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The issue on seismic analysis of asymmetric-plan structures with soil–structure interaction subjected to bidirectional ground motions has been investigated by many researchers Balendra et al. 1982; Sivakumaran and Balendra 1994. However, practic-ing engineers are usually not familiar with the complex-valued seismic analysis procedures in the frequency domain. Moreover, the equivalent modal damping estimated either by quantifying the dissipated energy in the soil, or by matching the approximate normal mode response with the rigorous solution is not readily available using a general purpose structural response analysis computer program. Thus, it is desirable to develop a simple and real-valued modal response history analysis procedure for engi-neering applications without the need for calculating the compli-cated equivalent modal damping. In this study, such a response history analysis procedure is proposed using the multi-degree-of-freedom MDOF modal equations of motion. The proposed method can be conveniently applied using any general purpose computer program with the capability of solving equation of motion.
This study consists of three main parts. First, MDOF modal equations of motion were derived from the equation of motion for the original SSI system. Next, the modal properties of the original SSI system were represented by the corresponding eight-degrees-of-freedom modal system resting on an elastic half-space. Finally, the efficiency of the proposed method was validated using nu-merical examples. Two extreme types of soil with shear wave velocity of 65 and 300 m/s, respectively, were used in the examples for investigating the SSI of a four-story asymmetric building.
1Associate Research Fellow, National Center for Research on Earth-quake Engineering, 200, Section 3, XinHai Rd., Taipei 106, Taiwan; formerly, NTU Ph.D. Student. E-mail: jllin@ncree.org
2Director, National Center for Research on Earthquake Engineering, 200, Section 3, XinHai Rd., Taipei 106, Taiwan; Professor, Dept. of Civil Engineering, National Taiwan Univ. corresponding author. E-mail: kctsai@ncree.org
3Associate Professor, Dept. of Civil and Environmental Engineering, Stanford Univ., Stanford, CA 94305. E-mail: emiranda@stanford.edu
Note. Associate Editor: Marvin W. Halling. Discussion open until July 1, 2009. Separate discussions must be submitted for inpidual papers. The manuscript for this paper was submitted for review and possible publication on February 26, 2007; approved on July 9, 2008. This paper is part of the Journal of Structural Engineering, Vol. 135, No. 2
Theoretical Background
A simplified beginning model for the SSI problem was used in this study. The system considered is an elastic N-story shear building resting on the surface of an elastic homogeneous half-space Fig. 1. The interaction forces at the soil–structure interface were simulated using frequency-independent spring-and-dashpot set in parallel Richart et al. 1970. In order to adopt the frequency-independent spring-and-dashpot set, a rectangular footing was simulated as a circular footing.
Equation of Motion of the SSI System
The equation of motion of an N-story asymmetric shear building resting on an elastic homogeneous half-space has been well es-tablished Balendra et al. 1982. For the sake of completeness, the noted equation of motion is briefly presented herein.
The mass of the building was assumed to be concentrated at the levels of the rigid floor decks which are supported by mass-less, axially inextensible columns and walls. The center of mass CM and the center of stiffness CR of each floor were assumed
o lie on two vertical lines. The system has 3N+5 degrees of freedom, consisting of 3N degrees of freedom of the superstruc-ture and 5 degrees of freedom due to interaction at the foundation. These 5 degrees of freedom include two horizontal translations, two rocking, and one twist Fig. 1. The equation of motion for the whole SSI system