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This paper presents a free vibration analysis of functionally graded Reissner-Mindlin circular plates with various supported boundaries in thermal environments. A FGM consisting of metal and ceramic was considered in the study. Based on the geometric equation, physical equation and equilibrium equation of thick plate, taking into account the transverse shearing deformation, the free vibration equation of the axisymmetric FGM moderately thick circular plates was derived in terms of the middle surface angles of rotation and lateral displacement. The material properties of the plate were assumed to vary continuously in the thickness direction according a power law. By using shooting method to solve the coupled ordinary differential equations with different boundary conditions, the natural frequencies of FGM thick circular plates were obtained numerically. The effects of material gradient property, thickness ratio and boundary conditions on the natural frequencies were discussed in detail.

Vibration and post-buckling of beams made from functionally graded materials (FGM) subjected to uniformly and tangentially compressing follower forces are studied in this paper. Based on the accurately and geometrically nonlinear theory for extensible beams, the dynamic governing equations for FGM beams under non-conservative load are formulated. By using a shooting method to solve the non-linearly differential equations numerically, the responses of post-buckling and free vibration in the vicinity of post-buckling configuration are obtained, in which the hinged-fixed boundary conditions of beam are considered. Effects of material gradient parameter on the critical buckling, post-buckling and lower frequencies of the FGM beam are discussed in details.

Based on the accurate geometrical theory for the extensible elastic beams, an exact mathematical model of post-buckling transverse free vibration of Euler beams subjected to a distributed tangential follower force along the central axis are established. By using shooting method, pre-buckling free vibrations of both simply supported and fixed Euler beam are solved and the responses of small amplitude vibration are obtained. The numerical results show that all the frequencies of unbuckled beams decrease continuously with the increment of the load parameters.

Based on the large deformation theory and considering the axial extension of the beam, the governing equations of post-buckling of a simply supported elastic FGM beam subjected to conservative and non-conservative distributed forces were established. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using shooting method, the nonlinear boundary-value problem was solved numerically and the equilibrium paths as well as the post- buckling configurations of the deformed beam were presented. A comparison between the results of conservative system and that of non-conservative systems were given. The results shows that the features of the equilibrium paths of the the functionally graded beam under non-conservative are evidently different from those to a conservative one.

By using shooting method, large deflection bending of the Euler-Bernoulli beam with fixed-free ends subjected to a tip follower force is investigated. Based on geometrically non-linear theory for extensible elastic beams, the governing nonlinear governing equations of elastic beam subjected to a tip-concentrated follower force along the central axis are established. They consist of a boundary value problem of ordinary differential equations with strong non-linearity, in which seven unknown functions are included and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and analytical continuation, the nonlinear boundary-value problem is numerically solved as well as the non-linear bending characteristic curves of the deformed beam versus the load are presented and the effects of load parameters on the deformation are also discussed. The results show that the load and the relevant physical quantities are evidently non-linearity.