fsdt laminated composite plates elastic foundation

8/20/2019 FSDT Laminated Composite Plates Elastic Foundation

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S. Seren Akavci, Huseyin R. Yerli, and Ali Dogan

October 2007 The Arabian Journal for Science and Engineering, Volume 32, Number 2B 341

THE FIRST ORDER SHEAR DEFORMATION THEORY

FOR SYMMETRICALLY LAMINATED COMPOSITE PLATES

ON ELASTIC FOUNDATION

S. Seren Akavci *

Department of Architecture, University of Cukurova

Adana, TURKEY

Huseyin R. Yerli and Ali Dogan

Department of Civil Engineering, University of Cukurova, 01330 Balcali, Adana,

TURKEY

1. INTRODUCTION

In the classical theory of plates (CPT), it is assumed that plane sections initially normal to the mid surface before

deformation remain plane and normal to that surface after deformation. This is the result of neglecting transverse shearstrains. However, non-negligible shear deformations occur in thick and moderately thick plates and the theory gives

inaccurate results for laminated plates. So, it is obvious that transverse shear deformations have to be taken into accountin the analysis. One of the well known plate theories is the Reissner and Mindlin model [1–3] which is a first order shear

deformation theory (FSDT) [4,5] and takes the displacement field as linear variations of midplane displacements. In this

theory, the relation between the resultant shear forces and the shear strains is affected by the shear correction factors.

This method has some advantages due to its simplicity and low computational cost. Some another plate theories, namely,

higher order shear deformation theories (HSDT), which include the effect of transverse shear deformations are;

Swaminathan [6], Ferreira [7] and Zenkour [8]. One of these theories, Reddy [9], provided a simple higher- order theorywhich accounts not only for transverse shear strains but also for a parabolic variation of the transverse shear strains

through the thickness, and there is no need to use shear correction coefficients in the computing of shear stresses.

*Address for correspondence:

Dr. S. Seren Akavci

Department of Architecture

University of Cukurova

01330 Balcali, Adana / Turkey

e-mail:[email protected]

Tel: 90.322.3387230; Fax: 90.322.3386126

Key Words: laminated, composite, plate, shear, Winkler, Pasternak

“This work has been supported by Cukurova University Scientific Research Project under Grant MMF.2006.BAP4.”

Paper Received 11 July 2005; Revised 23 February 2006; Accepted 9 June 2007


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The Arabian Journal for Science and Engineering, Volume 32, Number 2B October 2007 342

In some of the analyzes of the plates on elastic foundation, a single parameter k 0 is used to describe the foundation

behavior [10–12]. In this method it is assumed that there is a proportional interaction between pressure and deflection ofthe applied point in the foundation. This foundation is modelled by discrete vertical springs and does not take into

account the transverse shear deformations. Some other researchers have modeled the foundation with two parameters.

One of these models is the Pasternak model. This two parameter model takes into account effect of shear interaction

among the points in the foundation [13–17].

In this study the first–order shear deformation theory of Reisnner–Mindlin [1–3], has been studied for laminated

plates on an elastic foundation. The analytical solutions for bending deflections of symmetric cross-ply laminates have

been presented. The governing equations are derived by the principle of minimum total potential energy. The results of

the present analysis are compared with the Finite Element Method and found to be in good agreement. Then the effectsof foundation stiffness on bending of laminated plates are investigated.

2. GOVERNING EQUATIONS

The displacement field of a rectangular laminated plate, based on the classical plate theory and including the effect of

transverse shear deformations, can be expressed as [1-3]

00

0

0

0

( , , ) ( , )

( , , ) ( , )

( , , ) ( , )

x

y

wu x y z u x y z

x

w

v x y z v x y z y

w x y z w x y

α β θ

α β θ

∂⎡ ⎤= + +⎢ ⎥∂⎣ ⎦

⎡ ⎤∂

= + +⎢ ⎥∂⎣ ⎦=

(1)

where u0( x, y), v0( x, y) and w0( x, y) denote the corresponding midplane displacements in the x, y, z directions and θ x andθ y are the rotations of normals to midplane about the y and x axes. The above displacement field is the generaldisplacement field which gives both the CPT and FSDT theories, as:

• Classical plate theory (CPT): α = –1, β = 0

• First order shear deformation theory (FSDT): α = 0, β = 1.

