The bending solutions of rectangular thick plate with all four edges clamped (CCCC) were investigated in this study. The basic governing equations used for analysis are based on third-order shear deformation plate theory analysis under uniformly distributed load. Using a formulated total potential energy equation, the three coupled general governing differential equations for the determination of the out of plane displacement and shear deformations rotation along the direction of x and y coordinates were obtained. These equations as obtained are solved simultaneously after minimization to determine the coefficients of displacements of the plate and other the mentioned functions. By solving these equations, the analytic solutions of rectangular thick plate with all four edges clamped were derived. From the formulated expression, the formula for calculation of the maximum deflection, moment, stress and in-plane displacements were deduced. The proposed method obviates the need of shear correction factors, which is associated with Mindlin’s theory (FSDT) for the solution to the problem. Moreover, numerical comparison shows the correctness and accuracy of the results.

In this paper, an analytical three-dimensional (3D) bending characteristic of an isotropic rectangular thick plate with all edges simply supported (SSSS) and carrying uniformly distributed transverse load using the energy technique is presented. The three-dimensional constitutive relations which involves six stress components were used in the established, refined shear deformation theory to obtain a total potential energy functional. This theory obviates application of the shear correction factors for the solution to the problem. The governing equation of a thick plate was obtained by minimizing the total potential energy functional with respect to the out of plane displacement. The deflection functions which are in form of trigonometric were obtained as the solution of the governing equation. These deflection functions which are the product of the coefficient of deflection and shape function of the plate were substituted back into the energy functional, thereafter a realistic formula for calculating the deflection and stresses were obtained through minimizations with respect to the rotations and deflection coefficients. The values of the deflections and stresses obtained herein were tabulated and compared with those of previous 3D plate theory, refined plate theories and, classical plate theory (CPT) accordingly. It was observed that the result obtained herein varied more with those of CPT and RPT by 25.39% and 21.09% for all span-to-thickness ratios respectively. Meanwhile, the recorded percentage differences are as close as 7.17% for all span-to-thickness ratios, when compared with three dimensional plate analysis. This showed that exact 3D plate theory is more reliable than the shear deformation theory which are quite coarse for thick plate analysis. Doi: 10.28991/esj-2021-01320 Full Text: PDF

In this paper, direct variational calculus was put into practical use to analyses the three dimensional (3D) stability of rectangular thick plate which was simply supported at all the four edges (SSSS) under uniformly distributed compressive load. In the analysis, both trigonometric and polynomial displacement functions were used. This was done by formulating the equation of total potential energy for a thick plate using the 3D constitutive relations, from then on, the equation of compatibility was obtained to determine the relationship between the rotations and deflection. In the same way, governing equation was obtained through minimization of the total potential energy functional with respect to deflection. The solution of the governing equation is the function for deflection. Functions for rotations were obtained from deflection function using the solution of compatibility equations. These functions, deflection and rotations were substituted back into the energy functional, from where, through minimizations with respect to displacement coefficients, formulas for analysis were obtained. In the result, the critical buckling loads from the present study are higher than those of refined plate theories with 7.70%, signifying the coarseness of the refined plate theories. This amount of difference cannot be overlooked. However, it is shown that, all the recorded average percentage differences between trigonometric and polynomial approaches used in this work and those of 3D exact elasticity theory is lower than 1.0%, confirming the exactness of the present theory. Thus, the exact 3D plate theory obtained, provides a good solution for the stability analysis of plate and, can be recommended for analysis of any type of rectangular plates under the same loading and boundary condition. Doi: 10.28991/CEJ-2022-08-01-05 Full Text: PDF

This work present flexural analysis of rectangular plate subjected to uniform distributed transverse loads using displacement and third-order shear deformation theory. The aim of this study is to establish the formula’s for calculation of the critical lateral imposed load of the plate before deflection reaches the specified maximum specified limit q𝑖𝑤 and critical lateral imposed load before plate reaches an elastic yield point q𝑖𝑝. The essence is to ensure that deflection does not exceed specified maximum limit and the plate shear not exceeding the elastic yielding point. Furthermore, this approach overcomes the challenges of the conventional practice in the structural analysis/design which involves checking of deflection and shear; the process which is proved unreliable. Total potential energy equation of a thick plate was formulated from the static elastic theory of the plate. The formulated potential energy was in the same way used by the method of direct variation to obtain the coefficient of deflection and shear deformation. This expression was applied to solve bending problem of two different types of rectangular thick plates. The plates has one edge clamped and other three edges simply supported (CSSS). From the result obtained in this work among the two types of plate, it is observed that the value of q𝑖𝑝 if greater than that of q𝑖𝑤. It can be said that the failure of plate in q𝑖𝑤 is like a warning requesting maintenance whereas failure in q𝑖𝑝 means total failure and cannot be maintained. Hence, failure in deflection (q𝑖𝑤) is seen in the plate into consideration. The numerical analysis obtained, it is found that if the value of critical lateral imposed load (q𝑖𝑤 and q𝑖𝑝) increase as the specified thickness (t) of plate increases and decrease as the length to width ratio increases. This implies that as we increase the thickness and allowable deflection improve the safety in the plate, whereas an increase in the span (length) of the plate increases the failure tendency of the plate structure. Furthermore, effects of aspect ratio of the critical lateral load of isotropic plates are investigated and discussed. It is concluded that the values of critical lateral load obtained by this theory achieve accepted transverse shear stress to the thickness of plate variation and satisfied the transverse flexibility of the condition of the plate while predicting the be characteristics for the CSSS isotropic rectangular thin or thick plate.

This work studied the critical load analysis of rectangular plates, carrying uniformly distributed loads utilizing direct variational energy calculus. The aim of this study is to establish the techniques for calculating the critical lateral imposed loads of the plate before deflection attains the specified maximum threshold, qiw as well as its corresponding critical lateral imposed load before the plate reaches an elastic yield point. The formulated potential energy by the static elastic theory of the plate was minimized to get the shear deformation and coefficient of deflection. The plates under consideration are clamped at the first and second edges, free of support at the third edge and simply supported at the fourth edge (CCFS). From the numerical analysis obtained, it is found that the critical lateral imposed loads (qiw and qip) increase as the thickness (t) of plate increases, and decrease as the length to width ratio increases. This suggests that as the thickness increases, the allowable deflection improves the safety of the plate, whereas an increase in the span (length) of the plate increases the failure tendency of the plate structure.