Dynamic Analysis Of Functionally Graded Higher-Order Beams By The Finite Element Method

As a new kind of inhomogeneous composite materials, functionally graded materials (FGMs) are made for the requirement of modern materials science. The material properties of FGMs vary continuously along a certain direction or multiple directions of the structure. Due to the unique graded feature, FGMs have attracted considerable attention. By designing the material gradients, FGMs can be widely used in the aerospace engineering, biomedical engineering, civil engineering as well as many other fields.By assuming that the material properties of the beam vary continuously in thickness direction according to a power-law form, the dynamic characteristics behaviors of FGMs beam are analyzed by the finite element method (FEM). For the thin beam, C0-type element of FGMs Timoshenko beam results in the shear locking in the finite element analysis. Therefore, C1-type element is constructed to solve the shear locking problem. The accuracy and the versatility of the lower-order and higher-order C1-type element are also discussed. For the FGMs higher-order shear deformation beam, dynamic equations can be derived by applying the Darren Bell’s principle into the principle of virtual work. A higher-order beam element with four freedoms at a single node is derived for the FGMs higher-order shear deformation beam and a general procedure in the FEM is also presented. Based on ABAQUS, the corresponding program is developed for the derived element. Under the different slenderness ratios, material gradient index, element numbers, boundaries, etc. the fundamental frequencies of free vibration is analyzed by the higher-order beam element. With comparing with some results available in published literature, the derived element and the presented procedure are validated and the numerical results show the high accuracy and the applicability of the higher-order beam element.Finally, some examples are given to illustrate the effects of different material gradient index, slenderness ratios, slenderness ratios, elastic modulus and other parameters on the dynamic characteristic of the FGMs beam. The differences of numerical results between FGMs Timoshenko beam and FGMs higher-order beams are also analyzed.