Nonlinear Analysis Of Large Deformation Of Functionally Graded Material Curved Beams And Simple Frames

Functionally graded material (FGM) is a composite which are obtained by combing two or more different materials components according to certain rules in order to get a new desired material meet some requirements of special performance. The material gradient changes continuously in a certain space direction. The structure has no obvious interface, so the physical properties are considered to vary continuously. In this paper, a mathematical model of nonlinear large deformation of FGM curved beam and simple frames was established based on nonlinear Euler-Bernoulli beam theory and Timoshenko beam theory. By using numerical shooting method to solve the two-point boundary value problem with strong nonlinearity, large deformation response of the FGM structures subjected to thermal and mechanical loads were obtained and the numerical results by two different beam theories were compared. In this paper, the main work consists of the following three parts:1. Based on the geometric nonlinear theory, accurately considering the axial extension and the influence of initial curvature the large deformation equations of elastic FGM curved beam under the mechanical-thermal load are established. By using shooting method to solve transform the two-point boundary value problem with strong nonlinearity, numerical solutions for elastic stability of a semi-circular plane curved FGM beam into initial value problem to solve the bending problem under thermal load and the stability of a FGM curved beam with the two ends fixed, subjected to uniform distributed and concentrated loadings in the radial directions, were obtained. The corresponding equilibrium paths and configurations of the deformed curved beam were presented.2. On the basis of the governing equations derived for the FGM curved beams, by considering a simple plane frame as a plane structure connected rigidly piecewise by some curved or straight beam elements, governing equations for the large deformation of FGM simple frames were established, by which nonlinear bending problem of a simple closed FGM plane frame with the shape of track field. Large deformation response of the plane frame for some given values of the geometrical and physical parameters was analyzed. The equilibrium paths and equilibrium configurations for different loadings were presented.3. Considering the influence of the axial extension and shear angle, the geometrically nonlinear equations of FGM Timoshenko curved beam under the mechanical load are further established in the sense of Timoshenko beam theory. As a example, nonlinear bending problem of a closed circular FGM frame subjected to two opposite concentrated compressive loads The shooting method was solved by using shooting method. Comparisons between the numerical results based on the Timoshenko beam theory and those on Euler-Bernoulli beam theory were made. Euler-Bernoulli beam theory for slender beam is precise enough. For the analysis of the deep beam, it is need to adopt Timoshenko beam theory.