This paper presents a new methodology and algorithm for solving post bucklingproblems of a large deformed elastic beam. The total potential energy of thisbeam is a nonconvex functional, which can be used to model both pre- andpost-buckling phenomena. By using a canonical dual finite element method, a newprimal-dual semi-definite programming (PD-SDP) algorithm is presented, whichcan be used to obtain all possible post-buckled solutions. Applications areillustrated by several numerical examples with different boundary conditions.We find that the global minimum solution of the nonconvex potential leads to astable configuration of the buckled beam, the local maximum solution leads tothe unbuckled state, and both of these two solutions are numerically stable.However, the local minimum solution leads to an unstable buckled state, whichis very sensitive to external load, thickness of beam, numerical precision, andthe size of finite elements. The method and algorithm proposed in this papercan be used for solving general nonconvex variational problems in engineeringand sciences.
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