# A consistent Vlasov model for analysis of plates on elastic foundations using the finite element method

## Abstract

Plates supported by elastic foundations present very common technical problems in civil engineering. All structural loads must be transferred to the soil, and both the structure and the soil continuum must act together to support the loads. Developing more realistic foundation models and simplified methods to solve this complex soil-structure interaction problem are very important for safe and economical designs. In the past, many researchers have worked on this problem.

Generally, analysis of the bending of beams on an elastic foundation is developed on the assumption that the reaction forces of the foundation are proportional at every point to the deflection of the beam at that point. This assumption was first introduced by Winkler for the analysis of railroad tracks. The problem with the Winkler model applied for analysis of plates on elastic foundations is the necessity of the evaluation of the modulus of the subgrade reaction, k, which does not have a unique value for a particular soil or a particular loading on the plate. The major disadvantage of the Winkler model is that it gives a constant displacement of the plate for a uniformly distributed load which results in a zero bending moment and shear force in the plate, thus creating a non-conservative design criteria. However, the Winkler model has been used for everyday design by practicing engineers because of its simplicity.

Several researchers have tried to improve the Winkler model by considering the shear strain energy in the soil in addition to the strain energy due to normal strains as used in the Winkler model. Of all models, a two-parameter model by Vlasov using a variational method has attracted the attention of many engineers. The Vlasov model accounts for the effect of the neglected shear-strain energy in the soil and the shear forces on the plate edges that come from the surrounding soil. Vallabhan and Das (1987) developed an iterative technique to solve problems of beams on elastic foundations by introducing a modified Vlasov model. Straughan (1990) used the modified Vlasov model for the analysis of rectangular plates by the finite difference method.

The research herein develops an approximate numerical approach, based on the finite-element technique, using the modified Vlasov model. The plate on elastic foundation problem is a three-dimensional one in solid mechanics. All the energy of the plate and of the semi-infinite layered soil domain is considered here. However, the methodology reduces to an analysis to be performed only for the plate domain. The geometry of the plate is restricted to a geometrical figure that can be formed by rectangles since the plate is modelled using four-node rectangular finite elements. Special features are that the plate can have beams, rectangular openings, or cutouts, as one sometimes sees in practice. Another provision is that the modulus of elasticity of the soil can vary linearly from the top to the bottom of the soil domain. The modulus of subgrade reaction, k, and a soil shear parameter, t, are calculated iterativdy with respect to a deformation parameter, 7. Finally, a nondimensional parametric study is performed that aids in evaluating the modulus of subgrade reaction, k, to be used in a Winkler foundation model for concentrated loads at specific locations.