The principle of virtual displacements for a laminated plate resting on elastic foundation can be stated in

analytical form as:

( )

/ 2

20 0 1 0 0

/ 2

1( ) 0,

2

h

x x y y xy xy xz xz yz yz

h A A A

dAdz k w k w dA q wdAσ δε σ δε τ δγ τ δγ τ δγ δ −

+ + + + + + ∇ + =∫ ∫ ∫ ∫ (2)

in which w0 is vertical displacement of mid plane of the plate, k 0 is Winkler modulus, k 1 is the shear modulus of

foundation and2 2

2

2 2( )

x y

∂ ∂∇ = +

∂ ∂.

If Equation (2) is written in terms of stress and moment resultants and integrated by parts then collecting the

coefficients of δu, δv, δw, δθ x and δθ y, equilibrium equations are obtained as;

2

0 0 0 1 0

0

0

0

0

0

xy x

xy y

y x

xy x x

y xy

y

N N

x y

N N

x y

QQ q k w k w x y

M M Q

x y

M M Q

y x

∂∂+ =

∂ ∂

∂ ∂+ =

∂ ∂

∂∂ + + + − ∇ =∂ ∂

∂∂+ − =

∂ ∂

∂ ∂+ − =

∂ ∂

(3)

If the equations of (3) are rewritten in terms of displacements for cross-ply symmetrically laminated plates, the above

equations are obtained:


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2 2 2 2

0 0 0 011 12 662 2 2

0u v u v

A A A x x y x y

⎛ ⎞∂ ∂ ∂ ∂+ + + =⎜ ⎟

∂ ∂ ∂ ∂ ∂⎝ ⎠ (4)

2 2 2 2

0 0 0 066 12 222 2

0u v u v

A A A x y x x y y

⎛ ⎞∂ ∂ ∂ ∂+ + + =⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ (5)

2 2 2 2

0 0 0 055 44 0 0 0 12 2 2 2 0 (6) y

x

w w w w A A q k w k x x y y x y

θ θ β

⎡ ⎤∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂+ + + + + − + =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

2 22 2

011 12 66 552 2

0 y y x x

x

w D D D A

x x y y x y x

θ θ θ θ β θ

⎛ ⎞∂ ∂⎡ ⎤∂ ∂ ∂⎛ ⎞+ + + − + =⎜ ⎟⎢ ⎜ ⎟⎥⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎦⎣ ⎝ ⎠

(7)

2 22 2

012 22 66 442 2

0 y y x x

y

w D D D A

x y y x y x y

θ θ θ θ β θ

⎛ ⎞∂ ∂ ⎤⎡ ⎛ ⎞∂ ∂ ∂+ + + − + =⎜ ⎟ ⎥⎜ ⎟⎢ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦⎝ ⎠

(8)

3. NUMERICAL PROCEDURE

For the solution of Equations (4–8) the Navier method is used. The displacement and load functions are selected asthe following Fourier series:

1 1

1 1

1 1

1 1

1 1

1 1

( , ) cos sin

( , ) sin cos

( , ) sin sin

cos sin

sin cos

( , ) sin sin

mn

m n

mn

m n

mn

m n

x x mn

m n

y y mn

m n

mn

m n

m x n yu x y A

a b

m x n yv x y Ba b

m x n yw x y C

a b

m x n yT

a b

m x n yT

a b

m x n yq x y Q

a b

π π

π π

π π

π π θ

π π θ

π π

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

=

=

=

=

=

=

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

(9)

where;

0

2

0

16

, 1

, , 1,3,5,...

qmn

mn

q for sinusoidal load m nQ

for uniform load m nπ

= =⎧⎪= ⎨

=⎪⎩

Substituting Equations (9) in (4–8) we get below equation for any fixed value of m and n;

11 12 13 14 15

21 22 23 24 25

31 32 33 34 35

41 42 43 44 45

51 52 53 54 55

0

0

.

