Stress and displacement recovery for functionally graded conical, cylindrical shells and annular plates

Conical and cylindrical shells are the principal object of many numerical works in the literature background based on the shear deformation theories of composite shells[1]. As it is well known, the first order shear deformation theory of Reissner and Mindlin releases a five parameter displacement model. A third order expansion of the displacement field gives a quadratic variation of transverse strains and stresses, and require no “shear correction factor” compared to the first order theory[2]. In order not to violate the boundary condition while assuming a cubic expansion for the displacement field, the unconstrained model of the third order can become constrained, as suggested by Reedy[3]. Matsunaga [4] used the stress recovery technique for determining the accurate distributions of transverse displacement and stress - strain along the thickness direction for graded shells. In this work the authors focused on the static analyses of functionally graded conical, cylindrical panels and shells, by means of the generalized third order unconstrained theory with the contribution of the initial curvature, joined with the stress - strain and displacement recovery technique. The present theory starts with the definition of the displacement field which includes higher order terms and the components of strains are derived. The graded material via a four parameters power exponent law and the elastic engineering stiffness, the relations between stresses and strains, as well as between internal actions and generalized components of displacement and the definition of external applied loads are concerned in the formulation. Then, by applying the principle of virtual displacements seven indefinite equations of equilibrium are determined. Using the constitutive equations expressed in terms of generalized components of displacement, the fundamental equations are finally obtained. These equations are discretized via the G.D.Q technique and by means of the 2D numerical solution the membrane stress strain and displacement response of graded shells is determined along the thickness direction. The resting part of the mechanical response concerning with the transverse stress - strain and displacement distributions is accurately worked out via the recovery technique.