An analytical and numerical investigation is conducted for the lateral torsional buckling analysis of wooden beam with a mid-span lateral brace subjected to symmetrically distributed loading. Two models are developed; one for the case of a rigid brace and another one for the case of a flexible brace. The analytical solutions are based on the principle of stationary potential energy and a Fourier expansion of the buckling displacement fields and bending moments. The validity of both models are verified against 3D finite element analyses in ABAQUS. Where applicable, verifications were also conducted against available solutions from previous studies. Parametric studies were conducted to investigate the effect of geometric and material parameters on the critical moments. The results indicate the presence of two separate groups of potential buckling modes, symmetric and anti-symmetric, with fundamentally different behavioural characteristics. The governing buckling mode is shown to depend on the bracing height, load height and lateral brace stiffness. The study shows that beyond a certain threshold bracing height, the critical moment is governed by the antisymmetric mode of buckling. Also, above a certain optimum bracing stiffness, no increase is observed in the critical moments. The models developed are used to construct a comprehensive database of parametric investigations which are then employed for developing simplified equations for determining the threshold heights, associated critical moments, and optimum stiffness.

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 lateral-torsional 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 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.

Distortional lateral-torsional buckling theories assume that the flanges remain undistorted, while the web is free to distort as a thin plate. Most theories adopt a cubic polynomial distribution along the web height to relate the lateral displacement of the web to the displacements and angles of twist both flanges. The present study develops a family of finite element solutions for the distortional buckling of wide flange beams in which the flanges are assumed to remain undistorted. In contrast to past theories, the lateral displacement distribution along the web height is characterized by superposing (a) two linear modes intended to capture the classical non-distortional lateral-torsional behavior and (b) any number of user-specified Fourier terms intended to capture additional web distortion. In the longitudinal direction, all displacement fields characterizing the lateral displacements are taken to follow a cubic distribution. The first contribution of the thesis develops a finite element formulation that is able to replicate the classical non-distortional lateral torsional buckling solutions when the distortional modes are suppressed while enabling more accurate predictions for distortional lateral torsional buckling compared to those solutions based on the conventional cubic interpolation of the lateral displacement. The formulation is used to conduct an extensive parametric study to quantify the reduction in critical moments due to web distortion relative to the classical non-distortional predictions in the case of simply-supported beams, cantilevers, and beams with an overhang. The solution is then used to generate interaction curves for beams with an overhang subjected to various proportions of uniformly distributed and point loads. The second contribution of the thesis adds two additional features to the formulation (a) to capture the destabilizing effect due to the load height relative to the shear center and (b) a module that incorporates any number of user-defined multi-point kinematic constraints. The additional features are employed to investigate the effect of load height, bracing height, and combined effects thereof in practical design problems. A distortional indicator is then introduced to characterize the distribution of web distortion along the beam span as the beam undergoes distortional lateral buckling. A systematic design optimization technique is then devised to identify the location(s) along the span at which the addition of transverse stiffeners would maximize the critical moment capacity.

