Abstract
The aim of this work is to study the static bending of functionally graded beams accounting higher order of shear deformation theory. The governing equations, derived from the virtual work principle, are a set of ordinary differential equations describing a static bending of a thick beam. Thus, this paper presents the differential transform method used to solve the previous system of equations. The results obtained lay the foundation to determine the exact analytical solution for different boundary conditions and external loadings. The axial displacement and the bending and shear displacements, in the exact analytical form, of a thick clamped-clamped beam with functionally graded material under a uniform load will be fully developed. Moreover, normal and shear stresses will be analyzed. To confirm the efficiency of this work, a comparison with the numerical results provided by literature is performed. Through this work, the given analytical results provide engineers with an accurate tool to determine the analytical solution for the bending of plates and shells. In addition, the geometric and material parameters that appear clearly in the analytical results allow for a more optimized design of functionally graded material beams. This type of beams is frequently used in mechanical engineering fields such as aerospace engineering.
Highlights
Razouki and al (2019) [19] applied differential transform method(DTM) and gave the exact analytical solution to the bending functionally graded (FG) beam based on higher order shear deformat ion theory
106 A New M ethod of Resolution of the Bending of Thick FGM Beams Based on Refined Higher Order Shear Deformation Theory cross-section b x h, with b being the width and h being the height as shown in fig 1
Nondimensional shear bending displacement given by the exact analytical solution for clamped-clamped FG beam bending under uniform load
Summary
The virtual of the strain energy U of the FG beam is given by [6,16]: δU = ∫(σxδεx + τxy δγxy )dsdx (5). The virtual potential energy of the applied transverse load q(x) is given by [6,19]: δV = − ∫0L q(x)(δwb (x) + δws (x))dx (8). The displacement fields of various higher-order shear deformation beam theories are given in a general form as [6,14,15,17]. F(y) is a shape function indicating the distribution of the transverse shear strain and shear stress through the depth of the beam [6]. Hooke’s law, the stresses in the beam become σx = E(y)εx Axial normal stress (4a) τxy = G(y)γxy Shear stress (4b) where G(y) is the Shear modulus related to the Young’s modulus E(y) by: G(y)
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