Non-conservative dynamic stiffness analysis of thin-walled structures

Abstract

A dynamic stiffness method is introduced to analyse thin-walled structures under the influence of a follower axial force and an in-plane moment. When harmonic oscillation is concerned, both the spatial and temporal time discretization errors are eliminated to give an exact solution in the classical sense. When warping effects are included, the governing differential equations correspond to a matrix polynomial eigenproblem of third-order matrices and degree four. The determinant equation is expanded analytically to give a scalar polynomial equation of degree 12 providing 12 integration constants for the 12 nodal displacements of the thin-walled beam member (excluding the uncoupled axial displacements). The generalized nodal forces are related to the nodal displacements analytically resulting in the exact dynamic stiffness matrix. The natural boundary conditions are modified to cater for the non-conservative loads. Numerical examples show that the interaction diagram of natural frequency against the constant in-plane moment does not have a monotonic change of slope. This is due to the fact that the constant in-plane moment softens the flexural modes while hardening the torsional modes. In some cases, isola loops are possible in the interaction diagram. Examples on plane frames are also given.