Nonlinear free vibration of multi-stepped beams made of functionally graded triply periodic minimal surface materials with FG-GPLRC reinforcements

Stepped beams are crucial in various structural engineering applications. This investigation aims to explore linear and nonlinear vibrational behaviors of multi-stepped beams made of functionally graded triply periodic minimal surface materials with various patterns of graphene platelet (GPL) reinforcements through the thickness. The first order shear deformable theory coupled with von Kármán strains is employed to establish the equations of motion for describing linear and nonlinear vibrations of the beams. To solve the aforementioned problems, a numerical technique based on the generalized Ritz method cooperating with the Fourier sine functions and nodal Lagrangian polynomials is proposed to create the global system composed of several beam sections. The prime factors, such as number of beam steps, step ratio, porous and GPL distributions, boundary conditions, and others, which significantly influence the vibration of the beams, are rigorously investigated. According to the obtained results, in terms of the geometry of the beam, the increase in the number of beam steps and step ratio causes the frequency to rise.