, Volume 49, Issue 10, pp 2479–2489 | Cite as

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

  • Yepeng Xu
  • Tiantang Yu
  • Ding Zhou


This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.


Functionally graded beam Variable thickness Two-dimensional elasticity Analytical method 



The project supported by “The natural science foundation of Jiangsu Province” (No. BK20130822) and China Postdoctoral Science Foundation funded project (No. 2011M501158). The authors also acknowledge the support from the National Natural Science Foundation of China (No. 51179063).


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Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of Engineering MechanicsHohai UniversityNanjingPeople’s Republic of China
  2. 2.College of Civil EngineeringNanjing Tech UniversityNanjingPeople’s Republic of China

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