A higher order beam model for thin-walled structures with in-plane rigid cross-sections

Abstract

A higher order thin-walled beam model for the analysis of thin-walled structures considering the cross-section warping and shear deformation is presented in this paper. The formulation is derived from the corresponding elasticity governing equations by adopting an approximation of the beam displacement field over the cross-section by a set of linearly independent basis functions, which are refined according to the accuracy required for the representation of the three-dimensional behaviour of the structure. A set of beam-like equations is obtained through the integration over the cross-section of the corresponding elasticity equations, properly weighted by the cross-section approximation functions. A set of uncoupled beam deformation modes is obtained from a non-linear eigenvalue problem that stems directly from the homogeneous solution of the beam differential equilibrium equations, being reduced to a generalized eigenvalue problem for a transversely rigid cross-section. The criteria put forward for uncoupling the beam deformation modes is mathematically consistent and allows the interpretation of the involved structural phenomena. Two kinds of deformation modes are obtained: (i) classic deformation modes, being associated with a null eigenvalue, requiring an adequate computation of a Jordan chain and (ii) higher order modes corresponding to the non-null eigenvalues, allowing to measure the mode decay along the beam axis. Some examples are presented in order to verify the capability of the model to simulate the non classic effects associated with the higher order deformation modes of thin-walled structures.