AN ABSTRACT OF THE THESIS OF Sujan Shrestha, for the Master of Science degree in Civil Engineering, presented on March 3, 2015, at Southern Illinois University, Carbondale. TITLE: POST BUCKLING RESPONSE OF SWAY COLUMNS UNDER MECHANICAL AND THERMAL (FIRE) LOADS MAJOR PROFESSOR: Dr. Aslam Kassimali, Ph.D. The post-buckling response of sway columns under mechanical and thermal loadings are presented by using the geometrical nonlinear analysis. For thermal analysis, the columns are assumed to be fully restrained in their axial directions. The method of analysis uses Beam-Column theory which is based on an Eulerian (corotational) formulation. Numerical solutions are shown for the post-buckling response of sway columns. The numerical investigations of the geometrically nonlinear analysis of sway columns were carried out with three different boundary conditions of sway columns as suggested by AISC, under mechanical loading and temperature changes. The sway columns considered are the cases `c', `e' and `f' in Table C-A-7.1 of AISC Manual (2011). These were modeled and analyzed to predict the post-buckling response under the mechanical and thermal loads. Furthermore, for each of these support conditions, the effects of slenderness ratios on the post-buckling response were analyzed by considering the slenderness ratios of 50,125 and 200. Also, the effects on post-buckling strength were observed keeping the same slenderness ratios but varying rotational end conditions of sway columns. Many useful conclusions can be drawn from this study. The more important conclusions are: 1) As all unrestrained sway columns undergo excessive deformation under mechanical loading, they do not possess significant post-buckling strength once the loading reaches the buckling load. 2) All restrained sway columns undergo much smaller deformations under thermal loading as compared to mechanical loading; thus significant post-buckling strength is achieved even after reaching the buckling temperature. This post-buckling strength can be considered during the design of structures which will aid in safe and economical structures. 3) Slenderness ratios play an important role on post-buckling strength only in thermal analysis but not in mechanical analysis. Increase in slenderness ratio tends to decrease the post-buckling relative deformation of the columns under thermal loading but has no such effect under mechanical loading. 4) Rotational end conditions also play significant role on post-buckling strength during thermal analysis but not during mechanical analysis. Keeping the slenderness ratio constant and varying the rotational end conditions, the post-buckling strength of all sway columns remains same under mechanical loading but is different under thermal loading.

The main objective of this research was to study the buckling and post-buckling response of axially restrained beam-columns under thermal loading. Also the effects of slenderness ratios on pre-buckling and post-buckling behavior which is neglected in AISC specification was examined. The results of this study indicate that: a) Both the deflection and end moment amplification factors are significantly smaller for the restrained beam-columns subjected to temperature increase than the corresponding unrestrained beam-columns subjected to (mechanical) axial loads. b) The deflection amplification factors tend to decrease with decreasing ratio of end moments. However, reverse seems to occur for the moment amplification factors and as the moment amplification factors tend to increase with decreasing moment ratio particularly in the pre-buckling and the initial post-buckling range (0.1 < T/Tcr < 1.5). c) The thermal amplification factors tend to be smaller than the AISC values even in the pre-buckling range with those for the slender beam-columns significantly smaller than those for the shorter beam-columns.

Cette thèse s’inscrit dans le cadre de l’étude du post-flambement local des grandes struc- tures composites raidies. La simulation du post- flambement par la méthode des éléments- finis est aujourd’hui limitée par le coût du calcul en particulier pour les grandes structures. Seules des zones restreintes peuvent être étudiées, en négligeant les interactions global/local. L’objectif de cette thèse est de proposer une stratégie de calcul performante pour la simula- tion du post-flambement local des grandes structures raidies à partir des connaissances sur le comportement mécanique des structures en post-flambement et d’un découpage naturel le long des raidisseurs favorable au calcul parallèle.Dans la littérature, les méthodes de réduction de modèle adaptative ont démontré leur ca- pacité à réduire le nombre d’inconnues tout en maîtrisant l’erreur d’approximation de la solution des problèmes non-linéaires. Par ailleurs, les méthodes de décomposition de do- maine avec localisation non-linéaire sont particulièrement adaptées au calcul parallèle en mécanique des structures en présence de non-linéarités locales.Les travaux de thèse portent dans un premier temps sur une stratégie de réduction de modèle adaptative spécifique au cas du post-flambement. Dans le cas d’un flambement local d’une grande structure raidie une combinaison avec une méthode de décomposition de domaine primale est ensuite proposée. Toutes ces stratégies sont implémentées dans un code de re- cherche programmé pendant la thèse. / This thesis is part of the study of local post-buckling of large stiffened composite struc- tures. The finite element simulation of structures subjected to post-bucking still faces com- putational limits, especially for large structures. Only restricted area may be studied for now, neglecting global/local interactions.The aim of the thesis is to propose an efficient computational strategy for local post-buckling analysis of large stiffened structures from knowledge on mechanical behavior of post-buckling structures and a natural partitionning along stiffeners conducive to parallel computation. In litterature, the adaptive model reduction solving techniques have demonstrated their abi- lity to drastically reduce the number of unknowns as well as to control the approximation error of solving non-linear problems. Furthermore, domain decomposition methods with a non-linear local step are suited to parallel computation in structural mechanics in the pre- sence of local non-linearities.Our work deals first with an adaptive model reduction strategy dedicated to post-buckling problems. In order to adress larger stiffened structures subjected to local post-buckling, like an aircraft fuselage, partitioning is then performed. The model reduction, as well as the adap- tive procedure are written in the framework of the primal domain decomposition method with a non-linear local step. These strategies are implemented in a research code developed for the purpose of the thesis.

