Bending Solutions of FGM Reddy–Bickford Beams in Terms of Those of the Homogenous Euler–Bernoulli Beams

  • You-Ming Xia
  • Shi-Rong LiEmail author
  • Ze-Qing Wan


In this paper, correspondence relations between the solutions of the static bending of functionally graded material (FGM) Reddy–Bickford beams (RBBs) and those of the corresponding homogenous Euler–Bernoulli beams are presented. The effective material properties of the FGM beams are assumed to vary continuously in the thickness direction. Governing equations for the titled problem are formulated via the principle of virtual displacements based on the third-order shear deformation beam theory, in which the higher-order shear force and bending moment are included. General solutions of the displacements and the stress resultants of the FGM RBBs are derived analytically in terms of the deflection of the reference homogenous Euler–Bernoulli beam with the same geometry, loadings and end conditions, which realize a classical and homogenized expression of the bending response of the shear deformable non-homogeneous FGM beams. Particular solutions for the FGM RBBs under specified end constraints and load conditions are given to validate the theory and methodology. The key merit of this work is to be capable of obtaining the high-accuracy solutions of thick FGM beams in terms of the classical beam theory solutions without dealing with the solution of the complicated coupling differential equations with boundary conditions of the problem.


Functionally graded material beams Reddy–Bickford beam theory Euler–Bernoulli beam theory Bending solution Shear deformation 



This research was supported by the National Natural Science Foundation of China with Grant Numbers 11272278 and 11672260. The authors gratefully acknowledge the financial supports.


