Variation Method for Displacement and Stress Analysis of Trigonometric 3-D Shear Deformable Plate Under Distributed Lateral Loading

Abstract

In this work, variational calculus was applied to the analysis of stresses and displacement of a rectangular plate carrying a uniformly distributed lateral load. A 3-D trigonometric shear deformation model was developed using the elastic static principle and applied in the coupling the 3-D kinematics and constitutive relations from which the total potential energy equation was formulated. The formulated energy equation was transformed into the equilibrium equation which was used to obtain the shape function of the plate. An exact trigonometric deflection of the plate which is a product of its coefficient and shape function was obtained analytically through the principle of general variation. Furthermore, the formula for calculation of the displacements and stresses induced due to application of a lateral load in the plate was obtained by the direct variation of the total potential energy equation to produce a reliable solution for the statically bending analysis of the plate. The outcome of the numerical analysis revealed that increase in the span-thickness ratio led to the decrease in the value of displacement and stresses induced in the plate. On the other hand, as the longest-breadth ratio of the plate increased, the value of the displacement and stresses in the plate increases. The result showed that the present model developed gives distinct and satisfactory solution but still followed an identical pattern when compared with previous studies, this shows the credibility of the derived relationships. The percentage error analysis showed that the present model stress prediction for the analysis proved more reliable and can be used with confidence for the analysis of any type of rectangular plate compared to the approximate or 2-D model for the given edge condition.