There is a need for reliable methods to determine approximate solutions of nonlinear continuous systems. Recently, it has been proved that finite-degree-of-freedom Galerkin-type discretization procedures applied to some distributed-parameter systems may fail to predict the correct dynamics. By contrast, direct procedures yield reliable approximate solutions. Starting from these results and extending some of these concepts and procedures, we compare the outcomes of these two approaches (the Galerkin discretization and the direct application of a reduction method to the original governing equations) with experimental results. The nonlinear planar vibrations of a buckled beam around its first buckling mode shape are investigated. Frequency-response curves characterizing single-mode responses of the beam under a primary resonance are generated using both approaches and contrasted with experimentally obtained frequency-response curves. It is shown that discretization leads to erroneous quantitative as well as qualitative results in certain ranges of the buckling level, whereas the direct approach predicts the correct dynamics of the system. / Master of Science

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This study focuses on the improvement of the axial buckling capacity of elliptical composite cylinders through the use of a circumferentially-varying lamination sequence. The concept of varying the lamination sequence around the circumference is considered as a viable approach for off-setting the disadvantages of having the cylinder radius of curvature vary with circumferential position, the source of the reduced buckling capacity when compared to a circular cylinder with the same circumference. Post-buckling collapse behavior and material failure characteristics are also of interest. Two approaches to implementing a circumferential variation of lamination are examined. For the first approach the lamination sequence is varied in a stepwise fashion around the circumference. Specifically, each quadrant of the cylinder circumference is divided into three equal-length regions denoted as the crown, middle, and side regions. Eight different cylinders designs, whereby each region is constructed of either a quasi-isotropic or an axially-stiff laminate of equal thickness, are studied. Results are compared to the baseline case of an elliptical cylinder constructed entirely of a quasi-isotropic laminate. Since the thickness of the quasi-isotropic and axially-stiff laminates are the same, all cylinders weight the same and thus comparisons are meaningful. Improvements upwards of 18% in axial buckling capacity can be achieved with one particular stepwise design. The second approach considers laminations that vary circumferentially in a continuous fashion to mitigate the effects of the continuously-varying radius of curvature. The methodology for determining how to tailor the lamination sequence circumferentially is based on the analytical predictions of a simple buckling analysis for simply-supported circular cylinders. With this approach, axial buckling load improvements upwards of 30% are realized. Of all the cylinders considered, very few do not exhibit material failure upon collapse in the post-buckled state. Of those that do not, there is little, if any, improvement in bucking capacity. Results for the pre-buckling, buckling, post-buckling, and material failure are obtained from the finite-element code ABAQUS using both static and dynamic analyses. Studies with the code demonstrate that the results obtained are converged. / Ph. D.

