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

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



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


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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

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