Keywords: instability, buckling, column, shear effect, cell centred, finite volume.

In the past few years, the experiences gained with the finite volume method in the computational structural mechanics [1,2,3] show promising perspectives of the method. The method is simple in terms of concept and formulation; furthermore the shear locking problem is not observed in the analysis of thin Timoshenko beams [1] and also in the bending analysis of thin plates [2]. This paper presents a procedure for calculating the buckling load of columns with shear effects, which introduces a new scope of research for the instability analysis of structures by implementation of the finite volume method.

To obtain the buckling load of a prismatic and non-prismatic column with arbitrary cross section, it is discretised into a number of two-node line elements along its length. In the cell centred finite volume method each element considered as a control volume or cell having a centre, which is located at the element centre. A cell has connections to the nearest neighbouring cells located on the right and left sides of the cell. The centres of the cells are considered as the computational points with two degrees of freedom, i.e. transverse displacement and section rotation. To estimate unknown variables and their derivatives at the faces of the cells, which appear in the equilibrium equations, the idea of an interim element is used [3]. The interim elements are two-node isoparametric elements in which linear variations are assumed for the unknowns [4]. The end nodes of the interim elements are located at the centres of the two cells lying on either side of the face. To apply boundary conditions to the solution procedure, two point cells are assumed at the ends of the column. Three types of boundary conditions are possible at the boundaries: displacement boundary condition, force boundary condition and a mixed boundary condition. Equations associated with the internal cells and the point cells provide the eigenvalue equation with the standard form, which can be solved to calculating the buckling load. The smallest of the eigenvalues represents the critical buckling load of the column.

To verify the proposed formulation several test cases were studied by using a computer code developed based on the formulation presented. In these studies the results obtained were compared with the available analytical solutions. The test problems include columns as pinned-pinned, fixed-free, and fixed-fixed with rectangular constant section, a tapered fixed-free column and a pined-pined step column.

A monotonic convergence has been observed in all the problems studied. The accuracy of the results reveals the capability of the finite volume method in good predictions with the assumption of piece wise linear variation of displacement components.

The results obtained are encouraging and reveal good performance of the finite volume method for the instability analysis of columns. The formulation presented here provides a basic procedure that could be extended to the instability analysis of plate and shell structures.

1

N. Fallah, F. Hatami "A displacement formulation based on finite Volume method for analysis of Timoshenko beam", Accepted for publication in 7th International Congress on Civil Engineering, Tehran, Iran, 8-10 May 2006.

2

N.Fallah, "A cell vertex and cell centred finite volume method for plate bending analysis", Journal of computer methods in applied mechanics and engineering, 2004; 193: 3457-3470. doi:10.1016/j.cma.2003.08.005

3

N.Fallah, "Using shape function in cell centred finite volume formulation for two dimensional stress analysis", Lecture series on computer and computational sciences, ICCMSE 2005, Vol. 4A, 183-186.

4

K. J. Bathe, "Finite element procedures", Prentice-Hall, 1996.

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