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Combined Bending and Torsion of Steel I-Shaped Beams

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Combined Bending and Torsion of Steel I-Shaped Beams University of Alberta Department of Civil & Environmental Engineering Structural Engineering Report No. 276 Combined Bending and Torsion of Steel I-Shaped Beams by Bruce G. Estabrooks and Gilbert Y. Grondin January, 2008 Combined Bending and Torsion of Steel I-Shaped Beams by Bruce E. Estabrooks and Gilbert Y. Grondin Structural Engineering Report 276 Department of Civil & Environmental Engineering University of Alberta Edmonton, Alberta January, 2008 Abstract Tests were conducted on six simply supported wide flange beams of Class 1 and 2 sections. The primary experimental variables included class of section, bending moment to torque ratio, and the inclusion or omission of a central brace. The end restraint and loading conditions were designed to simulate simply supported conditions at both end supports and a concentrated force and moment at midspan. Initial bending moment to torque ratio varied from 5:1 to 20:1. A finite element model was developed using the finite element software ABAQUS. Although the initial linear response was predicted accurately by the finite element analysis, the ultimate capacity was underestimates. A comparison of the test results with the design approach proposed by Driver and Kennedy (1989) indicated that this approach is potentially non-conservative and should not be used for ultimate limit state design. A design approach proposed by Pi and Trahair (1993) was found to be a suitable approach for ultimate limit states design. Acknowledgements This research project was funded by the Steel Structures Education Foundation and the National Research Council of Canada. The test program was conducted in the I.F. Morrison Laboratory at the University of Alberta. The great efforts that Mr. Richard Helfrich has put in the design and erection of the test frame were invaluable to the success of the test program. Feedbacks and comments provided by Dr. Robert Driver of the University of Alberta at the early stage of this project are acknowledged with thanks. Table of Contents 1. Introduction.................................................................................................... 1 1.1 General............................................................................................... 1 1.2 Objectives and Scope......................................................................... 1 2. Literature Review........................................................................................... 2 2.1 General............................................................................................... 2 2.2 Elastic Torsion Theory....................................................................... 2 2.3 Steel Beams Subjected to Torsion Only ............................................ 4 2.4 Elastic Design Methods for Combined Bending and Torsion ........... 5 2.5 Inelastic Design Methods for Combined Bending and Torsion......... 8 3. Experimental Program................................................................................... 15 3.1 Test Program Objectives.................................................................... 15 3.2 Material Tests..................................................................................... 15 3.3 Test Set-Up and Procedure ................................................................ 16 3.3.1 End Conditions....................................................................... 16 3.3.2 Loading of the Test Specimens.............................................. 18 3.3.3 Lateral Bracing System.......................................................... 19 3.4 Instrumentation and Data Acquisition ............................................... 21 4. Experimental Results..................................................................................... 34 4.1 General............................................................................................... 34 4.2 Material Properties............................................................................. 34 4.3 Pre-Test Measurements...................................................................... 34 4.4 Test Data and Observations ............................................................... 35 4.4.1 Vertical Deflection Measurements ........................................ 35 4.4.2 Lateral Displacement Measurements..................................... 37 4.4.3 Warping at End Supports....................................................... 39 4.4.4 Torque.................................................................................... 41 4.4.5 Bending Moment ................................................................... 47 4.4.6 Strain Distributions................................................................ 48 5. Finite Element Analysis................................................................................. 67 5.1 General............................................................................................... 67 5.2 Description of the Model ................................................................... 67 5.2.1 Elements and Mesh................................................................ 67 5.2.2 Initial Conditions................................................................... 69 5.2.3 Material Model....................................................................... 69 5.2.4 Boundary Conditions............................................................. 70 5.3 Analysis.............................................................................................. 70 5.4 Analysis Results................................................................................. 71 5.5 Sensitivity Study ................................................................................ 72 5.5.1 Sensitivity to Material Strength ............................................. 73 5.5.2 Sensitivity to Level of Lateral Restraint ................................ 74 5.5.3 Sensitivity to Level of Torsional Restraint ............................ 75 5.6 Other Sources of Errors ..................................................................... 76 5.7 Summary of the Finite Element Analysis .......................................... 76 5.8 Parametric Study................................................................................ 77 6. Discussion...................................................................................................... 94 6.1 General............................................................................................... 94 6.2 Elastic Design.................................................................................... 94 6.3 Driver and Kennedy........................................................................... 94 6.4 Pi and Trahair..................................................................................... 96 6.5 Finite Element Analysis..................................................................... 99 6.6 Summary............................................................................................ 100 7. Conclusions.................................................................................................... 110 7.1 Summary and Conclusions ................................................................ 110 7.2 Recommendations for Future Research............................................. 110 List of References ...................................................................................................... 112 Appendix A – Results from Material Tests ............................................................... 114 Appendix B – Cross-Sectional Measurements and Properties .................................. 118 Appendix C – Calculations ........................................................................................ 123 List of Tables Table 3.1 Test program ...................................................................................... 24 Table 4.1 Material properties............................................................................. 50 Table 4.2 Average cross-sectional dimensions.................................................. 50 Table 4.3 Average cross-sectional properties .................................................... 50 Table 4.4 Discrepancies between applied and resisted torques ......................... 51 Table 4.5 Normal strains at south end support measured at maximum test load 51 Table 5.1 First phase of analysis........................................................................ 79 Table 5.2 Mesh refinement study results ........................................................... 79 Table 5.3 Comparison of experimental and predicted capacities ...................... 80 Table A.1 Material properties of Class 1 coupons.............................................. 117 Table A.2 Material properties of Class 2 coupons.............................................. 117 Table B.1 Cross-sectional measurements........................................................... 120 Table B.2 Cross-sectional properties.................................................................. 121 List of Figures Figure 2.1 Shear stress distribution due to St. Venant torsion ............................ 13 Figure 2.2 Flange shears due to warping torsion................................................. 13 Figure 2.3 Normal stress distribution due to warping torsion............................. 14 Figure 2.4 Flexural analogy................................................................................. 14 Figure 3.1 Test set-up.........................................................................................
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