Graph Products Applied to the Analysis of Regular Structures
In spite of considerable advances in computational capability of computers in recent years, efficient methods for more time-saving solutions of structures are of great interest. Large problems arise in many scientific and engineering problems. While the b
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Graph Products Applied to the Analysis of Regular Structures
8.1
Introduction
In spite of considerable advances in computational capability of computers in recent years, efficient methods for more time-saving solutions of structures are of great interest. Large problems arise in many scientific and engineering problems. While the basic mathematical ideas are independent of the size of the matrices, the numerical determination of the displacement and internal forces becomes more complicated as the dimensions of matrices increase and their sparsity decreases. The use of prefabrication in industrialised building construction often results in structures with regular patterns of elements exhibiting symmetry of various types, and special methods are beneficial for efficient solution of such problems. In the first part of this chapter, an efficient method is developed for the analysis of regular structures. A structure is called regular if its model can be formed by a graph product. Here, instead of direct solution of the equations corresponding to a regular structure or finding the inverse of the stiffness matrix directly, modal analysis is used, and eigenvectors are employed for calculating the displacements and then internal forces of the structures. For this purpose, first an efficient method is developed for calculating the eigenvectors of the product graphs, and then a method is presented for using these eigenvectors for evaluating the displacements of a structure [1]. In the second part, static analysis of structures with repeated patterns is presented. These structures are comprised of submodels each having different repeated pattern. As an example, considering a structure with two different repeated patterns, the nodal numbering is performed in such a manner that the resulting stiffness matrix of the structure contains two block-diagonal matrices. Thus, their inversion can easily be performed using regular matrices requiring smaller amount of computational time. In the second part, the modal analysis, free vibration and eigenfrequencies of such structures are studied. Here as well the stiffness and mass matrices are transformed into two block matrices forms and using dynamic
A. Kaveh, Optimal Analysis of Structures by Concepts of Symmetry and Regularity, DOI 10.1007/978-3-7091-1565-7_8, © Springer-Verlag Wien 2013
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8 Graph Products Applied to the Analysis of Regular Structures
condensation and the matrix inversion which is involved in this condensation, the eigensolution is performed on matrices of lower dimensions [2]. The presented examples consist of 2D and 3D structures in which in some stories, the stiffnesses are changed due to the addition of some members taking the structures out of regularity. Apart from these, the power transition towers often having additional bracings in some levels are investigated. Other applications correspond to calculating the buckling loads and natural frequencies of regular plates driven to irregular forms by having different support conditions and some added pa
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