Probabilistic spectrum superposition for response analysis including the effects of soil-structure interaction I. D. Gupta Earthquake Engineering Research Division, CWPRS, Pune-411024, lndia M. D. Trifunae Department of Civil Engineering, University of Southern California, Los Angeles, CA 90089, USA INTRODUCTION The soil-structure interaction may have significant influence on the dynamic response of structures, depending upon the material properties of the soil medium, the characteristics and shape of the foundation, nature of seismic excitation, and the characteristics of the structure. The translation and the rotation of the foundation associated with ground compliance may lead to a significant additional motion of the superstructure that may account for a major part of the total response. The inertial forces produced by such additional motion of the foundation may cause appreciable deformations in the structure (Wolf2S). For example, the analyses of the results of the forced vibration tests on the nine-storey reinforced concrete Millikan library building at California Institute of Technology have indicated that more than 30% of the total deflection at the roof is associated with the rigid-body motion of the foundation due to soil-structure interaction (Foutch et al. 6, Luco et al. 19, Wong et a/.27). Luco 16 showed that if inertial effects are also included, more than 90% of roof response can be attributed to soil-structure interaction. Results on structural identification are also influenced significantly by the interaction effects (McVerry 2°, Beck and Jennings a, Moslem and Trifunac 14, Luco eta/. 17A8 etc.). If the effects of the interaction are not considered, the fixed base natural frequencies of the structure are underestimated and the damping in the structure is overestimated. Thus, the soil-structure interaction is an important factor to be considered in the analysis of structural response to strong-motion earthquake excitation. For linear response analysis of structures with rigid base the mode superposition method is often preferred due to its simplicity. However, for deformable base, decomposition into classical normal modes only is not possible because of the dependence of foundation motion on excitation frequencies, and because of the presence of exponential mode shapes, which are not associated with energy transfer into the structure (Todorovska et al. 22, Todorovska and Lee 23, Todorovska and Trifunac24). Nevertheless, some possible applications of the modal analysis to interacting structure foundation systems have been suggested by several workers. Tajimi 2~ has Paper accepted October 1989. Discussion ends August 1990. presented the equations for a building on a foundation represented by frequency independent rotational and horizontal springs in terms of the normal modes of the building on a rigid-base. Jennings and Bielak 13 have given an approximate solution for the response of a multi-storey building with base disk having two degrees of freedom in terms of the modified responses of one-degree- of-freedom viscously damped linear oscillators resting on rigid ground. For forced vibrations in the neighbourhood of the first fixed-base natural frequency, Luco et al.18 have expressed the relative displacement response in terms of the fundamental fixed-base mode, and they have derived approximate solutions for different response components of interacting buildings. They have shown that the calculated response from these approximate relations for the Millikan library building is in excellent agreement with the observed response. Chopra and Gutierrez 5 have presented the methods for computing the frequency- response of multi-storey buildings including soil-structure interaction by expressing the relative response in terms of the first few modes of the fixed-base superstructure. Gupta and Trifunac 7'12 have presented a new stochastic approach for response spectrum superposition, which can be used to compute the total relative response of structures under simultaneous excitation by free-field translational and rocking earthquake ground motion. This approach can also provide the amplitudes of all the significant peaks of the response, which is not possible from other methods. In their study, the effects of the soil-structure interaction are not considered. In this paper, the formulation of Gupta and Trifunac 7'12 will be generalized to include the effects of the soil-structure interaction also. It has been assumed that the total relative response, including the effects of the additional translation and rocking of the foundation due to deformation of the soil, can be expressed in terms of the modes of the fixed-base structure. A simple three-storey structure has been analysed to illustrate the application of the presented method. The additional deformations at various levels of the structure due to the interaction have been evaluated for different properties of the foundation soil, storey heights of the building and the aspect ratio of the foundation slab. It is observed that the effects of the interaction are more prominent for low values of the © 1990Computational Mechanics Publications Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 9