Problem Definition and Formulation: Fid-Bau Portal

Problem Definition and Formulation

Spencer, B. F. Jr.

Abstract Of the models utilized to date to describe the hysteretic constitutive relation for a simple system, one of Che most versatile and tractable for dynamic applications is the modified Bouc model [19]. The original model was modified by Wen [127] in order that the smoothness of the transition between the pre-yield and post-yield regions of the force-deflection curve could be controlled. The constitutive relation for the modified Bouc hysteresis model can be stated for the SDOF oscillator in Figure 1.1 as $$Q(x(\tau ),\,\dot x(\tau ),\,0 < \,\tau \, < \,t;\,t) = M(2\zeta \omega \dot x\, + \,a{\omega ^2}x\, + \,(1 - a){\omega ^2}z)$$ (2.1) where z is an evolutionary variable described by the first order differential equation $$\dot z\, = \, - \,\gamma |\dot x|\,|z{|^{(n - 1)}}z\, - \,\beta \dot x|z{|^{n\,}} + \,A\dot x$$ (2.2) and ω is the undamped natural frequency of the oscillator when α = 1. The parameters α, β, γ, n and A are shape parameters of the hysteresis loops which can also be functions of time. The quantities $$Ma{\omega ^2}x\,and\,M(1 - a){\omega ^2}z$$ in Equation 2.1 are the linear and hysteretic portions of the total restoring force, respectively, and M(2ζωx) is the force associated with viscous dissipation. Baber and Wen [6] have shown that the hysteretic restoring force for the SDOF oscillator can be effectively modeled by Equations 2.1 and 2.2. Their report provided insight into the effect of each of the shape parameters on the hysteretic constitutive relation. These effects shall be discussed here for completeness.