Beam and frame structures are still widely used and very important question in these structures is estimation of the critical buckling loads. The commercial finite element codes offer many beam finite elements which use Hermite shape functions or isoparametric shape functions. For example, the ANSYS code [2] offers both possibilities. If we want to use these elements for material with a variation of material properties along the beam length, the beam has to be divided into small elements, where each element has different material properties. It has to be clear, that the accuracy of the results obtained is strongly dependent on the mesh density.

To overcome these mesh density problems, new beam finite element was developed, which can describe longitudinal variation of the material properties and also longitudinal variation of the cross-sectional geometric properties exactly in the case, when these variations are polynomials [3].

To adopt this beam finite formulation for FGM materials with a polynomial variation of the fibre and matrix volume fractions along the element length but with constant values of volume fractions through the cross-section, the mixture rules had to be used for calculation of effective material properties.

Our next step was extending this beam finite element formulation for symmetric sandwich beam element. Each symmetric pair of layers of the sandwich is made of the above mentioned FGM material with variations of the volume fractions along the element length.

In this contribution, the effective stiffness matrix for symmetric layered sandwich beam-column with non-constant cross-section is derived. Effective stiffnesses for axial loading and for bending are expressed as a sum of axial stiffnesses and bending stiffnesses of the individual layers, respectively. The coefficients in the effective stiffness matrix depend on the effective stiffnesses at the node i (the begining of the element) and on the transfer constants. The transfer constants for axial loading are calculated according to the first order beam theory, but the transfer constants for bending are calculated according to the second order beam theory - the influence of axial force is considered. This new beam-column finite element can be used for solution of critical buckling loads.

The accuracy and effectiveness of the new symmetric layered sandwich beam-column element with non-constant cross-section is evaluated in one numerical experiment. A layered beam with six layers and with non-constant cross-section with two modifications of boundary conditions is considered. Our new beam finite element is used to calculate the critical loads, the beam-column is modelled with only one new element.

1

A. Chakraborty, S. Gopalakrishnan, J.N. Reddy, "A new beam finite element for the analysis of functionally graded materials", International Journal of Mechanical Sciences, 45, 519-, 2003. doi:10.1016/S0020-7403(03)00058-4

2

ANSYS, version 11, Theory manual, 2007.

3

J. Murín and V. Kutiš, "Beam element with continuous variation of the cross-sectional area", Computers & Structures, 80, 329-338, 2002. doi:10.1016/S0045-7949(01)00173-0

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