Based on the assumption that shear modulus of foundation changes linearly along depth

the dynamic response of non-homogeneous foundation is studied in this paper by introducing differentiating operator of structure and using the stochastic vibration theory for continuous linear system. Then simplifying upside structures as multi-degrees of freedom system and foundation as shear beam

the stochastic earthquake response of interaction between foundation and structures is implemented. Using those results

the dynamic reliability parameters of foundation and structural maximal displacement are obtained. The response analysis is implemented by mode superposition method

which has a definite conception and simpler calculation procedures compared with ordinary separation variable method.