ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2017, Vol. 58, No. 2, pp. 354–361. c  Pleiades Publishing, Ltd., 2017. Original Russian Text c  S. Benguediab, A. Tounsi, H.H. Abdelaziz, M.A.A. Meziane. ELASTICITY SOLUTION FOR A CANTILEVER BEAM WITH EXPONENTIALLY VARYING PROPERTIES UDC 539.3; 539.5 S. Benguediab a , A. Tounsi a,b , H. H. Abdelaziz a,c , and M. A. A. Meziane c Abstract: An analytical solution of a plane stress problem for a cantilever beam made of a functionally graded material subjected to uniform loading is constructed. The material is assumed to be isotropic with constant Poisson’s ratio and exponentially varying Young’s modulus through the beam thickness. Expressions for displacements, strains, and stresses are obtained. Keywords: plane stress problem, stress function, exponential functionally graded material, ana- lytical solution. DOI: 10.1134/S0021894417020213 INTRODUCTION In recent years, functionally graded materials (FGMs) have attracted more and more attention. Due to their continuously varying material properties in space at the macroscopic scale, FGMs, therefore, are usually superior to conventional fiber-matrix materials in terms of their mechanical behavior. Now FGMs are widely used in various fields of science and engineering: electronics, chemistry, optics, biomedicine, etc. As the application of FGMs increases, new methodologies have to be developed to characterize them and also to design and analyze structural components made of these materials [1–5]. Publications on the FGM beam response to mechanical and other types of loading are limited. Shi and his colleagues [6–8] studied the response of FGM beams. Sankar and his colleagues [9–11] developed analytical methods for the thermomechanical and contact analysis of FGM beams and also for sandwich beams with FGM cores. In their studies, the thermomechanical properties of the FGM were all assumed to vary through the thickness in an exponential fashion. Zhu and Sankar [12] solved two- dimensional elasticity equations for an FGM beam subjected to transverse loading by means of a combined Fourier series–Galerkin method, in which the variation of Young’s modulus through the beam thickness was given by a polynomial and Poisson’s ratio was assumed to be constant. A new beam element based on the first-order shear deformation theory was developed to study the thermoelastic behavior of FGM beam structures by Chakraborty and Gopalakrishnan [13] and Chakraborty et al. [14]. In those papers, both exponential and power-law variations of material property distributions were employed. Two-dimensional analytical solutions for plates and beams were also presented by Ding et al. [15] and Huang et al. [16]. Based on the assumption of a higher-order variation of the axial displacement across the beam, various higher-order shear deformation theories were also developed [17–20]. Yaghoobi and Torabi [21] investigated the post-buckling and nonlinear free vibration responses of geometrically imperfect FGM beams resting on a nonlinear elastic foundation. Li et al. [22] considered vibrations of FGM a Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department, Algeria. b Algerian National Thematic Agency of Research in Science and Technology (ATRST), Algeria. c University Ibn Khaldoun, 14000 Tiaret, Algeria; benguediabs@yahoo.fr, tou abdel@yahoo.com, hadjhenni09@gmail.com, mohamed docs@hotmail.com. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 58, No. 2, pp. 193–201, March–April, 2017. Original article submitted January 8, 2015; revision submitted July 13, 2015. 354 0021-8944/17/5802-0354 c  2017 by Pleiades Publishing, Ltd.