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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.

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Formation of a suitable null basis is the main problem of finite elements analysis via force method. For an optimal analysis, the selected null basis matrices should be sparse and banded corresponding to sparse, banded and well-conditioned flexibility matrices. In this paper, an efficient method is developed for the formation of the null bases of finite element models (FEMs) consisting of tetrahedron elements, corresponding to highly sparse and banded flexibility matrices. This is achieved by associating special graphs with the FEM and selecting appropriate subgraphs and forming the self-equilibrating systems (SESs) on these subgraphs. Two examples are presented to illustrate the simplicity and effectiveness of the presented graph-algebraic method.