Vibration of symmetrically laminated composite plate

8/18/2019 Vibration of symmetrically laminated composite plate

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Vibration Studies for Simply Supported Symmetrically

Laminated Rectangular Plates

Manoj Kumar P.

Indian Institute Of Space Science and Technology

April 19, 2016

Manoj Kumar P. (IIST) Mechanics of Composite April 19, 2016 1 / 16

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Overview

1 IntroductionKinematic Relations

2 Equations of Motion

3 References


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Introduction

As laminated composite materials are increasingly used in structural

applications, there arises a need for more information on the behaviorof structural components, such as plates.

The rectangular plate having all its edges simply supported is an

important problem for study in structural mechanics.

Structural characteristics such as static deflections and bendingstresses, buckling loads, and vibration frequencies are easily andexactly found for symmetrically laminated cross-ply plates.

The primary purpose of the present work is to provide accurate andreasonable results for the free-vibration frequencies of symmetricallylaminated, simply supported plates, especially for cross ply.


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Kinematic Equations

Consider a thick rectangular plate of length a, width b and uniformthickness h.

The corresponding co-ordinate parameters are denoted by .r, y and z,respectively, while u, v and w represent the associated displacementcomponents.

It is assumed that the plate is made of an orthotropic material andthe principal material axes coincide with the axes of the adoptedrectangular coordinate system.


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The stresses σx , σy , σxy can be shown to be related to the strains andcurvatures at the reference surface by the following equation:


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The stress resultants and moment resultants are

Apply this to above one,we get


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Where Aij ,B ij ,D ij are

In matrix form Constitutive matrix can be written as,


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The sub-matrix [A] is an extensional stiffness matrix, [D] is a bendingstiffness matrix, and [B] is a bending-extension coupling the stiffnessmatrix.

Equation of motionAccording to Newton’s Second law,∂σx ∂ x

+ ∂τ xy ∂ y

+ q x = ρ∂ 2u ∂ t 2

∂τ xy ∂ x

+ ∂σy ∂ y

+ q y = ρ∂ 2v ∂ t 2

∂τ xz ∂ x +

∂τ yz ∂ y + q z = ρ

∂ 2w ∂ t 2

The Equation of motion in terms of Stress and Moment Resultant are,∂ N x ∂ x +

∂ N xy ∂ y + q x = I

∂ 2u 0∂ t 2

∂ N xy

∂ x + ∂ N y

∂ y + q x = I ∂ 2v 0∂ t 2

∂ 2M x ∂ X 2

+ 2∂ 2M xy ∂ x ∂ y +

∂ 2M y ∂ y 2

+ q z = I ∂ 2w 0∂ t 2

I =N

k =1 ρk (Z k − Z k −1)


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Where Lij is the differential operator

L11 = A11 ∂ 2

∂ x 2 + A66

∂ 2

∂ y 2 + 2A12

∂ 2

∂ x ∂ y

L12 = L21 = A16 ∂ 2

∂ x 2 + A26

∂ 2

∂ y 2 + (A12 + A66)

∂ 2

∂ x ∂ y

L22 = A66 ∂ 2

∂ x 2 + A22

∂ 2

∂ y 2 + 2A26

∂ 2

∂ x ∂ y

L13 = L31 = −B 11 ∂ 3

∂ x 3 − B 26

∂ 3

∂ y 3 − 3B 16

∂ 3

∂ 2x ∂ − (B 12 + 2B 66)

∂ 3

∂ x ∂ y 2

L23 = L32 = −B 16 ∂ 3

∂ x 3 −B 22

∂ 3

∂ y 3 −3B 26

∂ 3

∂ 2x ∂ −(B 12 + 2B 66)

∂ 3

∂ x 2∂ y

L33 = D 11 ∂ 4

∂ x 4 +4D 16

∂ 4

∂ x 3∂ y +2(D 12+2D 66)

∂ 4

∂ x 2∂ y 2 +4D 26

∂ 4

∂ x ∂ y 3 +D 22

∂ 4

∂ y 4


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Simply supported symmetric cross-ply rectangular plates

Cross-Ply Laminates and symmetric Laminates

A laminate is called cross-ply laminate if all the plies used to fabricatethe laminate are only 0◦ and 90◦. For example [0/90/0/90]

A laminate is called symmetric when the material, angle and thicknessof the layers are the same above and below the mid-plane. Forexample[30/45/0]s laminate

For an symmetric Angle-ply laminate,A

16 = A

26 = B

i , j = D

16 = D

26 = 0


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The differential parameters in the equations of motion Lij becomes,

L11 = A11 ∂ 2

∂ x 2 + A66 ∂ 2

∂ y 2

L12 = L21 = (A12 + A66) ∂ 2

∂ x ∂ y

L22 = A66 ∂ 2

∂ x 2 + A22

∂ 2

∂ y 2

L13 = L31 = −B 11 ∂ 3

∂ x 3 − (B 12 + 2B 66)

∂ 3

∂ x ∂ y 2

L23 = L32 = −B 22 ∂ 3

∂ y 3 − (B 12 + 2B 66)

∂ 3

∂ x 2∂ y

L33 = D 11 ∂ 4

∂ x 4 + 2(D 12 + 2D 66)

∂ 4

∂ x 2∂ y 2 + D 22

∂ 4

∂ y 4

Boundary conditions

N x = w 0 = v 0 = M x = 0 for the edges x=0,a.N y = W 0 = u 0 = M y = 0 for the edges y=0,b.


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The following solution satisfies the boundary conditions and the

equations of motion:u 0(x , y , t ) =

M m=0

N n=0 U nmcos (αmx )sin(β ny )sin(ωmnt )

v 0(x , y , t ) =M

m=0

N n=0 V nmsin(αmx )cos (β ny )sin(ωmnt )

w 0(x , y , t ) =M

m=0

N n=0 W nmsin(αmx )sin(β ny )sin(ωmnt )

Where

αm = mπa , β n =

nπb

ωmn is the natural frequency.


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Where


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The applied forces are zero for Free vibration.

The above solution reduces to a closed form one for symmetriccross-ply plates.

ω2mna4ρ

π4 = D 11m

4 + 2(D 12 + 2D 66)n2m2(a/b ) + D 22n

4(a/b )


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References

ArthurW. Leissa (1989)

Composite Structures

Vibration Studies for Simply Supported Symmetrically Laminated Rectangular Plates 12 (1989) 113-132

Mohamad S. Qatu

Vibration of laminated shells and plate.


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The End

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Vibration of symmetrically laminated composite plate