111

________________

Corresponding author: Sarmila Sahoo E-mail address: sarmila.sahoo@gmail.com Doi: http://dx.doi.org/10.11127/ijammc.2014.08.06

Copyright@GRIET Publications. All rights reserved.

Advanced Materials Manufacturing & Characterization Vol4 Issue 2 (2014)

Advanced Materials Manufacturing & Characterization

journal home page: www.ijammc-griet.com

Laminated composite stiffened saddle shells with cutouts under free vibration – a finite element approach Sarmila Sahoo Department of Civil Engineering, Heritage Institute of Technology, Kolkata 700107, India.

A R T I C L E I N F O Article history: Received: 26-05-2014 Accepted: 28-05-2014 Keywords: Laminated composites; Stiffened saddle shell; Cutout; Free vibration; Finite element

A B S T R A C T

Finite element method has been applied to solve free vibration problems of laminated composite stiffened saddle shells with cutouts employing the eight-noded curved quadratic isoparametric element for shell with a three noded beam element for stiffener formulation. Specific numerical problems of earlier investigators are solved to compare their results. Moreover, free vibration problem of stiffened saddle shells with different size and position of the cutouts with respect to the shell centre for different edge constraints are examined to arrive at some conclusions useful to the designers. The results are presented in the form of figures and tables. The results are further analyzed to suggest guidelines to select optimum size and position of the cutout with respect to shell centre considering the different practical constraints.

1. INTRODUCTION Finite element method has become an efficient tool to analyze complex structures. The dynamic analysis of shell structures, which may have complex geometry and arbitrary loading and boundary conditions, can be solved efficiently by the finite element method. Laminated composites are increasingly being used nowadays in aerospace, civil, marine and other related weight-sensitive engineering applications requiring high strength-to-weight and stiffness-to weight ratios. Among the different shell forms which are used as roofing units, saddle shells are one of them. Examples of such saddle roofs are: Warszawa Ochota railway station, Church Army Chapel, Blackheath, The Calgary, Saddledome, London Velopark. Quite often, to save weight and also to provide a facility for inspection, cutouts are provided in shell panels. In practice the margin of the cutouts must be stiffened to take account of stress concentration effects. Also, there can be some instruments directly fixed on these panels, and the safety of these instruments can be dependent on the vibration characteristics of the panels. Hence free vibration studies on saddle shell panels with cutouts are of interest to structural engineers.

Different computational models for laminated composites were proposed by Kapania [1], Noor and Burton [2], and Reddy [3]. Chao and Reddy [4] reported on the dynamic response of simply supported cylindrical and spherical shells. The transient response of spherical and cylindrical shells with various boundary conditions and loading was reported by Reddy and Chandrashekhara [5]. Chao and Tung [6] presented an investigation on the dynamic response of axisymmetric polar orthotropic hemispherical shells. Later free vibration study of doubly curved shells was done by Qatu [7], Liew and Lim [8,9], Chakravorty et al. [10-12], Shin [13] and Tan [14]. Kant et al. [15] solved problems of a clamped spherical and simply supported cylindrical cap under external pressure. Sathyamoorthy [16] reported the nonlinear vibration of moderately thick orthotropic spherical shells. Later in 1997, Gautham and Ganesan [17] reported free vibration characteristics of isotropic and laminated orthotropic spherical caps while Chia and Chia [18] reported nonlinear vibration of moderately thick anti-symmetric angle ply shallow spherical shell. Free vibration of curved panels with cutouts was reported by Sivasubramonian et. al. [19]. Qatu [20] reviewed the work done on the vibration aspects of composite shells between 2000-2009 and observed that most of the researchers dealt with closed cylindrical shells. Other shell geometries have also been investigated. Among those conical shells and shallow shells on rectangular, triangular, trapezoidal, circular, elliptical, rhombic or other planforms are receiving considerable attention. Shallow spherical shells also received

112

some attention. Wang et al [21] studied wave propagation of stresses in orthotropic thick-walled spherical shells. Lellep and Hein [22] did an optimization study on shallow spherical shells under impact loading. Dai and Wang [23] analyzed stress wave propagation in laminated piezoelectric spherical shells under thermal shock and electric excitation. Dynamic stability of spherical shells was studied by Ganapathi [24] and Park and Lee [25]. But, saddle shells on rectangular planform with cutout (stiffened along the margin) are far from the existing literature. Accordingly, the present endeavor focuses on the free vibration behavior of composite saddle shell with cutout (stiffened along the margin) with concentric and eccentric cutouts, and considers the shells to have various boundary conditions.

