general philosophy to Dowswell’s (2012b) for gusset plate and connection design. A study of his comments from nearly 30 years ago provides a number of potentially useful insights:

“The design of complex connections is not an exact science. Over the years, an intuitive approach to connection design has become widely accepted. This approach is based on the idea the structure (and parts thereof) will behave as the designer dictates, if he provides a path of adequate strength for the load (or loads) to follow. This ‘adequate strength path’ is determined from the principles of statics and strength of materials.

About 30 years ago, this intuitive load path method of design was put on a rigorous basis with the development of the Theorem of Limit Analysis – specifically the Lower Bound Theorem of Limit Analysis, which states: if a distribution of forces in the structure can be found which is in equilibrium with the applied loads, and if these forces everywhere within the structure are of such a magnitude that the yield stress (or yield criterion) is nowhere exceeded, then the applied loads are less than, or at most equal to, the loads required for collapse (unbounded yield deformations) to occur. Thus, if a load path is provided, the elements of which are in equilibrium with the applied loads, and if the stresses in these elements nowhere exceed the yield stress, a safe design will have been achieved. Also, the relative stiffness of the various connection elements should be considered to minimize the possibility of fracture.

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In addition to strength, the stiffness of the structure and its connections must be considered. For the Lower Bound Theorem to be valid, a structure must be stiff enough to preclude buckling before yielding occurs. In connection design, this requirement can usually be met by consideration of appropriate width/thickness ratios and related local buckling formulations which force the elements to yield before they buckle.”

Thornton (1984) went on to recommend a practical procedure for the design calculation of the buckling resistance of typical gusset plates in delivering axial compression from a brace diagonal to other adjacent framing. This approach is somewhat at odds with the above stated philosophy, unless one assumes that there is sufficient ductility in the gusset plate such that the approximate critical compressive stress can be sustained over sufficient inelastic deformations, such that the critical stress works essentially like an effective yield stress of an idealized elastic perfectly plastic material. Thornton basically proposed the use of the Whitmore effective width times the thickness of the gusset plate as the area of an effective column oriented along the line of action of the diagonal. The buckling strength of the gusset plate was taken as the compressive resistance of an imaginary column strip with a length taken as the average of the three lengths L1, Lmid, and L3 shown in Figure 1. Thornton recommended the use of an effective length factor of K = 0.65 with this imaginary column. This approach has become known as the “Thornton Method” (Cheng and Grondin, 1999) or the “Pseudo-Column Method” (Chambers and Ernst, 2005).

Figure 1. Geometry of the effective column assumed in the “Thornton Method.”

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Dowswell (2006 and 2012a) recently collected and analyzed the ultimate strength results from various test configurations in which the capacities were governed by the gusset plate buckling strength. Based on these studies, he recommends the following variations on the Thornton Method:

For rectangular corner gusset plates in which the closest offset distance from the diagonal fasteners to either of the supported edges, denoted by c in Figure 2, is smaller than

2

3mid

y

L t E

F

where t is the thickness of the gusset plate, set the nominal buckling stress to the full yield strength on the equivalent column, therefore neglecting stability effects.

For other rectangular corner gusset plates, use L = Lavg = (L1 + Lmid + L3)/3 and K = 1.0.

For “extended” corner gusset plates, typically where the brace has been offset by the distance 2t as shown in Figure 3a, use L = Lmid and K = 0.6.

For gussets in chevron type arrangements as illustrated in Figure 3b, or in Warren type truss systems, use L = Lmid and K = 0.65.

For “single brace” arrangements such as in Figure 3c, where the gusset plate is supported essentially only on one side, use L = Lmid and K = 0.7.

Figure 2. Offset distance c used by Dowswell (2006), along with Lmid, as a predictor of when stability effects may be neglected in the calculation of the gusset plate compressive

resistance.

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Offset

(a) Extended (b) Chevron (c) Single brace

Figure 3. Gusset plate configurations evaluated by Dowswell (2006 and 2012a).

The 1980’s and early 1990’s experienced a large amount of research activity focused on gusset plate behavior and design of gusset plates in braced building frames. Hardish and Bjorhovde (1985) focused on the behavior of gusset plates in tension. This research was the impetus for the current equations for checking block shear tearout in the AASHTO and AISC steel design specifications. Chakrabarti and Richard (1990) used finite element models to assess the buckling capacity of gusset plates in Warren truss joints and joints in vertical bracing systems of buildings. They modeled four of eight double gusset plate specimens that were tested experimentally by Yamamoto et al. (1988) and showed accurate predictions of the test results. The interface between the gusset plate and the chord was fixed against rotation and translation, and the out-of-plane displacements of the plate were constrained where the diagonal members connected to the plate in their models. The test results from Yamamoto et al. (1988) as well as the FEA results from Chakrabarti and Richard (1990) showed extensive yielding of the plates at the maximum strength limit.

Brown (1988) tested 24 half-scale vertical brace connections with single rectangular corner gusset plates in compression. Brown’s tests included three different plate thicknesses, six different bracing member orientations (26, 30, 35, 40, 45 and 55 degrees relative to the longer free edge of the plate), and two different bracing members (back-to-back C4x7.25 and back-to-back C8x11.5 members with a single row and with two rows of fasteners respectively). Brown observed that failure usually was preceded by out-of-plane movement along the gusset’s longer free edge. Based on these observations, she focused her attention on calculating the initial bifurcation buckling load of this edge as a measure of the useable load-carrying capacity. However, the ultimate strengths of Brown’s gusset plates were generally higher than these buckling strengths. Furthermore, subsequent research (Rabinovich and Cheng, 1993; Roeder et al., 2005) has shown little correlation between gusset plate maximum strength and free edge length. Unfortunately, Brown (1988) documented only two of her test geometries in sufficient detail to allow confirmation of her results, other than the calculation of full yielding on the Whitmore section. Her gusset plates generally failed at substantially smaller loads than the

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Whitmore section full yield capacity, but at larger loads than the theoretical out-of-plane buckling of the longer free edge.

Gross and Cheok (1988) and Gross (1990) discussed three tests conducted on nearly full-scale braced frame subassemblies representative of steel building construction. The gusset plate geometries in these tests were ones studied previously in the FEA studies by Chakrabarti and Richard (1990). One of the braces in the subassembly was loaded in tension and one was loaded in compression. The main parameters in of the studies were the gusset geometry, eccentricity of forces in the connections to the beam and column, and orientation of the column (strong-axis bending versus weak-axis bending). The strengths were governed by gusset plate buckling after extensive yielding in all but one of the tests. Thornton’s method for estimating the compressive strength of the gusset plate was found to be mildly to substantially conservative when used with a K factor of 0.5.

Astaneh (1992) conducted three cyclic experiments on gusset plates representing brace-to-beam connections in steel building chevron bracing systems. The third specimen failed by “dramatic and almost sudden buckling,” but the buckling load is predicted quite accurately by Thornton’s model with K = 0.5 (Ptest/Ppredicted = 1.01). The buckling was attributed to a high concentration of stress in the plate at the end of the bracing member plus the existence of a long horizontal free edge of the gusset.

From 1987 through 2002, Cheng and his associates at the University of Alberta conducted a substantial number of experimental tests and refined finite element simulation studies of gusset plates with a wide variety of geometries. This research is summarized below.

Hu and Cheng (1987) conducted 14 full-scale tests of six different rectangular, single-plane, corner gusset plates in compression. Their test program focused on the effects of plate slenderness, the position of splice plates connecting the diagonal member to the gusset plate within the plan geometry of the gusset plate, restraint and non-restraint of relative out-of-plane movement of the brace and the test fixture representing a corner beam and column, eccentricity of the brace member relative to the plane of the gusset plate, and reinforcement at the splice plate location. The gusset plates were all welded to the test fixture; hence, the test boundary conditions can be rationalized to be somewhat representative of some bridge single-gusset plate configurations supporting a compression diagonal. All of their tests exhibited maximum strengths substantially smaller than the yield load on the Whitmore section. In general, sway or local buckling modes were observed depending on whether the relative out-of-plane movement of the brace and the test fixture was allowed or not. The brace was held against out-of-plane movement in all of the tests. In the cases where free out-of-plane movement was allowed, the text fixture was permitted to roll in the out-of-plane direction. Thornton’s method with K = 0.5 substantially underestimates the capacities of these plates, with test to predicted strength ratios of 8.50 and 13.81 without and with out-of-plane restraint for the most slender plate tested.

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Yam and Cheng (1993) carried out a similar investigation with 19 stockier gusset plates that exhibited substantial inelastic behavior prior to failure. The parameters studied in their investigations were gusset plate thickness and size, brace angle, out-of-plane brace restraint conditions, moments in the framing members, and out-of-plane eccentricity of the brace load. Yam and Cheng found that the effect of beam and column moments on the compressive capacity of the specimens was small, although these moments affected the measured strain distributions in the plates. The capacities of these plates are again predicted very conservatively by Thornton’s method (using K = 0.5), with a minimum Ptest/Ppredicted of 1.91 and a maximum of 6.84.

Rabinovich and Cheng (1993) extended the above monotonic compression studies to investigate the cyclic behavior of gusset plate connections. They investigated the influence of plate slenderness, bolt slip, gusset plate geometry and free edge stiffeners. The combined beam-column/gusset plate assembly was free to slide out-of-plane while the brace was constrained to remain in the plane of the test. The brace had much greater bending stiffness than the gusset plate. As such, yielding and buckling of the gusset plate dissipated all of the energy introduced by the cyclic load. Rabinovich and Cheng found that cyclic loading caused an unstable drop in the compressive strength once overall plate buckling occurred, but had little effect on the tensile strength. Interestingly, the initial compressive strength for the rectangular specimens that did not have edge stiffeners is predicted accurately by Thornton’s model with K = 0.5; however, the compressive strengths after the unstable drop in the compressive capacity are not well predicted by this calculation. Furthermore, it is interesting that the addition of edge stiffeners had only a minor influence on the initial compressive strength, but allowed the specimens to maintain a stable post-buckling compressive strength with repeated cycling. Rabinovich and Cheng (1993) also showed that the provision of the additional “2t” offset with an extended (tapered) gusset plate type configuration as shown in Figure 3a resulted in an unnecessary reduction in the buckling load, and that the other gusset configurations were able to absorb substantial amounts of energy.

Walbridge et al. (1998) developed refined finite element simulation models of rectangular corner gusset plates that were validated with the experimental results from Yam and Cheng (1993) and Rabinovich and Cheng (1993). They applied these models to examine the interaction between the gusset plate and the brace member for several cyclic loading sequences. For cases in which the failure was by gusset plate compression buckling, the compressive strengths are predicted conservatively by Thornton’s model with K = 0.5. For the cases in which the brace member was the weaker element, the behavior of the assembly was dominated by the brace response with minor influence from the gusset plates.

Nast et al. (1999) conducted FEA simulation and experimental studies of the influence of free edge stiffeners and brace member - gusset plate interaction on the behavior of single-plane rectangular corner gusset plates subjected to cyclic loading. The experimental program involved four full-scale gusset plate - brace member assemblies. All four specimens included the same connection details and gusset plate geometry, except that two specimens included gusset plate

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free edge stiffeners. The other test variable was the slenderness of the brace member, which was adjusted by altering the member length. Two of the specimens were designed with the gusset as the weaker element under compression, and two were designed with the brace as the weaker element. The initial compressive strength of the specimen with the unstiffened gusset plate as the weaker element was predicted accurately by Thornton’s model with K = 0.5 (Ptest/Ppredicted = 1.08), and exhibited roughly 30 % loss in strength with repeated cycling.

Sheng et al. (2002) conducted a parametric FEA simulation study of the monotonic buckling capacity of single-plane corner gusset plates considering several different rectangular and tapered plate geometries, three different plate thicknesses, three different lengths of the connection of a splice member to the gusset plate, welded versus bolted connections to the gusset plate, stiffness of the splice member, the rotational restraint provided at the conjunction of the bracing member and the gusset plate, and various stiffening arrangements (stiffening directly at the end of the diagonal connection and stiffening of the gusset plate free edges). The supported edges of the gusset plate were assumed to be fully fixed. The brace member was unrestrained against out-of-plane movement relative to the plane of the gusset, but was assumed to be constrained from any rotation about any axis within the plane of the gusset. The results showed that Thornton’s model with K = 0.5 is generally conservative for all of the unstiffened plate cases studied, with Ptest /Ppredicted values ranging from 1.36 to 7.78. Similar to the above tests, the boundary conditions for these tests can be rationalized to be somewhat representative of some bridge single-gusset plates supporting a compression diagonal.

Recent braced building gusset plate research has focused on the balanced design of the brace – gusset plate system, that is, the yield mechanisms of the brace are balanced with the yield mechanisms of the gusset plate and undesirable failure modes to achieve a targeted yielding hierarchy and suppress the unwanted failure modes (Lehman, et al., 2008). One significant recommendation of this research is an “elliptical-clearance” requirement that specifies an elliptically-shaped zone with an offset of 8t of the end of the diagonal brace from the closest adjacent member, where t is the thickness of the gusset.

With the exception of the experimental studies by Yamamoto (1988) and the FEA studies of these tests by Chakrabarti and Richard (1990), all of the above compressive strength studies are focused predominantly on the behavior of gusset plates in steel building vertical bracing systems. Figures 4 through 8 show a number of steel truss bridge gusset plates that illustrate a number of distinct characteristics different than those of building-type gusset plates. Figure 4 shows an elevation view of the U10 joint of the I-35W Bridge. Figure 5 shows a view of buckled plates in the I-90 bridge over Grand River in Lake County, Ohio, which suffered a buckling failure in 1990 (Huckelbridge et al. 1997). In the case of the I-90 Bridge, significant corrosion loss reduced the capacity of the gusset plate at a lower truss joint adjacent to one of the main piers, allowing the plate to buckle. Fortunately, the end of the diagonal went into bearing against the compression chord, which prevented the collapse of the structure.

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Figure 4. Elevation rendering of the critical U10 joint of the I-35W Bridge.

Figure 5. Buckled gusset plates in the I-90 Bridge over Grand River in Lake County, Ohio (courtesy of FHWA).

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Figure 6. Constant thickness contours on the critical gusset plate in the I-90 Bridge over Grand River in Lake County, Ohio (ODOT, 2008).

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Figure 7. Photo of one of truss joints in the DeSoto Bridge, which carried Minnesota

Highway 23 over the Mississippi River (MnDOT, 2013).

Figure 8. Photo of gusset plates in the Interstate 40 Bridge over the French Broad River in

Jefferson County, TN (bridgehunter.com, 2013).

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The U10 gusset plate in the I-35W bridge also failed by buckling, although inadequate design (not corrosion), was the primary cause of the failure. Figure 6 shows an elevation sketch of the failed gusset plate in the I-90 Bridge with constant plate thickness contours superimposed on the drawing. It is apparent that the geometry of the connection and the corresponding load paths are relatively complex. Figure 7 shows an elevation view of one of the truss joints in the DeSoto Bridge, which carried Minnesota Highway 23 over the Mississippi River. This bridge was inspected after the I-35W disaster and found to be structurally sound. However, in March 2008, four gusset plates were observed to be bent. As a result, the bridge was closed indefinitely as a precaution and demolished in October 2008 (Wikipedia, 2013). One can observe the chamfering of one of the diagonals, which helps to avoid excessive free-edge lengths and relatively large distances between the adjacent members in certain areas of the gusset plates. Lastly, Figure 8 shows a photo of a number of gusset plates in the I-40 Bridge over the French Broad River in Jackson County, TN. A number of the gusset plates in this bridge have additional “shingle plates” over a portion of their geometry.

There are a number of characteristics of the steel truss bridge joints shown in the above figures that distinguish the gusset plate behavior in these joints from the joints evaluated in the above braced-building structures gusset plate studies:

1) There are two gusset plates, one on each side of the truss members. Furthermore, these gusset plates are bolted to the sides of the truss members parallel to the plane of the truss rather than being welded and/or bolted to sides of members that are perpendicular to the truss, as in typical building connections.

2) The arrangement or configuration of the members framing into the joint, and the resulting shape of the gusset plate, can be much more general and complex than typical corner gusset plates attached to a beam and a column in building vertical bracing systems.

3) The web members framing into the joint are often relatively large, and in some cases are substantially chamfered at the end of the diagonals to reduce the size of the gusset plates. This can result in substantial overlaps of the Whitmore section with other members and their Whitmore sections framing into the joint.

4) Bridge truss gusset plates often transfer a significant shear force across a plane along the bottom edge of the top chord members or the top edge of the bottom chord members. The gusset plate resistances potentially can be limited by the strength of these shear planes.

5) The chords in steel truss bridges are in many cases not continuous through the truss joint. As such, the gusset plates are designed commonly to participate with other chord splice plates to transfer the chord tension or compression across the joint. This adds to the complexity of the stress states within the gusset plates.

6) In some cases, additional “shingle plates” may be used over a portion of the gusset plate geometry, so that the gusset plates are effectively stepped to a larger thickness over a portion of the joint.

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7) Due to section loss (i.e., reduction in thickness) that has occurred to the gusset plate during the life of the bridge, the plate thickness may vary in a complex fashion throughout the geometry of the gusset plates, further complicating the rating assessment of the plate.

1.3 OBJECTIVE AND SCOPE

The objective of this research is to investigate the strength of steel truss bridge gusset plates using finite element parametric test simulations validated against full-scale experimental tests. A wide range of arrangements are studied by starting with several representative bridge geometries, then varying the locations in the bridge, and thus the corresponding joint loading scenarios, joint geometries including gusset thicknesses, member end details, member inclination angles, member connection lengths, free edge lengths, fastener strengths, corrosion patterns and retrofit scenarios. The parametric study results are summarized and recommendations are provided to improve current gusset plate strength calculations in design and rating procedures.

The final report of the parent NCHRP 12-84 project for this research (Ocel, 2013) provides a detailed explanation of the experimental testing program conducted in parallel with this FEA simulation research, as well as the reliability calibration of resistance factors for the various gusset plate limit states.

1.4 REPORT ORGANIZATION

Chapter 2 of this report addresses the design of the parametric test simulation studies. This portion of the research includes the selection of bridge configurations, loading scenarios, member design, and connection parameters.

Chapter 3 discusses the procedures used for test simulation using 3D nonlinear finite element analysis. Chapter 4 then discusses the test simulation results in detail.

Chapter 5 provides preliminary recommendations for calculation of the resistance of steel truss gusset plates based on the simulation results.

Chapter 6 offers an overall summary of the studies and resulting conclusions.

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2. PARAMETRIC STUDY DESIGN

2.1 OVERVIEW

Prior NCHRP 12-84 research (FHWA, 2009b; FHWA, 2010; Mentes, 2011) addressed the design, execution and interpretation of five initial full-scale experimental tests of steel truss gusset plates representing a range of practical gusset plate geometries. These tests were conducted early in the NCHRP project timeline, and the corresponding geometries were used as a starting point for the parametric test simulation studies addressed in this report. Section 2.2 reviews the experimental test geometries and explains several direct parametric variations on them considered in this work. Section 2.3 then describes the design of a broader suite of parametric study tests aimed at exploring a wide variety of gusset plate geometries, loadings, and material characteristics observed from bridge plans collected by NCHRP 12-84.

2.2 EXPERMENTAL TEST GEOMETRIES AND VARIATIONS ON EXPERIMENTAL TESTS

Figures 9 through 13 show the gusset plate geometries of five experimental tests conducted early in the NCHRP 12-84 research. The primary goal of these tests was to assess two key limit states:

1) Buckling of the gusset plates due to the forces transferred from compression members, and

2) Failure of the gusset plates in shear along a plane parallel to the chord,

in double-gusset Warren with vertical truss configurations. These tests were conducted to interpret and quantify the fundamental behavior as well as to validate the finite element models to be used for the parametric study simulations. The main parameters varied in these tests were:

1) The fastener type, which influences the length of the member connections to the gusset plates, hence influencing the free-edge lengths of the gusset plates,

2) The “stand-off” distance of the compression diagonal from the adjacent chord and vertical members, which has an important effect on the slenderness of the gusset plate between the end of the compression diagonal and the other framing within the truss joint, and

3) The length of the connection along the diagonal members, which has a direct effect on the free-edge lengths of the gusset plates.

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Figure 9. E1-U-307SS-WV test geometry.

Figure 10. E2-U-307LS-WV test geometry.

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Figure 11. E3-U-307SL-WV test geometry.

Figure 12. E4-U-490SS-WV test geometry.

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Figure 13. E5-U-490LS-WV test geometry.

An NCHRP 12-84 assessment of collected bridge plans (FHWA, 2009b) found that two types of fasteners are common in steel truss gusset plates: A502 rivets and A325 bolts. For practical reasons, A307 bolts were used instead of A502 rivets in the experimental tests, because of the difficulty and expense of installing rivets. Prior research indicates that A307 bolts have a similar load-deflection response to hot-driven rivets (Roeder et al. 1994). In addition, A490 high strength bolts were used instead of A325 bolts to produce the smallest bolt groups possible, in number of fasteners and in size, and the greatest contrast in the behavior relative to the use of A307 bolts.

The term “standoff distance” used in the above summary is defined as the smallest gap between the corner of the compression diagonal and the closest adjacent truss member. Since the chord members are typically wider than the vertical members within the plane of the truss, the standoff distance is typically with respect to the chord. Two standoff distances were selected, representing the smallest and largest values observed from bridge plans: 1 inch and 4.5 inches.

The third main parameter was varied by selecting the number of fasteners along the length of the compression diagonal from five in the shortest connection to nine in the longest connection. The number of required fasteners was determined from fastener shear strength design criteria. In some of the connections, fasteners were excluded from the interior of the fastener groups to balance the connection shear strength design with the force requirements.

The five base experimental designs are referred to in this document as E1-U through E5-U, where the symbol “U” stands for “Unchamfered.” Joints with chamfered members are consid-ered extensively in the subsequent parametric studies. In addition, the label “wwwXY” is ap-

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pended to each of the above test names, where “www” represents the fastener type, either “307” or “490” for A307 or A490 bolts, “X” refers to the standoff distance, either “S” for short or “L” for long, and “Y” stands for the length of the connection, either “S” for short or “L” for long. Lastly, the label “WV” is appended to the end of the test name, indicating that these are “Warren with Vertical” joint configurations. Therefore, the name of the first experimental test is “E1-U-307SS-WV,” indicating experimental test 1, having Unchamfered members, A307 bolts, a short standoff distance, a short connection length, and a Warren with Vertical truss joint configuration.

All of the above five physical tests used 3/8 inch diameter gusset plates. In addition, the orienta-tion of the diagonals was 45 degrees in all of these tests. In this report, the slenderness of the plates is varied, using the above experimental test geometries, by specifying thicknesses ranging from 1/4 to 5/8 inches in increments of 1/16 inch. This allows a maximum amount of informa-tion to be extracted from each of the detailed FEA simulation models, since the plate thicknesses and the positions of the plates in the direction out-of-the-plane of the truss can be varied with relative ease once the other attributes of the geometry have been addressed. Although the smallest of these thicknesses is somewhat less than would be expected as a nominal thickness in new steel truss bridge designs, smaller thicknesses can be encountered due to loss of section associated with corrosion. Also, it is important to exercise the ability of prediction models to estimate the strengths over a wide range of plate slenderness values to ensure the robustness of the approximate calculations.

In addition to varying the plate thicknesses, another variation on the experimental test geometries is achieved by removing the vertical member from the test. This gives a relatively extreme Warren without vertical test configuration and allows for an interesting assessment of the influence of the truss vertical member on the gusset plate capacity.

All of the other details of the above variations on the experimental tests, i.e., the truss member sizes and the boundary conditions for the simulation models, are the same as in the physical tests. The reader is referred to Mentes (2011) for this information. The specific joint loadings are applied in proportion to the loadings employed in the physical ultimate strength tests. The ultimate strength test forces are specified as the “reference” forces. These forces, which are then scaled and applied incrementally to the simulation models to load the parametric specimens through their ultimate capacities, are listed subsequently with the description of the parametric test results in Chapter 4 Sections 4.2 and 4.4.

2.3 PARAMETRIC STUDY TESTS

2.3.1 Joint Configurations

For the parametric studies conducted in this research, three main truss types were selected:

1) Warren with vertical members,

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2) Pratt, and

3) Warren without vertical members.

Examination of available bridge plans indicated that longer bridges with continuous spans generally have Warren configurations with or without vertical members (Mentes, 2011). In addi-tion, the NCHRP 12-84 project panel indicated that Warren trusses without verticals are more common in more recent construction, and that Pratt trusses are used mostly for shorter simple-span bridges. All of the cases considered are geometries representative of steel highway bridge trusses. All of the joints involve double gusset plates, one on each side of the truss members (i.e., on each face of the truss). Single-gusset geometries are not addressed in this research.

Given the above general truss configurations, representative joints were extracted from the following locations in hypothetical steel truss bridges:

1) Joints at mid-span,

2) Joints at a pier in continuous-span or cantilevered truss construction,

3) Joints near a pier in continuous-span or cantilevered truss construction,

4) Joints at an inflection point in continuous-span or cantilevered truss construction,

5) 90 degree angle corner joints at the simply-supported end of a span,

6) Top chord corner joints with a larger than 90 degree angle between the chord and the end diagonal member at the simply-supported end of a span, and

7) Bottom chord joints at the start of a haunch over an interior pier in continuous-span or cantilevered truss construction.

Cases (6) and (7) are referred to subsequently as joints with a positive and negative angle between the chord members respectively.

Figure 14 shows a summary of the parametric test geometries considered. With the exception of the 90o angle corner joints, all of the test configurations are two-panel subassemblies with the test joint in the middle (highlighted by a circle in the sketches). By using two-panel subassemblies, the loads from the bridge can be applied at the ends of the subassembly and realistic equilibrium and kinematic boundary conditions associated with the truss framing can be achieved at the test joint with relative ease.

In the following, Warren with and without vertical subassemblies are used to study the influence of the joint position within the span. By considering mid-span, pier, near pier and inflection point locations, the geometry conditions and force demands at the test joint can be varied substantially.

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The location of a joint in a given bridge span has a significant influence on the ratio of the total vertical shear versus the major-axis bending moment resisted by the overall truss cross-section. In addition, the depth of the truss impacts the chord forces given the truss major-axis bending moment. Typically, continuous and cantilevered bridge trusses are deeper in the vicinity of the interior piers and shallower within the spans. Also, the orientation of the diagonals impacts the magnitude of the shear force parallel to the line of the chords, which is transferred by the gusset plates to develop the chord forces. The orientation of the diagonals is of course determined by the ratio of the truss depth to the truss panel length along the span.

Figure 14. Truss subassembly configurations for gusset plate parametric studies.

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Figures 15 and 17 show the geometries and applied loadings for all the Warren subassemblies, with and without verticals, studied in this research. As in Figure 6, the circle indicates the location of the joint under study.

Figure 16 shows the Pratt configurations studied in this research. For the Pratt configurations, only joints located at approximately at the quarter-span of simply-supported truss bridges are considered. This is because, as mentioned above, Pratt truss bridges are typically shorter simple-span bridges. Furthermore, at the bridge midspan, Pratt truss joints have the same configuration as the corresponding joints in Warren trusses with a vertical shown in Figure 15, and Pratt joints at the ends of a bridge have the same configuration as the corner joints shown in Figure 18.

Figure 18 shows the “other” configurations corresponding to Figure 14(d) through (f).

It is impossible to address comprehensively all the potential joint geometries and loadings that can be encountered in design practice. However, the 20 different configurations shown in Figures 15 through 18 are believed to provide a reasonable representation of the wide range of potential cases. Table 1 gives a summary of the various relevant forces and dimensional parameters for the 20 parametric test configurations. The terms shown in this table are as follows:

Va = vertical shear force in the left-hand truss panel.

Vb = vertical shear force in the right-hand truss panel, or the center truss panel for Warren without vertical subassemblies with two vertical loads along the top chord (cases P13 and P14).

Vc = vertical shear force in the right-hand truss panel of Warren without vertical subassemblies with two vertical loads along the top chord (cases P13 and P14).

V = change in the vertical shear force across the test joint, equal to the vertical load applied at the panel point location corresponding to the test joint.

M1 = truss cross-section moment at left-hand end of subassembly.

M2 = truss cross-section moment at middle of subassembly, or generally, at the test joint.

M3 = truss cross-section moment at right-hand end of subassembly.

d1 = truss depth at left-hand end of subassembly.

d2 = truss depth at middle of subassembly, or generally, at the test joint.

d3 = truss depth at right-hand end of subassembly.

27

Figure 15. Warren with vertical configurations.

Case P1, Joint at midspan Case P2, Joint at midspan

Case P3, Joint at pier Case P4, Joint at pier

30003000

30003000

2500 2500500

5500

30 ft

55 ft 55 ft

60 ft

300

500

200

3000

3000

2900

2900

424 283

40 ft

40 ft

300 500 200

3000

3000

2900

2900

-424

-283

40 ft

40 ft

Case P5, Joint near pier Case P6, Joint near pier

Case P7, Joint at inflection point Case P8, Joint at inflection point

15003000

1500

3000

2500 500 2000

30 ft

60 ft

30 ft

1200 4800

4800

1000

1000

2400 1900500

40 ft

40 ft

20 ft

-226

02470

1600 500

2460-3580

1100

3200

3200

2200

2200

40 ft

20 ft

40 ft

500

27.5 ft40 ft

1100

1600

1600

16001600

1600

-2100 2100

40 ft40 ft

30003000

30003000

2500 25005000

30 ft

60 ft

28

Figure 16. Pratt configurations.

Figure 17. Warren without vertical configurations.

Case P9, Joint near pier Case P10, Joint near pier

360

30 ft

360

260 110

730

730

150

368156

30 ft 30 ft

T T

360

30 ft

360

260 110

730

730

150

368

156

30 ft 30 ft

Case P11, Joint at midspan Case P12, Joint at pier

Case P13, Joint near pier Case P14, Joint at inflection point

3000 40 ft

35 ft

3000

3000

3000

250 250

500

3000

60 ft

30 ft

50002500 2500

3000

3000

3000

1650

2500

500

30 ft

30 ft 30 ft

3000

3000

1280

1280

45 ft60 ft

350

750

1600

500

40 ft

40 ft

40 ft 40 ft

1350

1350

1350

135027.5 ft

350

29

Figure 18. Other configurations.

Case P15, 90o Corner Joint Case P16, 90o Corner Joint

Case P17, 116.5o Corner Joint Case P18, 135o Corner Joint

Case P19, Bottom Chord Joint at the Start of a Haunch

Case P20, Bottom Chord Joint at the Start of a Haunch

30

b = truss panel length along the span of the bridge.

Ca = force in chord member on the left-hand side of the test joint.

Cb = force in chord member on the right-hand side of the test joint.

C = change in chord force across the test joint, equal to the shear force transferred to the chord from the gusset.

Da = diagonal force on left-hand side of the test joint.

Db = diagonal force on the right-hand side of the test joint.

Vjcr = vertical shear force transferred on the critical side of the test joint, subtracting the contribution from the axial force in the bottom chord (if the bottom chord is sloped).

Dcr = diagonal force on the critical side of the test joint, i.e., the side governing the strength limit state of the joint.

Ccr = chord force on the critical side of the test joint.

The force values are negative for compression and positive for tension.

Table 2 shows the following ratios of the test joint forces most relevant to the gusset plate behavior:

Ccr /Vjcr = ratio of chord force to the joint vertical shear force on the critical side of the test joint, indicative of the potential impact of the chord axial tension or compression on the gusset plate response.

Ccr / Dcr = ratio of chord force to the diagonal force on the critical side of the joint, also indicative of the potential impact of the chord axial tension or compression on the gusset plate response.

C/ Ccr = ratio of the change in the chord force across the test joint (i.e., the shear force transferred to the chord along the plane parallel to the chord) to the chord force on the critical side of the joint, also indicative of the potential impact of the chord axial tension or compression on the gusset plate response.

C/ Vjcr = ratio of the change in the chord force (i.e., the shear force transferred to the chord along the plane parallel to the chord) to the joint vertical shear force on the critical side of the test joint, indicative of the impact of the shear force transfer to the chord on the gusset plate response.

31

Table 1 Relevant forces and dimensions for the 20 parametric test configurations selected for this research.

V a V b V c V M 1 M 2 M 3 d 1 d 2 d 3 b C a C b Da Db V jcr D cr C cr C(kip) (kip) (kip) (kip) (ft‐kip) (ft‐kip) (ft‐kip) (ft) (ft) (ft) (ft) (kip) (kip) (kip) (kip) (kip) (kip) (kip) (kip)

1 300 ‐200 NA 500 116,000 128,000 120,000 40 40 40 40 ‐2,900 ‐3,000 ‐424 ‐283 300 ‐424 ‐2,900 100

2 300 ‐200 NA 500 116,000 128,000 120,000 40 40 40 40 2,900 3,000 424 283 300 424 2,900 ‐100

3 ‐2,500 2,500 NA ‐5,000 ‐180,000 ‐255,000 ‐180,000 60 60 60 30 3,000 3,000 2,800 2,800 ‐2,500 2,800 3,000 0

4 ‐2,500 2,500 NA ‐5,000 ‐165,000 ‐240,000 ‐165,000 55 60 55 30 ‐3,041 ‐3,041 ‐2,240 ‐2,240 ‐2,500 ‐2,240 ‐3,041 0

5 2,500 2,000 NA 500 ‐180,000 ‐105,000 ‐45,000 60 45 30 30 3,000 1,500 ‐1,500 1,200 1,341 ‐1,500 3,000 1,500

6 2,400 1,900 NA 500 ‐192,000 ‐96,000 ‐20,000 40 30 20 40 4,800 1,000 ‐2,260 2,470 1,598 ‐2,260 4,800 3,800

7 1,600 1,100 NA 500 ‐64,000 0 44,000 20 20 20 40 3,200 ‐2,200 ‐3,580 2,460 1,600 ‐3,580 3,200 5,400

8 ‐1,100 ‐1,600 NA 500 44,000 0 ‐64,000 27.5 33.8 40 40 1,619 ‐1,619 ‐2,100 2,100 ‐1,354 ‐2,100 1,619 3,238

9 260 110 NA 150 10,800 18,600 21,900 30 30 30 30 ‐360 ‐620 ‐368 ‐156 260 ‐368 ‐360 260

10 260 110 NA 150 10,800 18,600 21,900 30 30 30 30 ‐620 ‐730 368 156 110 156 ‐730 110

11 250 ‐250 NA 500 105,000 115,000 105,000 35 35 35 40 ‐3,143 ‐3,143 ‐290 ‐290 250 ‐290 ‐3,143 0

12 ‐2,500 2,500 NA ‐5,000 ‐180,000 ‐255,000 ‐180,000 60 60 60 30 ‐3,625 ‐3,625 ‐2,580 ‐2,580 ‐2,500 ‐2,580 ‐3,625 0

13 ‐1,650 ‐2,000 ‐2,500 0 ‐57,600 ‐62,850 ‐180,000 45 52.5 60 30 ‐1,741 ‐2,611 ‐1,650 1,430 ‐2,000 ‐1,650 ‐1,741 870

14 ‐750 ‐1,100 ‐1,600 0 37,125 125 ‐54,000 27.5 33.8 40 40 732 ‐604 ‐1,400 1,190 ‐1,100 ‐1,400 732 1,336

15 NA 2,000 NA ‐2,000 NA 0 60,000 NA 60 60 30 NA ‐1,000 NA 2,240 2,000 2,240 ‐1,000 1,000

16 NA 2,000 NA ‐2,000 NA 0 80,000 NA 40 40 40 NA ‐2,000 NA 2,830 2,000 2,830 ‐2,000 2,000

17 2,000 1,500 NA 500 0 60,000 105,000 0 60 60 30 NA ‐1,750 ‐2,240 1,680 1,500 1,680 ‐1,750 1,750

18 2,000 1,500 NA 500 0 80,000 140,000 0 40 40 40 NA ‐3,500 ‐2,830 2,120 1,500 2,120 ‐3,500 3,500

19 ‐1,650 ‐2,150 NA 500 ‐48,000 ‐114,000 ‐200,000 20 20 40 40 ‐2,400 ‐5,600 ‐3,700 ‐780 ‐1,650 ‐3,700 ‐2,400 2,609

20 ‐1,650 ‐2,150 NA 500 ‐73,500 ‐123,000 ‐187,500 30 30 50 30 ‐2,450 ‐4,500 ‐2,330 ‐500 ‐1,650 ‐2,330 ‐2,450 1,294

Case

32

Table 2 Force ratios and location information for the 20 parametric test configurations selected for this research.

This table also indicates the location of the test joint within a hypothetical truss span, as well as the critical side of the test joint in each of the sketches pertaining to the gusset strength limit state. In several of the cases, several variations of the same joint are tested. In these cases, the critical side of the joint is indicated for each of the variations. In addition, several of the joints are critical within the chord splice unless the chord members are milled to bear and are assumed to transfer their compression forces directly via bearing of the chord areas. The joints that are critical within the chord splice are indicated by the phrase “(chord splice).” Lastly, in several of the joints, there is no diagonal member on the left-hand side of the test joint, or the diagonal there can be interpreted as a continuation of the chord (e.g., at a corner joint near the simply-supported end of a truss). These cases are indicated by the phrase “(no force on left).”

One can observe from Table 2 that the absolute value of the ratio Ccr /Vjcr ranges from 9.7 to 12.6 at the midspan joints, 1.2 to 1.5 at the pier joints, 0.9 to 3.0 at the near-pier joints in the continuous-span bridges, 1.4 to 6.6 at the near-pier joints in the simple-span Pratt bridges, and 0.7 to 2.0 for the joints located at the inflection points. In addition, C / Ccr ranges from zero to 0.3 for the midspan joints, it is equal to zero for all the pier subassemblies, it ranges from 0.4 to 2.4 for the near-pier joints, and it ranges from 1.2 to 3.4 for the joints at inflection points. These

1 ‐9.7 6.8 ‐0.03 0.3 Midspan Left (chord splice)

2 9.7 6.8 ‐0.03 ‐0.3 Midspan Left (chord splice)

3 ‐1.2 1.1 0 0 Pier Sym

4 1.2 1.4 0 0 Pier Sym

5 2.2 ‐2.0 0.5 1.1 Near Pier Left

6 3.0 ‐2.1 0.8 2.4 Near Pier Left

7 2.0 ‐0.9 1.7 3.4 Infl. Pt.  Left

8 ‐1.2 ‐0.8 2.0 ‐2.4 Infl. Pt.  Both or Left

9 ‐1.4 1.0 ‐0.7 1.0 Near Pier Left

10 ‐6.6 ‐4.7 ‐0.2 1.0 Near Pier Right (no force on left)

11 ‐12.6 10.8 0 0 Midspan Sym (chord splice)

12 1.5 1.4 0 0 Pier Sym

13 0.9 1.1 ‐0.5 ‐0.4 Near Pier Left

14 ‐0.7 ‐0.5 1.8 ‐1.2 Infl. Pt.  Left (chord splice) or left

15 ‐0.5 ‐0.4 ‐1.0 0.5 Corner Right (no force on left)

16 ‐1.0 ‐0.7 ‐1.0 1.0 Corner Right (no force on left)

17 ‐1.2 ‐1.0 ‐1.0 1.2 Corner Right (no force on left)

18 ‐2.3 ‐1.7 ‐1.0 2.3 Corner Right (no force on left)

19 1.5 0.6 ‐1.1 ‐1.6 Start of Haunch Left (chord splice) or left

20 1.5 1.1 ‐0.5 ‐0.8 Start of Haunch Chord splice or left

Critical Side of JointC /C crCase C cr/V jcr C cr/Dcr C /V jcr Location

33

are representative values from different locations in the steel truss bridges collected by the project.

In the case of the Warren with and without verticals configurations, the magnitudes of the key forces Va, V, and M2 are based on estimates for a the I-35 bridge in Minneapolis, MN, which was a continuous-span steel truss bridge with a 456 ft main span, 120 ft total of deck width and eight total lanes of traffic. The forces Va and V are selected at approximately 50 to 75 % of the estimated truss cross-section shear forces to accentuate the potential strength interactions associated with the gusset plate working as part of the chord splices. For the Pratt configurations, the magnitudes of these base forces are based on estimates for a 240 ft simple span with a narrow 20 ft deck and only one lane of traffic. This prototype is selected to accentuate the consideration of relatively light (smaller) members and smaller joints for the Pratt trusses.

The reader is referred to Appendix I-1 for a brief summary of coarse estimates of the internal forces for the I-35 Bridge and to Appendix I-2 for the 240 ft simple-span bridge. The various parametric study results for the above Cases 1 through 20 are presented subsequently in Sections 4.5 through 4.10.

2.3.2 Other Joint Parametric Considerations

2.3.2.1 Mill-to-Bear versus Non-Mill-to-Bear Compression Splices

From the above 20 parametric study cases, three connections were selected to study the effect of mill-to-bear conditions at compression chord splices. Since the above chord-splice joints have a gap between adjoining chord members, the joint fails via compression buckling or tension yielding of the gusset in the chord splice for the specimens labeled as “chord splice” in Table 2. Adding the mill-to-bear condition in selected joints where the splice is in compression allows for the next most critical failure mode to be identified. The selected connections are P1, P19 and P20. In the finite element models, the continuity between chord members is modeled by tie constraints between all the nodes of the cross section at the end of chord members. That is, all the displacements and rotations of the abutting chord member cross sections are constrained to be equal. The results for the above cases with mill-to-bear conditions are presented subsequently in Sections 4.5.1.2, 4.10.1.2, and 4.10.2.2.

2.3.2.2 Material Strength

All the above joints are studied assuming Grade 50 materials for the gusset and splice plates. To study the effect of high strength materials on the behavior of the connections, three test joints were analyzed using Grade 100 materials for the gusset and splice plates, P5, P7, and P8. For the first set of analyses, the gusset plates were given the same thickness as the initial designs but the material is changed to Grade 100. Then for the second set of analyses, the gusset plate thicknesses were reduced in inverse proportion to the increase in the material strengths. The thinner high-strength plates are subject to greater stability effects. The results for the above cases

34

with the use of high-strength plates are presented subsequently in Sections 4.5.5.3, 4.5.7.2, and 4.5.8.3.

2.3.2.3 Member Chamfer versus No Member Chamfer

The above joints were designed using chamfered members. In other words, all these joints have diagonals that are chamfered to reduce the overall size of the joints. In the most extreme cases, the members were chamfered up to the extent where there are only two fasteners in the last row of bolts at the member end. As shown in Figure 19, when the diagonals are chamfered, the areas between chamfered edges and the adjacent members are small. As a result, the gusset plate area is relatively small. After the initial set of 20 joints presented in Section 2.3.1 was analyzed, test joints P5, P6, P8, P13 and P14 were selected and redesigned using unchamfered members. In general, when the members are unchamfered, the areas between diagonals and chords and/or diagonals and vertical members are larger than the ones in joints with chamfered members. Also, the lengths of the free edges are larger (relative to the overall area of the plate). This allows for the study of the effect of larger distances between the members and longer free edges. The corresponding results are discussed in Sections 4.5.5.2, 4.5.6.2, 4.5.8.2, 4.7.3.2 and 4.7.4.2.

2.3.2.4 Shingle Plates

All the gusset plate joints were initially designed without shingle plates. However, the joints at a pier would likely use shingle plates, since otherwise the larger main gusset plates need to be significantly thicker to transfer the large member forces into the pier. By using shingle plates, the increased thickness can be localized to the smaller portion of the gusset plate that actually needs the large thickness. Therefore, several cases were selected to study the behavior of shingle plates. The goal of these studies was to assess the distribution of forces between the main plate and the shingle plates and to determine appropriate methods to estimate the capacity that can be developed by the combined plates. The results of these simulations are discussed in Section 4.11.

2.3.2.5 Edge Stiffening

One of common practices to retrofit gusset plate joints is to add edge stiffeners on the free edges to prevent the buckling of these edges. Engineers commonly add short angles on the inside of gusset plates between members, e.g., between a chord and a diagonal. Therefore, a number of cases were studied to address the effect of these and other types of stiffeners on the maximum capacities of gusset plate joints. Section 4.12 focuses on these results.

2.3.2.6 Corrosion

The effect of corrosion on the behavior of gusset plate joints was also studied in this research. A number of joints were selected to be modeled with corroded regions, including holes due to corrosion. These corroded test cases were then analyzed with shingle plates to investigate the benefits from adding shingle plates as a retrofit method for corroded gusset plate joints. Test

35

configurations with corroded gusset plates with and without shingle plates are discussed in Section 4.13.

2.3.3 Joint Design

Based on selected joint configurations from Section 2.3.1 and the member cross-sections presented subsequently in Section 2.3.4, an initial set of 20 parametric study gusset plate joints was designed. After this initial set of 20 joints were designed and analyzed, additional gusset plate joints were studied by varying a number of parameters as discussed above. Figure 19 shows a typical joint geometry. This particular joint is P5-C-WV-NP (addressed subsequently in Section 4.5.5.1).

The design of the gusset plates for the simulation studies followed typical design practices as documented by Kulicki and Reiner (2011), but with specific adaptations of the FHWA Guide (FHWA, 2009a) equations for the various limit state resistance checks. Generally speaking, adaptations of the FHWA Guide rules were implemented in the design process to arrive at design proportions in which the gusset plates would tend to fail either in diagonal compression or in shear along certain critical planes, or in which the truss chord splice resistance or other failure modes would govern the resistance, but with maximum potential interaction with the gusset diagonal compression and/or shear behavior. As such, the parametric test simulation designs would allow for close scrutiny of design assumptions associated with these limit states.

Several general design rules were used to ensure reasonable joint sizes and efficiency of the design calculations. These rules were:

1) The centerlines of the members always intersect at a common work point.

2) The minimum clearance between the members is 1 inch.

3) All the connections to the members have a minimum of six rows of fasteners along the member length.

4) The chord members have the same cross-section on each side of the test joint.

5) The truss chords are spliced at the middle of the truss joint. That is, the center of the splice is located at the work point of the truss members.

6) Web splice plates are attached inside the chord members on each side of the joint; additional splice plates are attached to the outside of the chord top and bottom flanges.

7) The geometry of the splice plates is symmetric about the work point at a given joint.

8) Chamfered members have at least two fasteners in the last row at the end of the member.

9) The fasteners are all 7/8 inch diameter.

36

10) A spacing and pitch of 3 inches and an edge distance of 1.5 inches is used to lay out a rectangular grid of fasteners within each connection. Fasteners are removed from selected locations within the rectangular grids to reduce the total number of fasteners where the additional capacity is not needed.

11) The maximum edge distance of the rectangular connection blocks is limited to approximately 2.5 inches. The maximum edge distance allowed by AASHTO (2010) is the minimum of 5 inches or 8 times of thickness of the thinnest outside plate. For a 0.5 inch thick gusset plate, the corresponding maximum edge distance is 4 inches.

Figure 19. Typical gusset plate joint design for initial set of tests (P5-C-WV-NP).

For the initial set of 20 parametric study joint designs, all the diagonals were chamfered as much as possible. It can be seen in Figure 19 that the centerlines of all the members intersect at a common work point, the chord members have the same cross-sections and both diagonals are chamfered to a maximum extent such that they have only two fasteners in the last row at the end of the members.

For the A307 and A490 bolts used in the experimental test configurations, bolt single-shear force shear-displacement curves were generated by calibrating to test results. Figure 20 shows the shear-force shear-displacement curves for the A307 and A490 bolts respectively. These shear-force shear-displacement curves are used in all the simulations of the experimental tests as well as in all the variations on these tests. However, a different connector strength and force-deformation response was used for the parametric study simulations as explained below.

1518

30,47

12

1,5

39

35,1

7

12

1,97

37

The required number of fasteners connecting each member to the gusset plates in the parametric study configurations was determined originally by dividing the design loads shown in Figures 15 through 18 by an assumed maximum rivet strength in single shear of Rn = 16 kips. This rivet strength value was obtained based on tension test results of A307 bolts mimicking rivets. At a later stage of the research, an updated rivet single-shear force shear-displacement curve was obtained by calibrating to tension test results for hot driven rivets. This shear-force shear-displacement curve for rivets, shown in Figure 21, indicates a maximum strength of 30.1 kips. The rivet shear-force shear-displacement curve shown in Figure 21 was used in modeling the fasteners in all the parametric test simulations.

Figure 20. Nonlinear shear-force shear-displacement curves for A307 and A490 bolts in single shear used in modeling the experimental tests and variations on these tests.

In traditional and current design practice, the decisions made in sizing the chord splice plates can vary widely. This is because, once the designer has determined the chord member forces that must be transmitted through the joint under consideration, he or she has to select a criterion for determining the percentage of these forces that are transferred by the different splice plates, as well as the splice force transfer contribution from the gusset plates. This system of plates and the different groups of connectors transferring forces between them and the members is highly inde-terminate. The actual distribution of the member forces to the splice plates and the gusset plates can vary substantially based on layout decisions made by the designer as well as the nonlinearity in the responses of the different components as the strength load levels are approached.

It is common for designers to assign the fraction of the forces that must be transferred by the splice plates at a given connection based on simplified assumptions, not realizing that this can have significant implications on the forces that must be transferred based on fundamental static

0

10

20

30

40

50

60

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Faste

ner

Fo

rce

in

Sin

gle

Sh

ea

r (

kip

s)

Displacement (in)

A490 7/8" Bolt

A307 7/8" Bolt

38

equilibrium within other components of the joint. Given an initial assumption about the force transfer at a given connection, it is necessary to then consider static equilibrium of the various components to determine a set of statically admissible forces throughout the joint. This can be illustrated using the P5-C-WV-NP joint as follows:

Figure 21. Nonlinear shear-force shear-displacement curves for hot driven rivets in single shear used in the parametric study test simulations.

1) First, assume that the left-side chord member force of 3000 kips tension (see Case P5 in Figure 7) is distributed to the plates in the joint in proportion to the ratio of the web and flange plate areas of this member. This is a reasonable assumption that many design engineers might make since the flange and web connection forces would then induce comparable axial stresses in both the flanges and the webs of the chord member.

2) Based on the relative web and flange areas of the chord member at P5-C-WV-NP, the force transferred by the combination of the splice plates and gusset plates attached to the two webs of this rectangular box section member is taken as 1772 kips. Correspondingly, the force transferred by the flange splice plates is taken as 1228 kips such that 1772 kips + 1228 kips = 3000 kips.

3) For the connections to the top and bottom flanges of the left-hand chord, all of the fasteners are in single shear, since splice plates are placed only on the outside of the flanges in the parametric study joint designs. Based on the above total force in the two chord flanges of 1228 kips, a force of 1228 kips / 2 = 614 kips must be transferred to each of the flange splice plates (one connected to the top flange and one to the bottom flange of the chord).

0

5

10

15

20

25

30

35

40

0.00 0.10 0.20 0.30 0.40 0.50

Load

(ki

ps)

Displacement (in)

Calibrated Rivet Strength for Test Simulation

Experimental Test Results

39

Using an assumed rivet strength of Rn = 16 kips, and assuming equal shear forces on each of the fastener shear planes, a total of 42 fasteners (seven rows of six fasteners) is selected for each of the top and bottom splice plates on the left- and right-hand sides of the joint (42 fasteners × 16 kips = 672 kips).

4) Given the above force of 1772 kips, which must be transferred from the two webs of the left-hand chord member in P5-C-WV-NP, the designer would naturally apportion half of this force to each web, maintaining symmetry of the force flow within all the components of the joint. Therefore, the web splice plate plus the gusset plate attached to each web must transfer 1772 kips / 2 = 886 kips.

5) Next, the designer needs to decide how much of the above 886 kip force should be taken by a web splice plate and how much should be transferred by the gusset plate connected to the web. At this stage, one may recognize that the web splice plates need only be provided over a short length on each side of the splice, and that the geometry of the gusset plates usually extends beyond the ends of the splice plates on each side of the splice. Furthermore, a common assumption used in determining the number and layout of the connectors between the chord web and the splice and gusset plates would be that the fasteners act in double-shear where they pass through three plates (i.e., the splice plate, the chord web plate and the gusset plate), whereas that act only in single-shear outside of the ends of the splice plate, where they only pass through the chord web plate and the gusset plate. After making these assumptions and decisions, the designer can determine the combined number of fasteners required to transfer the left-hand chord force. In the context of P5-C-WV-NP, a reasonable decision would be to use 24 fasteners in double-shear close to the middle of the splice. These fasteners can be placed in a 4 x 6 rectangular grid on the left-hand side of the splice, and the end of the web splice plate can be located at approximately 1.5 inch + 3 × 3 inches + 1.5 inch = 12 inches to the left of the middle of the splice. This end of the web splice plate is shown by a dashed line on the web of the left-hand chord member in Figure 11.

6) Using an assumed rivet strength of Rn = 16 kips, and assuming equal shear forces on each of the fastener shear planes, each web splice plate then needs to transfer 24 shear planes × 16 kips = 384 kips in the above design. Correspondingly, each of the gusset plates transfers 24 × 16 kips = 384 kips from these fasteners, which are assumed to be loaded in double shear.

7) In addition to the above fasteners in double shear, the designer might select an additional 24 fasteners in single shear between each gusset plate and webs of the left-hand chord member in P5-C-WV-NP. These fasteners are located to the left of the end of the web splice plates. These fasteners are capable of transferring an additional 24 × 16 kips = 384 kips to each of the gusset plates.

