buckling load of symmetric planar frames with semi-rigid joints using graph theory

the main aim of this paper is to extend the recently developed methods for calculating the buckling loads of planar symmetric frames to include the effect of semi-rigidity of the joints. this is achieved by decomposing a symmetric model into two submodels and then healing them in such a manner that the ::::union:::: of the eigenvalues of the healed submodels result in the eigenvalues of the entire model. thus the critical load of the frame is obtained using the eigenvalues of its submodels.