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The effects due to an arbitrary location of the load point on the buckling and post-buckling behaviour of elastic thin-walled beams are investigated. The governing differential equations and the corresponding stiffness matrices are derived, based on the virtual work equation of linearized finite displacement. Numerical examples are given to investigate the behaviour of I, channel and zee shaped beams. Location of load point greatly affects the post-buckling behaviour as well as the buckling loads. Although zee and channel sections show a large reserve strength as compared with I-section, this reserve strength corresponds to fairly large stresses and displacements.

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Beams having moments of inertia that differ when referred to the two principal axes may buckle in the direction of least stiffness when loaded in the· direction of greater stiffness. The author presents the results of a series of investigations into the lateral buckling of channel sections and Z-sections that are loaded in the direction of greater stiffness. The investigations included tests on beams and an analytical consideration of the problem which were aimed at devising a rational procedure for the design of such beams. Because of the resistance to twist, arising from the restraint to warping of the cross sections of the beams, the torsion-bending factor is introduced. This factor is shown to be a property of the cross section as vital to proper design as the commonly used section modulus and moment of inertia.

The lateral buckling of beams is a complicated phenomenon, the maximum elastic bending stress necessary to cause buckling being dependent upon the geometrical and material properties of the beam, the type and position of the loading, and also the support conditions. This paper presents a design procedure which enables all of these factors to be allowed for, in an accurate yet simple and direct manner, for 41 different cases of loading and support. The proposed method is both more accurate and more versatile than existing methods used in design codes.

The purpose of this study is to establish a non-iterative efficient computational scheme to trace the nonlinear finite displacement behaviour of space frames, using the tangent stiffness equation of linearized fininte displacement of a thin-walled elastic straight beam element. Direct solution of the tangent stiffness equation is used, imposing adequately small increments. Local coordinates are updated at each incremental step, utilizing a vector multiplication scheme. Numerical results for a wide variety of spatial structures are given, demonstrating the versatility of the present scheme.

The potential energy expression and the (14 by 14) stiffness matrix of a straight thin-walled beam element of open asymmetric cross section, subjected to initial axial force, initial bending moments, and initial bimoment, are derived. The transformation matrix relating the forces and displacements (including bimoment and warping parameter) at the adjacent end cross section of two elements meeting at an angle is deduced as the limiting case of a transfer matrix of a curved beam. Results of lateral post-buckling analysis of various beams are presented.

The explicit stiffness equations and the corresponding differential equations are formulated for a truss and a non-warping beam in the framework of the linearized finite displacement theory. The derivation is consistent with the theory of thin-walled members. One main objective is to show the exact correspondence between the stiffness equations and the differential equations with their boundary conditions. An alternative scheme of deriving the stiffness matrices is given as the direct modification of the already obtained matrix of thin-walled members.

The purpose of this study is to present a solution scheme for the problem of out-of-plane instability of thin-walled members. Based on the second order kinematic field, the stiffness equation of linearized finite displacements is formulated for thin-walled members, and given in a concise and explicit form. As a particular case, an important and practical application is made for the lateral-torsional buckling of in-plane beams and frames. Numerical examples are given for straight and curved members, and are compared with existing results. The analysis scheme presented is proved accurate, efficient and versatile.

With small applied loads thin-walled steel structures behave in a linear elastic manner until they start to buckle in an elastic or in an elasto-plastic mode. Finally beyond the maximum load they develop a plastic failure mechanism. Although there are many papers which deal with these three aspects separately there have been relatively few attempts to trace the history of a structure through all three stages. This paper describes a study of a simple channel cantilever with an eccentrically located end load.

The geometric nonlinear analysis of structures comprising members of asymmetric thin-walled open section is investigated. The geometric stiffness matrix for a thin-walled element is derived, incorporating second order nonlinear strain terms in the governing energy equations. The derived stiffness matrices are used to predict bifurcation loads of a number of structures. An updated Lagrangian formulation, coupled with the numerical arc-length technique, is employed to trace the non-linear load-deflection paths up to the maximum load. Numerical examples of the load-deflection behaviour of eccentrically loaded and restrained unequal angle beam-columns are presented.

Thin-Walled Elastic Beams, Israel Program for Scientific Translation

- V Vlasov

Vlasov, V.: Thin-Walled Elastic Beams, Israel Program for Scientific Translation, Jerusalem, 1961.

Large Deflection and Post Buckling Analysis of Two and Three Dimensional Spatial Frames

- J L Meek
- H S Tan

Meek, J. L. and Tan, H. S.: Large Deflection and Post Buckling Analysis of Two and Three Dimensional Spatial Frames,
Research Report No. CE49, Department of Civil Engineering, University of Queensland, Australia, Dec., 1983.