Dynamic And Static Analysis Of Elastic Rectangular Plate By Finite Integral Transform Method

The present paper focuses on the problems of elastic rectangular plates. The finite integral transform method is adopted to solve the static and dynamic problems of Kirchhoff thin plates, Mindlin plates and three-dimensional thick plates. Firstly, the finite integral transform is applied to the governing equations of elastic rectangular plates. High order partial differential equations are then transformed into a system of linear algebraic equations, by which the exact solutions of the problems are obtained via the corresponding inverse transform.Compared with the traditional superposition method and semi-inverse method, the finite integral transform method simplifies the solution process of the problems while the undetermined constants in the solution procedure have the obvious physical significance. The method eliminates the need to pre-determine the deflection functions and derives the exact solutions of plates under various boundary conditions theoretically because it starts from the governing equations; hence it is more reasonable than other available methods.Based on the solution of fully clamped plates and plates on elastic foundation, the generalized displacement functions for Kirchhoff thin plates and Mindlin plates with arbitrary boundary conditions are proposed, by which the generalized displacement functions can be directly obtained. The proposed functions greatly simplify the calculations and overcome the shortcomings of the complexity in theoretical derivation, providing obvious convenience in programming.Solution of elastic plates belongs to the three-dimensional problem but two-dimensional simplification is usually adopted for convenience. With the increased structure thickness in practical engineering and the vast application of composite laminated plates, the error of two-dimensional solution becomes increasingly significant. In addition, both the first-order and the high-order theories pre-determine the stress or deflection function, which induces the incompatibility of basic equations of elasticity as well as the loss of certain elastic constants. This indicates that the changes of some constants do not influence the calculation results, which is obviously not practical. Consequently, the real exact solutions of rectangular plates must be obtained from three-dimensional calculation.All assumptions about the stresses or the deflections in problems of Kirchhoff thin plates and Mindlin plates are eliminated in the last part of this paper. Based on three-dimensional fundamental equations of elasticity, the theory of state vector space is used in combination with the finite integral transform method for analytic solutions of fully clamped thick plates and laminated plates. In contrast with the traditional method with respect to the sixth-order matrix in elasticity, the basic equations about the stresses and the deflections are separated as two matrix differential equations, one second order and another fourth order. Because of the order reduction, the solution efficiency improves significantly.