A finite element procedure for vibration analysis of multi-span functionally graded material (FGM) beams subjected to a moving harmonic load is presented. The material properties of the beam are assumed to vary continuously in the thickness direction by a power-law distribution. The finite element formulation is derived by using the exact solution of the governing differential equations of an FGM Timoshenko beam segment to interpolate the displacements and rotation. The shift in the neutral axis position is taken into account in the formulation. The dynamic response of the beam is computed with the aid of the Newmark method. The numerical results show that the proposed formulation is capable to give accurate dynamic characteristics of the beams. A parametric study is carried out to highlight the effect of the material heterogeneity, number of spans and loading parameters on the dynamic response of the beams. The influence of the aspect ratio is also studied and highlighted.

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The effects of support conditions to the post-buckling behaviors of rod structures made of Axially Functionally Graded Material (AFGM) are presented. The material property of the rod member is assumed to vary linearly in the axis direction of the member. The non-linear material property of the rod element is formulated in the Finite Element context. The consistent shape functions for the rod element were developed to take into account the varying material property in the finite element formulation. The geometrically non-linear behavior of the rod element is formulated in the context of the updated co-rotational formulation. The non-linear equilibrium equations are solved by using the incremental and iterative procedures in combination with the arc-length control method. The influences of the material distribution on the post-buckling behaviors of the AFGM Williams’ toggle frames under various support conditions are highlighted. As a result, the graded between two materials can increase the post-buckling behaviors of the AFGM rod element regardless of the types of support conditions. The orientation of material distributions combined with the type of support condition can increase the performance of the rod element. The fixed-fixed support condition showed the highest performance of the AFGM rod element.

The dynamic response of non-uniform Timoshenko beams made of axially functionally graded materials subjected to multiple moving point loads is studied by using the finite element method. The material properties are assumed to vary continuously in the axial direction according to a power law. A beam element, taking the effects of shear deformation and cross-sectional variation into account, is formulated by using exact polynomials obtained from the governing differential equations of a homogenous Timoshenko beam element. The dynamic responses of the beams are computed by using the implicit Newmark method. The numerical results show that the dynamic characteristics of the beams are greatly influenced by the number of moving loads. The effects of the distance between the moving loads, material non-homogeneity, section profile as well as aspect ratio on the dynamic response of the beams are investigated in detail and highlighted.