A theoretical study is conducted for the lateral torsional buckling of wooden beam-deck assemblies consisting of twin beams braced by tongue-and-groove decking at the top. Two models are developed, each with a series of analytical and numerical solutions formulated. The first model targets twin-beam-deck assemblies where deck boards and other components are detailed to provide full continuous lateral restraint while the second model is built for situations where the beams are allowed to sway laterally and the relative lateral movement between the beams is partially restrained by the deck boards. In the first model, focus is on wind uplift while in the second model, both gravity and uplift loading scenarios are investigated. In the first model, an energy method is adopted and the principle of stationary potential energy is evoked to formulate closed-form solutions, energy-based solutions and a finite element solution. The validity of the present solutions is verified against a finite element based ABAQUS model. Similarly, a family of solutions is developed under the sway model and verified against the ABAQUS. Parametric studies are conducted for both models to examine the effects of various variables on the buckling capacity. A comparative investigation on the behavioral difference between the two models under ABAQUS is also presented. Overall, the restraining effects of deck boards bracing either on the beam compression or tension side is observed to have a significant influence on the lateral torsional buckling capacity of the twin-beam-deck assemblies.

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.

Engineered wood I-joists have grown in popularity as flooring and roofing structural systems in the past 30 years, replacing solid sawn lumber joists. Typical wood I-joists are manufactured with a very slender section, which is desirable to achieve higher flexural capacities and longer spans; however, this makes them susceptible to lateral torsional buckling failure. Continuous beam spans and uplift forces on roof uplift are potential scenarios where lateral instability can occur and reflects the need to investigate the lateral torsional buckling behavior of wood I-joists. Within this context, the present study conducts an experimental investigation on the material properties and the critical buckling load of 42 wood I-joist specimens. A 3D finite element model is built using the experimentally determined material parameters to effectively predict the observed buckling behavior of the specimens while also accounting for initial imperfections in the joists. The adequacy of other analytical models to predict the critical buckling load of wood I-joists are also investigated. It is demonstrated that the American design standard underestimates the critical buckling load of wood I-joists while the classical theory provides an adequate estimate of the buckling capacity. Furthermore, the effects of initial imperfections on the lateral torsional buckling behavior are discussed. The developed and verified FE model is used to reproduce the nonlinear buckling behavior of the wood I-joist and also to provide an accurate estimate of the lateral torsional buckling capacity using the linear buckling analysis.

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.

M. Tech. (Department of Mechanical Engineering, Faculty of Engineering and Technology), Vaal University of Technology. / Though aluminium (Al) is justifiably described as a green metal with an increasing rate of application in structures, designers still restrain themselves from its applications as a load-bearing skeleton in structure due to insufficient design guidelines. This insufficient information is more with channel sections that might experience lateral-torsional buckling (LTB) when used as a load-bearing skeleton in structures. This study investigates the effects on imperfections on LTB load-carrying stability for 6061-T6 Al alloy channel section arches and proposed design guidelines. The case study focused on freestanding circular fixed end arches subjected to a transverse point load at the shear centre. The software package Abaqus was used to study a total of 110 arch models from three separate channel sections with an additional 16 arch models for validation. Sixty-six channel arches were developed at a constant length, while the remaining 44 arches were formed at constant slender ratios using 11 discrete included angles. The FE analyses methods used for the investigation were validated with existing analytical methods and showed good agreement, despite the assumptions of the bilinear curve used for material nonlinearity, initial geometric imperfections and residual stresses that presented the imperfections of the models. The different investigated factors include slender ratios, change in cross-section area, imperfections, and angles. These factors were found to have substantial impacts on the prebuckling state, which turns to impact LTB behaviour and load-carrying capacity. From arches developed at constant span length, the arches with moderately included angles (50°≤2𝛼≤90°) were found suitable for the designs against LTB, followed by the shallow (2𝛼<50°) and deep arches (90°<2𝛼≤180°) respectively. For arches developed at constant slender ratios, the deep arches were found to be more suitable in the design against LTB, followed by the moderate and shallow arches, respectively. In addition, it was realised that the change in web-flange thickness, section depth and slender ratios, had significant effects on the LTB loads magnitudes and very insignificant effects on the general behaviour across the included angles. The same occurrence was also observed on the prebuckling analyses. All the investigated channel section arches showed the imperfections to have significant impacts on the LTB loads. Arches developed at constant span length showed the maximum elastic LTB loads to have overestimated the expected real LTB loads by approximately 48 percent. While the maximum elastic LTB loads of arches developed at 𝑆𝑟𝑥⁄= 60 and 90 showed that the real LTB loads were overestimated by about 39 and 14 percent, respectively. That said, the elastic LTB loads on average overestimated the real LTB loads by over 50 percent for the arches developed at the constant span length and by only 18 percent for arches developed at the constant slender ratios.

