Free vibration analysis of two-directional functionally graded material (2-D FGM) beams in thermal environment based on a new third-order shear deformation theory is presented. The material properties are assumed to be graded in both the thickness and longitudinal directions by a power law distribution, and they are considered to be temperature-dependent. Equations of motion, in which the shear rotation rather than the cross-sectional rotation is considered to be an independent variable, are constructed from Hamilton's principle. A finite element formulation is derived and employed to compute the vibration characteristics of the beams. The numerical results reveal that the developed formulation is accurate, and it is capable to give accurate natural frequencies by using a small number of elements. A parametric study is carried out to highlight the effects of material composition, temperature rise on the vibration characteristics of the beams.

This paper presents a finite element procedure for the large deflection analysis of functionally graded (FG) beams resting on a two-parameter elastic foundation. The material properties of the FG beams are assumed to vary continuously in the thickness direction by a power-law distribution. Based on the strain energy expression, a shear deformable beam element, taking the effect of the material non-homogeneity and the foundation support into account, is formulated and employed in the analysis. An incremental/iterative procedure in combination with the arc-length control method is used for solving nonlinear equilibrium equations. The numerical results show that the convergence of the formulated element is fast, and the large displacements of the beams can be accurately assessed by using a few numbers of the elements. A parametric study is carried out to highlight the effect of the material non-homogeneity and the foundation support on the large deflection behavior of the beams. The influence of the aspect ratio on the large deflection response of the beam is also examined and highlighted.

A Timoshenko beam element for large displacement analysis of planar beam and frame structures is formulated in the context of the co-rotational method. The shallow arch expression is adopted for the local strain, and cubic and quadratic polynomials obtained from the field consistence approach are respectively employed to interpolate the transversal displacement and rotation. The numerical examples show that the proposed element is capable of furnishing accurate results with a smaller number of elements as compared to the elements previously used in the examples. It has also shown that the nonlinear term in the expression of the local strain plays an important role in the accuracy of the element in the large displacement analysis of beam and frame structures.

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In this paper, the effects of shear deformation on the large deflection behavior of elastic frames is investigated by the finite element method. A two-node nonlinear beam element with the shear deformation is formulated and employed to analyze some frame structures. The element based on the energy method is developed in the context of the corotational approach. A bracketing procedure u ed the lowest eigenvalue of structural tangent stiffness matrix as indicating parameter is adopted to compute the critical loads. An incremental iterative procedure with the arc-length control technique is employed to trace the equilibrium paths. The numerical results show that the shear deformation plays an important role in the critical load and the large deflection behavior of the frames constructed from the components having low slenderness. A detail investigation is carried out to highlight the influence of slenderness on the behavior of the frames under large deflection.