Approximate Solutions for Bending of Beams and Buckling of Columns Made of Functionally Graded Materials

Abstract

In this research, approximate solutions are developed to solve functionally graded material (FGM) used in beams/columns. FGM is considered in the transverse direction of beams/columns using power/exponential/sigmoidal functions considering different material combinations. Approximate solutions are first developed for FGM beams using iterative techniques based on averaging the elasticity modulus in beam’s tension/compression zones to determine different structural outputs like deflection, slope, shear, and moment diagrams. Then outputs are compared to classical Euler bending theory to validate the adopted approximate methods. In addition to beam bending problems, approximate solutions for FGM column buckling problems are developed. The methods adopted in solution are Rayleigh’s quotient, Timoshenko’s quotient, and Rayleigh-Ritz method. These methods are compared to the analytical classical Euler buckling solution. MATLAB program is adopted in the solution. The results of this study shed light on the importance of approximate solutions in solving FGM bending/buckling problems. For bending problems, the approximate method resulted in trivial error (~ 0%) when compared to analytical solution. As for buckling problems, Rayleigh-Ritz method was the most accurate in calculating critical buckling load with error less than 0.75%, while Rayleigh method led to significant error (~ 22%).