Most thin-walled metallic structural members experience some extent of interactive buckling that corrodes the load carrying capacity. Current design methods predict the strength of thin-walled metallic structural members based on individual buckling limit-states and limited case of interactive buckling limit state. In order to develop design methods for most coupled buckling limit states, the interaction of buckling modes needs to be studied. This dissertation first introduces a generally applicable methodology for Generalized Beam Theory (GBT) elastic buckling analysis on members with holes, where the buckling modes of gross cross-section interact with those of net cross-section. The approach treats member with holes as a structural system consisting of prismatic sub-members. These sub-members are connected by enforcing nodal compatibility conditions for the GBT discretization points at the interfaces. To represent the shear lag effect and nonlinear normal stress distribution in the vicinity of a hole, GBT shear modes with nonlinear warping are included. Modifications are made to the GBT geometric stiffness because of the influence from shear lag effect caused by holes. In the following sections, the GBT formulation for a prismatic bar is reviewed and the GBT formulation for members with holes is introduced. Special aspects of analyzing members with holes are defined, namely the compatibility conditions to connect sub-members and the geometric stiffness for members with holes. Validation and three examples are provided. The second topic of this dissertation involves a buckling mode decomposition method of normalized displacement field, bending stresses and strain energy for thin-walled member displacement field (point clouds or finite element results) based on generalized beam theory (GBT). The method provides quantitative modal participation information regarding eigen-buckling displacement fields, stress components and elastic strain energy, that can be used to inform future design approaches. In the method, GBT modal amplitudes are retrieved at discrete cross-sections, and the modal amplitude field is reconstructed assuming it can be piece-wisely approximated by polynomials. The unit displacement field, stress components and strain energy are all retrieved by using reconstructed GBT modal amplitude field and GBT constitutive laws. Theory and examples are provided, and potential applications are discussed including cold-formed steel member design and post-disaster evaluation of thin-walled structural members. In the third part, post-buckling modal decomposition is made possible by development of a geometrically nonlinear GBT software. This tool can be used to assist understanding couple-buckling limit-states. Lastly, the load-deformation response considering any one GBT mode is derived analytically for fast computation and interpretation of structural post-buckling behavior. / Ph. D. / Here I present novel analytical methods to quantitatively decompose interactive buckling in the thin-walled structures. Interactive buckling, where multiple buckling modes are present to initiate structure failure, often controls the load-carrying capacity of thin-walled structures, e.g., the amount of load a column can withstand or the maximum acceleration a space shuttle can experience. In this research, based on Generalized Beam Theory, I describe in detail the analytical methods revealing how buckling modes are coupled and contribute to key quantities related to the structural failure, namely, displacement, stress, and strain energy. I obtain the algorithms by performing rigorous mathematical derivations based on thin-walled mechanics. The research not only facilitates better building design according to the simplified method in the current design standard, but also enables advanced, nonlinear modal decomposition analysis using the custom-made Finite Element program. These studies aim to provide the quantitative understanding of the coupled buckling mechanism and further the development of more powerful strength prediction methods for thin-walled structures.

The work of this thesis sets out to give a clearer in-depth understanding of the failure mechanics of thin-walled compression members which are associated with complex interactions between the different buckling modes during the loading process. This thesis employs the finite element method in order to examine the effect of the modelling techniques imposed at the section junctions of short struts and to investigate the influence of the local and global end conditions with regard to support and loading on the compressive response of various sections, i.e. I-sections, plain channel sections, box-sections, and lipped channel sections. The thesis also details appropriate finite element modelling strategies and solution procedures taking due account of the influence of material nonlinearity and geometrical imperfections for the determination of the coupled mode interactive response of thin-walled compression members. A detailed account of the complete loading history of the compression members from the beginning of loading through to final collapse is given in the thesis. This involves elastic local buckling, nonlinear elastic and elasto-plastic post-buckling interaction behaviour and yield propagation leading to the development of an appropriate failure mechanism which causes final collapse and unloading. A new finite element modelling strategy has been developed in the thesis with particular reference to being able to deal with the classical assumption of the stress-free in-plane boundary conditions existing at the section junctions of short length strut members during post-local buckling. Also, for fixed-ended columns, with particular reference to singly-symmetric plain channel sections, it has been shown that column deflections are initiated from the onset of local buckling for the case of the constituent plate elements of the section being locally rotationally constrained at their ends. Such columns should not therefore be considered as an overall bifurcation problem of the locally buckled member. In the case of the pinned and fixed-ended boundary conditions of the columns, the finite element simulations are shown to be able to accurately describe the rather different complex failure mechanics with a high degree of imperfection sensitivity being shown to be in evidence for the pin-ended case. Considerably good agreement has been shown to occur with the independent simulations of other researchers using the finite strip method of analysis, with the analytical solution procedures of others and with the findings from independent test work and this has provided confidence in the viability and usefulness of the modelling strategies and solution procedures developed in this thesis.

