Static analysis of functionally graded cantilever beam subjected to second-order polynomial load using the airy stress function

Functionally Graded Materials (FGMs) are advanced materials characterized by a continuous variation in properties due to a gradual change in composition or structure, enabling them to meet specific functional requirements. FGMs find applications in diverse fields such as aerospace, automotive, energy systems, and biomedical devices, where materials must withstand complex mechanical or thermal environments. Researchers employ various analysis methods to study FGMs, including theoretical modeling, numerical simulations like finite element analysis, and experimental validation, to understand their behavior and optimize their design for enhanced performance and durability. This paper investigates the static behavior of a functionally graded cantilever beam under a second-order polynomial load, unlike traditional studies that typically focus on uniform or linearly varying loads. The elasticity modulus is modeled as an exponential function through the thickness of the beam to represent the material gradation. The analysis employs the Airy stress function, expressed as a fourth-order polynomial along the longitudinal axis for a two-dimensional elasticity problem, to derive the stress and displacement fields. By solving the governing differential equations, integrating the resulting expressions, and applying the appropriate boundary conditions for the cantilever configuration, the tangential and normal stress components, as well as the transverse deflection of the beam, are obtained. The results demonstrate the efficiency of this approach in accurately capturing both the stress distribution and the deflection of the FG beam. Furthermore, the method's adaptability suggests its potential for analyzing other FG beams with varied boundary conditions and higher-order polynomial loads, offering a versatile tool for advanced structural analysis.