Analytic Solution of Timoshenko-Like Deformation in Bidirectional Functionally Graded Beams

Abstract

The variational asymptotic method (VAM) was used to analyze the deformation behavior of bidirectional functionally graded material (FGM) beams under transverse loading. Gradation was provided in the longitudinal as well as along the transverse direction of the beam. A number of gradation models available in the literature of FGMs were considered to derive the closed-form analytical solutions for various field variables, including Timoshenko-like shear deformations. A bidirectionally graded cantilever beam with a point load at the end was considered to verify the analytical results against finite-element analysis (FEA) solutions. Further, verification of the analytical results was also done against the available results in the literature for varied loading and boundary conditions. Application of VAM to this problem resulted in simpler governing equations and associated boundary conditions, thereby enabling closed-form analytical solutions without considering any ad hoc or a priori assumptions.