Analytical and semi-analytical 3D shell models for composite structures

This paper focuses on the mechanical analysis of multi-layered composite and sandwich plates and shells when subjected to several static loads. Two different 3D models are compared in order to show the advantages of both the methods and their respective limits. Both the 3D models consider the same mixed curvilinear and orthogonal coordinate system and the same 3D equilibrium equations written for spherical shells. The use of this coordinate system allows the equations to be reduced to those for plates and cylindrical shells with the advantage of a unique and comprehensive formulation, which automatically adapts to the considered geometry. Plates and shells are considered as simply supported imposing a harmonic form for the displacements, the stresses and the loads. This feature allows the 3D equilibrium equations to be analytically solved in the in-plane directions positioned in the mean reference surface and to be converted into partial differential equations in the thickness direction z. The system of partial differential equations is, then, analytically solved in the thickness direction by the first model, defined as 3D EM, which employs the Exponential Matrix (EM) technique. The computational cost is significantly reduced by means of the second model, referred to as 3D GDQ, which uses the Generalized Differential Quadrature (GDQ) method in place of the Exponential Matrix method. A layerwise approach is employed by both the models: the system of partial differential equation is solved using an appropriate number of mathematical layers in 3D EM model and the Chebyshev-Gauss-Lobatto grid distribution in 3D GDQ model. The considered geometries can be loaded on the top and bottom surfaces in different directions. A transverse normal load or transverse shear loads can be applied on the top and/or bottom external surfaces. All the proposed benchmarks show that the analytical model (3D EM) and the semi-analytical one (3D GDQ) are in agreement, with coinciding results for different material and lamination schemes, geometries and dimensions, thickness ratios and loads. The approximation introduced by the use of the GDQ method does not give any noticeable effect on the accuracy of the results when the points are distributed across the domain using a stable and accurate grid. The results will be proposed in terms of displacement, strain and stress amplitudes through the thickness direction. Both the models are able to show the typical zigzag form of displacements in multi-layered anisotropic structures. The equilibrium and compatibility conditions are also satisfied when transverse shear/normal stresses and displacements are continuous at each layer interface, respectively. Furthermore, transverse normal and transverse shear stresses exactly satisfy the boundary loading conditions on top and bottom external surfaces.