Semiexplicit Random Response and Sensitivity of Simple SSI System

For a simplified model of a structure-pile-soil system, semiexplicit forms of the responses to random earthquake inputs are derived together with sensitivity expressions of the random responses with respect to a certain model parameter. Originally this model has a non-tridiagonal stiffness and damping matrices with single earthquake input. It is shown for the first time that independent treatment of the free-field ground and multiple imposition of the free-field ground response into the structure-pile system allow for a transformation of the governing equations so that the final form has tridiagonal stiffness and damping matrices. Closed-form expressions of the inverse of the tridiagonal matrix lead to semiexplicit expressions of the random responses and their sensitivities with respect to a certain model parameter.