STABILITY OF LAMINATED COMPOSITE CURVED PANELS WITH CUTOUT USING FINITE ELEMENT METHOD

S. Dash*, A.V.Asha** and S. K. Sahu***

* M. Tech Student, email: reachsabita@yahoo.com

** Lecturer, Selection grade *** Assistant Professor, Department of Civil Engg, NIT, Rourkela-769008 (INDIA)

email: sksahu@nitrkl.ac.in

ABSTRACT: Cutouts are inevitable in structures due to practical consideration. The present study involves vibration and stability of laminated composite curved panels with cutouts using finite element method. The first order shear deformation (FSDT) is used to model the curved panels, considering the effects of transverse shear deformation and rotary inertia. Since the stress field is non-uniform due to cutout, plane stress analysis is carried out using the finite element method to determine the stresses and these are used to formulate the geometric stiffness matrix. The global matrices are obtained using skyline technique. The eigenvalues are obtained us ing subspace iteration scheme. The study reveals that the fundamental frequencies of vibration of an angle ply flat panel decrease with introduction of small cutouts but again rise with increase in size of cutout. However the higher frequencies of vibration continue to decrease up to a moderate size of cutout and then rise with further increase of size of cutout. The stability resistance decreases with increase in size of cutout in curved panels unlike the frequencies of vibration. The buckling loads of cross ply panels reduces drastically with increase in size of cutout compared to angle ply plates. The frequencies of vibration and the stability resistance increase with introduction of curvature in the flat panel with cutout. However, the effect of curvature is reduced with increase of size of cutout. NOTATION a, b : dimension of shell

c, d : dimension of the cutout

E : Young’s modulus

[K] : stiffness matrix

[M] : mass matrix

[Kg] : geometric stiffness matrix

{q} : load intensity

ν : Poisson’s ratio

ρ : mass density

θx, θy : slopes normal and transverse to the boundary

ω : frequency of vibration

λ : Buckling load

INTRODUCTION Laminated composite plates and shells are extensively used as structural parts in civil, aerospace, automotive and marine engineering structures. Cutouts are inevitable in these structures, mainly for practical considerations. Cutouts are commonly found as access ports for mechanical and electrical systems or simply to reduce weight. Cutouts are also needed to provide access for hydraulic lines, for damage inspection, to lighten the loads, provide ventilation and for altering the resonant frequency of the structures. In addition, the designers often need to incorporate cutouts or openings in a structure to serve as doors and windows. Despite the practical importance of the structures with

cutouts, the number of technical papers and reports dealing with the subjects are very limited due to complexity involved. Some researches have been done on vibration and buckling of isotropic plates/shells, with cutout. Few studies are available on vibration and buckling of composite plates with cutout. The free vibration of composite rectangular plates with rectangular cutouts has been studied by Lee, Lim and Chow (1987) employing Rayleigh-Ritz principle. Siva Kumar, Iyengar and Deb (1999) worked on the optimum design of laminated composite plates with cutouts using a genetic algorithm. An experimental and analytical study was carried out to examine the effect of circular cutouts on the resonant frequencies of thin cylindrical shells by Toda (1977). The free and forced vibration of laminated composite shells with small cutouts is studied by Chakravorty, Sinha & Bandyopadhyay (1998) employing finite element method. Almorth and Holmes (1972) studied the buckling of shells with cutouts, by experimental and analytical method. The behaviour of simply supported uniformly compressed rectangular plates with central holes, have been investigated theoretically and experimentally by Ritchie Rhodes (1975) using a combination of Rayleigh-Ritz and finite element methods. Nemeth (1988) has predicted the buckling of rectangular symmetrically laminated angle-ply plate with central circular hole using finite element and experiment. Nemeth (1996) has reviewed works on buckling and post buckling behavior of rectangular composite plates with cutout. Ko (1998) has investigated on the anomalous buckling characteristics of laminated metal-matrix composite (MMC) plate with central square holes using finite element method (FEM). The study of stability of composite curved panels is new. In the present study, the influence of cutout on the vibration and buckling behavior of laminated composite curved panels with cutout has been examined.

