Extension of Boley's method to functionally graded beams

The objective of this contribution is the computation of the Airy stress function for functionally graded beam-type structures subjected to transverse and shear loads. For simplification, the material parameters are kept constant in the axial direction and vary only in the thickness direction. The proposed method can be easily extended to material varying in the axial and thickness direction. In the first part an iterative procedure is applied for the determination of the stress function by means of Boley's method. This method was successfully applied by Boley for two-dimensional (2D) isotropic plates under plane stress conditions in order to compute the stress distribution and the displacement field. In the second part, a shear loaded cantilever made of isotropic, functionally graded material is studied in order to verify our theory with finite element results. It is assumed that the Young's modulus varies exponentially in the transverse direction and the Poisson ratio is constant. Stresses and displacements are analytically determined by applying our derived theory. Results are compared to a 2D finite element analysis performed with the commercial software ABAQUS. It is found that the analytical and numerical results are in perfect agreement.

Automated summary

This study focuses on the computation of the Airy stress function for functionally graded beam-type structures under transverse and shear loads. The proposed method, originally derived from Boley's method, is successfully applied to validate the theory with finite element results in a cantilever made of isotropic, functionally graded material. The analytical and numerical results show perfect agreement.