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Nonlinear dynamic response of isotropic thin rectangular plates on elastic foundations

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Summary

An approximate analytical solution of the large deflection dynamic response of isotropic thin rectangular plates resting on Winkler, Pasternak and nonlinear Winkler foundations is presented. Von Kármán type governing equations in terms of the transverse deflection and stress function are employed. The deflection is approximated by a one term shape function satisfying the boundary conditions. The Galerkin's method is used to get the differential equation for the deflection at the centre. Closed form solutions are presented for the nonlinear free vibration response and for the responses under uniformly distributed static and step function loads. Response under sinusoidal pulse load is also obtained. Clamped and simply supported plates with movable and immovable inplane conditions at the edges are considered.

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Abbreviations

A :

Amplitude

A i :

Coefficients in differential equation for ø

a, b, h :

Sides of plate alongx andy directions and its thickness

c,c :

Damping factor, dimensionless damping:c(γ * ha 4/D)1/2

D :

Flexural rigidity:Eh 3/[12(1-v 2)]

E, v :

Young's modulus, Poisson's ratio

F :

Stress function

g, k, k 1 :

Foundation parameters

G, K, K 1 :

Dimensionless foundation parameters:ga 2/D, ka 4/D, k 1 a 4 h 2/D

H :

Depth of single layer foundation

I, M :

Immovable and movable inplane conditions at the edges

N x ,N y ,N xy :

Inplane stress resultants

P x ,P y :

Resultant forces on an edge inx andy directions

q, Q, Q o :

Uniformly distributed load,Q=qa 4/Eh 4, step load

S, C :

Simply supported and clamped edges

t, τ:

Time, dimensionless time: τ = [D/γ * ha 4]1/2 t

T, T o :

Period for amplitudeA, linear period

u, v; w :

Inplane displacements; transverse displacement

x, y :

Rectangular Cartesian co-ordinates

α1, α3 :

Linear and cubic parameters of static response

γ, γ0; γ* :

Mass densities of plate and foundation; effective mass density

ε:

Nonlinearity parameter of nonlinear vibrations

λ:

Aspect ratio of the plate;a/b

×:

Damping ratio

φ, φ m :

Central deflection, maximum central deflection

ω 0 *,ω 0 :

Linear frequency, dimensionless linear frequency:ω 0 *[γ * ha 4/D]1/2

(·):

∂()/∂(τ)

References

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Authors and Affiliations

  1. Applied Mechanics Department, Indian Institute of Technology Delhi, 110016, New Delhi, India

    P. C. Dumir

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  1. P. C. Dumir
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Dumir, P.C. Nonlinear dynamic response of isotropic thin rectangular plates on elastic foundations. Acta Mechanica 71, 233–244 (1988). https://doi.org/10.1007/BF01173950

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