Study Of Static Bending Problem Of Functionally Graded Material Beams Based On Higher-order Shear Deformation Theory

Functionally graded material (FGM) is a new type of non-uniform composite materials, which is produced according to a specific technique from two or more kind materials with different properties through certain design procedure so that it can make the material components change continuously in the gradient direction and eliminate the non-continuity of the physical material properties, or avoid the internal interfaces. The study on the static and dynamic responses of functionally graded material beams is still one of the interesting and focused topics in the theory of structural mechanics as well as in engineering application. Firstly, based on the classical theory and the higher-order shear deformation theory, we studied the transition relationships between static bending solutions of the FGM Reddy beams and the corresponding homogenous beams by analytical approach when the material properties of the FGM Reddy beams varied continuously in the thickness direction. Then, by using differential quadrature method, static bending solutions of FGM Levinson beams and FGM Reddy beams were studied based on the higher-order shear deformation theory. Main work of this dissertation is formed of the following three parts:(1) Based on the higher-order shear deformation theory, transformative relation between the static bending solutions of FGM Reddy beams and the corresponding homogenous beams is analyzed. The geometric equations and physical equations of functionally graded material Reddy beam were derived by considering the material properties varied continuously in the thickness direction based on power law exponent, and beam’s equations of balance were derived by using the principle of minimum potential energy. By using mathematical similarity and load equivalence, the fourth-order ordinary differential equation which have three basic unknown functions of axial displacement, rotational angle and deflection can be transformed to a second-order ordinary differential equation with only one basic unknown function of shear force, giving the bending solutions of functionally graded material Reddy beams expressed in terms of the deflection of the corresponding homogenous Euler-Bernoulli beams with the same geometry, the same loadings and end constraints.(2) Based on the higher-order shear deformation theory, static analysis of FGM Levinson beams were performed by using the differential quadrature method. The non-dimensional governing equations of FGM beams were derived by considering the material properties varied continuously in the thickness direction based on power law exponent, provided the dimensionless governing equations and the dimensionless boundary conditions which were dispersed. For the specific FGM Levinson beam, the non-dimensional deflection of three common boundary conditions were expressed by considering the material properties varying with the power law exponent in the thickness direction while the slenderness ratio is ten firstly. Furthermore, the non-dimensional midway deflections of simply supported ends are also expressed under the varied slenderness ratio by considering the material properties varying with the power law exponent in the thickness direction.(3) Based on the higher-order shear deformation beam theory, static analysis of FGM Reddy beams were performed by using differential quadrature method. Considering the material properties varied continuously in the thickness direction based on power law exponent, we established the non-dimensional governing equations of FGM Reddy beams, defined the dimensionless coefficients of beams such as c,ca1,ca2,csl,cs2and eliminated the dimensionless axial displacement in all non-dimensional control equations and dimensionless internal forces, and expressed the non-dimensional control equations and dimensionless internal forces by the dimensionless coefficients of c, cal, ca2, cs1, cs2and the basic unknown function of (?) and W eventually. In the end, the non-dimensional deflection of three common boundary conditions were displayed by considering the material properties varying with the power law exponent in the thickness direction while the slenderness ratio is five.