MATHEMATICAL FORMULATION

The basic configuration of the problem considered here is a laminated composite doubly curved panel with cutout, as shown in Fig 1. The choice of double curved panel geometry as a basic configuration has been made so that depending on the value of curvature parameter, plate, cylindrical panel and different double curved panels such as spherical, hyperbolic parboiled and elliptic parboiled configurations can be considered as special cases. The governing equation for specified problems like plane stress, vibration and static stability are as follows:

Figure 1: Laminate composite doubly curved panel with cutout

1. Plane stress problem

Using the principle of stationery potential energy the equilibrium equation for plane

stress is expressed as:

[ ]{ } { }00 pqK p = (1)

2. Free Vibration

The governing equations for free vibration is

[ ]{ } [ ]{ } { }0=+ qKqM b&& (2)

3. Static stability or buckling

[ ] [ ][ ]{ } { }0=− qKPK gb (3)

The eigenvalues of the above equations give the natural frequencies and buckling

loads for different modes.

Finite Element Formulation An eight nodded isoparametric doubly curved element is used in the present analysis, with five degree of freedom u, v, w, xθ , yθ per node having two radii of curvatures, as shown in Fig 2. The deformation u & v are considered for the initial plane stress analysis. First order shear deformation theory (FSDT) is used & the shear correction coefficient has been employed to account for the non-linear distribution of shear strains through the thickness. The displacement field assumes that mid-plane normal remain straight but not necessarily normal after deformation, so that

),(),(uz)y,(x, 0 yxzyxu xθ+=

),(),(z)y,(x, 0 yxzyxvv yθ+= (4)

),(z)y,(x, yxww =

Where wvuwvu ,,&,, are the displacement components in the x, y, and z direction

at any section and at mid surface respectively.

Figure 2: Geometry and coordinates of shell element

Strain Displacement Relations

Green Lagrange’s strain displacement is used throughout the structural analysis. The linear part of the strain is used to derive the elastic stiffness matrix and the nonlinear part of the strain is used to derive the geometrical matrix. The linear strain displacement relations are:

xx

xl zkRw

xu

++∂∂

=ε

yyl zkRw

yv

y

++∂∂

=ε

xyxy

xyl zkR

wxv

yu

++∂∂

+∂∂

=2

γ (5)

xxzl xw

θγ +∂∂

=

yyzl yw

θγ +∂∂

=

Where the bending strains kj are express as ,

yxk yx

x ∂∂

=∂

∂=θθ

yk , (6)

xyk yx

xy ∂∂

+∂

∂=θθ

The linear strains can be described in terms of displacement as,

{ } [ ]{ } deB=ε (7)

Where{ } { }88888yx wv.....u.......... w v11111 yxude θθθθ= (8)

[ ] [ ] [ ] [ ][ ] ........................, 821 BBBB = (9)

[ ]

=

ii

ii

ii

i

i

ii

i

i

i

NyN

NxNxNyN

yN

xNxNyN

yNxN

B

000

000000

0000

0000000

00000000

,

,

,,

,

,

,,

,

,

(10)

Constitutive Relations The constitute relationship for the bending with transverse shear of doubly curved

shell become:

00

0

0ij

=

m

j

i

ij

ijij

ij

i

i

i

kS

DB

BA

QM

N

γ

ε

(11)

or

{F} = [D] { ε } (12)

Where Aij, B ij, Dij and Sij are the extensional bending stretching coupling, bending and transverse shear stiffness. They may be defined as

( ) ( )11

−=

−= ∑ kkk

n

kijij ZZQA

( ) ( )21

2

121

−=

−= ∑ kkk

n

kijij ZZQB (13)

( ) ( ) 6 2, 1, j i, ; 31 3

13

1

=−= −=

∑ kkk

n

kjiij ZZQD

( ) ( ) 5 4, j i, ; 11

=−= −=

∑ kkk

n

kijij ZZQkS

The accurate prediction for anisotropy laminates depends on the number of laminate properties and is also problem dependent. A Shear correction factor 5/6 is used in S ij in the present formulation for all numerical computations. Derivation Of Element Matrices

The element matrices are obtained as:

1. Element plane elastic stiffness matrix

[ ] [ ] [ ][ ]dABDBK ppT

pp ∫∫= (14)

2. Element elastic stiffness matrix [ ] [ ] [ ][ ]∫= dxdyBDBk T

b (15)

3. Consistent mass matrix

[ ] [ ] [ ][ ]∫∫= dydxNPNm Te (16)

where the shape function matrix

[ ]

=

i

i

i

i

i

NN

N

NN

N

00000000

0000

00000000

( (17)

[ ]

