Application Of Precise Integration Method To The Numerical Simulations Of Laminated Plates And Study On Boundary Conditions Of Semi-Analytical Finite Element Method

Before the computer appears, the simulations of laminated plates can be only solved by the analytical method. With the development of the computer technique, the numerical methods represented by the finite element method have been widely used. However, many researchers still devote their efforts to the exact solutions and have obtained fruitful achieves in providing a reference for the varieties of approximate solutions. The precise integration method proposed by Pro. Zhong Wanxie greatly enhanced the ability of the analytical method to solve the problems of laminated plates.Firstly, on the basis of the Hellinger-Reissner variational principle, the mixed three-dimensional state- space equations with initial stress state are derived for the isotropic laminated plates. The analytical solution is given by using the united theory of general solutions of systems of elasticity equations. Thus, the mixed three-dimensional state-space equations are transformed into two partial differential equations with two basis unknowns. The present approach can also be extended to solve the anisotropic laminated plates.However, the derivation of analytical solution is quite complex even for isotropic plates and it becomes more complex in anisotropic plates. Traditionally, the researchers use various numerical methods to solve the differential equations. It loses the advantages of high-precision of the analytical solution. The precise integration method provided an opportunity to solve this problem. The state-space equations of laminated plates are solved by using the precise integration method in the thickness direction while using analytical functions in the plane of the plate. The exact solutions for the vibration and stability of simply supported laminated plate are obtained. It turns out that the precise solution can be achieved by the application of the precise integration method in the thickness direction. The exact solutions of the vibration and stability of simply supported anisotropic angle-ply laminated plates and hybrid plates provide a reference for various numerical calculations in this field.The analytical method is often subjected to geometry and boundary conditions. The semi-analytical method with mixed state Hamiltonian element provided a new approach to analyze the solution of laminates. It is important to find a high-precision numerical integration method in the thickness direction. The precise integration method is applied here which can easily obtained the stress and displacement of nodes in the analytical direction, so it is more convenient to be programmed.The boundary condition is also important for the semi-analytical method. By means of Hellinger-Reissner's variational principle, we investigated the boundary conditions in the discrete direction. Considering two-dimensional problem as an example and using Hellinger-Reissner variational principle, the boundary conditions are obtained by values of state-space variables at two end nodes in discrete direction and by the differential equations satisfied with state-space variables. The known variables are substituted into the discrete mixed state differential equations of the end nodes which are obtained by Hellinger-Reissner's variational principle. The unknown variables of the end nodes are expressed by those of internal nodes and the known variables of the end nodes are substituted into the equations of internal nodes. This method can exactly satisfy the end boundary conditions. The clamped boundary, hinged boundary and free end boundary are investigated in detail. The stresses and displacements of the clamped-end, simply supported and cantilever beam subjected to various load conditions are obtained by the precise integration method. The present results show the accuracy of the integration method. Examples show that the first version precise integration method can obtain good results without self-locking phenomenon for the beam with middle depth-span ratio, even with very small depth-span ratio such as 1/10000. The second version precise integration method is suitable for the beams with arbitrary depth-span ratio.