Static analysis of functionally graded conical shells based on an unconstrained third order theory

Based on the Unconstrained Third Order Shear Deformation Theory (UTSDT) this paper focuses on the static behaviour of moderately thick functionally graded conical, cylindrical and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The treatment is developed within the theory of linear elasticity, when material is assumed to isotropic and inhomogeneous through the thickness direction. The two costituent functionally graded shell consist of ceramic and metal. These costituents are graded through the thickness, from one surface of the shell to the other. Two different four parameter power-law distributions [1] are considered for the ceramic volume fraction. For the first one, the bottom surface of the structure is ceramic rich, whereas the top surface can be metal rich, ceramic rich or made of a mixture of the two constituents. It is the opposite configuration for the second one. The homogeneous isotropic material can be inferred as a special case of functionally graded materials. An UTSDT is proposed [2] to analyze the bending of conical shells. This theory allows the presence of a finite transverse shear strain on the top and bottom surfaces of shell, releasing in this manner the additional constraint that must be imposed in the TSDT of Reddy [3]. The UTSDT involves seven displacement functions: two in plane displacements, one transverse displacement, two linear rotations and two cubic variations of the in plane displacements (higher order rotations). The governing equilibrium equations are expressed as a function of seven kinematic parameters, by using the constitutive and kinematic relations. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equation by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem. Numerical results concerning various types of shell structures illustrate the influence of the power law exponent, the power law distribution and the choice of the four parameters on the mechanical behaviour of shell structures considered.