Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames

A consistent finite element formulation is presented for the free vibration and spatial stability analysis of thin-walled tapered beams and space frames. The kinetic and potential energies are derived by applying the extended virtual work principle, introducing displacement parameters defined at the arbitrarily chosen axis and including second order terms of finite semitangential rotations. As a result, the energy functional corresponding to the semitangential rotation and moment is obtained, in which the elastic strain energy terms are coupled due to axial–flexural–torsional coupling effects but the potential energy due to initial stress resultants has a relatively simple expression. For finite element analysis, cubic polynomials are utilized as the shape functions of the two-noded Hermitian space frame element. Mass, elastic stiffness, and geometric stiffness matrices for the unsymmetric thin-walled cross-section are precisely evaluated, and load-correction stiffness matrices for off-axis concentrated and distributed loadings are considered. In order to illustrate the accuracy and practical usefulness of this formulation, finite element solutions for the free vibration and lateral–torsional buckling problems of thin-walled tapered beam-columns and space frames are presented and compared with available solutions.