Nonlinear bending analysis of functionally graded porous beams using the multiquadric radial basis functions and a Taylor series-based continuation procedure

Abstract

In this paper, the nonlinear bending analysis of Functionally Graded Porous (FGP) beams is investigated using an efficient numerical algorithm associating a meshless collocation technique uses the Multiquadric Radial Basis Function (MQRBF) approximation method and a higher-order Taylor series-based continuation procedure. Material properties of the FGP beams are described by adopting a modified power-law function taking into account the effect of porosities. Based on the First Order Shear Deformation Theory (FSDT) of beams with the von-Kármán kinematic hypothesis, the strong form of nonlinear equations is established. For an efficient application of the proposed numerical approach, a quadratic matrix strong form of the problem is presented. The resulting nonlinear equations are solved numerically with the proposed algorithm which leaned on the following three steps: a higher-order Taylor series expansion to transform the quadratic nonlinear equations into a sequence of linear ones, a meshless collocation technique based on MQRBF approximation method to solve numerically the resulting linear equations and a continuation procedure to get the whole solution branch. To demonstrate the robustness of the developed algorithm, convergence and validation studies have been carried out. Furthermore, the effects of power-law index, porosity volume fraction, Young’s modulus ratio, loads and boundary conditions are investigated.