Exact analytic solution for static bending of 3-D plate under transverse loading

In this study, an exact solution for the bending analysis of a three-dimensional (3-D) rectangular plate under transverse loading is presented using fundamentals of elasticity theory. The theoretical model whose formulation is based on static elastic principle considered transverse shear deformation and still obviate the need for shear correction factor effect which is associated to refined plate theory (RPT). As an improvement to RPT, the equations of equilibrium are obtained from elastic principle using 3-D kinematic and constitutive relations which is later converted to energy equation using general variation to get the deflection and rotation relationship. The solution of the equilibrium equation produced an exact trigonometric displacement function which is a product of the coefficient of deflection and shape function of the plate. By minimizing the general energy equation with respect to the coefficients of deflection and shear deformation rotation, a theoretical model for calculating the deflection, moment and stresses of thick rectangular plate are obtained. The comparative analysis between the present results and other theories shows that this 3-D predicts the vertical displacement, moments and the stresses more accurately than previous studies considered in this paper. It was observed that the present theory varied more with those of those of 2-D numeric analysis and 2-D HSDT with about 7.83% and 6.01%. Meanwhile, the recorded percentage differences showed that a derived 2-D HSDT predicted accurately the bending characteristics of the plate with 2.55%, proving that assumed deflection is coarser for the thick plate analysis. It is concluded that unlike an assumed function, a derived 2-D theory can give a close form solution, but a typical 3-D theory of elasticity is required for an exact solution of rectangular plate and can be recommended for the analysis of any type of rectangular plate with such loading and boundary condition.