An elasticity solution for functionally graded beams

An elasticity solution is obtained for a functionally graded beam subjected to transverse loads. The Youngʹs modulus of the beam is assumed to vary exponentially through the thickness, and the Poisson ratio is held constant. The exponential variation of the elastic stiffness coefficients allow an exact solution for the elasticity equations. A simple Euler–Bernoulli type beam theory is also developed on the basis of the assumption that plane sections remain plane and normal to the beam axis. The stresses and displacements are found to depend on a single non-dimensional parameter for a given variation of Youngʹs modulus in the functionally graded direction. It is found that the beam theory is valid for long, slender beams with slowly varying transverse loading. Stress concentrations occur in short or thick beams. The stress concentrations are less than that in homogeneous beams when the softer side of the functionally graded beam is loaded. The reverse is true when the stiffer side is loaded.