=

32

32

1

21

21

000000

0000

000000

PPPP

P

PPPP

P (18)

and

( ) ( ) ( )dzzzPPPn

k

Z

Zk

k

k

∑ ∫=

−

=1

2321

1

,,1 ρ (19)

The element mass matrix can be expressed in local natural coordinates of the element

as

[ ] [ ] [ ][ ]∫ ∫− −

=1

1

1

1

d ηξdJNPNmT

(20)

4. Geometric stiffness matrix

The element geometric stiffness matrix for the curved panel is derived using

the nonlinear in-plane Green’s strains with curvature component, using the procedure

explained by Cook, Malkus and Plesha(1989). Since the stress distribution is not

uniform due to the presence of cutout, the in-plane stress resultants N 0x , N 0

y and N0xy

at each Gauss point are obtained separately by plane stress analysis and the geometric

stiffness matrix is formed for these stress resultants. The strain energy becomes

U =2 { } [ ] [ ][ ]{ }dvGsG eTT

e δδ21

= { } [ ] { }eegT

e K δδ21

(21)

Where the element geometric stiffness matrix

[ ] [ ] [ ][ ] ηξddGsGK T

eg ∫ ∫− −

=1

1

1

1

(22)

i=1, 2…8

where [ ] =G

yN

xNyN

xN

yNxN

yNxN

yNxN

i

i

i

i

i

i

i

i

i

i

,

,

,

,

,

,

,

,

,

,

0000

00000000

0000

00000000

00000000

00000000

(23)

and [ ]=s

s

ss

s

s

0000

000000000000

0000

where (24)

[ ]=s

00

00

yxy

xyx

σττσ

=

00

001

yxy

xyx

NNNN

h (25)

Consistent Load Vector

The nodal load vector for an element when subjected to a distributed load

of intensity p(x, y) can be obtained by the expression

{ } [ ] qdxdyNp Tje ∫∫=

= [ ] ηξddJqN Tj∫ ∫

− −

1

1

1

1

(26)

where [ ]jN is the displacement function for the jth node

RESULTS AND DISCUSSIONS Extensive results of vibration and buckling of plates & shell structures with cutout for different panel geometries and size of cutouts have been presented using the formulation, given in chapter 3 and the following studies have been carried out. § Convergence study

§ Comparison with previous results

§ Numerical results.

Non-dimensionalisation of parameters

The non-dimensionalisation of different parameters like vibration, buckling are as

shown in Table 1.

Table 1: Non-dimensionalisation of vibration and buckling parameters

Sl no Parameter Plates/shells

1 Frequencies of vibration (ω ) 2

2

2

hEa ρϖ

2 Buckling load ( )λ 32

2

hEbN x

Convergence and Comparison with Previous Results

The convergence studies are carried out for non-dimensional frequencies of vibration

and buckling loads of rectangular plates with cutouts. Different mesh divisions are

considered and the converged results are compared with available literature.Based on

the convergence study (Table 2), 20X20 mesh has been employed to idealize the full

panel with cutout in subsequent analysis. The accuracy and the efficiency of the

present formulation are established through comparison of frequency parameters with

Reddy (1982) and buckling loads with Ko (1998) respectively. Good agreement exists

between the present FEM results with the literature as shown in Tables 2 and 3.

Table 2: Convergence and comparison of non-dimensional frequencies of a

simply supported square composite (0 0/900) plate with cutout (c/a=0.5).

b/h=100, 25.0=ν , E11/E22=40, G12 /E22=0.5, G13=G23=G12

Mesh division Non-dimensional frequencies

of composite plates with cutout

8X8 48.2535

12X12 48.0650

16X16 48.0222

20X20 48.0064

Reddy (1982) 48.4140

Table 3: Comparison of buckling loads Nx of a simply supported square panel with

(0/90/0/90) lamination for different cutout size, a=b=20 in, h=0.064 in, E1=27.72

X10 6 lb/in 2 , E2 = 18.09 X106 lb/in 2 , G12=8.15 X 106 lb/in 2 , 12ν =0.3

NON-DIMENSIONAL BUCKLING LOADS ( λ )

c/a=0.0 0.1 0.2 0.3 0.4 0.5

Present FEM 50.0166 47.8268 43.5535 40.2881 38.2646 36.9941

Ko(1998) 49.2286 46.9455 42.8393 39.6854 37.7255 36.4901

Numerical Results

After the convergence study and validating the formulation with the existing literature,

the results for vibration and buckling of laminated composite cross ply and angle ply

plates and shells with cutout for different cutout size, ply orientation, boundary

conditions have been presented. The geometrical dimensions of the panel are:

a = b = 0.5m, h = 0.005m,

The material properties are:

