Equivalent linear method considering frequency dependent characteristics of stiffness and damping

Abstract

Equivalent linear dynamic response analysis of ground is based on complex moduli and Fourier series expansion; therefore, it is not an equivalent method but an approximate method. Two deficiencies in the conventional equivalent linear method represented by SHAKE are described first. The maximum shear strength is overestimated, resulting in overestimation of the peak acceleration under a strong ground motion, and the amplification is underestimated at high frequency. The latter sometimes results in underestimation of the peak acceleration under weak ground shaking, and gives an incident wave with unrealistic large accelerations or a divergence of analysis in deconvolution analysis under strong ground motion. Both deficiencies are shown to come from the same cause, i.e. computing the effective strain as a constant fraction of the maximum strain. Since this is a key concept of the equivalent linear analysis, one cannot overcome both deficiencies at the same time in the conventional method. An apparent frequency dependence in stiffness and damping is shown to appear in the dynamic response, although soil itself does not show frequency dependent characteristics. Following this observation, the effective strain is expressed in terms of frequency from the similarity concept of the strain–frequency relationship between time domain and frequency domain. This enables the reduction of both deficiencies at the same time, resulting in a marked improvement in the equivalent linear analysis. The accuracy of the proposed method is examined by the simulations of three vertical array records during large earthquakes. The proposed method always gives much better prediction than conventional equivalent linear methods for both convolution and deconvolution analyses, and it is confirmed to be applicable at more than 1% shear strain.