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 the fundamentals of elasticity theory. The theoretical model whose formulation is based on static elastic principle considered transverse shear deformation and still obviates the need for the shear correction factor effect which is associated with 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 plate s coefficient of deflection and shape function. 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 were obtained. The result shows that, at a span - depth ratio between 4 and 30, deflection values vary between 0.0063 and 0.0054. At the span - depth ratio 30 and above, a constant value of 0.0054 is noticed, this value is equal to the value of the CPT. The value of transverse shear stress along y-z coordinates varies between 0.00198 and -0.0001. The value of transverse shear stress along y-z coordinates being less than or equal the value showed that this value is the same as that of CPT. It can be deduced that at a span - depth ratio between 4 and 30 the plate is regarded as thick. The plate s span - thickness ratio of 30 and beyond is regarded as thin or moderately thick because its value at this point coincides with the value of the CPT. 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 a rectangular plate and can be recommended for the analysis of any type of rectangular plate with such loading and boundary condition.