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PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and M. Papadrakakis
Planar Truss Structures with Multi-Symmetry
A. Kaveh and L. Shahryari
Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
A. Kaveh, L. Shahryari, "Planar Truss Structures with Multi-Symmetry", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 163, 2008. doi:10.4203/ccp.88.163
Keywords: multi-symmetry, trusses, decomposition, eigenfrequencies, graph.
Many eigenvalue problems arise in many scientific and engineering problems. While the basic mathematical ideas are independent of the size of matrices, the numerical determination of eigenvalues and eigenvectors becomes more complicated as the dimensions of matrices increase. Special methods are beneficial for the efficient solution of such problems, especially when their corresponding matrices are highly sparse.
Methods are developed for decomposing and healing the graph models of structures, in order to calculate the eigenvalues of matrices and graph matrices with special patterns. The eigenvectors corresponding to such patterns for the symmetry of Form I, Form II and Form III are studied in references [1,2], and the applications to vibrating mass-spring systems and frame structures are developed in [3,4], respectively. These forms are also applied to calculating the buckling load of symmetric frames .
Consider a structural system with two translational degrees of freedom (DOFs) per node which has two axes of symmetry. Suppose each DOF is parallel to one of the axes and is perpendicular to the other axis. One can find matrices in canonical forms, and using the symmetry relationships twice, one can find four submatrices. The union of the eigenvalues for these four submatrices results in the eigenvalues of the original matrix.
In this paper, the region in which the structural system is situated is divided into upper, lower, left and right subregions. The stiffness matrix of the entire system is formed and then using the existing direct and reverse symmetries, relationships between the entries of the matrix are established.
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