Аңдатпа
This paper presents a refined analytical theory for the bending and free vibration analysis of functionally graded (FG) beams accounting for transverse shear deformation. The material properties are assumed to vary continuously through the thickness according to a power-law distribution. Based on the adopted kinematic assumptions, the governing differential equations and effective stiffness relations are derived for both static and dynamic problems. Analytical solutions for the static bending of an FG beam under a uniformly distributed load and for the natural frequencies of free vibrations are obtained. A parametric study is performed to evaluate the influence of the material gradient index ???? and the beam slenderness ratio ????/h. The results show that the dimensionless mid-span deflection increases from 6.2586 to 9.6986 when the gradient index changes from ???? = 1 to ???? = 5 for ????/ℎ = 5. The corresponding dimensionless natural frequency decreases from 3.9710 to 3.4025. Comparisons with the Timoshenko beam theory, Reddy’s higher-order theory, and other refined models demonstrate very good agreement, with differences typically not exceeding 3-5%. The proposed formulation provides a realistic distribution of transverse shear stresses and converges to classical beam solutions for slender beams. The developed model can be effectively applied in the analysis and design of engineering structures made of functionally graded materials.