Consistent FEM-Vlasov model for slabs on elastic foundations

The deformation of a slab on ground is often analyzed considering the interaction between the slab and the soil. Sometimes, these slabs are stiffened with beams in both directions which makes the analysis more complicated. Moreover, in many practical situations, the soil underneath can be layered with variable material properties. Using assumptions of Vlasov and Leontev and finite element theory, four noded rectangular finite elements with 12 degrees of Freedom are employed to model the slab and the soil along with a 4 degrees of freedom element for beams that stiffen the slab. The model consists of unique stiffness matrices representing the soil that are consistent with energy principles.