, Volume 49, Issue 1, pp 191–213 | Cite as

Nonlinear finite element analysis of laminated composite spherical shell vibration under uniform thermal loading

  • S. K. Panda
  • T. R. Mahapatra


In this article, nonlinear free vibration behavior of laminated composite shallow shell under uniform temperature load is investigated. The mid-plane kinematics of the laminated shell is evaluated based on higher order shear deformation theory to count the out of plane shear stresses and strains accurately. The nonlinearity in geometry is taken in Green-Lagrange sense due to the thermal load. In addition to that, all the nonlinear higher order terms are taken in the mathematical model to capture the original flexure of laminated panel. A nonlinear finite element model is proposed to discretise the developed model and the governing equations are derived using Hamilton’s principle. The sets of governing equations are solved using a direct iterative method. In order to validate the model, the results are compared with the available published literature and the limitations of the existing models have been discussed. Finally, some numerical experimentation has been done using the developed nonlinear model for different parameters (thickness ratio, curvature ratio, modular ratio, support condition, lamination scheme, amplitude ratio and thermal expansion coefficient) and their effects on the responses are discussed in detail.


Laminated shells Nonlinear vibration Green-Lagrange nonlinearity Nonlinear FEM 



Cartesian co-ordinate axes

u, v, w

Displacements corresponding to x, y and z direction respectively

R1, R2, R12

principal radii of curvature of spherical shell panel in 1, 2 direction and the in-plane curvature


the rotations with respect to y and x direction

ψ1, ψ2, θ1, θ2

higher order terms of Taylor series expansion

a, b, h

length, width and thickness of the shell panel

{εL}, {εNL}

linear and nonlinear strain vectors


displacement vector

\({[ \bar{Q} ]_{k}}\)

transformed reduced elastic constant


Young’s modulus


shear modulus


Poisson’s ratio

[KL], [KNL1], [KNL2]

Linear and nonlinear stiffness matrices


Geometry matrix


Thermal load vector

[H], [f]

function of thickness coordinate


strain energy


kinetic energy


External work done


maximum deflection at the center of the shell panel


Amplitude ratio


Aspect ratio


Modular ratios


Thickness ratio


curvature ratio

ωL, ωNL

linear and nonlinear frequency

\(\bar{\omega}_{L}\), \(\bar{\omega}_{NL}\)

Nondimensional linear and nonlinear frequency


  1. 1.
    Doxsee LE Jr (1989) A higher-order theory of hygrothermal behaviour of laminated composite shells. Int J Solids Struct 25(4):339–355 CrossRefMATHGoogle Scholar
  2. 2.
    Liu CF, Huang CH (1996) Free vibration of composite laminated plates subjected to temperature changes. Comput Struct 60(1):95–101 CrossRefMATHGoogle Scholar
  3. 3.
    Zhang W, Hao Y, Guo X, Chen L (2012) Complicated non-linear responses of a simply supported FGM rectangular plate under combined parametric and external excitations. Meccanica 47:985–1014 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Shariyat M (2012) A general nonlinear global-local theory for bending and buckling analyses of imperfect cylindrical laminated and sandwich shells under thermomechanical loads. Meccanica 47:301–319 CrossRefMathSciNetGoogle Scholar
  5. 5.
    Amabili M (2010) Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass. J Sound Vib 329:4501–4514 ADSCrossRefGoogle Scholar
  6. 6.
    Sai Ram KS, Sinha PK (1992) Hygrothermal effects on the free vibration of laminated composite plates. J Sound Vib 158(l):133–148 ADSCrossRefMATHGoogle Scholar
  7. 7.
    Patel BP, Ganapathi M, Makhecha DP (2002) Hygrothermal effects on the structural behaviour of thick composite laminates using higher-order theory. Compos Struct 56:25–34 CrossRefGoogle Scholar
  8. 8.
    Wang YG, Shi JL, Wang XZ (2009) Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations. Appl Math Model 33:3523–3532 CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Naidu NVS, Sinha PK (2007) Nonlinear free vibration analysis of laminated composite shells in hygrothermal environments. Compos Struct 77:475–483 CrossRefGoogle Scholar
  10. 10.
    Ohnabe H (1993) Non-linear vibration of heated orthotropic sandwich plates and shallow shells. Int J Non-Linear Mech 30(4):501–508 CrossRefGoogle Scholar
  11. 11.
    Bhimaraddi A (1993) Nonlinear vibrations of heated anti-symmetric angle-ply laminated plates. Int J Solids Struct 30(9):1255–4268 CrossRefMATHGoogle Scholar
  12. 12.
    Youzera H, Meftah SA, Challamel N, Tounsi A (2012) Nonlinear damping and forced vibration analysis of laminated composite beams. Composites, Part B, Eng 43:1147–1154 CrossRefGoogle Scholar
  13. 13.
    Ganapathi M, Patel BP, Makhecha DP (2004) Nonlinear dynamic analysis of thick composite/sandwich laminates using an accurate higher-order theory. Composites, Part B, Eng 35:345–355 CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Huang XL, Shen HS (2005) Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators. Int J Mech Sci 47:187–208 CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Chang WP, Jen SC (1986) Non-linear free vibration of heated orthotropic rectangular plates. Int J Solids Struct 22(3):267–281 CrossRefMATHGoogle Scholar
  17. 17.
    Nejad FB, Bideleh SM (2012) Nonlinear free vibration analysis of prestressed circular cylindrical shells on the Winkler/Pasternak foundation. Thin-Walled Struct 53:26–39 CrossRefGoogle Scholar
  18. 18.
    Hong CC, Jane KC (2003) Shear deformation in thermal vibration analysis of laminated plates by the GDQ method. Int J Mech Sci 45:21–36 CrossRefMATHGoogle Scholar
  19. 19.
    Majak J, Pohlak M (2010) Optimal material orientation of linear and non-linear elastic 3D anisotropic materials. Meccanica 45:301–319 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Brischetto S, Carrera E (2013) Static analysis of multilayered smart shells subjected to mechanical, thermal and electrical loads. Meccanica 48:1263–1287 CrossRefMathSciNetGoogle Scholar
  21. 21.
    Amabili M, Carra S (2009) Thermal effects on geometrically non-linear vibrations of rectangular plates with fixed edges. J Sound Vib 321:936–954 ADSCrossRefGoogle Scholar
  22. 22.
    Panda SK, Singh BN (2009) Nonlinear free vibration of spherical shell panel using higher order shear deformation theory—a finite element approach. Int J Press Vessels Piping 86(6):373–383 CrossRefGoogle Scholar
  23. 23.
    Reddy JN (2004) Mechanics of laminated composite plates and shells, 2nd edn. CRC Press, Boca Raton Google Scholar
  24. 24.
    Cook RD, Malkus DS, Plesha ME, Witt RJ (2000) Concepts and applications of finite element analysis, 3rd edn. Willey, Singapore Google Scholar
  25. 25.
    Sundaramoorthy R, David W, Murray M (1973) Incremental finite element matrices. J Struct Div 99:2423–2438 Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNIT RourkelaRourkelaIndia
  2. 2.School of Mechanical EngineeringKIIT UniversityBhubaneswarIndia

Personalised recommendations