8) Given the above layout of the fasteners, 384 kips can be transferred to the web splice plates and 384 kips x 2 = 768 kips can be transferred to the gusset plates. Considering both webs

40

of the chord, the sum of the connection capacities is 2 × 384 kips + 2 × 768 kips = 2304 kips, which is greater than the above required 1772 kips force. Therefore, there is adequate fastener shear capacity to transfer the total forces from the webs of the left-hand chord member, based on the above assumptions about how the forces distribute. The assumption that all of the fastener shear planes are loaded equally at the fastener shear capacities is a very broad one. This is based on the general premise that, as the connections approach their ultimate strength, the connection components are able to deform inelastically to the extent that any non-uniform fastener forces are redistributed.

9) For the further assessment of the gusset plates, the designer might commonly assume a uniform distribution of the fastener forces to all their shear planes, although this assumes that redistribution occurs at a force level less than the ultimate strength of all the fasteners in shear. This is a reasonable assumption based on the lower-bound theorem of plastic design. That is, assuming that all the various components of the joint are sufficiently ductile, the designer can assume any internal force distribution that is statically admissible and does not violate any of the component strengths. Based on this assumption, the two gusset plates resist 2 × 591 kips and the two web splice plates resist 2 × 295 kips, giving the total force of 2 × 591 kips + 2 × 295 kips = 1772 kips that must be transferred from the webs of the left-hand chord member.

10) Figure 22 shows a free-body diagram of the end of the left-hand chord member illustrating the force transfer to the flange splice plates, the web splice plates, and the gussets based on the above design assumptions.

Figure 22. Free-body diagram of end of left-hand chord member in joint P5-C-WV-NP,

based on design assumption of equal average stress in the chord flanges and webs associated with the web and flange forces transferred to the joint.

11) Given the above assumptions for the transfer of the 3000 kip force from the left-hand chord member to the gusset plates, the web splice plates and the flange splice plates at the joint, one can sketch a free-body diagram of an individual gusset plate. This is shown in Figure

41

23. It can be observed that even though a tension force is transferred from the right-hand chord member to the joint overall, which is directed toward the right in a free-body diagram of the joint, horizontal equilibrium of the gusset plate requires the transfer of a 159 kip force to the left from the fasteners attached to the right-hand chord member. This is necessary to balance the 333 + 417 = 750 kip horizontal shear force transferred from the diagonal members to each of the gussets (i.e., 750 kips – 591 kips = 159 kips).

Figure 23. Free-body diagram of an individual gusset plate in joint P5-C-WV-NP.

12) In the above, basically the designer started the sequence of calculations by assuming a reasonable distribution of forces transferred from the flange and web connections to the left-hand chord. This was followed by a reasonable apportioning of the forces that must be transferred from the left-hand chord webs to the web splice plates and the gussets. However, the above decisions by the designer lead to a somewhat odd transfer of forces that must occur in the connection to the right-hand chord, all based on satisfying fundamental static equilibrium of the gusset plates. The corresponding forces transferred to the webs and flanges of the right-hand chord are not as well balanced, and the fasteners that have two shear planes do not work in double-shear, given the statically admissible gusset plate forces shown in Figure 15.

13) In the design of the P5-C-WV-NP joint, 36 fasteners were selected to connect the gusset plate to the right-hand chord member within the length of the member to the right of the web splice plate and 18 fasteners were selected to connect the gusset plate to the right-hand chord member within the length of the splice plate to the right of the splice. This is of

42

course more than adequate to develop the above 159 kip force into the right-hand chord member. This number of fasteners was provided to “stitch” the gusset plate to the chord member on the right-hand side of the joint. It can be observed from Figure 19 that the interior bolts in this fastener group are spaced at 5 inches longitudinally and 2.5, 5 or 7.5 inches transversely.

14) Figure 24 shows a free-body diagram of the right-hand chord member based on the above approximations. One can observe that a total tension force of 1818 kips = 1500 kips + 159 kips × 2 must be transferred from the right-hand chord member to the flange and web splice plates based on the above assumed distributions. This force is balanced exactly by the flange splice plate forces of 2 × 614 kips and the web splice plate forces of 2 × 295 kips shown in Figure 14.

Figure 24. Free-body diagram of end of right-hand chord member in joint P5-C-WV-NP.

15) One can observe that given the initial assumption that the 3000 kip force in the left-hand chord member was distributed to the flange splice plates and the combined gusset and web splice plates in proportion to the flange and web areas of the chord member, and given the assumption of equal forces in all the fasteners, one arrives at a required distribution of forces on the right-hand chord member (based on static equilibrium of the gusset plate) that is not proportional to the relative flange and web areas on this member. This is not a problem. In fact, the designer can make other valid assumptions as long as the flow of forces throughout the joint is accomplished satisfying equilibrium. For instance, if it is assumed that the shear capacity of the fasteners of 672 kips and 384 kips is developed in transferring the forces to the flange and web splice plates from the left-hand chord member, one obtains the free-body diagram shown in Figure 25, in which only 444 kips is transferred into the gusset plate on the left-hand side of the joint. However, this requires a force of 306 kips to be transferred from each gusset plate to the right-hand chord member in the free-body diagram of the gussets shown in Figure 26. In turn, this means that the flange splice plates and web splice plates must develop at total force of 2112 kips = 1500 kips + 306 kips × 2 from the right-hand chord member based on the free-body diagram in

43

Figure 19. This matches with the total force of 672 kips x 2 + 384 kips x 2 = 2112 kips used as the above alternate initial assumption.

Figure 25. Free-body diagram of left-hand chord member in joint P5-C-WV-NP, based on the assumption that the fasteners in the flange and web splice plates are loaded to their shear

capacity.

Figure 26. Alternate free-body diagram of an individual gusset plate in joint P5-C-WV-NP,

based on the assumption that the fasteners in the flange and web splice plates are loaded to their shear capacity.

44

Figure 27. Alternate free-body diagram of right-hand chord member in joint P5-C-WV-NP,

based on the assumption that the fasteners in the flange and web splice plates are loaded to their shear capacity.

16) In the design of P5-C-WV-NP, instead of using either of the above sets of equilibrium force diagrams, it was decided to layout the fasteners attached to the chord on the right-hand side of the joint such that the gusset plate resists a small amount of tension from the right-hand chord and all of the fastener shear forces are in the opposite direction from the 1500 kip tension force in the right-hand chord. Using this approach, twenty-eight fasteners were provided on the right-hand side of the flange splice plates (to the right of the splice) to develop a total force of 28 x 16 kips = 448 kips and 18 fasteners were provided on the right-hand side of the web splice plates to develop a total force of 18 x 16 kips = 288 kips. Assuming these fastener shear forces are developed, then an additional 14 kips must be transferred to each of the gusset plates as shown in Figure 28. This results in the gusset free-body diagram shown in Figure 29 and the left-hand chord member free-body diagram shown in Figure 30. Thirty-six additional bolts are provided between the gusset plate and the right-hand chord member to the right of the web splice plate, to stitch the gusset plate to the right-hand chord as discussed in step (10).

17) Furthermore, it was decided to layout the fasteners for the left-hand chord of P5-C-WV-NP based on the above steps (1) through (7). Given the free-body diagrams in Figures 20 through 22, one can observe that the total shear capacity of the 48 fasteners connecting the gusset plate and the left-hand chord (24 in single shear, outside of each of the web splice plate, and 24 also transferring a shear force from the web splice plate), equal to 48 x 16 kips = 768 kips is larger than the required shear force from Figure 29 of 764 kips. In addition, the total shear that can be transferred by the 24 fasteners attached to the web splice plates is 24 x 16 = 384 kips, which is larger than the required force of 288 kips shown in Figure 20. Lastly, the required shear force of 448 kips shown in Figure 22, which needs to be developed into each of the flange splice plates, is smaller than the shear capacity of the 42 fasteners provided between the flange splice plates and the left-hand chord.

45

Figure 28. Second alternate free-body diagram of right-hand chord member in joint P5-C-

WV-NP, based on developing the shear capacity of 28 fasteners to the flange splice plates and 18 fasteners to the web splice plates on the right-hand side of the joint.

Figure 29. Second alternate free-body diagram of an individual gusset plate in joint P5-C-

WV-NP, based on developing the shear capacity of 28 fasteners to the flange splice plates and 18 fasteners to the web splice plates on the right-hand side of the joint.

46

Figure 30. Second alternate free-body diagram of left-hand chord member in joint P5-C-WV-

NP, based on developing the shear capacity of 28 fasteners to the flange splice plates and 18 fasteners to the web splice plates on the right-hand side of the joint.

As mentioned above, the rivet strength curve with Rn = 30.1 kips was used in the simulation studies of the joints. As a result, the joints were designed generally with a larger number of rivets than required to transfer the truss member design loads. As a result, the fastener responses were essentially elastic at the strength limit. Since the “actual” rivet strengths (Rn = 30.1 kips) were significantly larger than the originally assumed fastener shear strengths of 16 kips in single shear in the test designs considered in this research, failure of the fasteners was encountered in only one of the studies. This exceptional case is discussed in Section 4.5.1.2.

Given the above statically admissible force distributions, the web and flange splice plates may be designed. The flange splice plates were designed to each transfer 448 kips in tension and the web splices were designed to each transfer 288 kips in tension in P5-C-WV-NP. In general, statically admissible force distributions have been developed as a first step in sizing the splice plates in each of the parametric test joints studied in this research. In other joints similar to this example, but where a splice plate was in compression, the plate was sized based on compression buckling of the splice plate length between the last rows of fasteners in each of the chord members at the splice, using K = 1.2.

For all of the parametric study joint designs, the widths of the web splice plates were assumed to be 0.5 inches smaller than the depth of chord members. In addition, the widths of top and bottom flange splice plates were assumed to be same as the widths of flange plates of the chord members. Given these widths, the required plate thicknesses for the splice plates were calculated. For P5-C-WV-NP, the final flange splice plate cross-section dimensions were 19.625 inches x 0.775 inches and the final web splice plate cross-section dimensions were 17.5 inches x 0.508 inches.

One design philosophy for sizing the splice plates would be to proportion them such that their capacities are smaller than the shear capacity of the fasteners on either side of a splice, thus ensuring that the splice plates would fail before the fasteners fail in shear. In fact, one could take

47

a philosophy that the splice plates also should fail by tension yielding and not tension rupture. This type of capacity design approach was utilized in effect for the fastener design in this research by the selection of the required number of fasteners based on Rn = 16 kips, but then using a fastener force-displacement curve with a maximum strength or Rn = 30.1 kips in the test simulations. In addition, due to the significantly larger fastener strengths used in the simulation models, the distribution of forces within the joints is not influenced any fastener nonlinearity associated with reaching a maximum fastener strength limit. Rather, the distribution of forces is affected by the elasticity of the fasteners and the plates, as well as the onset of inelastic response in the plates as the governing strength conditions of the joints are approached.

The initial design gusset plate thicknesses for the parametric study joints were selected using the FHWA Guide (FHWA, 2009a), but using liberal assumptions of:

K = 0 for design of the gusset plate for the force delivered from the compression diagonal in cases of diagonal members with chamfered ends (i.e., stability effects were neglected in selecting the gusset plate thickness in these cases). The general model for the compression buckling from the FHWA Guide is referred to in the subsequent discussions as the Whitmore buckling model.

K = 1.2 for design of the gusset plate for the compression diagonal force in cases where the diagonal members do not have chamfered ends. In some cases, these were also sized with K = 0.

= 1 for the shear strength of the gusset plate along the edge of the truss chord.

These liberal assumptions were selected to arrive at initial designs where the gusset plate would likely be critical in compression and/or shear at the reference load level, and thus the various attributes of the gusset plate compression and shear strength models could be scrutinized. Generally, for checking of gusset buckling due to compression from truss vertical members, Whitmore buckling with K = 1.2 was assumed. As noted previously, K = 1.2 also was used for sizing of splice plates in compression, but in this case, the full cross-section area of the splice plates generally was used in the calculation.

As mentioned previously, in addition to the above “initial” or “base” gusset plate thicknesses, the gusset plate thicknesses generally were varied over a wide range to maximize the useful data for each of the joint geometries. For the tests with other gusset plate thicknesses, the splice plate thicknesses were held constant at the values determined in the initial design. Specific design thicknesses are summarized for the various problems with the presentation of the simulation results in Chapter 4.

The above design procedures were used only to arrive at representative joint configurations to be evaluated in the research studies. Chapter 5 addresses preliminary assessment of potential improved design calculations.

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2.3.4 Member Design

Two sets of box- and I-section members were designed for the parametric study truss subassemblies. For the larger bridges shown in Figures 15, 17 and 18, five box-sections and three I-sections were designed to resist the ranges of compression and tension forces obtained from the subassembly geometries. In as the companion experimental NCHRP 12-84 research, box-sections were used for all the chord members, compression diagonals, and vertical members subjected to compression. Conversely, I-sections were used for tension diagonals and for vertical members subjected to tension.

Based on the example continuous-span truss bridge used to calculate the maximum chord and shear forces, the total out-of-plane depth of all the members was taken as 21 inches. For the Pratt truss joints shown in Figure 16, two box-sections and one I-section were designed. For these members, the full out-of-plane depth was set to 12 inches. Tables 3 and 4 show the plate dimensions for each of the member cross-sections, the corresponding factored tension yielding and tension rupture strengths, and the factored compressive strengths for a range of member lengths from 360 inches to 805 inches (30 feet to 67 feet). This is the range of member lengths used in the subassembly configurations shown in Figures 15 to 18.

Table 3 Member designs for all configurations other than Pratt joints.

(i) Box Sections

Box Sections

Dimensions Tension Yielding (y = 0.95)

Tension Rupture

(u = 0.80)

Compressive Strength (c = 0.9) web cover plate

bi ti bo to yPn uPn L cPn (in) (in) (in) (in) (kips) (kips) (in) (kips)

B1 12.00 0.750 20.250 0.8125 2418 2374

360 - 805

964 - 1927 B2 15.00 0.875 20.125 0.875 2920 2798 1512 - 2451 B3 18.00 1.375 19.625 0.875 3983 3609 2270 - 3408 B4 21.00 1.500 19.5 1.25 5308 4856 3521 - 4683 B5 24.00 2.0 19 1.5 7267 6500 5133 - 6492

(ii) I Sections

I-Sections

Dimensions Tension Yielding (y = 0.95)

Tension Rupture

(u = 0.80) Flange web

bf tf D tw yPn uPn (in) (in) (in) (in) (kips) (kips)

I1 12 0.875 19.25 0.75 1683 1524 I2 18 1.375 18.25 1.125 3326 2891 I3 21 1.75 17.5 1.5 4738 4231

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Table 4 Member designs for Pratt joints.

(i) Box Sections

Box Sections

Dimensions Tension Yielding (y = 0.95)

Tension Rupture

(u = 0.80)

Compressive Strength (c = 0.9) web cover plate

bi ti bo to yPn uPn L cPn (in) (in) (in) (in) (kips) (kips) (in) (kips)

B1p 8.00 0.375 11.625 0.5 837 826 360 - 805

114 - 538 B2p 12.00 0.625 11.375 0.625 1388 1292 464 - 1067

(ii) I Sections

I-Sections

Dimensions Tension Yielding (y = 0.95)

Tension Rupture

(u = 0.80) flange web

bf tf Dw tw yPn uPn (in) (in) (in) (in) (kips) (kips)

I1p 8 0.5625 10.875 0.5 686 648

It should be noted that nominal resistances were used in determining the gusset plate proportions, whereas factored strengths were used in the above member designs.

50

51

3. TEST SIMULATION PROCEDURES

3.1 FINITE ELEMENT MODELS

Figure 31 shows a typical test simulation model used in this research. In all subsequent discussions, the joints in the two-panel subassemblies are always referred to as U1 through U3 on the upper chords and L1 through L3 on the lower chords. In all cases with the exception of the corner joint configurations of Cases P15 and P16, the gusset plate test joint is located at the U2 or the L2 location. For example, for Case P5 shown in Figure 31, the gusset plate joint is located at U2.

Figure 31. Typical test simulation model (shown for P5-C-WV-NP).

The test simulations conducted in the current research were performed using Abaqus (Simulia, 2010). In all the cases, the gusset plates, splice plates, and a partial length of all the members in the vicinity of the test joint were modeled using four-node shell elements referred to as the S4R element in Abaqus. The length along the members initially targeted for modeling using shell elements was a minimum of three times the connection length for any given truss member (d1 + d2 ≥ 3d1, see Figure 32). However, to simplify the generation of the finite element models, the partial member lengths modeled using shell elements were set finally to 200 inches in all cases

52

unless noted otherwise, which approximately satisfies the above requirement. All the parts in Figure 31 not shown as lines were modeled using S4R elements.

The remaining lengths of the truss members outside of the 200 inch limit, as well as the other truss members not connected to the gusset plate joints, were modeled using 2-node linear-order beam elements (the B31 element in Abaqus). Multi-point constraints were used to connect the member cross-section modeled with shell elements to the corresponding end node of a beam element at the transition between the element types.

Figure 32. Length from work point within which the members are modeled using shell finite elements.

Figure 33 shows typical loading and boundary conditions used in this research. All the end nodes of the truss members that are on the outside perimeter of two-panel system (L1 to L3, U1, and U3

for P5-C-WV-NP) are restrained in the out-of-plane direction. The truss subassemblies are supported within the plane of the truss. For the case shown in Figure 33, horizontal and vertical displacements are restrained at U1 and the vertical displacement is restrained at U3. In addition, an out-of-plane constraint is applied at one node at the center of the flange splice plate on the outside of the test joint to prevent overall out-of-plane movement of the test joint. The tendency for this overall out-of-plane movement is largest for the joints located at the midspan of the prototype bridge span. However, the out-of-plane reaction at this constraint was found to be small in all the study joints.

As shown in Figure 33, loads are applied at the nodes at the left and right end of the two-panel system (L1 to L3, U1, and U3 in this figure) except for the 500-kip load applied at the vertical member. Whenever there was a load applied at the truss cross-section corresponding to the test joint, this load was applied at the location where the shell-element cross-section is attached by a multi-point constraint to the beam element model. This decision was based on the fact that in actual bridges, this load is transferred from a floor beam which is typically attached at some

53

location away from the joint. By applying this load at the end node of a beam element, any potential issues associated with stress concentrations from a point load applied to the shell elements are avoided.

Figure 33. Typical loading and boundary conditions (shown for P5-C-WV-NP).

3.2 GEOMETRIC IMPERFECTIONS

Early studies conducted within the NCHRP 12-84 research (Mentes, 2011; FHWA, 2010) indicated that:

Gusset plate compressive strengths can be affected substantially by small geometric imperfections; however, the sensitivity to the magnitude of the geometric imperfections becomes small once the magnitude becomes only a small fraction of various allowable maximum tolerances that may be considered as appropriate fabrication and erection tolerances in practice.

For the double-gusset plate geometries focused on in this research, the slightest lack of symmetry in the imperfection patterns about the plane of the truss leads to gusset plate compression diagonal failure modes, when diagonal compression governs the resistance, in which the end of the compression diagonal attached to the gusset plate moves out of the plane of the truss and the gusset plates effectively fail in a sidesway mode.

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Various detailed gusset plate geometric imperfection patterns have little effect on the overall resistance of the gusset plates to diagonal compression, as long as the patterns include some out-of-plane displacement at the end of the compression diagonal that is a small fraction of allowable maximum tolerances.

These initial findings led the research team to focus on using the following approach for seeding geometric imperfections into finite element models for all of the simulation tests.

Figure 34 shows a typical geometric imperfection shape used in the simulations. The geometric imperfections include both out-of-flatness and general out-of-plane displacement of both of the gusset plates, as well as an out-of-alignment of the compression diagonal created by an out-of-plane displacement of the compression diagonal member (the left diagonal member in this case) at the gusset plate. These imperfections are generated by using a separate linear elastic analysis of the test joint model. In this separate analysis, pressure loads are applied to the surface of the gusset plates in the out-of-plane direction to cause the gusset plate out-of-flatness in the vicinity of the compression diagonal side, and a specified out-of-plane displacement is applied at the end of the compression diagonal to create an initial out-of-alignment of this member in the out-of-plane direction. The pressure loadings were applied to the entire “free surface area” of the gusset plates (i.e., the surface areas not in contact with the structural members) on both faces of the truss in the vicinity of the compression diagonal.

To produce the geometric imperfections for the simulation analysis, the deflections from the above pre-analysis were scaled such that the maximum magnitudes of the out-of-flatness of the gusset plates and the out-of- plumbness of a diagonal member match the following imperfection limits: (1) Lmax/150 for the maximum out-of-flatness of the gusset plate, where Lmax is the length of longest free edge adjacent to the compression diagonal and (2) 0.1Lgap for the maximum out-of-plane displacement of the compression diagonal at the gusset plate, where Lgap is the smallest length of the gap between the compression diagonal and the adjacent members. For P5-C-WV-NP shown in Figure 34, the vertical free edge between the left chord and the compression diagonal gives Lmax = 35.2 inches. Therefore, the maximum out-of-flatness of the gusset plate was set to 0.235 inches. In addition, Lgap is 1 inch between the compression diagonal and the vertical member. Therefore, a maximum out-of- plane displacement of the compression diagonal of 0.1 inch was specified. The deformations from the pre-analysis were scaled accordingly to obtain the geometric imperfections for the simulation analysis.

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Figure 34. Typical geometric imperfection shapes on gusset plate joints (shown for P5-C-WV-NP).

3.3 MATERIAL PROPERTIES

Figure 35 shows true stress-strain curves for Grade 50 and Grade 100 steel used in the test simulations. The yield strength of the material was determined based on a bias of 1.10 of the true yield strength from the nominal specified and a reduction of 2 ksi to account for the difference between the 0.2 % offset and static yield strengths. The 2 ksi reduction was based on an evaluation of tensile coupon data by NCHRP 12-84. Therefore, for Grade 50 steel, the static yield strength, Fy, was selected as 50 ksi × 1.10 – 2 ksi = 53 ksi.

The Grade 50 true stress-strain curve shown in Figure 35 is based on a curve-fit to tension cou-pon data for the experimental test specimen E2-U-307LS(3/8)-WV. This specimen has a yield strength of 48.2 ksi. The stress-strain curve shown in Figure 35 was obtained by scaling the curve-fit data so that the true yield stress is 53.1 ksi. This 53.1 ksi true stress corresponds to the targeted engineering yield stress of 53 ksi. In the test simulations, all the data points shown in Figure 35 are input explicitly from a plastic strain of 0.0 up to the peak of the true stress-strain response. Abaqus assumes a flat plateau after this data point. Generally, this strain magnitude is beyond the maximum strain required for assessment of the joint behavior in the simulation models.

As discussed in Section 2.3.2.2, Grade 100 steel was used in the test simulations for selected cases (tests P5, P7, and P8). The static yield strength of Grade 100 steel is estimated as described above, and as such, the engineering yield stress was taken as Fy = 100 ksi × 1.10 – 2 ksi = 108

56

ksi. The true stress-strain curve shown in Figure 35 is based on fit-curve data to a Grade 100 steel coupon test that has an actual yield strength of 110 ksi. The Grade 100 steel stress-strain curve data was specified for Abaqus as described above. The descending portion of the true stress-strain curve subsequent to approximately 7 % strain was not modeled in Abaqus. However, the specimens generally did not experience any maximum local strains larger than 7 % prior to reaching “failure” as discussed in Section 4.1.

Figure 35. True stress-strain curves for Grade 50 and Grade 100 steel.

3.4 FASTENER STRENGTHS

For all the test simulations, the fasteners are modeled using the “CARTEISAN” plus “ALIGN” connector element with the force-deformation characteristics fit to nonlinear shear force-shear deformation responses in Abaqus. Figures 36 and 37 show the nonlinear shear-force and shear-displacement curves for A307 and A490 bolts and hot driven rivets respectively. It should be noted that these are the same curves as shown previously in Figures 20 and 21. The fastener properties are modeled such that the strength curves shown in Figures 36 and 37 are applied to the square root of the sum of squares of the shear loads within the fastener element. Where the fasteners connect three or more plates, the fasteners have two or more layers of connector elements. For the out-of-plane component for the fastener force-deformation response (i.e., the fastener axial tension force-elongation response), elastic behavior is assumed with a stiffness of EA/L, where E is Young’s modulus, A is the area of the fastener shank, and L is the total length of the fastener. The relative movements between the ends of the connector elements (located at

0

20

40

60

80

100

120

140

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Tru

e S

tres

s (k

si)

Plastic Strain (in/in)

Grade 50 steel (Fy=53 ksi)

Grade 100 steel (Fy=108 ksi)

57

the mid-thickness of the plates) are modeled such that any relative rotation of the plates is restrained but relative displacements between the plates are allowed.

Early project studies showed that the joint response is insensitive to the fastener type when different fastener types are used with the same connection geometry. Of course, different fastener design resistances can lead to significantly different connection sizes and geometries. In this research, this is the most important effect of the selected fastener characteristics.

Figure 36. Nonlinear shear-force shear-displacement curve for A307 and A490 bolts in single shear.

3.5 MODELING OF TENSION RUPTURE OR SHEAR RUPTURE RESISTANCES

The finite element simulation models employed in this research are targeted at capturing general yielding and buckling resistances of gusset plates. They do not have the resolution capable of modeling tension rupture or shear rupture strength conditions. The above fastener models represent the broad phenomenological response associated with the local combined shear deformation of the fasteners and/or the bearing deformation of the plates at each of the fastener locations. The fastener holes are not explicitly modeled. Rather, a “region of influence” equal to a radius intermediate between the fastener head maximum radius and the radius of the shank of the fastener is modeled as a rigid zone within the plates. The displacements of any plate nodes within the region of influence are constrained to deflect as if connected to a rigid body. Actual modeling of detailed plate bearing failure at a fastener location, or rupture of a plate through the fastener holes would require a substantially more refined FEA simulation model within the specific regions of the bearing or rupture failure. The values for the plate equivalent plastic

0

10

20

30

40

50

60

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fas

ten

er F

orc

e in

Sin

gle

Sh

ear

(ki

ps)

Displacement (in)

A490 7/8" Bolt

A307 7/8" Bolt

58

strains (PEEQ) (see the discussion in Section 4.1) are monitored as an indication of the likelihood of potential rupture conditions.

Figure 37. Nonlinear shear-force shear-displacement curve for hot driven rivets in single shear.

0

5

10

15

20

25

30

35

0.00 0.05 0.10 0.15 0.20 0.25

Load

(ki

ps)

Displacement (in)

59

4. TEST SIMULATION RESULTS

This chapter presents detailed descriptions of all the test configurations studied in this research and provides the corresponding test simulation results. The loading scenarios and the general gusset plate joint design procedures are summarized in Chapter 2. The reader is referred to Chapter 3 for a description of the modeling procedures, the non-linear material properties used for the plates and the fasteners, and the geometric imperfections used in the test simulations.

4.1 DEFINITION OF FAILURE AND DETERMINATION OF FAILURE MODES

In the test simulations discussed in this chapter, the maximum load capacity is defined as the load level at which a test first reaches either:

(A) 4% equivalent plastic strain (PEEQ) over a length equal to the plate thickness at the mid-surface of any of the plates (gusset, shingle, and splice), or

(B) The peak load on the load-displacement curve.

unless noted otherwise. The 4% PEEQ criterion is a reasonable but somewhat arbitrary limit beyond which potential rupture of the physical material may become suspect. This is also a limit commonly used in the published literature at which the strains are considered as “large.” This type of criterion is necessary since the resolution of the FEA models is not sufficient to capture the physical localization in strain associated with rupture. In general, the 4% criterion governed in situations where extensive plasticity occurred, such as in specimens with a shear-dominant failure mode. The peak load criterion is a natural choice for buckling-dominated behavior, since this is a clear and unambiguous definition of the maximum load capacity. That is, the maximum load is reached, followed by a post-peak unloading response of the specimen. In cases where a specimen unloaded but then subsequently started to support increasing loads, the first peak is chosen as the failure load. In some cases, post-buckling strength causes some subsequent increases in the load capacity; however, this tends to occur at relatively large deformations.

For developing design recommendations, it is convenient to classify the performance of each specimen under a single “observed” or “governing” failure mode. In this research, three basic failure mechanisms were observed: shear yielding and/or buckling along a critical section of the plate, out-of-plane buckling of the plate along with out-of-plane movement of the compression diagonal, or extensive plasticity within a short length across a chord splice. To ascertain the observed failure mode, two different approaches were pursued within the project. One approach, described by Ocel (2013), involved plotting the displacement paths of various points on the gusset plates from the position at the start of the test up through the maximum load capacity of the test. These displacement vectors generally were magnified, to facilitate inspection, and plotted on an elevation view of the gusset plate. Based on these plots, failure modes involving

60

gusset plate buckling could be identified generally as cases where the displacement vectors in the vicinity of the end of the compression diagonal were clearly larger than the other displacement vectors and were directed approximately parallel to the length of the diagonal and toward the work point of the joint. Conversely, failure modes involving shearing actions generally were dominated by corresponding displacement vectors throughout the gusset plate that were parallel to the direction of the shearing. Nevertheless, these “shearing” displacements often had a significant component involving diagonal compression in the vicinity of a compression diagonal, or diagonal tension in the vicinity of a tension diagonal.

A second approach that the Georgia Tech research team found to give similar results to the above, but avoided ambiguity in certain cases and could be applied in an automated fashion more easily is as follows. Basically, in cases where a peak load was reached in the test simulation prior to achieving the above 4% PEEQ definition of the load capacity (4% PEEQ at the mid-surface of the plates), the failure mode was identified generally as a “Buckling” failure mode. If the load capacity was defined by the 4% PEEQ criterion, then the failure was interpreted as an in-plane mode of failure. Given these broad classifications, the gusset plate strains were evaluated to determine the location and type of the largest deformations. This allowed for a clear assessment of the nature of the out-of-plane buckling failure, or the type of in-plane failure.

In a few situations, the peak load limit occurred essentially at the same time as reaching the 4% PEEQ criterion at the gusset plate mid-surface. In these cases, a “combined” failure mode is indicated.

All of the above discussions pertain to the definition of “observed” failure modes. However, it is important to understand that in many situations, the distinction between one type of failure or another is subtle and subject to interpretation. However, generally once a given set of resistance equations is identified for characterizing the various applicable limit states, the definition of the “governing" limit state from these resistance equations is unambiguous (although in many situations, the resistances determined from the equations for different limit states may be nearly the same; therefore, small changes in the test characteristics or in the applicable resistance equations can result in a different categorization of the failure mode).

The following developments refer to two “Methods,” or basically the strength evaluations using two sets of resistance equations developed as part of this research. In the subsequent summary assessments discussed in Chapter 5, the various test cases are grouped by the governing failure modes identified by the calculations from each of these methods. In the view of the authors, this provides the most correct assessment of the professional factor, i.e., the ratio of the test strength to the predicted strength, and the dispersion associated with the predictions, represented by the coefficient of variation (COV) on the professional factor. If the different test cases are grouped for assessment based on “observed” failure modes, the professional factors and their COV values do not actually correspond with the way the selected limit state equations for a given method are being applied. That is, for example, it is asserted that once the limit state equations for say a

61

diagonal buckling limit state and a shear yielding limit state have been defined (supposing for purposes of illustration that these are the only limit state checks), then all the tests that are governed by the diagonal buckling equations should be grouped together to evaluate the diagonal buckling equations and all the tests that are governed by the shear yielding equations should be grouped together to evaluate the shear yielding checks. If the tests are grouped by “observed” failure modes, then used to evaluate the selected limit states design equations, one does not obtain a correct evaluation of how the different limit states equations are actually working to assess the overall strengths.

4.2 OVERVIEW

For each study case in this research, the results from the FEA simulations were synthesized into the following output:

The test configuration with applied loads and boundary conditions,

The gusset plate geometry and the reference design loads transferred from the truss members,

The required plate thicknesses estimated from the adaptations of the FHWA Guide procedures discussed in Section 2.3.3,

Load-deformation curves,

von Mises stress contours at key load conditions, and

Equivalent plastic strain (PEEQ) contours at key load conditions.

Although repetitive in nature, this approach facilitates direct comparisons of the various cases. For some of the test simulations, additional data is provided to further clarify the behavior. Readers should pay particular attention to the fact that the specific load-deflection responses cannot be compared directly between different cases. This is because the locations where the deformations are taken generally differ from one case to another.

To evaluate the shear strength behavior, the gusset plates were sectioned along two planes in various cases: (a) a horizontal section taken along the edge of the top chords, and (b) a two-part vertical section extending through the chord splice and along the side of the vertical member.

Tables 5 through 14 summarize all of the test cases considered in this research along with key results pertaining to the test capacities. The name of each of the configurations tested is shown in the first column. A range of gusset plate thicknesses is considered for most of the tests. These thicknesses are shown in the second column of the table. The third column of the table gives the failure mode observed for each test case. The criterion for identifying a buckling failure versus a yielding failure is described in the above section. The fourth column gives the design reference load for each test, Rref, which is generally taken as the applied load most relevant to the failure mode. Free-body diagrams of each truss joint are provided in the subsequent sections. The reference load can be readily identified on these diagrams. Next, the fifth column gives the

62

applied load fraction (ALF) at the defined test capacity. The value of the key applied load at the failure condition in the simulation models can always be determined by multiplying the ALF with the load Rref. That is, the capacity can be determined as RFEA = ALF Rref.

The last four columns in Tables 5 through 14 summarize the computed failure mode and the ratio of the test simulation capacity to the governing calculated nominal resistance, RFEA/Rn, using the potential improved procedures discussed subsequently in Chapter 5. RFEA/Rn is referred to commonly as the professional factor. The results for two nominal strength prediction models are shown in the tables:

Method 1: This streamlined model was recommended for integration into the AASHTO Specification provisions in the early manuscript of the NCHRP 12-84 final report as of December 2011 (FHWA, 2011), and

Method 2: This is a slightly more elaborate model aimed at providing some improvement in the predictions of Method 1.

The final recommendations for Method 1 as of January 2013 are the same as those evaluated in this report with the exception of several minor modifications, some of which are based on the findings pertaining to Method 2 in this report.

It is important to understand the naming convention for the various tests shown in column 1 of Tables 5 through 14. The naming convention for the variations on the experimental test geometries is described in Section 2.2. The naming convention rules for the parametric study tests are as follows:

The name starts with the designation “P1,” “P2,” etc. corresponding to the number of the parametric study configuration shown in Figures 15 through 18.

The second part of the name is either the symbol “C” for a configuration with chamfered diagonals or “U” for a configuration with unchamfered diagonals.

This if followed by one of the following designations:

o “CCS” for compression chord splice,

o “MTB” for a chord splice in which the members are modeled as mill-to-bear,

o “TCS” for tension chord splice,

o “HS(x.xx)” for high-strength (Grade 100) steel with “x.xx” representing the gusset plate thickness,

63

o “SP(x.xx:x.xx)” for joints with a shingle plate, with “x.xx:x.xx” representing the ratio of the thicknesses between the gusset plate and the shingle plates,

o “SES” or “EES” for joints reinforced with a short edge stiffener or an extended edge stiffener,

o “C1” or “C2” for joints with corrosion patterns “1” or “2” on both gusset plates, or “COS” for gusset plates with corrosion only on one of the two gusset plates (i.e., “COS” stands for “Corrosion on One Side of the joint”),

o “C1-SP(x.xx:x.xx)” or “COS-SP(x.xx:x.xx)” for joints with the above corrosion patterns and reinforced by shingle plates, or

o No additional designation (in some cases, the name is simply continued with the next designation below).

The third or fourth designation is then one of the following:

o “WV” for Warren with vertical truss configurations,

o “P” for Pratt configurations,

o “W” for Warren configurations without verticals,

o “CJ” for the 90o corner joints,

o “POS” for the corner joints with a greater than 90o angle between the main members at the joint, i.e., a “positive” angle between the top chord of the truss and the end diagonal,

o “NEG” for joints at the start of a haunch in the bottom chord of the truss, referred to a “negative” angle between the bottom chord members.

The final designation, indicating the location of the joint and given as:

o “M” for midspan,

o “P” for pier,

o “NP” for near pier,

o “INF” for inflection point, or

o No additional designation is used here in the case of the “CJ,” “POS” and “NEG” configurations.

64

Table 5 Test simulations for Warren with vertical experimental configurations, unchamfered members

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2

E1-U-307SS-WV

0.2500 DB 716 0.53 PSPY 1.06 DB 1.06

0.3125 DB 716 0.74 PSPY 1.18 DB 1.10

0.3750 DB 716 0.94 PSPY 1.25 DB 1.11

0.4375 DB 716 1.14 PSPY 1.30 DB 1.13

0.5000 DB 716 1.36 PSPY 1.36 DB 1.16

0.6250 FSPY 716 1.91 PSPY 1.51 DB 1.27

E2-U-307LS-WV

0.2500 DB 796 0.50 DB 0.86 DB 1.04

0.3125 DB 796 0.82 DB 0.99 DB 1.12

0.3750 DB 796 1.10 PSPY 1.07 DB 1.13

0.4375 DB 796 1.26 PSPY 1.14 DB 1.13

0.5000 DB 796 1.64 PSPY 1.19 DB 1.14

0.6250 DB 796 2.14 PSPY 1.25 DB 1.13

E3-U-307SL-WV

0.2500 DB 946 0.60 PSPY 1.07 DB-TWS 1.10

0.3125 DB 946 0.78 PSPY 1.12 FSPY 1.02

0.3750 FSPY 946 1.03 PSPY 1.23 FSPY 1.13

0.4375 FSPY 946 1.21 PSPY 1.24 FSPY 1.13

0.5000 FSPY 946 1.40 PSPY 1.25 FSPY 1.15

0.6250 FSPY 946 1.76 PSPY 1.26 FSPY 1.15

E4-U-490SS-WV

0.2500 DB 728 0.61 PSPY 1.05 DB 1.17

0.3125 DB 728 0.82 PSPY 1.13 DB 1.14

0.3750 DB-FSPY 728 1.05 PSPY 1.21 DB 1.15

0.4375 FSPY 728 1.28 PSPY 1.26 DB 1.17

0.5000 FSPY 728 1.47 PSPY 1.27 DB 1.15

0.6250 FSPY 728 1.84 PSPY 1.27 FSPY 1.14

E5-U-490LS-WV

0.2500 DB 478 0.62 DB 0.96 DB 1.01

0.3125 DB 478 0.96 DB 1.01 DB 1.04

0.3750 DB 478 1.31 DB 1.05 DB 1.07

0.4375 DB 478 1.68 DB 1.09 DB 1.10

0.5000 DB 478 2.04 DB 1.12 DB 1.13

0.6250 DB 478 2.67 PSPY 1.13 DB 1.13

DB = Diagonal Buckling PSPY = Partial Shear Plane Yielding DB-FSPY = 4 % PEEQ reached at the limit load associated with DB DB-TWS = Diagonal Buckling - Truncated Whitmore Section FSPY = Full Shear Plane Yielding

65

Table 6 Test simulations for Warren without vertical variations on experimental configurations, unchamfered members

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2

E1-U-307SS-W

0.2500 DB 716 0.45 DB 1.13 DB 1.02

0.3125 DB 716 0.65 DB 1.11 DB 1.04

0.3750 DB 716 0.83 DB 1.09 DB 1.04

0.4375 DB 716 1.01 DB 1.07 DB 1.04

0.5000 DB 716 1.23 PSPY 1.11 DB 1.08

0.6250 FSPY 716 1.72 PSPY 1.25 DB 1.17

E2-U-307LS-W

0.2500 DB 796 0.41 DB 1.50 DB 1.17

0.3125 DB 796 0.68 DB 1.34 DB 1.14

0.3750 DB 796 0.95 DB 1.25 DB 1.12

0.4375 DB 796 1.21 DB 1.20 DB 1.11

0.5000 DB 796 1.44 DB 1.15 DB 1.08

0.6250 DB 796 1.89 DB 1.09 DB 1.05

E3-U-307SL-W

0.2500 DB 946 0.53 DB 1.27 DB-TWS 1.16

0.3125 DB 946 0.77 DB 1.16 DB-TWS 1.11

0.3750 DB 946 1.00 DB 1.10 FSPY 1.09

0.4375 DB 946 1.21 PSPY 1.09 FSPY 1.13

0.5000 FSPY 946 1.38 PSPY 1.08 FSPY 1.13

0.6250 FSPY 946 1.73 PSPY 1.09 FSPY 1.13

E4-U-490SS-W

0.2500 DB 728 0.55 DB 1.41 DB 1.24

0.3125 DB 728 0.78 DB 1.31 DB 1.20

0.3750 DB 728 0.98 DB 1.22 DB 1.16

0.4375 DB 728 1.20 DB 1.20 DB 1.15

0.5000 FSPY 728 1.44 DB 1.21 DB 1.17

0.6250 FSPY 728 1.83 DB 1.17 DB 1.14

E5-U-490LS-W

0.2500 DB 478 0.53 DB 1.23 DB 1.16

0.3125 DB 478 0.85 DB 1.16 DB 1.11

0.3750 DB 478 1.23 DB 1.18 DB 1.15

0.4375 DB 478 1.59 DB 1.17 DB 1.15

0.5000 DB 478 1.96 DB 1.19 DB 1.17

0.6250 DB 478 2.61 DB 1.17 DB 1.16

DB = Diagonal Buckling PSPY = Partial Shear Plane Yielding DB-TWS = Diagonal Buckling – Truncated Whitmore Section FSPY = Full Shear Plane Yielding

66

Table 7 Test simulations for Warren with vertical parametric configurations

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2

P1-C-CCS-WV-M

0.2500 CSEC 3200 0.97 CSEC 1.05 CSEC 0.98

0.3125 CSEC 3200 1.07 CSEC 1.32 CSEC 1.21

0.3750 CSEC 3200 1.37 CSEC 1.32 CSEC 1.20

0.4000 CSEC 3200 1.37 CSEC 1.38 CSEC 1.24

0.4375 CSEC 3200 1.47 CSEC 1.40 CSEC 1.25

0.5000 CSEC 3200 1.56 CSEC 1.41 CSEC 1.24

0.6250 CSEC 3200 1.67 CSEC 1.49 CSEC 1.29

P1-C-MTB-WV-M

0.3500 DB 424 1.96 PSPY 1.07 DB-TWS 1.06

P2-C-TCS-WV-M

0.2500 CSET 3200 1.13 CSET 1.14 CSET 1.06

0.3125 CSET 3200 1.23 CSET 1.22 CSET 1.12

0.3750 CSET 3200 1.33 CSET 1.26 CSET 1.14

0.4000 CSET 3200 1.33 CSET 1.30 CSET 1.17

0.4375 CSET 3200 1.34 CSET 1.32 CSET 1.18

0.5000 CSET 3200 1.43 CSET 1.37 CSET 1.21

0.6250 CSET 3200 1.61 CSET 1.45 CSET 1.26

P3-C-WV-P

0.2500 CBS 2800 0.45 PSPY* 1.06 DT-TWS 0.92

0.3125 CBS 2800 0.58 PSPY* 1.09 DT-TWS 0.95

0.3750 CBS 2800 0.71 PSPY* 1.11 DT-TWS 0.97

0.4375 CBS 2800 0.85 PSPY* 1.14 DT-TWS 0.99

0.5000 CBS 2800 0.98 PSPY* 1.15 DT-TWS 1.00

0.6250 CBS 2800 1.27 PSPY* 1.19 DT-TWS 1.04

P4-C-WV-P 0.8000 CSEC 4001 0.97 CSEC 1.64 CSEC 1.02

CBS = Compression Block Shear CSEC = Chord Splice Eccentric Compression CSET = Chord Splice Eccentric Tension DB-TWS = Diagonal Buckling – Truncated Whitmore Section DT-TWS = Diagonal Tension – Truncated Whitmore Section PSPY = Partial Shear Plane Yielding PSPY* = Partial Shear Plane Yielding (Yielding along both edges of the vertical member)

67

Table 7. Test simulations for Warren with vertical parametric configurations (continued)

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2

P5-C-WV-NP

0.2500 DB 2190 0.52 PSPY 1.13 DB-TWS 0.94

0.3125 DB 2190 0.70 PSPY 1.21 DB-TWS 0.98

0.3750 DB 2190 0.87 PSPY 1.26 DB-TWS 0.99

0.4000 DB 2190 0.94 PSPY 1.27 DB-TWS 1.00

0.4375 DB 2190 1.06 PSPY 1.31 DB-TWS 1.02

0.5000 FSPY 2190 1.26 PSPY 1.36 DB-TWS 1.03

0.6250 FSPY 2190 1.65 PSPY 1.43 DB-TWS 1.07

P5-U-WV-NP

0.2500 DB 1500 0.35 DB 0.89 DB-TWS 1.20

0.3125 DB 1500 0.52 DB 0.84 DB-TWS 1.08

0.3750 DB 1500 0.70 PSPY 0.88 DB-TWS 1.02

0.4000 DB 1500 0.78 PSPY 0.92 DB-TWS 1.02

0.4375 DB 1500 0.90 PSPY 0.97 DB-TWS 1.02

0.5000 DB 1500 1.09 PSPY 1.03 DB-TWS 1.01

0.6250 DB 1500 1.43 PSPY 1.08 DB-TWS 0.98

P5-C-HS(0.4)-WV-NP

0.4000 DB 1500 1.69 PSPY 1.12 DB-TWS 0.92

P5-C-HS(0.2)-WV-NP

0.2000 DB 1500 0.64 PSPY 0.85 DB-TWS 0.82

P6-C-WV-NP

0.2500 DB 2260 0.36 PSPY 0.98 DB-TWS 0.94

0.3125 DB 2260 0.51 PSPY 1.11 DB-TWS 0.95

0.3750 DB 2260 0.67 PSPY 1.21 DB-TWS 0.98

0.4375 DB 2260 0.82 PSPY 1.27 FSPY 1.01

0.5000 FSPY 2260 0.98 PSPY 1.33 FSPY 1.05

0.6250 FSPY 2260 1.29 PSPY 1.40 FSPY 1.11

P6-U-WV-NP

0.2500 DB 2260 0.23 PSPY 0.58 DB-TWS 0.96

0.3125 DB 2260 0.35 PSPY 0.71 DB-TWS 0.89

0.3750 DB 2260 0.49 PSPY 0.83 DB-TWS 0.89

0.4375 DB 2260 0.64 PSPY 0.93 DB-TWS 0.91

0.5000 DB 2260 0.80 PSPY 1.01 DB-TWS 0.94

0.6000 DB 2260 1.06 PSPY 1.12 DB-TWS 0.97

0.6250 DB 2260 1.12 PSPY 1.13 DB-TWS 0.98

DB = Diagonal Buckling DB-TWS = Diagonal Buckling – Truncated Whitmore Section FSPY = Full Shear Plane Yielding

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Table 7 Test simulations for Warren with vertical parametric configurations (continued)

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2

P7-C-WV-INF

0.2500 DB 3580 0.34 PSPY 1.00 DB-TWS 1.09

0.3125 DB 3580 0.46 PSPY 1.08 DB-TWS 1.13

0.3750 DB 3580 0.59 PSPY 1.15 DB-TWS 1.18

0.4375 FSPY 3580 0.73 PSPY 1.22 DB-TWS 1.23

0.5000 FSPY 3580 0.88 PSPY 1.29 DB-TWS 1.27

0.6250 FSPY 3580 1.12 PSPY 1.31 DB-TWS 1.24

0.7000 FSPY 3580 1.28 PSPY 1.34 FSPY 1.26

P7-C-HS(0.7)-WV-INF

0.7000 FSPY 3580 2.28 PSPY 1.17 DB-TWS 1.15

P7-C-HS(0.35)-WV-INF

0.3500 DB 3580 0.99 PSPY 1.02 DB-TWS 1.11

P8-C-WV-INF

0.2500 DB 2100 0.42 PSPY 1.01 DB-TWS 1.08

0.3125 FSPY 2100 0.58 PSPY 1.12 FSPY 1.16

0.3750 FSPY 2100 0.72 PSPY 1.15 FSPY 1.20

0.4375 FSPY 2100 0.84 PSPY 1.15 FSPY 1.20

0.5000 FSPY 2100 0.97 PSPY 1.17 FSPY 1.21

0.6250 FSPY 2100 1.20 PSPY 1.15 FSPY 1.20

P8-U-WV-INF 0.5000 DB-FSPY 2100 0.94 PSPY 0.96 DB-TWS 1.17

P8-C-HS(0.5)-WV-INF

0.5000 FSPY 2100 1.79 PSPY 1.06 FSPY 1.10

P8-C-HS(0.25)-WV-INF

0.2500 DB 2100 0.77 PSPY 0.91 DB-TWS 1.21

DB = Diagonal Buckling PSPY = Partial Shear Plane Yielding DB-TWS = Diagonal Buckling – Truncated Whitmore Section FSPY = Full Shear Plane Yielding

69

Table 8 Test simulations for Pratt parametric configurations

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2 P9-C-P-NP 0.2500 DB 368 0.96 PSPY 0.81 DB-TWS 1.11

P10-C-P-NP

0.2000 CSEC 620 1.73 CSEC 1.00 CSEC 0.90

0.2500 CSEC 620 2.06 CSEC 1.13 CSEC 0.98

0.3125 CSEC 620 2.43 CSEC 1.23 CSEC 1.04

0.3750 CSEC 620 2.77 CSEC 1.31 CSEC 1.09

0.4375 CSEC 620 3.01 CSEC 1.38 CSEC 1.12

0.5000 CSEC 620 3.31 CSEC 1.44 CSEC 1.15

CSEC = Chord Splice Eccentric Compression DB = Diagonal Buckling DB-TWS = Diagonal Buckling – Truncated Whitmore Section PSPY = Partial Shear Plane Yielding

70

Table 9 Test simulations for Warren without vertical parametric configurations

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2

P11-C-W-M

0.2500 CSEC 3290 0.80 CSEC 0.89 CSEC 0.82 0.3125 CSEC 3290 1.17 CSEC 1.23 CSEC 1.12 0.3750 CSEC 3290 1.30 CSEC 1.32 CSEC 1.18 0.4500 CSEC 3290 1.41 CSEC 1.37 CSEC 1.20 0.5000 CSEC 3290 1.50 CSEC 1.42 CSEC 1.23

P12-C-W-P 1.0000 CC 5000 1.05 CC 1.03 CC 1.03

P13-C-W-NP

0.2500 DB 1650 0.52 PSPY 1.09 DB-TWS 1.13 0.3125 DB 1650 0.71 PSPY 1.20 DB-TWS 1.15 0.3750 DB 1650 0.91 PSPY 1.28 DB-TWS 1.18 0.4000 DB-CSEC 1650 0.99 PSPY 1.30 DB-TWS 1.19 0.4375 CSEC 1650 1.11 PSPY 1.34 DB-TWS 1.21 0.5000 CSEC 1650 1.28 PSPY 1.35 DB-TWS 1.20 0.6250 CSEC 1650 1.60 PSPY 1.35 DB-TWS 1.17

P13-U-W-NP

0.2500 DB 1650 0.39 PSPY 0.80 DB-TWS 1.29 0.3125 DB 1650 0.58 PSPY 0.95 DB-TWS 1.19 0.3750 DB 1650 0.77 PSPY 1.05 DB-TWS 1.13 0.4000 DB 1650 0.85 PSPY 1.08 DB-TWS 1.13 0.4375 DB 1650 0.96 PSPY 1.12 DB-TWS 1.11 0.5000 DB 1650 1.16 PSPY 1.18 DB-TWS 1.11 0.6250 DB 1650 1.53 PSPY 1.25 DB-TWS 1.09

P14-C-W-INF

0.2500 DB 1400 0.46 PSPY 0.81 DB-TWS 0.98 0.3125 DB 1400 0.65 PSPY 0.91 DB-TWS 0.94 0.3750 DB 1400 0.85 PSPY 0.99 FSPY 0.96 0.4375 DB 1400 1.03 PSPY 1.03 FSPY 0.99 0.5000 DB-FSPY 1400 1.22 PSPY 1.07 FSPY 1.03 0.6250 FSPY 1400 1.60 PSPY 1.12 FSPY 1.08

P14-U-W-INF

0.2500 DB 1400 0.39 PSPY 0.68 DB-TWS 0.97 0.3125 DB 1400 0.58 PSPY 0.81 DB-TWS 0.91 0.3750 DB 1400 0.79 PSPY 0.92 DB-TWS 0.91 0.4375 DB 1400 0.99 PSPY 0.99 FSPY 0.95 0.5000 DB 1400 1.18 PSPY 1.03 FSPY 1.00 0.6250 DB-FSPY 1400 1.54 PSPY 1.08 FSPY 1.04

CC = Column Compression CSEC = Chord Splice Eccentric Compression DB = Diagonal Buckling DB-FSPY = 4 % PEEQ reached at the limit load associated with DB DB-TWS = Diagonal Buckling – Truncated Whitmore Section FSPY = Full Shear Plane Yielding, PSPY = Partial Shear Plane Yielding

71

Table 10 Test simulations for other base parametric configurations

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2 P15-C-CJ 0.50 FSPY 2000 1.15 FSPY 1.08 FSPY 1.19

P16-C-CJ 0.85 CPM-

BMTM 2830 1.22 FSPY 0.78a FSPY 0.87a

P17-C-POS 0.60 CPM 2139 1.47 DT 0.84 DT-TWS 1.10

P18-C-POS 0.60 CPM 2499 1.30 DT 0.77 DT-TWS 1.14

P19-C-CCS-NEG

0.60 DB-CSEC 6068 0.89 CSEC 1.09 DB-TWS 1.01

P19-C-MTB-NEG

0.60 DB 3700 1.01 PSPY 1.16 DB-TWS 1.15

P20-C-CCS-NEG

0.60 DB-CSEC 2230 1.03 CSEC 1.12 CSEC 0.96

P20-C-MTB-NEG

0.60 DB-

BMTM 2330 0.22 DB 1.16 DB-TWS 1.09

a Premature failure due to buckling of the main truss members BMTM = Buckling of Main Truss Members CSEC = Chord Splice Eccentric Compression DB = Diagonal Buckling DB-TWS = Diagonal Buckling – Truncated Whitmore Section DT = Diagonal Tension DT-TWS = Diagonal Tension – Truncated Whitmore Section FSPY = Full Shear Plane Yielding PSPY = Partial Shear Plane Yielding CPM = Coupled Plastic Mechanism

72

Table 11 Test simulations for parametric configurations with shingle plates

Configuration Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2 P3-C-SP(0.4:0.2)-WV-P CBS 2800 0.90 PSPY 1.07 DT-TWS 0.94

P3-C-SP(0.5:0.25)-WV-P CBS 2800 1.17 PSPY 1.11 DT-TWS 0.97

P3-C-SP(0.3:0.3)-WV-P CBS 2800 0.79 PSPY 1.18 DT-TWS 1.03

P5-C-SP(0.3:0.2)-WV-NP DB 1500 0.95 PSPY 1.17 DB-TWS 0.98

P5-C-SP(0.4:0.25)-WV-NP DB 1500 1.34 PSPY 1.26 DB-TWS 1.01

P5-C-SP(0.3:0.3)-WV-NP DB 1500 1.14 PSPY 1.21 DB-TWS 0.99

P12-C-SP(0.5:0.5)-W-P CC 5000 0.71 CC 0.75 CC 0.75

CC = Column Compression CBS = Compression Block Shear DB = Diagonal Buckling DT-TWS = Diagonal Tension – Truncated Whitmore Section DB-TWS = Diagonal Buckling – Truncated Whitmore Section PSPY = Partial Shear Plane Yielding

73

Table 12 Test simulations for parametric configurations with edge stiffeners

Configuration tg (in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2 E4-U-490SS-

SES-WV 0.375 DB 1000 0.72 PSPY 1.14 DB 1.09

E4-U-490SS-EES-WV

0.375 FSPY 1000 0.84 PSPY 1.33 FSPY 1.19

E5-U-490LS-SES-WV

0.375 DB 1000 0.60 DB 1.00 DB 1.02

E5-U-490LS-EES-WV

0.375 FSPY 1000 0.81 PSPY 1.19 DB 1.10

P5-U-EES-WV-NP

0.400 DB 1500 0.97 PSPY 1.15 DB-TWS 0.89

P14-C-EES-W-INF

0.500 DB-FSPY 1400 1.32 PSPY 1.16 FSPY 1.11

DB = Diagonal Buckling DB-FSPY = 4 % PEEQ reached at the limit load associated with DB DY = Diagonal Yielding DY-TWS = Diagonal Yielding – Truncated Whitmore Section FSPY = Full Shear Plane Yielding PSPY = Partial Shear Plane Yielding

74

Table 13 Test simulations for configurations with corroded gusset plates

Configuration tg

(in.)

Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2 E1-U-307SS-

C1-WV 0.375 DB 1000 0.43 DB 1.54 DB 1.90

E1-U-307SS-C2-WV

0.375 FSPY 1000 0.73 DT 2.19 DT 2.19

P8-C-C1-WV-INF

0.500 FSPY 3249 0.61 PSPY 1.67 FSPY 1.35

P8-C-C2-WV-INF

0.500 FSPY 2100 0.76 DB 1.53 FSPY 1.25

P8-C-COS-WV-INF

0.500 FSPY 3249 0.83 PSPY 1.39 FSPY 1.32

P14-U-C1-W-INF

0.500 DB 1400 0.94 PSPY 1.58 DB-TWS 1.42

P14-U-C2-W-INF

0.500 DB 1400 0.95 PSPY 1.60 DB-TWS 1.28

P14-U-COS-W-INF

0.500 DB 1400 1.07 PSPY 1.23 FSPY 1.06

DB = Diagonal Buckling DB-TWS = Diagonal Buckling – Truncated Whitmore Section FSPY = Full Shear Plane Yielding PSPY = Partial Shear Plane Yielding

75

Table 14 Test simulations for parametric configurations with corroded gusset plates reinforced by shingle plates

Configuration Observed Failure Mode

Rref (kips)

ALF at Failure

Method 1 Failure Mode

RFEA/Rn Method

1

Method 2 Failure Mode

RFEA/Rn Method

2 P8-C-C1-SP(0.5:0.25)-

WV-INF FSPY 3249 0.93 PSPY 1.49 FSPY 1.28

P8-C-COS-SP(0.5:0.25)-WV-INF

FSPY 2100 0.95 PSPY 1.31 FSPY 1.24

P14-U-C1-SP(0.5:0.25)-W-INF

DB 1400 1.22 PSPY 1.19 DB-TWS 1.16

P14-U-COS-SP(0.5:0.25)-W-INF

DB 1400 1.20 PSPY 1.11 DB-TWS 1.02

DB = Diagonal Buckling DB-TWS = Diagonal Buckling – Truncated Whitmore Section FSPY = Full Shear Plane Yielding FSPY-DB = FSPY of corroded gusset plate, DB of uncorroded gusset plate and shingle plate PSPY = Partial Shear Plane Yielding Each of the above designations is delimited by a dash (i.e., “-”). Hence, the name “P1-C-CCS-WV-M” represents parametric test configuration 1, with chamfered members, focused on the compression chord splice response in a Warren configuration with verticals, and with the joint located at the midspan of the prototype truss. Various failure modes are activated by the different test simulations. In summary, these modes are:

“DB” = buckling of the gusset plate associated with diagonal compression, including generally an out-of-plane movement of the compression diagonal,

“DT” = diagonal tension,

“FSPY” = full shear plane yielding,

“CSEC” = chord splice eccentric compression,

“CSET” = chord splice eccentric tension,

“CBS” = compression block shear,

“PSPY” = partial shear plane yielding,

“DB-TWS” = diagonal buckling with a truncated Whitmore section,

“DT-TWS” = diagonal tension with a truncated Whitmore section,

76

“CC” = column compression,

“DB-CSEC” = combined diagonal buckling and chord-splice eccentric compression,

“DB-FSPY” = combined diagonal buckling and full shear plane yielding,

“BMTM” = buckling of main truss members,

“CPM”= coupled plastic mechanism,

“DY” = general yielding of the gusset in the diagonal tension direction,

“DY-TWS” = general diagonal yielding with a truncated Whitmore section, and

“FSPY-DB” = full shear plane yielding of a corroded gusset plate combined with diagonal buckling of an uncorroded gusset plate and shingle plate.

Some of the specific implications of these names are clarified in the subsequent discussions of the test simulation results and the prediction of the test simulation strengths.

As noted in Section 2.3.3, the study joints were designed using specific adaptations of the FWHA Guide (FHWA, 2009a). Key controlling thicknesses arrived at using these procedures are presented in the following sections. Chapter 5 addresses the potential improved gusset plate strength calculations.

4.3 WARREN WITH VERTICAL EXPERIMENTAL TEST CONFIGURATIONS, UNCHAMFERED MEMBERS

This section summarizes key characteristics and sample results for the Warren with vertical experimental test configurations using unchamfered members, as well as variations on these physical tests. The corresponding results are summarized in Table 5 of Section 4.2.

4.3.1 E1-U-307SS-WV

Specimen E1-U-307SS-WV is a joint with unchamfered members, A307 bolts having a response representative of hot-driven rivets, a short standoff distance for the compression diagonal, and a short connection length to the compression diagonal. Figure 38(a) shows key dimensions of the gusset plate and its members. It should be noted that a minimum fastener spacing of 2.5 inches and an edge distance of 1.5 inches is used in this test as well as in the other experimental tests. Figure 38(b) shows the reference member forces for this joint.

Figure 39 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement curves is nearly elastic-perfectly plastic. Figure 40 shows contours of the von Mises stresses at the limit load of the subassembly for the

77

case of a 3/8 inch thick gusset plate. These contours, and all the subsequent contours, correspond to the mid-thickness of the plates unless noted otherwise. The lighter grey contours correspond approximately to the regions where the yield strength of the steel is breached. Figure 41 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. The darkest areas are entirely linear elastic. Based on the more lightly shaded area in the gusset at the end of the compression diagonal and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

(a) Gusset plate geometry

(b) Design forces

Figure 38. Gusset plate geometry and design forces for E1-U-307SS-WV.

78

Figure 39. In-plane vs. out-of-plane displacements, E1-U-307SS(3/8)-WV.

Figure 40. von Mises stress response contours at the limit load (ALF=0.94), E1-U-

307SS(3/8)-WV

00.10.20.30.40.50.60.70.80.9

1

0.00 0.05 0.10 0.15 0.20 0.25

App

lied

Loa

d Fr

actio

n

Displacements at Point A on GP307-SS3 Specimen (in.)

Out-of-Plane Displacement Horizontal Displacement

Displacements at Point A (in.)

79

Figure 41. Equivalent plastic strain response contours at the limit load (ALF=0.94), E1-U-

307SS(3/8)-WV

4.3.2 E2-U-307LS-WV

Specimen E2-U-307SS-WV is a joint with unchamfered members, A307 bolts having a response representative of hot-driven rivets, a long standoff distance (4.5 inches) for the compression diagonal, and a short connection length to the compression diagonal. Figure 42(a) shows key dimensions of gusset plate and its members. Figure 42(b) shows the reference member forces for this joint.

Figure 43 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that out-of-plane displacement of point A is much larger than the horizontal in-plane displacement as the joint fails. Figure 44 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 45 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the compression diagonal, one can observe that the gusset is yielding because of the buckling of the gusset including the out-of-plane movement of the diagonal.

80

(a) Gusset plate geometry

(b) Design forces

Figure 42. Gusset plate geometry and design forces for E2-U-307LS-WV.

81

Figure 43. In-plane vs. out-of-plane displacements, E2-U-307LS(3/8)-WV.

Figure 44. von Mises stress response contours at the limit load (ALF=1.10), E2-U-

307LS(3/8)-WV.

0

0.2

0.4

0.6

0.8

1

1.2

-0.05 0.00 0.05 0.10 0.15 0.20 0.25

App

lied

Loa

d Fr

actio

n

Displacements at Point A on GP307-LS3 Specimen (in.)

Out-of-Plane Displacement Horizontal Displacement

Displacements at Point A (in.)

82

Figure 45. Equivalent plastic strain response contours at the limit load (ALF=1.10, E2-U-

307LS(3/8)-WV.

4.3.3 E3-U-307SL-WV

Specimen E3-U-307SL-WV is a joint with unchamfered members, A307 bolts having a response representative of hot-driven rivets, a short standoff distance (1.0 inch) for the compression diagonal, and a long connection length to the compression diagonal. Figure 46(a) shows key dimensions of gusset plate and its members. Figure 46(b) shows the reference member forces for this joint.

Figure 47 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that horizontal in-plane displacement of point A continuously increases while the out-of-plane displacement is essentially zero until the joint reaches its limit load. Figure 48 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 49 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the both diagonals and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. Figure 50 shows the equivalent plastic strain contours of the gusset at the PEEQ > 4 % strength limit. One can observe that the yielding in the

83

gusset along the “full shear plane” is already substantial at this limit. Figure 51 shows the shear stress contours of the gusset at the PEEQ > 4 % strength limit. For a static yield of 46.6 ksi, the shear yield limit is approximately 27 ksi. The shear stress contours in Figure 51 indicates that most of the full shear plane of the gusset reaches its shear yield limit at the PEEQ > 4 % strength limit.

(a) Gusset plate geometry

(b) Design forces

Figure 46. Gusset plate geometry and design forces for E3-U-307SL-WV.

706620

12

84

Figure 47. In-plane vs. out-of-plane displacements, E3-U-307SL(3/8)-WV.

Figure 48. von Mises stress response contours at the limit load (ALF=1.06), E3-U-

307SL(3/8)-WV.

0

0.2

0.4

0.6

0.8

1

1.2

-0.50 -0.30 -0.10 0.10 0.30 0.50

App

lied

Loa

d Fr

actio

n

Displacements at Point A on GP307-SL3 Specimen (in.)

Out-of-Plane Displacement Horizontal Displacement

Displacements at Point A (in.)

85

Figure 49. Equivalent plastic strain response contours at the limit load (ALF=1.06), E3-U-

307SL(3/8)-WV.

Figure 50. Equivalent plastic strain response contours at the 4% PEEQ Limit (ALF=1.02),

E3-U-307SL(3/8)-WV

86

Figure 51. Shear stress response contours at the 4% PEEQ Limit (ALF=1.02), E3-U-

307SL(3/8)-WV.

4.3.4 E4-U-490SS-WV

Specimen E4-U-490SS-WV is a joint with unchamfered members, A490 high-strength bolts, a short standoff distance (1.0 inch) for the compression diagonal, and a short connection length to the compression diagonal. This specimen is similar to E1-U-307SS-WV except the type of fasteners. Figure 52(a) shows key dimensions of gusset plate and its members. Figure 52(b) shows the reference member forces for this joint.

Figure 53 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement curves is nearly elastic-perfectly plastic. Figure 54 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 55 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the compression diagonal and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

87

(a) Gusset plate geometry

(b) Design forces

Figure 52. Gusset plate geometry and design forces for E4-U-490SS-WV.

Figure 53. In-plane vs. out-of-plane displacements, E4-U-490SS(3/8)-WV.

0

0.2

0.4

0.6

0.8

1

1.2

0.00 0.10 0.20 0.30 0.40 0.50

App

lied

Loa

d Fr

actio

n

Displacements at Point A on GP490-SS3 Specimen (in.)

Out-of-Plane Displacement Horizontal Displacement

Displacements at Point A (in.)

88

Figure 54. von Mises stress response contours at the limit load (ALF=1.04), E4-U-

490SS(3/8)-WV.

Figure 55. Equivalent plastic strain response contours at the limit load (ALF=1.04), E4-U-

490SS(3/8)-WV

89

4.3.5 E5-U-490LS-WV

Specimen E5-U-490LS-WV is a joint with unchamfered members, A490 high-strength bolts, a long standoff distance (4.5 inch) for the compression diagonal, and a short connection length to the compression diagonal. This specimen is similar to E2-U-307LS-WV except the type of fasteners. Figure 56(a) shows key dimensions of gusset plate and its members. Figure 56(b) shows the reference member forces for this joint. It should be noted that this joint has a 67 % larger force in the tension diagonal compared to the compression diagonal.

Figure 57 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the out-of-plane displacement is significantly larger than the horizontal in-plane displacement. Figure 58 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 59 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the tension diagonal and along the gusset just above the chord from the corner end of the compression diagonal to the end of the gusset on the right, one can observe that a substantial area of the gusset is yielding along the shear plane just above the chord and around the tension diagonal when the limit load is reached. The substantial yielding around the tension diagonal is expected because the tension diagonal transfers a 67 % larger force compared the compression diagonal. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

90

(a) Gusset plate geometry

(b) Design forces

Figure 56. Gusset plate geometry and design forces for E5-U-490LS-WV.

91

Figure 57. In-plane vs. out-of-plane displacements, E5-U-490LS(3/8)-WV

Figure 58. von Mises stress response contours at the limit load (ALF=0.97), E5-U-

490LS(3/8)-WV.

0

0.2

0.4

0.6

0.8

1

1.2

-0.50 -0.30 -0.10 0.10 0.30 0.50

App

lied

Loa

d Fr

actio

n

Displacements at Point A on GP490-LS3 Specimen (in.)

Out-of-Plane Displacement Horizontal Displacement

Displacements at Point A (in.)

92

Figure 59. Equivalent plastic strain response contours at the limit load (ALF=0.97), E5-U-

490LS(3/8)-WV.

4.4 WARREN WITHOUT VERTICAL VARIATIONS ON EXPERIMENTAL TEST CONFIGURATIONS, UNCHAMFERED MEMBERS

This section summarizes the key characteristics and sample results for the Warren without vertical variations on the experimental test configurations using unchamfered members. The corresponding results are summarized in Table 6 of Section 4.2.

4.4.1 E1-U-307SS-W

Specimen E1-U-307SS-W is a joint with unchamfered members, A307 bolts having a response representative of hot-driven rivets, a short standoff distance for the compression diagonal, a short connection length to the compression diagonal, and no vertical member. Specimen E1-U-307SS-W is the same as E1-U-307SS-WV except that the vertical member is removed in E1-U-307SS-W. All the specimens shown in this section have the same geometry as the corresponding joints shown in Section 4.3 except the fact that the vertical member is removed in the specimens shown in this section. The key dimensions of the gusset plate and its members are shown in Figure 38(a). Figure 60 shows the reference member forces for E1-U-307SS-W.

93

Figure 60. Design forces for E1-U-307SS-W.

Figure 61 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement curves is nearly elastic-perfectly plastic similar to E1-U-307SS-WV. Figure 62 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 63 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the compression diagonal and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

Figure 61. Load-displacement plot, E1-U-307SS(3/8)-W.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25

Ap

plie

d L

oa

d F

rac

tio

n

Displacements at Point A (in)

Horizontal Displacement

Out-of-plane Displacement

94

Figure 62. von Mises stress response contours at the limit load (ALF=0.83), E1-U-307SS(3/8)-W (DSF = 5).

Figure 63. Equivalent plastic strain response contours at the limit load (ALF=0.83), E1-U-307SS(3/8)-W (DSF = 5).

95

4.4.2 E2-U-307LS-W

Specimen E2-U-307LS-W is a joint with unchamfered members, A307 bolts having a response representative of hot-driven rivets, a long standoff distance (4.5 inches) for the compression diagonal, a short connection length to the compression diagonal, and no vertical member. The key dimensions of the gusset plate and its members are shown in Figure 42(a). Figure 64 shows the reference member forces for E2-U-307LS-W.

Figure 64. Design forces for E2-U-307LS-W.

Figure 65 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement plot of the horizontal in-plane displacement is approximately elastic even though the magnitude of the horizontal displacement is larger than the out-of-plane displacement. The out-of-plane displacement is not substantial until the limit load is reached. Figure 65 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 66 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in Figure 66, one can observe that yielding in the gusset occurs within the small area along with the gusset just above the chord in the middle of the gusset between the compression and tension diagonals when the limit load is reached. However, the predominant mode of failure of this joint is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

96

Figure 65. Load-displacement plot, E2-U-307LS(3/8)-W.

Figure 66. von Mises stress response contours at the limit load (ALF=0.95), E2-U-307LS(3/8)-W (DSF = 5).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

Ap

plie

d L

oa

d F

rac

tio

n

Displacements at Point A (in)

Horizontal Displacement

Out-of-plane Displacement

97

Figure 67. Equivalent plastic strain response contours at the limit load (ALF=0.95), E2-U-307LS(3/8)-W (DSF = 5).

4.4.3 E3-U-307SL-W

Specimen E3-U-307SL-W is a joint with unchamfered members, A307 bolts having a response representative of hot-driven rivets, a short standoff distance (1.0 inch) for the compression diagonal, a long connection length to the compression diagonal, and no vertical member. Figure 46(a) shows key dimensions of gusset plate and its members. Figure 68 shows the reference member forces for this joint.

Figure 68. Design forces for E3-U-307SL-W.

Figure 69 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that horizontal in-plane displacement of point A continuously increases while the out-of-plane displacement is essentially zero until the joint reaches its limit load. Figure 70 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 71 shows the equivalent plastic strain contours at the mid-

98

thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the both diagonals and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. However, the maximum equivalent plastic strain (PEEQ) of this joint has not reached the 4 % limit. The predominant failure of this joint is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

Figure 69. Load-displacement plot, E3-U-307SL(3/8)-W.

0

0.2

0.4

0.6

0.8

1

1.2

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Ap

plie

d L

oa

d F

rac

tio

n

Displacements at Point A (in)

Horizontal Displacement

Out-of-plane Displacement

99

Figure 70. von Mises stress response contours at the limit load (ALF=1.0), E3-U-307SL(3/8)-W (DSF = 5).

Figure 71. Equivalent plastic strain response contours at the limit load (ALF=1.0), E3-U-307SL(3/8)-W (DSF = 5).

100

4.4.4 E4-U-490SS-W

Specimen E4-U-490SS-W is a joint with unchamfered members, A490 high-strength bolts, a short standoff distance (1.0 inch) for the compression diagonal, a short connection length to the compression diagonal, and no vertical member. This specimen is similar to E1-U-307SS-W except the type of fasteners. Figure 52(a) shows key dimensions of gusset plate and its members. Figure 72 shows the reference member forces for this joint

Figure 72. Design forces for E4-U-490SS-W.

Figure 73 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement curves is nearly elastic-perfectly plastic. Figure 74 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 75 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the compression diagonal and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

101

Figure 73. Load-displacement plot, E4-U-490SS(3/8)-W.

Figure 74. von Mises stress response contours at the limit load (ALF=0.98), E4-U-490SS(3/8)-W (DSF = 5).

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15

Ap

plie

d L

oa

d F

rac

tio

n

Displacements at Point A (in)

Horizontal Displacement

Out-of-plane Displacement

102

Figure 75. Equivalent plastic strain response contours at the limit load (ALF=0.98), E4-U-490SS(3/8)-W (DSF = 5).

4.4.5 E5-U-490LS-W

Specimen E5-U-490LS-W is a joint with unchamfered members, A490 high-strength bolts, a long standoff distance (4.5 inch) for the compression diagonal, a short connection length to the compression diagonal, and no vertical member. This specimen is similar to E2-U-307LS-W except the type of fasteners. Figure 56(a) shows key dimensions of gusset plate and its members. Figure 76 shows the reference member forces for this joint. Unlike E5-U-490LS-WV, this joint has the same member force in the tension and compression diagonals because there is no vertical member.

Figure 76. Design forces for E5-U-490LS-W.

Figure 77 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the out-of-plane displacement is significantly larger than the horizontal in-plane displacement. Figure 78 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 79 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. One can observe in

103

Figures 78 and 79 that most of the gusset plate area remains elastic except the corners of the end of the compression diagonal and the corner of the end of the tension diagonal just above the chord when the limit load is reached. The predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal

Figure 77. Load-displacement plot, E5-U-490LS(3/8)-W.

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0 0.05 0.1 0.15 0.2 0.25

Ap

plie

d L

oa

d F

rac

tio

n

Displacements at Point A (in)

Horizontal Displacement

Out-of-plane Displacement

104

Figure 78. von Mises stress response contours at the limit load (ALF=1.23), E5-U-490LS(3/8)-W (DSF = 5).

Figure 79. Equivalent plastic strain response contours at the limit load (ALF=1.23), E5-U-490LS(3/8)-W (DSF = 5).

105

4.5 WARREN WITH VERTICAL PARAMETRIC TEST CONFIGURATIONS

This section summarizes the key characteristics and sample results for the Warren with vertical parametric test configurations. The corresponding results are summarized in Table 7 of Section 4.2.

4.5.1 P1-C-WV-M

4.5.1.1 P1-C-CCS-WV-M

This is an upper-chord joint at the midspan location of a Warren truss bridge with vertical members. Figure 80 shows the configuration of the two panel system and the U2 connection under study. The geometry for this connection is shown in Figure 81(a). The design forces for the connecting members are shown in Figure 81(b). The compressive forces (2900 kips and 3000 kips) in the chords are relatively large compared to the compressive forces (424 kips and 283 kips) in the diagonals and the tensile force (500 kips) in the vertical member. Based on the design forces and the unbraced lengths of the connecting members, this joint uses the box section B3 for its chord members, the box section B1 for both diagonals and the I section I1 for its vertical member.

Figure 80. Applied loads and boundary conditions for P1-C-CCS-WV-M.

106

18

12

12

30

39

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 81. Gusset plate geometry and design forces for P1-C-CCS-WV-M.

Table 15 summarizes the results from the key FHWA Guide design checks, as well as from a supplementary pseudo-plastic section analysis used to determine the required gusset thickness for the given loading condition. The pseudo-plastic section procedure is used to determine the gusset plate thickness necessary for transfer of the chord forces with the gusset working as a part of the chord splice. Section 5.1 provides a detailed discussion of this procedure, which is referred to in this work as Method 2. Based on this procedure, the required gusset thickness is 0.41 inch, while Whitmore buckling of the right-hand chord requires a 0.37 inch gusset. A gusset plate thickness of 0.40 inches, a flange splice plate thickness of 0.775 inches, and a web splice plate thickness of 0.509 inches are selected for the simulation shown below.

107

Table 15 Key required gusset plate thicknesses from the FHWA Guide and from a supplementary pseudo-plastic section analysis of the compression splice for P1-C-CCS-WV-M.

Limit State tg.req (in)

Pseudo plastic section – compression splice 0.41

Whitmore buckling at right-hand chord 0.37 (K = 1.2)

Edge slenderness check 0.34

A load-displacement plot for P1-C(0.4)-CCS-WV-M is shown in Figure 82. The ordinate of the plot is the applied load fraction (ALF), which is the fraction of the specified design loads applied to the joint. The abscissa is the vertical deflection at U2 relative to the fixed vertical supports at L1 and L3. Figure 82 shows that this joint reaches its load capacity at an ALF of 1.37 i.e., at 137% of the applied reference loads when it reaches the PEEQ > 4% strength limit. Note that the simulation continues well beyond this point (up to an ALF of about 1.75), but these results are deemed unreliable and are thus ignored.

Figure 83 shows the von Mises stress contours at an ALF of 1.37. The deformation scale factor (DSF) in the figure is 5; that is, the displacements are magnified by a factor of 5.0 in the rendering of the deformed geometry. The second contour from the top indicates areas that are close to the yield strength and the grey areas have exceeded the yield limit. The equivalent plastic strain (PEEQ) contours provided in Figure 84 show a sharp increase in the magnitude of the plastic strains in the gusset plate within the length between the end fasteners of the chord members at the chord splice. The splice plates also are substantially yielded at this stage (not shown in the figure). Any PEEQ contour other than the darkest grey is a region where the mid-surface of the plate is yielded.

108

Figure 82. Load-displacement plot for P1-C-CCS(0.4)-WV-M.

Figure 83. von Mises stress contours for P1-C-CCS(0.4)-WV-M at the PEEQ > 4 % strength condition occurring at an ALF of 1.37 (DSF = 5).

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Ap

pli

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oa

d F

rac

tio

n,

P/P

refe

ren

ce

Vertical Deflection at U2 (in)

PEEQ > 4% at ALF = 1.37

109

Figure 84. Equivalent plastic strain contours for P1-C-CCS(0.4)-WV-M at the PEEQ > 4 % strength condition occurring at an ALF of 1.37 (DSF = 5).

4.5.1.2 P1-C-MTB-WV-M

A gusset plate thickness of 0.40 inches is used for P1-C-MTB-WV-M, to address the behavior of the above joint with the chord splice in compression but with the chord members idealized to be in perfect bearing over their cross-section areas. In addition, a second simulation is shown using a gusset plate thickness of 0.35 inches required to satisfy the Whitmore buckling criteria at the right-hand chord member.

Figure 85 shows load-displacement curves for P1-C-MTB(0.40)-WV-M and P1-C-MTB(0.35)-WV-M. When the plate thickness is 0.40 inches, the load capacity of the configuration is reached at an ALF of 2.29, controlled by the ultimate strength of the fasteners between the compression diagonal and the gusset plate. This is the only case in the studies presented in this report where the fastener strength governs the overall resistance of the joint. Hence, the discussions at the beginning of the chapter do not mention this limit state as a potential controlling strength criterion for the joint. When the plate thickness is 0.35 inches the configuration reaches its load capacity at an ALF of 1.96 by buckling of the gusset plate under the diagonal compression. Figures 86 and 87 illustrate the von Mises and equivalent plastic strain contours for P1-C-MTB(0.35)-WV-M at an ALF of 1.96. Unlike the P1-C-CCS(0.4)-WV-M joint where the yielding is highly concentrated at the chord splice, the gusset plates of P1-C-MTB(0.35)-WV-M are plastified within a wide area around the end of the compression diagonal and the vertical, and

110

at the chord splice. In addition, the out-of-plane movement of the compression diagonal is evident.

Figure 85. Load-displacement curves for P1-C-MTB(0.35)-WV-M and P1-C-MTB(0.4)-WV-M.

0

0.5

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1.5

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2.5

-2.5 -2 -1.5 -1 -0.5 0

Ap

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oad

Fra

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P/P

refe

ren

ce

Vertical Deflection at U2 (in)

Case 1-a with tg=0.40"

Case 1-a with tg=0.35"

P1-C-MTB(0.4)-WV-M

P1-C-MTB(0.35)-WV-M

111

Figure 86. von Mises stress contours for P1-C-MTB(0.35)-WV-M at the limit load occurring at an ALF of 1.96 (DSF = 5).

Figure 87. Equivalent plastic strain contours for P1-C-MTB(0.35)-WV-M at the limit load occurring at an ALF of 1.96 (DSF = 5).

112

4.5.2 P2-C-TCS-WV-M

This is a lower-chord joint at a midspan location of a Warren truss bridge with vertical members. Figure 88 shows the configuration of the two panel system and the L2 connection under study. The geometry for this connection is shown in Figure 89(a) and the design forces in the connecting members of the gusset are shown in Figure 89(b). The tensile forces (2900 kips and 3000 kips) in the chords are relatively large compared to the tensile forces (424 kips and 283 kips) in the diagonals and the compressive force (500 kips) of the vertical member. Based on the design forces and lengths of the connecting members, the following sections are used: box section B3 for the chord members, box section B1 for both diagonals and box section B1 for the vertical member. The overall configuration for this test is simply flipped upside down relative to P1-C-CCS-WV-M and the loading is the same except that all the member loads are tensile in this test whereas they are compressive in P1-C-CCS-WV-M.

Figure 88. Applied loads and boundary conditions of P2-C-TCS-WV-M.

113

18

12

12

30

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 89. Gusset plate geometry and design forces for P2-C-TCS-WV-M.

Table 16 summarizes the key required gusset thicknesses from the FHWA Guide for the given loading condition, plus the supplementary Method 2 calculation for the participation of the gusset plate in the chord splice. The left chord block shear check requires a gusset thickness of 0.45 inch. A gusset plate thickness of 0.40 inches, a flange splice plate thickness of 0.775 inches, and a web splice plate thickness of 0.509 inches are selected for the simulation shown below.

A load-displacement plot for P2-C-TCS(0.4)-WV-M is shown in Figure 90, using the vertical deflection at L2 as the abscissa. This configuration reaches its load capacity at 133% of the applied reference loads (ALF=1.33) as shown in Figure 90. The strength of the connection is controlled by the 4% PEEQ criterion. Figure 91 shows the von Mises stress contours at an ALF of 1.33. The equivalent plastic strain contours are provided in Figure 92. Due to the large tensile forces in the chords, substantial yielding of the gusset occurs within the length between the end fasteners of the chord members at the chord splice. The splice plates also are substantially yielded in this region (not shown in the figure).

114

Table 16 Key required gusset plate thicknesses from the FHWA Guide and from a supplementary pseudo-plastic section analysis of the tension splice for P2-C-TCS-WV-M.

Limit State tg.req (in)

Block shear at right-hand chord 0.45 Pseudo plastic section – tension splice 0.35

Figure 90. Load-displacement plot for P2-C-TCS(0.4)-WV-M.

0

0.2

0.4

0.6

0.8

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1.2

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1.6

1.8

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Ap

pli

ed L

oad

Fra

ctio

n,

P/P

refe

ren

ce

Vertical Deflection at L2 (in)

PEEQ > 4% at ALF = 1.33

115

Figure 91. von Mises stress contours for P2-C-TCS(0.4)-WV-M at the PEEQ > 4 % strength condition occurring at an ALF of 1.33 (DSF = 5).

Figure 92. Equivalent plastic strain contours for P2-C-TCS(0.4)-WV-M at the PEEQ > 4 % strength condition occurring at an ALF of 1.33 (DSF = 5).

116

4.5.3 P3-C-WV-P

This is an upper-chord joint at the pier location of a Warren truss bridge with vertical members. Figure 93 shows the configuration of the two panel system and the U2 connection under study. The geometry for this connection is shown in Figure 94(a), and the design forces are shown in Figure 94(b). As can be seen from the figures, the diagonal members are relatively steep. This requires substantial chamfering of the diagonal members, and results in the presence of a long vertical free edge. A large compressive force (5000 kips) is transferred to the hypothetical support at L2 by the vertical member. The tensile forces (3000 kips and 2800 kips) in the chord and diagonal members are also relatively large, resulting in a large shear force on the vertical plane and zero net shear on the horizontal plane of the gusset plate. Based on the design forces and the unbraced lengths of the connecting members, the following member sections are used: the box section B3 for the chord members, the I section I2 for both diagonals and the box section B5 for the vertical member.

Figure 93. Applied loads and boundary conditions for P3-C-WV-P.

117

18

43

54

24

1

3

57.818

(a) Gusset plate geometry

(b) Design forces

Figure 94. Gusset plate geometry and design forces for P3-C-WV-P.

Table 17 shows the key required gusset thicknesses from the FHWA Guide for the given loading condition. The Whitmore buckling checks for the vertical member require a gusset thickness of 0.73 inches, while a 0.53-inch gusset is required to resist shear on the vertical plane. A gusset thickness of 0.5 inches, a flange splice plate thickness of 0.786 inches, and web splice plate thickness of 0.525 inches are selected for this simulation.

118

A load-displacement plot for P3-C-WV-P is shown in Figure 95. The displacement is the vertical deflection at L2 (essentially equal to the vertical deflections of L1 and L3 in the prototype). Figure 95 shows that this configuration reaches the 4% PEEQ failure criterion at an ALF of 0.96.

Figures 96 and 97 show the von Mises stress contours and equivalent plastic strain contours at the PEEQ > 4 % strength condition. The equivalent plastic strain contours in Figure 97 show a sharp increase in the plastic strains within the small width of the gusset plate between the chamfered end of the diagonals and the side of the vertical member. Figure 98 shows the normal, shear and von Mises stresses on one of the vertical planes of the gusset along the side of the vertical member. This figure shows that shear yielding is developed on these planes essentially over the full height of the gusset except for the portion of the gusset that overlaps with the chord. Out-of-plane movement of the vertical member is noticeable in Figures 96 and 97. Given the load capacity reached using the 0.5 inch thick gusset, it appears that the larger thickness obtained using the FHWA Guide checks with K = 1.2 is conservative.

Table 17 Key required gusset plate thicknesses from the FHWA Guide for P3-C-WV-P.

Limit State tg.req (in)

Whitmore buckling at vertical member 0.73 (K = 1.2) Vertical plane shear 0.53 (Ω = 1) Block shear at tension diagonals 0.50

119

Figure 95. Load-displacement plot for P3-C(0.5)-WV-P.

Figure 96. von Mises stress contours for P3-C(0.5)-WV-P at the PEEQ > 4 % strength condition occurring at an ALF of 0.98 (DSF = 10).

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0.8

1

1.2

1.4

0 2 4 6 8 10

Ap

pli

ed L

oad

Fra

ctio

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P/P

refe

ren

ce

Vertical Deflection at L2 (in)

PEEQ > 4% at ALF = 0.98

120

Figure 97. Equivalent plastic strain contours for P3-C(0.5)-WV-P at the PEEQ > 4 % strength condition occurring at an ALF of 0.98 (DSF = 10).

121

Figure 98. von Mises, normal, and shear stresses in the gusset plate along the vertical plane at the right-hand edge of the vertical member in P3-C(0.5)-WV-P, generated at the PEEQ > 4 %

strength condition occurring at an ALF of 0.98.

4.5.4 P4-C-WV-P

This is a lower-chord joint at the pier location of a Warren truss bridge with vertical members. Figure 99 shows the configuration of the two panel system and the L2 connection under study. The geometry for this connection is shown in Figure 100(a), and the design forces are shown in Figure 100(b). In this case, the vertical member is supported directly by the bearing. Also, as can be seen from the figures, the diagonal members are relatively steep. This requires substantial chamfering of the diagonals and also results in the presence of a long vertical free edge in the gusset plate. A large compressive force (5500 kips) is transferred from the bearing in this problem. In addition, the compressive forces in the chord and diagonal members are relatively large (3041 kips and 2240 kips), resulting in a large shear force on the planes on each side of the vertical member as well as large opposing horizontal forces. The total horizontal shear force is zero in this problem since the loading and geometry are symmetric about the vertical member. Based on the design forces and the unbraced lengths of the connecting members, the following sections are used: the box section B3 for the chord members and the vertical, and the box section B4 for the diagonal members.

0

10

20

30

40

50

60

70

80

90

-80 -60 -40 -20 0 20 40 60 80

Ver

tica

l Pla

ne

Dis

tan

ce f

rom

Bo

tto

m

(in

)

Stress (ksi)

Normal Stress

Shear Stress

Von Mises StressD

ista

nce

fro

m b

ott

om

of

gu

set

(in

) von Mises Stress

Shear Stress

Normal Stress

122

Figure 99. Applied loads and boundary conditions of P4-C-WV-P.

123

85.4

21.8

18

2118

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 100. Gusset plate geometry and design forces for P4-C-WV-P.

Table 18 presents the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. Whitmore buckling at the chords requires a gusset plate thickness of 0.84 inches while a 0.52-inch gusset is required to resist vertical plane shear. A gusset thickness of 0.8 inches is selected for the simulation shown below.

124

A load-displacement plot for P4-C(0.8)-WV-P is shown in Figure 101. The displacement is the vertical deflection at U3. Figure 101 shows that the P4-C(0.8)-WV-P configuration reaches the PEEQ > 4 % strength condition at an ALF of 0.97. However, the peak of the load-deflection curve for P4-C-WV-P (i.e., the limit load for this connection) also occurs essentially at an ALF of 0.97.

Figures 102 and 103 show the von Mises stress and equivalent plastic strain contours at the above strength limit. The equivalent plastic strain contours in Figure 103 indicate a sharp increase in the plastic strains over the short width of the gusset plate between the end bolt rows of the chord members and the adjacent bolt rows in the vertical. The deformed shape and the contours in Figure 102 indicate buckling of chord members and partial vertical plane shear failure. Figure 104 shows the normal, shear and von Mises stresses on one of the vertical planes along the side of the vertical member and through the bottom chord in this problem. The upper part of the gusset plate on this plane is dominated by shear whereas the lower part is dominated by the axial compression from the chord member. In this particular design, no continuity plates are provided through the vertical to the two chords, hence this connection is susceptible to out-of-plane movement of the chords.

Table 18 Key required gusset plate thicknesses from the FHWA Guide for P4-C-WV-P.

Limit State tg.req (in)

Edge slenderness check 0.85 Whitmore buckling at chords 0.84 (K = 1.2) Vertical plane shear 0.52 (Ω = 1)

125

Figure 101. Load-displacement plot for P4-C(0.8)-WV-P.

Figure 102. von Mises stress contours for P4-C(0.8)-WV-P at the PEEQ > 4 % strength condition occurring at an ALF of 0.97 (DSF = 10).

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0.2

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1

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-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Ap

pli

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oad

Fra

ctio

n,

P/P

refe

ren

ce

Vertical Deflection at U3 (in)

PEEQ > 4% at ALF = 0.97

126

Figure 103. Equivalent plastic strain contours for P4-C(0.8)-WV-P at the PEEQ > 4 % strength condition occurring at an ALF of 0.97 (DSF = 10).

Figure 104. von Mises, normal, and shear stresses along the vertical plane of P4-C-WV-P at the PEEQ > 4 % strength condition occurring at an ALF of 0.97.

0

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30

40

50

60

70

80

90

100

-80 -60 -40 -20 0 20 40 60 80

Ver

tica

l Pla

ne

Dis

tan

ce f

rom

Bo

tto

m

(in

)

Stress (ksi)

Normal Stress

Shear Stress

Von Mises Stress

Dis

tan

ce f

rom

bo

tto

m o

f g

use

t (i

n)

von Mises Stress

Shear Stress

Normal Stress

127

4.5.5 P5-WV-NP

4.5.5.1 P5-C-WV-NP

This is an upper-chord joint near a pier location of a Warren truss bridge with vertical members. Figure 105 shows the configuration of the two panel system and the U2 connection under study. The geometry for this connection is shown in Figure 106(a), and the design forces are shown in Figure 106(b). As can be seen from the figures, the compression diagonal is relatively steep and has a moderate compressive force (1500 kips). Large tension forces (3000 kips and 1500 kips) are present in the chords resulting in a moderate shear force across the horizontal plane of the gusset. Based on the design forces and the unbraced lengths of the connecting members, the following sections are used: the box section B3 for the chord members, the I section I1 for both the tension diagonal and the vertical and the box section B2 for the compression diagonal.

Figure 105. Applied loads and boundary conditions of P5-C-WV-NP.

128

15

18

30.5

12

39

35.2

12

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 106. Gusset plate geometry and design forces for P5-C-WV-NP.

Table 19 shows the key required gusset thicknesses from the FHWA Guide for the given loading condition. Whitmore buckling of the compression diagonal requires a gusset thickness of 0.34 inches while a 0.43-inch gusset is required to resist vertical plane shear and a 0.40-inch gusset is required to resist horizontal plane shear. A gusset thickness of 0.4 inches, a flange splice plate thickness of 0.775 inches, and a web splice plate thickness of 0.509 inches are selected for the simulation presented below.

A load-displacement plot for P5-C-WV-NP is shown in Figure 107. The abscissa is the vertical deflection at U2. Figure 107 shows that the P5-C-WV-NP configuration reaches its load capacity at an ALF of 0.94 (at the first peak), based on the limit load criterion. While the ALF subsequently reaches above 1.00, this value is not taken as the strength since it requires the development of large deformations.

Figures 108 and 109 show the von Mises stress and equivalent plastic strain contours at the limit load in this test. The equivalent plastic strain contours in Figure 109 show extensive yielding

129

around the entire periphery of the compression diagonal at the strength limit, with the largest concentration of plastic strain occurring at the chamfered and non-chamfered ends of the compression diagonal. In addition, out-of-plane movement of the compression diagonal and of the long free edge of the gusset plate is evident at the strength limit. Figure 110 shows the variation of the normal, shear and von Mises stresses on the vertical plane at the edge of the vertical member on the compression diagonal side of the gusset. One can observe that the gusset plate is essentially fully yielded in shear except for where it is bolted to the truss chord. Figure 111 illustrates the variation of the shear stress on the horizontal plane just below the chord in this problem. This plane is also nearly fully yielded, but there is significant interaction between the normal and shear stresses in the vicinity of the small offset of the compression diagonal from the chord.

Table 19 Key required gusset plate thicknesses from the FHWA Guide for P5-C-WV-NP.

Limit State tg.req (in)

Edge slenderness check on compression diagonal side

0.73

Vertical plane shear 0.43 (Ω = 1) Block shear at tension diagonal 0.40 Horizontal plane shear 0.40 (Ω = 1) Whitmore buckling at compression diagonal 0.34 (K = 0)

130

Figure 107. Load-displacement plot for P5-C(0.40)-WV-NP.

Figure 108. von Mises stress contours for P5-C(0.40)-WV-NP at the limit load occurring at an ALF of 0.94 (DSF = 5).

0

0.2

0.4

0.6

0.8

1

1.2

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Ap

pli

ed L

oad

Fra

ctio

n,

P/P

refe

ren

ce

Vertical Deflection at U2 (in)

131

Figure 109. Equivalent plastic strain contours for P5-C(0.40)-WV-NP at the limit load occurring at an ALF of 0.94 (DSF = 5).

Figure 110. von Mises, normal, and shear stresses along the vertical plane of P5-C(0.40)-WV-NP at the limit load occurring at an ALF of 0.94.

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30

40

50

60

70

-60 -40 -20 0 20 40 60

Ver

tica

l Pla

ne

Dis

tan

ce F

rom

Bo

tto

m

(in

)

Stress (ksi)

Normal Stress

Shear Stress

Von Mises Stress

Dis

tan

ce f

rom

bo

tto

m o

f g

uss

et (

in)

von Mises Stress

132

Figure 111. von Mises, normal, and shear stresses along the horizontal plane of P5-C(0.40)-WV-NP at the limit load occurring at an ALF of 0.94.

4.5.5.2 P5-U-WV-NP

The configuration of the two panel system for P5-U-WV-NP is the same as that shown in Figure 105. The geometry for this connection is shown in Figure 112(a), and the design forces are shown in Figure 112(b). Since the members are unchamfered, the gusset plate of P5-U-WV-NP has a significantly longer vertical free edge between the top left chord member and the left compression diagonal. The free edge of P5-U-WV-NP is 40.6 inches whereas the corresponding free edge of P5-C-WV-NP is only 32.3 inches. The tension diagonal on the right-hand side of P5-C-WV-NP, however, only has a minor chamfer (see Figure 106). As a result, the free edges between the tension diagonal and the adjacent members of P5-U-WV-NP are only slightly longer than those of P5-C-WV-NP. As can be seen from Figure 112(b), the member forces for P5-U-WV-NP are the same as those for P5-C-WV-NP.

-80

-60

-40

-20

0

20

40

60

80

0 20 40 60 80

Str

ess

(ks

i)

Horizontal Plane Distance from Left (in)

Normal Stress

Shear Stress

Von Mises Stressvon Mises Stress

133

33.61

45

61.

5

18

12

1512

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 112. Gusset plate geometry and design forces for P5-U-WV-NP.

Table 20 gives the gusset plate thickness requirements for P5-U-WV-NP based on the FHWA Guide for the given loading condition. As expected, the edge slenderness check requires significantly thicker gusset plates because of the long vertical edge between the chord and compression diagonal members. If K = 1.2 is used, the Whitmore buckling check at the compression diagonal requires a gusset thickness of 0.65 inches. The shear checks of the horizontal and vertical planes of P5-U-WV-NP are less critical; the required gusset plate thicknesses are 0.38 and 0.35 inches respectively for these limit states (using = 1). For the FEA test simulation below, a gusset thickness of 0.40 inches is selected (this is the same as the base thickness used in P5-C-WV-NP).

134

Figure 113 shows a load-displacement plot for P5-U-WV-NP. The abscissa is the vertical deflection at U2. This plot shows that the P5-U-WV-NP joint reaches its limit load at an ALF of 0.78 whereas an ALF of 0.94 is reached for P5-C-WV-NP (also at its limit load). For P5-C-WV-NP and P5-U-WV-NP, the maximum PEEQ is 2 % and 0.4 % respectively at the limit load.

Figures 114 and 115 show the von Mises stress and equivalent plastic strain contours for P5-U-WV-NP at the limit load for this problem, using a displacement scale factor (DSF) of 5 on the deformed geometry. These contours show the gusset plates are yielded slightly within the area below the top left chord member and at the end of the compression diagonal. Also, the vertical free edge between the top chord and compression diagonal member is buckled out-of-plane. In addition, in this problem, the compression diagonal starts moving out-of-plane significantly just after significant out-of-plane movement of the free edge occurs. Figures 116 and 117 show the von Mises and equivalent plastic strain contours on the deformed geometry of the joint (DSF = 5) within the post-peak range of the response (at ALF = 0.55). These images show significant out-of-plane sway buckling of the gusset plate. It should be noted that the peak load at ALF = 0.78 is reached using tg = 0.4 inches. This thickness is substantially smaller than the FHWA Guide requirements for edge slenderness and Whitmore buckling strength along the compression diagonal, corresponding to tg.req = 0.65 and 0.89 respectively. Although the results are not shown here, when the gusset plate thickness is increased to 0.5 inches, the peak ALF increases to 1.09. This indicates that the edge slenderness check and the Whitmore buckling calculation with K = 1.2 are significantly conservative for P5-U-WV-NP.

135

Table 20 Key required gusset plate thicknesses from the FHWA Guide for P5-U-WV-NP.

Limit State tg.req (in)

P5-U P5-C

Edge slenderness check on compression diagonal side

0.89 0.73

Whitmore buckling at compression diagonal 0.65 (K = 1.2) 0.34 (K = 0)

Block shear tear out at tension diagonal 0.41 0.40

Tension rupture check for left chord 0.40 0.43

Vertical plane shear along left side of vertical 0.38 ( = 1) 0.43 (Ω = 1)

Horizontal plane shear along chords 0.35 ( = 1) 0.40 (Ω = 1)

Figure 113. Load-displacement plot for P5-U(0.4)-WV-NP.

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Figure 114. von Mises stress contours for P5-U(0.4)-WV-NP at the limit load occurring at an ALF of 0.78 (DSF = 5).

Figure 115. Equivalent plastic strain contours for P5-U(0.4)-WV-NP at the limit load occurring at an ALF of 0.78 (DSF = 5).

137

Figure 116. von Mises stress contours for P5-U(0.4)-WV-NP at a post-peak ALF of 0.55 (DSF = 5).

Figure 117. Equivalent plastic strain contours for P5-U(0.40)-WV-NP at a post-peak ALF of 0.55 (DSF = 5).

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4.5.5.3 P5-C-HS(0.4)-WV-NP and P5-C-HS(0.2)-WV-NP

Figure 118 shows the load-displacement curves for P5-C-HS(0.4)-WV-NP and P5-C-HS(0.2)-WV-NP. The load-displacement curve for the original configuration, P5-C(0.4)-WV-NP is also shown in this figure for comparison purposes. The abscissa is the vertical displacement of node U2. Figure 118 shows that by using the Grade 100 material, the limit load of P5-C-HS(0.4)-WV-NP is increased by 79.8 % (ALF = 1.69) compared to the limit load of P5-C-WV-NP (ALF = 0.94). With the thinner gusset plates, P5-C-HS(0.2)-WV-NP reaches its load capacity at an ALF of 0.64. This is 31.9 % smaller than P5-C(0.4)-WV-NP and 62.1 % smaller than P5-C-HS(0.4)-WV-NP.

Figures 119 and 120 show the von Mises stress and equivalent plastic strain contours at the peak load of P5-C-HS(0.4)-WV-NP. These figures illustrate that the failure mode of P5-C-HS(0.4)-WV-NP is not much different from that of P5-C(0.4)-WV-NP except for the tension splice region. In both configurations, the horizontal plane below the chord members and the vertical plane on the left side of the vertical member are plastified significantly. However, the tension splice of P5-C(0.4)-WV-NP is yielded more significantly at the limit load than in P5-C-HS(0.4)-WV-NP. In addition, significant out-of-plane deformations are observed along the vertical free edge next to the compression diagonal. The gusset plates at the end of the compression diagonal also are deflected significantly out-of-plane.

Figures 121 and 122 show the von Mises stress and equivalent plastic strain contours at the limit load of P5-C-HS(0.2)-WV-NP. These contours illustrate that the gusset plates of this configuration are plastified only partially within the regions experiencing significant yielding in P5-C-HS(0.4)-WV-NP. In addition to exhibiting less yielding at the strength limit, P5-C-HS(0.2)-WV-NP shows significant out-of-plane movement of the gusset at the end of the compression diagonal and along the adjacent vertical free edge at an ALF of 0.64.

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Figure 118. Load-displacement plot of P5-C(0.4)-WV-NP, P5-C-HS(0.4)-WV-NP, and P5-C-HS(0.2)-WV-NP.

Figure 119. von Mises stress contours of P5-C-HS(0.4)-WV-NP at the limit load occurring at an ALF of 1.69 (DSF = 5).

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Vertical Deflection at U2 (in)

P5-C(0.4)-WV-NP

P5-C-HS(0.4)-WV-NP

P5-C-HS(0.2)-WV-NP

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Figure 120. Equivalent plastic strain contours of P5-C-HS(0.4)-WV-NP at the limit load occurring at an ALF of 1.69 (DSF = 5).

Figure 121. von Mises stress contours of P5-C-HS(0.2)-WV-NP at the limit load occurring at an ALF of 0.64 (DSF = 5).

141

Figure 122. Equivalent plastic strain contours of P5-C-HS(0.2)-WV-NP at the limit load occurring at an ALF of 0.64 (DSF = 5).

4.5.6 P6-WV-NP

4.5.6.1 P6-C-WV-NP

This is an upper-chord joint at a near pier location of a Warren truss bridge with vertical members. Figure 123 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 124(a), and the design forces are shown in Figure 124(b). As can be seen from the figures, the angle of the tension diagonal is relatively shallow and this member has a large tensile force (2470 kips). Due to its shallow angle, the tension diagonal requires substantial chamfering. In addition, the shallow angle produces a long horizontal free edge at the bottom of the gusset plate. A large tensile force (4800 kips) is present on the left-hand chord, resulting in a large shear force on the horizontal plane of the gusset below the chord. Based on the design forces and the unbraced lengths of the connecting members, the following sections are used: the box section B5 for chord members, the I sections I2 and I1 for the tension diagonal and for the vertical respectively, and the box section B3 for the compression diagonal.

142

Figure 123. Applied loads and boundary conditions of P6-C-WV-NP.

51 76.1

18

18

1

12

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 124. Gusset plate geometry and design forces for P6-C-WV-NP.

143

Table 21 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. The edge slenderness check on the compression diagonal side requires a gusset thickness of 0.68 inches, whereas the horizontal plane shear check requires a thickness of 0.54 inches and block shear at the tension diagonal requires a 0.49-inch thick gusset. A gusset thickness of 0.5 inches is selected based on these results along with a top and bottom splice plate thickness of 1.067 inches, and a web splice plate thicknesses of 0.636 inches.

A load-displacement plot for P6-C-WV-NP is shown in Figure 125. The abscissa is the horizontal displacement at L1. Figure 125 shows that the P6-C-WV-NP configuration reaches its limit load at an ALF of 0.98. For P6-C-WV-NP, the maximum PEEQ becomes larger than 4 % immediately after the test reaches its limit load, but the 4 % PEEQ criterion does not control the capacity.

