Bending and torsional resonances in structures subject to ground motion

Abstract

A perturbation technique is used to analyse the bending and torsional responses of a single-storey structure subject to ground excitations. The structure response is governed by a system of nonlinearly coupled second-order ordinary differential equations. For weak nonlinearities and weak damping it is shown that when the frequency of the lateral excitation is nearly identical to the natural frequency of bending, the steady state amplitude of the bending mode exhibits a cusp catastrophe as a function of the lateral excitation frequency. The steady state amplitude of the bending mode increases with the lateral excitation amplitude, decreases when the damping term increases, and is independent of the structure nonlinearities and torsional excitation amplitude. The lateral excitation frequency at which the steady state maximum bending amplitude occurs decreases as the torsional excitation amplitude, the structural nonlinearities and the lateral excitation amplitude increase. It is shown that when the lateral excitation frequency is nearly identical to the natural frequency of bending, subharmonic, superharmonic, and internal resonances occur in the torsion mode. Some of these resonances result in aperiodic structural responses. When the structure is in resonance in both the torsional and bending modes, it is shown that no steady solution exists when the lateral excitation frequency is nearly identical to the torsional excitation frequency and nearly identical to the natural frequencies of bending and torsion.