Detailed three-dimensional FEM models have been used for seismic analysis with the growth of computer performance. In seismic analysis using the FEM models, Rayleigh damping and element Rayleigh damping have been commonly used; however, it is not necessarily clear how to model those damping in performing seismic analysis of a system consisted of structures with multiple damping properties. The purpose of this study is (i) to propose modeling method for element Rayleigh damping applied to the above-mentioned system and (ii) to examine effectiveness of the proposed method. In the proposed method, coefficients of element Rayleigh damping are determined by using weighted least-squares method, and the weighting factors are based on products of modal participation function and response spectrum of input seismic motion. Additionally, in order to examine effectiveness of the proposed method, seismic analyses were performed by using lumped-mass analysis model with the proposed method and the conventional methods. Analysis results were compared with the ones given by the strain energy proportional damping, and error rates were evaluated. As a result, average error rate of the proposed method was 6 %, and that was smaller than those of the conventional methods. Thus, it is concluded that the proposed method is effective for seismic analysis of the system consisted of structures with multiple damping.

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Damping modeling is important for the accurate evaluation of the seismic response of structures. Our group previously reported a damping modeling method using element Rayleigh damping and evaluated the effectiveness using a simple lumped-mass model with multiple damping properties; however, the effectiveness of the method was not evaluated for three-dimensional (3D) finite element method (FEM) models with multiple damping properties. Moreover, further studies showed that the method needed to be improved to be applied to 3D FEM models. Therefore, the method has been improved to enable application to the seismic analysis of 3D FEM models, and the effectiveness of the method has been evaluated. The proposed method uses a weighted least-squares method to automatically determine the coefficients of element Rayleigh damping. The weighted least-squares method minimizes the differences between the modal damping ratios to be modeled and those given by element Rayleigh damping. Although all modal damping ratios in a simple lumped-mass model were used for damping modeling in our previous study, obtaining them for 3D FEM models is impractical because these models have more natural modes than simple lumped-mass models. Therefore, we used modal damping ratios below a cut-off frequency. The effectiveness of the proposed method was evaluated by comparing it with conventional methods in terms of the modeling errors related to the modal damping ratios and the maximum absolute acceleration. The proposed method tended to have lower errors than the conventional methods and is concluded to be more effective for the seismic analysis of 3D FEM models with multiple damping properties. The proposed method can automatically determine the coefficients of element Rayleigh damping and can more accurately model the damping properties of analysis models, indicating that the proposed method is a powerful tool for the seismic analysis of 3D FEM models with multiple damping properties.

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