A new method for formulation of dynamic responses of shallow foundations in simple general form

A single uniform rectangular area, either homogeneous or heterogeneous, is considered in a soil medium (fundamental cell). Two governing ordinary differential equations in special form are developed for the fundamental cell. The ground supporting a partially embedded foundation is divided into a number of coarse rectangular areas (secondary cells). Each secondary cell is treated as either a single fundamental cell for homogeneous ground or a stack of fundamental cells for inhomogeneous ground. Differential equations for the assembly of secondary cells are formed with those for the fundamental cells. These equations lead to the soil responses in each cell expressed in simple closed form. They also lead to the convenient treatment of soil with appropriate Winkler-type models along the foundation faces and concentrated forces acting at the foundation corners. With them, the foundation responses are finally expressed in simple closed form. The approach is demonstrated for various cases and confirmed to produce the results reasonable enough for civil engineering use.