The objective of this work is to analyze the behavior beams functionally graded, simply supported, under different conditions such as bending, buckling, and vibration and this by use shear deformation theories a two-dimensional (2D) and quasi-three-dimensional (quasi-3D). The proposed theories take into account a new field of displacement which includes indeterminate whole terms and contains fewer unknowns, compared to other theories of the literature; by taking account of the effects of the transverse shears and the thickness stretching. In this theory, the distribution of the transverse shear stress is hyperbolic and satisfies the boundary conditions on the upper and lower surfaces of the beam without the need for a shear correction factor. In this type of beam the properties of the materials vary according to a distribution of the volume fraction, the Hamilton principle is used to calculate the equations of motion, and in order to check the accuracy of the theory used comparison is made with the studies existing in the literature.

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