Elastostatic analysis of tapered FGM beams with spatially varying material properties

In this article an effective method for elastostatic analysis of tapered beams made of functionally-graded material (FGM) is presented. The spatially variable stiffness of the beam is the consequence of the continuous longitudinal variability of the cross-sectional dimension, accompanied by the variability of the material properties in three orthogonal directions. The longitudinally varying effective stiffnesses of the homogenized FGM beam for tension-compression, biaxial Timoshenko bending, and uniform torsion are determined, using the Reference Beam Method (RBM). For computation of primary quantities, such as internal forces and moments as well as displacements and angles of cross-sectional rotation, a novel tapered FGM finite beam element is developed. The evaluation of the normal and shear stresses in the cross-sections of the FGM beam requires relationships that consider the variability of the material properties and of the cross-sectional parameters. FGM beams of variable stiffness can be modeled efficiently, using only one finite element. The mathematical models are applied to the elastostatic analysis of cantilever beams with longitudinally variable, quadratic cross-sections, considering the aforementioned variability of the material properties. The proposed algorithm is verified by means of three-dimensional continuum mechanics and, alternatively, by very fine discretizations with solid finite elements. The accuracy of the presented method is excellent, and the computational effort is very small compared to other approaches.