A 3D layer-wise model for the correct imposition of transverse shear/normal load conditions in FGM shells

A general elasticity 3D layer-wise shell model is proposed for the static investigation of plates and shells including functionally graded material (FGM) layers. A closed-form solution is used considering simply-supported sides and harmonic forms for displacements and loads. The partial differential equations obtained from the 3D equilibrium relations developed in curvilinear orthogonal coordinates are solved using the exponential matrix methodology. These equations have constant coefficients because an opportune number of mathematical layers has been introduced in order to calculate the parametric coefficients including the radii of curvature for the shells and the elastic coefficients that are variable through the thickness direction in the case of functionally graded materials. The main aim of the present work is to fill the gap found in the literature where the 3D elasticity theories always give solutions for FGM plates or shells in the only case of a transverse normal load positioned at the top or at the bottom surfaces. The present work proposes an exhaustive static analysis where the load boundary conditions have been appropriately rewritten in order to allow the use of several transverse normal and transverse shear loads separately or simultaneously positioned at top and/or bottom surfaces. One-layered and sandwich FGM plates, cylinders, cylindrical shells and spherical shells are analyzed changing the material laws and properties, the applied loads and the thickness ratios. The importance of the zigzag features, the interlaminar continuity in terms of compatibility and equilibrium requirements, the boundary load requirements, the considerations about the symmetry, the thickness ratio effect and the three-dimensional behavior have been opportunely discussed. Advantages connected with the use of FGM layers have also been analyzed. These new 3D exact results will allow the validation of recent advanced 2D shell models in the literature for the static investigation of FGM structures subjected to different load conditions. The proposed 3D model is general for several geometries (plates and shells) and materials (classical ones, composites and FGMs) and it allows a unique 3D exact solution for a large variety of structures.