This article investigates the solution of exponentially graded (EG) thick rectangular plates resting on two-parameter elastic foundations according to a trigonometric plate theory (TPT). This theory includes the effect of both shear and normal strains thickness without needing to any shear correction factor. The displacement fields contains initial terms of a power series across plate thickness as well as additional trigonometric terms. The material properties of plate is graded such that Lamé’s coefficients convert exponentially in a given constant orientation. Equilibrium equations according to the EG plate resting on Pasternak’s foundations are derived. The solution is obtained by using Navier’s technique. Numerical results for the EG thick plate on elastic foundations are presented, and compared with those available in the literature. The influences of Winkler’s and Pasternak’s parameters, side-to-thickness ratio, inhomogeneity parameter and aspect ratio on the bending responses of EG plates are investigated.
The vibration and stability analysis of uniform beams supported on two-parameter elastic foundation are performed. The second foundation parameter is a function of the total rotation of the beam. The effects of axial force, foundation stiffness parameters, transverse shear deformation and rotatory inertia are incorporated into the accurate vibration analysis. The work shows very important question of relationships between the parameters describing the beam vibration, the compressive force and the foundation parameters. For the free supported beam, the exact formulas for the natural vibration frequencies, the critical forces and the formula defining the relationship between the vibration frequency and the compressive forces are derived. For other conditions of the beam support conditional equations were received. These equations determine the dependence of the frequency of vibration of the compressive force for the assumed parameters of elastic foundation and the slenderness of the beam.