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The study and manufacture of functionally graded (FG) shallow shells with the variable thickness are essential for the actual application of these types of structures. Previously, many researchers have developed different approaches to analyzing the mechanical behaviors of the FG structures, in which the thickness parameter has been assumed to be constant. Moreover, studies on these types of structures with the variable thickness are still quite limited. For this reason, this paper provides an effective and simple approach via mathematical formulations for analyzing the free vibrational responses of FG shallow shells with the variable thickness. Also, the presence of porosities that can occur during the fabrication of FG materials is considered. One of the highlights here is to consider the influence of the physical neutral surface. Before conducting the analysis of these FG structures, firstly, it is important to establish governing equations based on the classical shell theory. These essential equations are used for the FG shallow shells. Then, by setting the Navier solution with the help of simply supported boundary conditions, the natural frequencies are obtained. Finally, the numerical examples will be given to illustrate the effects of porosity coefficient, material and geometry parameters as well as the physical neutral surface on the natural frequencies. In addition, two numerical examples for verification will also be performed and thereby show the accuracy of the current method and model. It is hoped that the main results of this study can be useful not only to designers and researchers but also to be extended to more complex problems. [ABSTRACT FROM AUTHOR]

This paper deals with geometrically nonlinear buckling analysis of functionally graded carbon nanotube reinforced (FG-CNTR) cylindrical panels. The FG-CNTR cylindrical panel is assumed to be rested on the Winkler–Pasternak elastic foundation and subjected to uniform pressure. In the FG-CNTR cylindrical panel model, uniform and three distributions of carbon nanotubes, which are graded in the thickness direction of the panel, are considered. Effective properties of materials of the panels reinforced by single-walled carbon nanotubes are estimated through a micromechanical model based on the extended rule of mixtures. Governing equilibrium equations of the FG-CNTRC cylindrical panel are obtained based on the classical shell theory and considering the von Karman geometrically nonlinearity and initial geometric imperfection. A closed form of the resulting stability equations is established via the Galekin procedure to obtain the buckling load–deflection relations in case of simply supported boundary condition. In the numerical results section, the exactness of formulation is validated by comparing the obtained results with those reported in the open database. Then, a comprehensive investigation into the influence of carbon nanotube volume fraction, carbon nanotube distribution rule, imperfection parameter, elastic foundation as well as the geometry parameters on the nonlinear buckling behaviors of the FG-CNTRC cylindrical panels is discussed in detail. [ABSTRACT FROM AUTHOR]