Figures 126 and 127 show the von Mises stress and equivalent plastic strain contours at the limit load in this problem. The equivalent plastic strain contours in Figure 127 show a concentration in plastic strain essentially along the entire length of the gusset plate underneath the top chord with the largest plastic strains occurring at the chamfered end of the compression diagonal. The gusset plate is yielded within the grey areas in Figure 126 and the areas not in the darkest grey in Figure 127. One can observe that the compression diagonal is buckled out-of-plane in these figures. However, the largest out-of-plane movement is along the vertical free edge next to the compression diagonal. It is apparent that the transfer of the force from the compression diagonal to the joint is predominantly within gusset area around the end of the diagonal.

Table 21 Key required gusset plate thicknesses from the FHWA Guide for P6-C-WV-NP.

Limit State tg.req (in)

Edge slenderness check on compression diagonal side

0.68

Horizontal plane shear 0.54 (Ω = 1) Block shear at tension diagonal 0.49 Vertical plane shear 0.48 (Ω = 1) Whitmore buckling at compression diagonal 0.42 (K = 0)

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Figure 125. Load-displacement plot for P6-C(0.5)-WV-NP.

Figure 126. von Mises stress contours for P6-C(0.5)-WV-NP at the limit load occurring at an ALF of 0.98 (DSF = 10).

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Figure 127. Equivalent plastic strain contours for P6-C(0.5)-WV-NP at the limit load occurring at an ALF of 0.98 (DSF = 10).

4.5.6.2 P6-U-WV-NP

The gusset plate geometry and the design forces in the connecting members of P6-U-WV-NP are shown in Figure 128. Because of the shallow angle of the tension diagonal, the free edges of the gusset plates between the vertical and the tension diagonal member and the horizontal shear plane under the top chords are significantly longer in both P6-C-WV-NP and P6-U-WV-NP. The free edge lengths between the vertical and tension diagonal member are 63.1 and 72.1 inches and the lengths of horizontal shear plane are 128 and 138 inches for P6-C-WV-NP and P6-U-WV-NP respectively. As shown in Figure 124, both diagonals of P6-C-WV-NP are highly chamfered, resulting in a small area of the gusset plate between the top chord and diagonal members. Conversely, the gusset plates of P6-U-WV-NP have a large area of the gusset plate at the end of the diagonals as shown in Figure 128.

146

51 87

18

12

18

24

(a) Gusset plate geometry (units = inches)

(b) Design Forces

Figure 128. Gusset plate geometry and design forces for P6U-WV-NP.

Table 22 shows the key required gusset plate thicknesses from the FHWA Guide for P6-U-WV-NP. Similar to other cases, the free edge slenderness check on the compression diagonal side requires the largest thickness for the gusset plate, tg.req = 0.66 inches. By using K = 1.2, the Whitmore buckling check at the compression diagonal requires tg.req = 0.64 inches. Because the lengths of the free edges on the compression diagonal side of P6-U-WV-NP are relatively short, the required gusset plate thickness tg.req based on the edge slenderness check is only slightly larger than the Whitmore buckling check in this case. Block shear tear out at the tension diagonal gives tg.req = 0.61 inches while the horizontal and vertical plane shear checks give tg.req = 0.50 and 0.46 respectively using = 1.0. For direct comparison with the analysis results of P6-C(0.5)-WV-NP, a gusset thickness of 0.5 inches is selected for the simulation shown below.

Figure 129 shows a load-displacement plot for P6-U(0.5)-WV-NP. The abscissa is the horizontal displacement at the L1 node. This plot demonstrates that the maximum capacity of this connection is reached at an ALF of 0.80, which is 18 % smaller than the ALF of 0.98 for P6-

147

C(0.5)-WV-NP (see Figure 125). Figures 130 and 131 show the von Mises and equivalent plastic strain contours for P6-U(0.5)-WV-NP at the limit load. Figures 130 and 131 show that the gusset plates in the area between the end of the compression diagonal and the top chord are substantially yielded while they are only slightly yielded at the end of the tension diagonal. Figures 132 and 133 illustrate von Mises and equivalent plastic strain contours for P6-U(0.5)-WV-NP at a post-peak ALF of 0.59. These figures show significant out-of-plane movement of the compression diagonal. It appears that there is a loss of load transfer at the end of the compression diagonal in the post-peak range of the response (see Figure 132), after this region is extensively plastified at the limit load (see Figure 130). Figure 132 shows that this is combined with substantial redistribution of force to the adjacent gusset plate areas along each side of the compression diagonal. Although the results are not shown here, when the gusset plate thickness is increased to 0.6 inches, P6-U(0.6)-WV-NP attains an ALF of 1.06. This indicates that the Whitmore buckling check using a K = 1.2 is a good predictor for this case, although it is slightly conservative.

Table 22 Key required gusset plate thicknesses from the FHWA Guide for P6-U-WV-NP.

Limit State tg.req (in)

P6-U-WV-NP P6-C-WV-NP

Edge slenderness check on compression diagonal side

0.66 0.68

Whitmore buckling for compression diagonal 0.64 (K = 1.2) 0.42 (K = 0)

Block shear check at tension diagonal 0.61 0.49

Horizontal plane shear check 0.50 ( = 1) 0.54 (Ω = 1)

Vertical plane shear check 0.46 ( = 1) 0.48 (Ω = 1)

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Figure 129. Load-displacement plot for P6-U(0.5)-WV-NP.

Figure 130. von Mises stress contours for P6-U(0.5)-WV-NP at the limit load occurring at an ALF of 0.80 (DSF = 5).

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Horizontal Displacement at L1 (in)

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Figure 131. Equivalent plastic strain contours for P6-U(0.5)-WV-NP at the limit load occurring at an ALF of 0.80 (DSF = 5).

Figure 132. von Mises stress contours for P6-U(0.5)-WV-NP at a post-peak ALF of 0.59 (DSF = 5).

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Figure 133. Equivalent plastic strain contours for P6-U(0.5)-WV-NP at a post-peak ALF of 0.59 (DSF = 5).

4.5.7 P7-WV-INF

4.5.7.1 P7-C-WV-INF

This is an upper-chord joint at the inflection point of a Warren truss bridge with a vertical member. Figure 134 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 135(a), and the design forces are shown in Figure 135(b). As can be seen from the figures, the diagonal members are at a shallow angle and have a high compression force (3580 kips) and moderate tensile force (2460 kips), respectively. The critical horizontal plane of the gusset plate supports a large shear force. This requires substantial chamfering, and results in the presence of a long horizontal free edge. Based on the design forces and the unbraced lengths of the connecting members, the following sections are used: the box section B3 for the chord members, the I sections I1 and I2 for the vertical and the tension diagonal respectively, and the box section B4 for the compression diagonal.

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Figure 134. Applied loads and boundary conditions of P7-C-WV-INF.

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 135. Gusset plate geometry and design forces for P7-C-WV-INF.

66.775.5

12

18

21 18

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Table 23 shows the key required gusset thicknesses from the FHWA Guide for the given loading condition. Gusset thickness values of 1.29 inches, 0.69 inches and 0.70 inches are required by the edge slenderness check on the compression diagonal side, horizontal plane shear, and Whitmore buckling of the compression diagonal respectively. The simulation shown below uses plate thicknesses of 0.7 inches for the gusset, 0.775 inches for the flange splice plates, and 0.509 inches for the web splice plates.

A load-displacement plot for P7-C(0.7)-WV-INF is shown in Figure 136. The abscissa is the horizontal displacement at L1. Figure 136 shows that this configuration reaches its load capacity at an ALF of 1.24 by reaching the PEEQ > 4 % strength condition.

Figures 137 and 138 show the von Mises stress and equivalent plastic strain contours at the PEEQ > 4 % strength limit. The equivalent plastic strain contours (Figure 138) show that essentially the entire plane of the gusset plate is fully yielded just below the chord, with a sharp increase in the plastic strains particularly at the chamfered end of the compression diagonal and with high plastic strains extending across to the chamfered end of the tension diagonal. There is very little out-of-plane movement of the compression diagonal or the free edge in this problem, probably because of the highly chamfered compression diagonal.

Table 23 Key required gusset plate thicknesses from the FHWA Guide for P7-C-WV-INF.

Limit State tg.req (in)

Edge slenderness check on compression diagonal side

1.29

Whitmore buckling for compression diagonal 0.70 (K = 1.2) Horizontal plane shear 0.69 (Ω = 1)

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Figure 136. Load-displacement plot for P7-C(0.7)-WV-INF.

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Horizontal Displacement at L1 (in)

PEEQ > 4% at ALF = 1.28

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Figure 137. von Mises stress contours for P7-C(0.7)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 1.28 (DSF = 5).

Figure 138. Equivalent plastic strain contours for P7-C(0.7)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 1.28 (DSF = 5).

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4.5.7.2 P7-C-HS(0.7)-WV-INF and P7-C-HS(0.35)-WV-INF

Figure 139 shows the load-displacement curves for P7-C-HS(0.7)-WV-INF and P7-C-HS(0.35)-WV-INF, which investigate the behavior for high strength steel. The load-displacement curve for P7-C(0.7)-WV-INF is also shown in this figure for comparison purposes. The abscissa is the horizontal displacement of the node L1. Figure 139 shows that the configuration of P7-C-HS(0.7)-WV-INF reaches its load capacity at an ALF of 2.28, controlled by the 4% PEEQ criterion. This is 83.9 % higher than the load capacity of the P7-C-WV-INF configuration, which is at an ALF of 1.24, also controlled by the 4% PEEQ criterion. P7-C-HS(0.35)-WV-INF reaches its load capacity at an ALF of 0.99.

Figures 140 and 141 show the von Mises stress and equivalent plastic strain contours for P7-C-HS(0.7)-WV-INF at the PEEQ > 4 % strength limit. These figures demonstrate that the failure mode of P7-C-HS(0.7)-WV-INF is not much different from that of P7-C(0.7)-WV-INF. P7-C-HS(0.7)-WV-INF also exhibits extensive shear yielding in the horizontal plane under the chord members as seen in P7-C(0.7)-WV-INF (compare Figures 140 and 141 to Figures 137 and 138).

Figures 142 and 143 show the von Mises stress and equivalent plastic strain contours at the peak load of P7-C-HS(0.35)-WV-INF. Unlike the configurations discussed above (i.e., tests P7-C(0.7)-WV-INF and P7-C-HS(0.7)-WV-INF), the horizontal plane of this configuration is not fully plastified. Instead, the gusset plates are yielded mostly within the area between the chord and the compression diagonal. The maximum PEEQ is approximately 2.4 % at the limit load ALF of 0.99. This joint fails by reaching a limit load, as shown by the plot in Figure 139.

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Figure 139. Load-displacement plot for P7-C(0.7)-WV-INF, P7-C-HS(0.7)-WV-INF, and P7-C-HS(0.35)-WV-INF.

Figure 140. von Mises stress contours for P7-C-HS(0.7)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 2.28 (DSF = 5).

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Horizontal Displacement at L1 (in)

P7-C(0.7)-WV-INF

P7-C-HS(0.7)-WV-INF

P7-C-HS(0.35)-WV-INF

PEEQ > 4% at ALF = 2.28

PEEQ > 4% at ALF = 1.28

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Figure 141. Equivalent plastic strain contours for P7-C-HS(0.7)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 2.28 (DSF = 5).

Figure 142. von Mises stress contours for P7-C-HS(0.35)-WV-INF at limit load occurring at an ALF of 0.99 (DSF = 5).

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Figure 143. Equivalent plastic strain contours for P7-C-HS(0.35)-WV-INF at the limit load occurring at an ALF of 0.99 (DSF = 5).

4.5.8 P8-WV-INF

4.5.8.1 P8-C-WV-INF

This is a lower-chord joint at the inflection point of a Warren truss bridge with vertical members. Figure 144 shows the configuration of the two panel system and the L2 connection under study. The geometry of this connection is shown in Figure 145(a), and the design forces are shown in Figure 145(b). As can be seen from the figures, the diagonal members carry moderate to high tensile and compression forces (2100 kips), generating a relatively large shear force on the plane of the gusset just above the bottom chord. The inclination of the compression diagonal is shallow relative to the chord, making the chamfering of the diagonal member desirable. Based on the design force and length of the connecting members, the following sections are used: box section B2 for the chord members, I-section I2 for the tension diagonal and B1 for vertical, and B3 for the compression diagonal.

159

Figure 144. Applied loads and boundary conditions of P8-C-WV-INF.

160

0.554

15

18

12

39.6

18

(a) Gusset plate geometry

(b) Design forces

Figure 145. Gusset plate geometry and design forces for P8-C-WV-INF.

Table 24 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. A gusset plate thickness of 0.68 inches is required to satisfy the edge slenderness check and block shear checks, 0.54 inches is required to resist the applied horizontal plane shear, and 0.47 inches is required for Whitmore buckling check with K = 0. A gusset thickness of 0.5 inch is selected along with flange splice plate thicknesses of 0.557 inches, and web splice plate thicknesses of 0.293 inches for the simulation shown below.

A load-displacement plot for P8-C(0.5)-WV-INF is shown in Figure 146. The abscissa is the horizontal displacement at U1. This configuration reaches its load capacity at an ALF of 0.97 by reaching the PEEQ > 4% strength criterion. Figures 147 and 148 show the von Mises stress and equivalent plastic strain contours at this load level. The equivalent plastic strain contours in Figure 148 indicate a sharp increase in the plastic strains within the length between the start of the chamfer of the compression diagonal on the left and the start of the chamfer on the tension

161

diagonal on the right. Both Figures 147 and 148 highlight that the entire plane of the gusset plate just above the chord is yielded. The behavior of this specimen is similar to that seen in E5-U-490LS(3/8)-WV (see Section 4.3.5). Initially, there is extensive plastic deformation along the horizontal shear plane. This is followed by large gusset plate deformations at the ends of the diagonals in tension and compression. Although one can see the beginnings of a block shear failure on the tension side and buckling in the compression side at the ultimate load, these responses occur near the end of the loading. The connection appears to fail predominantly in shear.

Table 24 Key required gusset plate thicknesses from the FHWA Guide for P8-C-WV-INF.

Limit State tg.req (in)

Block shear at tension diagonal 0.68 Edge slenderness check on compression diagonal side

0.68

Horizontal plane shear 0.54 ( = 1) Whitmore buckling at compression diagonal 0.47 (K = 0)

Figure 146. Load-displacement plot for P8-C(0.5)-WV-INF.

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Horizontal Displacement at U1 (in)

PEEQ > 4% at ALF = 0.97

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Figure 147. von Mises stress contours for P8-C(0.5)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 0.97 (DSF = 10).

Figure 148. Equivalent plastic strain contours for P8-C(0.5)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 0.97 (DSF = 10).

4.5.8.2 P8-U-WV-INF

The gusset plate geometry and the design forces for the members of P8-U-WV-INF are shown in Figure 149. Because of the relatively shallow angle between the compression diagonal and bottom chord member of 31.3 degrees, the free edge between the vertical and compression diagonal member is moderately long at 38.3 inches, whereas it is 30.4 inches in P8-C-WV-INF. As shown previously, the use of chamfered diagonals for P8-C-WV-INF results in a smaller area

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of the gusset plate between the bottom chord and diagonal members. Because P8-U-WV-INF uses an unchamfered compression diagonal, it has a large triangular area of the gusset plate at the end of the compression diagonal. Despite the tension diagonal being unchamfered as well, the area between the vertical, tension diagonal, and chord members is still relatively small in P8-U-WV-INF. This is because the angle between the vertical and the chord is larger than 90o on the tension diagonal side of the joint, and the tension diagonal frames into the joint at approximately an equal angle with respect to the vertical and the chord.

61.6

15

0.540.5

18

12

18

(a) Gusset plate geometry (unit = inches)

(b) Design forces

Figure 149. Gusset plate geometry and design forces for P8-U-WV-INF.

Table 25 shows the key required gusset plate thicknesses from the FHWA Guide for P8-U-WV-INF. The edge slenderness check on the compression diagonal side requires a large thickness for the gusset plate of tg.req = 0.85 inches. By using K = 1.2, the Whitmore buckling check at the compression diagonal requires tg.req = 0.68 inches. Block shear tearout at the tension diagonal and the horizontal and vertical shear plane checks with = 1.0 require similar values of tg.req (0.54 to

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0.56 inches). For the test simulation, a gusset plate thickness of 0.5 inches is used as in P8-C(0.5)-WV-INF.

Figure 150 shows a typical load-displacement plot for P8-U(0.5)-WV-INF. The abscissa is the horizontal displacement at the U1 node. This plot demonstrates that the limit load of this joint is reached at an ALF of 0.94, slightly smaller than the ALF of 0.97 for P8-C(0.5)-WV-INF (see Figure 146).

Figures 151 and 152 show the von Mises stress and equivalent plastic strain contours for P8-U(0.5)-WV-INF at the limit load in this problem. These figures demonstrate that even though gusset plate area in front of the compression diagonal is relatively large, the strength limit of P8-U(0.5)-WV-INF is mostly driven by a shear failure of the horizontal plane along the chord members. Figure 152 illustrates substantial yielding throughout the shear plane along the top of the chord members. In addition, the corner area at the end of the tension diagonal and the triangular area between the compression diagonal and chord members are yielded substantially. Figures 153 and 154 illustrate the von Mises stress and equivalent plastic strain contours for P8-U(0.5)-WV-INF at a post-peak ALF of 0.70. These figures show that the horizontal plane yielding above the chords is followed by out-of-plane sway buckling of the compression diagonal. The overall movement of the compression diagonal is mobilized to a somewhat greater extent than the overall shear deformation of the gusset plate in the post-peak range of the response.

Based on the FHWA Guide requirements shown in Table 25, P8-U-WV-INF needs thick gusset plates to provide sufficient buckling capacity at the compression diagonal. However, the test simulation results indicate that the strength limit of P8-U(0.5)-WV-INF is actually driven more by the shear failure along the chord members. This indicates that the design criteria need to be improved to better capture the anticipated failure modes for this type of gusset plate connection.

Table 25 Key required gusset plate thicknesses from the FHWA Guide for P8-U-WV-INF.

Limit State tg.req (in)

P8-U-WV-INF P8-C-WV-INF

Edge slenderness check on compression diagonal side

0.85 0.68

Whitmore buckling at compression diagonal 0.68 (K = 1.2) 0.47 (K = 0)

Block shear at tension diagonal 0.56 0.68

Horizontal plane shear check 0.56 ( = 1) 0.54 ( = 1)

Vertical plane shear check 0.54 ( = 1) 0.53 ( = 1)

165

Figure 150. Load-displacement plot for P8-U(0.5)-WV-INF.

Figure 151. von Mises stress contours of P8-U(0.5)-WV-INF at the limit load occurring at an ALF of 0.94 (DSF = 5).

0

0.2

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0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Ap

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refe

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Horizontal Displacement at U1 (in)

166

Figure 152. Equivalent plastic strain contours of P8-U(0.5)-WV-INF at the limit load occurring at an ALF of 0.94 (DSF = 5).

Figure 153. von Mises stress contours of P8-U(0.5)-WV-INF at a post-peak ALF of 0.70 (DSF = 5).

167

Figure 154. Equivalent plastic strain contours of P8-U(0.5)-WV-INF at a post-peak ALF of 0.70 (DSF = 5).

4.5.8.3 P8-C-HS(0.5)-WV-INF and P8-C-HS(0.25)-WV-INF

Figure 155 shows the load-displacement curves for P8-C-HS(0.5)-WV-INF and P8-C-HS(0.25)-WV-INF, which investigate the behavior for high strength steel. The load-displacement curve of P8-C(0.5)-WV-INF is also shown in this figure for comparison purposes. The abscissa is the horizontal displacement of the node U1. Figure 155 demonstrates that P8-C-HS(0.5)-WV-INF reaches its load capacity at an ALF of 1.79 by reaching the PEEQ > 4% strength criterion. This is 84.5 % higher than the load capacity of the P8-C(0.5)-WV-INF configuration, which is at an ALF of 0.97, also controlled by the PEEQ > 4% strength criterion. P8-C-HS(0.25)-WV-INF reaches its load capacity at an ALF of 0.77 by reaching the limit load in the test simulation.

Figures 156 and 157 show the von Mises stress and equivalent plastic strain contours for P8-C-HS(0.5)-WV-INF at its PEEQ > 4 % strength limit. These figures demonstrate that the behavior of P8-C-HS(0.5)-WV-INF is not much different from that of P8-C(0.5)-WV-INF except that the gusset plates between the vertical member and the tension diagonal member are plastified slightly more in P8-C(0.5)-WV-INF. In both configurations, the failure is driven by the shear yielding in the horizontal plane above the chord members. The areas around the end of the tension diagonal are yielded as well in both P8-C(0.5)-WV-INF and P8-C-HS(0.5)-WV-INF.

Figures 158 and 159 show the von Mises stress and equivalent plastic strain contours at the limit load of P8-C-HS(0.25)-WV-INF. These contours illustrate that this configuration also shows yielding within the horizontal plane. However, the maximum PEEQ is significantly smaller than in P8-C(0.5)-WV-INF and P8-C-HS(0.5)-WV-INF. The maximum PEEQ in Figure 159 is about 0.8 %. In Figures 148 and 157, the maximum PEEQ exceeds 4 %. Also, there are larger out-of-

168

plane displacements of the gusset plates in the vicinity of the compression diagonal in Figures 158 and 159 compared to the results for P8-C(0.5)-WV-INF and P8-C-HS(0.5)-WV-INF.

Figure 155. Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-HS(0.5)-WV-INF, and P8-C-HS(0.25)-WV-INF.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Ap

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P/P

refe

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Horizontal Displacement at U1 (in)

P8-C-HS(0.5)-WV-INF

P8-C(0.5)-WV-INF

P8-C-HS(0.25)-WV-INF

PEEQ > 4% at ALF = 1.79

PEEQ > 4% at ALF = 0.97

169

Figure 156. von Mises stress contours for P8-C-HS(0.5)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 1.79 (DSF = 5).

Figure 157. Equivalent plastic strain contours for P8-C-HS(0.5)-WV-INF at the PEEQ > 4 % strength limit occurring at an ALF of 1.79 (DSF = 5).

170

Figure 158. von Mises stress contours for P8-C-HS(0.25)-WV-INF at the limit load occurring at an ALF of 0.77 (DSF = 5).

Figure 159. Equivalent plastic strain contours for P8-C-HS(0.25)-WV-INF at the limit load occurring at an ALF of 0.77 (DSF = 5).

171

4.6 PRATT PARAMETRIC TEST CONFIGURATIONS

This section summarizes the key characteristics and sample results for the Pratt parametric study test configurations. The corresponding results are summarized in Table 8 of Section 4.2.

4.6.1 P9-C-P-NP

This is an upper-chord joint near a pier location of a Pratt truss bridge. Figure 160 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 161(a), and the design forces are shown in Figure 161(b). This is a smaller truss configuration with smaller connection lengths. As can be seen from the figures, the diagonal carries a compression force of 368 kips whereas the vertical member carries a tension force of 260 kips. This results in a small shear force across the vertical plane and inconsequential shear force on the horizontal plane. Based on the design force and length of the connecting members, the following sections are used: box section B2p for the chord members, I-section I1p for the vertical, and B2p for the compression diagonal.

Figure 160. Applied loads and boundary conditions of P9-C-P-NP.

172

1.5

27 18

42

12

12

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 161. Gusset plate geometry and design forces for P9-C-P-NP.

Table 26 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. A gusset plate thickness of 0.50 inch is required to satisfy the edge slenderness criteria, 0.18 inches is required to satisfy the Whitmore buckling check at the right chord using K = 1.2, and 0.16 inches is required to satisfy the Whitmore buckling check at the compression diagonal using K = 0 as well as for the block shear tearout check at the vertical member A gusset plate thickness of 0.2 inch is selected along with 0.344 inch flange splice plates, and 0.285 inch web splice plates.

A load-displacement plot for P9-C-P-NP is shown in Figure 162. The abscissa is the vertical displacement of node U2. The figure shows that this configuration reaches its load capacity at an ALF of 0.96, which is the limit load in the test simulation.

Figures 163 and 164 show the von Mises stress and equivalent plastic strain contours at the limit load in this problem. The equivalent plastic strain contours in Figure 164 indicate the onset of yielding mainly around the end of the compression diagonal. In this problem, the limit load corresponds to the onset of significant out-of-plane movement of the compression diagonal along with some additional “bulging” of the gusset plate along the two free edge lengths adjacent to the

173

compression diagonal member. It appears that this “bulging” of the free edge lengths is mostly due to compatibility of deformations. The largest portion of compression force transfer occurs near the end of the diagonal, but when this region yields and reaches the limit of its ability to transfer force due to compression buckling, the area of the gusset near the free edge is unable to withstand the corresponding redistribution of the force. As shown in Table 26, Whitmore buckling with K = 0 at the slightly chamfered compression diagonal requires a 0.16-inch gusset. However, the FEA analysis with a 0.20-inch gusset reaches its limit load at an ALF of 0.96 (Figure 162). This indicates that the buckling check with K = 0 can be too optimistic for members that are lightly chamfered or unchamfered.

Table 26 Key required gusset plate thicknesses from the FHWA Guide for P9-C-P-NP.

Limit State tg.req (in)

Edge slenderness check along compression diagonal

0.50

Whitmore buckling at right-hand chord 0.18 (K =1.2) Whitmore buckling at compression diagonal 0.16 (K = 0) Block shear at vertical member 0.16

Figure 162. Load-displacement plot for P9-C(0.2)-P-NP.

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0.2

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0.6

0.8

1

1.2

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

Ap

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P/P

refe

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Vertical Displacement at U2 (in)

174

Figure 163. von Mises stress contours for P9-C(0.2)-P-NP at the limit load occurring at an ALF of 0.96 (DSF = 5).

Figure 164. Equivalent plastic strain contours for P9-C(0.2)-P-NP at the limit load occurring at an ALF of 0.96 (DSF = 5).

175

4.6.2 P10-C-P-NP

This is an upper-chord joint near a pier location of a Pratt truss bridge. Figure 165 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 166(a), and the design forces are shown in Figure 166(b). This is also a smaller truss configuration with small connection lengths. In this connection the diagonal is in tension and carries a small load of 156 kips. In addition, there is no chamfering of this member. The vertical member carries a compression force of 110 kips resulting in equal shears on both the vertical and horizontal planes. Based on the design force and length of the connecting members, the following sections are used: box section B2p for the chord members, I-section I1p for the diagonal, and B1p for the vertical member.

Figure 165. Applied loads and boundary conditions for P10-C-P-NP.

176

18

12

27

88

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 166. Gusset plate geometry and design forces for P10-C-P-NP.

Table 27 gives the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. A 0.50-inch thick gusset is required to satisfy the edge slenderness check and 0.17 inches for the Whitmore buckling check at the right-hand chord with K = 1.2. A gusset plate thickness of 0.20 inches is selected along with a flange splice plate thickness of 0.344 inches, and a web splice plate thickness of 0.285 inches.

A load-displacement plot for P10-C(0.2)-P-NP is shown in Figure 167. The abscissa is the vertical displacement of U2. This configuration reaches its load capacity at an ALF of 1.73 controlled by the PEEQ > 4% strength criterion. Figures 168 and 169 show the von Mises stress and equivalent plastic strain contours at an ALF of 1.73. The equivalent plastic strain contours in Figure 169 show a significant spike in the plastic strains within the small length of the gusset plate between the end fasteners of the two chord members at the middle of the splice. These plastic strains are particularly large at the top of the splice, apparently due to bending within the plane of the gusset plate. In addition, yielding is evident around the entire end of the tension diagonal at this stage of the loading.

177

Table 27 Key required gusset plate thicknesses from the FHWA Guide for P10-C-P-NP.

Limit State tg.req (in)

Edge slenderness check on compression diagonal side

0.50

Whitmore buckling at left-hand chord 0.17 (K =1.2) Whitmore buckling at vertical 0.17 (K = 0)

Figure 167. Load-displacement plot for P10-C(0.2)-P-NP.

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0.5

1

1.5

2

2.5

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Ap

pli

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Fra

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P/P

refe

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Vertical Displacement at U2 (in)

PEEQ > 4% at ALF = 1.37PEEQ > 4% at

ALF = 1.73

178

Figure 168. von Mises stress contours for P10-C(0.2)-P-NP at the PEEQ > 4 % strength limit occurring at an ALF of 1.73 (DSF = 2).

Figure 169. Equivalent plastic strain contours for P10-C(0.2)-P-NP at the PEEQ > 4 % strength limit occurring at an ALF of 1.73 (DSF = 2).

179

4.7 WARREN WITHOUT VERTICAL PARAMETRIC TEST CONFIGURATIONS

This section summarizes the key characteristics and sample results for the Warren without vertical parametric study test configurations. The corresponding results are summarized in Table 9 of Section 4.2.

4.7.1 P11-C-W-M

This is an upper-chord joint near a midspan location of a Warren truss bridge without vertical members. Figure 170 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 171(a), and the design forces are shown in Figure 171(b). The connection lengths are relatively small for this gusset. As can be seen from the figures, both diagonal members carry small compression forces of 290 kips and the inclination of the diagonals can be categorized as moderate. The diagonals also are moderately chamfered. The compressive force of 3000 kips in the chord members are large compared to the diagonals. Based on the design force and length of the connecting members, the following sections are used: box section B4 for the chord members and B1 for the diagonals.

Figure 170. Applied loads and boundary conditions of P11-C-W-M.

180

48

21

12 12

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 171. Gusset plate geometry and design forces for P11-C-W-M.

Table 28 gives the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. The gusset plate thickness is required to be 0.57 inches to satisfy the edge slenderness requirements and 0.45 inches to resist Whitmore buckling at the chord members with K = 1.2. A gusset plate thickness of 0.45 inches is selected along with a flange splice plate thickness of 0.816 inches, and a web splice plate thickness of 0.392 inches for the following simulation.

A load-displacement plot for P11-C(0.45)-W-M is shown in Figure 172. The abscissa is the horizontal displacement of L1. This plot shows this configuration reaches its load capacity at an ALF of 1.41, controlled by the PEEQ > 4% strength criterion. Figures 173 and 174 show the von Mises stress and equivalent plastic strain contours at this load level. The equivalent plastic strain contours in Figure 174 indicate a sharp increase in the plastic strains in the gusset plate within the short length between the last line of fasteners in each of the chords at the middle of the splice. These strains are largest at the top of the splice, apparently due to bending within the plane of the

181

gusset plate. The buckling check at the chord members with K = 1.2 appears to be conservative for this problem since an ALF significantly larger than 1.0 is reached prior to encountering a PEEQ of 4 % at the top gusset plate at the middle of the splice.

Table 28 Key required gusset plate thicknesses from the FHWA Guide for P11-C-W-M.

Limit State tg.req (in)

Edge slenderness check 0.57 Whitmore Buckling at chord members 0.45 (K = 1.2) Pseudo Plastic Section – compression splice 0.45 ( = 0.9)

Figure 172. Load-displacement plot for P11-C(0.45)-W-M.

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Horizontal Displacement at L1 (in)

PEEQ > 4% at ALF = 1.41

182

Figure 173. von Mises stress contours for P11-C(0.45)-W-M at the PEEQ > 4 % strength limit occurring at an ALF of 1.41 (DSF = 5).

Figure 174. Equivalent plastic strain contours for P11-C(0.45)-W-M at the PEEQ > 4 % strength limit occurring at an ALF of 1.41 (DSF = 5).

183

4.7.2 P12-C-W-P

This is a lower-chord joint at the pier location of a Warren truss bridge without vertical members. Figure 175 shows the configuration of the two panel system and the L2 connection under study. The geometry of this connection is shown in Figure 176(a), and the design forces are shown in Figure 176(b). In this connection, the most direct load path to transfer the large diagonal forces to the pier is by transfer of these forces to the gusset plates from the diagonals in single shear, and then by direct compression of the gusset plates on the bearing (since the diagonals are stopped above the bottom chord). The chord compression forces are larger than the diagonal forces, and hence, the chord members are continued through the joint. The bottom part of the gusset plates is supported over the pier and transfers 5000 kips compressive force from the bearing. As can be seen from the figures, the diagonal members are relatively steep, requiring substantial chamfering. This results in the presence of a long vertical free edge on the gusset plate. Each diagonal member is carrying 2580 kips of compressive force. The 3630 kip compressive forces in the chords also are large. Based on the design force and length of the connecting members, the following sections are used: box section B4 for the chord members, and B4 for the diagonals.

Figure 175. Applied loads and boundary conditions of P12-C-W-P.

184

60

21

70.

54

21

(a) Gusset plate geometry (units = inches)

(b) Design Forces

Figure 176. Gusset plate geometry and design forces for P12-C-W-P.

Table 29 gives the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. Transfer of 5000 kips to the bearing, based on a bearing width of 48 inches, requires a gusset thickness of 1.28 inches . The edge slenderness check requires a 0.91-inch gusset while a 0.50-inch gusset is required to resist Whitmore buckling of the compression diagonal, using K = 0. A thickness of 0.64 inches is required to resist the vertical plane shear (using = 1). A gusset thickness of 1.0 inch, a flange splice plate of 0.924 inches, and a web splice plate of 0.392 inches are selected for the simulation below.

185

A load-displacement plot for P12-C-W-P is shown in Figure 177. The abscissa is the vertical displacement at L3. This configuration reaches its load capacity at an ALF of 1.05 corresponding to the limit load of the load-deflection curve.

Figure 178 shows the von Mises stress contours and Figure 179 shows the equivalent plastic strain contours, both at an ALF of 1.05. Due to the large compressive force being transferred from the two diagonals into the gusset, the gusset appears to fail in buckling as the two diagonals move significantly out-of-plane. Yielding of the bottom of the gusset is noticeable since it bears directly on the pier.

Table 29 Key required gusset plate thicknesses from the FHWA Guide for P12-C-W-P.

Limit State tg.req (in)

Transfer of 5000 kips to the bearing 1.0 Edge slenderness check 0.86 Vertical plane shear 0.64 ( = 1) Whitmore buckling at compression diagonals 0.50 (K = 0)

Figure 177. Load-displacement plot for P12-C(1)-W-P.

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0.2

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1

1.2

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

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Vertical Deflection at L3 (inch)

186

Figure 178. von Mises stress contours for P12-C(1)-W-P at the limit load occurring at an ALF of 1.05 (DSF = 5).

Figure 179. Equivalent plastic strain contours for P12-C(1)-W-P at the limit load occurring at an ALF of 1.05 (DSF = 5).

187

4.7.3 P13-W-NP

4.7.3.1 P13-C-W-NP

This is a lower-chord joint at a pier location for a Warren truss bridge without vertical members. Figure 180 shows the configuration of the two panel system and the L2 connection under study. The geometry of this connection is shown in Figure 181(a), and the design forces are shown in Figure 181(b). The tapered truss system shown in the figures uses inclined chord members, where both the chords are in moderate compression with forces of 1750 and 2610 kips, respectively. The diagonal members are relatively steep but relatively minor chamfering is required for the compression diagonal which carries a moderate force of 1650 kips. No chamfering is required for the tension diagonal. The existence of moderate unbalanced forces between the chords produces a net shear on the horizontal plane. Based on the design forces and length of the connecting members, the following sections are used: box section B3 for the chord members, I-section I1 for the tension diagonal, and B2 for the compression diagonal.

Figure 180. Applied loads and boundary conditions of P13-C-W-NP.

188

18

15

33

30

12

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 181. Gusset plate geometry and design forces for P13-C-W-NP.

Table 30 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. A gusset plate thickness of 0.66 inches is required to satisfy the edge slenderness criteria, 0.44 inches to resist the shear along the right-hand diagonal using = 1.0, 0.39 inches for Whitmore buckling at the compression diagonal with K = 0, and 0.37 inches for block shear at the tension diagonal. A gusset plate thickness of 0.40 inches is selected along with s flange splice plate thickness of 0.669 inches and a web splice plate thickness of 0.394 inches.

A load-displacement plot for P13-C-W-NP is shown in Figure 182. The abscissa is the horizontal displacement at U1. Figure 182 shows that this configuration reaches the limit load at an ALF of 0.99. It should be noted that the 4% PEEQ is reached at the limit load as well.

Figures 183 and 184 show the von Mises stress and equivalent plastic strain contours at the PEEQ > 4 % strength limit. The equivalent plastic strain contours in Figure 184 show that the gusset plate is fully yielded along the left-hand side of the vertical with a sharp increase in the

189

plastic strains along the chamfered edge of the compression diagonal. In addition, it is apparent in Figure 184 that a line of peak plastic strain turns slightly to the right near the bottom of the vertical and then continues through the middle of the chord splice to the bottom of the chord. Both Figures 183 and 184 show significant out-of-plane movement of the compression diagonal at the strength limit of this joint. Figure 185 shows a plot of the normal, shear and von Mises stresses in the gusset plate along the left-hand edge of the vertical member and then straight through to the bottom of the chord. This plot shows that the gusset plate is yielded in shear along this entire plate until the top of the chord is reached. Below this point, the gusset plate picks up significant compression from the chord and the shear stress tapers off. The FEA results with a gusset thickness of 0.40 inches show that the FHWA Guide shear strength prediction with = 1.00 works well, but also that the FHWA Guide edge slenderness limit is highly conservative for this problem.

Table 30 Key required gusset plate thicknesses from the FHWA Guide for P13-C-W-NP.

Limit State tg.req (in)

Edge slenderness check 0.66 Shear along right-hand diagonal 0.44 ( = 1) Buckling at compression diagonal 0.39 (K = 0) Block shear at tension diagonal 0.37

190

Figure 182. Load-displacement plot for P13-C(0.4)-W-NP.

Figure 183. von Mises stress contours for P13-C(0.4)-W-NP at the limit load occurring at an ALF of 0.99 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load level in this

problem).

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0 0.5 1 1.5 2 2.5 3 3.5

Ap

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Horizontal Displacement at U1 (in)

PEEQ > 4% at ALF = 0.99

191

Figure 184. Equivalent plastic strain contours for P13-C(0.4)-W-NP at the limit load occurring at an ALF of 0.99 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load

level in this problem).

Figure 185. von Mises, normal, and shear stresses in the gusset plate along the vertical plane at the left-hand edge of the vertical member of P13-C(0.4)-W-NP at the limit load occurring at

an ALF of 0.99.

0

10

20

30

40

50

60

70

80

-60 -40 -20 0 20 40 60 80

Ver

tica

l Pla

ne

Dis

tan

ce f

rom

Bo

tto

m

(in

)

Stress (ksi)

Normal Stress

Shear Stress

Von Mises Stress

Dis

tan

ce f

rom

bo

tto

m o

f g

uss

et (

in)

von Mises Stress

Shear Stress

Normal Stress

192

4.7.3.2 P13-U-W-NP

The gusset plate geometry and the design forces in the members of P13-U-W-NP are shown in Figure 186. Due to the test configuration, the angle between the tension diagonal and bottom chord members is large (88 degrees), which makes the geometry of this joint similar to that of a Pratt joint with a compression diagonal. As a result, the tension diagonal connection is the same as in P13-C-W-NP. With the unchamfered compression diagonal, P13-U-W-NP has a 39.9-inch long free edge between the compression diagonal and bottom chord members where it is only 32.7 inches for P13-C-W-NP. In addition, it should be noted that there is a short area where two diagonals of P13-C-W-NP are close to each other (see Figure 181). In P13-U-W-NP, this area is eliminated by removing the chamfer from the compression diagonal.

66.3

57

19.8

15

12

(a) Gusset Plate geometry (units = inches)

(b) Design Forces

Figure 186. Gusset plate geometry and design forces for P13-U-W-NP.

193

Table 31 shows the key required gusset plate gusset plate thicknesses from the FHWA Guide for P13-U-W-NP. The edge slenderness check on the compression diagonal side requires a large thickness of 0.95 inches. The second governing check is the vertical shear check along the tension diagonal with = 1.0, which requires 0.58 inches. By using K = 1.2, the Whitmore buckling check at the compression diagonal side requires 0.40 inches. The block shear check at the tension diagonal and horizontal shear check with = 1.0 requires 0.38 and 0.33 inches respectively. A gusset plate thickness of 0.40 inches is selected for P13-U-W-NP, the same as for the simulation shown for P13-C-W-NP.

Figure 187 shows the load-displacement plot for P13-U(0.4)-W-NP. The load capacity of P13-U-W-NP is reached at an ALF of 0.85, whereas it is 0.99 for P13-C(0.4)-W-NP. Figures 188 and 189 show the von Mises stress and equivalent plastic strain contours for P13-U(0.4)-W-NP at an ALF of 0.85. The contours show that even with the unchamfered diagonal, the vertical plane between the two diagonals is yielded as well as the corner area between compression diagonal and the left chord member. As a result, the compression diagonal moves out-of-plane immediately after the joint reaches its limit load, indicating this connection failed via buckling of the gusset plate. Although it appears that the yielding concentrated along the vertical plane could allow for mobilization of significant vertical shear displacements, the experience in the lab experiments showed higher levels of strain are needed for mobilization to occur in shear, and therefore buckling typically precedes the shear failure. The edge slenderness check shown in Table 31 requires tg.req = 0.95 inches, which is more than twice the gusset plate thickness used in the test simulation. The analysis results of P13-U(0.4)-W-NP demonstrate that the FHWA Guide edge slenderness check is substantially conservative. However, the P13-U(0.4)-W-NP test does not reach an ALF of 1.0, and therefore, a gusset plate thickness larger than 0.4 inches is needed in this problem.

Table 31 Key required gusset plate thicknesses from the FHWA Guide for P13-U-W-NP.

Limit State tg.req (in)

P13-U-W-NP P13-C-W-NP

Edge slenderness check on compression diagonal side

0.80 0.66

Vertical plane shear along tension diagonal 0.58 ( = 1) 0.44 ( = 1)

Whitmore buckling at compression diagonal 0.54 (K = 1.2) 0.39 (K = 0)

Block shear tear out at tension diagonal 0.38 0.37

Horizontal plane shear check 0.33 ( = 1) 0.26 ( = 1)

194

Figure 187. Load-displacement plot for P13-U(0.4)-W-NP.

Figure 188. von Mises stress contours for P13-U(0.4)-W-NP at the limit load occurring at an ALF of 0.85 (DSF = 10).

0

0.1

0.2

0.3

0.4

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0.6

0.7

0.8

0.9

0 1 2 3 4 5

Ap

pli

ed L

oad

Fra

ctio

n, P

/Pre

fere

nce

Horizontal Displacement at U1 (in)

195

Figure 189. Equivalent plastic strain contours for P13-U(0.4)-W-NP at the limit load occurring at an ALF of 0.85 (DSF = 10).

4.7.4 P14-W-INF

4.7.4.1 P14-C-W-INF

This is a lower-chord joint at an inflection point location of a Warren truss bridge without vertical members. Figure 190 shows the configuration of the two panel system and the L2 connection under study. The geometry of this connection is shown in Figure 191(a), and the design forces are shown in Figure 191(b). The tapered truss system shown in the figures uses inclined chord members where both chords are carrying relatively low forces of 590 and 730 kips, respectively. As can be seen from the figures, the diagonal members are not significantly chamfered and one carries a moderate compression force 1400 kips while the other carries a moderate tensile force of 1190 kips. Based on the design force and length of the connecting members, the following sections are used: box section B1 for the chord members, I-section I1 for the tension diagonal, and B1 for the compression diagonal.

196

Figure 190. Applied loads and boundary conditions of P14-C-W-INF.

197

12

1212

36.121

(a) Gusset plate geometry (unit = inches)

(b) Design forces

Figure 191. Gusset plate geometry and design forces for P14-C-W-INF.

Table 32 shows the key required gusset plate thicknesses from the FHWA Guide for this problem. A gusset plate thickness of 0.66 inches is required for the edge slenderness check, 0.45 inches for block shear at the tension diagonal, 0.44 inches for vertical plane shear (perpendicular to the longitudinal axis of chords) with = 1, and 0.42 inches for shear along the plane just above the chords. A gusset plate thickness of 0.5 inches is selected along with a flange splice plate thickness of 0.315 inches, and a web splice plate thickness of 0.285 inches.

A load-displacement plot for P14-C-W-INF is shown in Figure 192. The abscissa is the horizontal displacement at U1. This connection reaches its limit load at an ALF of 1.22, at which the maximum PEEQ also exceeds 4 %.

Figures 193 and 194 show the von Mises stress and equivalent plastic strain contours at an ALF of 1.22. The equivalent plastic strain contours in Figure 194 indicate a significant concentration in plastic strain from the start of the chamfered edge of the compression diagonal on the left and over to the chamfered corner of the tension diagonal closest to the chord on the right. However, there are also two lines of larger plastic strain that extend vertically from the top of the chord and

198

up just past the inside chamfered edges of the two diagonal members. From both figures, out-of-plane movement of the compression diagonal is apparent. In addition, similar to other problems, it appears that the largest portion of the force transfer from the compression diagonal occurs near the end of the diagonal member. However, once this region yields and then reaches its strength limit due to the axial compression, the areas of the gusset on each side of the compression diagonal are unable to accept any substantial redistribution of the force transfer. The free edges of the gusset plate “bulge” out of plane largely due to the compatibility of deformations associated with the onset of significant movement of the compression diagonal in the direction of the compression. Based on the FEA analysis, the gusset free edge slenderness check is conservative in this problem. Figure 195 shows a plot of the normal, shear and von Mises stresses along the top of the chord at the strength limit in this problem. The magnitude of the von Mises stress confirms that the gusset plate is fully yielded along this entire plane. In addition, one can observe that there is significant interaction between the axial force from the compression diagonal and the shear stress. However, the interaction between the normal stress coming from the tension diagonal and the shear stress appears to be negligible. This may be due to stability effects on the compression diagonal side of the joint.

Table 32 Key required gusset plate thicknesses from the FHWA Guide for P14-C-W-INF.

Limit State tg.req (in)

Edge slenderness check 0.66 Block shear at tension diagonal 0.45 Vertical plane shear (perpendicular to the longitudinal axis of the chord)

0.44 ( = 1)

Horizontal plane shear along chord 0.42 ( = 1)

Whitmore buckling at compression diagonal 0.33 (K = 0)

199

Figure 192. Load-displacement plot for P14-C(0.5)-W-INF.

Figure 193. von Mises stress contours for P14-C(0.5)-W-INF at the limit load occurring at an ALF = 1.22 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load level in this

problem).

0

0.2

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0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ap

plie

d L

oa

d F

rac

tio

n, P

/Pre

fere

nc

e

Horizontal Displacement at U1 (in)

PEEQ > 4% at the peak load(ALF = 1.24)

PEEQ > 4% at the peak load, ALF = 1.22

200

Figure 194. Equivalent plastic strain contours for P14-C(0.5)-W-INF at the limit load occurring at an ALF of 1.22 (DSF = 5) (the PEEQ > 4 % strength limit also occurs at this load

level in this problem).

Figure 195. von Mises, normal, and shear stresses along the horizontal plane of P14-C(0.5)-W-INF at the limit load occurring at an ALF of 1.22 (the PEEQ > 4 % strength limit also occurs

at this load level in this problem).

-80

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-20

0

20

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60

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0 10 20 30 40 50 60

Str

ess

(ksi

)

Horizontal Plane Distance from Left (in)

Normal Stress

Shear Stress

Von Mises Stressvon Mises Stress

Shear Stress

Normal Stress

Distance from the left end of gusset (in)

201

4.7.4.2 P14-U-W-INF

The gusset plate geometry and the design forces in the connecting members of P14-U-W-INF are shown in Figure 196. Since the diagonal members of the original P14-C-W-INF are not severely chamfered, the geometry for P14-U-W-INF is similar to that of P14-C-W-INF. However, due to the chamfered diagonals, P14-C-W-INF is shorter along the bottom chord members.

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 196. Gusset plate geometry and design forces for P14-U-WV-INF.

The key required gusset plate thicknesses from the FHWA Guide for both P14-U-W-INF and P14-C-W-INF are shown in Table 33. As expected, the FHWA Guide calculations for P14-U-W-INF are similar to those for P14-C-W-INF. It should be noted that the vertical shear check is calculated for a plane along an axis perpendicular to the longitudinal axis of bottom chord members. For the test simulation, a gusset plate thickness of 0.5 inches is used for P14-U-W-INF.

12

36,1121

12

12

202

Figure 197 shows a load-displacement plot for P14-U(0.5)-W-INF. This plot shows that the limit load of this configuration occurs at an ALF of 1.14, which is slightly smaller than ALF = 1.22 obtained by P14-C(0.5)-W-INF. Figures 198 and 199 illustrate the von Mises stress and equivalent plastic strain contours for P14-U(0.5)-W-INF at the peak load using a displacement scale factor of 10.0. It can be seen that the horizontal plane along the bottom chord members has completely yielded at this stage. Also the triangular areas between two diagonals and between the compression diagonal and left chord members have been significantly plasticized. It appears that the out-of-plane movement of the compression diagonal and the buckling of free edges adjacent to the compression diagonal are secondary behavior following the substantial yielding of the gusset plate area around the compression diagonal. Based on the analysis results, it is again found that the edge slenderness check is significantly conservative. In addition, although there is a slight reduction in the maximum capacity due to the use of unchamfered members, it is observed that the overall behavior of P14-U(0.5)-W-INF is similar to that of P14-C(0.5)-W-INF.

Table 33 Key required gusset plate thicknesses from the FHWA Guide for P14-U-W-INF and P14-C-W-INF.

Limit State tg.req (in)

P14-U-W-INF P14-C-W-INF

Edge slenderness check on compression diagonal side

0.75 0.66

Vertical plane shear check (perpendicular to the longitudinal axis of chords) 0.43 ( = 1) 0.44 ( = 1)

Horizontal plane shear along chords 0.42 ( = 1) 0.42 ( = 1)

Block shear at tension diagonal 0.41 0.45

203

Figure 197. Load-displacement plot for P14-U(0.5)-W-INF.

Figure 198. von Mises stress contours for P14-U(0.5)-W-INF at the limit load occurring at an ALF of 1.14 (DSF = 10).

0

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1

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Ap

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/Pre

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Horizontal Displacement at U1 (in)

204

Figure 199. Equivalent plastic strain contours for P14-U(0.5)-W-INF at the limit load occurring at an ALF of 1.14 (DSF = 10).

4.8 CORNER JOINT PARAMETRIC TEST CONFIGURATIONS

This section summarizes the key characteristics and sample results for the parametric study test configurations involving corner joints. The corresponding results are summarized in Table 10 of Section 4.2.

4.8.1 P15-C-CJ

This is a corner joint of a truss bridge, either Warren or Pratt. Figure 200 shows the configuration of the single panel system and the U1 connection under study for this joint. The geometry of this connection is shown in Figure 201(a), and the design forces are shown in Figure 201(b). The inclination of the tension diagonal is steep and carries a moderate to large force of 2240 kips. The vertical member transfers a moderate to large compression force of 2000 kips and consequently a shear force of 2000 kips exists on the vertical plane of the gusset on the right-hand side of the vertical. Based on the design force and length of the connecting members, the following sections are used: box section B1 for the chord member, I-section I2 for the diagonal, and B3 for the vertical member.

205

Figure 200. Applied loads and boundary conditions of P15-C-CJ.

206

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 201. Gusset plate geometry and design forces for P15-C-CJ.

Table 34 shows the key required gusset thicknesses from the FHWA Guide for the given loading condition. Due to the existence of a long vertical free edge, a gusset plate thickness of 1.01 inches is required for the edge slenderness. A thickness of 0.67 inches is required for horizontal plane shear using = 1.0, 0.51 inches to resist vertical plane shear with = 1.0, and 0.52 inches for Whitmore section rupture at the tension diagonal. A gusset plate thickness of 0.50 inches is selected for the simulation study shown below. The gusset plate is the only plate connecting the members in this joint.

207

A load-displacement plot for P15-C-CJ is shown in Figure 202. The abscissa is the horizontal displacement at U2. This plot shows this configuration reaches its load capacity at an ALF of 1.15 by reaching the PEEQ > 4 % strength criterion. Figures 203 and 204 show the von Mises stress and equivalent plastic strain contours at an ALF of 1.15. The equivalent plastic strain contours in Figure 204 indicate that the vertical plane of the gusset plate along the right-hand edge of the vertical is yielded along the full gusset height, including through the splice to the horizontal chord member, with a substantial spike in the plastic strains in the area of the gusset between the chamfered end of the tension diagonal and the right-hand edge of the vertical. Figure 205 provides plots of the normal, shear and von Mises stresses along this plane. This plot indicates that the response along this plane is dominated by shear, although there are significant tensile stresses from the tension diagonal in the lower regions of the plate as well as significant compressive stresses associated with the force transfer to the top chord within the region of the top chord splice. Figure 206 plots the normal, shear and von Mises stresses along the horizontal plane of the gusset plate just below the top chord. This plot shows that the gusset is fully yielded within the length from the right-hand edge of the vertical to the free right-hand vertical edge of the gusset plate. Significant plastic interaction between normal tension stresses and the shear stresses is evident along this length. The above results indicate that the FHWA Guide shear yielding check is slightly conservative even with =1.00 since the guidance checks required a 0.51-inch thick plate, yet the joint reached an ALF of 1.15 using a 0.50 inch thick plate.

Table 34 Key required gusset plate thicknesses from the FHWA Guide for P15-C-CJ.

