Fig. 2. a Front elevation of the lumped-parameter model of the original SSI system; b the conventional nth SDOF modal system with equivalent damping ratio; and c the front elevation of the pro-posed nth MDOF modal system resting on an elastic base

sponse histories, Dnt, were obtained using the direct integration method in solving Eq. 13. Thus, the total response histories of the original SSI system were obtained as

 Same as the conventional modal response history analysis, only the first few modal responses need to be included in the above-presented summation to obtain satisfactory analytical results. This will be shown in the following numerical examples.

Physical Meaning of the MDOF Modal Equations

The front elevation of the lumped-parameter model of the original SSI system is illustrated in Fig. 2a. By calculating the equiva-lent modal damping for each vibration mode Roesset et al. 1973; Novak and Hifnawy 1983; Tsai 1974; Balendra et al. 1982, the original SSI system can be decomposed into conventional single-degree-of-freedom SDOF modal systems as shown in Fig. 2b. In contrast to the conventional SDOF modal equation of motion represented by a SDOF modal system  Fig. 2b, the MDOF modal equation of motion shown in Eq. 13 can be representedby a MDOF modal system resting on an elastic base as illustrated in Fig. 2c. The properties of the MDOF modal system as shown

in Fig. 2b, i.e., ˜M, ˜C, and ˜K, are expected to represent those of the nth MDOF modal equation of motion as shown in Eq. 14. As Fig. 2c can be regarded as the lumped-parameter model of a one-story asymmetric building with soil–structure interaction, the

expressions of ˜M, ˜C, and ˜K will be those defined in Eqs. 2b–2d with N=1. By comparing the elements of Mn, Cn, and Kn, as shown in Appendix I with the corresponding elements shown in Eqs. 2b–2d with N=1, the elements of ˜M, ˜C, and were obtained as

With the notations defined in Eqs. 17a–17j, Eq. 14 can be expressed in a similar form to those of Eqs. 2b–2d with N=1: