, Volume 51, Issue 4, pp 921–937 | Cite as

Three dimensional static and free vibration analysis of cross-ply laminated plate bonded with piezoelectric layers using differential quadrature method

  • M. Feri
  • A. Alibeigloo
  • A. A. Pasha Zanoosi


This paper illustrates static and free vibration analysis of a cross-ply laminated composite plate embedded in piezoelectric layers based on three dimensional theory of elasticity. In this approach, a semi-analytical solution is presented for the hybrid plate with arbitrary boundary condition. For analysis a differential quadrature method (DQM) is used in two directions and state space method is employed along the thickness direction. The method is validated by comparing numerical results with the one obtained in the literature and also the results of DQM are compared with the results of analytical solution obtained by Fourier series solution. In parametric study, the effect of boundary condition, mechanical load, electric voltage, length to thickness ratio and piezoelectric thickness on both vibration and static behavior of plate are investigated.


Static Free vibration Three dimensional Laminated Piezoelectric DQM 

List of symbols

a, b, h

Plate dimensions in x, y and z directions

\( C_{ij} \,\left( {{\text{i}},{\text{j}} = 1,2, \ldots ,6} \right) \)

Material elastic constants

\( D_{x} ,{\text{D}}_{\text{y}} \,{\text{and}}\,{\text{D}}_{\text{z}} \)

Electric displacement components

\( E_{x} ,{\text{E}}_{\text{y}} \,{\text{and}}\,{\text{E}}_{\text{z}} \, \)

Electric field components


Young’s modulus composite along the y-direction

\( \eta_{i} \,\left( {i = 1,2,3} \right) \)

Dielectric constants


Density of sensor, actuator and composite layers

\( \sigma_{i} (i = x,y,z) \)

Normal stresses


Shear stresses


Piezoelectric constant


Applied electric voltage

\( {{\updelta}} \)

State vector

hc, hp

Thicknesses of the composite and piezoelectric layers


Displacements in the x-, y- and z-direction, respectively


Shear strains

\( {{\upvarepsilon}}_{i} \,(i = x,y,z) \)

Normal strains

\( \omega \)


\( {{\updelta}}_{c} \), \( {{\updelta}}_{p} \)

