Study On Exact Solution For Laminated Plates Based On Laplace Transform

Due to the need of engineering practice, the problem of solving the composite laminated plates has been paid close attention by many scholars in recent years, especially for the complex boundary condition problem. Conventional displacement finite element method or semi-analytical method based on state vector equation has no difficulty in dealing with the boundary conditions of the plate in principle. However, the analytic solution of the plate/shell structures under complex boundary conditions has been a challenging problem.The research work of this paper mainly includes the following three parts:Firstly, the basic theory of the state space was briefly introduced, and the basic equations of elasticity were listed. The state vector equation of the elastic body with inhomogeneous term was derived based on the modified H-R(Hellinger-Reissner) variation principles. The related theory of the Laplace transform was given. By using the symbolic function of Mathematica?, the Laplace transformation equations corresponding to the non-homogeneous state vector equations were derived. The more important was that derived inversion formula corresponding to the Laplace transformation equation was very simple and convenient, especially, the inverse transform was analytic. The inverse transform is one of the most important innovations in this paper, which laid a solid foundation for the successful numerical example analysis.Secondly, according to relevant literature, eight basic boundary conditions and boundary displacement functions were introduced, the inhomogeneous term and solution-determining conditions for the state vector equation in the eight boundary conditions were derived in detail. For the process of inhomogeneous term, the expression of the nonlinear form along the thickness direction was adopted. The correctness of the method was verified by the numerical examples of the two-ply and the three-ply of the angle-ply laminated plates. At the same time, the distribution of stress and displacement along the thickness direction of the laminated plates was displayed in the form of drawing and analyzed in depth.Finally, in view of frequently encountered problem that the laminated plates subjected to the different boundary conditions along the thickness direction in the engineering practice, the method for processing different boundary conditions of different layers was given based on the principle that the boundary conditions of each layer can be treated independently and consistency between interfaces. The numerical examples were given to verify the correctness of numerical solutions. Meanwhile, the distribution of stress and displacement along the thickness in different boundary conditions of different layers were analyzed.In this paper, a simple and convenient inverse transformation formula was established, on the other hand, the processing methods of different layers of different boundary conditions were proposed, which provided a simple and reliable numerical analysis strategy for practical engineering application. The results of this paper can be used as a benchmark for other numerical methods.