Chapter 22 FEM with Floquet Theory for Non-slender Elastic Columns Subject to Harmonic Applied Axial Force Using 2D and 3D Solid Elements

摘要

The Rayleigh-Ritz formulation of finite element method using solid elements is implemented for a 2D and 3D clamped-clamped column which is subject to a periodically applied axial force. Non-linear strain is considered. A mass element matrix and two stiffness matrices are obtained. After assembly by elements, the calculated natural frequencies and buckling loads are compared to Euler-Bernoulli beam theory predictions. For 2D triangular and 3D cuboid elements, a large number of degrees of freedom are required for sufficient convergence which adds particular computational costs to applying Floquet theory to determine stability of the harmonically forced column. A method popularised by Hsu et al. is used to reduce the computational load and obtain the full monodromy matrix. The Floquet multipliers are discussed in relation to their bifurcations. The versatile 2D and 3D elements used allows for the discussion of non-slender columns. In addition, the stability of a 3D steel column comprised of impure materials or with changed aspect ratio are investigated.