2D Stress Analysis of Functionally Graded Beam Under Static Loading Condition

  • Sandeep S. PendhariEmail author
  • Tarun Kant
  • Yogesh Desai
Conference paper


Two dimensional (2D) stress analysis is performed in this paper for functionally graded (FG) beam under the plane stress condition of elasticity by using semi analytical approach developed by Kant et al. [6]. Modulus of elasticity is assumed to be varied exponentially through the thickness of beam. The mathematical model consists in defining a two-point boundary value problem (BVP) governed by a set of coupled first-order ordinary differential equations (ODEs) in the beam thickness direction. Elasticity solutions presented by Sankar [9] is used to show the accuracy, simplicity and effectiveness of present semi analytical solution. It is observed from the numerical investigation that the present mixed semi analytical model predicts structural response as good as the one given by the elasticity solution, which in turn proves the robustness of the presented formulation.


Functionally graded material Plane stress Laminated composite Sandwich materials Semi analytical method Transfer matrix method 


  1. 1.
    Bian ZG, Chen WQ, Lim CW, Zhang N (2005) Analytical solution for single and multi span functionally graded plates in cylindrical bending. Int J Solids Struct 42:6433–6456CrossRefzbMATHGoogle Scholar
  2. 2.
    Chakraborty A, Gopalakrishnan S (2003) A new beam finite element for the analysis of functionally graded materials. Int J Mech Sci 45:519–539CrossRefzbMATHGoogle Scholar
  3. 3.
    Chakraborty A, Gopalakrishnan S (2003) A spectrally formulated finite element for wave propagation analysis in functionally graded beams. Int J Solids Struct 40:2421–2448CrossRefzbMATHGoogle Scholar
  4. 4.
    Ching HK, Chen JK (2006) Thermomechanical analysis of functionally graded composites under laser heating by the MLPG methods. Comput Model Eng Sci 13(3):199–218MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kant T, Ramesh CK (1981) Numerical integration of linear boundary value problems in solid mechanics by segmentation method. Int J Numer Meth Eng 17:1233–1256CrossRefzbMATHGoogle Scholar
  6. 6.
    Kant T, Pendhari SS, Desai YM (2007) A general partial discretization methodology for interlaminar stress computation in composite laminates. Comput Model Eng Sci 17(2):135–161zbMATHGoogle Scholar
  7. 7.
    Koizumi M (1993) The concept of FGM-ceramic transactions. Functionally Gradient Mater 34:3–10Google Scholar
  8. 8.
    Nguyen TK, Sab K, Bonnet G (2008) First-order shear deformation plate models for functionally graded materials. Compos Mater 83:25–36Google Scholar
  9. 9.
    Sankar BV (2001) An elasticity solution for functionally graded beam. Compos Sci Technol 61:689–696CrossRefGoogle Scholar
  10. 10.
    Sankar BV, Tzeng JT (2002) Thermal stresses in functionally graded beams. AIAA J 410(6):1228–1232CrossRefGoogle Scholar
  11. 11.
    Sladek J, Slasek V, Zhang Ch (2005) The MLPG method for crack analysis in anisotropic functionally graded materials. Struct Integrity Durability 1(2):131–144Google Scholar
  12. 12.
    Soldatos KP, Liu SL (2001) On the generalised plane strain deformations of thick anisotropic composite laminated plates. Int J Solids Struct 38:479–482CrossRefGoogle Scholar
  13. 13.
    Thomson WT (1950) Transmission of elastic waves through a stratified solid medium. J Appl Phys 21:89–93CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Yamanouchi M, Koizumi M, Shiota T (1990) In: Proceedings of the first international symposium on functionally gradient materials, Sendai, Japan, pp 1228–1232Google Scholar

Copyright information

© Springer India 2015

Authors and Affiliations

  • Sandeep S. Pendhari
    • 1
    Email author
  • Tarun Kant
    • 2
  • Yogesh Desai
    • 2
  1. 1.Structural Engineering DepartmentVeermata Jijabai Technological InstituteMumbaiIndia
  2. 2.Department of Civil EngineeringIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations