In this paper, the analytical solution for static and vibration analysis the cross-ply laminated composite doubly curved shell panels with stiffeners resting on Winkler-Pasternak elastic foundation is presented. Based on the first-order shear deformation theory, using the smeared stiffeners technique, the motion equations are derived by applying the Hamilton's principle. The Navier's solution for shell panel with the simply supported boundary condition at all edges is presented. The accuracy of the present results is compared with those in the existing literature and shows good achievement. The effects of the number of stiffeners, stiffener's height-to-width ratio, and number of layers of cross-ply laminated composite shell panels on the fundamental frequencies and deflections of stiffened shell with and without the elastic foundation are investigated.

In this paper, free vibration analysis of rotating functionally graded cylindrical shells with orthogonal stiffeners is presented. Based on Love's first approximation theory and smeared stiffeners technique, the governing equations of motion which take into account the effects of initial hoop tension and also the centrifugal and Coriolis forces due to rotation are derived. The influence of the power law index, the stiffener's height-to-width ratio, the circumferential wave numbers, the shell length-to-radius ratio, and the shell radius-tothickness ratio on the natural frequencies of the simply supported rotating stiffened functionally graded cylindrical shell are investigated. To validate the present analysis, comparisons are made with those available in the literature for particular cases; very good agreements are achieved.

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In this paper, the analytical solution for the cross-ply laminated composite double curved shell panels with stiffeners is presented. Based on the smeared stiffeners technique and the first shear deformation theory (FSDT), the motion equations are derived by applying the Hamilton’s principle. The Navier’s solution for the simply supported boundary condition for all edges is presented. The numerical results are verified with the known results in the literature. The effects of the number of stiffeners, dimensions of stiffeners, and lamination scheme of cross-ply laminated composite doubly curved shell panels on the natural fundamental frequencies are investigated.

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.