| Due to the advantages of better thermal resistance of the ceramic phase and exhibiting stronger mechanical performance, functionally graded materials of metals and ceramics have been increasingly used in aerospace, automobile and defence industries. In this thesis, we have established a higher-order theory of circular cylindrical beams. Based on the theory, we have discussed the bending,vibration and buckling behaviors of homogeneous beams and circular cylindrical radially functionally graded beams. Different from other classical beam theories, such as Euler-Bernoulli beam and Timoshenko beam theories, the obtained solutions have satisfied the stress-free condition at the circumferential surface. Moreover, we present an analytical integral method for dealing with the free vibration and buckling of axially functionally graded Euler-Bernoulli beams. In the first part, we have obtained the two coupled governing equations for analyzing the bending,vibration and buckling behaviors of the circular cylindrical radially functionally graded beams starting from the elastic theory. Next we have mainly discussed the free vibration and buckling of functionally graded beams with varying materials properties along the axial direction. Based on the introduced methods, we have given the calculation of related issues in each chapter and compared the results with closed-form solutions as well as the existing numerial results, which show the effectiveness of the method for dealing with circular cylindrical functionally graded beams.Specific contents are as follows:(1) We have established the higher-order theory of a cicular beam. Based on the traction-free condition at the circumferential surface of the beam, expression for the axial displacements is constructed, then two coupled equations for the deflection and rotation are deduced, and they are used to investigate the bending and vibration of the homogeneous circular cylindrical beam. Different from the Euler-Bernoulli beam and Timoshenko beams, we not only take into account the effects of the rotary inertia and shear deformation, but also need not assume the shear stress uniform or introduce the revised shear correction factor. Moreover, the shear stress-free surface condition is satisfied at the circumferential surface. This higher-order theory is an extension to circular cylindrical beams of the Levinson higher-order beam theory suitable for rectangular beams. The results obtained by the proposed higher-order theory show very good agreement with the results of the three dimensional elastic analysis.(2) Based on the higher-order theory, we have studied the bending and free vibration of the circular cylindrical functionally graded beam, where the graded material properties are assumed to vary arbitrarily in the radial direction. For different boundary conditions, we have obtained the bending solutions, such as deflection and stress distribution, and charateristc equations of natural frequencies as well as the characteristic mode equations. The effects of gradient variation on deflection, stress distribution and natural frequencies are analyzed. Moreover, we apply this model to analyze the free vibration of an isotropic circular cylindrical shell.(3) We have exploited the higher-order theory to discuss the buckling of radial circular cylindrical functionally graded beam. The relationship of critical buckling loads, which is calculated through different beam theories, has been given. The effects of shear deformation and gradient variation critical buckling loads are analyzed. Furthermore, we have investigated the buckling behaviors of double-walled carbon nanotubes based on the higher-order theory.(4) We have analyzed the dynamic behaviors of graded non-uniform beams, where the material properties are assumed to vary arbitrarily in the axial direction. Combining with the boundary condition as well as the integral method, we present a novel approach for transforming the governing equation with varying coefficients to Fredholm integral equations. Then by expanding the mode shapes as power series, the natural frequencies can be determined by the polynomial characteristic equation.(5)We have investigated the buckling behaviors of axially graded non-uniform beams, and the buckling of axially non-uniform columns with elastic restrained has also been studied. Through the new method, critical buckling loads can be determined by the polynomial characteristic equation. Based on this method, an suboptimal design of varying cross-section simply supported beam is illustrated for a cylindrical bar. It has been found that there is an optimal design such that the bar achieves its maximum load-carrying capacity.
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