How to easily model doubly curved shells with variable radii of curvature

The studies of shell structures have always been a special topic in structural mechanics due to the complexity of shell geometries. Historically speaking, the first studies involved shallow shells, which include a small curvature effect. This theory can only be applied for shells of rectangular planforms and very high curvature with respect to the shell thickness. Later on, other researchers introduced the differential geometry to investigate thin shells, but generally this mathematical tool is introduced only to present a general framework of a wider problem. In fact, only shells with constant curvature are involved in most of the papers published in the last 70 years. It is very unusual to find works in which doubly-curved shells with variable radii of curvature are presented, mainly because their geometric description is not easy to implement with standard tools. This work aims to present a general framework which has its basis in the mathematical description of a doubly-curved thick body and can be easily solved by using a fast, accurate and reliable numerical tool termed Generalized Differential Quadrature (GDQ) method. The fundamental equations, which are needed to describe the geometry of doubly-curved shells with variable radii of curvature, will be presented. The same equations are discretized and numerically evaluated in order to obtain the geometric parameters of the same shell structures. All these details are essential for obtaining the equations of motions of such structures. It is remarked that no approximation is given on the normal vector (only limited to the discretization used for the GDQ method), because the shell geometric parameters are evaluated in all the discrete point on the actual shell surface. Therefore, the authors are able to mechanically investigate doubly-curved shell structures with variable radii of curvature. The structural theory employed in all the presented numerical applications is a higher order unified formulation, which is able to easily model any kind of first and higher order shear deformation theory for laminated composite shell structures.