E11=141.0 Gpa, E22 = 9.23 Gpa, G12 = G13 = 5.95 Gpa, G23 = 2.96 Gpa, =12ν 0.313

The variation of non-dimensional frequencies for plates with different cutout ratios are shown in Figure 3. it is observed that the fundamental frequencies of vibration tends to decrease with increase in cutout size from c/a=0.1 to c/a=0.2 and then increases with size of cutout. For higher modes, the decrease of frequency occurs with increase in cutout upto c/a=0.6 and then increases. This may be due to reduction of stiffness initially and subsequent reduction of mass of the structure.

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

CUTOUT SIZE c/a

NO

N-D

IME

NS

ION

AL

FR

EQ

UE

NC

Y

Mode 1

Mode 2Mode 3

Mode 4

?

Figure 3: Variation of non-dimensional frequency with size of cutout for 4 - layered antisymmetric angle ply plates The variation of non-dimensional frequencies of curved panel with cutout is as shown in fig 4. The frequencies of vibration increases with introduction of curvatures in the flat panel for smaller cutout sizes c/a = 0.1 The effect of curvature is reduced with increase in size of cutout Non Dimensional Buckling Loads For Plates With Cutout The variation of non-dimensional buckling loads for plates with cutouts for different cutout ratios, ply orientations; layers of laminates, boundary conditions are studied. The non-dimensional buckling loads for a simply supported laminated square plate with various cutout sizes has been found out and presented in Table 4.

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

SIZE OF CUTOUT c/a

NO

N-D

IME

NS

ION

AL

FR

EQ

UE

NC

YMode 1

Mode 2

Mode 3Mode 4

Figure 4: Variation of non-dimensional frequency with size of cutout for 4- layered antisymmetric angle ply cylindrical shells Table 4: Variation of non dimensional buckling loads with size of cutout of a 4 layered simply supported laminated square plates with cutout a/b =1, b/h =100, Rx= Ry =0.1x1020

NON-DIMENSIONAL BUCKLING LOADS(λ ) SIZE OF CUTOUT c/a

ANTISYMMETRIC ANGLE PLY PLATES

200 ]45/45[ −

SYMMETRIC ANGLE PLY PLATES [ ]S

00 45/45 −

ANTISYMMETRIC CROSS PLY PLATES [ ]290/0

SYMMETRIC CROSS PLY PLATES [ ]s90/0

0.1 21.8732 19.5633 13.0066 14.7561 0.2 20.1880 17.6853 10.9706 12.3673 0.3 19.3759 16.6604 9.1971 10.3240 0.4 19.1018 16.0736 7.9041 8.7894 0.5 18.8699 15.4907 6.9816 7.6144 0.6 18.7432 14.9854 6.2872 6.6607 0.7 18.8689 14.7343 5.6960 5.8562 0.8 19. 1482 14.7822 5.1686 5.2009 As shown in Table 4, the buckling load decreases with increase in size of cutout up to a cutout size c/a=0.6 and then increases for anti-symmetric plates, where as the load decreases up to c/a=0.7 and then increases for symmetr ic angle ply plates. However for cross ply plates the buckling load decreases with increase in size of cutout. The variation of non-dimensional buckling loads with size of cutout of a 4-layered simply supported laminated cylindrical shell with cutout is shown in Table 5. It is observed that the buckling load decreases with size of cutout for angle and cross ply shells for both symmetric and antisymmetric lamination. The buckling loads for antisymmetric angle ply cylindrical shells are the highest for all cutout sizes.