While moment frames are a popular lateral-force resisting system, their constant cross-section can lead to inefficiencies in energy absorption and stiffness. By tapering the flange width linearly toward the center of the beam length, the energy absorption efficiency can be increased, leading to a better elastic response from the beam and more elastic stiffness per pound of steel used. Lateral-torsional buckling is an important failure mode to be considered for tapered-flange moment frame shapes. No closed-form or finite element solutions have yet been developed for tapered-flange I-beams with a non-uniform, linear moment gradient and intermediate bracing conditions. In this study, finite element analysis is used to find the buckling stress of each W-shape in the AISC Steel Construction Manual with both a standard straight-flange and the proposed tapered-flange at several lengths and with three intermediate lateral bracing conditions (no bracing, mid-span bracing, and third-span bracing). Plots are generated for each shape at each bracing condition as the buckling stress versus length for both beams and columns. Overall, the results indicate that lateral-torsional buckling of tapered-flange I-beams is not a problem that would prohibit the wide-scale use of this configuration in moment frames. Also, the buckling capacity tapered-flange moment frame shapes can be reasonably estimated as 20% of the corresponding straight-flange moment frame shape.

Corrugated web plate girders (CWPGs) have become an increasingly popular structural member in Canada in recent years. This is because of their economic efficiency over standard wide flange members. Although the flexural performance of such has been increasingly studied in recent years there is still advancements that can be made in their design. No research has been completed in Canada on the subject of lateral torsional buckling (LTB) strength and very minimal research has been published on sinusoidal CWPGs. In order to examine the LTB strength of a CWPG with a sinusoidally shaped web, nine specimens were loaded and failed in simply supported arrangement that favours lateral torsional buckling. Specimens were chosen to observe the difference in strength due to web thickness, web depth and variation in identical beams. All of the specimens recorded strengths that exceeded the theoretical design strengths confirming that the current design procedure is conservative. A trend of ultimate capacity increasing was observed with the increase of web thickness. The depth of the web had no significant effect on the torsional strength besides what is gained from the increased flange distance. An equivalent web thickness equation was formulated based on the results for the purpose of calculating LTB strength. To test the proposed equation a numerical analysis was run on a wider range of beams and compared with the testing results. It was determined the physical testing results can be effectively captured by the proposed equation among more than just the tested beams. Two additional analyses were prepared to lay the foundation for further investigation of the proposed equation. The first was a Monte Carlo simulation to test the risk of using the proposed equation which requires additional data. Secondly, a preliminary finite element analysis (FEA) model was developed and presented for future use to expand this research. / Thesis / Master of Applied Science (MASc) / Corrugated web plate girders (CWPG) have grown in popularity due to their economic efficiency. No research has been presented in Canada and very minimal research has been published on the lateral torsional strength of CWPGs with sinusoidally corrugated webs. This research studied the lateral torsional buckling (LTB) strength of CWPGs through the experimental testing of physical members and a new equation for the calculation of the LTB strength is proposed. This equation and design process was then numerically tested to determine its viability as a design process.

