The plate bending problem is one of the most fundamental problems in structural analysis. Since analytical methods are limited to relatively simple geometries, boundary conditions and loadings, numerical methods have been applied to this problem. Among those methods the finite element method has attracted much attention because of its general application to complex problems. However, the difficulty in satisfying the slope compatibility condition in a usual displacement-based finite element method has been a barrier to a plate bending problem. In this research the new mixed finite element equation is formulated using the Galerkin method to overcome the difficulty mentioned above. Unknowns are moments and a lateral displacement. This eliminates the difficulty of slope compatibility. Moments are obtained accurately at nodal points. This formulation includes shear deformations for thick plate problems and is extended for anisotropic and layered composite materials. In addition, this study includes an elasto-plastic analysis of plate bending problems. The elasto-plastic formulation utilizes moment-based yield criterion and a nonlayered approach. An initial-stress type technique is developed for a nonlinear solution. A beam equation is degenerated from the plate equation and applied to elastic and elasto-plastic problems.