State vectors of the elastic and piezoelectric layers


Electric voltage


  1. 1.
    Tiersten HF (1969) Linear piezoelectric plate vibrations. Plenum Press, New YorkCrossRefGoogle Scholar
  2. 2.
    Nowacki W (1983) Electromagnetic effects in deformable solids. Polska Akademia Nauk (Polish Academy of Sciences), Warsaw (In Polish) Google Scholar
  3. 3.
    Srinivas S, Rao AK (1970) Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 6:1463–1481CrossRefMATHGoogle Scholar
  4. 4.
    Bisegna P, Maceri F (1996) An exact three-dimensional solution for simply supported rectangular piezoelectric plates. J Appl Mech 6:628–638CrossRefMATHGoogle Scholar
  5. 5.
    Heyliger P (1997) Exact solutions for simply supported laminated piezoelectric plates. J Appl Mech 64:299–306CrossRefMATHGoogle Scholar
  6. 6.
    Vel SS, Batra RC (2000) Three-dimensional analytical solution for hybrid multilayered piezoelectric plates. J Appl Mech 67:558–567CrossRefMATHGoogle Scholar
  7. 7.
    Heyliger P, Brooks S (1995) Free vibration of piezoelectric laminates in cylindrical bending. Int J Solids Struct 32:2945–2960CrossRefMATHGoogle Scholar
  8. 8.
    Heyliger P, Saravanos DA (1995) Exact free-vibration analysis of laminated plates with embedded piezoelectric layers. J Acoust Soc Am 98:1547–1557ADSCrossRefGoogle Scholar
  9. 9.
    Chen WQ, Xu RQ, Ding HJ (1998) On free vibration of a piezoelectric composite rectangular plate. J Sound Vibr 218:741–748ADSCrossRefGoogle Scholar
  10. 10.
    Bisegna P, Caruso G (2001) Evaluation of higher-order theories of piezoelectric plates in bending and in stretching. Int J Solids Struct 38:8805–8830CrossRefMATHGoogle Scholar
  11. 11.
    Rogacheva NN (1994) The theory of piezoelectric shells and plates. CRC Press, Boca RatonMATHGoogle Scholar
  12. 12.
    Mitchell JA, Reddy JN (1995) A refined hybrid plate theory for composite laminates with piezoelectric laminae. Int J Solids Struct 32:2345–2367CrossRefMATHGoogle Scholar
  13. 13.
    Tzou HS (1992) Active piezoelectric shell continua. In: Tzou HS, Anderson GL (eds) Intelligent structures and their applications. Kluwer Academic Publishers, AmsterdamGoogle Scholar
  14. 14.
    Fernandes A, Pouget J (2001) Accurate modelling of piezoelectric plates: single-layered plate. Arch Appl Mech 71:509–524CrossRefMATHGoogle Scholar
  15. 15.
    Krommer M, Irschik H (2000) A Reissner—Mindlin type plate theory including the direct piezoelectric and the pyroelectric effect. Acta Mech 141:51–69CrossRefMATHGoogle Scholar
  16. 16.
    Benjeddou A, Deu JF (2002) A two-dimensional closed-form solution for the free-vibration analysis of piezoelectric sandwich plates. Int J Solids Struct 39:1463–1486CrossRefMATHGoogle Scholar
  17. 17.
    Bellman R, Casti J (1971) Differential quadrature and long-term integration. J Math Anal Appl 34:235–238MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Bellman R, Kashef BG, Casti J (1972) Differential quadrature: a technique for the rapid solution of non-linear partial differential equations. J Comput Phys 10:40–52ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bert CW, Jang SK, Striz AG (1989) Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature. Comput Mech 5:217–226CrossRefMATHGoogle Scholar
  20. 20.
    Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28ADSCrossRefGoogle Scholar
  21. 21.
    Chen WQ, Lv CF, Bian ZG (2003) Elasticity solution for free vibration of laminated beams. Compos Struct 62:75–82CrossRefGoogle Scholar
  22. 22.
    Chen WQ, Ying J, Cai JB, Ye GR (2004) Benchmark solution of imperfect angle-ply laminated rectangular plates in cylindrical bending with surface piezoelectric layers as actuator and sensor. Comput Struct 82:1773–1784CrossRefGoogle Scholar
  23. 23.
    Chen WQ, Lu CF (2005) 3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported. Compos Struct 69:77–87CrossRefGoogle Scholar
  24. 24.
    Lu CF, Lim CW, Chen WQ (2009) Semi-analytical analysis for multi-directional functionally graded plates: 3-D elasticity solutions. Int J Numer Methods Eng 79:25–44MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhou YY, Chen WQ, Lu CF (2010) Semi-analytical solution for orthotropic piezoelectric laminates in cylindrical bending with interfacial imperfections. Compos Struct 92:1009–1018CrossRefGoogle Scholar
  26. 26.
    Alibeigloo A, Madoliat R (2009) Static analysis of cross-ply laminated plates with integrated surface piezoelectric layers using differential quadrature. Compos Struct J 88:342–353CrossRefGoogle Scholar
  27. 27.
    Ng TY, Lia H, Lam KY (2003) Generalized differential quadrature for free vibration of rotating composite laminated conical shell with various boundary conditions. Int J Mech Sci 45:567–587CrossRefMATHGoogle Scholar
  28. 28.
    Pervez T, Seibi AC, Al-Jahwari FKS (2005) Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory. Compos Struct 71:414–422CrossRefGoogle Scholar
  29. 29.
    Wang R, Han Q, Pan E (2010) An analytical solution for a multilayered magneto-electro-elastic circular plate under simply supported lateral boundary conditions. Smart Mater Struct 19:065025ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentBu Ali Sina UniversityHamedanIran
  2. 2.Mechanical Engineering Department, Faculty of EngineeringTarbiat Modares UniversityTehranIran
  3. 3.Faculty of Industrial and Mechanical Engineering, Qazvin BranchIslamic Azad UniversityQazvinIran

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