The research summarized the experimental analysis and finite element modeling of the lateral and rotational response of unbraced wood composite I-joists to worker loads. All experimentation and modeling was conducted on simply supported I-joists varying from 11-7/8 inches to 14 inches in depth and 20 feet to 24 feet in length. I-joists were subjected to static and dynamic loads. The deflections of the top and bottom flanges, as well as the rotation, were measured or calculated at both one-half and one-quarter the span length. The overall goal of this project is to accurately model the lateral and rotational displacements caused by human load effects. I-joists were first tested statically by subjecting each joist to a three point bending test, free from all lateral restraints. This test was necessary to prove that the performance of the joists was repeatable. Lateral and rotational stiffness of the joist were calculated at one-half and one-quarter of the span length. The static experimental tests results were statistically analyzed using an analysis of variance (ANOVA) test. The results from this analysis indicated no difference between repetitions of the same joist; however, the test did indicate that there was a significant difference between joists of the same manufacture and size. Dynamic testing was then conducted. Dynamic loads were induced by having test subjects traverse each I-joist. The resulting loads induced at the top and bottom flanges were recorded for use in the finite element model. The lateral deflections and induced loads were compared to the static weight of the test subject and analyzed with an ANOVA test. The results indicated an increase in both the induced load and resulting deflection with an increase in weight. The analysis also indicated an increase in load and deflection with a decrease in lateral and rotational joist stiffness. The recorded load values from the dynamic test were used as inputs into a finite element model. The resulting lateral deflections of the midpoint and quarter point were generated. The rotation of the beam was calculated from the difference between the top and bottom flange. Experimental results and finite element model results were compared by calculating a running average of the error between the acquired data and the finite element model. The model was said to be valid until the average model error reached 10 percent of the maximum acquired test value. All six deflection readings were analyzed in this manner. The percent of beam at which the model no long represented the test data was determined for each data set. This point was averaged across all deflection readings of similar joists and across all data sets of the same joist type. The model predicted the 20 foot long 11-7/8 and 14 inch deep joists until 54.5 percent and 51.2 percent, respectively, of the beam completed by the test subject. However, the 24 foot long 11-7/8 inch deep joist was only accurate to 31.2 percent of the beam completed by the test subject. Differences in peak values, and the time at which the peak values occurred were also analyzed using an ANOVA test. There was a significant difference between the peak values of the acquired test data and the deflections generated with the finite element model. However, there was no significance within the time that the peak values occurred between the model and experimental results. A simplified pseudo dynamic analysis was conducted using a constant percentage of the test subject's static weight applied to the top and bottom flange. This approximation proved adequate for the lateral displacement and rotation of the 11-7/8 inch and 14 inch deep and 20 foot long I-joists. However, the model became un-conservative for the 11-7/8 inch deep and 24 foot I-joists. / Master of Science

Metal building frames are typically designed using welded prismatic and web-tapered members with doubly-symmetric and/or singly-symmetric cross sections. Until recently, the base U.S. provisions for design of frames with web-tapered members were provided in the AISC ASD (1989) and LRFD (1999) Specifications. Unfortunately, these previous AISC provisions address only a small range of practical designs. As a result, metal building manufacturers have tended to develop their own methods for design of the wide range of nonprismatic member geometries and configurations encountered in practice. This research develops new design procedures for design of frames using general prismatic members and web-tapered members. An equivalent prismatic member concept utilized in prior research and the prior AISC provisions is generalized to accommodate the broad range of member types and configurations commonly used in metal building industry. Furthermore, the new design procedures incorporate many of the improvements achieved in the AISC (2005&2010) Specifications to metal building frame design. These improvements include a new stability design method, the direct analysis method, more complete considerations of different column buckling limit states (flexural, torsional and flexural-torsional buckling), and improved axial load and flexural resistance provisions. This research develops practical design-based procedures for simplified calculation of the elastic buckling resistances of prismatic and web-tapered members to facilitate the application of the proposed design methods. In addition, this research performs a relatively comprehensive assessment of beam lateral torsional buckling (LTB) behavior and strength of prismatic and web-tapered members using refined virtual test simulation. It is demonstrated that web-tapered members behave in a comparable fashion to prismatic members. Based on the virtual simulation study, recommendations for potential improvement of the AISC LTB resistance equations are provided. Lastly, the strength behavior of several representative metal building frames is studied in detail using the same virtual test simulation capabilities developed and applied for the assessment of the beam LTB resistances.