Finite Element Approach of Bending and Buckling Analysis of FG Beams Based on Refined Zigzag Theory

Abstract

This is the first attempt to utilize the refined zigzag theory (RZT) to study the bending and buckling behavior of a functional graded (metal/ceramic) thick beam. RZT, which has been exploited for the analysis of multilayered composite and sandwich beams does not employ shear correction factor. Furthermore, the number of kinematics variables of the RZT is not dependent to the number of layers in comparison with the layerwise theory. With regarding to the numerical solution, RZT, also requires C0 continuity interpolation, which leads to the development of this theory in finite element method. It is assumed that the mechanical properties of the beam varies through the thickness. According to the volume fraction of metal and ceramic, it is discretized across the thickness; consequently, the functionally graded beam (FGB) is modeled as a multi-layered one. The beam subjected to uniformly transverse and axial loadings. The equilibrium equations are established using the principle of virtual work. Using the shape functions of the first and second order forms, the non-isoparametric finite element consisting of three nodes and nine degrees of freedom are extracted. To confirm the excellent accuracy of the present approach, some numerical examples are provided and compared with those available in the literature that reveals that RZT is a trusty and validated theory to analyze the FG thick beams.

A New Method of Resolution of the Bending of Thick FGM Beams Based on Refined Higher Order Shear Deformation Theory

“…(8) into Eq. (9) and integrating by parts space variables, and collecting the coefficients of δu, δw b , and δw s , the follo wing equations of equilibriu m of the functionally graded beam are obtained [6,19]:…”

“…(3) and (4) into Eqs. (7), the constitutive equations for the stress resultants are obtained as follows [6,19]:…”

“…Assuming that the solutions of the shear w s ( x ) , bending w b ( x ) , axial u ( x ) displacements and external load q ( x ) are a polyno mial forms (power series) as follows [19]:…”

“…Vo, Thuc and al [6] presented finite element solution with Hermite interpolation based on refined shear deformation theory. Farhatnia and al (2019) [18] developed a fin ite elementary approach to study the bending and buckling FG beam based on refined Zigzag Theory. Razouki and al (2019) [19] applied differential transform method(DTM) and gave the exact analytical solution to the bending FG beam based on higher order shear deformat ion theory (simply supportedsimply supported beam case).…”

“…In this paper, DTM is applied to the governing equations, which are obtained fro m the principle of virtual work, to give a general solution [19]. The general exact solution form for a bending FG beam with various higher-order shear deformation beam theories is fully developed.…”