The objective of this study is to investigate the behavior of the static and free vibration analyses of axially functionally graded elliptical planar curved beams using a mixed finite element method (MFEM) based on the Timoshenko beam theory. A two-noded curved mixed finite element has 12 field variables at each node. These variables denote three displacements, three cross-sectional rotations, three forces, two bending moments, and torque, respectively. The functionally graded material is composed of ceramic-particle material and metal-matrix material. The volume fraction of ceramic and metal materials varies along the beam axis. The effective material properties (modulus of elasticity, Poisson's ratio, and density) of the functionally graded material are determined according to the rule of mixture. It is aimed in the benchmark examples to present the influence of ceramic-particle material and non-homogeneity index of material gradation, the minimum radius of the elliptical beam, and boundary condition to the results of static and free vibration analysis in detail.