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 a 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 finite element analysis 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 priori assumptions and (b) the ordered warping solutions result 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.

Automated summary

A closed-form analytical solution for a planar inhomogeneous beam under transverse loading is developed using the variational asymptotic method (VAM). The VAM decouples the problem into cross-sectional and along-the-length analyses, resulting in a set of ordinary differential equations. These equations, along with boundary conditions, have been solved to obtain closed-form analytical solutions. The present formulation has been validated against 3D finite element analysis and previous literature, showing excellent agreement. Key contributions include obtaining solutions without ad hoc assumptions and providing novel closed-form expressions for transverse shear strain and stress in higher-order solutions. The effect of inhomogeneity on various field variables has also been analyzed and discussed.