The aging infrastructure worldwide and the typical increase in service loads relative to original design loads make it essential to develop effective techniques for strengthening and rehabilitating existing structures, to enhance their resistance. An effective method for strengthening existing steel I-beams is to weld either one or two cover plates to the flange(s). In many cases, it is not feasible to completely unload the beam before carrying out the strengthening procedure. In these conditions, operators resort to strengthen beams while under loading. In such scenarios, it becomes a challenging task to assess the lateral torsional buckling (LTB) capacity of the member under present steel design standards (e.g., CAN/CSA-S16 2019 and ANSI/AISC360 2022) which do not consider the effect of pre-strengthening loads on LTB resistance. Within this context, the present study investigates the effects of pre-strengthening loads on the critical moment capacity by developing a series of solutions, ranging from elaborate and accurate to simplified but approximate, to predict the elastic LTB capacity of beams strengthened with cover plate(s) while under load. In this respect, the study contributes to the existing body of knowledge through four aspects: In the first contribution, a shell-based finite element (FE) study is developed to analyze the effect of various geometric and loading parameters on the LTB capacity of doubly symmetric beams strengthened symmetrically with two cover plates. The study carefully simulates the entire history, including the application of pre-existing loads, clamping forces to align the initially straight steel cover plates with the bent beam configuration, the rebound effect arising after clamping force removal, the contact at the interfaces between cover plates and flanges induced by welding, and the application of post-strengthening loads up to the point of elastic LTB initiation for the strengthened system, as determined by eigenvalue analysis. A simplified design equation is then proposed to quantify the post-strengthening critical moment capacity. The validity of the equation is assessed against FE results and its merits and limitations are discussed. The study shows that web distortional effects play a crucial role in reducing the elastic critical moment capacity. Practical recommendations are provided to mitigate such distortional effects and hence maximize the elastic critical moment capacity of the strengthened beams. The second contribution formulates a variational principle for the LTB analysis of doubly symmetric beams strengthened symmetrically with identical steel cover plates. The formulation considers the full sequence of loading and strengthening and captures the effects of pre-strengthening loads and the beneficial effects of pre-buckling deformation (PBD). The study examines the effect of geometry, partial strengthening schemes, presence of different pre- and post-strengthening load patterns, and load height effects. The variational principle is subsequently used to develop a FE formulation, culminating in a quadratic eigenvalue problem. The validity of the FE formulation is assessed through comparisons with other numerical techniques predictions as well as experimental results by others, and subsequently used to conduct a parametric study to characterize the gain in elastic critical moment capacity attained by cover plate strengthening. For beams partly strengthened with cover plates along their spans, the study identifies the optimum locations for cover plates that maximize the critical moments. The third contribution builds upon the variational principle developed by formulating a simple and approximate energy-based design-oriented solution to quantify the LTB resistance of simply supported I-beams strengthened with cover plates. The solution captures the detrimental effect of loads acting on the beam before strengthening and the beneficial effects resulting from PBD, pre- and post-strengthening load heights, as well as moment gradient effects. The potential use of the equations developed in practical applications involving beam strengthening is illustrated through design examples. The fourth contribution expands the variational formulation to include beams with monosymmetric cross-sections and/or symmetric beams with unsymmetric cover plate geometries. The modified variational principle is used to develop a thin-walled beam FE formulation, which is subsequently employed to predict the non-distortional LTB capacity of monosymmetric strengthened beams. Comparative analyses with shell models confirm the validity of the proposed solutions, and practical design recommendations for suppressing web distortion are provided. The effects of various design parameters on the total elastic critical moment capacity are evaluated in a systematic parametric study. The study identifies the loading conditions under which the magnitude of pre-strengthening loads significantly influences the predicted total critical moments. The solutions developed in the present study equip structural designers and analysts with novel techniques that reliably quantify the LTB strength of steel beams strengthened with cover plates, thus enabling them to optimize strengthening strategies for beams whose strengths are governed by LTB modes of failure.

The flexural capacity of the new hollow flange steel section known as LiteSteel beam (LSB) is limited by lateral distortional buckling for intermediate spans, which is characterised by simultaneous lateral deflection, twist and web distortion. Recent research based on finite element analysis and testing has developed design rules for the member capacity of LiteSteel beams subject to this unique lateral distortional buckling. These design rules are limited to a uniform bending moment distribution. However, uniform bending moment conditions rarely exist in practice despite being considered as the worst case due to uniform yielding across the span. Loading position or load height is also known to have significant effects on the lateral buckling strength of beams. Therefore it is important to include the effects of these loading conditions in the assessment of LSB member capacities. Many steel design codes have adopted equivalent uniform moment distribution and load height factors for this purpose. But they were derived mostly based on data for conventional hot-rolled, doubly symmetric I-beams subject to lateral torsional buckling. In contrast LSBs are made of high strength steel and have a unique crosssection with specific residual stresses and geometrical imperfections along with a unique lateral distortional buckling mode. The moment distribution and load height effects for LSBs, and the suitability of the current steel design code methods to accommodate these effects for LSBs are not yet known. The research study presented in this thesis was therefore undertaken to investigate the effects of nonuniform moment distribution and load height on the lateral buckling strength of simply supported and cantilever LSBs. Finite element analyses of LSBs subject to lateral buckling formed the main component of this study. As the first step the original finite element model used to develop the current LSB design rules for uniform moment was improved to eliminate some of the modelling inaccuracies. The modified finite element model was validated using the elastic buckling analysis results from well established finite strip analysis programs. It was used to review the current LSB design curve for uniform moment distribution, based on which appropriate recommendations were made. The modified finite element model was further modified to simulate various loading and support configurations and used to investigate the effects of many commonly used moment distributions and load height for both simply supported and cantilever LSBs. The results were compared with the predictions based on the current steel code design rules. Based on these comparisons, appropriate recommendations were made on the suitability of the current steel code design methods. New design recommendations were made for LSBs subjected to non-uniform moment distributions and varying load positions. A number of LSB experiments was also undertaken to confirm the results of finite element analysis study. In summary the research reported in this thesis has developed an improved finite element model that can be used to investigate the buckling behaviour of LSBs for the purpose of developing design rules. It has increased the understanding and knowledge of simply supported and cantilever LSBs subject to non-uniform moment distributions and load height effects. Finally it has proposed suitable design rules for LSBs in the form of equations and factors within the current steel code design provisions. All of these advances have thus further enhanced the economical and safe design of LSBs.