Limit State tg.req (in)

Edge slenderness check 1.01 Whitmore section rupture at tension diagonal 0.52 Vertical plane shear 0.51 ( = 1)

Whitmore buckling at the vertical 0.48 (K = 1.2)

Horizontal plane shear 0.38 ( = 1)

208

Figure 202. Load-displacement plot for P15-C(0.5)-CJ.

Figure 203. von Mises stress contours for P15-C(0.5)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.15 (DSF = 2).

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1.6

-7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0

Ap

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Fra

ctio

n,

P/P

refe

ren

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Horizontal Displacement at U2 (in)

PEEQ > 4% at ALF = 1.15

209

Figure 204. Equivalent plastic strain contours for P15-C(0.5)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.15 (DSF = 2).

Figure 205. von Mises, normal, and shear stresses in the gusset plate along the vertical plane at the right-hand edge of the vertical member in P15-C(0.5)-CJ at the PEEQ > 4 % strength

condition occurring at an ALF of 1.15.

210

Figure 206. von Mises, normal, and shear stresses in the gusset plate along the plane at the bottom of the horizontal chord in P15-C(0.5)-CJ at the PEEQ > 4 % strength limit occurring at

an ALF of 1.15.

4.8.2 P16-C-CJ

This is a corner joint of a truss bridge with a shallower framing angle of the diagonal than P15-C-CJ. Figure 207 shows the configuration of the single panel system and the U1 connection under study. The geometry of this connection is shown in Figure 208(a), and the design forces are shown in Figure 208(b). In this joint, the angle of inclination of the tension diagonal is moderate and the diagonal carries relatively large force of 2830 kips. The vertical member transfers a moderate to large compression force of 2000 kips, which in turn induces a 2000 kip shear on the vertical and horizontal planes just inside the vertical and horizontal members. Based on the design force and length of the connecting members, the following sections are used: box section B2 for the chord member, I-section I2 for the diagonal, and B2 for the vertical member.

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-40

-20

0

20

40

60

80

0 10 20 30 40 50

Str

ess

(ks

i)

Distance from the Left End of Gusset (in)

Normal Stress

Shear Stress

von Mises Stress

211

Figure 207. Applied loads and boundary conditions of P16-C-CJ.

212

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 208. Gusset plate geometry and design forces for P16-C-CJ.

Table 35 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading configuration. A gusset plate thickness of 0.85 inches is required to resist the horizontal plane shear using = 1.0, 0.68 inches for the edge slenderness check, 0.61 inches for the block shear check at the tension diagonal, and 0.85 inches for vertical plane shear using = 1.0. A gusset plate thickness of 0.85 inches is selected for the simulation shown below.

Both the horizontal and vertical shear planes in this problem are taken only over the partial lengths to the right and below the truss members. If the full horizontal and vertical widths of the plates are used, one obtains a required gusset plate thickness of 0.60 inches. However, initial simulation studies conducted with a gusset plate thickness of 0.60 inches showed that the joint was significantly under-strength. Section 5.7 provides further discussion of this behavior.

A load-displacement plot for P16-C-CJ is shown in Figure 209. The abscissa is the horizontal displacement at U2. It shows this configuration reaches its load capacity at an ALF of 1.16

213

controlled by the PEEQ > 4% strength criterion. Figures 210 and 211 show the von Mises stress and equivalent plastic strain contours at an ALF of 1.16. The grey contours in Figure 210 indicate areas that have exceeded the yield stress. Essentially the entire gusset plate is yielded with the exception of the regions that are directly attached to the members. The equivalent plastic strain contours in Figure 211 show the development of several predominant yield lines in the gusset plate near the 4 % PEEQ strength limit, i.e., yield lines along (1) the inside edges of the vertical and the horizontal chord, (2) each side of the tension diagonal, and (3) along an intermediate line within the gusset plate between each of the above lines. Figure 212 shows a plot of the normal, shear and von Mises stress on the vertical plane in the gusset plate at the right-hand edge of the vertical member at ALF = 1.16. The curves in this plot show substantial interaction between the shear and normal stresses along this plane. The gusset plate is fully yielded along this plane except for the lower region of the splice directly between the horizontal chord and the vertical member. It appears that the plate is strain hardened in tension near the bottom of this vertical plane. In addition, the plate is subjected to significant membrane compression in the vicinity of the bottom of the horizontal chord. This appears to be due to the bending of the gusset plate due to the bending deformation of the vertical member to the right as can be seen in Figures 210 and 211. Also, there is significant compression normal stress somewhat lower along this vertical plane, just below the tension diagonal, apparently for the same reason. Figure 213 shows a plot of the normal, shear and von Mises stresses along the horizontal plane just below the horizontal chord. Again, there is significant interaction between the normal and shear stresses along this plane. Similar to the response shown in Figure 212, the gusset plate is strain-hardened in tension at its right-hand edge. Also, a significant membrane compression occurs just above the tension diagonal. There is no second “spike” in the normal stresses within the horizontal length that cuts across the width of the vertical though. This appears to be because there is no splice at this position along the horizontal plane, whereas the vertical plane cuts through the splice of the horizontal chord to the vertical member.

Table 35 Key required gusset plate thicknesses from the FHWA Guide for P16-C-CJ.

Limit State tg.req (in)

Horizontal plane shear 0.85 ( = 1) Edge slenderness check 0.68 Block shear check at tension diagonal 0.61 Vertical plane shear 0.85 ( = 1)

214

Figure 209. Load-displacement plot for P16-C(0.85)-CJ.

Figure 210. von Mises stress contours for P16-C(0.85)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.16 (DSF = 2).

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Ap

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fere

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Horizontal Displacement at U2 (in)

PEEQ > 4% at ALF = 1.22

215

Figure 211. Equivalent plastic strain contours P16-C(0.85)-CJ at the PEEQ > 4 % strength limit occurring at an ALF of 1.16 (DSF = 2).

Figure 212. von Mises, normal, and shear stresses in the gusset plate along the vertical plane on the right-hand side of the vertical member in P16-C(0.85)-CJ at the PEEQ > 4 % strength

limit occurring at an ALF of 1.16.

0

10

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40

50

60

70

80

-80 -60 -40 -20 0 20 40 60 80

Dis

tan

ce

fro

m B

ott

om

of

Gu

sset

(in

)

Stress (ksi)

Normal Stress Shear Stress von Mises Stress

216

Figure 213. von Mises, normal, and shear stresses in the gusset plate along the plane at the bottom of the horizontal chord in P16-C(0.85)-CJ at the PEEQ > 4 % strength limit occurring at

an ALF of 1.16.

4.9 PARAMETRIC TEST CONFIGURATIONS WITH A POSITIVE ANGLE BETWEEN THE CHORD MEMBERS

This section summarizes the key characteristics and sample results for the parametric study test configurations involving a positive angle between the chord members. The corresponding results are summarized in Table 10 of Section 4.2.

4.9.1 P17-C-POS

This is an upper-chord joint of a truss bridge in which the chords form a positive angle (<180 degrees). Figure 214 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 215(a), and the design forces are shown in Figure 215(b). The left-hand diagonal member supports a large compression force of 2240 kips and the chord member supports a moderate compressive force of 1750 kips. The tension diagonal has slight chamfering at its end and transfers 1680 kips tensile force to the joint whereas the heavily chamfered vertical member transfers a small tensile force of 500 kips. Based on the design force and length of the connecting members, the following sections are used: box section B3 for the chord members, I-section I1 for the vertical member, and I2 for the tension diagonal.

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0

20

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60

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0 10 20 30 40 50 60 70

Str

ess

(ks

i)

Distance from the Left End of Gusset (in)

Normal Stress Shear Stress von Mises Stress

217

Figure 214. Applied loads and boundary conditions of P17-C-POS.

218

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 215. Gusset plate geometry and design forces for P17-C-POS.

Table 36 presents the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. A gusset plate thickness of 0.61 inches is required for the edge slenderness check and 0.60 inches is required to resist block shear tearout at the tension diagonal. A gusset thickness of 0.60 inches is selected along with a flange splice plate thickness of 0.715 inches and a web splice plate thickness of 0.449 inches.

A load-displacement plot for P17-C(0.6)-POS is shown in Figure 216. The abscissa is the horizontal displacement at U3. This configuration reaches its load capacity at an ALF of 1.47, controlled by the PEEQ > 4 % strength criterion. Figures 217 and 218 show the von Mises stress and equivalent plastic strain contours at this load level. The equivalent plastic strain contours in Figure 218 indicate a concentration in plastic strain along the short widths between the chamfered edges of the two tension members and the outside diagonal member. In addition, it can be seen that there is a fully yielded zone between the two tension members and the outside the compression chord member. Therefore, at the strength limit in this problem, it appears that a

1750

2240500

1680

219

plastic mechanism is developed in the gusset plate, limiting the development of further force into the tension members.

Table 36 Key required gusset plate thicknesses from the FHWA Guide for P17-C-POS.

Limit State tg.req (in)

Edge slenderness check 0.61

Block shear at tension diagonal 0.60

Figure 216. Load-displacement plot for P17-C(0.6)-POS.

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-7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0

Ap

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P/P

refe

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Horizontal Displacement at U3 (in)

PEEQ > 4% at ALF = 1.47

220

Figure 217. von Mises stress contours for P17-C(0.6)-POS at the PEEQ > 4 % strength limit occurring at an ALF of 1.47 (DSF = 5).

Figure 218. Equivalent plastic strain contours for P17-C(0.6)-POS at the PEEQ > 4 % strength limit occurring at an ALF of 1.47 (DSF = 5).

221

4.9.2 P18-C-POS

This is an upper-chord joint of a truss bridge in which the chords formed a positive angle (< 180 degrees). Figure 219 shows the configuration of the two panel system and the U2 connection under study. The geometry of this connection is shown in Figure 220(a), and the design forces are shown in Figure 220(b). The left-hand diagonal carries a large compression force of 2830 kips where the angle of inclination of this member is moderate; the chord member also carries a large compressive force of 3500 kips. A heavily chamfered tension diagonal supports a moderate tensile force of 2120 kips while the vertical member carries 500 kips tensile force. Based on the design force and length of the connecting members, the following sections are used: box section B4 for the chord members, I-section I1 for the vertical member, and I3 for the tension diagonal.

Figure 219. Applied loads and boundary conditions of P18-C-POS.

222

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 220. Gusset plate geometry and design forces for P18-C-POS.

Table 37 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. To resist block shear tearout at the tension diagonal, a gusset thickness of 0.55 inch is required, whereas a 0.48-inch gusset is required by the edge slenderness check. A gusset thickness of 0.6 inches, a flange splice plate of 0.849 inches, and a web splice plate of 0.416 inches are selected for the simulation shown below.

A load-displacement plot for P18-C(0.6)-POS is shown in Figure 221. The abscissa is the horizontal displacement at U3. The plot shows that this joint reaches its load capacity at an ALF of 1.30, controlled by the PEEQ >4 % strength criterion. Figures 222 and 223 show the von Mises stress and equivalent plastic strain contours at this load level. The equivalent plastic strain contours in Figure 223 show a spike in the plastic strains over the short width between the chamfered end of the larger tension member and the bottom of the horizontal chord member. This figure also shows that a fully-yielded zone between the two tension members and the

223

outside compression members is nearly developed at the strength limit in this problem. However, a fully-yielded path has not quite developed on the left-hand side of the smaller tension member at the strength limit. In addition, there is significant plasticity within the chord splice due to the high compressive forces that need to be transferred between the left-hand diagonal and the horizontal chord (since these two members are assumed not to be in contact at the joint). The strength limit in this problem is associated with the transfer of the combined forces from the vertical and the tension diagonal members to the outside compression members by the gusset plates. Although not seen in the figures, there also are minor plastic strains in the splice plates.

Table 37 Key required gusset plate thicknesses from the FHWA Guide for P18-C-POS.

Limit State tg.req (in)

Block shear at right-hand diagonal 0.55 Edge slenderness check 0.48

Figure 221. Load-displacement plot for P18-C(0.6)-POS.

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-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

Ap

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Fra

ctio

n, P

/Pre

fere

nce

Horizontal Displacement at U3 (in)

PEEQ > 4% at ALF = 1.30

224

Figure 222. von Mises stress contours for P18-C(0.6)-POS at the PEEQ > 4 % strength limit occurring at an ALF of 1.30 (DSF = 10).

Figure 223. Equivalent plastic strain contours for P18-C(0.6)-POS at the PEEQ > 4 % strength limit occurring at an ALF of 1.30 (DSF = 10).

225

4.10 PARAMETRIC TEST CONFIGURATIONS WITH A NEGATIVE ANGLE BETWEEN THE CHORD MEMBERS

This section summarizes the key characteristics and sample results for the parametric study test configurations involving a negative angle between the chord members. The corresponding results are summarized in Table 10 of Section 4.2.

4.10.1 P19-NEG

4.10.1.1 P19-C-CCS-NEG

This is a lower-chord joint of a truss bridge in which the chords form a negative angle (i.e., the subtended angle is larger than180 degrees). Figure 224 shows the configuration of the two panel system and the L2 connection under study. The geometry of this connection is shown in Figure 225(a), and the design forces are shown in Figure 225(b). The right chord is under a large compression force of 5600 kips and it has a relatively shallow angle of inclination. As can be seen from the figures, inclination of both the diagonals is shallow, consequently leading the left diagonal to be highly chamfered and also creating a long horizontal free edge of the gusset plate. The right diagonal and the vertical member carry smaller compressive forces of 780 and 500 kips respectively. The left diagonal supports a large compressive force of 3700 kips. Based on the design force and length of the connecting members, the following sections are used: box section B5 for the chord members, B1 for the vertical and right diagonal members, and B4 for the left diagonal.

Figure 224. Applied loads and boundary conditions of P19-C-CCS-NEG.

226

(a) Gusset plate geometry (units = inches)

(b) Design forces

Figure 225. Gusset plate geometry and design forces for P19-C-CCS-NEG.

Table 38 gives the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. Due to the presence of a long free edge, the edge slenderness check requires a gusset plate thickness of 1.35 inches. A plate thickness of 0.70 inches is required to resist Whitmore buckling at the horizontal L1L2 chord using K = 1.2; 0.66 inches is required if K = 0 is used for the U1L2 compression diagonal Whitmore buckling. A gusset thickness of 0.6 inches is selected along with flange splice plate thicknesses of 1.28 inches, and web splice plate thicknesses of 0.42 inches.

227

A load-displacement plot for P19-C-CCS(0.6)-NEG is shown in Figure 226. The abscissa is the horizontal displacement at L3. It shows that this configuration reaches its load capacity at an ALF of 0.86, controlled by the 4% PEEQ criterion. Figures 227 and 228 show the von Mises stress and equivalent plastic strain contours at this load level. The equivalent plastic strain contours in Figure 228 indicate that the largest concentration in plastic strain in this problem occurs in the gusset plate at the middle of the splice between the chord members. In this problem, the gusset has not plastified over the full Whitmore section at the chord. The primary plasticity at the splice occurs only within the width of the chord member. In addition, the full width of the Whitmore section at the end of the compression diagonal on the left side of the joint does not appear to be developed either. Figure 228 shows that the gusset plate is yielded around the end of this compression diagonal, particularly on the side that is closest to the horizontal chord member.

Table 38 Key required gusset plate thicknesses from the FHWA Guide for P19-C-CCS-NEG.

Limit State tg.req (in)

Edge slenderness check 1.35

Whitmore buckling at right-hand bottom chord

0.70 (K = 1.2)

Whitmore buckling at left-hand diagonal 0.66 (K = 0)

228

Figure 226. Load-displacement plot for P19-C-CCS(0.6)-NEG.

Figure 227. von Mises stress contours for P19-C-CCS(0.6)-NEG at the PEEQ > 4 % strength limit occurring at an ALF of 0.89 (DSF = 10).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

Ap

plie

d L

oad

Fra

ctio

n,

P/P

refe

ren

ce

Horizontal Displacement at L3 (in)

PEEQ > 4% at ALF = 0.89

229

Figure 228. Equivalent plastic strain contours for P19-C-CCS(0.6)-NEG at the PEEQ > 4 % strength limit occurring at an ALF of 0.89 (DSF = 10).

4.10.1.2 P19-C-MTB-NEG

Figure 229 illustrates a load-displacement for P19-C-MTB(0.6)-NEG, which studies the behavior of the joint when the chord members are mill-to-bear. The abscissa is the horizontal displacement at L3. The limit load of P19-C-MTB(0.6)-NEG is reached at an ALF of 1.01, whereas without the mill-to-bear condition, the chord splice fails at an ALF of 0.86 controlled by the PEEQ > 4 % strength limit for P19-C-CCS(0.6)-NEG. Figures 230 and 231 show the von Mises and equivalent plastic strain contours for P19-C-MTB(0.6)-NEG. With the mill-to-bear condition at the chord splice, the failure mode of the joint is diagonal buckling. The gusset plate is plastified significantly within the triangular area between the compression diagonal and the left-hand chord member. In addition, unlike P19-C-CCS(0.6)-NEG, there is no stress concentration observed at the chord splice except for a small region at the bottom of the splice. At a post-peak ALF of 0.93, significant out-of-plane movements of the compression diagonal and the adjacent free edges are observed (see Figures 232 and 233). At this point, the area around the end of the compression member is almost completely yielded.

230

Figure 229. Load-displacement plot for P19-C-MTB(0.6)-NEG.

Figure 230. von Mises stress contours for P19-C-MTB(0.6)-NEG at the limit load occurring at an ALF of 1.01 (DSF = 10).

0

0.2

0.4

0.6

0.8

1

1.2

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Ap

pli

ed L

oad

Fra

ctio

n, P

/Pre

fere

nce

Horizontal Displacement at L3 (in)

231

Figure 231. Equivalent plastic strain contours for P19-C-MTB(0.6)-NEG at the limit load occurring at an ALF of 1.01 (DSF = 10).

Figure 232. von Mises stress contours for P19-C-MTB(0.6)-NEG at a post-peak ALF of 0.93 (DSF = 10).

232

Figure 233. Equivalent plastic strain contours for P19-C-MTB(0.6)-NEG at a post-peak ALF of 0.93 (DSF = 10).

4.10.2 P20-NEG

4.10.2.1 P20-C-CCS-NEG

This is a lower-chord joint of a truss bridge in which the chords form a negative angle (i.e., the subtended angle is larger than180 degrees). Figure 234 shows the configuration of the two panel system and the L2 connection under study. The geometry of this connection is shown in Figure 235(a), and the design forces are shown in Figure 235(b). Inclined with a moderately steep angle, the right chord is carrying a large compression force of 4500 kips. The left-hand chord supports a moderate compressive force of 2450 kips. The U1L2 diagonal, which has a minor chamfer, is carrying 2330 kips of compressive force whereas the unchamfered vertical and right-hand diagonal each carry a small compressive force of 500 kips. Based on the design forces and the unbraced length of the connecting members, the following sections are used: box section B5 for the chord members, B1 for the vertical member and right diagonal, and B3 for the left diagonal.

233

Figure 234. Applied loads and boundary conditions of P20-C-CCS-NEG.

234

(a) Gusset plate geometry (unit = inches)

(b) Design forces

Figure 235. Gusset plate geometry and design forces for P20-C-CCS-NEG.

Table 39 shows the key required gusset plate thicknesses from the FHWA Guide for the given loading condition. The gusset plate thickness of 0.68 inches is required for Whitmore buckling of the U1L2 member using K = 0, 0.63 inches for Whitmore buckling for the L2L3 chord using K = 1.2, and 0.712 inches for the edge slenderness criterion. A gusset plate thickness of 0.6 inches is selected along with a flange splice plate thickness of 1.06 inches and a web splice plate thickness of 0.325 inches for the simulation shown below.

A load-displacement plot for P20-C-CCS(0.6)-NEG is shown in Figure 236. The abscissa is the horizontal displacement at L3. It shows this configuration reaches its load capacity at an ALF of 1.03 by reaching the PEEQ > 4% criterion. Figures 237 and 238 show the von Mises stress and equivalent plastic strain contours at this load level. The equivalent plastic strain contours in Figure 238 indicate that yielding in the gusset plate is particularly large in the short length

235

between the end fasteners of the chord members at the chord splice. However, the gusset plate is mildly yielded near the ends of all the web members of the truss and above the bottom chord members. Similar to the behavior for P19-C-CCS(0.6)-NEG, the plasticity only extends over the width of the chord at the splice, not the Whitmore area.

Table 39 Key required gusset plate thicknesses from the FHWA Guide for P20-C-CCS-NEG.

Limit State tg.req (in)

Edge slenderness check 0.71

Whitmore buckling at left-hand diagonal 0.68 (K = 0) Whitmore buckling at right-hand bottom chord 0.63 (K = 1.2)

Figure 236. Load-displacement plot for P20-C-CCS(0.6)-NEG.

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

Ap

pli

ed L

oad

Fra

ctio

n, P

/Pre

fere

nce

Horizontal Displacement at L3 (in)

PEEQ > 4% at ALF = 1.03

236

Figure 237. von Mises stress contours for P20-C-CCS(0.6)-NEG at the PEEQ > 4 % strength limit occurring at an ALF of 1.03 (DSF = 10).

Figure 238. Equivalent plastic strain contours for P20-C-CCS(0.6)-NEG at the PEEQ > 4 % strength limit occurring at an ALF of 1.03 (DSF = 10).

237

4.10.2.2 P20-C-MTB-NEG

Figure 239 shows a load-displacement plot for P20-C-MTB(0.6)-NEG, which examines the behavior of the P20-C joint with the chords fabricated mill-to-bear. The abscissa is the horizontal displacement at L3. The load capacity of this configuration is reached at an ALF of 1.22, whereas it is reached at ALF = 1.03 for P20-C-CCS(0.6)-NEG when the chord splice fails.

Figures 240 and 241 show the von Mises and equivalent plastic strain contours for P20-C-MTB(0.6)-NEG. With the mill-to-bear condition at the chord splice, the failure mode of the joint is diagonal buckling. P20-C-CCS(0.6)-NEG also shows partial yielding of this horizontal plane but this is minor compared to the yielding at the chord splice. The post-peak response contours at an ALF of 1.01 are shown in Figures 242 and 243, showing significant out-of-plane movements of the compression diagonal and vertical members.

It is important to note that in P20-C-MTB(0.6)-NEG, the compression diagonal is attached so close to the adjacent members that yielding is spread out across the bottom of the vertical as well. At the post-peak ALF of 1.01, both the compression diagonal and vertical members move out-of-plane. On the other hand, in P19-C-MTB(0.6)-NEG, there is a gap of 11.0 and 3.4 inches between the compression diagonal and vertical members and between the vertical and left chord members. Because the P19-C-MTB(0.6)-NEG joint has larger dimensions compared to the P20-C-MTB(0.6)-NEG joint on the compression side, the yielding of horizontal plane is concentrated only between the compression diagonal and the left chord at the peak load (see Figure 231). In the post-peak range of the response of P19-C-MTB(0.6)-NEG, the primary movement is the out-of-plane displacement of the compression diagonal (see Figure 233).

238

Figure 239. Load-displacement plot for P20-C-MTB(0.6)-NEG.

Figure 240. von Mises stress contours for P20-C-MTB(0.6)-NEG at the limit load occurring at an ALF of 1.22 (DSF = 10).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

Ap

pli

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oad

Fra

ctio

n, P

/Pre

fere

nce

Horizontal Displacement at L3 (in)

239

Figure 241. Equivalent plastic strain contours for P20-C-MTB(0.6)-NEG at the limit load occurring at an ALF of 1.22 (DSF = 10).

Figure 242. von Mises stress contours for P20-C-MTB(0.6)-NEG at a post-peak ALF of 1.01 (DSF = 10).

240

Figure 243. Equivalent plastic strain contours for P20-C-MTB(0.6)-NEG at a post-peak ALF of 1.01 (DSF = 10).

4.11 PARAMETRIC TEST CONFIGURATIONS WITH SHINGLE PLATES

This section summarizes the key characteristics and sample results for the parametric study test configurations in which shingle plates have been added to the base test geometry. The corresponding results are summarized in Table 11 of Section 4.2.

4.11.1 P3-C-SP(0.4:0.2)-WV-P

This joint has the same geometry as P3-C-WV-P but shingle plates are added to study the effect of shingle plates on the resistance of gusset plate connections. The geometry of this connection is shown in Figure 244. The design forces are the same as shown in Figure 94(b). Table 40 shows the key required gusset and shingle plate thicknesses from the FHWA Guide. Whitmore buckling at the vertical member requires a gusset thickness of 0.65 inches and a shingle plate thickness of 0.32 inches. Block shear tearout at the tension diagonal requires a gusset thickness of 0.465 inches and a shingle plate thickness of 0.08 inches. A gusset thickness of 0.4 inches and shingle plate thickness of 0.2 inches is selected for analysis; i.e., tg:tsh = 2:1.

241

43

1

43

50.

7

85

.7

Figure 244. Gusset and shingle plate geometries for P3-C-SP(0.4:0.2)-WV-NP (units = inches).

A load-displacement plot for this test is shown in Figure 245 where the plot of original P3-C-WV-P without a shingle plate is also shown. The original P3-C-WV-P without a shingle plate reaches its strength at an ALF of 0.96 whereas P3-C-SP(0.4:0.2)-WV-NP reaches its strength at an ALF of 0.89, controlled by the PEEQ > 4 % strength criterion in both problems. It should be noted that for the original P3-C-WV-P, a gusset thickness of 0.5 inches is used while a 0.4-inch gusset plus a 0.2-inch shingle plate are used for P3-C-SP(0.4:0.2)-WV-NP . The reason that a single gusset of 0.5 inches attains a comparable strength is because the connection is governed by buckling of the vertical member. As a result, the moment of inertia of the gusset and shingle plates significantly influences the strength of the connection. The moment of inertia of the gusset plate of 0.5 inches is comparable to a combined moment of inertia of two plates with thicknesses of 0.4 inches and 0.2 inches (0.01 in3 vs. 0.006 in3).

The equivalent plastic strain contours are shown in Figure 246 at an ALF of 0.89. The same contours are shown in Figure 247 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. The contours indicate that a significant portion of the shingle plate is plastified in a manner similar to the main gusset plate. This is an indication that substantial forces are being developed into the shingle plates from the connected members.

Figure 248 shows the distribution of vertical member forces between the gusset and shingle plates. The lighter solid line with rectangular marks represents the force on the gusset plate

242

determined from the FEA and the lighter dashed line represents the force on the gusset plate calculated based on the following assumptions: (1) a member force is distributed to the gusset and shingle plates based on the relative areas and (2) each shear plane of fasteners transfers an equal shear. The darker solid line with circular marks corresponds to the shingle plate force from the FEA and the darker dashed line represents the calculated force on the shingle plate using the above assumptions. For the vertical member of this joint, the calculated forces on the gusset and shingle plates are determined as follows:

1) The vertical member has 96 fasteners connecting the member to the gusset plate and 72 fasteners connecting the member to the shingle and gusset plates. When the vertical member is loaded with a 2000-kip force (at ALF = 0.4), each fastener on one side of the joint connecting the vertical transfers 1000 kips / (96 + 72) = 5.95 kips.

2) Where only the gusset plate is connected to the vertical, each fastener transfers a 5.95-kip force to the gusset plate.

3) Where the shingle and gusset plates are connected to the vertical, the portion of the 5.95-kip force that is transferred to the gusset plate is calculated as 5.95 kips × (0.4 / 0.6) = 3.97 kips. Therefore, the force transferred to the shingle plate is 1.98 kips.

4) The force transferred by the gusset plate is 96 × 5.95 kips + 72 × 3.97 kips = 857 kips. Similarly, the force transferred by the shingle plate is 72 × 1.98 kips = 143 kips. The total force transferred by the shingle and gusset plates on one side of the joint is 857 + 143 = 1000 kips.

Figure 248 indicates the calculated force distributions between the gusset and shingle plates represent the FEA results well. It appears that the error between the FEA results and the calculated force distributions increases after the PEEQ > 4 % strength limit is reached. Similarly, Figure 249 shows the distribution of right-hand diagonal forces between the gusset and shingle plates. The behavior shown by this plot is the same as that for Figure 248.

Table 40 Key required gusset plate and shingle plate thicknesses from the FHWA Guide for P3-C-SP(0.4:0.2)-WV-NP .

Limit State tg.req (in) tsh.req (in)

Whitmore buckling at vertical member 0.65 0.32

Block shear at tension diagonal 0.465 0.08

243

Figure 245. Load-displacement plot for P3-C-SP(0.4:0.2)-WV-NP.

Figure 246. Equivalent plastic strain contours for P3-C-SP(0.4:0.2)-WV-NP at the PEEQ > 4 % strength limit occurring at an ALF of 0.90 (DSF=10).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Ap

pli

ed L

oad

Fra

ctio

n, P

/Pre

fere

nce

Vertical Deflection at L2 (inch)

P3-C(0.5)-WV-P

P3-C-SP(0.5:0.25)-WV-P

PEEQ > 4% at ALF = 0.90

PEEQ > 4% at ALF = 0.96

P3-C-SP(0.4:0.2)-WV-P

244

Figure 247. Equivalent plastic strain contours for P3-C-SP(0.4:0.2)-WV-NP at the PEEQ > 4 % strength limit occurring at an ALF of 0.90 (DSF=10), shingle plate removed to show the

contours on the gusset plate.

Figure 248. Distribution of vertical member force between main gusset and shingle plate for P3-C-SP(0.4:0.2)-WV-NP .

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1 1.2

Fo

rce

(kip

)

Applied Load Fraction, P/Preference

Gusset Force (FEA)

Gussset Force (Predicted)

Shingle Force (FEA)

Shingle Force (Predicted)

PEEQ>4% at ALF=0.89PEEQ > 4 % at ALF = 0.90

245

Figure 249. Distribution of right-hand diagonal force between main gusset and shingle plate for P3-C-SP(0.4:0.2)-WV-NP.

4.11.2 P3-C-SP(0.5:0.25)-WV-P

The geometry of this connection is the same as shown in Figure 244. The design forces are the same as shown in Figure 94(b). Table 40 shows the key required gusset and shingle plate thicknesses from the FHWA Guide. A gusset thickness of 0.5 inches and shingle plate thickness of 0.25 inches is selected for analysis; i.e., tg:tsh = 2:1, same as P3-C-SP(0.4:0.2)-WV-P.

A load-displacement plot for the gusset assembly with a shingle plate is shown in Figure 250. The line with rectangular marks represents the load-displacement plot for P3-C-SP-(0.5:0.25)-WV-P. By increasing the thicknesses of the gusset and shingle plates, the load capacity of the joint is increased approximately by 16.4%.

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1 1.2

Fo

rce

(kip

)

Applied Load Fraction, P/Preference

Gusset Force (FEA)

Gussset Force (Predicted)

Shingle Force (FEA)

Shingle Force (Predicted)

PEEQ>4% at ALF=0.89PEEQ > 4 % at ALF = 0.90

246

Figure 250. Load-displacement plot for P3-C-SP(0.5:0.25)-WV-P.

4.11.3 P3-C-SP(0.3:0.3)-WV-P

The geometry of this connection is the same as shown in Figure 244. The design forces are the same as shown in Figure 94(b). Table 40 shows the key required gusset and shingle plate thicknesses from the FHWA Guide. To investigate the accuracy of the assumption that the fastener force is distributed into the gusset and shingle plates based on the relative areas, a different ratio of tg to tsh is used in this case using a gusset thickness of 0.3 inches and shingle plate thickness of 0.3 inches, i.e., tg:tsh = 1:1.

A load-displacement plot for P3-C-SP(0.3:0.3)-WV-NP is shown in Figure 251. The load-displacement plot of P3-C-SP(0.4:0.2)-WV-NP is also shown in this figure for comparison purposes. One can observe that the load capacity of P3-C-SP(0.4:0.2)-WV-NP (ALF = 0.90) is larger than P3-C-SP(0.3:0.3)-WV-NP (ALF = 0.79).

Figure 252 shows the distribution of vertical member forces between the gusset and shingle plates. The solid lines with marks represent the forces on the gusset and shingle plates determined from the FEA. The dashed lines represent the forces on the gusset and shingle plates calculated using the same procedures described in Section 4.11.1. Figure 252 indicates that when the joint remains elastic, the calculated force distributions predict the force distributions in the gusset and shingle plates determined from FEA well. When the gusset and shingle plates start yielding, the differences between the calculated forces and the forces determined from FEA become larger. Similarly, Figure 253 shows the distribution of tension diagonal forces between

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5

Ap

pli

ed

Lo

ad

Fra

cti

on

, P/P

refe

ren

ce

Vertical Deflection at L2 (inch)

P3-C(0.5)-WV-P

P3-C-SP(0.5:0.25)-WV-P

PEEQ > 4% at ALF = 1.17

PEEQ > 4% at ALF = 0.98

247

the gusset and shingle plates. The behavior in this figure is essentially the same as that in Figure 252.

Figure 251. Load-displacement plot for P3-C-SP(0.3:0.3)-WV-P.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Ap

plie

d L

oad

Fra

ctio

n, P

/Pre

fere

nc

e

Vertical Deflection at L2 (inch)

P3-C-SP(0.4:0.2)-WV-P

P3-C-SP(0.3:0.3)-WV-P

PEEQ > 4% at ALF = 0.90

PEEQ > 4% at ALF = 0.79

248

Figure 252. Distribution of the vertical member force between main gusset and shingle plate for P3-C-SP(0.3:0.3)-WV-P.

Figure 253. Distribution of the right-hand diagonal force between main gusset and shingle plate for P3-C-SP(0.3:0.3)-WV-P.

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1 1.2

Fo

rce

(kip

)

Applied Load Fraction, P/Preference

Gusset Force (FEA)

Gussset Force (Predicted)

Shingle Force (FEA)

Shingle Force (Predicted)

PEEQ>4% at ALF=0.78

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1 1.2

Fo

rce

(kip

)

Applied Load Fraction, P/Preference

Gusset Force (FEA)

Gussset Force (Predicted)

Shingle Force (FEA)

Shingle Force (Predicted)

PEEQ>4% at ALF=0.78

PEEQ > 4 % at ALF = 0.79

PEEQ > 4 % at ALF = 0.79

249

4.11.4 P5-C-SP(0.3:0.2)-WV-NP

This joint has the same geometry as P5-C-WV-NP but shingle plates are added to study the effect of shingle plates on the resistance of gusset plate connections. The geometry of this connection is shown in Figure 254. The design forces are the same as shown in Figure 106(b). 0 shows the key required gusset and shingle plate thicknesses from the FHWA Guide. Block shear tearout at the tension diagonal requires a gusset thickness of 0.32 inches and a shingle plate thickness of 0.13 inches. Whitmore buckling at the compression diagonal requires a gusset thickness of 0.32 inches and a shingle plate thickness of 0.10 inches. A gusset thickness of 0.3 inches and shingle plate thickness of 0.2 inches is selected for analysis; i.e., tg:tsh = 1.5:1.

30.5 39

18

15 12

12

45.3

55.6

Figure 254. Gusset and shingle plate geometry for P5-C-SP(0.3:0.2)-WV-NP (units = inches).

250

Table 41 Key required gusset plate and shingle plate thicknesses from the FHWA Guide for P5-C-SP(0.3:0.2)-WV-NP.

Limit State tg.req (in) tsh.req (in)

Edge slenderness check on compression diagonal side

0.73 0.43

Block shear at tension diagonal

0.32 0.13

Whitmore buckling at compression diagonal

0.32 (K = 0) 0.10 (K = 0)

Horizontal Plane shear 0.32 ( = 1) 0.08 ( = 1)

Left-hand chord tension 0.26 0.17

A load-displacement plot is shown in 0 where the plot of original P5-C-WV-NP without a shingle plate is also shown. The original P5-C-WV-NP without a shingle plate reaches its limit load at an ALF of 0.94 whereas P5-C-SP(0.3:0.2)-WV-NP reaches its limit load at an ALF of 0.95. It should be noted that for the original P5-C-WV-NP, a gusset thickness of 0.4 inches is used while a 0.3-inch gusset plate plus a 0.2-inch shingle plate are used for P5C-SP1-1.5:1. The reason that a single gusset of 0.4 inches attains a comparable strength is because the connection is governed by buckling of the compression diagonal. As a result, the moment of inertia the gusset and shingle plates significantly influences the strength of the connection. The moment of inertia of the gusset plate with 0.4 inches is comparable to a combined moment of inertia of two plates with thicknesses of 0.3 inches and 0.2 inches (0.0053 in3 vs. 0.0029 in3).

The equivalent plastic strain contours are shown in Figure 256 at an ALF of 0.95 and the same contours are shown in Figure 257 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. The contours indicate that a significant portion of the shingle plate is plastified in a manner similar to the main gusset plate. This is an indication that significant forces are being developed into the shingle plate from the connected members.

Figure 258 shows the distribution of compression diagonal forces between the gusset and shingle plates. The lighter solid line with rectangular marks represents the force on the gusset plate determined from the FEA and the lighter dashed line represents the force on the gusset plate calculated based on the assumptions discussed in Section 4.11.1. The darker solid line with circular marks corresponds to the shingle plate force from the FEA and the darker dashed line represents the calculated force on the shingle plate. The vertical line in the Figure 258 indicates the limit load of the connection at an ALF of 0.95. Figure 258 indicates the calculated force distributions between the gusset and shingle plates are essentially same as the distributions

251

determined from FEA while the plates remain elastic. The differences between the FEA results and the calculated force distributions increases when the joint is loaded more than an ALF of 0.7. Similarly, Figure 259 shows the distribution of tension diagonal forces between the gusset and shingle plates. The calculated force distributions are essentially the same as the FEA results for the entire analysis for the tension diagonal.

Figure 255. Load-displacement plot for P5-C-SP(0.3:0.2)-WV-NP.

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Vertical Deflection at L2 (inch)

P5-C(0.4)-WV-NP

P5-C-SP(0.3:0.2)-WV-NP

252

Figure 256. Equivalent plastic strain contours for P5-C-SP(0.3:0.2)-WV-NP at the limit load occurring at an ALF of 0.95 (DSF=5).

Figure 257. Equivalent plastic strain contours for P5-C-SP(0.3:0.2)-WV-NP at the limit load occurring at an ALF of 0.95 (DSF=5), shingle plate removed to show the contours on the gusset

plate.

253

Figure 258. Distribution of compression diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.2)-WV-NP

Figure 259. Distribution of tension diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.2)-WV-NP

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254

4.11.5 P5-C-SP(0.3:0.3)-WV-NP

The geometry of this connection is the same as shown in Figure 254. The design forces are the same as shown in Figure 106(b). 0 shows the key required gusset and shingle plate thicknesses from the FHWA Guide. To investigate the accuracy of the assumption that the fastener force is distributed into the gusset and shingle plates based on the relative areas, a different ratio of tg to tsh is used in this case using a gusset thickness of 0.3 inches and shingle plate thickness of 0.3 inches, i.e., tg:tsh = 1:1.

A load-displacement plot for P5-C-SP(0.3:0.3)-WV-NP is shown in Figure 260. The load-displacement plot of P5-C-SP(0.3:0.2)-WV-NP is also shown in this figure for comparison purposes. One can observe that the limit load of P5-C-SP(0.3:0.3)-WV-NP is larger than P5-C-SP(0.3:0.2)-WV-NP.

The von Mises contours are shown in Figure 261 at an ALF of 1.14 and the same contours are shown in Figure 262 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. Similar to P5-C-SP(0.3:0.2)-WV-NP, these contours indicate that a significant portion of the shingle plate is plastified in a manner similar to the main gusset plate. This is an indication that significant forces are being developed into the shingle plate from the connected members.

Figure 263 shows the distribution of the compression diagonal forces between the gusset and shingle plates. The solid lines with marks represent the forces on the gusset and shingle plates determined from the FEA. The dashed lines represent the forces on the gusset and shingle plates calculated using the same procedures described in Section 4.11.1. Figure 263 indicates that when the joint remains elastic, the calculated force distributions predict the force distributions determined from FEA well. When the gusset and shingle plates start yielding, the differences between the calculated forces and the forces determined from FEA become larger. Similarly, Figure 264 shows the distribution of tension diagonal forces between the gusset and shingle plates. The calculated force distributions are essentially the same as the FEA results for the entire analysis for the tension diagonal.

255

Figure 260. Load-displacement plot for P5-C-SP(0.3:0.2)-WV-NP and P5-C-SP(0.3:0.3)-WV-NP.

Figure 261. von Mises contours for P5-C-SP(0.3:0.3)-WV-NP at the limit load occurring at an ALF of 1.14 (DSF=5).

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Vertical Deflection at L2 (inch)

P5-C-SP(0.3:0.3)-WV-NP

P5-C-SP(0.3:0.2)-WV-NP

256

Figure 262. von Mises contours for P5-C-SP(0.3:0.3)-WV-NP at the limit load occurring at an ALF of 1.14 (DSF=5), shingle plate removed to show the contours on the gusset plate.

Figure 263. Distribution of compression diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.3)-WV-NP.

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257

Figure 264. Distribution of tension diagonal force between main gusset and shingle plate for P5-C-SP(0.3:0.3)-WV-NP.

4.11.6 P12-C-SP(0.5:0.5)-W-P

This joint has the same geometry as P12-C-W-P but shingle plates are added to study the effect of shingle plates on the resistance of gusset plate connections. The geometry of this connection is shown in Figure 265. The design forces are the same as shown in Figure 176(b). Table 42 shows the key required gusset and shingle plate thicknesses from the FHWA Guide. Transfer of 5000 kips to the bearing, based on a bearing width of 48 inches, requires a gusset thickness of 0.50 inches and a shingle plate thickness of 0.50 inches. Vertical plane shear requires a gusset thickness of 0.64 inches and a shingle plate thickness of 0.49 inches. Whitmore buckling at the compression diagonals requires a gusset thickness of 0.25 inches and a shingle plate thickness of 0.25 inches. A gusset thickness of 0.5 inches and shingle plate thickness of 0.5 inches is selected for analysis; i.e., tg:tsh = 1:1.

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rce

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)

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Gusset Force (FEA)Gussset Force (Predicted)Shingle Force (FEA)Shingle Force (Predicted)Peak at ALF=1.14

258

60

70.5

46.6

42.8

22.7

Figure 265. Gusset plate and shingle plate geometry for P12-C-SP(0.5:0.5)-W-P (units = inches).

Figure 266 shows a load-displacement plot of P12-C-SP(0.5:0.5)-W-P. The load-displacement plot of the original P12-C-W-P without shingle plates is also shown in Figure 266 for comparison purposes. One can observe in Figure 266 that the limit load of P12-C-SP(0.5:0.5)-W-P (ALF = 0.71) is significantly smaller than the original P12-C-W-P with a gusset plate of 1.0 inch (ALF = 1.05). The reason that a single gusset of 1.0 inch attains a substantially larger strength is because the connection is governed by buckling of the compression diagonals. As a result, the moment of inertia the gusset and shingle plates significantly influences the strength of the connection. The moment of inertia of the gusset plate with 0.1 inch is approximately four times larger than a combined moment of inertia of two plates with thicknesses of 0.5 inches (0.083 in3 vs. 0.010 in3).

Figure 267 shows the equivalent plastic strain contours of P12-C-SP(0.5:0.5)-W-P at the limit load occurring at an ALF of 0.71. The same contours are shown in Figure 268 with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. The contours indicate that substantial forces are developed into the shingle plate from the connected members.

259

Table 42 Key required gusset plate and shingle plate thicknesses for P12-C-SP(0.5:0.5)-W-P.

Limit State tg.req (in) tsh.req (in)

Transfer of 5000 kips to the bearing 0.50 0.50

Edge slenderness check 0.86 0.46

Vertical plane shear 0.64 ( = 1) 0.49 ( = 1)

Whitmore Buckling at compression diagonals

0.25 (K = 0) 0.25 (K = 0)

Figure 266. Load-displacement curves for P12-C(1.0)-W-P and P12-C-SP(0.5:0.5)-W-P.

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ion

, P/P

refe

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Vertical Deflection at L3 (inch)

P12-C(1.0)-W-P

P12-C-SP(0.5:0.5)-W-P

260

Figure 267. Equivalent plastic strain contours for P12-C-SP(0.5:0.5)-W-P at the limit load occurring at an ALF of 0.71 (DSF=5).

Figure 268. Equivalent plastic strain contours for P12-C-SP(0.5:0.5)-W-P at the limit load occurring at an ALF of 0.71 (DSF=5) ), shingle plate removed to show the contours on the

gusset plate.

261

4.12 TEST CONFIGURATIONS WITH EDGE STIFFENERS

Edge stiffeners are often used as one of retrofit measures to reinforce gusset plates around compression diagonals, or to provide lateral support to free edges that exceed specification edge slenderness criteria. In this section, to study the effect of the edge stiffeners on compression capacities of gusset plate joints, two experimental specimens, E4-U-490SS-WV and E4-U-490LS-WV and two parametric test configurations, P5-U-WV-NP and P14-C-W-INF are modified to include edge stiffeners on the free edges next to the compression diagonal. The results from these studies are summarized in Table 12 of Section 4.2.

4.12.1 E4-U-490SS(3/8)-SES-WV

Specimen E4-U-490SS(3/8)-SES-WV is a joint with unchamfered members, A490 high-strength bolts, a short standoff distance (1.0 inch) for the compression diagonal, a short connection length to the compression diagonal, and short edge stiffeners (SES) attached inside the gusset plates. Figure 269 shows the configuration of E4-U-490SS(3/8)-SES-WV. This is a typical configuration with edge stiffeners used in current practice. In this configuration, short L angles are attached to the inside of the gusset plates next to the compression diagonal. For E4-U-490SS(3/8)-SES-WV, L3 × 3 × ½ angles are used for the edge stiffeners. In the analyses of experimental specimens with edge stiffeners, the edge stiffeners are modeled using the same shell elements as the gusset plates. Figure 270 shows the reference member forces of E4-U-490SS(3/8)-SES-WV. It should be noted that the reference member forces shown in Figure 270 are not the same as the original E4-U-490SS-WV.

Figure 271 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A, which is located at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement curves is nearly elastic-perfectly plastic. When the original E4-U-490SS(3/8)-WV without stiffeners is analyzed with the reference forces shown in Figure 270, its limit load is reached at an ALF of 0.71 (not shown). The limit load of E4-U-490SS(3/8)-SES-WV is reached at an ALF of 0.72. It appears that the short edge stiffeners are not effective to increase the compression capacities of the joint.

Figure 272 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 273 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the lightly shaded area in the gusset at the end of the compression diagonal and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load is reached. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

262

Figure 269. Configuration of E4-U-490SS(3/8)-SES-WV with short angle stiffeners attached to the inside of the gusset plates.

Figure 270. Reference member forces of E4-U-490SS(3/8)-SES-WV.

263

Figure 271. Load-displacement plot for E4-U-490SS(3/8)-SES-WV.

Figure 272. von Mises stress contours for E4-U-490SS(3/8)-SES-WV at the limit load occurring at an ALF of 0.72 (DSF = 5).

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Horizontal Displacement

Out-Of-Plane Displacement

264

Figure 273. Equivalent plastic strain contours for E4-U-490SS(3/8)-SES-WV at the limit load occurring at an ALF of 0.72 (DSF = 5).

4.12.2 E4-U-490SS(3/8)-EES-WV

Specimen E4-U-490SS(3/8)-EES-WV is the same test as E4-U-490SS(3/8)-SES-WV except extended edge stiffeners (EES) are attached outside the gusset plates. The meaning of the term “extended edge stiffeners” is that the edge stiffeners are positively connected to the truss members so that the stiffeners are engaged in resisting the overall sidesway buckling of the gusset plates via frame action of the stiffeners between the truss members. The same angle section (L3 × 3 × ½) is used for the edge stiffeners as in E4-U-490SS(3/8)-SES-WV. The reference member forces used for this joint are shown in Figure 270.

Figure 274 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. The load capacity of E4-U-490SS(3/8)-EES-WV is reached at an ALF of 0.84 by reaching the PEEQ > 4 % strength criterion. The load capacity of E4-U-490SS(3/8)-EES-WV is increased by 18 % compared to the original E4-U-490SS-WV without stiffeners analyzed with the same reference forces. This illustrates that the edge stiffeners typically need to be extended and connected to the truss members to increase the compression capacities of the joint. The increase in the strength appears to be due mainly to the restraint that the stiffeners provide to the out-of-plane movement

265

of the compression diagonal. In this joint, it appears that these stiffeners also provide some additional resistance to shear along the horizontal plane at the top of the bottom chord.

Figure 275 shows contours of the von Mises stresses at the PEEQ > 4 % strength limit of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 276 shows the equivalent plastic strain contours at the mid-thickness of the plate at the PEEQ > 4 % strength limit. Based on the more lightly shaded area in the gusset at the end of both diagonals and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the PEEQ > 4 % strength limit is reached. One can also observe that a substantial area of the edge stiffener connecting the chord and the compression diagonal is also yielded when the limit load is reached.

Figure 274. Load-displacement plot for E4-U-490SS(3/8)-EES-WV.

266

Figure 275. von Mises stress contours for E4-U-490SS(3/8)-EES-WV at the PEEQ > 4 % strength limit occurring at an ALF of 0.84 (DSF = 5).

Figure 276. Equivalent plastic strain contours for E4-U-490SS(3/8)-EES-WV at the PEEQ > 4 % strength limit occurring at an ALF of 0.84 (DSF = 5).

267

4.12.3 E5-U-490LS(3/8)-SES-WV

Specimen E5-U-490LS(3/8)-SES-WV is a joint with unchamfered members, A490 high-strength bolts, a long standoff distance (4.5 inch) for the compression diagonal, a short connection length to the compression diagonal, and short edge stiffeners (SES) attached inside the gusset plates. The configuration of the edge stiffeners for this joint is similar to E4-U-490SS(3/8)-SES-WV shown in Figure 269. For E5-U-490LS(3/8)-SES-WV, L3 × 3 × ½ angles are used for the edge stiffeners. The reference member forces of this joint is the same as E4-U-490SS(3/8)-SES-WV shown in Figure 270.

Figure 277 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A at the mid-width of the end of the compression diagonal. One can observe that the shape of the load-displacement curves is nearly elastic-perfectly plastic. When the original E5-U-490LS(3/8)-WV without stiffeners is analyzed with the reference forces shown in Figure 270, its limit load is reached at an ALF of 0.58 (not shown). The limit load of E5-U-490LS(3/8)-SES-WV is reached at an ALF of 0.60. It appears that the short edge stiffeners increase the compression capacities of the joint, but only very slightly.

Figure 278 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 279 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the compression and tension diagonals, one can observe that a little area of the gusset is yielding when the limit load is reached. The predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

268

Figure 277. Load-displacement plot for E5-U-490LS(3/8)-SES-WV.

Figure 278. von Mises stress contours for E5-U-490LS(3/8)-SES-WV at the limit load occurring at an ALF of 0.60 (DSF = 5).

269

Figure 279. Equivalent plastic strain contours for E5-U-490LS(3/8)-SES-WV at the limit load occurring at an ALF of 0.60 (DSF = 5).

4.12.4 E5-U-490LS(3/8)-EES-WV

Specimen E5-U-490LS(3/8)-EES-WV is the same as the above joint except with extended edge stiffeners (EES) attached outside the gusset plates. The configuration of the edge stiffeners for this joint is shown in Figure 280. For E5-U-490LS(3/8)-EES-WV, the same angle section (L3 × 3 × ½) is used for the edge stiffeners as in E5-U-490LS(3/8)-SES-WV. The reference member forces of this joint are the same as in E4-U-490SS(3/8)-SES-WV shown in Figure 270.

Figure 281 shows the overall applied load fraction (ALF) versus the out-of-plane and horizontal in-plane displacement of point A, which is located at the mid-width of the end of the compression diagonal. One can observe that the out-of-plane displacement is significantly larger than the horizontal in-plane displacement. The load capacity of E5-U-490LS(3/8)-EES-WV is reached at an ALF of 0.81 based on the limit load criterion. It should be noted that the PEEQ > 4 % strength limit is also reached at an ALF of 0.81. The load capacity of E5-U-490LS(3/8)-EES-WV is increased by 40 % compared to Specimen E5-U-490LS(3/8)-WV analyzed with the same member forces. Compared to Specimen E5-U-490LS(3/8)-SES-WV, the load capacity is increased by 35 % by using extended edge stiffeners. This indicates that the edge stiffeners should be extended and connected to the truss members to increase the compression capacities of the joint efficiently as observed previously in Sections 4.12.1and 4.12.2. Again, the increase in

270

the resistance obtained by adding the edge angles is primarily due to the frame action achieved by attaching these components to the main truss members.

Figure 282 shows contours of the von Mises stresses at the limit load of the subassembly for the case of a 3/8 inch thick gusset plate. Figure 283 shows the equivalent plastic strain contours at the mid-thickness of the plate at the specimen limit load. Based on the more lightly shaded area in the gusset at the end of the compression and tension diagonals and along the width of the gusset just above the chord, one can observe that a substantial area of the gusset is yielding along the “full shear plane” just above the chord when the limit load and the PEEQ > 4 % strength limit are reached. However, the predominant mode of failure is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

Figure 280. Long stiffening angles applied externally on the horizontal and vertical edges of E5-U-490LS(3/8)-EES-WV.

271

Figure 281. Load-displacement plot for E5-U-490LS(3/8)-EES-WV.

272

Figure 282. von Mises stress contours for E5-U-490LS(3/8)-EES-WV at limit load and the PEEQ > 4 % strength limit occurring at an ALF of 0.81 (DSF = 5).

Figure 283. Equivalent plastic strain contours for E5-U-490LS(3/8)-EES-WV at the limit load and the PEEQ > 4 % strength limit occurring at an ALF of 0.81 (DSF = 5).

