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 post-buckling 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.