[pt] O presente trabalho tem por objetivo estudar o comportamento crítico e pós-crítico de placas dobradas. Placas dobradas, tanto de aço quanto de concreto, têm sido usadas com grande freqüência em obras de engenharia, particularmente como paredes portantes e coberturas de grandes vãos. Neste último caso, placas dobradas têm uma aplicação semelhante a de cascas esbeltas, sendo, porém, mais simples de construir, dado que são compostas de superfícies planas. Neste trabalho utiliza-se o software ABAQUS para análise de placas dobradas compostas de duas placas formando um dado ângulo entre si. Estuda-se o uso destas placas como paredes portantes sob compressão, sendo conduzido um estudo paramétrico para determinar a influência dos diversos parâmetros geométricos e condições de contorno nas cargas críticas e modos críticos da placa dobrada. A seguir são determinados os caminhos pós-críticos e estuda-se a influência dos diversos parâmetros na rigidez pós-crítica e sensibilidade a imperfeições. Finalmente, estuda-se o comportamento não-linear destas placas sob flexão, sendo investigados os fenômenos de perda de estabilidade associados a este problema. / [en] The aim of the present work is to study the buckling and post-buckling behavior of folded plates. Folded plates, made of steel or concrete, have been widely used in engineering construction as load-bearing walls designed to carry floor and roof loads or roofs for large spaces. In the last case, folded plates work as a thin shell, being however easier to build since they are formed by flat surfaces joined along the fold lines. In this work the software ABAQUS is used to study the behavior of V-shaped folded plates. Initially it is analyzed the behavior of these plates when used as load-bearing walls under axial compression. A detailed parametric analysis is conducted to identify the influence of the geometrical parameters and boundary conditions on critical loads and modes. The influence of these parameters on the post-buckling response, post-critical stiffness and imperfection sensitivity is also investigated. Finally, the non-linear response of V-shaped folded plates under flexure is investigated and the possible instabilities under these loading conditions are analyzed.

La modélisation des défauts de planéité apparaissant en ligne en laminage à froid des tôles minces est abordée comme un problème de flambage de tôles minces sous contraintes résiduelles. Celles-ci sont les contraintes engendrées au-delà de l'emprise par le laminage lui-même. Pour cela, un modèle de flambage et post-flambage de tôle (forme, amplitude, contraintes) fondé sur la méthode multi-échelle de Damil et Potier-Ferry, et nommé MSBM pour Multi-Scale Based Method, a été développé. En entrée, on y introduit une carte de contraintes post-emprise venant d'un calcul de laminage. Les hypothèses simplificatrices du modèle de flambement permettent de ramener sa résolution à un ensemble de problèmes éléments finis 1D, mais de ce fait restreignent l'analyse aux défauts de type bord long ou centre long. Dans sa version découplée, ce modèle a été comparé avec succès à des résultats de la littérature. Il permet d'effectuer des études paramétriques d'intérêt pratique, comme l'influence du frottement ou de la force de contre-flexion des cylindres sur l'état de contrainte et la géométrie de la tôle.Dans un second temps, ce modèle est introduit comme modèle de flambage intégré dans le logiciel éléments finis de laminage Lam3/Tec3. Comme dans le modèle précédent implémenté par Abdelkhalek en 2010, MSBM calcule un champ de déformation lié spécifiquement aux déplacements hors-plan caractérisant le flambage, champ de déformation qui est ajouté à la décomposition élastique – plastique et réactualisé à chaque itération du calcul éléments finis. Des comparaisons ont été effectuées avec les deux modèles couplés précédemment implantés par Abdelkhalek. Elles montrent les insuffisances du présent modèle de flambage, unidirectionnel, qui ne permet pas de traiter toutes ensemble les diverses instabilités, d'orientations différentes, qui ont lieu après la sortie d'emprise et se révèlent fortement couplées entre elles. Des pistes d'amélioration sont proposées en conséquence. / Modelling of on line manifest flatness defects in thin strip cold rolling is addressed as a problem of buckling under residual stresses. The latter are stresses built beyond the roll bite by the rolling process itself. To this aim, a buckling / post buckling model has been developed, giving strip shape, amplitude and stresses, based on Damil and Potier-Ferry's method and hereafter named MSBM like Multi-Scale Based Method. Its input is a post-bite stress map computed by a rolling model. Simplifications of the buckling model make it amenable to a series of 1D FEM solutions, but restrict its application to simple flatness defects such as wavy edges or wavy centre. In a decoupled version, it has been successfully compared with literature results. It allows parametric studies of practical interest, such as the influence of friction or work roll bending force on post-buckled strip shape and stress.In a second stage, this model is implemented as the internal buckling model in the FEM software Lam3/Tec3. As the previous one, implemented by Abdelkhalek in 2010, MSBM computes a strain field strictly due to the out-of-plane displacement which characterizes buckling. This strain field is introduced into the elastic – plastic decomposition and updated at each iteration of the finite element computation. Comparisons have been performed with the two models previously coupled to Lam3/Tec3 by Abdelkhalek. They show the limits of the present unidirectional buckling model, which cannot deal with all instabilities together, which have different orientations and take place after roll bite exit, which furthermore prove to be strongly interacting. Ideas for future generalization of the coupled model are proposed accordingly.