The steady state responses of s.d.o.f. viscous elasto-plastic oscillator under sinusoidal loadings

Depending on the values of the parameters identified, the long-term behaviour of s.d.o.f. viscous elasto-plastic oscillator under sinusoidal loadings may be periodic with or without intermediate unloading, or elastic shakedown (including purely elastic). A new phase-plane estimation method for the steady state elasto-plastic solutions is presented. We identify three dimensionless ratios, namely damping ratio ζ, force ratio rf and frequency ratio rw, as well as an elastic-phase duration variable y. The new estimate offers closed-form formulae for the force ratio rf and for the ductility ratio μ in terms of ζ, rw and y. Applying numerical method to the function rf=rf(ζ,rw,y), we can obtain the inverse function y=y(ζ,rw,rf), such that the variation of μ in terms of ζ, rw and rf can be evaluated. Alternatively, for a given ductility ratio, we can solve μ=μ(ζ,rw,y) numerically for y and then obtain the curves of rf versus rw to meet specified ductility ratio. The proposed method can estimate the steady state responses for any applied load and forcing frequency. The results calculated are in very good agreement with the exact time-marching solutions. A simple criterion of parameters values for elastic shakedown is derived, by which we can calculate the maximum driving force amplitude to avoid structures oscillating in the plastic range. It is found that there exists a best driving force amplitude for maximizing dissipation efficiency. The distribution of periodic points with and without intermediate unloading in the parametric plane (rw,1/rf) for given damping ratio is clarified. The intermediate unloading motion region locates within the range of rw⩽1−ζ2/3, and its size decreases when damping ratio increases.