The steady‐state responses of a bilinear elastoplastic oscillator under sinusoidal loading

Abstract

In this paper a new phase‐plane estimation method for periodic elastoplastic solutions is presented for the steady‐state responses of the single‐degree‐of‐freedom bilinear elastoplastic oscillator under sinusoidal loadings. Three dimensionless ratios, namely stiffness ratio α, force ratio rf and frequency ratio rw , as well as an elastic‐phase duration variable y are identified. The new estimate offers closed‐form formulae for rf and the nondimensionalized maximum displacement ?/xy in terms of α, rw and y, where xy is the yield displacement. Applying the half‐interval numerical method to the function rf=rf (α, rw, y) one can build up the inverse relation y=y(α, rw, rf ), and hence the variation of ?/xy in terms of α, rw and rf can be evaluated. The validity and accuracy of the formulae developed are confirmed by comparing with exact time‐marching solutions. A simple criterion of parameters’ values for elastic shakedown is derived, by which the minimum driving force amplitude needed to prevent the oscillator vibrating in the elastic range can be estimated. It is found that there exists a best driving force amplitude for maximizing dissipation efficiency.