Exact Elasticity Solution for the Density Functionally Gradient Beam by Using Airy Stress Function
In this paper the problem of a density-functionally gradient beam subjected to uniform load is studied. Airy stress function methodology is used to obtain a set of analytical solutions for simply supported and clamped beams subjected to uniform load. A stress function in the form of polynomial is proposed and determined. The treatment for fixed-end boundary conditions is the same as that presented by Timoshenko and Goodier (1970). By this method all of the analytical plane-stress solutions can be obtained for a uniformly loaded isotropic beam with rectangular cross section under different types of classical boundary conditions.
A. R. Daneshmehr et al., "Exact Elasticity Solution for the Density Functionally Gradient Beam by Using Airy Stress Function", Applied Mechanics and Materials, Vols. 110-116, pp. 4669-4676, 2012