273

4.12.5 P5-U-EES-WV-NP

Joint P5-U-EES-WV-NP has the same geometry as P5-U-WV-NP except that edge stiffeners are added to P5-U-EES-WV-NP. Figure 284 shows the edge stiffener detail for P5-U-EES-WV-NP. It is shown in Sections 4.12.1 to 4.12.4, that the extended edge stiffeners are required to increase the compression capacities of a joint. Therefore, for P5-U-EES-WV-NP, the edge stiffener is connected from the top of the left chord member, where the stiffener is connected to the chord by four fasteners, through the first fastener of the compression diagonal on the left. It should also be noted that the same edge stiffeners are used on both gusset plates of this joint. Because the free edge between the compression diagonal and the vertical member is relatively short, edge stiffeners are placed only on the vertical free edges of both gusset plates.

To determine a required stiffness of the edge stiffeners necessary to achieve different strength levels, the size of angles is varied from L1 × 1 × 1/8 to L4 × 4 × ⅝. The edge stiffeners are modeled using 3D linear-order beam elements. It should be noted that these beam elements are attached to the gusset plates at the fastener locations only. By doing this, the physical attachment between edge stiffeners and the gusset plates is better represented. Between the physical attachments, the stiffener is modeled with four beam elements.

2.111.89

2.011.99

61.4

6

Figure 284. Edge stiffener geometry for P5-U-EES-WV-NP (units = inches).

The design checks for P5-U-EES-WV-NP are the same as P5-U-WV-NP as they would not account for the presence of the edge stiffeners (see Table 20). First a gusset plate thickness of 0.4 inches, which is same as P5-U-WV-NP, is used in the finite element analysis. Then gusset plate thicknesses of 0.3 and 0.5 inches are used to study the effect of edge stiffeners with gusset plates of different thicknesses.

274

Figure 285 illustrates the load-displacement plot of P5-U-EES-WV-NP with edge stiffeners of L3 × 3 × ½. With this stiffener size, the limit load of P5-U-EES-WV-NP is reached at an ALF of 0.97, which is an increase of 24 % compared to P5-U-WV-NP, which has a limit load at an ALF of 0.78. Figures 286 and 287 show the von Mises and equivalent plastic strain contours for P5-U-EES-WV-NP with L3 × 3 × ½ stiffeners. Unfortunately, Abaqus 6.10, which is used for the analysis, does not have the capability of rendering beam elements with contours. Therefore, the contours for the stiffeners are not shown in either of these figures. It can be observed that most of the gusset plate on the compression diagonal side is yielded except the area near the vertical free edge where the stiffener is attached. In addition, the tension chord splice as well as the triangular area at the end of the tension diagonal is yielded. After this joint reaches its limit load, the compression diagonal moves in the out-of-plane direction as shown in Figures 288 and 289.

Figure 285. Load-displacement plot for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ edge stiffener.

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P5-U-EES(0.4)-WV-NP

P5-U(0.4)-WV-NP

275

Figure 286. von Mises stress contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ edge stiffener at the limit load occurring at an ALF of 0.97 (DSF = 5).

Figure 287. Equivalent plastic strain contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ edge stiffener at the limit load occurring at an ALF of 0.97 (DSF = 5).

276

Figure 288. von Mises stress contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ stiffener at a post-peak ALF of 0.73 (DSF = 5).

Figure 289. Equivalent plastic strain contours for P5-U-EES(0.4)-WV-NP with an L3 x 3 x ½ edge stiffener at a post-peak ALF of 0.73 (DSF = 5).

277

Figure 290 demonstrates the relationship between a percentage increase of the limit load and the ratio of the moment of inertia of an edge stiffener to the moment of inertia of a unit width of the gusset plate, Istiffener/Ig. The values of Istiffener and Ig are calculated as shown in Figure 291. The moment of inertia of the edge stiffeners is calculated about the surface of the gusset plates where the angles are attached. The angle legs are idealized as rectangular plates, neglecting the fillet between the legs and the rounded edges of the legs present in physical angles. The axes that the moments of inertia are calculated about are indicated by the dashed lines in Figure 291. Several different gusset plate thicknesses (tg = 0.5, 0.4, and 0.3 inches) are considered in Figure 290. The results for the different thicknesses are denoted by the different symbols, as shown in the legend of the plot.

Figure 290. Percentage increase in joint capacity vs. relative stiffness of edge stiffener to gusset plates (Istiffener/Ig) for P5-U-EES-WV-NP.

It can be observed in Figure 290 that a small value of Istiffener /Ig can increase the limit load of the joint significantly. For example, L1½×1½×¼ angles increase the limit load for the joint with tg = 0.4 inches by 12 %. It is also important to note that the limit load of a joint does not increase significantly with the increase of the angle sizes after a certain value of Istiffener/Ig has been established. For P5-U-EES-WV-NP, the increase in the limit load is relatively small with further increases in the stiffener moment of inertia Is when Istiffener/Ig is roughly greater than 500 regardless of what tg is used.

0%

5%

10%

15%

20%

25%

30%

35%

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Per

cen

tag

e In

crea

se in

Pea

k L

oad

Istiffener/Igp

Case 5B, tg = 0.3"

Case 5B, tg = 0.4"

Case5B, tg = 0.5"

Istiffener / Ig

278

(a) Calculation of Ig. (b) Calculation of Istiffener.

Figure 291. Calculation of Istiffener and Ig.

The percentage increase in the limit load is somewhat different with different gusset plate thicknesses. After close inspection, it is apparent that the maximum percentage increase in the limit load is controlled by the ratio of the fully plastic capacity of the gusset plates to the capacity of the gusset plates without stiffening. In other words, once the gusset plates are sufficiently stiffened by making the extended edge angles large enough, they are able to develop their fully-plastic capacities for the governing diagonal buckling limit state.

4.12.6 P14-C(0.5)-EES-W-INF

Joint P14-C(0.5)-EES-W-INF has the same geometry as P14-C-(0.5)-W-INF except that extended edge stiffeners are added. The same edge stiffeners are used on both of the gusset plates of this joint. Similar to P5-U-EES-WV-NP, the edge stiffeners are extended and attached to the adjoining members. Figure 292 shows the edge stiffener detail: (1) from the left diagonal to the bottom of the left chord member and (2) from the left diagonal to the right diagonal. To determine a required stiffness of the angles, the size of angles is varied from L2 × 2 × ¼ to L4 × 4 × ½.

Figure 293 illustrates the load-displacement plot of P14-C(0.5)-EES-W-INF with L3½ × 3½ × ⅜ edge stiffeners. The maximum capacity is reached at an ALF of 1.32 and is governed by the PEEQ > 4 % strength criterion. This is an 8.2% increase from the maximum capacity of P14C(0.5)-INF, which is reached at an ALF of 1.22 and is governed by the limit load criterion. Figures 294 and 295 show the von Mises and equivalent plastic strain contours for P14-C(0.5)-EES-W-INF with the L3½ × 3½ × ⅜ stiffeners. As mentioned in the previous sections, the

3 3( )

3 3stiffener

b t t b tI

3(1in)

12g

g

tI

279

stiffeners are not shown in the contour plots since Abaqus cannot render beam elements. It can be observed that the majority of the gusset plate areas are yielded at the PEEQ > 4 % strength limit. This is especially the case for the shear plane just above the chord as well as the bottom half of the area between the two diagonals, which show significantly large PEEQ values from 1.2% to 4%. The overall von Mises stress and equivalent plastic strain contours of P14-C(0.5)-EES-W-INF are similar to those of P14-C(0.5)-W-INF. However, P14-C(0.5)-W-INF shows significant out-of-plane movements of the compression diagonal at an ALF of 1.22. Conversely, P14-C(0.5)-EES-W-INF shows relatively small out-of-plane movements at an ALF of 1.32.

39.4

40.9

2.3

2.4

Figure 292. Edge stiffener geometry for P14-C-EES-W-INF (units = inches).

Figure 296 shows the relationship between the percentage increase the maximum load capacity and the, Istiffener/Ig. The values of Istiffener and Ig are calculated as discussed in Section 4.5.5.2. By comparing Figures 290 and 296, one can observe that the percentage increase in the maximum capacity of P14-C-EES-W-INF is smaller than P5-U-EES-WV-NP. This smaller increase is due to:

1) The dominant failure mode of P5-U-WV-NP is buckling of the compression diagonal. Therefore, the addition of the edge stiffeners next to the compression diagonal (P5-U-EES-WV-NP) has a significant impact on the joint’s load capacity.

2) However, the dominant failure mode in P14-C-W-INF involves substantial yielding along the full shear plane just above the chord. It appears that the addition of edge stiffeners is more effective at providing additional out-of-plane buckling strength rather than increasing the shear yielding resistance. Therefore, the effect of the edge stiffeners is smaller in P14-C(0.5)-EES-W-INF compared to P5-U-EES(0.4)-WV-NP.

280

3) In addition, there is substantially more yielding associated with the diagonal compression resistance in P14-C-W-INF compared to P5-U-WV-NP. The maximum diagonal compression resistance achieved due to edge stiffening appears to be associated with essentially a fully-plastic diagonal compression strength. The diagonal compression response is already close to a fully-plastic condition at the strength limit in P14-C-W-INF. Therefore, one would not expect to develop significantly more strength by edge stiffening in this case.

Figure 296 also shows that the maximum capacity of the joint does not increase significantly with an increase of the stiffener sizes after a certain value of Istiffener/Ig has been established. The maximum capacity of P14-C(0.5)-EES-W-INF is relatively constant when Istiffener/Ig is larger than roughly 500, as shown in Figure 296.

Figure 293. Load-displacement curves for P14-C(0.5)-W-INF and P14-C-EES(0.5)-W-INF.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Ap

pli

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Fra

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/Pre

fere

nce

Horizontal Displacements at U1 (in)

P14-C-EES(0.5)-W-INF

P14-C(0.4)-W-INF

PEEQ > 4% at 

ALF = 1.32

P14‐C(0.5)‐W‐INF 

281

Figure 294. von Mises stress contours for P14-C-EES(0.5)-W-INF with L3½ x 3½ x ⅜ stiffeners at the PEEQ > 4 % strength limit occurring at an ALF of 1.32 (DSF = 5).

Figure 295. Equivalent plastic strain contours for P14-C-EES(0.5)-W-INF with L3½ x 3½ x ⅜ stiffeners at the PEEQ > 4 % strength limit occurring at an ALF of 1.32 (DSF = 5).

282

Figure 296. Percentage increase in joint capacity vs. relative stiffness of edge stiffeners and gusset plate (Istiffener/Ig) for P14-C-EES(0.5)-W-INF.

4.13 PARAMETRIC TEST CONFIGURATIONS WITH CORRODED GUSSET PLATES

Corrosion on gusset plates is common in many older steel truss bridges. Figures 297 and 298 show examples of corrosion in gusset plates from two different bridges. In both cases there is significant section loss of different amounts in different parts of the gusset plate, including some locations with holes. Corrosion is usually found in locations where water and debris can accumulate and/or pond. There are endless scenarios involving the effect of corrosion on the strength of gusset plates given all the potential locations and different magnitudes of corrosion. In this research, two parametric test configurations, P8-C-WV-INF and P14-U-W-INF, are selected and modified to include corrosion in the gusset plates. The results from these tests are summarized in Table 14 of Section 4.2.

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

0 200 400 600 800 1000 1200

Per

cen

tag

e In

crea

se in

Ap

plie

d L

oad

R

atio

at

PE

EQ

> 4

%

Istiffener/IgpIstiffener / Ig

283

Figure 297. Corrosion of a gusset plate shown with the percentage loss in the gusset plate thickness highlighted at several locations (courtesy of Mn/DOT).

284

Figure 298. Corrosion on gusset plate just above the chord including holes just below the end of the diagonal (courtesy of Illinois DOT).

4.13.1 P8-C-C1-WV-INF and P8-C-C2-WV-INF

Joints P8-C-C1-WV-INF and P8-C-C2-WV-INF have the same geometry as P8-C(0.5)-WV-INF except that corrosion is included in P8-C-C1-WV-INF and P8-C-C2-WV-INF. The loading and overall geometry of the connection is the same as that shown in Figure 144. Two different corrosion patterns C1 and C2 are used. A schematic of P8-C-C1-WV-INF is shown in Figure 299. This pattern has partial and complete section losses concentrated just above the chord members. The same corrosion pattern is applied to the gusset plates on both sides of the joint. Pattern C2 only has holes at the locations shown as having holes in Figure 299, but has no section loss in the other areas.

285

Figure 299. Corroded gusset plate geometry for P8-C-C1-WV-INF (units = inches).

Load-displacement plots for P8-C-C1-WV-INF and P8-C-C2-WV-INF are shown in Figure 300. A load-displacement plot of the original P8-C(0.5)-WV-INF is also shown for comparison purposes. It should be noted that for the corroded joints the maximum equivalent plastic strain value exceeds 4 % soon after the loadings are applied because a stress concentration occurs near the corroded region, especially near the holes. Therefore, the maximum capacity of the corroded joints is taken at a load level when the PEEQ values along the entire corroded region (i.e., the area just above the chord for P8-C-C1-WV-INF and P8-C-C2-WV-IN) have exceeded the 4 % PEEQ limit. The load capacities of P8-C-C1-WV-INF and P8-C-C2-WV-INF are reached at an ALF of 0.61 and 0.76 respectively, which are 37 % and 22 % smaller than the original P8-C(0.5)-WV-INF.

Figures 301 and 302 show equivalent plastic strain contours for P8-C-C1-WV-INF at an ALF of 0.61 and for P8-C-C2-WV-INF at an ALF of 0.76, both at the load level when the PEEQ > 4 % limit is reached along the entire corroded region as discussed above. One can observe from the contours that the “full shear plane” just above the chord and the area at the end of the tension diagonal is yielding when the PEEQ > 4 % limit is reached along the entire corroded region.

18

43

0.5

0.87

50.

5

15

1.5Holetg = 0.5 in.tcorroded-1 = 0.35 in.tcorroded-2 = 0.25 in.

286

Figure 300. Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-C2-WV-INF.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5

Ap

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Fra

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/Pre

fere

nce

Horizontal Displacement at U1 (inch)

P8-C(0.5)-WV-INF

P8-C-C1-WV-INF

P8-C-C2-WV-INF

PEEQ > 4% at 

ALF = 0.97

PEEQ > 4% at 

ALF = 0.76

PEEQ > 4% at 

ALF = 0.61

287

Figure 301. Equivalent plastic strain contours for P8-C-C1-WV-INF at the PEEQ > 4 % limit occurring at an ALF of 0.61 (DSF = 1).

Figure 302. Equivalent plastic strain contours for P8-C-C2-WV-INF at the PEEQ > 4 % limit occurring an ALF of 0.76 (DSF = 1).

288

4.13.2 P8-C-COS-WV-INF

Through discussions with various state DOTs, it was identified that in many cases, one of the two gusset plates is significantly corroded while the other may have little damage. Therefore, it was decided to analyze P8-C(0.5)-WV-INF with a corroded gusset plate with the corrosion pattern C1 in one plate and with an uncorroded gusset plate on the other side. This joint is P8-C-COS-WV-INF.

A load-displacement plot for P8-C-COS-WV-INF is shown in Figure 303. The maximum capacity of P8-C-COS-WV-INF is reached at an ALF of 0.83 reaching the PEEQ > 4 % strength limit along the corroded region. The maximum capacity of P8-C-COS-WV-INF is 14 % smaller than the original P8-C(0.5)-WV-INF.

The equivalent plastic strain contours for the corroded gusset plate of P8-C-COS-WV-INF are shown in Figure 304 at an ALF of 0.83. Figure 305 shows the equivalent plastic strain contours for the uncorroded gusset plate. Based on the lighter area just above the chord and at the end of tension diagonal, one can observe that a substantial area of the gusset plate is yielding along the “full shear plane” just above the chord and at the end of the tension diagonal.

Figure 303. Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-COS-WV-INF.

0

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1

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1.4

0 1 2 3 4 5

Ap

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/Pre

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Horizontal Displacement at U1 (inch)

P8-C(0.5)-WV-INF

P8-C-C1-WV-INF

P8-C-COS-WV-INF

P8‐C(0.5)‐WV‐INF           

PEEQ > 4% at ALF = 0.97 

PEEQ > 4% at 

ALF = 0.83

PEEQ > 4% at 

ALF = 0.61

289

Figure 304. Equivalent plastic strain contours for P8-C-COS-WV-INF for the corroded gusset plate at the PEEQ > 4 % strength limit occurring at an ALF of 0.83 (DSF = 1).

Figure 305. Equivalent plastic strain contours for P8-C-COS-WV-INF for the gusset plate without corrosion at the PEEQ > 4 % strength limit occurring at an ALF of 0.83 (DSF = 1).

290

4.13.3 P14-U-C1-W-INF and P14-U-C2-W-INF

Joints P14-U-C1-W-INF and P14-U-C2-W-INF are corroded joints derived from the original P14-C(0.5)-W-INF test. The gusset plate geometry and reference member forces are the same as shown in Figure 196. The geometries of P14-U-C1-W-INF and P14-U-C2-W-INF are shown in Figures 306 and 307. Pattern C1 has partial and complete section losses just above the chord and at the end of the compression diagonal. Pattern C2 has the same section losses just above the chord but the hole at the end of the compression diagonal is removed in this pattern.

Figure 306. Corroded gusset plate geometry for P14-U-C1-W-INF (units = inches).

0.75

1.13

0.75

6

910.5

42.

3

1.5

291

Figure 307. Corroded gusset plate geometry for P14-U-C2-W-INF (units = inches).

Load-displacement plots for P14-U-C1-W-INF and P14-U-C2-W-INF are shown in Figure 308. A load-displacement plot for P14-U(0.5)-W-INF is also shown in this figure. Because stress concentrations occur at the corroded region, the maximum equivalent plastic strain value exceeds 4 % at an early stage of loading. However, before the limit load of these joints is reached, only partial areas of the corroded regions exhibit PEEQ values larger than 4 %. Therefore, the maximum capacities of P14-U-C1-W-INF and P14-U-C2-W-INF are determined based on the limit load criterion. The maximum capacities of P14-U-C1-W-INF and P14-U-C2-W-INF are reached at an ALF of 0.94 and 0.95 respectively based on the limit load criterion, which are 20 % and 19 % smaller than the limit load of P14-U(0.5)-W-INF, which is reached at an ALF of 1.18. It appears that the additional hole in P14-U-C1-W-INF has negligible effect on the maximum capacity of the joint.

Equivalent plastic strain contours for P14-U-C1-W-INF and P14-U-C2-W-INF are shown in Figures 309 and 310 respectively. Based on the lighter areas just above the chord, at the end of both diagonals and just below the compression diagonal on the left free edge, one can observe that a substantial area of the gusset plate is yielding when the limit load is reached for both P14-U-C1-W-INF and P14-U-C2-W-INF. The predominant failure mode is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

0.75

1.13

0.75

6

910.5

Holetg = 0.5 in.tcorroded-1 = 0.35 in.tcorroded-2 = 0.25 in.

292

Figure 308. Load-displacement plots for P14-U(0.5)-W-INF, P14-U-C1-W-INF and P14-U-C2-W-INF.

Figure 309. Equivalent plastic strain contours for P14-U-C1-W-INF at the limit load occurring at an ALF of 0.94 (DSF = 1).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ap

pli

ed

Lo

ad

Fra

cti

on

, P/P

refe

ren

ce

Horizontal Displacement at U1 (inch)

P14-U(0.5)-W-INF

P14-U-C1-W-INF

P14-U-C2-W-INF

293

Figure 310. Equivalent plastic strain contours for P14-U-C2-W-INF at the limit load occurring at an ALF of 0.95 (DSF = 1).

4.13.4 P14-U-COS-W-INF

Joint P14-U-COS-W-INF is a joint with a corroded gusset plate on one side and an uncorroded gusset plate on the other side. The corrosion pattern of P14-U-COS-W-INF is the same as P14-U-C1-W-INF shown in Figure 306.

A load-displacement plot for P14-U-COS-W-INF is shown in Figure 311. In this figure, load-displacement plots for P14-U(0.5)-W-INF and P14-U-C1-W-INF are shown as well. The limit load of P14-U-COS-W-INF is reached at an ALF of 1.07, which is 9 % smaller than the original P14-U(0.5)-W-INF and 14 % larger than P14-U-C1-W-INF.

The equivalent plastic strain contours for the corroded gusset plate of P14-U-COS-W-INF are shown in Figure 312 at the limit load. Similar to P14-U-C1-W-INF, a substantial area of the gusset plate just above the chord, at the end of both diagonals, and just below the compression diagonal on the left free edge is yielding at the limit load. The predominant failure mode is the buckling of the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

294

Figure 311. Load-displacement curves for P14-U(0.5)-W-INF, P14-U-C1-W-INF, and P14-U-COS-W-INF.

Figure 312. Equivalent plastic strain contours for P14-U-COS-W-INF at the limit load occurring at an ALF of 1.07.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ap

pli

ed

Lo

ad F

ract

ion

, P/P

refe

ren

ce

Horizontal Displacement at U1 (inch)

P14-U(0.5)-W-INF

P14-U-C1-W-INF

P14-U-COS-W-INF

295

4.14 PARAMETRIC CONFIGURATIONS WITH CORRODED GUSSET PLATES REINFORCED BY SHINGLE PLATES

This section summarizes key characteristics and sample results for the resistance of several selected gusset plates having significant section loss and subsequently reinforced by shingle plates. The corresponding results are summarized in Table 14 of Section 4.2.

4.14.1 P8-C-C1-SP(0.5:0.25)-WV-INF

To study the effect of shingle plates on the capacity of a joint with corroded gusset plates, a shingle plate of 0.25 inches is added on the gusset plates on each side of Joint P8-C-C1-WV-INF. This joint is designated as P8-C-C1-SP(0.5:0.25)-WV-INF. The geometries of the gusset and shingle plates are shown in Figure 313. The shingle plate is present from the beginning of the analysis and is not truly modeled as a retrofit by including the initial stress state in the main gusset. However, due to the excessive yielding at the limit-state, the effects of the initial stress state are marginal.

64.394.1

34.7

38.2

45.2

51.8

71.5

Figure 313. Gusset and shingle plate geometry for P8-C-C1-SP(0.5:0.25)-WV-INF (units = inches).

A load-displacement plot for P8-C-C1-SP(0.5:0.25)-WV-INF is shown in Figure 314. Also shown are the load-displacement plots for P8-C(0.5)-WV-INF and P8-C-C1-WV-INF. The maximum capacity of P8-C-C1-SP(0.5:0.25)-WV-INF is determined in the same way as described in Section 4.13.1. That is, the maximum capacity is determined at a load level when the PEEQ > 4 % strength limit is reached along the entire corroded region. For P8-C-C1-

296

SP(0.5:0.25)-WV-INF, the PEEQ > 4 % strength limit is reached at an ALF of 0.93, which is 4 % smaller than P8-C(0.5)-WV-INF and 52 % larger than P8-C-C1-WV-INF. One can observe that the load-displacement plot of P8-C-C1-SP(0.5:0.25)-WV-INF is essentially same as the original P8-C(0.5)-WV-INF.

The equivalent plastic strain contours for the P8-C-C1-SP(0.5:0.25)-WV-INF are shown in Figure 315 at an ALF of 0.93. The same contours are shown in Figure 316 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. It can be seen in Figure 315 that the shingle plate has been plastified just above the chord members. This is an indication that substantial forces are being developed into the shingle plates from the connected members. One can observe in Figure 316 that the “full shear plane” of the gusset plate just above the chord members and the area at the end of the tension diagonal is yielding when the PEEQ > 4 % limit is reached.

Figure 314. Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-C1-SP(0.5:0.25)-WV-INF.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5

Ap

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Horizontal Displacement at U1 (inch)

P8-C(0.5)-WV-INF

P8-C-C1-WV-INF

P8-C-C1-SP(0.5:0.25)-WV-INF

P8‐C(0.5)‐WV‐INF           

PEEQ > 4% at ALF = 0.97 

P8‐C‐C1‐SP‐WV‐INF         

PEEQ > 4% at ALF = 0.93 

PEEQ > 4% at 

ALF = 0.61

297

Figure 315. Equivalent plastic strain contours for P8-C-C1-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.93 (DSF = 1).

Figure 316. Equivalent plastic strain contours for P8-C-C1-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.93 (DSF = 1), shingle plate removed to show the

contours on the gusset plate.

298

4.14.2 P8-C-COS-SP(0.5:0.25)-WV-INF

Joint P8-C-COS-SP(0.5:0.25)-WV-INF is a joint with a shingle plate added on the corroded gusset plate for P8-C-COS-WV-INF. The geometry of the shingle plate is the same as shown in Figure 313 but it is only applied to the corroded gusset plate on one side of the joint. The thickness of the shingle plate is 0.25 inches.

A load-displacement plot for P8-C-COS-SP(0.5:0.25)-WV-INF is shown in Figure 317 along with the load-displacement plots for the original P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, and P8-C-COS-WV-INF.The maximum capacity of P8-C-COS-SP(0.5:0.25)-WV-INF is reached at an ALF of 0.95 based on the PEEQ > 4 % strength limit. The strength limit of P8-C-COS-SP(0.5:0.25)-WV-INF is 2 % smaller than P8-C(0.5)-WV-INF and 55 % and 14 % larger than P8-C-C1-WV-INF and P8-C-COS-WV-INF respectively.

The equivalent plastic strain contours for P8-C-COS-SP(0.5:0.25)-WV-INF are shown in Figure 318 at an ALF of 0.95. The same contours are shown in Figure 319 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. It can be seen in Figure 318 that the shingle plate has been plastified just above the chord members. This is an indication that substantial forces are being developed into the shingle plates from the connected members. One can observe in Figure 319 that the “full shear plane” of the gusset plate just above the chord members and the area at the end of the tension diagonal is yielding when the PEEQ > 4 % limit is reached

299

Figure 317. Load-displacement curves for P8-C(0.5)-WV-INF, P8-C-C1-WV-INF, P8-C-COS-WV-INF, and P8-C-COS-SP(0.5:0.25)-WV-INF.

Figure 318. Equivalent plastic strain contours for P8-C-COS-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.95 (DSF = 1).

P8‐C(0.5)‐WV‐INF           

PEEQ > 4% at ALF = 0.97 

P8‐C‐COS‐SP‐WV‐INF        

PEEQ > 4% at ALF = 0.95 

PEEQ > 4% at 

ALF = 0.83

PEEQ > 4% at 

ALF = 0.61

300

Figure 319. Equivalent plastic strain contours for P8-C-COS-SP(0.5:0.25)-WV-INF at the PEEQ > 4 % strength limit at an ALF of 0.95 (DSF = 1) ), splice plate removed to show the

contours on the gusset plate.

4.14.3 P14-U-C1-SP(0.5:0.25)-W-INF

Joint P14-U-C1-SP(0.5:0.25)-W-INF is a joint with a shingle plate added on each corroded gusset plate of P14-U-C1-W-INF. The shingle plate has a thickness of 0.25 inches. The geometries of the gusset and shingle plates are shown in Figure 320.

Figure 321 shows a load-displacement plot for P14-U-C1-SP(0.5:0.25)-W-INF as well as the load-displacement plots for P14-U(0.5)-W-INF and P14-U-C1-W-INF.The limit load of P14-U-C1-SP(0.5:0.25)-W-INF is reached at an ALF of 1.22. This is 3 % and 30 % larger than the limit loads of P14-U(0.5)-W-INF and P14-U-C1-W-INF respectively.

The equivalent plastic strain contours for P14-U-C1-SP(0.5:0.25)-W-INF are shown in Figure 322 at an ALF of 1.22. The same contours are shown in Figure 323 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. It can be seen in Figure 322 that the shingle plate has been plastified just above the chord members along approximately ¾ of the full width and at the end of compression diagonal. This is an indication that substantial forces are being developed into the shingle plates from the connected members. One can observe in Figure 323 that almost the full width of the gusset plate just above the chord members and the area at the end of the compression diagonal is yielding when the limit load is reached. However, yielding in the gusset plate does not extend to the full width of the gusset plate as observed in P14-U-C1-W-INF. The predominant failure mode is the buckling of the

301

gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

57.1

42

40.3

28

42

28.1

36.6

21

Figure 320. P14-U-C1-SP(0.5:0.25)-W-INF shingle plate geometry.

302

Figure 321. Load-displacement curves for P14-U(0.5)-W-INF, P14-U-C1-W-INF, and P14-U-C1-SP(0.5:0.25)-W-INF.

Figure 322. Equivalent plastic strain contours for P14-U-C1-SP(0.5:0.25)-W-INF at the limit load occurring at an ALF of 1.22 (DSF = 1).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ap

pli

ed

Lo

ad F

ract

ion

, P/P

refe

ren

ce

Horizontal Displacement at U1 (inch)

P14-U(0.5)-W-INF

P14-U-C1-W-INF

P14-U-C1-SP(0.5:0.25)-W-INF

303

Figure 323. Equivalent plastic strain contours for P14-U-C1-SP(0.5:0.25)-W-INF at the limit load occurring at an ALF of 1.22 (DSF = 1) ), shingle plate removed to show the contours on the

gusset plate.

4.14.4 P14-U-COS-SP(0.5:0.25)-W-INF

Joint P14-U-COS-SP(0.5:0.25)-W-INF is a joint with a shingle plate added only on the corroded gusset plate of P14-U-COS-W-INF. The shingle plate geometry is the same as shown in Figure 320 and a thickness of 0.25 inches is used.

A load-displacement plot for P14-U-COS-SP(0.5:0.25)-W-INF is shown in Figure 324 as well as the load-displacement plots for P14-U(0.5)-W-INF and P14-U-COS-W-INF. The limit load of P14-U-COS-SP(0.5:0.25)-W-INF is reached at an ALF of 1.20, which is 2 % and 12 % larger than the limit loads of P14-U(0.5)-W-INF and P14-U-COS-W-INF respectively.

The equivalent plastic strain contours for P14-U-COS-SP(0.5:0.25)-W-INF are shown in Figure 325 at an ALF of 1.20. The same contours are shown in Figure 323 but with the shingle plate removed so that the extent of plasticity in the main gusset plate can be observed. It can be seen in Figure 325 that the shingle plate has been plastified just above the chord members along approximately ¾ of the full width and at the end of compression diagonal. This is an indication that substantial forces are being developed into the shingle plates from the connected members. One can observe in Figure 323 that almost the full width of the gusset plate just above the chord members and the area at the end of the compression diagonal is yielding when the limit load is reached. However, yielding in the corroded gusset plate does not extend to the full width of the gusset plate as observed in P14-U-COS-W-INF. The predominant failure mode is the buckling of

304

the gusset plate at the end of the compression diagonal, including the out-of-plane movement of the diagonal.

Figure 324. Load-displacement curves for P14-U(0.5)-W-INF, P14-U-COS-W-INF, and P14-U-COS-SP(0.5:0.25)-W-INF.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ap

pli

ed

Lo

ad F

ract

ion

, P/P

refe

ren

ce

Horizontal Diaplacement at U1 (inch)

P14-U(0.5)-W-INF

P14-U-COS-W-INF

P14-U-COS-SP(0.5:0.25)-W-INF

305

Figure 325. Equivalent plastic strain contours for P14-U-COS-SP(0.5:0.25)-W-INF at the limit load occurring at an ALF of 1.20 (DSF = 1).

Figure 326. Equivalent plastic strain contours for P14-U-COS-SP(0.5:0.25)-W-INF at the limit load occurring at an ALF of 1.20 (DSF = 1), splice plate removed to show the contours on

the gusset plate.

306

307

5. DISCUSSION OF RESULTS AND DEVELOPMENT OF DESIGN RECOMMENDATIONS

5.1 OVERVIEW

As noted in Sections 4.1 and 4.2, two design evaluation methods were developed during the process of the NCHRP 12-84 research. Each of these methods implements a set of equations that define all the limit states needed to properly evaluate a gusset plate. Method 1 is a set of streamlined equations recommended for integration into the AASHTO Specifications. A number of minor modifications have been made to Method 1 as of this writing (January 2013) and are reported in (Ocel, 2013); however, the major concepts in Method 1 as evaluated in this document are unchanged in the January 2013 AASHTO proposed provisions.

As demonstrated in the following, Method 1 has a number of areas where the Georgia Tech research team felt it was desirable to pursue potential improvements in the accuracy of its models while still retaining simplicity of application to the maximum extent possible. The resulting set of models is referred to here as Method 2.

This section first provides an overview of the prediction accuracy obtained by the Method 1 and Method 2 equations for each of the relevant limit states being evaluated. The various test simula-tions summarized in Chapter 4 are used for this purpose. This is followed by a presentation of the equations for each of the methods as well as useful examples from various test simulations to illustrate the application of the equations and their prediction of the behavior. The examples generally are selected to highlight specific qualities and limitations of the different calculations.

Table 43 summarizes the Method 1 and Method 2 predictions for seven parametric tests gov-erned by chord-splice eccentric tension and 21 tests governed by chord-splice eccentric compres-sion limit states. Method 1 is an elastic analysis based calculation, whereas Method 2 is a pseudo-plastic analysis based calculation. As such, Method 2 gives a mean professional factor Rtest/Rn closer to 1.0 and a slightly smaller coefficient of variation (COV). However, the minimum Rtest

/Rn is 0.82 in Method 2 compared to 0.89 in Method 1. Figures 327 and 328 show the distribu-tion of Rtest/Rn for these calculations versus the gusset plate thickness and indicate that P10-C-P(0.2)-P-NP and P11-C-HS(0.25)-W-M have the smallest professional factors for both methods.

Table 43 Summary assessment of the professional factor from chord-splice eccentric tension and chord-splice eccentric compression prediction equations, 28 parametric tests

Method 1 Method 2 Mean 1.29 1.12

COV 0.13 0.11

Max 1.64 1.29

Min 0.89 0.82

308

Figure 327. Method 1 professional factor for cases governed by chord-splice eccentric

compression or chord-splice eccentric tension

Figure 328. Method 2 professional factor for cases governed by chord-splice eccentric compression or chord-splice eccentric tension

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0

Professional Factor Rtest /Rn

tg (in)

P11‐C‐HS(0.25)‐W‐M

P4‐C(0.8)‐WV‐P

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0

Professional Factor Rtest /Rn

tg (in)

P11‐C‐HS(0.25)‐W‐M

P10‐C‐P(0.2)‐P‐NP

309

Method 1 shows a slight trend toward being more conservative for cases with thicker gusset plates. This trend is exhibited by both the tension and compression chord splice tests. Method 2 does not show any significant dependency on the gusset plate thickness with the exception of the two tests exhibiting the smallest professional factors.

Table 44 summarizes the Method 1 and Method 2 predictions for all the other gusset plate limit states evaluated in this research. With Method 1, only one of the 148 tests (P15-C-CJ) was governed by Full Shear Plane Yielding (FSPY). This test had chamfered members and an Rtest/Rn = 1.08. Method 2 generally gives a significantly smaller COV as well as minimum and maximum Rtest/Rn values much closer to 1.0 compared to Method 1. (P16-C-CJ was also governed by FSPY, but was excluded from consideration because of a premature buckling failure of the main truss members.) The predictions using Method 2 are especially good for the No Chamfer cases. Figures 329 and 330 show the distributions of Rtest/Rn for Method 1 and Method 2 versus the normalized slenderness parameters 0.5(Fy/E)0.5 (Lavg/tg) and 0.35(Fy/E)0.5 (Lmid/tg) respectively for the No Chamfer cases governed by Diagonal Buckling (DB) (the coefficients 0.5 and 0.35 are implicit gusset plate K factors for these cases in Methods 1 and 2 respectively). One can observe that there is substantial dispersion of the data for Rtest/Rn for Method1, and that there is a large number of data points with Rtest/Rn significantly less than 1.0. However, the data points for Method 2 are clustered in a relatively tight band mostly between Rtest/Rn of 1.01 and 1.20. Clearly, Method 2 provides a significantly better prediction for cases that are governed by the DB limit state checks.

Table 44 Summary assessment of professional factors for Method 1 combined diagonal buckling or partial shear plane yielding, Method 2 diagonal bucking, Method 1 or Method 2 diagonal

tension yielding, and Method 1 or Method 2 full shear plane yielding.

Method Method 1 Method 2 Method 1

Method 2

Geometry No Chamfer

Chamfer No Chamfer

No Chamfer

No Chamfer Chamfer Chamfer All Cases

All Cases

Limit States

DB/PSPY or TY

DB/PSPY or TY

DB FSPY DB-TWS & TY-TWS

DB-TWS & TY-TWS

FSPY

Count 88 59 47 13 28 46 14 148 148

Mean 1.11 1.14 1.12 1.09 1.04 1.07 1.11 1.12 1.09

COV 0.15 0.14 0.05 0.06 0.10 0.10 0.08 0.14 0.09

Max 1.53 1.43 1.27 1.15 1.29 1.27 1.26 1.53 1.29

Min 0.58 0.77 1.01 0.95 0.89 0.82 0.96 0.58 0.82

Thirteen tests are governed by the Full Shear Plane Yielding (FSPY) check in Method 2. Figure 331 shows the distribution of professional factor for these tests versus the gusset plate thickness tg. The Rtest/Rn for these tests is between 1.0 and 1.15, with the exception of two tests, P10-C-P(0.2)-P-NP and P11-C-HS(0.25)-W-M, where the gusset plates were extraordinarily thin.

310

Figure 329. Method 1 professional factor for cases with no chamfer governed by combined

diagonal buckling (DB) or Partial Shear Plane Yielding (PSPY)

Figure 330. Method 2 professional factor for cases with no chamfer governed by diagonal buckling (DB)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Professional Factor Rtest /Rn

0.5(Fy/E)0.5 (Lavg/tg)

P6‐U(0.25)‐WV‐NP

P14‐U(0.25)‐W‐INF

P6‐U(0.3125)‐WV‐NP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Professional Factor Rtest/R

n

0.35(Fy/E)0.5 (Lmid/tg)

311

Figure 331. Method 2 professional factor for cases with no chamfer governed by full shear plane yielding (FSPY)

Figure 332 shows the distribution of Rtest/Rn versus 0.35(Fy/E)0.5 (Lmid/tg) for 28 No Chamfer cases governed by the Method 2 Diagonal Buckling limit state check in which the Whitmore section is truncated due to overlaps with adjacent framing. The mean professional factor for these cases is slightly smaller (1.04) and the COV is slightly larger (0.10) compared to the other Method 2 predictions above, as might be expected (see Table 44). A significant number of the tests at intermediate plate slenderness values have an Rtest/Rn close to 0.9. However, the results still are generally better than those shown for Method 1 in Figure 329. The smallest Rtest/Rn value is 0.89 for in Figure 332 versus a minimum of 0.58 and five different tests with a wide range of slenderness values having Rtest/Rn values close to 0.60 in Figure 329.

Figures 333 through 335 show the results from Methods 1 and 2 for the Chamfered test cases. Figure 333 shows the distribution of Rtest/Rn versus 0.5(Fy/E)0.5 (Lavg/tg) for 59 tests governed either by the Method 1 Diagonal Buckling (DB), Diagonal Tension Yielding (TY), or Partial Shear Plane Yielding (PSPY) limit state checks, Figure 334 shows the professional factor for 46 tests governed by the Method 2 Diagonal Buckling with a Truncated Whitmore Section (DB-TWS) or Diagonal Tension Yielding with a Truncated Whitmore Section (TY-TWS) versus 0.35(Fy/E)0.5 (Lmid/tg), and Figure 335 shows the results versus 0.35(Fy/E)0.5 (Lmid/tg) for 14 tests that were governed by the Method 2 Full Shear Plane Yielding (FSPY) limit state calculations. Again, Method 2 gives somewhat better predictions, with mean Rtest /Rn closer to 1.0, smaller COV on Rtest/Rn, and larger minimum Rtest /Rn values (also see Table 44). Figure 333 indicates that there are five Chamfered tests with a wide range of slenderness values that have an Rtest /Rn

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Professional Factor Rtest/R

n

0.35(Fy/E)0.5 (Lmid/tg)

312

close to 0.8 (minimum value of 0.77), while Figures 334 and 335 show that Method 2 has only one test, P5-C-HS(0.2)-WV-NP, that has low Rtest /Rn of 0.82 with all the other tests exhibiting Rtest /Rn > 0.90.

Figure 332. Method 2 professional factor for cases with no chamfer governed by diagonal

buckling with a truncated Whitmore section (DB-TWS)

Figure 333. Method 1, chamfered cases governed by diagonal buckling, diagonal yielding, or

partial shear plane yielding (DB, TY or PSPY)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Professional Factor Rtest/R

n

0.35(Fy/E)0.5 (LM/t)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Professional Factor Rtest/R

n

0.5 (Fy/E)0.5 (Lavg/tg)

P5‐C‐HS(0.2)‐WV‐NP

P17‐C‐POS & P18‐C‐POS

P14‐C(0.25)‐W‐INFP9‐C(0.20)‐P‐NP

313

Figure 334. Method 2 professional factor for chamfered cases governed by diagonal buckling or diagonal yielding with a truncated Whitmore section (DB-TWS or DY-TWS)

Figure 335. Method 2 professional factor for chamfered cases governed by full shear plane yielding (FSPY)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0

Professional Factor Rtest/R

n

0.35(Fy/E)0.5 (LM/tg)

P5‐C‐HS(0.2)‐WV‐NP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0

Professional Factor Rtest /Rn

0.35(Fy/E)0.5 (LM/tg)

314

5.2 CHORD SPLICE ECCENTRIC COMPRESSION

This section explains the Method 1 and Method 2 calculations for checking a gusset plate that is acting with other splice plates to transfer the chord tension or compression force across a truss joint. Method 1 has been modified producing a simpler, more easily understood, but slightly more conservative set of calculations in the January 2013 AASHTO provisions proposed by the NCHRP 12-84 project (Ocel, 2013). The differences between the Method 1 calculations recommended here and those recommended for the AASHTO provisions are explained below.

5.2.1 Method 1

Method 1 in this research uses the “effectiveness factor,” Ef, as discussed by Kulicki and Reiner (2011) to account for the eccentric application of the chord load to the gusset plate when checking the capacity of a chord splice. The effectiveness factor

totgPtotg

go

totgPtotg

gof SeA

A

SPeAP

APE

...:. //1

/1

//

/

Eq. (1)

is simply the uniform axial stress calculated on the portion of the gusset plate width that overlaps the depth of the chord divided by the maximum theoretical elastic normal stress due to the eccentric application of the chord axial load to the full cross-section of the gusset plate. These two corresponding theoretical stress distributions, as well as the gusset plate widths bw and bg associated with them, are illustrated in Figures 336 and 337. These figures are also used as part of the example calculations provided in Section 5.2.1.1.

The area associated with the stress term in the numerator of Eq. (1) is

Ago = bwtg

where bw is the width of the chord web associated with the splice cross-section at which the gusset plate stresses are being evaluated. Similarly, the area associated with the stress term in the denominator of Eq. (1) is

Ag.tot = bgtg

where bg is the full depth of the gusset plate at the subject cross-section. The eccentricity of the chord axial force with respect to the centroid of the full gusset plate cross-section (i.e., the full gusset plate mid-depth is

eP = bg / 2 – bw / 2

if the center of the splice is located at the work point for the joint, in which case, the line of action of the axial force applied to the gusset plate is located at the centroid of the chord

315

members. If the splice is offset from the work point, one generally should determine the location of the resultant of the gusset plate axial forces from the truss chord and web members at the splice location. An example of this calculation is provided below in Section 5.2.1.2. The gusset plate elastic section modulus associated with the stress calculation in the denominator of Eq. (1) is of course

Sg.tot = bg2tg/6

Therefore, for the joint shown in Figures 336 and 337, the effectiveness factor is simply the ratio of the idealized uniform axial stress represented by the shaded rectangle in Figure 336 to the idealized maximum elastic normal stress due to the combined axial load plus the eccentric bending about the mid-depth of the full gusset plate, labeled at the bottom of the shaded stress distribution in Figure 337. For a compression splice, as shown by Kulicki and Reiner (2001), the effectiveness factor Ef is then applied to the gusset plate axial capacity based on a uniform stress applied to the partial width bw, Fcr.g Ago, such that one is actually forcing the maximum theoretical elastic stress shown in Figure 337 to be less than or equal to Fcr.g.

Figure 336. Method 1 base stress distribution in gusset plate for evaluation of chord splice

eccentric compression, shown on P19-C-CCS(0.6)-NEG.

 

goAP /

2400 kips

3700 kips

500 kips

780 kips

5600 kips

bw

316

Figure 337. Method 1 elastic stress distribution in gusset plate for evaluation of chord splice

eccentric compression, shown on P19-C-CCS(0.6)-NEG.

This gusset plate contribution to the chord splice resistance is then summed with the contributions from all the other splice plates, assuming uniform axial compression on the other splice plates and assuming that these plates are centered about the chord. The resulting total splice capacity for the case with two gusset plates (one on each side of the truss members) is

gogcrffspocrfspofspofspicrfspifspiwspcrwspwspn AFEFtbFtbFtbP ....2 Eq. (2)

where:

bwsp, twsp and Fcr.wsp are the width, thickness and critical compressive stress for the web splice

plates (attached to the chord members on the insides of the chord webs),

bfspi, tfspi and Fcr.fspi are the width, thickness and critical compressive stress for the flange splice plates attached to the chord members on the insides of the chord flanges, and

bfspo, tfspo and Fcr.fspo are the width, thickness and critical compressive stress for the flange splice plates attached to the chord members on the outsides of the chord flanges,

 

totgwgtotg SbbPAP .. /)2/2/(/

2400 kips

3700 kips

500 kips

780 kips

5600 kips

bg

317

Each of the above pairs of splice plates is assumed to be the same on each side of the chord cross-section. If any one set of splice plates is not provided at a given joint, the corresponding term in Eq. (2) is taken equal to zero.

It should be noted that, depending on the specifics of bw compared to bg, Ef can be either greater than or less than 1.0. If the maximum elastic stress shown in Figure 337 is smaller than the uniform stress shown in Figure 336, Ef is greater than one.

Lastly, for the calculation of the compressive strengths of the gusset plate and the splice plates in the above, each plate is treated as a column with a length L equal to the distance between the last row of fasteners on each side of the splice and an effective length factor K = 0.5. In most cases, the corresponding equivalent column slenderness KL/r is less than 25. When this occurs, the calculations borrow a rule from Section J4.4 of the AISC (2010) Specification and take Fcr = Fy.

As noted at the beginning of Section 5.2, the January 2013 NCHRP 12-84 procedures proposed for the AASHTO Specifications have been modified relative to this Method 1 procedure in an effort to make them simpler at the expense of some additional conservatism. The Method 1 procedure proposed to AASHTO involves taking the gusset plate plus all the splice plates as a combined composite elastic cross-section. This entire cross-section is assumed to act as an eccentrically loaded beam-column, and the corresponding maximum axial compression is required to be smaller than the Fcr values of the gusset plate and all of the splice plates. Conversely, the above Method 1 calculation ignores the effect of the chord eccentricity when calculating the resistance of the splice plates; the above Method 1 only considers the effect of the chord eccentricity in estimating the contribution of the gusset plate to the splice resistance. This is consistent with the application of the effectiveness factor by Kulicki and Reiner (2011).

5.2.1.1 Method 1 Example, Chord Splice Eccentric Compression – P19-C-CCS(0.6)-NEG

The Method 1 chord splice eccentric compression resistance recommended by this research is illustrated here by considering the compression splice in P19-C-CCS(0.6)-NEG. In this joint, we have

bg = 55.0 in, tg = 0.6 in

bwsp = 23.5 in, twsp = 0.42 in

bfspo = 19 in, tfspo = 1.28 in,

bfspi = 0 in, tfspi = 0 in,

(that is, this joint has splice plates only on the outside of the chord flanges)

Fcr.g = Fcr.wsp = Fcr.fsp1 = Fy = 53 ksi

318

since KL/r < 25 for the gusset plate and all of the splice plates, and

bw = 24 in / cos (13.2o) = 24.7 in.

(see Figure 336). The above dimensional parameters give

Ago = 24.7 x 0.6 = 14.8 in2

Ag.tot = 55.0 x 0.6 = 33.0 in2

ep = 55.0 / 2 – 24.7 / 2 = 15.2 in

and

32

. in 3026

0.556.0

totgS

As a result, the effectiveness factor for this joint is

1 / 14.80.828

1 / 33.0 15.5 / 302fE

Upon combining all of the above quantities into Eq. (2), one obtains

Pn = 2 x (23.5 x 0.42 x 53 + 19 x 1.28 x 53 + 0.828 x 53 x 14.4) = 4888 kip

The corresponding maximum strength of this joint, obtained from the test simulation, is

Ptest = 0.89 x [780 cos(26.6 + 13.3) + 5600 cos(26.6 – 13.3)] = 5384 kip

(Note that both the chord axial compression of 5600 kips and the diagonal axial compression of 780 kips contribute to the total compressive force that is transferred across the splice.) The resulting professional factor for the chord splice eccentric compression in this case is

Ptest/Pn = 5384 / 4888 = 1.10

5.2.1.2 Method 1 Example, Chord Splice Eccentric Compression – P4-C(0.8)-WV-P

Test P4-C(0.8)-WV-P is a slightly more complex case compared to the above in that the gusset plates contribute to the compression splice resistance at a location that is offset from the work point of the truss joint. In addition, in this case, there are no flange or web splice plates. The gusset plate provides the only force transfer for the horizontal axial compression delivered from the bottom chord and from the compression diagonal on either side of the center vertical member.

319

Figure 338 illustrates the base uniform axial stress distribution associated with the gusset plate area Ago = bwtg and Figure 339 shows the idealized elastic stress distribution due to the axial compression coming from the chord and diagonal members plus the eccentricity of the resultant horizontal force from these two members.

Figure 338. Method 1 base stress distribution in gusset plate for evaluation of chord splice

eccentric compression, shown on P4-C(0.8)-WV-P.

The dimensional parameters in this example are

bg = 87.9 in, tg = 0.8 in

bwsp = 0 in, twsp = 0 in, bfspo = 0 in, tfspo = 0 in, bfspi = 0 in, tfspi = 0 in

and

bw = 18 / cos (9.46) = 18.2 in,

and the base compressive strength of the gusset plates (with KL/r < 25) is

Fcr.g = Fy = 53 ksi

320

The reader should note that the base spacing between all the connectors in this problem, as well as in all the other parametric test problems, is 3 inches (the spacing used in the experimental tests and in the variations on the experimental tests is 2.5 inches).

Figure 339. Method 1 elastic stress distribution in gusset plate for evaluation of chord splice

eccentric compression, shown on P4-C(0.8)-WV-P.

The above dimensional parameters give the gusset plate areas

Ago = 18.2 x 0.8 = 14.6 in2

and

Ag.tot = 87.9 x 0.8 = 70.3 in2

Furthermore, the eccentricity of the resultant horizontal force, generated by the diagonal plus the bottom chord, relative to the mid-depth of the full gusset plate is

in 5.31)4.63cos(2240)46.9cos(3041

5.7)4.63tan(5.7)4.63cos(2240)46.9cos(

2/18)46.9cos(3041

2

9.87

Pe

totgPtotg SPeAP .. //

321

and the elastic section modulus of the full gusset cross-section is

32

. in 10306

9.878.0

totgS

The combination of all of the above parameters gives an effectiveness factor in this problem of

53.11030/5.313.70/1

6.14/1

fE

In this case, by virtue of the substantial depth of the gusset plate compared to the depth of the chord web, the effectiveness factor is significantly greater than one. The resulting capacity of the two gusset plates is

Pn = 2 x (1.53 x 53 x 14.6) = 2368 kip

This result can be compared to the corresponding force achieved in the test simulation of

Ptest = 0.97 x [3041 cos(9.46) + 2240 cos(63.4)] = 3883 kip

such that the professional factor for this problem is

Ptest/Pn = 3883 / 2368 = 1.64

5.2.2 Method 2

The Method 2 calculations for the chord splice eccentric compression resistance use the pseudo-plastic cross-section model shown in Figure 340. The compressed region of the gusset plate and all the splice plates are assumed to be in uniform compression at a stress equal to their respective Fcr values. Similar to the Method 1 calculations discussed in Section 5.2.1, the plate Fcr values are taken equal to Fy if the KL/r of the compressed plates is smaller than 25, where L is taken as the distance between the last row of fasteners on each side of the splice and the effective length factor K is taken equal to 0.5. In most situations, all the Fcr values are equal to Fy. The plastic neutral axis of the gusset plate subjected to the combined axial compression and bending moment due to the chord eccentricity is located at the distance bt from the “inside” edge of the gusset plate at the splice cross-section (see Figure 340). The gusset plate stresses normal to the splice are assumed to be in axial tension at the yield strength Fy throughout the width bt. The eccentric moment developed in the gusset plates (one gusset on each side of the truss members) is equal to the total axial force transferred by the gusset plates times the eccentricity of the chord relative to the mid-depth of the gusset at the splice location, which is

eP = bg / 2 – bw / 2

322

(see Figure 340) if the splice is located at the work point of the joint, in which case, the line of action of the axial force in the gusset plate is located at the centroid of the chord members. Otherwise, one must determine the location of the resultant of the gusset plate axial forces from the combination of the truss chord and the web members at the splice location. This can be done by subtracting the axial force contribution provided by the chord splice plates from the total forces transferred across the splice, and then focusing just on the forces transferred by the gusset plate.

