STRAIN–DISPLACEMENT EXPRESSIONS AND THEIR EFFECT IN DEFLECTION OF RECTANGULAR PLATE
Authors: IBEARUGBULEM O. M. , ONYEKA F. C. , BALOGUN V.A. , WASIU JOHN. , NNOCHIRI S. E. 
Published: July 2020

Due to enormous application of thick plate and its relevance in engineering, various theories for plate analysis have been developed using linear strain–displacement expressions. It is proven from previous studies that results obtained using linear strain–displacement expressions may be unreliable for nonlinear stress and bending analyses. In the present paper, nonlinear strain– displacement expressions are employed for rectangular plates subjected to uniform distributed loads to suggest a more reliable refined plate theory that satisfies the continuity of all of the transverse stress components. This theory, which is based on traditional third-order shear deformation theory of plate is presented and applied in a bending analysis of rectangular thick plate. Governing equations and associated boundary conditions of the theory are obtained using the principle of variational calculus. From the formulated expression, the formula for calculation of the actual critical lateral imposed load, q𝑖𝑤, on the plate before deflection reaches the specified maximum specified limit and critical lateral imposed load, q𝑖𝑝, before plate reaches an elastic yield stress were obtained. By solving using the formulated expression, the effect of deflection and crack in a mild steel rectangular plates with opposite edge clamped and the other edge simply supported (CSCS) and simply supported at first and fourth edge and clamped at second edge and free of support at third edge (SCFS) were analysed and compared. 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. In the result of CSCS plate, the positive value of the critical lateral imposed load, q𝑖𝑤(between 31.08735 N/mm to 155.4414 N/ mm) before deflection reaches the maximum specified limit and the critical lateral imposed load, q𝑖𝑝 (between 193.8246 N/mm to 193.8246 N/mm) before mild steel plate reaches the elastic yield stress, reveals that the plate neither failed in q𝑖𝑤 nor in q𝑖𝑝 for plate span (a) of 1000mm at allowable deflection, wa of 1mm to 5mm. Also, the positive value of critical lateral imposed load q𝑖𝑤 (between 16.23992 N/mm to 81.20424 N/mm) 𝑎𝑛𝑑 q𝑖𝑝(between 115.3523 N/ mm to 115.3523 N/mm) reveals that the plate neither fail in q𝑖𝑤 nor in q𝑖𝑝 for plate span (a) of 1000mm at allowable deflection, wa of 1mm to 5mm for SCFS. This means that the plate structure is safe. It is observed that the value of q𝑖𝑝 is constant at any value of wa for SCFS plate. This means that change in specified deflection limit does not affect the overall performance of SCFS rectangular plate unlike CSCS plate. Hence, it also reveals that the values of critical lateral imposed loads q𝑖𝑤 𝑎𝑛𝑑 q𝑖𝑝 decrease as the length-width ratio increases. This continues until failure occurs. This means that increase in plate width increases the chance of failure in a plate structure.It is concluded that the values of critical lateral load obtained by this theory gives realistic variation of transverse shear stress through the thickness of plate and satisfied the transverse flexibility of the rectangular plate’s condition while predicting the bending behaviour of isotropic thick rectangular plate. Therefore, using this theory it is possible to predict actual load that cause the bending behaviour of isotropic rectangular plate.