Coupled axial–torsional vibration of thin-walled Z-section beam induced by boundary conditions

The coupled axial–torsional vibration of thin-walled Z-section beam induced by the boundary conditions is investigated. The value of the warping function is not zero at centroid for Z-section beam. If the axial displacement of the pin end is restrained at the centroid of the Z-section for thin-walled Z-section beam, the axial vibration and torsional vibration may be coupled. The governing equations for linear axial and torsional vibration of a thin-walled Z-section beam are derived by the d’Alembert principle and the virtual work principle. The bending vibration is uncoupled from axial and torsional vibrations and is not dealt with in this paper. For harmonic vibration, the general solution of these equations with undetermined constant coefficients may be obtained. Substituting the general solution into the displacement and force boundary conditions, a set of homogeneous equations can be obtained. The natural frequencies and the coefficients of the general solution may be obtained by solving the homogeneous equations using the bisection method.Numerical examples are studied to verify the accuracy of the proposed method and to investigate the effect of boundary conditions and the value of warping function at centroid on the coupled axial and torsional natural frequency of Z-section beam.