Generalized flexural rigidity of laterally functionally graded material cross sections and its application to cantilever beam elasticas

The generalized flexural rigidity of laterally functionally graded material (LFGM) cross sections and their application to the nonlinear cantilever beam problem were investigated. The main focus here was to explicitly formulate the flexural stiffness of LFGM cross sections. Young’s modulus of the LFGM was symmetrically graded about the neutral axis along the sectional depth using a power-law function. The flexural rigidity of the cross sections was derived as a function of the modular ratio and exponential index of the LFGM. In practical numerical examples, rectangular and elliptical cross sections were considered. The flexural rigidity results have been presented in graphical and numerical formats for ease of use in bending problems. As an example of the application of flexural rigidity for structural analysis, a nonlinear cantilever beam subjected to combined and separated loads, such as a uniformly distributed load, free-end load, and free-end couple, was considered. Nonlinear differential equations with relevant boundary conditions governing largely deformed beams were derived based on the Euler–Bernoulli beam theory. The differential equations were solved numerically using the Runge–Kutta and Regula–Falsi methods to obtain largely deformed elasticas. As numerical examples, the elastica behaviors in dimensionless and dimensional forms are reported in graphical charts along with various beam parameters. In particular, the bending strain/stress along the sectional depth is reported, which is highly dependent on the beam parameters.