A theory for the finite displacement of a thin-walled Bernoulli–Euler beam and lateral post-buckling analysis

Abstract

The state vector equation for lateral buckling in finite displacement theory is formulated using only the hypothesis of the Bernoulli-Euler beam. By using an appropriate orthogonalization of the warping functions, the normally complicated calculation has been processed systematically using only matrix notation. As a numerical analysis, the lateral buckling load on the cantilever receiving a concentrated end load on the upper flange was calculated using the coefficient matrix of the first-order increment; the postbuckling behaviour was investigated with increasing load. Since the state vector equation is a higher order nonlinear equation, the original coefficient matrix was fixed with an arbitrary initial value and the solution was provided by the Runge-Kutta transfer matrix method. Subsequent calculations were pursued in the same way with the solution obtained via Runge-Kutta methods as a new initial value and then shifted to the next load condition. This theory and analysis method does not employ an assumed displacement function, such as the Ritz's method; it is therefore useful for the finite displacement analysis of a beam with arbitrary boundary conditions and intermediate support conditions.

Analysis of flexural natural vibrations of thin‐walled box beams using higher order beam theory

“…Abundant theoretical researches have been performed by various researchers on different beam theories in this area. Many studies are based on Euler–Bernoulli's and Timoshenko's beam theory which greatly simplify the deformation mode of the section . Higher order theory were developed by an enrichment of the corresponding displacement field approximation over the cross section, in order to capture higher order effects such as higher order shear deformations and shear lag .…”

The cantilever beams analysis by the means of the first-order shear deformation and the Euler-Bernoulli theory

“…The Euler-Bernoulli theory has been successfully applied in engineering practice. This theory served as the base to formulate a theory for the finite displacement in the beam and post-buckling analysis [1]. Ghosh used this theory as a base for developing and implementing a solution for shape memory polymers [2].…”

Postbuckling analysis of beams of arbitrary cross section using BEM

“…Mohri et al in [19] presented analytical solutions for simply supported beam column elements of bi-symmetric sections under combined bending and axial forces based on a nonlinear stability model and using the Galerkin method, while for the estimation of the postbuckling behavior a non-linear FEM software employing shell elements has also been used. Finally, Usuki in [20] formulated the state vector equation for lateral buckling in finite displacement theory using only the hypothesis of the Bernoulli-Euler beam, while employing an appropriate orthogonalization of the warping functions, the normally complicated calculation has been processed systematically using only matrix notation. Nevertheless, in all of the aforementioned research efforts the analysis is not general, since either the analysis is restricted to the thin-walled theory assumptions or to monoor doubly-symmetric cross sections, or the beam boundary conditions are not general, or the approximation for the deflection-curvature relation is not so ''accurate'' or the Wagner's coefficients and the shortening effect [21][22][23][24] have been ignored.…”