, Volume 49, Issue 5, pp 1039–1068 | Cite as

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

  • Niranjan L. Shegokar
  • Achchhe Lal


In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C 0 nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.


Piezoelectric FGMs beam Stochastic analysis Nonlinear free vibration Probability density function Random material properties 


  1. 1.
    Koizumi M (1997) FGM activities in Japan. Composites, Part B, Eng 28B:1–4 CrossRefGoogle Scholar
  2. 2.
    Koizumi M (1993) The concept of FGM. In: Proceedings of the second international symposium on FGM, vol 34, pp 3–10 Google Scholar
  3. 3.
    Birman V, Byrd LW (2007) Modeling and analysis of functionally graded materials and structures. Appl Mech Rev 60(5):195–216 CrossRefADSGoogle Scholar
  4. 4.
    Talha M, Singh BN (2011) Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method. Finite Elem Anal Des 47(4):394–401 CrossRefGoogle Scholar
  5. 5.
    Kapania RK, Raciti S (1989) Recent advances in analysis of laminated beams and plates, part I: shear effect and buckling. AIAA J 27(7):923–934 CrossRefMATHMathSciNetADSGoogle Scholar
  6. 6.
    Ke LL, Yang J, Kitipornchai S (2010) An analytical study on the nonlinear vibration of functionally graded beams. Meccanica 45(6):743–752 CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Shooshtari A, Rafiee M (2011) Nonlinear forced vibration analysis of clamped functionally graded beams. Acta Mech. doi: 10.1007/s00707-011-0491-1 MATHGoogle Scholar
  8. 8.
    Yang J, Chen Y (2008) Free vibration and buckling analysis of functionally graded beams with edge cracks. Compos Struct 83:48–60 CrossRefGoogle Scholar
  9. 9.
    Xiang HJ, Yang J (2008) Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Composites, Part B, Eng 39:292–303 CrossRefGoogle Scholar
  10. 10.
    Kitipornchai S, Ke LL, Yang J, Xiang Y (2009) Nonlinear vibration of edge cracked functionally graded Timoshenko beams. J Sound Vib 324(3–5):962–982 CrossRefADSGoogle Scholar
  11. 11.
    Sina SA, Navazi HM, Haddadpour H (2009) An analytical method for free vibration analysis of functionally graded beams. Mater Des 30:741–747 CrossRefGoogle Scholar
  12. 12.
    Pradhan KK, Chakraverty S (2013) Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method, Composites Part B. Engineering 51:175–184 Google Scholar
  13. 13.
    Vo TP, Thai HT, Nguyen TK, Inam F (2013) Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica 1-14. doi: 10.1007/s11012-013-9780-1
  14. 14.
    Aydogdu M (2005) Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Int J Mech Sci 47:1740–1755 CrossRefMATHGoogle Scholar
  15. 15.
    Simsek M (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl Eng Des 240:697–705 CrossRefGoogle Scholar
  16. 16.
    Thai H-T, Vo PT (2012) Bending and vibration of functionally graded beam using various higher-orders shear deformation theories. Int J Mech Sci 62(1):57 CrossRefGoogle Scholar
  17. 17.
    Wattanasakulpong N, Gangadhara Prusty B, Kelly DW (2011) Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. Int J Mech Sci 53:734–743 CrossRefGoogle Scholar
  18. 18.
    Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Methods Eng 47:663–684 CrossRefMATHGoogle Scholar
  19. 19.
    Huang XL, Shen HS (2006) Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. J Sound Vib 289:25–53 CrossRefADSGoogle Scholar
  20. 20.
    Kapuria S, Bhattacharya M, Kumar AN (2008) Bending and free vibration response of layered functionally graded beam: a theoretical model and its experimental validation. Compos Struct 82:390–402 CrossRefGoogle Scholar
  21. 21.
    Li S-r, Su H-d, Cheng C-j (2009) Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment. Appl Math Mech 30(8):969–982 CrossRefMATHGoogle Scholar
  22. 22.
    Fu Y, Jianzhe W, Yiqi M (2011) Nonlinear analysis of bucklng, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment. Appl Math Model 39(9):4324 Google Scholar
  23. 23.
    Ying J, Lu CF, Chen WQ (2008) Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Compos Struct 84:209–219 CrossRefGoogle Scholar
  24. 24.
    Fallah A, Aghdam MM (2011) Nonlinear free vibration and post buckling analysis of functionally graded beams on nonlinear elastic foundation. Eur J Mech A, Solids 30:571–583 CrossRefMATHGoogle Scholar
  25. 25.
    Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 15:1031–1051 CrossRefADSGoogle Scholar
  26. 26.