Figure 340. Method 2 plastic stress distribution in gusset plate and splice plates for

evaluation of chord splice eccentric compression.

The key concept invoked to determine the depth of the gusset plate in tension, bt, in the above model is that the resultant moment in the gusset plates must be zero about the centerline of the chord (assuming that all the other splice plates are placed symmetrically about the chord centerline). Thus, after writing an equation for the total moment in the gusset plates about the chord centerline, one can solve for bt to obtain

2

.

. .

2 2

2 4

g pg cr gt P g P

y g cr g

b eb Fb e b e

F F

Eq. (3)

WP

bw/2

bw/2

bc

bt

bg

bg x tg

bfspi x tfspi

bfspo x tfspo

bwsp x twsp Fcr.wsp

Fcr.fspi

Fcr.fspo

Fcr.fspi

Fcr.fspo

Fcr.g

Fy.g

323

Once bt has been determined, the overall capacity of the splice may be calculated as

gygtgcrgtgfspocrfspofspofspicrfspifspiwspcrwspwspn FtbFtbbFtbFtbFtbP .....2 Eq. (4)

from the statical contributions of the stress blocks in Figure 340.

5.2.2.1 Method 2 Example, Chord Splice Eccentric Compression – P19-C-CCS(0.6)-NEG

The Method 2 chord splice eccentric compression resistance recommended by this research is illustrated here using the same compression splice example, P19-C-CCS(0.6)-NEG, as used to demonstrate the Method 1 calculations in Section 5.2.1.1. As such, the characteristics of the two calculations can be compared and contrasted. In the P19-C-CCS(0.6)-NEG joint, we have

bg = 55.0 in, tg = 0.6 in

bwsp = 23.5 in, twsp = 0.42 in

bfspo = 19 in, tfspo = 1.28 in,

bfspi = 0 in, tfspi = 0 in,

(that is, this joint has splice plates only on the outside of the chord flanges)

Fcr.g = Fcr.wsp = Fcr.fsp1 = Fy = 53 ksi

since KL/r < 25 for the gusset plate and all of the splice plates, and

bw = 24 in / cos (13.2o) = 24.7 in

Given these parameters, the eccentricity of the gusset plate mid-depth relative to the chord centerline is

eP = 55.0 / 2 – 24.7 / 2 = 15.2 in

at the splice (which is centered about the work point of the truss joint). The above parameters may then be substituted into Eq. (3) to obtain

255.0 2 15.255.0 2 53

15.2 55.0 15.2 11.3 in2 4 53 53tb

and

bc = bg – bt = 55.0 – 11.3 = 43.7 in

324

(see Figure 341). With the dimensions bt and bc determined, the contributions of all the plates to the chord splice eccentric compression capacity may be calculated and summed to obtain

2 23.5 0.42 53 19 1.28 53 55.0 11.3 0.6 53 11.3 0.6 53 5685 kipnP

The corresponding maximum strength of this joint, obtained from the test simulation, is

Ptest = 0.89 x [780 kip cos(26.6 + 13.3) + 5600 kip cos(26.6 – 13.3)] = 5384 kip

(Note that the both the chord axial compression of 5600 kips and the diagonal axial compression of 780 kips contribute to the total compressive force that is transferred across the splice.) The resulting professional factor for the chord splice eccentric compression in this case is

Ptest/Pn = 5685 / 5384 = 0.95

Figure 341. Method 2 plastic stress distribution in gusset plate for evaluation of chord splice

eccentric compression, shown on P19-C-CCS(0.6)-NEG.

5.2.2.2 Method 2 Example, Chord Splice Eccentric Compression – P4-C-(0.8)-WV-P

Test P4-C-(0.8)-WV-P is used to demonstrate the application of the Method 2 Chord Splice Eccentric Compression calculations to a slightly more complex case in which the gusset plates provide the compression splice resistance at a location offset from the work point of the truss joint.

 

Fcr.g

Fy.g

bt

bc

2400 kips

3700 kips

500 kips

780 kips

5600 kips

325

Figure 342 illustrates the idealized normal stress distribution across the splice cross-section at the side of the vertical member in this problem. The dimensional parameters in this example are

bg = 87.9 in, tg = 0.8 in

bwsp = 0 in, twsp = 0 in, bfspo = 0 in, tfspo = 0 in, bfspi = 0 in, and tfspi = 0 in

Figure 342. Method 2 plastic stress distribution in gusset plate for evaluation of chord splice eccentric compression, shown on P4-C-(0.8)-WV-P.

(there are no other splice plates in this example) and the base compressive strength of the gusset plates is

Fcr.g = Fy = 53 ksi

The eccentricity of the resultant horizontal force, from the diagonal plus the bottom chord, relative to the mid-depth of the full gusset plate is

in 5.31)4.63cos(2240)46.9cos(3041

5.7)4.63tan(5.7)4.63cos(2240)46.9cos(

2/18)46.9cos(3041

2

9.87

Pe

326

Given the above dimensions, the gusset plate width in tension may be calculated from Eq. (3) as

in 4.215.319.87

5353

532

4

5.3129.875.31

2

9.87 2

tb

and thus

bc = 87.9 – 21.4 = 66.5 in

Therefore, using Eq. (4), the calculated splice capacity is

kip 3824538.04.21538.0)4.219.87(2 nP

The corresponding force reached in the test simulation is Ptest = 0.97 x [3041 cos(9.46) + 2240 cos(63.4)] = 3883 kip

resulting in a professional factor of

Ptest/Pn = 3883 / 3824 = 1.02

5.3 CHORD SPLICE ECCENTRIC TENSION

The calculation of the chord splice eccentric tension capacity is conceptually the same as the calculation of the eccentric compression capacity except that: (1) the compression resistance on the splice cross-section is replaced by the minimum of the tension yielding and tension rupture resistances, and, where needed, (2) the small tension area on the gusset plate in the prior case is now handled using an appropriate axial compression resistance.

5.3.1 Method 1

The Method 1 chord splice eccentric tension resistance equation that parallels Eq. (2) for the eccentric compression resistance is:

),min(),min(

),min(),min(2

........

........

gugeogyggoffspoufspoefspoyfspog

fspiufspiefspiyfspigwspuwspewspywspgn

FAFAEFAFA

FAFAFAFAP

Eq. (5)

As noted above, the compressive resistance of each of the plates is replaced by the corresponding minimum of the tension yielding and tension rupture resistances.

327

5.3.1.1 Method 1 Example, Chord Splice Eccentric Tension – P2-C-TCS(0.4)-WV-M

Test P2-C-TCS(0.4)-WV-M, shown in Figures 343 and 344, allows for a useful illustration of the chord splice eccentric tension calculations, including a subtle attribute involving the appropriate location of the resultant axial force in the gusset plate. Strictly speaking, the critical cross-section for the tension chord splice is located at the distance bsplice on one side or the other of the center of the splice, which is located at the work point of the truss joint in this problem. As such, at the critical cross-section, the resultant horizontal force transferred from the chord plus the diagonal is located slightly above the centerline of the chord. Furthermore, the location of this resultant is different on each of the potential critical cross-sections of the tension splice. In addition, the problem is further complicated by the fact that the splice plates are typically located symmetrically about the centerline of the chord. Therefore, the resultant of the horizontal force transferred by the gusset plate must be such that, when combined with the splice plate resistances, the total resultant force is at the same depth within the joint as the resultant from the applied loads in the chord plus the diagonal on the side of the joint under consideration (the web splice plate stresses would typically be assumed to be uniform and the top and bottom flange splice plate forces would typically be assumed to be equal, i.e., any eccentric bending is neglected in considering the splice plate responses).

Figure 343. Method 1 base stress distribution in gusset plate for evaluation of chord splice

eccentric tension, shown on P2-C-TCS-WV-M.

It is recommended that the above offset, bsplice, be neglected in the calculation of the splice resistance. Therefore, for this problem, the eccentricity of the axial force transmitted by the splice with respect to the mid-depth of the gusset plate may be taken as

 

18

12

12

30

328

eP = bg/2 – bw /2

as discussed previously in Section 5.2.1 for a splice that is centered about the truss joint work point. Note that if the center of the chord splice is physically offset from the truss joint work point, the above complications are encountered.

Figure 344. Method 1 elastic stress distribution in gusset plate for evaluation of chord splice

eccentric tension, shown on P2-C-TCS-WV-M.

The dimensional parameters in this example are

bg = 39.0 in, tg = 0.4 in,

bwsp = 17.5 in, twsp = 0.509 in,

bfspo = 19.6 in, tfspo = 0.775 in, bfspi = 0 in, tfspi = 0 in

and

bw = 18 in.

Furthermore, the material parameters are

Fy = 53 ksi and Fu = 80 ksi.

Given the above dimensions, the gusset plate areas are

Ago = 18 in x 0.4 in = 7.2 in2

 

18

12

12

30. ./ /g tot P g totP A Pe S

329

and

Ag.tot = 39.0 in x 0.4 in = 15.6 in2

and the gross section modulus of the gusset plate is

Sg.tot = 101 in3

It is recommended that the gusset plate effectiveness factor should be calculated using only the gross cross-section properties of the gusset plate. Given

eP = 39.0/2 – 18/2 = 10.5 in

the effectiveness factor is

1 / 7.20.828

1 /15.6 10.5 / 101fE

Assuming punched standard size holes in the plates, the effective diameter of the holes used in

determining the plate net tension areas is

dh = 7/8 in + 1/8 in = 1 in

Hence, the various additional gross and net areas of the splice plates and gusset plates are

Ag.wsp = 17.5 x 0.509 = 8.91 in2

Ae.wsp = [ 17.5 – 6 x 1 ] x 0.509 = 5.85 in2

Ag.fspo = 19.6 x 0.775 = 15.2 in2

Ae.fspo = [ 19.6 – 4 x 1 in ] x 0.775 = 12.1 in2

Ago.g = 18 x 0.4 = 7.2 in2

and

Aeo.g = [ 18 – 6 x 1 ] x 0.4 = 4.8 in2

The resulting Eq. (5) calculation is then

330

2 min(8.91 53, 5.85 80) min(15.2 53,12.1 80) 0.828 min(7.2 53, 4.8 80)

2 min(472, 468) min(806,968) 0.828 min(382,384) 3180 kip

nP

This may be compared to the corresponding force from the test simulation of

Ptest = 1.29 x [3000 kip + 283 kip cos(45)] = 4130 kip

giving a professional factor of

Ptest/Pn = 1.30

5.3.2 Method 2

The Method 2 chord splice eccentric tension equations that parallel Eqs. (3) and (4) of Section 5.2.2 are

Pg

gcrgy

gypgP

gc eb

FF

Febe

bb

..

.2 2

4

2

2

Eq. (6)

and

gcrgcgugtegygcgfspoufspoefspoyfspog

fspiufspiefspiyfspigwspuwspewspywspgn

FtbFtbFtbbFAFA

FAFAFAFAP

.......

........

,min),min(

),min(),min(2

Eq. (7)

where

eP = bg / 2 – bw / 2

if the center of the splice is located at the work point of the truss joint, in which case, the line of action of the axial force in the gusset plate is located at the centroid of the chord members. If this is not the case, one must determine the location of the resultant of the gusset plate axial forces from the truss chord and web members at the splice location. This can be done by subtracting the axial force contribution from the splice plates from the total forces transferred across the splice, and then working with the resulting force from the gusset plates. The corresponding idealized normal stress distributions in the plates, assuming that all the plate capacities are governed by tension yielding, are shown in Figure 345. Generally, it is recommended that the gusset plate strength in compression, Fcr.g in Figure 345, should be determined based on the lengths between fastener locations along the line of these normal stresses within the depth of the gusset plate subjected to compression, bc. Section 5.3.2.1 provides an example Method 2 calculation for a typical chord splice in eccentric tension.

331

Figure 345. Method 2 plastic stress distribution in gusset plate and splice plates for

evaluation of chord splice eccentric tension, shown for a case in which all the plates are governed by tension yielding.

5.3.2.1 Method 2 Example, Chord Splice Eccentric Tension – P2-C-TCS(0.4)-WV-M

Figure 346 shows the Method 2 plastic stress distribution in the gusset plate for test P2-C-TCS(0.5)-WV-M. The reader is referred to Section 5.3.1.1 for the dimensional and material parameters of this joint. As noted previously in Section 5.3.1.1, the critical sections for the chord splice tension resistance calculations in this joint are actually offset from the center of the splice. However, it is recommended that this small offset should be neglected in the calculations. As such, the depth of the gusset plate in compression may be calculated as

in 85.74

18

5353

53190.390.39

2

180.39

2

cb

which then gives the width in tension as

bt = 39.0 – 7.85 = 31.2 in

WP

bw/2

bw/2

bt

bc

bg

bg x tg

bfspi x tfspi

bfspo x tfspo

bwsp x twsp Fy.wsp

Fy.fspi

Fy.fspo

Fy.fspi

Fy.fspo

Fy.g

Fcr.g

332

After deducting the holes within this tension width, one obtains a net tension width of

bte = 31.2 – 8 x 1 = 23.2 in

Upon substituting all of the dimensional and material parameters into Eq. (7), on obtains

kip 35401666608064682534.085.7

804.02.23,534.085.70.39min)801.12,532.15min()8085.5,5391.8min(2

nP

for the chord splice eccentric tension capacity. The corresponding force from the test simulation is

Ptest = 1.29 x [3000 kip + 283 kip cos(45)] = 4130 kip

which gives a resulting professional factor of

Ptest/Pn = 1.17

Figure 346. Method 2 plastic stress distribution in gusset plate for evaluation of chord splice

eccentric tension, shown on P2-C-TCS-WV-M.

 18

12

12

30Fy.g on Ag orFu.g on Ae

Fcr.g

bt

bc

2900 kips

424 kips

500 kips

283 kips

3000 kips

333

5.3.3 Importance of considering splice eccentricity

The NCHRP 12-84 project final report (Ocel, 2013) indicates a significantly unconservative nature of the traditional Whitmore section checks if these checks are applied to evaluate the contribution of the gusset plates to a truss chord splice tension or compression capacity. Figure 347 shows the distribution of the professional factors for the chord splice tests conducted in this research if the contribution of the gusset plates to the chord splice resistance is based simply on the Whitmore section of the gusset plates. The mean for these Rtest /Rn values is 1.02 and the COV is 0.11. However, five of these 28 tests have professional factors less than 0.90. This plot may be compared to Figures 327 and 328, where only one test had an Rtest /Rn less than 0.90 for both Methods 1 and 2. The cause of the lower Rtest /Rn values, when the chord splice capacity is evaluated using the Whitmore section for the gusset plate, is that this approach completely neglects the eccentricity of the Whitmore section relative to the centerline of the chord members. The NCHRP 12-84 project ultimately decided to recommend a more conservative form of Method 1 for the chord splice eccentric tension or compression checks, permitting the use of ordinary resistance factors with the resistance calculations. If the Whitmore section approach were used, it is likely that significantly smaller resistance factors would be required with the resulting design calculations to achieve the targeted level of reliability.

Figure 347. Professional factors for Whitmore section calculation of chord splice eccentric compression and chord-splice eccentric tension resistance.

334

Nevertheless, it can be observed that the Whitmore section method gives a mean resistance closest to 1.0 and a COV that is essentially the same as that for Method 2. Therefore, it is of some interest to illustrate how the Whitmore section based checks work for a representative test joint P2-C-TCS(0.4)-WV-M. The corresponding Whitmore section for the gusset plate contribution to the chord splice is illustrated in Figure 348.

Figure 348. Whitmore section model of chord splice neglecting gusset eccentricity, shown on

P2-C-TCS(0.4)-WV-M.

The width of the Whitmore section in this problem is

bWhitmore = 27 tan (30o) + 16.5 = 32.1 in

(note that basic spacing of the fasteners is 3 inches in this and the other parametric study test problems), giving a gross Whitmore area of

Ag.Whitmore = 32.1 x 0.4 x 2 = 25.7 in2

Therefore, the corresponding yield capacity is

Fy Ag.Whitmore = 1360 kip

Deducting the area of eight holes in each of the two gusset plates, the net area of the Whitmore section in this problem is

An.Whitmore = Ag.Whitmore – 2 x 8 x 1 x 0.4 = 19.3 in2

giving a tension rupture capacity of

 

18

12

12

30

335

Fu An.Whitmore = 1540 kip

The tension yield capacity of the gusset plates governs, and when added with the governing tension rupture capacity of the each of the web splice plates (468 kip) and the governing tension yield capacity of the flange splice plates (806 kips), the total chord splice tension capacity is obtained as

Pn = 2 x (468 kip + 806 kip) + 1360 kip = 3910 kip

This prediction may be compared to the tension transferred by the splice at the maximum strength condition in the test simulation of

Ptest = 1.29 x [3000 kip + 283 kip cos(45)] = 4130 kip

giving a professional factor of

Ptest/Pn = 1.06

5.4 CONCENTRIC TENSION OR COMPRESSION

One of the parametric tests studied in this research, P12-C(1.0)-W-P, had a distinctly different behavior compared to all the other tests. In this test, the large forces from two heavily chamfered diagonals were delivered to a bearing at an interior pier location where the joint transfers substantial compression from the chord members from each side (see Figure 349). As such, to maintain the continuity of the chords through the joint, and to avoid the use of interior diaphragm plates within the chord members, the gusset plates were designed to transfer the vertical resultant of the diagonal compression forces to the bearing over a 48 inch width of the bearing. As shown in Section 4.7.2, the vertical compression in the gusset plates at the top of the chord is critical in this problem. Essentially, since the total width at the bottom of the diagonals is less than the 48 inch width of the bearing, the simulation studies indicate that it is unconservative to use a width larger than 48 inches for the effective column section in this problem. Using a width for the two gusset plates of

bgp = 48 in

at the critical section, an equivalent column length of

L = 8.4 in

and K = 0.5, the gusset plates have a KL/r = 14.5, which is smaller than the limit of 25 at which the gusset plate compressive resistance Fcr may be taken equal to Fy = 53 ksi. Upon multiplying this stress by the gusset plate areas, one obtains

Pn = 2 x 53 x 48 x 1.0 = 5088 kip

336

The corresponding strength reached in the test simulation was

Ptest = 1.05 x 5000 kip = 5250 kip

which gives a professional factor of

Ptest /Pn = 5250 / 5088 = 1.03

Figure 349. Stress distribution for concentric compression check, shown on P12-C(1.0)-W-P.

60

21

70.5

4

21

337

5.5 FULL SHEAR PLANE YIELDING

This section explains the Method 1 and Method 2 calculations for checking the “Full Shear Plane Yielding” (FSPY) limit state. Several examples are provided from test simulation cases where the capacity was governed by these checks. The calculation of the strength of the full shear plane using a theoretical expression for the fully-plastic strength from beam theory also is considered.

5.5.1 Method 1

The Method 1 calculation of the FSPY resistance is simply

gFSPyn tLFV 58.02 Eq. (8)

where LFSP = length of the full shear plane. The application of this check to the single test of the parametric simulation studies where this limit state governed is shown below. It should be noted that this limit state was close to governing in many other test cases; however, the conservatism of the separate Method 1 Partial Shear Plane Yielding (PSPY) check eliminated all but one test from this category in the Method 1 calculations.

5.5.1.1 Method 1 Example, Full Shear Plane Yielding – P15-C(0.5)-CJ

The application of the Method 1 FSPY check to the test P15-C-(0.5)-CJ is illustrated in Figure 350. This check only applies the vertical plane highlighted by the bold line in this figure, since the horizontal plane along the bottom of the top chord is intersected by the cross-section of the truss vertical. However, the highlighted vertical plane passes through the intersection of the top chord and the vertical member, where only the gusset plate provides a connection between the two members (i.e., there are no other splice plates in this joint).

Based on Figure 350, the FSPY resistance may be calculated as

kips 21435.07.695358.02 nV

(note that the fasteners are spaced at 3 inches on center in this and all of the other parametric study test simulation models). The corresponding strength achieved in the test simulation is

Vtest = 1.15 x 2000 = 2300 kips

and therefore, the professional factor for this case is

Vtest/Vn = 2300 / 2143 = 1.08

The reader is reminded that the simulation studies in this research are not focused on predicting tension rupture or shear rupture conditions (see Section 3.5). Therefore, the calculation of the shear rupture resistance of the gusset plates is not considered here.

338

Figure 350. Full shear plane and joint free-body diagram for P15-C-CJ.

5.5.1.2 Method 1 Example, Full Shear Plane Yielding – E3-U-307SL(3/8)-WV and E3-U-307SL(3/8)-W

Figure 351 shows the forces acting on the joint in test E3-U-307SL(3/8)-WV at the maximum load achieved in the FEA test simulation. This load level is 1.03 time the reference loads, which for this and the other experimental “WV” (Warren truss with Vertical) test joints, are the actual forces in each of the members at the maximum load level reached in the experimental test. Although the Method 1 resistance calculations for this joint indicate a Diagonal Buckling (DB) mode of failure (see Table 5), the FSPY resistance is nearly the same as the DB resistance in this case. Therefore, it is informative to observe how the FSPY checks work for this problem. It is useful to also consider the test E3-U-307SL(3/8)-W (i.e., the same test simulation, but with the vertical member removed from the joint), since the FSPY resistances of these tests are the same regardless of where the vertical is present or not.

 

WP

2300 kips

2576 kips

1150 kips

14.9 in

69.7 in

6 in

Full Shear Plane

339

Figure 351. Full shear plane and joint free-body diagram for E3-U-307SL-WV.

For these test cases, the calculated Method 1 FSPY resistance is

kips 1348375.05.666.4658.02 nV

where the yield strength of the gusset plates is 46.6 ksi and the width of the full shear plane, highlighted in Figure 351, is 66.5 in. The corresponding shear force reached in the test simulation is

Vtest = 1.03 x (706 + 620) = 727 + 639 = 1366 kips for E3-U-307SL(3/8)-WV

giving a

Vtest/Vn = 1366 / 1348 = 1.01 for E3-U-307SL(3/8)-WV

and

Vtest = 1.00 x (706 + 620) = 1326 kips for E3-U-307SL(3/8)-W

giving a

Vtest/Vn = 1326 / 1348 = 0.98 for E3-U-307SL(3/8)-W

340

It is important to note that for the governing Method 1 limit states for these tests (DB), the actual professional factors are Vtest/Vn = 1.23 and 1.10 respectively. Hence, the above Vtest/Vn values of 1.01 and 0.98 are somewhat misleading. The governing Method 1 resistance is more conservative than the resistance calculated by assuming that this test is governed by the shear resistances. When taken to the extreme, if one performs a calculation for a design limit state that is not anywhere near the governing limit state calculation for a given problem, one can obtain a very small test strength to nominal strength ratio. This does not mean that the limit state prediction is unconservative. If another limit state governs for the calculated resistance, it is likely that the calculated limit state with the very small ratio does not actually have any relevance to the mode of failure encountered in the test.

5.5.2 Theoretical Plastic Strength Interaction on Full Shear Plane, Elastic-Plastic Material Idealization

It is informative to consider the prediction of shear resistance of the full shear plane in the two example problems considered in Section 5.5.1 using a representative theoretical plastic strength interaction equation to account for the plastic interaction between the shear stresses and the normal stresses acting along the critical shear plane. The following subsections present these calculations.

5.5.2.1 Plastic Strength Interaction Check on Full Shear Plane – P15-C(0.5)-CJ

The shear plane in Figure 350 is actually subjected to combined shear and moment,

Mtest = 14.9 x 1.15 x 1000 = 17,140 in-kips

Vtest = 2000 x 1.15 = 2300 kips

at the strength limit in the test simulation, although the total axial force transferred across this plane is

Ptest = 0 kips

The Method 1 FSPY resistance is the fundamental fully-plastic shear resistance, and therefore, we can write

Vp = Vn = 2143 kips

Furthermore, the fully-plastic flexural resistance of the two gusset plates at the full-shear plane is

Mp = 2 x 53 x 0.5 x 69.72 / 4 = 64,370 in-kips

and although there is no contribution from a net axial load across the full-shear plane in this problem, the fully-plastic axial resistance may be written as

341

Py = 2 x 53 x 0.5 x 69.7 = 3694 kips

If the fully-plastic strength interaction equation recommended by Astaneh (1998) (see Section 1.2) is considered for this problem, then the fully-plastic strength interaction is

42

p

test

y

test

p

test

V

V

P

P

M

M0.266 + 0 + 1.0734 = 1.46

The maximum shear force corresponding to the development of the fully-plastic cross-section capacity, considering the above interaction, is

Vmax.CPM = 1859 kips

and therefore, the professional factor one would obtain, assuming that this is the governing strength check, is

Vtest / Vmax.CPM = 2300 / 1859 = 1.24

One can observe that the test simulation indicates a substantially greater strength than predicted using the above plastic strength interaction equation. This is largely due to the well-known fact that strain-hardening can influence the resistance substantially in cases where there is a substantial stress or strain gradient. The above so-called “rigorous” plastic strength check is based on the assumption of shear deformable beam theory, i.e., plane sections remain plane but not necessarily normal to the axis of the member, and the assumption that the material response is elastic-perfectly plastic. The gusset plate geometry in this problem is very “squat” compared to the typical dimensions where one might expect beam theory to be an accurate representation of reality. In addition, physical steel materials are not elastic-perfectly plastic.

5.5.2.2 Plastic Strength Interaction Check on Full Shear Plane – E3-U-307SL(3/8)-WV

A number of additional insights regarding the application of fully-plastic beam strength interaction equations can be gained by considering the application of this approach to the full shear plane in test E3-U-307SL(3/8)-WV. In this case

Mtest = 5 x 1.03 x 946 x sin(45) + 5 x 1.03 x 929 x sin(45) = 6830 in-kips

Vtest = 1.03 x (620 + 706) = 1366 kips

and

Ptest = 0 kips

whereas the corresponding resistances are

342

kips 1348375.05.666.4658.02 pV

Mp = 2 x 46.6 x 0.375 x 66.52 / 4 = 38,640 in-kips

Py = 2 x 46.6 x 0.375 x 66.5 = 2324 kips

This gives a strength interaction check of

42

p

test

y

test

p

test

V

V

P

P

M

M0.177 + 0 + 1.0134 = 1.23

The maximum shear force corresponding to the development of the fully-plastic cross-section capacity, considering the above interaction, is

Vmax.CPM = 1288 kips

such that

Vtest / Vmax.CPM = 1366 / 1288 = 1.06

This is slightly more conservative than the value of Vtest/Vp = 1.01 obtained based on the Method 1 FSPY strength check. However, the governing Method 1 limit state check for this problem is Diagonal Buckling (DB) with a professional factor of 1.23 (see Table 5).

With respect to plastic strength interaction, it is important to emphasize again that the above interaction equation assumes that the gusset plate responds according to beam theory and that the material response is elastic perfectly plastic. However, it should be noted that in this problem, the gusset plate contains significant 11

components of stress (where the 1-1 axis is the horizontal axis in Figure 351). The above so-called “rigorous” strength interaction equation does not consider the influence of the 11 stress components on the plasticity behavior at the critical shear plane. If the plate has significant tension stress normal to the shear plane (22), then compressive 11 stresses would theoretically reduce the plastic resistance along the shear plane, while if the plate has significant compressive 22 normal to the shear plane, then tensile 11 stresses would theoretically reduce the plastic resistance along the shear plane. Conversely, compressive 11 would tend to increase the compressive 22 values causing yielding, and tensile 11 would tend to increase the tensile 22 values associated with yielding. There are no simple hand calculation procedures that easily capture these interactions. However, the FEA simulations do, and they also capture the benefits of strain hardening and do not impose artificial beam theory constraints on the kinematics of the deformation.

343

5.5.3 Method 2

The Full Shear Plane Yield (FSPY) strength calculations for Method 2 are the same as those for Method 1 except that the nominal resistance is taken as 0.9 of the fully-plastic shear resistance.

gFSPygFSPyn tLFtLFV 52.0258.09.02 Eq. (9)

The 0.9 coefficient is essentially a calibration factor associated with the development of the Method 2 FSPY and DB (Diagonal Buckling) models. It was found that the use of Eq. (9) along with the subsequently discussed Method 2 DB models led to the smallest overall COV for the combined checks from these two limit states. The corresponding Method 2 calculation of the FSPY strengths for P15-C(0.5)-CJ, E3-U-307SL(3/8)-WV and E3-U-307SL(3/8)-W are shown below. As shown in Tables 10, 5 and 6 respectively, the FSPY check governs the Method 2 calculations for all of these tests.

5.5.3.1 Method 2 Example, Full Shear Plane Yielding – P15-C(0.5)-CJ

The Method 2 FSPY check for test P15-C(0.5)-CJ is

kips 19215.07.695352.02 nV

(see Figure 350 for the problem dimensions and an illustration of the critical shear plane). As noted previously,

Vtest = 1.15 x 2000 = 2300 kips

and therefore,

Vtest/Vn = 2300 / 1921 = 1.19

This is the governing (largest) Vtest/Vn for this test, as indicated in Table 10.

5.5.3.2 Method 2 Example, Full Shear Plane Yielding – E3-U-307SL(3/8)-WV and E3-U-307SL(3/8)-W

The Method 2 FSPY check for E3-U-307SL(3/8)-WV and E3-U-307SL(3/8)-W is

kips 1210375.05.666.4652.02 nV

(see Figure 351 for the problem dimensions and an illustration of the critical shear plane). As noted previously,

Vtest = 1.03 x (706 + 620) = 727 + 639 = 1366 kips for E3-U-307SL(3/8)-WV

344

such that

Vtest/Vn = 1366 / 1210 = 1.13 for E3-U-307SL(3/8)-WV

and

Vtest = 1.00 x (706 + 620) = 1326 kips for E3-U-307SL(3/8)-W

giving

Vtest/Vn = 1326 / 1210 = 1.10 for E3-U-307SL(3/8)-W

As noted above, these are the governing (largest) Method 2 strength ratios for these tests.

5.6 DIAGONAL BUCKLING, FULL WHITMORE SECTION

This section introduces the Method 1 and Method 2 equations for calculation of the gusset plate diagonal compression buckling resistance for cases in which the strength is governed by the full Whitmore section. For Method 1, this is all cases unless the Whitmore section intersects a free edge or a plane of symmetry. As noted previously, Method 2 uses a truncated Whitmore section for these cases as well as whenever this section intersects the fastener lines of adjacent members.

5.6.1 Method 1

The Method 1 Diagonal Buckling (DB) resistance calculations can be explained using the test E1-U-307SS-WV shown in Figure 352. The Whitmore width is calculated as

WWhitmore = Wconn + 2 Lconn tan(30) Eq. (10)

where Wconn is the width of the fastener group connecting the compression diagonal to the gusset plate and Lconn is the length of the connection from the first to the last fastener along this connection group. This is the “standard” Whitmore section width used traditionally with the “Thornton method” discussed in Section 1.2. The Whitmore area of the two gusset plates, that is, the area of the equivalent column used to evaluate the compression capacity of the gusset plate, is therefore

Ag = 2WWhitmore tg Eq. (11)

In Method 1, the length of the equivalent column is taken as

L = Lavg = (L1 + L2 + L3)/3 Eq. (12)

where the lengths L1, L2 and L3 are taken as the distances along the orientation of the compression diagonal from the Whitmore plane at the end of the compression diagonal to the

345

adjacent fastener lines as shown in Figure 351. The Method 1 effective length factor for the equivalent column is taken as

K = 0.5 Eq. (13)

Given this effective length factor, the length Lavg, and the radius of gyration of a general rectangular area

12gtr Eq. (14)

the theoretical Euler elastic buckling stress of the above equivalent column may be written as

22

2

)/(

29.3

)/( gavgavge tL

E

rKL

EF

Eq. (15)

Finally, the value of Fe obtained from Eq. (15) may be substituted into the standard AASHTO LRFD steel column strength equations

otherwise 877.0

25.2/ if658.0 /.

ge

eygyFF

Whitmoren

AF

FFAFP ey

Eq. (16)

to determine the Diagonal Buckling resistance for Method 1.

It should be noted that, as of this writing (January 2013), the Method 1 calculations recom-mended for integration into the AASHTO Specifications have been changed to the use of Lmid as a simplification. However, the use of Lavg is retained in the Method 1 calculations evaluated in this research. The Method 1 calculations exhibit significantly greater overall accuracy with the use of Lavg rather than Lmid. Table 45 compares the summary statistics for the two different equivalent column length approximations. One can observe that the results for the cases with chamfered members are only slightly more conservative overall with the use of Lmid rather than Lavg. However, for the cases involving No Chamfer, the predictions are significantly more conservative overall with the use of Lmid. The minimum professional factor for the No Chamfer cases is increased to 0.77 with the use of Lmid however, versus 0.58 with the use of Lavg.

346

Figure 352. Method 1 gusset plate diagonal buckling model with full Whitmore section, shown on E1-U-307SS-WV.

Table 45 Summary assessment of Method 1 professional factors for the cases governed either by diagonal tension yielding or by the combined diagonal buckling and partial shear plane

yielding limit state check with Lmid versus Lavg used for the equivalent column length in the diagonal buckling calculation.

Method Method 1 with Lavg for the DB check

Method 1 with Lmid for the DB check

Geometry No Chamfer Chamfer No Chamfer Chamfer

Count 88 59 88 59

Mean 1.11 1.14 1.30 1.17

COV 0.15 0.14 0.23 0.14

Max 1.53 1.43 3.03 1.62

Min 0.58 0.77 0.79 0.77

347

The resistance of the gusset plates to the compression from a diagonal member tends to be over-predicted using the above Diagonal Buckling (DB) equations particularly for joints with tightly-spaced members. However, these predictions are improved by combining them with a separate “Partial Shear Plane Yielding” (PSPY) check. This separate check is based on the observation that early shear yielding occurs between the compression diagonal and one or both of the adjacent members in situations where the DB equations tend to over-predict the resistance. The corresponding reduction in the gusset plate stiffnesses appears to have an influence on their stability. The PSPY check first requires the identification of the critical partial shear plane of the two planes along the fastener lines in the two adjacent members. These candidate planes are illustrated by the lines Lpsp1 and Lpsp2 in Figure 353 for E1-U-307SS-WV.

Figure 353. Method 1 partial shear planes on E1-U-307SS-WV.

The critical partial shear plane is defined as the one that:

Parallels the chamfer in the compression diagonal, for cases where only one side of the compression diagonal is chamfered, or has the greater chamfer, if both sides are chamfered,

Lpsp1 = 25.8

Lps

p2 =

24.

0

psp1

psp2

psp1 = psp2 = 45o

348

Has the smaller framing angle between the compression diagonal and the adjacent members, if neither side of the compression diagonal is chamfered, or

Has the smallest cross-sectional shear area, if the compression diagonal is unchamfered and the framing angles with the adjacent members are equal.

The “critical partial shear plane” is usually the plane that has the largest average shear stress developed due to the component of the diagonal compression tangent to the plane. However, in some cases, these rules define the plane with the smaller average shear stress as the critical plane. This typically occurs when one side is heavily chamfered or has a significantly smaller offset relative to the adjacent member than the other side, leading to a significantly shortened length of the shear plane on the opposite side. In these cases, the side selected by the above rules tends to be the most critical. It appears that this is the case because of the relative stiffness of the two sides (larger force tends to be attracted to the side with the larger stiffness), as well as the nature of the forces in these types of joints.

Once the critical partial shear plane is identified, the PSPY resistance is basically taken as the compression diagonal axial force that is developed by assuming its component tangent to the critical plane is equal to the shear yield strength of the partial plane, i.e.,

cos

58.02.

garPartialSheyarPartialShen

tLFP Eq. (17)

This partial shear plane equation is not applied for cases involving diagonal tension. This is because shear yielding of the critical partial plane is a behavioral attribute that tends to precipitate diagonal buckling. However, when the diagonal is in tension, shear yielding of the partial plane does not precipitate any limit state condition corresponding to the diagonal. Method 1 does require checking of the combined total resistance from the shear planes on each side of truss vertical members. This check ensures that truss verticals subjected to compression can develop their compressive resistance into the gusset plate.

Given the above calculation associated with the critical partial shear plane from Eq. (17), the nominal resistance in diagonal compression is taken as the minimum of the strengths from DB and from PSPY:

Pn = min(Pn.Whitmore, Pn.PartialShear) Eq. (18)

The following sections illustrate the application of the Method 1 combined DB and PSPY checks.

349

5.6.1.1 Method 1 Example, Diagonal Buckling, Full Whitmore Section – E1-U-307SS(3/8)-WV

The application of the Method 1 combined Diagonal Buckling (DB) and Partial Shear Plane Yielding (PSPY) checks to E1-U-307SS(3/8)-WV (see Figure 352) is as follows. The yield strength of the gusset plates in this physical test was

Fy = 36.4 ksi

and based on the lengths L1, L2 (= Lmid) and L3 in Figure 352,

Lavg = 6.2 in

The full Whitmore section width in this problem is

WWhitmore = 10 + 2 x 12.5 x tan(30) = 24.4 in

which gives an equivalent column area of

Ag = 2 x 24.4 x 0.375 = 18.3 in2

Based on Eq. (15), the theoretical elastic bucking resistance of the equivalent Whitmore column is

ksi 349)375.0/2.6(

000,2929.32

eF

and therefore

104.0349

4.36

e

y

F

F

This results in a Whitmore column buckling resistance of

kips 6393.184.36957.03.184.36658.0 104.0. WhitmorenP

The length of the governing partial shear plane in this problem, shown in Figure 353, is

LPartialShear = Lpsp2 = 24.0 in

therefore giving a diagonal compression force of

kips 537)45cos(

375.00.244.3658.02.

arPartialShenP

350

corresponding to the partial shear plane yielding limit state. Similar to a majority of the gusset plates evaluated in this research, the PSPY strength governs:

Pn = min(639, 537) = 537 kips

The corresponding diagonal compression force reached in the test simulation is

Ptest = 0.94 x 716 = 673 kips

giving

Ptest/Pn = 673/537 = 1.25

5.6.1.2 Method 1 Example, Diagonal Buckling, Full Whitmore Section – E1-U-307SS(3/8)-W

It is insightful to observe how the Method 1 combined DB and PSPY checks change in the above problem if the diagonal is removed, giving test E1-U-307SS(3/8)-W (see Figure 354).

Figure 354. Method 1 gusset plate diagonal buckling model with full Whitmore section, shown

on E1-U-307SS-W.

WWhitmore = 24.4

L2 = 13.2FcrL1 = 0.96

L3 = 25.4

30o30o

30o

Lavg = (L1 + L2 + L3)/3 = 13.2

Wconn = 10 Lconn = 12.5

493 kips

716 kips

716 kips

520 kips

351

In this case, the yield strength is still taken as

Fy = 36.4 ksi

but the average of the lengths L1, L2 and L3 is changed to

Lavg = 13.2 in

because of the absence of the vertical. However, the Whitmore section with is still

WWhitmore = 10 + 2 x 12.5 x tan(30) = 24.4 in

and the corresponding area is still

Ag = 2 x 24.4 x 0.375 = 18.3 in2

Given the larger length Lavg, the elastic buckling stress of the equivalent Whitmore column is now

ksi 0.77)375.0/2.13(

000,2929.32

eF

giving

473.00.77

4.36

e

y

F

F

and

kips 5473.184.36820.03.184.36658.0 473.0. WhitmorenP

The partial shear planes for this problem are illustrated in Figure 355. The length along the bottom chord governs, resulting in a PSPY strength of

kips 593)45cos(

375.05.264.3658.02.

arPartialShenP

As such, the DB strength of the gusset plates now governs.

Pn = min(547, 593) = 547 kips

The diagonal compression force at the strength limit in the test simulation for this problem is

352

Ptest = 0.83 x 716 = 594 kips

which gives

Ptest/Pn = 593/547 = 1.09

Figure 355. Method 1 partial shear planes on E1-U-307SS-W.

5.6.1.3 Method 1 Example, Diagonal Buckling, Full Whitmore Section – E3-U-307SL(3/8)-WV

Test E3-U-307SL(3/8)-WV represents a case where the Method 1 Diagonal Buckling resistance is based on the full Whitmore section, without any truncation (plus the partial shear plane yielding checks). However, in the subsequent illustrations of the Method 2 calculations, the Diagonal Buckling strength involves some minor truncation due to the intersection of the Whitmore section with the bolt lines in one of the members adjacent to the compression diagonal (see Section 5.7.2.2). The Method 1 Whitmore section model is shown in Figure 356.

The yield strength in this physical test was

Fy = 46.6 ksi

and based on the geometry shown in the figure,

Lavg = (0 + 13.2 + 0.2) / 3 = 4.4 in

353

and

WWhitmore = 10 + 2 x 20 x tan(30) = 33.1 in

Therefore, the area of the equivalent Whitmore column is

Ag = 2 x 33.1 x 0.375 = 24.8 in2

and this column’s theoretical elastic buckling stress is

ksi 693)375.0/4.4(

000,2929.32

eF

This gives

067.0693

6.46

e

y

F

F

and an equivalent column buckling resistance of

kips 11238.246.46972.08.246.46658.0 067.0. WhitmorenP

The governing partial shear plane in this problem is again the plane along the side of the connection of the vertical to the gussets, similar to the result for E1-U-307SS-WV shown in Figure 353:

LPartialShear = Lpsp2 = 27.7 in

Therefore, the PSPY resistance is

kips 794)45cos(

375.07.276.4658.02.

arPartialShenP

which like many cases, governs over the Diagonal Buckling resistance,

Pn = min(1123, 794) = 794 kips

The corresponding Diagonal Buckling strength from the test simulation is

Ptest = 1.03 x 946 = 974 kips

which gives

354

Ptest/Pn = 974/794 = 1.23

Figure 356. Method 1 gusset plate diagonal buckling model with full Whitmore section, shown on E3-U-307SL-WV.

5.6.1.4 Method 1 Example, Diagonal Buckling, Full Whitmore Section – E3-U-307SL(3/8)-W

Similar to the investigation of E1-U-307SS(3/8)-WV and E1-U-307SS(3/8)-W in Sections 5.6.1.1and 5.6.1.2, it is useful to understand how the capacity predictions change if the vertical member is removed from the joint. The corresponding geometry for E3-U-307SL(3/8)-W is shown in Figure 357.

The yield strength used in this simulation study is the same as that in the above physical test,

Fy = 46.6 ksi

but

Lavg = (0 + 13.2 + 29.7) / 3 = 14.3 in

355

is now significantly larger, due to the length L3. It should be noted that if the Method 1 strengths are based solely on Lmid, as in Ocel (2013), there is no difference in the Diagonal Buckling strength calculation relative to that above for E3-U-307SL(3/8)-WV. In addition, since the controlling partial shear plane has a larger length in this problem compared to E3-U-307SL(3/8)-WV, one obtains the paradoxical result that the strength of the gusset is increased by taking the vertical member out.

As shown in both Figures 356 and 357,

WWhitmore = 10 + 2 x 20 x tan(30) = 33.1 in

and thus the area of the equivalent Whitmore column section is

Ag = 2 x 33.1 x 0.375 = 24.8 in2

However, given the larger equivalent column length in Figure 357,

ksi 6.65)375.0/3.14(

000,2929.32

eF

and

710.06.65

6.46

e

y

F

F

which gives an inelastic Diagonal Buckling resistance of

kips 8598.246.46743.08.246.46658.0 710.0. WhitmorenP

The governing partial shear plane in this problem falls along the bottom chord and is similar to that shown in Figure 355. Its length is

LPartialShear = 31.5 in

resulting in a diagonal compression force corresponding to the PSPY limit state of

kips 903)45cos(

375.05.316.4658.02.

arPartialShenP

The Diagonal Buckling resistance governs, giving

Pn = min(859, 903) = 859 kips

356

This may be compared to Pn = 794 kips for E3-U-307SL(3/8)-WV. Paradoxically, the predicted strength is increased by 8 % by removing the vertical member. However, in the test simulation, the strength for E3-U-307SL(3/8)-W is

Ptest = 1.00 x 946 = 946 kips

whereas it was 974 kips for E3-U-307SL(3/8)-WV, a 3 % decrease. The corresponding professional factor for E3-U-307SL(3/8)-W is

Ptest/Pn = 946/869 = 1.10

Figure 357. Method 1 gusset plate diagonal buckling model with full Whitmore section, shown

on E3-U-307SL-W.

5.6.1.5 Method 1 Example, Diagonal Buckling, Full Whitmore Section – P6-U(0.25)-WV-NP

The parametric study test P6-U(0.25)-WV-NP gives the smallest professional factor of all of the Method 1 combined Diagonal Buckling (DB) and Partial Shear Plane Yielding (PSPY) limit state calculations. Method 1 gives a disturbingly small professional factor of only 0.58 for this problem. Therefore, it is important to lay out the Method 1 calculations and scrutinize them for this problem. The geometry of the full Whitmore section Diagonal Buckling model for P6-U(0.25)-WV-NP is shown in Figure 358, and the geometry of the partial shear planes is shown in Figure 359 for this problem.

357

Figure 358. Method 1 gusset plate diagonal buckling model with full Whitmore section, shown on P6-U-WV-NP.

Figure 359. Method 1 partial shear planes on P6-U-WV-NP.

As noted previously, the yield strengths are taken as

Fy = 53 ksi

in all the parametric study test problems in this research, unless noted otherwise. From Figure 358, the Method 1 length of the equivalent Whitmore column is

Lavg = (0 + 18.7 + 3.9) / 3 = 7.5 in

and the Whitmore section width is

358

WWhitmore = 27 + 2 x 15 x tan(30) = 46.2 in

which gives an equivalent column area of

Ag = 2 x 46.2 x 0.25 = 23.1 in2

Given the above short length of the equivalent Whitmore column, the column theoretical elastic buckling stress is

ksi 106)25.0/5.7(

000,2929.32

eF

giving

500.0106

53

e

y

F

F

and a column Diagonal Buckling strength of

kips 9931.2353811.01.2353658.0 500.0. WhitmorenP

The length of the controlling partial shear plane is

LPartialShear = 41.2 in

from Figure 359, which gives a PSPY resistance of

kips 895)45cos(

25.02.415358.02.

arPartialShenP

Taking the smaller of the two above strengths as the governing compressive resistance of the gusset plate along the direction of the diagonal, we have

Pn = min(993, 895) = 895 kips

Interestingly, the corresponding resistance from the test simulation is only

Ptest = 0.23 x 2260 = 520 kips

giving the professional factor

Ptest/Pn = 520/895 = 0.58

359

The main reasons for the excessive over-prediction of the test simulation resistance in this problem appear to be:

KLavg = 0.5Lavg is too small for this problem (Lmid is more than double Lavg, and therefore, the use of Lmid rather than Lavg improves the results for this problem,

WWhitmore is too large for this problem. There is a substantial overlaps of the Whitmore section with the bolt lines within the top chord of this truss.

The partial shear plane limit state check does not restrict the above errors sufficiently for this problem.

The subsequently demonstrated Method 2 professional factor for this problem (see Section 5.7.2.5) is 0.96.

5.6.1.6 Method 1 Example, Diagonal Tension Yielding, Full Whitmore Section – P18-C(0.6)-POS

This section presents a final example of the Method 1 calculations using the full Whitmore section, but in this case, the diagonal member is in tension rather than in compression. The problem is P18-C(0.6)-POS, and the geometry used in the calculation of its Diagonal Tension capacity is shown in Figure 360. This problem is a case where it is easy for a “thinking Engineer” to become very confounded by the complexity of what sort of interaction may exist between the strengths associated with the load transfer of the tension from the two web tension members to the chord members.

As noted previously, the yield strength assumed in the parametric study problems is

Fy = 53 ksi

unless noted otherwise. Based on Figure 360, the Whitmore section width for the larger diagonal member is

WWhitmore = 18 + 2 x 33 x tan(30) = 56.1 in

giving a gross area of the Whitmore section of

Ag = 2 x 56.1 x 0.6 = 67.3 in2

It is assumed here that tension rupture does not govern the resistance, since the FEA simulations focus on overall gross yielding in tension in cases like this one, rather than tension rupture. Hence, the Diagonal Tension resistance is calculated as

kips 35673.6753. WhitmorenP

360

Since the diagonal is in tension, not compression, in this problem, Method 1 does not employ any partial shear plane yielding check for the reasons discussed at the beginning of Section 5.6. The corresponding diagonal tension at the strength limit in the simulation model is

Ptest = 1.3 x 2120 = 2756 kips

which gives a professional factor of

Ptest/Pn = 2756/3567 = 0.77

Figure 360. Method 1 gusset plate diagonal tension yielding model with full Whitmore section,

shown on P18-C(0.6)-POS

There are other cases somewhat like this one, particularly cases with simpler Warren with vertical configurations, where there appears to be some interaction between the compression diagonal and compression vertical strengths in the gusset plate. The predictions are most flawed for cases with small gusset plates near the middle of a bridge span, if the mill-to-bear condition of the chord is counted upon in the design (which is not generally allowed by AASHTO).

Fy

WWhitmore = 56.1

2830 kips

3500 kips

2120 kips

500 kips

361

Method 1 tends to over-predict the resistance particularly for these types of problems. However, although the Method 1 professional factor above is 0.77, the truncated Whitmore section model of Method 2 gives a professional factor of 1.14 for this case (see Section 5.7.2.6). One should note that there is substantial overlap of the Whitmore sections for the two web members, and hence “double counting” of the material in the Method 1 calculations for this type of problem.

5.6.2 Method 2

This section summarizes the recommended Method 2 approach for cases where the Whitmore section is not truncated by the fastener lines in the adjacent members. As noted previously in Section 5.1 and shown in Table 44, the mean professional factor for the 47 test cases where the Method 2 calculations take this form is 1.12, and the coefficient of variation on the professional factor is only 0.05.

For cases where the Whitmore section is not truncated by any adjacent fastener lines, the Method 2 calculation of the Diagonal Buckling resistance is very straightforward and simple. The Whitmore section width is taken as

WWhitmore = Wconn + 2 Lconn tan(30) Eq. (19)

and the corresponding area is

Ag = 2WWhitmore tg Eq. (20)

Furthermore, the equivalent Whitmore column length is determined using just Lmid, but with the effective length factor taken as

0.35 for Warren trusses with verticals

0 44 otherwise

K

.

Eq. (21)

These effective length factors are based on calibration of the basic Diagonal Buckling model to the test simulation resistances determined in this research. It should be emphasized that the fact these values are smaller than K = 0.5 should be no cause for concern. The gusset plate not an actual column. Its geometry, boundary conditions and detailed stability behavior are far different than an axially loaded prismatic column.

The radius of gyration of the equivalent Whitmore column is generally

12gtr Eq. (22)

Hence, if we substitute this into the basic equivalent column elastic flexural buckling equation, we obtain

362

otherwise)/(

25.4

icals with vertssesWarren tru)/(

71.6

)/(

2

22

2

gmid

gmidmide

tL

E

tL

E

rKL

EF

Eq. (23)

for the equivalent column theoretical elastic buckling stress. This theoretical stress is then substituted into the steel column strength equations

otherwise 877.0

25.2/ if658.0 /

ge

eygyFF

n

AF

FFAFP ey

Eq. (24)

to determine the gusset plate Diagonal Buckling strength.

5.6.2.1 Method 2 Example, Diagonal Buckling, Full Whitmore Section – E1-U-307SS(3/8)-WV

Figure 361 shows the Method 2 calculations for E1-U-307SS(3/8)-WV, which was addressed earlier in Section 5.6.1.1 using Method 1. The yield strength of the gusset plates in this physical test was

Fy = 36.4 ksi

The Whitmore equivalent column length by Method 2 is

Lmid = 13.2 in

and the Whitmore section width is

WWhitmore = 10 + 2 x 12.5 x tan(30) = 24.4 in

as illustrated in the figure. The corresponding equivalent column area is

Ag = 2 x 24.4 x 0.375 = 18.3 in2

and the corresponding theoretical elastic buckling stress is

ksi 157)375.0/2.13(

000,2971.62

eF

Thus

363

232.0157

4.36

e

y

F

F

and the diagonal buckling strength of the gusset plates is estimated as

kips 6043.184.36907.03.184.36658.0 232.0 nP

This corresponds to the diagonal compression strength of

Ptest = 0.94 x 716 = 673 kips

in the test simulation, and a professional factor of

Ptest/Pn = 673/604 = 1.11

Figure 361. Method 2 gusset plate diagonal buckling model with full Whitmore section, shown

on E1-U-307SS-WV.

This Method 2 professional factor can be compared to a professional factor of 1.25 for the Method 1 calculations in Section 5.6.1.1. The primary reason for the more conservative calculation of the Diagonal Buckling (DB) resistance in Section 5.6.1.1 is that this resistance is governed by the Partial Shear Plane Yielding (PSPY) check in the previous section. However, from the results of Method 2, it is clear that:

364

1) Lmid is the better choice for the equivalent column length, and

2) The PSPY check is not really needed to control any unconservative errors

in the cases where the Whitmore section is not truncated significantly by fastener lines in the adjacent members (see Table 44).

5.6.2.2 Method 2 Example, Diagonal Buckling, Full Whitmore Section – E1-U-307SS(3/8)-W

This section shows the Method 2 calculations for E1-U-307SS(3/8)-W, that is, the above gusset plate problem with the vertical member removed. This problem was addressed previously in Section 5.6.1.2 using Method 1. The geometry associated with the Method 2 Diagonal Buckling calculations is shown in Figure 362. As stated above, the yield strength of the gusset plates in this problem is

Fy = 36.4 ksi

For Method 2, there is no difference in the equivalent column length for this problem compared to E1-U-307SS(3/8)-WV. That is,

Lmid = 13.2 in

Also, the width of the Whitmore section is the same as in E1-U-307SS(3/8)-WV:

WWhitmore = 10 + 2 x 12.5 x tan(30) = 24.4 in

The corresponding equivalent column area is

Ag = 2 x 24.4 x 0.375 = 18.3 in2

and the theoretical elastic buckling stress of the Whitmore equivalent column is

ksi 5.99)375.0/2.13(

000,2925.42

eF

This gives

366.05.99

4.36

e

y

F

F

and

kips 5713.184.36858.03.184.36658.0 366.0 nP

365

The smaller calculated strength in this problem, compared to that in the previous section, is due to the use of K = 0.44 for cases other than Warren truss configurations with verticals in Method 2.

