The stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads are studied. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of plane stress problem, the general expressions of displacements, which satisfy the governing differential equations and the boundary conditions at two ends of the beam, can be deduced. The unknown coefficients in the general expressions were then determined by using the Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence property. The numerical results being compared with those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.
Ding H J, Huang D J, Chen W Q. Elasticity solutions for plane anisotropic functionally graded beams[J].International Journal of Solids and Structures,2007,44(1):176-196. doi: 10.1016/j.ijsolstr.2006.04.026