This paper is devoted to study the buckling response of axially compressed rectangular thick plate based on the exact polynomial potential functional. The governing and equilibrium equation of an isotropic plate was derived based on the three-dimensional (3-D) static theory of elasticity, to get the relations between the rotations and deflection. These equations are solved in the form of polynomial analytically to obtain the exact displacements and stresses that are induced due to uniaxial compressive load action on the plate. By incorporating deflection and rotation function into the fundamental equation and minimized with respect to deflection coefficient, a new expression of the determination of the critical buckling load was established. This expression was applied to solve the buckling problem of a clamped thick rectangular plate which was simply supported at the first and freely supported at the third edge (SCFC). A graphic representation of results showed that, as the aspect ratio of the plate increases, the value of critical buckling load decreases while as critical buckling load increases as the length to breadth ratio increases. This implies that an increase in plate width increases the chance of failure in a plate structure. This theory obviates the numerical approximations in the thickness direction thereby guaranteeing accuracy in the solution of the displacement along the direction of thickness axis of the plate, hence, a significant lessening of the cost of computation.
A 3-D total potential energy was developed. A derived shape function of the plate was obtained. An expression for stress and moment of the plate was formulated.An exact solution for the bending attributes of a thick rectangular plate under transverse loading is modeled herein using three-dimensional (3-D) elasticity plate theory and fourth-order polynomial shear deformation function. Precluding coefficients of shear correction, this model captured the effect of shear deformation along with the transverse normal strain stress. The expression for total potential energy was derived from a 3-D kinematic and constitutive relation the equilibrium equation was developed and employed from the energy functional transformation to get the relationship between slope and deflection. Exact polynomial functions were obtained from the outcome of the equilibrium equation and with the aid of the direct variation approach, the coefficient of deflection of the plate was generated from the governing equation. The expression for computing the displacement, bending moments, and stress components along the three axes of the plate was established from these solutions for the assessment of the bending characteristics of a rectangular plate. The result of a simply supported at one edge, free at one edge and clamped at the two outer edges (SCFC) was evaluated using the obtained functions in this study. The report of this study confirms the exactness and consistency of the 3-D model unlike the refined plate theories applied by previous authors in the available literature. The value of 8.05% is the comprehensive average percentage variation of the values for center deflection obtained by Onyeka and Okeke (2020) and Gwarah (2019). It is established that at the 92 % confidence level, this model is worthy of adoption for safe, cost-effective and accurate bending analysis of thick plates of any support condition.
In this work, a new 3-D modified trigonometric displacement model was used to study the structural behavior and bending analysis of rectangular thick plate which was clamped in one edge and other three edges simply supported. The theoretical model whose formulation is based on static elastic principle as already reported in the literature are presented herein, obviating the shear correction coefficients while considering shear deformation effect and transverse normal strain/stress in the analysis. The equilibrium equations are obtained using 3-D kinematic and constitutive relations. An exact solution of deflection and rotation are obtained from the equilibrium equation using the general variational principle. The minimization, energy equation yields the general equation which was used to obtain the 3-D trigonometric displacement model of the plate. The percentage difference between the present work and those of 2-D Refined Plate Theory (RPT) with an assumed displacement and 2-D Refined plate theory (RPT) with derived function is 1.43% and 5.15% respectively. More so, percentage difference between the present work and those using polynomial shape function is 3.29%. The result showed that the 3-D trigonometric model for the present work predicts the vertical displacement and the stresses more accurately than RPT and polynomial displacement model. It is concluded that the 3-D trigonometric model gives an exact solution unlike polynomial and can be used with confidence in the analysis of thick plate under the particular initial condition.
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