A Closed Form Earthquake Response Analysis of Multistory Building on Compliant Soil*

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Abstract
A transfer matrix formulation is used to analyze structural response of an W-story building to earthquake excitation, taking in to account soil compliancy under the footing. The free-field ground acceleration is modeled as an evolutionary random process. Analysis is simplified by assuming that the superstructure is composed of N identically constructed story units and that soil behavior is characterized by a known impedance matrix or a compliance matrix. Closed form solutions are obtained for the frequency response functions of the system, permitting new insights to be gained into various parameters that affect the response. Ft is found that if the superstructure and the footing remain the same, the effect of soil-foundation interaction is determined by four frequency dependent functions: KHH/K, KHM/Kh KMH/mhω2, and KMM/mh2ω2 where the K's are the elements of the soil impedance matrix, m is the unit story mass of the superstructure, and h is the unit story height of the superstructure. A numerical example is given for an 8-story building, showing that soil compliancy reduces the levels of both displacement- and force-type response variables, as previously reported by other investigators. However, the additional degrees of freedom of footing rocking and translation do not necessarily add more peaks in the response

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