Theoretical Analysis And Application Research Of Multilayer Functionally Graded Straight And Curved Beams

In this dissertation, the elastic bending of multilayer functionally graded straight and curved beams is studied. Elasticity and approximate solutions are derived for cantilever, simply supported, simply-fixed, and fixed-end beams, respectively. Based on the solutions, some applications are discussed. The main works and some conclusions are listed as follows:(1) Using the Airy stress function, elasticity solutions are derived for both the bi-material isotropic straight beam with a graded intermediate layer and multilayer orthotropic functionally graded straight beam. The beams are subjected to a uniform load on upper surfaces, in which Poisson’s ratio of the isotropic layer is kept a constant, and its Young’s modulus and the elastic compliance parameters of the orthotropic layer are both assumed to be arbitrary functions of the thickness coordinate.(2) The form of the Airy stress function is selected for the multilayer functionally graded circular curved beam subjected to a uniform load on its outer surface, and the process to obtain the elasticity solution is given. The solutions are derived for the multilayer isotropic functionally graded curved beam with Young’s modulus respectively being Ei0(r+βi) and Eiormeλr and Poisson’s ratio being a constant and the multilayer orthotropic functionally graded curved beam with the elastic compliance parameters being Sklirm (r+βkli,) and Sklirmeλr, respectively.(3) The above elasticity solutions are demonstrated to be correct and effective by the FEM (finite element method) and the known solutions. The obtained solutions can be degenerated into different forms and also have many applictions.(4) In multilayer functionally graded beam, neither the thickness nor the material property of the graded layer influences the bearing stress, but they influence the maximum bending and shear stresses a little. The material property of the graded layer obviously affects the bending stress in the graded layer. When the ratio of span to thickness is not less than ten, the cross section of the beam will remain plane. However, the section will become zigzag in a sandwich beam for the compliance ratio between the face-sheet and core is too big.(5) The type of description for the fixed end partly affects the stresses and displacements. In the elasticity solutions, the constraint condition BC1(an element of the axis of the beam being fixed at the fixed end) is stronger than the real case while the condition BC2(a vertical element of the cross section being fixed at the fixed end) weaker than the real one, but the averages of the results for BC1and BC2are close to the FEM ones.(6) Based on the Euler-Bernoulli and Timoshenko beam theories, basic differential equations are deduced for multilayer functionally graded straight and curved beams, and the approximate solutions are obtained for the beam subjected to a uniform load on the upper/outer surface. The precision of the bending stress of these solutions can meet the engineering needs; here, the precision of the Timoshenko beam theory with BC2is higher than the others in the straight beam while the precisions of the Euler-Bernoulli and Timoshenko beam theories are similar in the curved beam. For the bending deflection, the only one of the straight beam, obtained by the Timoshenko beam theory with BC2, is close to the FEM result.