Nonlinear Response Analysis of Structures Subjected to Random Ground Motions

Abstract eng:
The explicit expression of random ground motion is incorporated into the probability density evolution method to implement stochastic seismic analysis. Recently, two kinds of stochastic harmonic function (SHF) representations of stochastic processes have been proposed. For instance, the first kind (SHF-I) representation of the ground motion acceleration takes the form X_g(t)=∑^N_i=1A(Ω_i)cos(Ω_it+∅_i) where WΩ_i's are independent random frequencies, of which the probability densities are consistent with the target PSD function in the sub-interval, ∅_i's are mutually independent random variables uniformly distributed over [0, 2π] (Chen & Li, 2011). Inserting Eq.(1) into the equation of motion of a nonlinear MDOF structure subjected to random ground motion yields MŸ+CY+f(Y)=-MIX_g(Θ,t) Thus, based on the probability density evolution method the following generalized density evolution equation could be obtained ∂p_zΘ(z,θ,t)/∂t+∑^m_j=1Ż_j(θ,t)∂p_zΘ(z,θ,t)/∂z_j=0 where p_zΘ(z,θ,t) is the joint PDF of the augmented state (Z,Θ) and Z(t) is the the physical quantities of interest, e.g. displacements, stresses, and internal forces, etc (Li & Chen, 2008; 2009). Then a 9-story shear frame exhibiting strong nonlinearity under random ground motion is studied to illustrate the methodology. The Clough-Penzein spectrum is adopted as the target power spectrum density, which could be reproduced by SHFs exactly (Figure 1). The results validate that the proposed method could achieve tradeoffs between accuracy and computation efforts (Figures 2 and 3).

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