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Stochastic nonlinear free vibration of laminated composite plates resting on elastic foundation in thermal environments

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Abstract

This paper presents the nonlinear free vibration analysis of laminated composite plates resting on elastic foundation with random system properties in thermal environments. System parameters are modeled as basic random variables for accurate prediction of system behavior. A C 0 nonlinear finite element based on HSDT in von Karman sense is used to descretize the laminate. A direct iterative method in conjunction with first-order perturbation technique is outlined and applied to solve the stochastic nonlinear generalized eigenvalue problem. The developed stochastic procedure is successfully used for thermally induced nonlinear free vibration problem with a reasonable accuracy. Numerical results for various combinations of boundary conditions, geometric parameters, amplitude ratios, foundation parameters and thermal loading have been compared with those available in literature and an independent MCS. Some new results are also presented which clearly demonstrate the importance of the randomness in the system parameters on the response of the structures.

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Abbreviations

a, b:

plate length and breadth

b i :

basic random system parameters

E11, E22:

longitudinal and Transverse elastic moduli

G12, G13, G23:

shear moduli

h :

thickness of the plate

K l :

linear bending stiffness matrix

K nl :

nonlinear bending stiffness matrix

K g :

thermal geometric stiffness matrix

M :

consistent mass matrixs

NE, N :

number of elements, number of layers in the laminated plate

NN:

number of nodes per element

φ i :

shape function of ith node

\({\left\{{\overline{{Q}}_{ij}}\right\}_k}\) :

reduced elastic material constants

Λ, {Λ}(e) :

vector of unknown displacements displacement vector of eth element

\({\overline{{u}}, \overline{{v}}, \overline{{w}}}\) :

displacements of a point (x, y, z)

u, v, w:

displacement of a point on the mid plane of plate

\({\{\sigma\}, \{\varepsilon\}}\) :

stress vector, Strain vector

ψ y , ψ x :

rotations of normal to mid plane about the x and y axis respectively

θx, θy, θk:

two slopes and angle of fiber orientation wrt x-axis for kth layer

x, y, z:

cartesian coordinates

ρ, λ, Var(.):

mass density, eigenvalue, variance

ω l , ϖ l :

fundamental linear frequency and its dimensionless form

ω nl , ϖ nl :

fundamental nonlinear frequency and its dimensionless form

δT :

Difference in temperatures

α1, α2, α12:

thermal expansion coefficients along x and y directions

α l , α t :

thermal coefficients in longitudinal and transverse directions of the fibre

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Authors and Affiliations

  1. Department of Mechanical Engineering, SVNIT, Surat, 395007, India

    Achchhe Lal

  2. Department of Aerospace Engineering, IIT, Kharagpur, 721302, India

    B. N. Singh

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  1. Achchhe Lal
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  2. B. N. Singh
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Correspondence to Achchhe Lal.

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Lal, A., Singh, B.N. Stochastic nonlinear free vibration of laminated composite plates resting on elastic foundation in thermal environments. Comput Mech 44, 15–29 (2009). https://doi.org/10.1007/s00466-008-0352-5

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