Coupled Rocking and Translating Vibrations of a Buried Foundation

The plane strain model of a rigid building foundation embedded below the surface of an elastic half space is treated. The foundation is assumed to be vibrating freely with arbitrary transient time-dependences U(t) and V(t) in the horizontal and vertical directions, respectively, and to be rocking about its mass center with magnitude Ω(t). The total restraining reactions exerted upon the foundation by the surrounding medium are determined exactly during the initial time period for a P-wave to traverse the foundation base width. Thereafter the results become approximate. In this manner the coupled equations of motion for free vibrations of the foundation are obtained. A numerical example presents the impulse response matrix for U, V, and Ω. Peak responses are found to occur during the early time period where the results are exact.