Distortional Static And Buckling Analysis Of Wide Flange Steel Beams

Distortional Static and Buckling Analysis of Wide Flange Steel Beams Payam Pezeshky Under the supervision of Dr. Magdi E. Mohareb Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For the Ph.D. degree in Civil Engineering Civil Engineering Department Faculty of Engineering University of Ottawa Payam Pezeshky, Ottawa, Canada, 2017

To my beloved parents, Soroush & Mahnaz Pezeshky ii

Acknowledgement Acknowledgement I would like to express my deepest gratitude to my supervisor, Dr. Magdi Mohareb, whose extensive knowledge, valuable advice, creative thought, technical and financial support, and commitment to the highest professional standards inspired me throughout this research. I am also thankful to the members of my thesis proposal committee Dr. Khoo, Dr. Martín-Pérez, and Dr. Dragomirescu for their thoughtful advice and encouragement. I would like to thank my dear friends Haleh and Farhood Nowzartash and their sweet daughters Leva and Ava for their moral support. My highest appreciations go to my dear parents, Soroush and Mahnaz, my beloved sisters, Sanaz and Sarir, my kindhearted brother Parham, my adorable niece Ma’va and my lovely nephew Vala for their unconditional and continuous love, patience and support. Special thanks go to the Baha’i Institute for Higher Education (B.I.H.E.) staff and colleagues in Iran for their sincere self-dedicated commitment towards the development of knowledge and growth of education. Indeed, without their sacrifice, I could not have made it up to this point. Last but not least, I would like to thank the Baha’i community of Ottawa, my wonderful circle of friends and my officemates Amir and Leonardo for their company and support. iii

Abstract Abstract Existing design provisions in design standards and conventional analysis methods for structural steel members are based on the simplifying kinematic Vlasov assumption that neglects cross-sectional distortional effects. While the non-distortional assumption can lead to reasonable predictions of beam static response and buckling strength in common situations, past work has shown the inadequacy of such assumption in a number of situations where it may lead to over-predicting the strength of the members. The present study thus develops a series of generalized theories/solutions for the static analysis and buckling analysis of steel members with wide flange cross-sections that capture distortional effects of the web. Rather than adopting the classical Vlasov assumption that postulates the cross-section to move and rotate in its own plane as a rigid disk, the present theories assume the web to be flexible in the plane of the cross-section and thus able to bend laterally, while both flanges to move as rigid plates within the plane of the cross-section to be treated as Euler-Bernouilli beams. The theories capture shear deformation effects in the web, as well as local and global warping effects. Based on the principle of minimum potential energy, a distortional theory is developed for the static analysis of wide flange steel beams with mono-symmetric cross-sections. The theory leads to two systems of differential equations of equilibrium. The first system consists of three coupled equilibrium differential equations that characterize the longitudinal-transverse response of the beam and the second system involves four coupled equilibrium differential equations of equilibrium and characterizes the lateral-torsional response of the beam. Closed form solutions are developed for both systems for general loading. Based on the kinematics of the new theory, two distortional finite elements are then developed. In the first element, linear and cubic Hermitian polynomials are employed to interpolate displacement fields while in the second element, the closed-form solutions developed are adopted to formulate special shape functions. For longitudinal-transverse response the elements consist of two nodes with four degree of freedom per node for longitudinal-transverse response and for lateraltorsional response, the elements consist of two nodes with eight degrees of freedom per node. The solution is able to predict the distortional deformation and stresses in a manner similar to shell solutions while keeping the modeling and computational effort to a minimum. Applications of the new beam theory include (1) providing new insights on the response of steel beams under torsion whereby the top and bottom flanges may exhibit different angles of twist, (2) capturing the response of steel beams with a single restrained flange as may be the case when a concrete slab iv

Abstract provides lateral and/or torsional restraint to the top flange of a steel beam, and (3) modelling the beneficial effect of transverse stiffeners in reducing distortional effects in the web. The second part of the study develops a unified lateral torsional buckling finite element formulation for the analysis of beams with wide flange doubly symmetric cross-sections. The solution captures several non-conventional features. These include the softening effect due to web distortion, the stiffening effect induced by pre-buckling deformations, the pre-buckling nonlinear interaction between strong axis moments and axial forces, the contribution of pre-buckling shear deformation effects within the plane of the web, the destabilizing effects due to transverse loads being offset from the shear centre, and the presence of transverse stiffeners on web distortion. Within the framework of the present theory, it is possible to evoke or suppress any combination of the features and thus isolate the individual contribution of each effect or quantify the combined contributions of multiple effects on the member lateral torsional capacity. The new solution is then applied to investigate the influence of the ratios of beam span-to-depth, flange width-to-thickness, web height-to-thickness, and flange width-to-web height on the lateral torsional buckling strength of simply supported beams and cantilevers. Comparisons with conventional lateral torsional buckling solutions that omit distortional and pre-buckling effects quantify the influence of distortional and/or pre-buckling deformation effects. The theory is also used to investigate the influence of P-delta effects of beam-columns subjected to transverse and axial forces on their lateral torsional buckling resistance. The theory is used to investigate the load height effect relative to the shear centre. Comparisons are made with load height effects as predicted by non-distortional buckling theories. The solution is adopted to quantify the beneficial effect of transverse stiffeners in controlling/suppressing web distortion in beams and increasing their buckling resistance. v

