A 3D linear elastic static analysis of functionally graded structures subjected to transverse shear loads

This work proposes an exact three-dimensional shell model for the linear elastic static analysis of plates and shells embedding Functionally Graded Material (FGM) layers. This general shell model allows static analysis of plates, cylinders, cylindrical and spherical shell panels subjected to transverse normal and transverse shear loads applied at the external surfaces. The application of transverse shear loads is the main novelty introduced by the present work. Exact solutions are possible because simply supported boundary conditions are considered and harmonic forms for mechanical loads are applied. The 3D shell model is based on a layer-wise approach which allows the zigzag form of displacements through the thickness direction. Displacements and transverse stresses will be considered as continuous through the thickness direction of the structures by means of the direct imposition of compatibility and equilibrium conditions in the proposed model, respectively. Such conditions can be included in the considered 3D layer-wise model without any difficulty. The equilibrium equations are written in general orthogonal curvilinear coordinates valid for spherical shells. These equations automatically degenerate in those for plates, cylinders and cylindrical shell panels by means of opportune considerations made on the radii of curvature. The proposed 3D equilibrium equations are a system of second order differential equations in z. These equations can be reduced to a system of first order differential equations in z simply redoubling the number of variables. After this reduction, the obtained system can be solved using the exponential matrix method. An opportune number of mathematical layers is used for the correct evaluation of curvature terms and for the appropriate determination of functionally graded material laws through the thickness direction of the structure. Results proposed at the conference will show the preliminary study for the correct choice of the appropriate number M of mathematical layers for the curvature and FGM law evaluations, and for the appropriate choice of the order of expansion N for the correct calculation of the exponential matrix. Moreover, new results will be shown for the cases of loads applied at the top and/or bottom of the structure in the x, y and z directions. Results proposed in tabular and graphical forms will show the 3D capability of the proposed shell model for the determination of the accurate stress and displacement states of one-layered and multi-layered structures embedding functionally graded layers. The proposed stress and displacement states will confirm the zigzag behaviour of multi-layered structures and the continuity of displacements and transverse stresses through the thickness direction as given by the correct imposition of compatibility and equilibrium conditions. In general, in-plane stresses can be discontinuous through the thickness direction of typical multi-layered structures. The presence of FGM layers in multi-layered structures guarantees the continuous variation of elastic properties through the thickness direction. Therefore, this feature allows in-plane stresses which are continuous through the thickness direction of the FGM structures. The proposed results can be considered as reference benchmarks for the validation of new refined 2D numerical shell/plate models developed for the analysis of composite and sandwich structures embedding FGM layers. Future developments will consider the application of concentrated or distributed loads obtained using the appropriate Fourier series analytical solutions. Moreover, opportune failure analyses can be carried out by means of the implementation of the most known failure parameters for classical and advanced materials.