Free vibration steady state response of geometrically nonlinear shallow arches

Abstract

A study of the large amplitude, free vibration of geometrically non-linear shallow arches is presented. To obtain the non-linear steady state response a combination of finite elements and finite difference methods is employed. Finite elements idealization is used to express the geometrically non-linear and incremental stiffness matrices and the rest of the terms appearing in the equation of motion are expressed by finite difference equations relating displacement, velocity and acceleration. The resulting recurrence formula is then solved by assuming that for the first time interval, the transverse displacement away from the centre of the arch, is in the form of a sinusoidal equation. The complex relationships between frequency and large amplitude for shallow arches having various ratios of radius to thickness are thus obtained and the results are discussed in relation to the elastic geometrically non-linear load-deflection curves which are generally multi valued exhibiting local maximum and minimum and resulting into snap-through between various equilibrium states. These elastic geometrically non-linear curves are based on an analytical solution in which the shape of the arch is expressed in terms of an infinite Fourier series and the thrust in the shallow arch is assumed to be constant.