Static Analysis with Normal and Shear Stress Recovery for Anisotropic Doubly-Curved Shell Panels Using the Differential Geometry and the GDQ Method

Anisotropic doubly-curved shells have been widespread in many fields of engineering. The deflection and interlaminar state of stress on these structures caused by different forces can have serious consequences for their strength and safety. Therefore, an accurate deflection and interlaminar state of stress determination are of considerable importance for the technical design of these structural elements. The aim of this work is to study the static behavior of doubly-curved shell structures, which are very common structural elements. There are two different ways to study anisotropic shell structures. The first one is based on the 3D elasticity [1] and the second one is deduced from the three-dimensional theory in order to reduce the initial 3D problem to a 2D problem defined on the reference surface of the shell [2]. The mechanical model used in this work is based on the so called First-order Shear Deformation Theory (FSDT) [1,2,3] with curvature effect included [4]. The description of the shell geometry is made using the reference surface that can be defined using the powerful tools of the Differential Geometry [2]. Staring from the definition of the position vector that represents the reference surface of the shell all the geometric quantities, such as the tangential vectors to the two orthogonal co-ordinate lines, the normal vector, the Lamè parameters and the radii of curvatures, can be directly obtained. In fact, the reference surface is numerically defined through the co-ordinates of each point and the derivations of the position vector determine the other geometrical quantities. The Differential Quadrature rule [5] allows to numerically evaluate all the derivatives that are needed to completely describe the doubly-curved shell. Then, the 2D governing equilibrium equations are expressed as functions of five kinematic parameters, by using the constitutive and the kinematic relationships [6-10]. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Referring to the formulation of the 2D equilibrium equations of doubly-curved shells, the system of second-order linear partial differential equations is solved using the Generalized Differential Quadrature (GDQ) Method [6-10]. Based on the fact that the 2D shell problem derives from the 3D elasticity and that the initial problem has been simplified using the well-defined hypotheses of the FSDT, the 2D numerical approximated solution well captures the real behavior of the shell. Since the 3D elasticity equations are always valid for the shell problem under consideration, it is possible to use the 2D approximated solution obtained via the GDQ method to evaluate some quantities such as in-plane stresses and their derivatives and to infer others quantities of interest such as shear and normal stresses by solving the 3D elasticity equilibrium equations. After the solution of the fundamental system of equations in terms of displacements, the generalized strains and stress resultants can be evaluated by applying the Differential Quadrature rule [5] to the displacements. Finally, the transverse shear and normal stress profiles through the laminate thickness are reconstructed a posteriori by solving the local three-dimensional equilibrium equations in each point of the reference surface. By using GDQ procedure through the thickness, the reconstruction procedure needs only to be corrected to properly account for the boundary equilibrium conditions. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained using analytical and numerical solutions for composite laminate shells. Very good agreement is observed.