Free Vibration analysis of simply supported FG cylindrical shells for four boundary conditions is performed. Fist different methods for manufacturing FGM materials is discussed, then shell theories are mentioned including theories considering thickness, shear stress and rotary iteria Zero radial displacement at both ends and four different sets of in-plane boundary conditions based on Zero displacement or Zero tension are considered. Material properties of ingredients are assumed to be temperature-dependant and metal and ceramic volume fractions are gradually changed in the thickness direction of the shell. First static analysis is performed, axial and radial displacements are determined. To verify the thesis axial stress is increased and buckling of the shell for one example is found and compared with buckling results, and it shows satisfactory agreement. The effects of static deflection due to thermal effects and internal pressure on the dynamic response are investigated. Distribution of temperature across the shell thickness is found from steady state heat conduction, only in the thickness direction, for two kind of FGM and different volume fractions the temperature distribution is shown . The equations of motion are based on the Love’s shell theory and the von Karman–Donnell-type of kinematic nonlinearity, Hamilton principle is used to derive static and dynamic equations of motion. The total nonlinear equations are linearized by separating static and dynamic deflection. Using exact solution, solving two uncoupled equations, iteration method is used to find axial unknown stress, static analysis is performed to determine the pre-stressed and pre-deflected state induced by the thermal and pressure loadings, Since circumferential displacement is decoupled only axial and radial displacements are mentioned. and then the equations of motion are solved by the Galerkin’s method. The equations are discriticized based on comparison functions, circumferential wave number and frequency term. The effects of circumferential wave-number is considered and optimum value to minimize the frequency is determined. To solve the equations we use m(number of axial wave-number), this gives of 3m equation, and related frequencies and modeshapes. By increase of m we can find higher frequencies which were not calculated before. Convergence of natural frequency by increase of axial wave-number is investigated, Axial, Circumferential and radial frequencies are calculated, To find the lowest frequency, Modeshapes for minimum axial, circumferential and radial frequencies are found and shows they satisfy boundry conditions . To increased the code run time two different methods to find mass and stiffness matrix are presented. How to apply extended Galerkin method for different Boundry conditions to find minimum natural frequencies is mentioned. Quantity of rotary-inertia is evaluated for different thicknesses, and due to its smallness it is ignored.The effects of power law index and different geometrical parameters on the natural frequencies and corresponding mode shapes for different circumferential wave number in a thermal environment are investigated. Effects of temperature increase on decline of natural frequencies is investigated. Effects of static deflection on the vibration behavior of the shell are also studied. The study shows the the effects of different geometries, different kinds of FGM(based on variation of metal ceramic from in to out surface and vice versa). At the end conclusion and new proposals to continue the work are mentioned. Keywords : Functionally graded material; Cylindrical shells; Free Vibration; Pre-deflection, Thermal effects, Internal pressure.