Table 5: Variation of non-dimensional buckling loads with size of cutout of a 4-layered simply supported laminated cylindrical shell with cutout a/b =1, b/h =100, Rx =0.1x1020 , Ry =2.0

NON-DIMENSIONAL BUCKLING LOADS ( λ ) SIZE OF

CUTOUT

c/a

ANTISYMMETRIC ANGLE PLY CYLINDRICAL SHELL

200 ]45/45[ −

SYMMETRIC ANGLE PLY CYLINDRICAL SHELL

[ ]S00 45/45 −

ANTISYMMETRIC CROSS PLY CYLINDRICAL SHELL [ ]290/0

SYMMETRIC CROSS PLY CYLINDRICAL SHELL [ ]s90/0

0.1 59.2697 52.4831 47.6511 49.5332 0.2 51.5906 43.8957 42.8047 45.2165 0.3 42.5786 35.3980 33.8836 38.0347 0.4 33.5829 27.7356 23.8078 28.7522 0.5 26.0811 21.1889 16.2210 19.5060 0.6 22.0501 17.4507 11.0541 13.2083 0.7 20.3407 15.6966 7.7497 9.0003 0.8 19.9036 15.0868 5.7895 6.3117

From Table 4 and 5, it can be shown that the buckling loads are comparatively higher in case of cylindrical shells than flat plates. This may be due to the introduction of curvatures in flat plates. The effect of curvatures on the buckling loads of a simply supported antisymmetric angle ply 4-layered laminated ( 0000 45/45/45/45 −− ) shell with cutout is as shown in Table 6. It is observed that the buckling load increases with introduction of curvature of panel with cutout. The buckling load for a cylindrical shell is lowest where as hyperbolic paraboloid shows the highest buckling load. Table 6: Variation of non-dimensional buckling loads with curvature of a 4-layered simply supported laminated shell with cutout, a/b =1, b/h =100, c/a=0.5

SHELLS BUCKLING LOADS( λ ) CYLINDRICAL SHELL (Ry/Rx=0)

26.0812

SPHERICAL SHELL (Ry/Rx=1) 30.6803 HYPERBOLIC PARABOLOID (Ry/Rx= -1)

33.3537

ELLIPTIC PARABOLOID (Ry/Rx=1.5)

31.5003

CONCLUSION

The conclusions from the study on the vibration and stability behavior of composite

panels with cutout are summarized as given below.

1. The fundamental frequencies of vibration of the angle -ply flat panel decease

with introduction of cutouts but again rise with increase of size of cutouts.

2. The buckling loads decrease with increase in size of cutout in a panel.

3. The buckling loads increase significantly with introduction of curvature in the

panel. However, the effect of curvature is reduced with increase in size of

cutout.

4. The hyperbolic paraboloid with cutout shows highest buckling load out of all

curved panels.

From the above investigation it can be concluded that the stability

behaviour of laminated composite panels and shells are greatly influenced by the size

of cutout and curvature in curved panels. This can be used to the advantage of

tailoring during design of composite structures.

REFERENCES

1. Lee, H.P, Lim, S.P. and Chow, S.T(1987), Free vibration of composite rectangular plates with

rectangular cutouts, Composite Structures , 8, 63 -81.

2. Sivakumar, Iyengar and Deb (1999), Free vibration of laminated composite plates with cutout,

Journal of Sound and Vibration, 221(3), 443-470.

3. Toda S (1977), Vibration of circular cylindrical shells with cutouts, Journal of Sound and

Vibration, 52, 497-510.

4. Chakravorty, D, Sinha, P.K. and Bandyopadhyay, J.N (1998), Finite element free vibration

analysis of doubly curved laminated composite shells, Journal of Sound and Vibration, 191

(4), 491-504.

5. Almorth, B.O. Holmes and A.M.C -Buckling of shells with cutouts, experiment and analysis.

International Journal of Solids and Structures , 8 (8), 1057-1068, 1972

6. Ritchie, D. and Rhodes, J. -Buckling and post buckling behaviour of plates with holes.

Aeronautical Quarterly, 24, 281-296, 1975.

7. Nemeth -Buckling behaviour of compression loaded symmetrically laminated angle-ply plates

with holes.AIAA Journal , 26, 330-336, 1988.

8. Nemeth-Buckling and post Buckling behaviour of laminated composite plates with a cutout.

NASA Technical paper -3587, 1-23, 1996.

9. Ko, W.L- Mechanical and thermal buckling vibration of rectangular plates with different

central cutout. NASA/TM Technical paper , 206592, 1-13, 1998.

10. Cook, R, Malkus, D.S. and Plesha, M. E.(1989), Concepts and Applications of Finite Element

Analysis , John Wiley & Sons, U.S.A.

11. Reddy, J.N-Large amplitude flexural vibration of layered composite plates with cutouts.

Journal of Sound and Vibration, 83 (1), 1 -10, 1982.