Three-dimensional closed form solutions and exact thin plate theories for isotropic plates

A Navier-type method for finding the exact three-dimensional solution for isotropic thick and thin rectangular plates is presented. The method uses the Mixed Form of Hooke’s Law (MFHL) which leads one to write the boundary conditions on the top and bottom surfaces of the plate directly in terms of transverse stresses. The solution is found by solving a first order system of differential equations in the unknown amplitudes of the displacements and stresses. This leads to an eigenvalue problem in which only two (over a total of six) eigenvalues are distinct. Therefore, a basis of eigenvectors is not available and two generalized eigenvectors have to be found. The solution is a combination of the eigenvectors and generalized eigenvectors multiplied by functions of the out-of-plane coordinate z. The paper also presents exact closed form expressions (function of the geometry and material properties) for the displacements and stresses for a simply supported rectangular plate with sinusoidal pressure on the top surface. Three Thin Plate Theories are obtained by expanding the exact solution in Taylor series with respect to the parameter h/a, which is a measure of the accuracy of the two-dimensional theories. For small ratios h/a the Thin Plate Theory of order zero (the Classical Plate Theory CPT) works very well, but for thick plates higher order terms of the series have to be taken into account in order to have good accuracy. Finally, a plate subjected to two pressures on both the top and bottom surfaces is analyzed and the exact solution is compared with the quasi-3D results obtained by adopting the mixed assiomatic theories of order 5 and 10 presented here for the first time.