The corresponding diagonal buckling capacity in the test simulation is

Ptest = 0.83 x 716 = 594 kips

and therefore,

Ptest/Pn = 594/571 = 1.04

In Section 5.6.1.2, Method 1 gave a professional factor of 1.09, slightly more conservative than the Method 2 calculations here. The primary reasons for the more conservative prediction in Section 5.6.1.2 are:

1) The use of K = 0.5, and

2) The use of Lavg

in Section 5.6.1.2

Figure 362. Method 2 gusset plate diagonal buckling model with full Whitmore section, shown on E1-U-307SS-W.

366

5.7 DIAGONAL BUCKLING, TRUNCATED WHITMORE SECTION

5.7.1 Method 1

In Method 1, the Whitmore section is truncated at intersections with free edges of the gusset plate, and at planes of symmetry. After all, one cannot count on thin air to transfer force, and in addition, the Whitmore section is suspect if it crosses a plane of symmetry, since one would then be double-counting from the two sides. However, otherwise, the full Whitmore section is used regardless of any intersections with adjacent fastener lines or intersections with the Whitmore section of adjacent members. The following sections show several examples of the Method 1 calculations for cases where the Whitmore section is truncated by the free edge of a plate. Comparable calculations for Method 2 are shown subsequently.

5.7.1.1 Method 1 Example, Diagonal Buckling, Truncated Whitmore Section – P7-C(0.375)-WV-INF

Figure 363 shows a case where the Whitmore section is truncated by the top edge of the gusset plates. As with the other parametric study test cases,

Fy = 53 ksi

unless noted otherwise. The truncated width of the Whitmore plane is

WWhitmore = 57.0

as shown in the figure, giving

Ag = 2 x 57.0 x 0.375 = 42.8 in2

Given that

Lmid = 13.7 in

but L1 = L3 = 0,

Lavg = 13.7 in / 3 = 4.6 in

Therefore, for this case, the theoretical elastic buckling resistance is

ksi 634)375.0/6.4(

000,2929.32

eF

for the gusset plate,

367

084.0634

53

e

y

F

F

and the Diagonal Buckling strength is

kips 21908.4253966.08.4253658.0 084.0. WhitmorenP

Figure 363. Method 1 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P7-C-WV-INF.

Figure 364 illustrates the governing partial shear plane for this problem. Since

LPartialShear = Lpsp2 = 71.0 in

the diagonal compression corresponding to the PSPY limit state is

kips 1830)6.26cos(

375.00.715358.02.

arPartialShenP

As with the majority of other cases, the diagonal compression resistance is governed by the PSPY check, i.e.,

Pn = min(2190, 1830) = 1830 kips

The capacity achieved in this test simulation is

Ptest = 0.59 x 3580 = 2112 kips

and therefore,

Ptest/Pn = 2112/1830 = 1.15

Fcr

Lmid = 13.7 WWhitmore = 57.0

L1 = L3 = 0

3200 kips 2200 kips

3580 kips

500 kips

2460 kips

368

Section 5.7.2 investigates the Method 2 calculations for this example.

Figure 364. Method 1 partial shear planes on P7-C-WV-INF.

5.7.1.2 Method 1 Example, Diagonal Buckling, Truncated Whitmore Section – P5-C-HS(0.2)-WV-NP

P5-C-HS(0.2)-WV-NP is a case where the Whitmore section intersects the free edge of the gusset plate on one side only slightly (see Figure 365 and L1 = 0). Furthermore, on the other side of the compression diagonal, we do not have a plane of symmetry, but the Whitmore section substantially overlaps the truss vertical to the point that the starting location for L3 is already on the opposite side of the vertical from the compression diagonal. Therefore, L3 is also taken equal to zero.

The yield strength in this problem is

Fy = 108 ksi

since this is one of the high-strength steel plate study cases. As shown in Figure 365,

WWhitmore = 49.7 in

giving

Ag = 2 x 49.7 x 0.2 = 19.9 in2

Also, given that L1 = L3 = 0, the average length used for the equivalent Whitmore column is

Lavg = (0 + 10.2 + 0) / 3 = 3.4 in

369

and the corresponding theoretical elastic buckling resistance is

ksi 330)2.0/4.3(

000,2929.32

eF

This gives

327.0330

108

e

y

F

F

and

kips 18749.19108872.09.19108658.0 327.0. WhitmorenP

Figure 365. Method 1 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P5-C-HS-WV-NP.

Figure 366 shows the controlling partial shear plane for the Method 1 calculation of the diagonal compression strength in this problem.

370

Figure 366. Method 1 partial shear planes on P5-C-HS-WV-NP.

Given

LPartialShear = 40.2 in

the PSPY strength is

kips 1126)6.26cos(

2.02.4010858.02.

arPartialShenP

which governs substantially compared to the Diagonal Buckling strength:

Pn = min(1874, 1126) = 1126 kips

Nevertheless, for this thin high-strength steel gusset plate problem, the test simulation resistance

Ptest = 0.64 x 1500 = 960 kips

is still overpredicted by a substantive margin, giving a professional factor of

371

Ptest/Pn = 960/1126 = 0.85

This test is studied further using Method 2 in Section 5.7.2.4.

5.7.2 Method 2

Figures 367 and 368 show how Method 2 works with respect to the truncation of the Whitmore section by the fastener lines in adjacent members. The detailed considerations in the Method 2 “Truncated Whitmore Section” (TWS) model are as follows:

The truncated Whitmore section width is subdivided generally into three parts, WL, WM, and WR.

The width between the points where the Whitmore section intersects the adjacent fastener lines, WM, is referred to as the main or “M” part of the Whitmore section. The critical stress on this length, FcrM, is calculated in the same way as for the cases where the Whitmore section is not truncated by the adjacent fastener lines, except this strength is based on LM, the distance to the adjacent fastener line from the middle of WM.

The other widths, WL and WR are the left- and right-hand projects of the fastener lines that truncate the Whitmore section onto the Whitmore plane. We calculate separate lengths LL and LR as the perpendicular distances from the fastener lines that truncate the Whitmore section to the closest fastener lines in the diagonal member.

K = 0.35 is used with the lengths LL and LR in calculating FcrL and FcrR, which are taken as base normal stresses acting on the widths WL and WR.

We use 0.9FcrL and 0.9FcrR as the nominal axial stresses on WL and WR at the maximum strength condition.

Figure 368 illustrates that the use of 0.9FcrL and 0.9FcrR on the projected widths WL and WR is physically equivalent to the assumption of a stress state along the portions of the fastener lines that truncate the Whitmore section that involves a normal stress of 0.9FcrL

or 0.9FcrR on a cut perpendicular to the axis of the compression diagonal, zero normal stress on a cut parallel to the axis of the member, and zero shear stress on both of these cuts. This plane stress state then actually corresponds to an assumed state of shear and compression stress on a cut parallel to the fastener lines on the adjacent members.

The separate calculation of FcrL and FcrR based on LL and LR often does not give us much im-provement over doing something much simpler, such as, say using FCRM on WL + WM + WR. However, doing this gives noticeable improvements for a number of the more extreme cases. Note that we do not concern ourselves with the lack of satisfaction of moment equilibrium by the loads on the different portions of the truncated Whitmore section. The eccentricities caused by

372

the traditional truncation of the Whitmore section at free edges or planes of symmetry have never been a worry either. The calibrations with the simulation results tend to be sufficient without the need to address this complexity.

Figure 367. Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P7-C-WV-INF.

Figure 368. Method 2 gusset plate diagonal buckling model with truncated Whitmore section, shown on P7-C-WV-INF – illustration of assumed state of stress at the bolt lines truncating the

Whitmore Section.

0.9FcrL

FcrMLM = 6.7

WL = 19.9

WM = 26.8

WR = 9.7

0.9FcrRLR = 48.9

LL = 5.73200 kips 2200 kips

3580 kips2460 kips

500 kips

373

The following subsections illustrate the application of this TWS model to various example problems.

5.7.2.1 Method 2 Example, Diagonal Buckling, Truncated Whitmore Section – P7-C(0.375)-WV-INF

Figure 367 shows the various parameters relevant to the Method 2 calculation of the diagonal buckling resistance for P7-C(0.375)-WV-INF. The Method 1 resistance was evaluated for this problem in earlier Section 5.7.1.1. The yield strength of the gusset plates is

Fy = 53 ksi

in this problem. The contribution to the resistance from the left-hand side is determined as follows:

WL = 19.9 in; LL = 5.7; ksi 842)375.0/7.5(

000,2971.62

eLF ; 063.0

842

53

eL

y

F

F

kips 69346.753877.02375.09.1953658.09.02 063.0 nLP

Similarly, the contribution to the resistance from the right-hand side is

WR = 9.7 in; LR = 48.9; ksi 4.11)375.0/9.48(

000,2971.62

eRF ; 65.4

4.11

53

eR

y

F

F

kips 66375.07.94.11877.09.02 nRP

and the contribution from the “Main” part of the Whitmore section is

WM = 26.8 in; LM = 6.7; ksi 610)375.0/7.6(

000,2971.62

eMF ; 087.0

610

53

eM

y

F

F

0.0872 0.658 53 26.8 0.375 2 0.964 53 10.0 1022 kipsnMP

These three capacities are summed to obtain

Pn = PnL + PnR + PnM = 693 + 66 + 1022 = 1781 kips

The corresponding diagonal buckling resistance obtained in the test simulation is

Ptest = 0.59 x 3580 = 2112 kips

374

giving

Ptest/Pn = 2112/1781 = 1.18

This is slightly more conservative than the professional factor of 1.15 obtained from the Method 1 procedure for this problem.

5.7.2.2 Method 2 Example, Diagonal Buckling, Truncated Whitmore Section – E3-U-307SL(3/8)-WV

The strength for test E3-U-307SL(3/8)-WV was evaluated using Method 1 earlier in Section 5.6.1.3. Using Method 2, this test has some minor truncation of the Whitmore section on its left-hand side (see Figure 369). Therefore, this is a good example to explore the differences in the calculations and the results for the two different methods for a problem involving a minor amount of truncation. The yield strength of the gusset plates in this physical test is

Fy = 46.6 ksi

Figure 369. Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on E3-U-307SL-WV.

375

Based on Figure 370, the contribution of the left-hand truncated portion to the resistance is

WL = 2.4 in; LL = 13.0 in; ksi 162)375.0/0.13(

000,2971.62

eLF ; 288.0

162

6.46

eL

y

F

F

kips 67900.06.46886.09.02375.04.26.46658.09.02 288.0 nLP

and the contribution from the “Main” portion of the Whitmore section is

WM = 29.5 in; LM = 15.0 in; ksi 122)375.0/0.15(

000,2971.62

eMF ; 382.0

122

6.46

eM

y

F

F

kips 8811.116.46852.02375.05.296.46658.02 382.0 nMP

Therefore, the summation of the contributions to the total resistance is

Pn = PnL + PnR + PnM = 67 + 0 + 881 = 948 kips

The corresponding maximum load observed in the test simulation is

Ptest = 1.03 x 946 = 974 kips

for this problem. Therefore, the professional factor associated with Diagonal Buckling (DB) is

Ptest/Pn = 974/948 = 1.03

Interestingly, (DB) is not the governing limit state in this test when using Method 2. The Full Shear Plane Yielding (FSPY) check gives a larger professional factor with Vtest /Vn = 1.13 (see Section 5.5.3.2). Therefore, this test is categorized as failing by FSPY by Method 2. However, when evaluated using Method 1, the governing limit state is the combined Diagonal Buckling, Partial Shear Plane Yielding check, and Ptest /Pn = 1.23 (see Section 5.6.1.3). The Method 1 Vtest / Vn is 1.01 (see Section 5.5.1.2).

5.7.2.3 Method 2 Example, Diagonal Buckling, Truncated Whitmore Section – E3-U-307SL(3/8)-W

It is useful to also consider the above test in the case where the vertical is removed from the joint. The corresponding geometry corresponding to the Method 2 Diagonal Buckling checks is shown in Figure 370. As noted above, the yield strength is

Fy = 46.6 ksi

376

Figure 370. Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on E3-U-307SL-W.

The strength determined from the left-hand portion of the truncated Whitmore Section is the same in this case as in E3-U-307SL-WV:

WL = 2.4 in; LL = 13.0 in; ksi 162)375.0/0.13(

000,2971.62

eLF ; 288.0

162

6.46

eL

y

F

F

kips 67900.06.46886.09.02375.04.26.46658.09.02 288.0 nLP

However, the “Main” section resistance is slightly smaller for this case, due to the use of K = 0.44 for general problems versus K = 0.35 for the prior Warren truss with verticals:

WM = 29.5 in; LM = 15.0 in; ksi 0.77)375.0/0.15(

000,2925.42

eMF ; 605.0

0.77

6.46

eM

y

F

F

kips 8031.116.46776.02375.05.296.46658.02 605.0 nMP

Summing the separate capacities, we have

Pn = PnL + PnR + PnM = 67 + 0 + 803 = 870 kips

The corresponding test simulation resistance for this problem is

377

Ptest = 1.00 x 946 = 946 kips

such that

Ptest/Pn = 946/870 = 1.09

Interestingly, similar to E3-U-307SL-WV, the Method 2 resistance in this test is governed by FSPY with Vtest /Vn = 1.10 (see Section 5.5.3.2). When Method 1 is used, both the combined DB and PSPY checks give a governing professional factor of 1.10 for this case (see Section 5.6.1.4). The Method 1 FSPY professional factor for this problem is 0.98 (see Section 5.5.1.2).

5.7.2.4 Method 2 Example, Diagonal Buckling, Truncated Whitmore Section – P5-C-HS(0.2)-WV-NP

The parametric study test P5-C-HS(0.2)-WV-NP was considered using Method 1 in Section 5.7.1.2. As discussed in that section, the traditional Whitmore section has a substantial overlap with the truss vertical in this problem, which is believed to be a key factor resulting in a low Ptest/Pn of 0.85. Hence, it is useful to investigate the behavior of the Truncated Whitmore Section (TWS) model of Method 2 for this problem. Figure 371 shows the geometry associated with the Diagonal Buckling calculations for P5-C-HS(0.2)-WV-NP.

Unfortunately, the Method 2 TWS model does not provide any improvement on the low professional factor from Method 1 for this problem. In fact, the Ptest/Pn is slightly smaller at 0.82. However, the is the smallest professional factor from the Method 2 TWS model for all of the tests considered in this research. As such, this is another important reason to consider the calculations for this test.

As noted previously, the yield strength in this problem is

Fy = 108 ksi

Based on Figure 371, the left-hand segment of the truncated Whitmore section contributes as follows to the resistance:

WL = 4.0 in; LL = 34.7; ksi 46.6)2.0/7.34(

000,2971.62

eLF ; 7.16

46.6

108

eL

y

F

F

kips 82.00.446.6877.09.02 nLP

Similarly, the right-hand segment calculations are:

WR = 9.7 in; LR = 4.0; ksi 486)2.0/0.4(

000,2971.62

eRF ; 222.0486

108

eR

y

F

F

378

kips 34494.1108820.022.07.9108658.09.02 222.0 nRP

and the “Main” segment contribution is

WM = 23.9 in; LM = 6.3; ksi 196)2.0/3.6(

000,2971.62

eMF ; 551.0196

108

eM

y

F

F

0.5512 0.658 108 23.9 0.2 2 0.794 108 4.78 820 kipsnMP

Figure 371. Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P5-C-HS-WV-NP.

Upon summing these three contributions, the total Diagonal Buckling capacity is obtained as

Pn = PnL + PnR + PnM = 8 + 344 + 820 = 1172 kips

Unfortunately, the maximum resistance obtained from the test simulation is only

LM = 6.3

LL = 34.7

LR = 4.0

WL = 4.0

WM = 23.9

WR = 9.7

0.9FcrL

FcrM

0.9FcrL 1500 kips

3000 kips

1500 kips

1200 kips

500 kips

379

Ptest = 0.64 x 1500 = 960 kips

thus giving the professional factor

Ptest/Pn = 960/1172 = 0.82

It appears that both Methods 1 and 2 have some limitations in their ability to capture the strengths of very thin high-strength steel plates.

5.7.2.5 Method 2 Example, Diagonal Yielding in Tension, Truncated Whitmore Section – P6-U(0.25)-WV-NP

Another challenging example previously considered using Method 1 in Section 5.6.1.5 is P6-U(0.25)-WV-NP. This problem gives the smallest governing professional factor of all the Method 1 combined Diagonal Buckling (DB) and Partial Shear Plane Yielding (PSPY) limit state calculations. Therefore, it is important to investigate how Method 2 works for this problem. The geometries associated with the Method 2 Diagonal Buckling calculations are shown in Figure 372.

As noted previously,

Fy = 53 ksi

for P6-U(0.25)-WV-NP. Based on Figure 372, the left-hand portion of the Truncated Whitmore Section (TWS) gives the following contribution to the resistance

WL = 2.8 in; LL = 17.8 in; ksi 4.38)25.0/8.17(

000,2971.62

eLF ; 38.1

4.38

53

eL

y

F

F

kips 38700.053561.09.0225.08.253658.09.02 38.1 nLP

Also, the “Main” portion of of the TWS gives

WM = 41.7 in; LM = 20.8 in; ksi 1.28)25.0/8.20(

000,2971.62

eMF ; 89.1

1.28

53

eM

y

F

F

kips 5004.1053453.0225.07.4153658.02 89.1 nMP

Upon summing the resistances, we have

Pn = PnL + PnR + PnM = 38 + 0 + 500 = 538 kips

This is relatively close to the test simulation resistance of

380

Ptest = 0.23 x 2260 = 520 kips

such that the Method 2 professional factor is

Ptest/Pn = 520/538 = 0.96

Figure 372. Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P6-U-WV-NP.

It appears that the Method 1 usage of Lavg in this problem resulted in an excessively low value of the equivalent column length and a resulting high prediction of the Diagonal Buckling (DB) resistance. Although the PSPY strength governs in Method 1 (see Section 5.6.1.5), the PSPY resistance doesn’t sufficiently limit the estimated capacity of the gusset plates to provide a good prediction of the test simulation capacity. However, with Method 2, the above length LM = 20.8 in gives a reduced capacity of the gusset plates that matches well with the simulation results. It should be noted that when Method 1 is used with Lmid = 18.7 inches in this problem, and K = 0.5, its professional factor becomes significantly conservative at Ptest/Pn = 1.48.

5.7.2.6 Method 2 Example, Diagonal Yielding in Tension, Truncated Whitmore Section – P18-C(0.6)-POS

The last basic example selected to illustrate the Method 2 TWS calculations is P-18-C(0.6)-POS, which was considered previously with Method 1 in Section 5.6.1.6. The geometry associated with the TWS calculations for this Diagonal Tension problem is shown in Figure 373.

381

Figure 373. Method 2 gusset plate diagonal tension yielding model with truncated Whitmore section, shown on P18-C(0.6)-POS.

The yield strength in this test is

Fy = 53 ksi

The contributions to the resistance in these problem are

WL = 11.5 in

kips 6586.05.11539.02 nLP

from the left-hand truncated Whitmore section width,

WR = 15.6 in

kips 8936.06.15539.02 nRP

from the right-hand truncated Whitmore section width, and

WM = 13.5 in

0.9Fy

0.9Fy

Fy

11.5

13.5

15.6

2830 kips

3500 kips

2120 kips

1000 kips

382

kips8596.05.13532 nMP

from the “Main” Whitmore section width. Upon summing these three capacities, the total resistance corresponding to the larger tension diagonal is taken as

Pn = PnL + PnR + PnM = 658 + 893 + 859 = 2410 kips

Given a test simulation resistance of

Ptest = 1.3 x 2120 = 2756 kips

this results in a professional factor of

Ptest/Pn = 2756/2410 = 1.14

This is a much improved prediction relative to the professional factor of 0.77 obtained using Method 1 in Section 5.6.1.6. A similar calculation would be made to check the ability of the gusset plate to transfer the tension from the vertical.

It should be noted that there is still substantial “overlap” or “double-counting” of material in the TWS calculations for this problem. However, by limiting the extent of the Whitmore section in the proposed fashion, we are able to keep the design calculations relatively simple while still obtaining reasonable accuracy for cases where the Whitmore section is clearly “running wild” in terms of overlaps and potential double-counting. Alternately, for problems of this nature in general, direct FEA calculations may be merited to ensure that an accurate estimate of the gusset plate capacity is obtained.

5.8 INFLUENCE OF SHINGLE PLATES

This section illustrates how the proposed Methods 1 and 2 are applied for gusset plates reinforced by shingle plates. The research studies indicate that as long as there is a sufficient number of fasteners to develop the force into the shingle plates via shear of the fasteners, one can calculate the total resistance of the combined gusset and shingle plates for a given strength limit state by applying the calculations as described in the previous sections separately to the gusset plates and to the shingle plates, then summing the resistances. That is, one may assume that all of the components are stressed at their nominal capacities when estimating the ultimate strength of an assembly. Table 11 of Section 4.2 provides a summary of the parametric study results. With the exception of test P12-C-SP(0.5:0.5)-W-P, both Methods 1 and 2 give reasonable results. The professional factor from Method 1 ranges from 1.07 to 1.26, whereas it ranges from 0.94 to 1.03 for Method 2. Test P12-C-SP(0.5:0.5)-W-P may be considered as an outlier, since this test involved a failure of the gusset and shingle plates under diametrically opposed axial forces applied over a small width from steep chamfered diagonals and from the bridge bearings at an interior support in a continuous-span Warren truss arrangement.

383

The following examples emphasize the calculation of the gusset plate resistances. They do not illustrate the equally important evaluation that the fasteners attaching the shingle plates to the gusset plates are adequate to develop the strength of the shingle plates.

5.8.1 Method 1

Figures 374 and 375 show the geometry associated with the calculation of the Method 1 Diagonal Buckling (DB) and Partial Shear Plane Yielding (PSPY) resistance contributions from the shingle plate in test P5-C-SP-WV-NP. The specific example calculations shown below are for P5-C-SP(0.3:0.3)-WV-NP. The yield strength of the all the plates is taken as

Fy = 53 ksi

in this problem. The shingle plate in this test has a uniform thickness of

tsp = 0.3 in

Figure 374. Method 1 shingle plate diagonal buckling model with full Whitmore section,

shown on P5-C-SP-WV-NP.

WWhitmore = 29.3

Fcr

1500 kips

3000 kips

1500 kips

1200 kips

500 kips

Lmid = 10.2

L3 = 0

L3 = 2.5

Lavg = (L1 + L2 + L3)/3 = 4.2

384

Figure 375. Method 1 splice plate partial shear plane for P5-C-SP-WV-NP.

From Figure 374, one can observe that

WWhitmore = 29.3 in

giving an area of the equivalent column in the shingle plate of

Ag = 2 x 29.3 x 0.3 = 17.6 in2

Based on the lengths L1, Lmid and L3 shown in the figure, the length of the equivalent Whitmore column is taken as

Lavg = (0 + 10.2 + 2.5) / 3 = 4.2 in

which gives a theoretical elastic buckling resistance of

1500 kips

3000 kips

1500 kips

1200 kips

500 kips

LPartialShear

= 28.5

385

ksi 487)3.0/2.4(

000,2929.32

eF

and

109.0487

53

e

y

F

F

Therefore, the contribution of the two shingle plates (one on each face of the truss) to the DB resistance is

kips 8916.1753955.06.1753658.0 109.0. WhitmorenP

Given the controlling partial shear plane length

LPartialShear = 28.5 in

shown in Figure 375, the diagonal compression resistance associated with the PSPY limit state is

kips 588)6.26cos(

3.05.285358.02.

arPartialShenP

The splice plate diagonal compression resistance calculation is completed by taking the smaller of the two above resistances as the governing resistance:

Pn.sp = min(891, 588) = 588 kips

From the gusset plates, similar to the calculations in Section 5.7.1.2 but using tg = 0.3 in and Fy = 53 ksi, we obtain the contribution of the gusset plates to the diagonal compression resistance (again, governed by the PSPY limit state) as

Pn.g = 830 kips

Therefore, the total resistance from the combined assembly of the gusset plates and the splice plates is

Pn = Pn.g + Pn.sp = 830 + 588 = 1418 kips

The test simulation indicates a maximum capacity for this problem of

Ptest = 1.14 x 1500 = 1710 kips

giving a professional factor of

386

Ptest/Pn = 1710/1418 = 1.21

Generally, all of the potential strength limit states must be evaluated to ensure they are sufficient to develop the required strengths associated with a statically admissible set of load paths throughout a given joint. Figure 376 shows the failure plane for calculation of the Full Shear Plane Yielding (FSPY) limit state in the shingle plates of P5-C-SP-WV-NP. The calculations below focus solely on the full shear plane yielding limit state. The shear rupture resistance of the net section along the line of fasteners at the bottom of the chord in this problem also must be checked. However, as noted previously, this research does not evaluate the assessment of shear or tension rupture of the steel.

Figure 376. Full shear plane on gusset and shingle plates for P5-C-SP-WV-NP.

The Method 1 contribution from the shingle plates to the FSPY resistance in this problem is simply

kips 12823.05.695358.02. spnV

Similarly,

1500 kips

3000 kips

1500 kips

1200 kips

500 kips

Full Shear Plane

387

Vn.g = 1282 kips

since the gusset plates have the same thickness, the same width of the failure plane, and the same yield strength in this example.

Therefore, the combined Method 1 shear capacity of the gusset plates plus the shingle plates is

Vn = Vn.g + Vn.sp = 2564 kips

The corresponding shear on these failure planes is

Vtest = 1.14 x 1519 = 1732 kips

at the maximum load capacity in the test simulation. This gives a professional factor of

Vtest/Vn = 1732 / 2564 = 0.68

It is important to recognize that this low professional factor simply means that the above shear planes are not loaded anywhere near to their capacity in the test simulation. The above diagonal compression strength check governs with a Ptest / Pn = 1.21. Generally, the largest professional factor determined from the various checks is the governing one. Furthermore, this is the only professional factor that can be legitimately compared against the test resistances to assess the ability of the strength models to predict the limit states behavior.

5.8.2 Method 2

Figure 377 shows the geometry associated with the calculation of the Method 2 Diagonal Buckling (DB) resistance for P5-C-SP(0.3:0.3)-WV-NP. As noted above, the yield strength of all the plates is taken as

Fy = 53 ksi

and the thickness of the shingle plates is taken as

tsp = 0.3 in

in this problem. From Figure 377, one can observe that the Whitmore section is truncated substantially by the truss vertical on the right-hand side of the compression diagonal, resulting in the following shingle plate contributions:

WR = 4.9 in; LR = 4.0; ksi 1042)3.0/1.4(

000,2971.62

eRF ; 051.01042

53

eR

y

F

F

kips 13747.153881.023.09.453658.09.02 051.0 nRP

388

Correspondingly, the “Main” portion of the truncated Whitmore section in the shingle plates contributes the following resistance:

WM = 19.0 in; LM = 7.4; ksi 320)3.0/4.7(

000,2971.62

eMF ; 166.0320

53

eM

y

F

F

kips56470.553933.023.00.1953658.02 166.0 nMP

Therefore, summing all the shingle plate contributions, the total shingle plate resistance to the diagonal compression is

Pn.sp = PnL + PnR + PnM = 0 + 137 + 564 = 701 kips

Figure 377. Method 2 shingle plate diagonal buckling model with truncated Whitmore section,

shown on P5-C-SP-WV-NP.

For the gusset plate, similar to the calculations in Section 5.7.2.4 but using tg = 0.3 in and Fy = 53 ksi, we obtain the contribution of the gusset plates to the diagonal compression resistance as

WM = 19.0

WR = 4.9

0.9FcrL

FcrM

1500 kips

3000 kips

1500 kips

1200 kips

500 kips

LM = 7.4

LR = 4.0

389

Pn.g = 1022 kips

Therefore, the total resistance from the combined assembly of the gusset plates and the splice plates is

Pn = Pn.g + Pn.sp = 1022 + 701 = 1723 kips

Given the test simulation capacity of

Ptest = 1.14 x 1500 = 1710 kips

noted also in the previous section, the professional factor for the Method diagonal compression resistance is

Ptest/Pn = 1710/1723 = 0.99

Although this is the governing Method 2 professional factor for test P5-C-SP(0.3:0.3)-WV-NP, it is informative to also consider the calculation of the Method 2 FSPY resistance for this problem. The critical shear plane is the same as that shown in Figure 376 for Method 1, but the Method 2 FSPY resistance is taken as 90 % of the value used in Method 1:

kips 11503.05.695352.02. spnV

Similarly,

Vn.g = 1150 kips

since the gusset plates have the same thickness, the same width of the failure plane, and the same yield strength in this example.

Therefore, the combined Method 2 shear capacity of the gusset plates plus the shingle plates is

Vn = Vn.g + Vn.sp = 2300 kips

The corresponding shear on these failure planes is

Vtest = 1.14 x 1519 = 1732 kips

at the maximum load capacity in the test simulation. This gives a professional factor of

Vtest/Vn = 1732 / 2300 = 0.75

It is important to recognize that this low professional factor simply means that the above shear planes are not loaded anywhere near to their capacity in the test simulation. The above diagonal

390

compression strength check governs with a Ptest / Pn = 0.99. Generally, the largest professional factor determined from the various checks is the governing one. Furthermore, this is the only professional factor that can be legitimately compared against the test resistances to assess the ability of the strength models to predict the limit states behavior.

5.9 INFLUENCE OF EDGE STIFFENERS

This section is included here to comment briefly on the influence of edge stiffening on the strength of gusset plates. Section 4.12 of this report explains several study cases in which stiffeners were placed at the free edges of the gusset plates. These studies essentially show that:

If the edge stiffeners are extended such that they connect into the cross-section of the truss diagonal and into the cross-section of the adjacent members, the edge stiffeners are very effective at permitting the gusset plates to develop substantial yielding prior to reaching their maximum resistance. This was the case even with gusset plates that were relatively thin (tg values as small as 0.3 inches were considered) and relatively slender such that the failure mode involved lateral buckling of the gussets with only a small amount of plasticity.

However, if the edge stiffeners were not connected into the truss diagonal and adjacent members, they had very little effect on the gusset plate capacity. It is apparent that the primary mechanism by which the “extended” edge stiffeners developed increased gusset plate strength is by restraining out-of-plane buckling such that a highly plastified condition could be attained in the gusset plates prior to their strength limit.

Table 12 in Section 4.2 summarizes the strengths calculated using Methods 1 and 2 for cases where the size of the “Extended Edge Stiffeners” was increased beyond the point where the strengths are no longer effected by the edge stiffener properties. The Method 1 and Method 2 strengths are then calculated for these cases assuming no influence of stability effects, i.e., using Fcr = Fy in the calculation of the Diagonal Buckling resistances. For the “Short Edge Stiffener” (SES) cases, the Method 1 and Method 2 strengths are calculated assuming no effect of the stiffeners. This decision is based on the observation in Section 4.12 that the edge stiffening is ineffective in the SES cases.

From Table 12, one can observe that both Methods 1 and 2, assuming the development of a fully-yielded equivalent Whitmore column, provide reasonable predictions of the above EES test simulations. The professional factors range from 1.15 to 1.33 for Method 1 and 0.89 to 1.19 for Method 2. The smallest professional factor occurs for the test P5-U-EES-WV-NP, for which the gusset plates are relatively very slender.

Since the Method 1 and Method 2 calculations based on the assumption of full yielding (i.e., Fcr = Fy for all the equivalent Whitmore column lengths) is relatively simple, no additional examples are provided here to demonstrate these calculations.

391

Normalized relationships were provided for the percentage strength gain versus the extended edge stiffener moment of inertia for several cases in Section 4.12 (see Figures 290 and 296). Unfortunately, the studies conducted here are not sufficient to permit the definition of an extended edge stiffener stiffness necessary to develop the full yield capacity of the gusset plate. It is apparent that this stiffness is a function of the slenderness of the equivalent Whitmore column representing the diagonal buckling resistance. If the equivalent Whitmore column is stocky enough such that its resistance is already close to the full yield resistance, only a small amount of additional stiffness is needed from the Extended Edge Stiffeners. However, the additional capacity that can be developed also is small in this case. In addition, it is recommended that further studies are needed involving gusset plate geometries that are extremely slender due to section loss. These studies are necessary to ensure that it is always possible to develop the fully-yielded strength of the gusset plates with sufficient edge stiffening.

5.10 HANDLING OF CORROSION EFFECTS

This section illustrates how the Diagonal Buckling (DB) and Full Shear Plane Yield (FSPY) strengths may be calculated for corroded gusset plates, using the test simulation problem P14-U-C2-W-INF. Only the Method 2 calculations are shown. The basic concept is the same for the Method 1 calculations. This concept is as follows:

At each point along the relevant Whitmore section width or widths outside of the width of the member’s fastener group, determine the smallest plate thickness along a line parallel to the compression diagonal that extends from the lines that fan out at 30o to define the Whitmore section to the intersection with the adjacent member fastener lines.

At each point along the width of the member’s fastener group, determine the smallest plate thickness along a line from the end of the fastener group to the intersection with the adjacent member fasterner lines.

Take the effective Whitmore column thickness as the average projeted thickness for any of the relevant portions of the Method 1 or Method 2 calculations, and take the equivalent Whitmore column area as the area based on a projection of all of these smallest plate thicknesses onto the Whitmore section plane (i.e., the Whitmore plane at the end of the member).

The bold black dashed lines in Figure 378 are examples of the above lines. The test simulation results summarized previously in Tables 13 and 14 of Section 4.2 demonstrate that this approach tends to give an accurate to conservative assessment of the influence of nonuniform section loss on the gusset plate resistance, even for cases where the corrosion exists only on a gusset plate on one face of the truss. The professional factors range from 1.23 to 1.67 for Method 1 and 1.06 to 1.42 for Method 2 for the cases studied for corroded gusset plates without any reinforcement.

392

They range from 1.11 to 1.49 for Method 1 and 1.02 to 1.28 for Method 2 for corroded gusset plates reinforced by shingle plates.

The above rule, or any other approximate rule to estimate the capacities must be applied with careful judgment, given the substantial complexity of the problem. For instance, suppose that the plate were essentially corroded all the way through the thickness along the entire right-hand side of the fastener group in the diagonal, indicated by the bold grey dashed line in Figure 378. In this case, it would be difficult to develop any significant contribution to the Diagonal Buckling capacity on the right-hand side of the diagonal member. It would be wise in this case to count only on the portion of the Whitmore section to the left of this corroded line.

Figure 378. Method 2 gusset plate diagonal buckling model with truncated Whitmore section,

shown on P14-U-C1-WV-INF.

Given the corrosion pattern for P14-U-C1-WV-INF shown previously in Figure 306 (see Section 4.13.3), the above simple rule can be applied as follows to estimate the capacity of the gusset plates. Figure 378 shows the Method 2 Truncated Whitmore Section (TWS) for this problem.

As with all the other parametric study test problems,

Fy = 53 ksi

393

unless noted otherwise. This is the value assumed in this test case. The Whitmore section is truncated slightly by the top fastener line on the bottom chord, such that

WL = 5.0 in

and

LL = 19.9 in

Based on the corrosion pattern detailed in Figure 306, 2.86 inches of WL does not have any section loss along a line parallel to the axis of the diagonal, whereas 2.14 inches of WL has a minimum projected thickness along the corresponding line (illustrated by the bold black dashed line on the left-hand side of the diagonal member) equal to 0.25 inches. Therefore, the effective thickness of the left-hand side of the Truncated Whitmore Section (TWS) in this problem is

in 393.00.5

25.014.25.086.2.

effLt

This thickness is substituted along with LL into the appropriate elastic buckling equation to obtain

ksi 9.75)393.0/9.19(

000,2971.62

eLF

or

698.09.75

53

eL

y

F

F

which indicates that inelastic buckling governs for the left-hand portion of the TWS. As such, the contribution of the left-hand side portion of the TWS to the Diagonal Bucking resistance is

kips 14096.153747.09.02393.00.553658.09.02 698.0 nLP

(note that tL.eff is used with the corresponding WL as the calculation of the equivalent column area of this portion of the TWS).

For the main portion of the Truncated Whitmore Section (TWS),

WM = 36.4 in

and

394

LM = 15.0 in

as shown in Figure 378. The projection of the smallest reduced thicknesses from the corroded sections indicated in Figure 306 onto the Whitmore plane gives a length of 2.46 inches in which the minimum thickness is 0.25 inches, 6.94 and 3.09 inch lengths over which the thickness is zero, a 5.01 inches of length over which the minimum thickness is 0.25, and a 18.90 inches of length over which there is no section loss. Therefore, the effective thickness for the main portion of the Whitmore section is

.

2.46 0.25 6.94 0 3.09 0 5.01 0.25 18.90 0.50.311 in

36.4M efft

This gives a theoretical elastic buckling strength of the main portion of

2

4.25 29,00053.0 ksi

(15.0 / 0.311)eMF

531.00

53.0y

eM

F

F

and a contribution to the overall diagonal compression resistance of

1.002 0.658 53 36.4 0.311 2 0.658 53 11.3 788 kipsnMP

Upon adding the above resistances together, the total diagonal compression resistance of the gusset plates is obtained as

Pn = PnL + PnR + PnM = 140 + 0 + 788 = 928 kips

This may be compared to the strength of

Ptest = 0.94 x 1400 = 1316 kips

from the test simulation, giving a professional factor of

Ptest/Pn = 1316/928 = 1.42

It is informative to also determine the Full Shear Plane Yield (FSPY) strength in P14-U-C1-WV-INF considering the reduced gusset plate thicknesses detailed in Figure 306. In general, it is recommended that the shear area for FSPY should be taken as the area using the minimum thickness within a small depth adjacent to the fastener line on the chord. In P14-U-C1-WV-INF, this gives an effective thickness of

395

in 349.01.57

5.06.3100.925.05.16

efft

Therefore, the Method 2 FSPY resistance may be calculated as

kips 1098349.01.575352.02 nV

The corresponding force developed on the critical shear plane at the strength of the simulated test is

Vtest = 0.94 x (590 + 730) = 555 + 686 = 1241 kips

where 590 kips and 730 kips are the reference bottom chord axial forces (see Figure 196). This gives a professional factor of

Vtest/Vn = 1241 / 1098 = 1.13

Since this professional factor is smaller than the above value (1.42), DB-TWS with a Ptest /Pn = 1.42 is the controlling limit state. As discussed in the previous sections, the value of Vtest /Vn of 1.13 is not a relevant value for assessing the FSPY shear strength, since the forces that can be developed by the joint in this problem are limited by the DB-TWS check first.

396

397

6. SUMMARY AND CONCLUSIONS

Based on the results of the studies conducted in this research, the following conclusions may be drawn:

Basic beam and column idealizations of the gusset plate responses can be used to obtain an accurate to somewhat coarse calculation of the strengths for design and rating.

For characterizing the gusset diagonal compression resistance, modifications of the traditional equivalent Whitmore column model are needed in general to ensure adequate predictions. Problems occur when the traditional Whitmore section has substantial overlaps with other members within a joint. These problems are addressed in two different ways in this work, termed Methods 1 and 2:

o In Method 1, an additional Partial Shear Plane Yielding (PSPY) check is imposed along a critical fastener line in one of the adjacent members. This check is based on the observation that significant shear yielding along one of the adjacent boundaries can precipitate an overall diagonal compression buckling of the gusset plate. The critical PSPY check governs over the traditional equivalent Whitmore column check in most cases where the gusset plate geometry is reasonably compact.

o In Method 2, the traditional equivalent Whitmore section is truncated at the fastener lines in the adjacent members, and an alternate Truncated Whitmore Section (TWS) calculation is applied. This TWS calculation is reasonably successful at restricting over-predictions by the traditional equivalent Whitmore column model even for the most demanding situations involving substantial overlaps of multiple diagonal member Whitmore sections for diagonals both in compression and in tension. The Method 1 calculations are somewhat unsuccessful at restricting the over-predictions for some of the more demanding cases.

Aside from the above modifications, the following attributes of the traditional equivalent Whitmore column model are found to work well:

o The dispersion angle for calculation of the Whitmore section is 30o

o The length of the equivalent Whitmore column may be taken as Lmid, the length along the direction parallel to the diagonal from the end of the fastener group in the diagonal to the closest adjacent fastener line. Method 1, as implemented in this report, utilizes the average of three lengths, Lavg. This approach is referred to

398

in the literature as the “Thornton” method after Thornton (1984). For the test simulations considered in this research, Lmid is simpler and tends to give better overall results. As such, the use of Lmid has been adopted in the final Method 1 model recommended to AASHTO by NCHRP 12-84.

o The effective length factor K providing the best correlation with the test simulation strengths is generally smaller than 0.5. In Method 2, K = 0.35 provides the best prediction of the mean strengths with the smallest dispersion in the professional factor for Warren truss configurations having vertical members. For other cases, Method 2 recommends K = 0.44. Method 1 recommends K = 0.5. However, K = 0.5 is generally an artificial and unnecessary limit for characterizing gusset plate strengths using the equivalent Whitmore column model. Separate calculations of the resistances for the various single gusset plate tests collected by Dowswell (2006 & 2012a) show adequate Method 1 and 2 predictions as well for most cases. The Method 1 model employed in this research gives a professional factor Ptest/Pn ranging from 18.4 to 0.85, and the Method 2 model gives a professional factor Ptest /Pn ranging from 11.8 to 0.78. The larger professional factors here correspond to extremely slender rectangular corner gusset plates, and the smaller values correspond to slender Chevron gusset plate configurations.

For characterizing the Full Shear Plane Yielding (FSPY) resistance such as calculated along the inside fastener line of a truss chord, this research shows that there are some minor strength reductions due to stability effects. However, this shear resistance is predicted well based on the assumption of fully-plastic behavior in shear. Method 1 bases its FSPY strength directly on the fully-plastic shear yield condition. Method 2 finds that the dispersion in the overall prediction of the different strength is minimized using 90 % of the fully-plastic shear yield condition for this limit state.

Approaches in the literature that recommend the application of a beam-theory based elastic-perfectly plastic strength interaction equation to quantify the plastic resistance of the full shear plane are ill founded. The normal stress in the direction parallel to the “cross-section cut” can be significant in steel truss gusset plates, due to the action of the gusset plate in transferring chord forces. Furthermore, structural steel is not physically elastic-perfectly plastic, and it is well known that strain hardening can have a positive influence on steel strengths in situations involving stress and strain gradients.

For characterizing the contribution of the gusset plates to chord splice resistances, the studies conducted in this research indicate that an approach referred to in the literature as the “effectiveness factor” (Kulicki and Reiner, 2011) generally provides a conservative assessment of the eccentricity of the chord relative to the gusset plate areas. The Method

399

1 calculations presented in the current research use this approach. Method 1 calculations that have been finalized for recommendation to AASHTO introduce some additional conservatism by handling the complete gusset plate and splice plate assembly as a composite beam cross-section. The calculations recommended by Kulicki and Reiner (2011) treat the gusset plate as an eccentric element, but treat the splice plates as separate concentrically-loaded elements.

The Method 2 calculations recommended in this research for calculation of the contribution of gusset plates to the chord splice resistance are based on a pseudo-plastic cross-section analysis applied to the gusset plates. This approach gives some improvements over the above Method 1 procedure, but has not been recommended to AASHTO because it deviates substantially from traditional practice. Further research studies, including experimental testing, may be merited prior to implementing this type of significant change.

For characterizing the contribution of shingle plates to the various resistances, this research finds that, as long as a sufficient number of fasteners is provided to develop the targeted shingle plate resistance at the ultimate strength level, it is acceptable to independently calculate the shingle plate contributions using the same models, and then simply sum the shingle plate contributions with the gusset plate contributions to determine the total resistance.

For characterizing the effect of edge stiffeners on the gusset plate resistance, the studies in this work indicate the following:

o If the edge stiffeners are not attached positively to the truss diagonal and the adjacent member, such that they would be engaged in resisting the out-of-plane movement of the diagonal, the edge stiffeners are generally ineffective at increasing the capacity of the gusset plates.

o If the edge stiffeners are attached to the diagonal and the adjacent member to develop frame action of the stiffeners in resisting out-of-plane movement of the diagonal, the edge stiffeners can effectively develop the equivalent Whitmore column full yield capacity in diagonal compression, if they are stiff enough.

o Sufficient research studies have not been conducted at this time to fully quantify the stiffness of the edge stiffeners necessary to achieve the increase in strength from the gusset diagonal compression capacity without edge stiffeners to the full yield capacity in diagonal compression.

The current research indicates very little correlation between the slenderness of the free edge of the gusset plates and the gusset plate limit state capacities. The gusset plate

400

diagonal compression capacity appears to be based predominantly on the force transfer from the end of the compression diagonal (i.e., within the “interior” region of the gusset). However, it is considered wise to continue to restrict the free edge slenderness to the current AASHTO free edge slenderness limit as a measure to avoid excessive distortion of the gusset plates.

For characterizing the strength of corroded gusset plates, the current research shows that accurate to conservative calculations of the diagonal compression resistance can be obtained by projecting the minimum thicknesses in the gusset plate onto the Whitmore section from lines parallel to the diagonal direction. Similarly, regarding the calculation of the Full Shear Plane Yielding (FSPY) resistance, one should use the minimum thickness within a small depth of the gusset plate adjacent to the critical shear plane.

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APPENDIX I-1. REPRESENTATIVE INTERNAL SHEAR FORCES AND MOMENTS IN A CONTINUOUS-SPAN WARREN TRUSS

The forces Va, V, and M2 in Warren truss joint configurations, with and without diagonals, are based on the following continuous-span truss bridge configuration:

Main span length, Lm = 456 ft.

Two end span lengths, La = 266 ft.

Number of traffic lanes per truss, nlanes = 4 (eight total lanes of traffic on the bridge).

Total weight per unit deck area, wD = 150 psf.

Width of deck per truss, bdeck = 60 ft.

Truss panel dimensions along the length of the bridge, bpanel = 38 ft.

AASHTO LRFD Strength I factored dead weight per truss panel point, 1.25 wD bdeck bpanel = 430 kips.

AASHTO LRFD multiple presence factor for eight lanes, mp = 0.65.

AASHTO LRFD lane load, qlane = 0.64 kips/ft.

AASHTO LRFD Strength I factored lane load per truss panel point, 1.75 qlane nlanes mp b-

panel = 110 kips.

AASHTO LRFD HS-20 truck load, QL = 72 kips.

AASHTO LRFD dynamic impact factor, IM = 1.33.

AASHTO LRFD Strength I factored truck loads at the truss panel points of 0.75 1.75 QL IM nlanes mp = 330 kips and 0.25 1.75 QL IM nlanes mp = 110 kips for the load cases causing maximum shear effects (Figure 379 shows this loading distribution for a single truck). For the load case causing maximum moment effects (see Figure 380), the corresponding panel point loads are 0.156 1.75 QL IM nlanes mp = 68 kips, 0.806 1.75 QL IM nlanes mp = 351 kips, and 0.038 1.75 QL IM nlanes mp =17 kips.

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Figure 379. Distribution of HS20 truck loads to truss panel points for load cases causing maximum shear effects in the representative continuous- or cantilever span truss bridge.

Figure 380. Distribution of HS20 truck loads to truss panel points for load cases causing maximum moment effects in the representative continuous- or cantilever span truss bridge.

The reader may find it interesting to note that the above parameters correspond approximately to the I-35 Bridge in Minneapolis, MN.

Given the above geometry and loading parameters, the moment-to-shear ratios at several locations within the main span of this design may be estimated coarsely by analyzing the main span of the bridge as an idealized prismatic fixed-fixed beam.

The maximum panel vertical shear force and the corresponding major-axis bending moment at the mid-span of the truss may be determined using the truss panel point loads illustrated in Figure 381. Similarly, the maximum moments at the mid-span and over the bridge piers may be estimated using the truss panel point loads illustrated in Figure 382, Figure 383 produces the maximum panel vertical shear force at the right-hand pier, and Figure 384 produces the maximum panel vertical shear force in the third panel from the right-hand support, in the vicinity of the inflection point within the span.

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Figure 381. Factored loads causing the maximum panel shear force at the mid-span of the representative continuous- or cantilever-span truss bridge.

Figure 382. Factored loads causing the maximum mid-span or pier-section moment for the representative continuous- or cantilever-span truss bridge.

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Figure 383. Factored loads causing the maximum shear force in the right-hand end truss panel for the representative continuous- or cantilever-span truss bridge.

Figure 384. Factored loads causing the maximum shear force in the third truss panel from the right-hand support for the representative continuous- or cantilever-span truss bridge.

Based on the above cases, the midspan, pier, and inflection point internal forces shown in Table 46 are obtained, where Vm is the larger of the two shears in the panels adjacent to the truss panel point being considered, Vadj is the shear force in the other adjacent panel, V is the change in the shear force between the two panels (equal to the applied vertical load at the targeted truss panel point), and Mm is the largest truss cross-section internal moment within the panels on each side of the targeted truss panel point.

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Table 46 Internal forces in the representative continuous- or cantilever-span truss bridge.

Case Vm

(kips)Vadj

(kips) V

(kips) Mm

(ft-kips) Mm/Vm

Midspan Vmax 546 -269 815 135,000 250 Midspan Mmax 465 -426 891 148,000 320 Pier Mmax 3,200 -2,400a 5,600 270,000 84 Pier Vmax 3,400 -2,400a 5,800 261,000 77 Inflection Point Vmax 2,214 1,399 815 84,000b 38

a Calculated based on 5quLa/8 for the end span, using the factored dead and lane loads. b Calculated as Vm bpanel based on the assumption that the inflection point falls at the truss

panel point

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APPENDIX I-2. REPRESENTATIVE INTERNAL SHEAR FORCES AND MOMENTS IN A SIMPLE-SPAN PRATT TRUSS

The Pratt subassemblies studied parametrically in this research are based on the following simple-span truss bridge configuration:

Span length, Ls = 240 ft.

Number of traffic lanes per truss, nlanes = 0.5 (one total lane of traffic on the bridge).

Total weight per unit deck area, wD = 150 psf.

Width of deck per truss, bdeck = 10 ft.

Truss panel dimensions along the length of the bridge, bpanel = 30 ft.

AASHTO LRFD Strength I factored dead weight per truss panel point, 1.25 wD bdeck bpanel = 56.2 kips.

AASHTO LRFD multiple presence factor for one lane, mp = 1.2.

AASHTO LRFD lane load, qlane = 0.64 kips/ft.

AASHTO LRFD Strength I factored lane load per truss panel point, 1.75qlane nlanes mp bpanel = 20.2 kips.

AASHTO LRFD HS20 truck load, QL = 72 kips.

AASHTO LRFD dynamic impact factor, IM = 1.33.

AASHTO LRFD Strength I factored truck loads at the truss panel points of 0.69 1.75 QL IM nlanes mp = 69.4 kips and 0.31 1.75 QL IM nlanes mp = 31.2 kips for the load cases causing maximum shear effects (Figure 385 shows this loading distribution for a single truck). For the load case causing maximum moment effects (see Figure 386), the corresponding panel point loads are 0.207 1.75 QL IM nlanes mp = 20.8 kips, 0.741 1.75 QL IM nlanes mp = 74.5 kips, and 0.038 1.75 QL IM nlanes mp = 3.8 kips.

Figure 385. Distribution of HS20 truck loads to truss panel points for load cases causing maximum shear effects in the representative simple-span truss bridge.

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Figure 386. Distribution of HS20 truck loads to truss panel points for load cases causing maximum moment effects in the representative simple-span truss bridge.

The focus for the Pratt truss joint studies is on the truss panel point located at the quarter-span of the representative simple-span structure. This is because the Warren truss with verticals joint configurations are believed to be sufficiently representative of the midspan configuration for the Pratt trusses. The joint configuration at a midspan location is essentially the same in a Pratt truss or a Warren truss with verticals. In addition, subsequent cases are considered that focus on the joint responses at and near a simply-supported end of a bridge. Based on the above geometry and loading parameters, the loading shown in Figure 387 produces the maximum shear force at the simple-span truss bridge quarter point, and the loading shown in Figure 388 produces the maximum major-axis bending moment at this location.

Figure 387. Factored loads causing the maximum panel shear force second panel from the support of the representative simple-span truss bridge.

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Figure 388. Factored loads causing the maximum moment at the second panel point from the support of the representative simple-span truss bridge.

Based on the above cases, the one-quarter point internal forces shown in Table 47 are obtained, where Vm is the larger of the two shears in the panels adjacent to the truss panel point being considered, Vadj is the shear force in the other adjacent panel, V is the change in the shear force between the two panels (equal to the applied vertical load at the targeted truss panel point), and Mm is the largest truss cross-section internal moment within the panels on each side of the targeted truss panel point.

Table 47 Internal forces in the representative simple-span truss bridge.

Case Vm

(kips)Vadj

(kips) V

(kips) Mm

(ft-kips) Mm/Vm

Quarter Point Vmax 264 118 146 21,400 81 Quarter Point Mmax 259 108 151 21,200 82