  1. 1.
    Naebe M, Shirvanimoghaddam K. Functionally graded materials: a review of fabrication and properties. Appl Mater Today. 2016;5:223–45.CrossRefGoogle Scholar
  2. 2.
    Jha DK, Kant T, Singh RK. A critical review of recent research on functionally graded plates. Compos Struct. 2013;96:833–49.CrossRefGoogle Scholar
  3. 3.
    Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E. Stress, vibration and buckling analyses of FGM plates—a state-of-the-art review. Compos Struct. 2015;120:10–31.CrossRefGoogle Scholar
  4. 4.
    Şimşek M, Kocatür T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct. 2009;90:465–73.CrossRefGoogle Scholar
  5. 5.
    Khalili SMR, Jafari AA, Eftekhari SA. A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads. Compos Struct. 2010;92:2497–511.CrossRefGoogle Scholar
  6. 6.
    Alshorbagy AE, Eltaher MA, Mahmoud FF. Free vibration characteristics of a functionally graded beam by finite element. Appl Math Model. 2011;35:412–25.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Yang J, Chen Y. Free vibration and buckling analysis of functionally graded beams with edge cracks. Compos Struct. 2011;93:48–60.CrossRefGoogle Scholar
  8. 8.
    Li X-F. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vib. 2008;318:1210–29.CrossRefGoogle Scholar
  9. 9.
    Sina SA, Navazi HM, Haddadpour HMH. An analytical method for free vibration analysis of functionally graded beams. Mater Des. 2009;30:741–7.CrossRefGoogle Scholar
  10. 10.
    Murin J, Aminbaghai M, Hrabovsky J, Kutiš V, Kugler S. Modal analysis of the FGM beams with effect of the shear correction function. Compos Part B. 2013;45:1575–82.CrossRefGoogle Scholar
  11. 11.
    Pradhan KK, Chakarverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos Part B. 2013;51:175–84.CrossRefGoogle Scholar
  12. 12.
    Esfahani SE, Kiani Y, Eslami MR. Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations. Int J Mech Sci. 2013;69:10–20.CrossRefGoogle Scholar
  13. 13.
    Ansari R, Gholami R, Shojaei MF, Mohammadi V, Sahmani S. Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Compos Struct. 2013;100:385–97.CrossRefGoogle Scholar
  14. 14.
    Levinson M. A new rectangular beam theory. J Sound Vib. 1981;74:81–7.CrossRefzbMATHGoogle Scholar
  15. 15.
    Bickford WB. A consistent higher-order beam theory. Dev Theor Appl Mech. 1982;11:137–50.Google Scholar
  16. 16.
    Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech. 1984;51:745–52.CrossRefzbMATHGoogle Scholar
  17. 17.
    Touratier M. An efficient standard plate theory. Int J Eng Sci. 1991;1991(29):901–16.CrossRefzbMATHGoogle Scholar
  18. 18.
    Soldatos KP. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech. 1992;94:195–220.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Karama M, Afaq KS, Mistou S. Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct. 2003;40:1525–46.CrossRefzbMATHGoogle Scholar
  20. 20.
    Aydogdu M. A new shear deformation theory for laminated composite plates. Compos Struct. 2009;89:94–101.CrossRefGoogle Scholar
  21. 21.
    Kadoli R, Akhtar K, Ganesan N. Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model. 2008;32:2509–23.CrossRefzbMATHGoogle Scholar
  22. 22.
    Benatta MA, Tounsi A, Mechab I, Bouiadjra MB. Mathematical solution for bending of short hybrid composite beams with variable fibers spacing. Appl Math Comput. 2009;212:337–48.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sallai BO, Tounsi A, Mechab I, Bachir MB, Meradjah MB, Adda EA. A theoretical analysis of flexional bending of AI/AI2O3 S-FGM thick beams. Comput Mater Sci. 2009;44:1344–50.CrossRefGoogle Scholar
  24. 24.
    Kapuria S, Bhattacharyya M, Kumar AN. Bending and free vibration response of layered functionally graded beams: a theoretical model and its experiment validation. Compos Struct. 2008;88:390–402.CrossRefGoogle Scholar
  25. 25.
    Thai H-T, Vo TP. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int J Mech Sci. 2012;62:57–66.CrossRefGoogle Scholar
  26. 26.
    Vo T-P, Thai HT, Nguyen T-K, Inam F. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccnica. 2014;49:155–68.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vo T-P, Thai HT, Nguyen T-K, Inam F, Lee J. Static behavior of functionally graded sandwich beams using a quasi-3D theory. Compos Part B Eng. 2015;68:59–74.CrossRefGoogle Scholar
  28. 28.
    Filippi M, Carrera E, Zenkour AM. Static analysis of FGM beams by various theories and finite elements. Compos Part B Eng. 2015;72:1–9.CrossRefGoogle Scholar
  29. 29.
    Aydogdu M, Tashkin V. Free vibration analysis of functionally graded beams with simply supported edges. Mater Des. 2007;28:1651–6.CrossRefGoogle Scholar
  30. 30.
    Şimşek M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl Eng Des. 2010;240:697–705.CrossRefGoogle Scholar
  31. 31.
    Pradhan KK, Chakraverty S. Effects of different shear deformation theories on free vibration of functionally graded beams. Int J Mech Sci. 2014;82:149–60.CrossRefGoogle Scholar
  32. 32.
    Mahi A, Bedia EAA, Tounsi A, Mechab I. An analytical method for temperature-dependent free vibration analysis of functionally graded beams. Compos Struct. 2010;92:1877–87.CrossRefGoogle Scholar
  33. 33.
    Shen H-S, Lin F, Xiang Y. Nonlinear bending and thermal postbuckling of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations. Eng Struct. 2017;140:89–97.CrossRefGoogle Scholar
  34. 34.
    Sankar BV. An elasticity solution for functionally graded beams. Compos Sci Technol. 2001;61:689–96.CrossRefGoogle Scholar
  35. 35.
    Zhong Z, Yu T. Analytical solution of cantilever functionally graded beam. Compos Sci Technol. 2007;67:481–8.CrossRefGoogle Scholar
  36. 36.
    Ding H-J, Huang D-J, Chen W-Q. Elastic solution for plane anisotropic functionally graded beams. Int J Solids Struct. 2007;44:176–96.CrossRefzbMATHGoogle Scholar
  37. 37.
    Wang CM. Timoshenko beam-bending solutions in terms of Euler–Bernoulli solutions. J Eng Mech ASCE. 1995;121:763–5.CrossRefGoogle Scholar
  38. 38.
    Reddy JN, Wang CM, Lim GT, Ng KH. Bending solutions of Levinson beams and plates in terms of the classical theory. Int J Solids Struct. 2001;38:4701–20.CrossRefzbMATHGoogle Scholar
  39. 39.
    Reddy JN, Wang CM, Lee KH. Relationships between bending solutions of classical and shear deformation beam theories. Int J Solids Struct. 1997;26:3373–84.CrossRefzbMATHGoogle Scholar
  40. 40.
    Reddy JN, Wang CM, Lee KH. Shear deformable beams and plates-relationship with classical solutions. Amsterdam: Elsevier; 2000.zbMATHGoogle Scholar
  41. 41.
    Li S-R, Cao D-F, Wan Z-Q. Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams. Appl Math Model. 2013;37:7077–85.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Li S-R, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Compos Struct. 2013;95:5–9.CrossRefGoogle Scholar
  43. 43.
    Li S-R, Wang X, Wan Z-Q. Classical and homogenized expressions for buckling solutions of functionally graded material Levinson beams. Acta Mech Solida Sin. 2015;28:592–604.CrossRefGoogle Scholar
  44. 44.
    Li S-R, Wan Z-Q, Wang X. Homogenized and classical expressions for static bending solutions for FGM Levinson Beams. Appl Math Mech. 2015;36:895–910.CrossRefzbMATHGoogle Scholar
  45. 45.
    Groh RMJ, Weaver PM. Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams plates and shells. Compos Struct. 2015;120:231–45.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics 2019

Authors and Affiliations

  1. 1.School of Civil Science and EngineeringYangzhou UniversityYangzhouChina

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