0

0

mn

mn

mn

mn

mn mn

x

y

Aa a a a a

Ba a a a a

C a a a a a Q

T a a a a a

a a a a a T

⎧ ⎫⎡ ⎤ ⎧ ⎫⎪ ⎪⎢ ⎥ ⎪ ⎪⎪ ⎪⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥ =⎨ ⎬ ⎨ ⎬

⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎩ ⎭⎣ ⎦ ⎩ ⎭

(10)

Where

( )

( ) ( )

( )( )

( )

2 2

11 11 66

12 12 66

13 14 15 23 24 25

2 222 66 22

2 2 2 2

33 55 44 0 1

34 55

35 44

2 2

44 55 11 66

45 12 66

2 2

55 44 66 22

0, 0

a A A

a A A

a a a a a a

a A A

a A A k k

a A

a A

a A D D

a D D

a A D D

λ µ

λ µ

λ µ

β λ µ λ µ

β λ

β µ

β λ µ

β λµ

β λ µ

= +

= +

= = = = = =

= +

= + + + +

=

=

= + +

= +

= + +

(11)

(6)


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0

1000

2000

3000

4000

5000

6000

7000

8000

0 2 4 6 8 10 12x (m)

andm n

and a b

π π α β = = .

4. NUMERICAL RESULTS

A computer program has been prepared for the analytical solution of bending of laminated plates resting on anelastic foundation. First, a numerical example is solved to verify the computer program and the results are compared with

those obtained from the Finite Element Method (Example 1). Then additional examples are solved to search the effect offoundation stiffness and the fiber orientations on the bending of laminated plates resting on elastic foundation (Examples2–3).

Example 1. In this example a symmetrically laminated (0/90/0)s and simply supported square plate resting on a Winkler

foundation (k 0=100 Pa/m, k 1=0) is considered. All the layers are of equal thickness and their material properties are:

E 1=181×106 kN/m

2, E 2=10.3×10

6 kN/m

2, G12=7.17×10

6 kN/m

2, ν12=0.28. The side length of plate is 12 m. The plate is

subjected to a uniformly distributed load, q=1 kN/m2. For verification, the problem is solved by using the present method

and finite element method (ANSYS computer program) and the results are compared. In the finite element analysis the

12×12 finite element mesh is used. The foundation is modeled by LINK3D element and the plate is modeled by

SHELL91 element.

The displacement and stress distributions along the x-axis of the plate are given in Figures 1–3 for two different

foundation stiffnesses.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 2 4 6 8 10 12

x (m)

w ( m m )

(a) a/h = 10 (b) a/h = 100

Figure 1. The displacement distributions along the x-axis for different foundation stiffnesses

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

x (m)

(a) a/h = 10 (b) a/h = 100

Figure 2. σ x distribution along the x axis at the upper layer of the plate

σ x

( k N / m 2 )

σ x (

k N / m 2 )

0

2

4

6

8

10

12

0 2 4 6 8 10 12

x (m)

w ( m m )

In figures;

k 0 =0, k 1=0

k 0 =100 Pa/m, k 1=0Present analysis

ANSYS

In figures;

k 0 =0, k 1=0

k 0 =100 Pa/m, k 1=0Present analysis

ANSYS


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0

100

00

00

00

00

0 2 4 6 8 10 12

x (m)

(a) a/h = 10 (b) a/h = 100

Figure 3. σ y distribution along the x axis at the upper layer of the plate

It is seen from the figures that, when the plate thickness ratio a/h is increased, the effect of the foundation stiffness is

evident. While a/h = 10, distributions of displacement and stress are almost same for different foundation stiffnesses. Butwhen a/h =100, the distributions of displacement and stress are different for different foundation stiffnesses.

Example 2. A simply supported, (0/90/0) degree cross-ply laminate with which dimensions of a and b, in x and y

directions respectively, is considered as subjected to a sinusoidally distributed load and resting on a Pasternak

foundation. The lamina properties are:6 6

1 2

6 6

12 13 23 12 13

25 10 psi, 10 psi

0.5 10 psi, 0.2 10 psi, 0.25.

E E

G G G ν ν

= × =

= = × = × = =

In graphics:

3 2 222

4 2 2

0 0 0

0 0

10 , , , , , , ,2 2 2 2 2 6

,0,0 , 0, ,02 2

x y x y

yz xz yz xz

h E a b h h a b h hw w

q a q a q a

a h b h

q a q a

σ σ σ σ

τ τ τ τ

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

The nondimensionalized stress and deflection distributions along the thickness of laminate, which are calculated by

the formulas given above, for two different foundation stiffnesses and thickness ratios (a/h) are given in Figures 4–5.

It is seen from the Figures 4–5 that although the foundation stiffness is not effective on bending when a/h=10, itsinfluence increases with increasing a/h. When this value is equal to 100, the effect of foundation stiffness is evident.

