Keywords: buckling load, symmetry, planar frames, graph theory, eigenvalues, decomposition, healing, factors.

Symmetry has been widely studies in science and engineering [1,2,3,4]. Large eigenvalue problems arise in many scientific and engineering problems [5,6,7]. 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 and the sparsity of matrices increase. Special methods are needed for efficient solution of such problems.

Methods are developed for decomposing the graph models of structures in order to calculate the eigenvalues of matrices with special patterns, Kaveh and Sayarinejad [8]. The eigenvectors corresponding to such patterns are studied in reference [9]. The application of these methods in extended to the vibration of mass-spring systems [10], and free vibration of frames [11].

In this paper, two special cases are studied based on factoring the graph models of the frame structures. Here, a brief description of different forms of symmetry is presented. The main objective of this paper is to develop a methodology for an efficient calculation of buckling loads for symmetric planar frame structures in order to reduce the size of the eigensolution problems involved. This is achieved by decomposing the symmetric model into submodels. The operations performed after decomposition is called the healing of substructures. The submodels obtained after the decomposition and healing are known as the factors of the structural model.

Healing for different types of symmetry is performed using different operations. The buckling load of the entire structure is then obtained by calculating the buckling loads of its factors. Many simple examples are provided to illustrate the simplicity and efficiency of the present method.

Factoring the symmetric structures has the following advantages:

1. The DOF of the problem is reduced.
2. The computational effort is decreased.
3. The accuracy of the calculations is increased.

Though the examples are selected from small structures, however, the method shows its potential more when applied to large-scale structures.

Here, only simple types of symmetry are studied. The method can be extended to other cases when more than one axis of symmetry is present.

1

Hargittai, I, Symmetry; Unifying Human Understanding, Pergamon Press Ltd, UK, 1986.

2

Gruber, B., Symmetries in Science VIII, Plenum Press, NY, 1995.

3

Glockner, P.G., "Symmetry in structural mechanics", ASCE, Journal of Structural Division, 99:71-89, 1973.

4

Zingoni, A., Pavlovic, M.N., Zlokovic G.M., "A symmetry-adapted flexibility approach for multi-storey space frames: General outline and symmetry-adapted redundants", Structural Engineering Review, 7:107-119, 1995.

5

Livesley, R.K., Mathematical Methods for Engineers, Ellis Horwood Series in Mathematics and its Applications, UK, 1989.

6

Jennings A., McKeown, J.J., Matrix Computation, John Wiley and Sons, UK, 1992.

7

Bathe, K.J., Wilson, E.L., Numerical Methods for Finite Element Analysis, Englewood Cliffs, Printice Hall, USA, 1976.

8

Kaveh, A., Sayarinejad, M.A., "Eigensolutions for matrices of special patterns", Communications in Numerical Methods in Engineering, 19:125-136, 2003. doi:10.1002/cnm.576

9

Kaveh, A., Sayarinejad, M.A., "Eigensolutions for factorable matrices of special patterns", textitCommunications in Numerical Methods in Engineering, No. 2, 20, 133-146, 2004. doi:10.1002/cnm.656

10

Kaveh, A., Sayarinejad, M.A., "Graph symmetry in dynamic systems", Computers and Structures, in press, 2004. doi:10.1016/j.compstruc.2004.03.066

11

Kaveh, A., Salimbahrami, B., "Free vibration of frames and symmetry", Communications in Numerical Methods in Engineering, in press, 2004.

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