Elastic analytical modeling of functionally graded beams subjected to cubic loading profiles

Functionally graded materials (FGMs) constitute a category of advanced composites characterized by a continuous variation of mechanical and thermal properties, resulting from gradual changes in composition or microstructure. This gradation enables the design of components with tailored performance for specific engineering applications. In this study, an elasticity-based analytical solution is developed for the static analysis of a functionally graded cantilever beam under a cubic load distribution. In contrast to most existing studies that primarily address uniformly distributed or simple load cases, the present work introduces a new analytical framework capable of capturing the response of graded beams under higher-order distributed loading. The formulation uses the Airy stress function to predict the internal stress fields within the beam, ensuring that both equilibrium and compatibility conditions are rigorously satisfied. By applying the boundary conditions consistent with the cantilever configuration, closed-form expressions for the stress components and deflection are derived. The obtained results show the capability of the proposed approach to accurately capture the stress distribution and deformation response of functionally graded beams under cubic loading. This highlights not only the efficiency of the method but also its novelty in extending elasticity-based solutions beyond conventional load assumptions, providing new insights for analyzing graded structures under higher-order load distributions.