A Numerical Approach Based on the GDQ Method for the Linear Static Analysis of Laminated Composite Shells Subjected to Point and Line Loads

The numerical analysis of laminated composite doubly-curved shells represents a challenging topic in the computational mechanics field due to the difficulties related to the description of their peculiar geometries. Those issues are even more evident when concentrated forces, such as point and line loads, are applied on the external surfaces of the shells. For the sake of clarity, examples of these configurations are shown in Figure 1. It is well-known that concentrated forces represent strong mechanical discontinuities which can negatively affect the static solution when a numerical approach is employed, as highlighted by many researchers in the pertinent literature. The present study aims to investigate the linear static response in terms of displacements, stresses and strains along the shell thickness. For this purpose, a recovery procedure based on the three-dimensional equilibrium equations is developed by the authors. As far as the theoretical framework is concerned, a two-dimensional structural theory able to deal with several Higher-order Shear Deformation Theories (HSDTs) in a unified manner is considered. These refined models could provide more accurate results especially when particular mechanical configurations, such as lamination schemes with inner soft-core, are taken into account. The numerical solutions are obtained by means of the Generalized Differential Quadrature (GDQ) method, due to their outstanding features to solve this kind of structural problem [1]. Since the present technique is able to approximate the partial differential derivatives, which appear in the governing equations, the strong formulation of the fundamental system of equations obtained through the Hamilton’s variational principle is solved. In the present research, the concentrated loads are modeled by means of the well-known Dirac-delta function applied to a two-dimensional domain, due to its property to assume a zero value everywhere in the domain except in the application point of the force. Alternatively, the Gaussian function could be used for the same aim. As highlighted in the papers [2, 3], an integral statement is solved in the point (or along the line) in which these forces are applied. For this purpose, the Generalized Integral Quadrature (GIQ) method is employed for the numerical implementation of these concentrated loads. For the sake of completeness, it should be noted that these concentrated forces can be described by variable orientations; analogously, they can be combined to obtain cross loads or different load cases. The validity of the current approach is tested by means of the comparison with several examples of laminated shell structures subjected to point and line loads and with the semi-analytical solutions for composite plates as well. These validation procedures are carried out in terms of both displacements and through-the-thickness stress profiles. New findings are shown for doubly-curved shells with more complex geometries, by varying boundary conditions, lamination schemes and applied loads. Finally, the present research can be considered as the proof that the GDQ method can deal accurately with these structural problems by solving the strong form of the governing equations.