Table of Contents Table of Contents Acknowledgement . iii Abstract . iv Table of Contents . vi List of Figures . xiii List of Tables . xvi Glossary . xvi Chapter 1 . 1 Introduction . 1 General . 1 Deformations in lateral torsional buckling . 2 Motivation . 4 Limitations of conventional beam theories in static analysis applications . 4 1.3.1.1. Beams with restraints at one of the flanges . 4 1.3.1.2. Effect of transverse stiffeners on web distortion . 4 1.3.1.3. Wide flange beams under twisting moments . 6 1.3.1.4. Beams with overhangs under twisting moments . 7 Limitations of conventional beam theories in buckling applications . 8 1.3.2.1. Effects of web distortion and PBD . 9 1.3.2.2. Effect of load height . 10 1.3.2.3. P-delta effect in pre-buckling analysis . 10 1.3.2.4. Shear deformation in pre-buckling analysis . 10 1.3.2.5. Effect of transverse stiffeners . 11 1.3.2.6. Features of the Sought distortional buckling theory . 11 Overview of the thesis . 12 References . 13 Chapter 2 . 16 vi

Table of Contents Literature review . 16 General . 16 Fundamentals of calculus of variation and Variational principles. 16 Principle of stationary total potential energy . 19 Kinematics of beam theories . 21 Euler-Bernoulli beam theory . 21 Timoshenko beam theory . 22 Higher order beam theories . 23 Vlasov thin-walled beam theory. 23 Gjelsvik thin-walled beam theory . 25 Geometrically nonlinear pre-buckling analysis . 27 Background . 27 Overview of eigen-value buckling solutions . 28 2.5.2.1. Conventional approach based on linear pre-buckling analysis: . 28 2.5.2.2. Accounting for non-linear effects in pre-bucking analysis: . 29 2.5.2.3. Modifications to account for pre-bucking deformation effects: . 29 References . 30 Chapter 3 . 32 Distortional theory for the analysis of wide flange steel beams (a) . 32 Abstract . 32 Introduction and literature review . 33 Statement of the problem . 38 Assumptions and kinematics. 41 Displacements in the plane of the cross-section . 41 Longitudinal displacement . 42 vii

Table of Contents Expressions for strains and stresses . 44 Energy formulation . 45 Field equations and boundary conditions . 48 Closed form solution . 51 Homogenous solution for the longitudinal-transverse response . 51 Homogenous lateral-torsional response . 53 Particular solution. 54 Numerical examples. 55 Example 1: Longitudinal-transverse response . 55 Example 2: Cantilever under torsion . 57 Example 3: Cantilever under various types of twisting moments . 59 Example 4: Cantilever beam with restrained top flange . 61 Example 5: Cantilever under transverse uniform distributed load . 62 Summary and conclusions . 65 Appendix A . 66 Appendix B . 68 List of symbols . 70 References . 73 Chapter 4 . 76 Distortional Theory for the Analysis of Wide Flange Steel Beams –Finite Element Formulation(a) . 76 Abstract . 76 Introduction and Literature Review . 77 Statement of the problem . 78 Assumptions and Kinematics . 79 viii

Table of Contents Energy Formulation . 80 Finite Element Formulation 1- Based on Polynomial Interpolation (FE1). 82 Longitudinal-transverse response . 83 4.6.1.1. Displacement fields in terms of Nodal Displacement . 83 4.6.1.2. Equilibrium Conditions . 85 Lateral-torsional response . 85 4.6.2.1. Displacement fields in terms of Nodal Displacement . 85 4.6.2.2. Equilibrium Conditions . 87 Finite Element Formulation 2- Based on Exact Shape Functions (FE2) . 87 Longitudinal-transverse response . 89 4.7.1.1. Displacement fields in terms of Nodal Displacement . 89 4.7.1.2. Equilibrium Conditions . 92 Lateral torsional response . 92 4.7.2.1. Displacement fields in terms of Nodal Displacement . 92 4.7.2.2. Equilibrium Conditions . 95 Numerical Examples . 95 Example 1: Cantilever under transverse loading . 95 Example 2: Mesh Sensitivity Analysis . 96 Example 3: Beam with Overhangs under Torsion . 98 Summary and Conclusion . 102 Appendix A- Closed form solution for lateral torsional response . 102 Notations . 104 Subscripts . 107 References . 107 Chapter 5 . 109 Geometric nonlinear analysis of shear-deformable beam-columns. 109 Introduction and motivation . 109 ix

Table of Contents Statement of the problem . 109 Geometry and coordinates . 109 Assumptions and kinematics. 110 Formulation . 111 Finite element formulation . 113 Total potential energy in terms of nodal displacement . 114 Discretized equilibrium conditions . 116 Coordinate transformations . 117 Details of the iterative technique . 117 Numerical examples. 119 Example 1: Long-span simply supported member . 119 Example 2: Short span cantilever . 120 References . 121 Chapter 6 . 122 Lateral torsional buckling of wide flange sections including distortional and pre-buckling effects. 122 Abstract . 122 Introduction and motivation . 123 Literature review . 124 Distortional buckling effect: . 124 Pre-buckling deformation effect: . 125 Statement of the problem . 127 Assumptions . 130 Kinematics . 131 Pre-buckling Analysis . 132 x