The derivation of a thick and thin plate formulation without ad hoc assumptions

For the plate formulation considered in this paper, appropriate three-dimensional elasticity solution representations for isotropic materials are constructed. No a priori assumptions for stress or displacement distributions over the thickness of the plate are made. The strategy used in the derivation is to separate functions of the thickness variable z from functions of the coordinates x and y lying in the midplane of the plate. Real and complex 3-dimensional elasticity solution representations are used to obtain three types of functions of the coordinates x, y and the corresponding differential equations. The separation of the functions of the thickness coordinate can be done by separately considering homogeneous and nonhomogeneous boundary conditions on the upper and lower faces of the plate. One type of the plate solutions derived involves polynomials of the thickness coordinate z. The other two solution forms contain trigonometric and hyperbolic functions of z, respectively. Both bending and stretching (or in-plane) solutions are included in the derivation.