Asymptotically Accurate Analytical Solution for Timoshenko-like Deformation of Functionally Graded Beams

A closed-form analytical solution is developed for a planar inhomogeneous beam subjected to transverse loading, using Variational Asymptotic Method (VAM). The VAM decouples the problem into a cross-sectional and an along-the-length analysis, leading to a set of ordinary differential equations. These equations along with associated boundary conditions have been solved to obtain the closed-form analytical solutions. Three distinct gradation models have been used to validate the present formulation against 3D FEA and few prominent results from the literature. Excellent agreement has been obtained for all the test cases. Key contributions of the present work are (a) the solutions have been obtained without any ad-hoc and a-prior assumptions (b) the ordered warping solutions results in Euler-Bernoulli type deformation in the zeroth-order, whereas the higher-order solutions provide novel closed-form expressions for transverse shear strain and stress. Finally, the effect of inhomogeneity on various field variables has been analyzed and discussed.