Exact statement of the instability problem for frames and its direct numerical solution

Abstract

A general approach for the systematic evaluation of the critical buckling load and the determination of the buckling mode is presented. The Navier-Bernoulli beam model is considered, having the possibility of variable cross-section under any type of load (including pressures and thermal loading). With this purpose, the equilibrium equations of each beam element in its deformed configuration under the hypothesis of infinitesimal strains and displacements is considered, resulting in a system of differential equations with variable coefficients for each element. To obtain the nonlinear response of the frame, one should impose the compatibility of displacements and the equilibrium of forces and moments in each beam-end, also in the deformed configuration. The solution is obtained by requiring that the total variation of potential energy is zero at the instant of buckling. The objective of this work is to develop a systematic method to determine the critical buckling load and the bucklingmode of any frame without using the common simplifications usually assumed in matrix analysis or finite element approaches. This way, precise results can be obtained regardless of the discretization done.