    Vanmarcke E, Grigoriu M (1983) Stochastic finite element analysis of simple beams. J Eng Mech 109(5):1203–1214 CrossRefGoogle Scholar
  27. 27.
    Kaminski M (2001) Stochastic second-order perturbation approach to the stress-based finite element method. Int J Solids Struct 38:3831–3852 CrossRefMATHGoogle Scholar
  28. 28.
    Locke JE (1993) Finite element large deflection random response of thermally buckled plates. J Sound Vib 160:301–312 CrossRefMATHADSGoogle Scholar
  29. 29.
    Chang TP, Chang HC (1994) Stochastic dynamic finite element analysis of a non uniform beam. Int J Solids Struct 31:587–597 CrossRefMATHGoogle Scholar
  30. 30.
    Navaneetha Raj B, Iyengar NGR, Yadav D (1998) Response of composite plates with random material properties using FEM and Monte Carlo simulation. Adv Compos Mater 7(19):219–237 CrossRefGoogle Scholar
  31. 31.
    Singh BN, Yadav D, Iyengar NGR (2001) Natural frequencies of composite plates with random material properties using higher order shear deformation theory. Int J Mech Sci 43:2193–2214 CrossRefMATHGoogle Scholar
  32. 32.
    Singh BN, Yadav D, Iyengar NGR (2003) A C0 element for free vibration of composite plates with uncertain material properties. Adv Compos Mater 11:331–350 CrossRefGoogle Scholar
  33. 33.
    Onkar AK, Yadav D (2005) Forced nonlinear vibration of laminated composite plates with random material properties. Compos Struct 70:334–342 CrossRefGoogle Scholar
  34. 34.
    Kitipornchai S, Yang J, Liew KM (2006) Random vibration of functionally graded laminates in thermal environments. Comput Methods Appl Mech Eng 195:1075–1095 CrossRefMATHADSGoogle Scholar
  35. 35.
    Shaker A, Abdelrahman W, Tawfik M, Sadek E (2008) Stochastic finite element analysis of the free vibration of functionally graded material plates. Comput Mech 41:707–714 CrossRefMATHGoogle Scholar
  36. 36.
    Shaker A, Abdelrahman W, Tawfik M, Sadek E (2008) Stochastic finite element analysis of the free vibration of laminated composite plates. Comput Mech 41:493–501 CrossRefMATHGoogle Scholar
  37. 37.
    Lal A, Singh BN (2009) Stochastic nonlinear free vibration response of laminated composite plates resting on elastic foundation in thermal environments. Comput Mech 44:15–29 CrossRefMATHGoogle Scholar
  38. 38.
    Lal A, Singh BN, Kumar R (2007) Natural frequency of laminated composite plate resting on an elastic foundation with uncertain system properties. Struct Eng Mech 27:199–222 CrossRefGoogle Scholar
  39. 39.
    Yang J, Liew KM, Kitipornchai S (2005) Stochastic analysis of compositionally graded plates with system randomness under static loading. Int J Mech Sci 47:1519–1541 CrossRefMATHGoogle Scholar
  40. 40.
    Jagtap KR, Lal A, Singh BN (2011) Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment. Compos Struct 93:3185–3199 CrossRefGoogle Scholar
  41. 41.
    Heiliger PR, Reddy JN (1988) A higher order beam finite element for bending and vibration problems. J Sound Vib 126(2):309–326 CrossRefADSGoogle Scholar
  42. 42.
    Shegokar NL, Lal A (2013) Stochastic nonlinear bending response of piezoelectric functionally graded beam subjected to thermoelectromechanical loadings with random material properties. Compos Struct 100:17–33 CrossRefGoogle Scholar
  43. 43.
    Javaheri R, Eslami MR (2002) Thermal buckling of functionally graded plates. AIAA J 40:162–169 CrossRefADSGoogle Scholar
  44. 44.
    Fakhari V, Ohadi A, Yousefian P (2011) Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment. Compos Struct 93:2310–2321 CrossRefGoogle Scholar
  45. 45.
    Lal A, Choski P, Singh BN (2012) Stochastic nonlinear free vibration analysis of piezolaminated composite conical shell panel subjected to thermoelectromechanical loading with random material properties. J Appl Mech 79:1–17 Google Scholar
  46. 46.
    Klieber M, Hien TD (1992) The stochastic finite element method. Wiley, New York Google Scholar
  47. 47.
    Lal A, Jagtap KR, Singh BN (2013) Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties. Appl Math Model 37:2900–2920 CrossRefMathSciNetGoogle Scholar
  48. 48.
    Nguyen TK, Vo TP, Thai HT (2013) Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory, Composites Part B. Engineering 55:147–157 Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringS.V. National Institute of TechnologySuratIndia
  2. 2.Department of Aerospace and Ocean EngineeringVirginia TechBlacksburgUSA

Personalised recommendations