Example 3. A simply supported, (0/90/90/0) degree square cross-ply laminate resting on a Pasternak foundation and

subjected to sinusoidally distributed transverse load is considered. Material properties are same as for the second

example. The nondimensionalized stress and deflection distributions along the thickness of laminate, which arecalculated by the formuleas given in second example, for two different foundation stiffnesses and thickness ratios (a/h)

are given in Figures 6–7.

In this example, it can be seen from figures that effect of foundation stiffness is evident when a/h ratio increases as

in the previous example.

5. RESULTS AND DISCUSSION

The bending analysis of cross-ply rectangular thick plates resting on elastic foundation has been done by using first

order shear deformation theory. A computer program has been developed from the present analysis. For verifying of present analysis, a numerical example has been solved and compared with finite element method and found in good

agreement. Some other numerical examples have been solved for observing the influence of the elastic foundation on the

bending response of symmetrically laminated thick plates.

The results of examples show that the mid-plane deflections and stresses of the laminated plate are significantly

influenced by foundation stiffness. While the plate thickness ratio a/h decreases, the effect of foundation stiffness

decreases and when a/h = 10, distribution of deflections and shear stresses for both of foundation stiffnesses are nearly

same. But when a/h = 100, distribution of deflections and shear stresses for both of foundation stiffnesses are

significantly different.

0 2 4 6 8 10 12

x (m)


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w

Figure 4. Stress and deflection distributions of 0/90/0 laminate resting on elastic foundation for thickness ratio of a/h=10

w

Figure 5. Stress and deflection distributions of 0/90/0 laminate resting on Pasternak foundation for thickness ratio of a/h=100


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-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6. Stress and deflection distributions of 0/90/90/0 laminate resting on Pasternak foundation for thickness ratio of a/h=10

-0.5

-0.3

-0.1

0.1

0.3

0.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 7. Stress and deflection distributions of 0/90/90/0 laminate resting on Pasternak foundation for thickness ratio of a/h=100

In figures;

k 0 =0, k 1=0

k 0 =300 Pa/m, k 1=300 N/m

In figures;

k 0 =0, k 1=0

k 0 =300 Pa/m, k 1=300 N/m

z / h

z / h

z / h z

/ h

z / h z

/ h

z / h

z / h

x/a

x/a

w

w

xσ

xσ

σ

yσ

xz τ yz τ

xz τ yz τ


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REFERENCES

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pp. 69–77.

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Mech., 18(1951), pp. 31–38.

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[6] K. Swaminathan, and D. Ragounadin, “Analytical Solutions Using a Higher-Order Refined Theory for the Static

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Composites Part B, 34(2003), pp. 627–636.

[8] A. M. Zenkour, “Exact Mixed-Classical Solutions for the Bending Analysis of Shear Deformable Rectangular Plates”,

Applied Mathematical Modelling , 27(2003), pp. 515–534.

[9] J. N. Reddy, “A Simple Higher Order Theory for Laminated Composite Plates”, J. Appl. Mech., 51(1984), pp. 745–752.

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pp. 1086–1105.

[11] G. Bezine, “A New Boundary Element Method for Bending of Plates on Elastic Foundations”, Int. J. Solids Struct., 24

(6), (1988), pp. 557–565.

[12] A. El-Zafrany, S. Fadhil, and K. Al-Hosani, “A New Fundamental Solution for Boundary Element Analysis of this

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Gosudarstvennogo Izdatelstoo, Literatury po Stroitelstvu i Architekture, Moskau, 1954, 1–56.

[14]

R. Buczkowski and W. Torbacki, “Finite Element Modelling of Thick Plates on Two-Parameter Elastic Foundation”, Int. J. Num. Anal. Meth. Geo., 25(2001), pp. 1409–1427.

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[16] J. Sladek, V. Sladek, and H. A. Mang, “Meshless Local Boundary Integral Equation Method for Simply Supported and

Clamped Plates Resting on Elastic Foundation”, Comp. Meth. In Appl. Mech. and Eng., 191(2002), pp. 5943–5959.

[17] H. S. Shen, “Nonlinear Bending of Simply Supported Rectangular Reissner–Mindlin Plates under Transverse and In-

Plane Loads and Resting on Elastic Foundations”, Engineering Strutures, 22(2000), pp. 847–856.

fsdt laminated composite plates elastic foundation

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