2. MATHEMATICAL FORMULATION A laminated composite saddle shell of uniform thickness h (Fig.1) and radius of curvature Rx and Ry is considered. Keeping the total thickness the same, the thickness may consist of any number of thin laminae each of which may be arbitrarily oriented at an angle with reference to the X-axis of the co-ordinate system. The constitutive equations for the shell are given by (a list of notations is already given):

Fig. 1 Saddle shell with a concentric cutout stiffened along the margins

F=E (1) where,

Tyxxyyxxyyx QQMMMNNNF ,,,,,,,

S

DB

BA

E

00

0

0

,

Tyzxzxyyxxyyx kkk 00000 ,,,,,,, .

The force and moment resultants are expressed as

2/

2/

,,.,.,.,,,

,,,,,,,

h

h

T

yzxzxyyzxyyx

T

yxxyyxxyyx

dzzzz

QQMMMNNN

(2) The submatrices [A], [B], [D] and [S] of the elasticity matrix [E] are functions of Young’s moduli, shear moduli and the Poisson’s ratio of the laminates. They also depend on the angle which the

individual lamina of a laminate makes with the global X-axis. The detailed expressions of the elements of the elasticity matrix are available in several references including Vasiliev et al. [26] and Qatu [27]. The strain-displacement relations on the basis of improved first order approximation theory for thin shell (Dey et al.[28]) are established as

Tyzxzxyyx

T

yzxzxyyx

T

yzxzxyyx kkkkkz ,,,,,,,,,,,, 00000

(3) where, the first vector is the mid-surface strain for a saddle shell and the second vector is the curvature. 3. FINITE ELEMENT FORMULATION 3.1 Finite Element Formulation for Shell An eight-noded curved quadratic isoparametric finite element is used for saddle shell analysis. The five degrees of freedom taken into consideration at each node are u, v, w, , . The following expressions establish the relations between the displacement at any point with respect to the co-ordinates and and the nodal degrees of freedom.

i

i

iuNu

8

1 i

i

i vNv

8

1 i

i

i wNw

8

1

i

i

iN

8

1 i

i

iN

8

1 (4) where the shape functions derived from a cubic interpolation polynomial [22] are: Ni =(1+i)(1+i)( i+i-1)/4, for i=1,2,3,4 Ni =(1+i)(1-2)/2, for i=5,7 Ni =(1+i)(1-2)/2, for i=6,8 (5) The generalized displacement vector of an element is expressed in terms of the shape functions and nodal degrees of freedom as:

edNu (6)

i.e.,

8

1i

i

i

i

i

i

i

i

i

i

i

w

v

u

N

N

N

N

N

w

v

u

u

Element Stiffness Matrix The strain-displacement relation is given by

edB, (7)

Where

113

,

,

8, ,

1

,

,

, ,

,

,

0 0 0

0 0 0

20 0

0 0 0 0

0 0 0 0

0 0 0

0 0 0

0 0 0

ii x

x

ii y

y

ii y i x

xy

i

i x

i y

i y i x

i x i

i y i

NN

R

NN

R

NN N

RB

N

N

N N

N N

N N

(8) The element stiffness matrix is

dxdyBEBKT

e (9)

Element Mass Matrix The element mass matrix is obtained from the integral

dxdyNPNMT

e , (10)

where,

8

1

0000

0000

0000

0000

0000

i

i

i

i

i

i

N

N

N

N

N

N ,

8

1

0000

0000

0000

0000

0000

i

I

I

P

P

P

P ,

in which

np

k

z

z

k

k

dzP1

1

and

np

k

z

z

k

k

dzzI1

1

(11)

3.2 Finite Element Formulation for Stiffener of the Shell Three noded curved isoparametric beam elements (Fig.2) are used to model the stiffeners, which are taken to run only along the boundaries of the shell elements. In the stiffener element, each node has four degrees of freedom i.e. usx, wsx, sx and sx for

Fig. 2 (a) Eight noded shell element with isoparametric coordinates (b) Three noded stiffener elements (i) x-stiffener (ii) y-stiffener

X-stiffener and vsy, wsy, sy and sy for Y-stiffener. The generalized force-displacement relation of stiffeners can be expressed as:

X-stiffener: sxisxsxsxsxsx BDDF

;

Y-stiffener: syisysysysysy BDDF

(12)

where, Tsxxzsxxsxxsxxsx QTMNF ;

Txsxsxxsxxsxxsxsx wu ....

and Tsyyzsyysyysyysy QTMNF ;

Tysysyysyysyysysy wv ....

The generalized displacements of the x-stiffener and the shell are

related by the transformation matrix sxi xT

where

1

0 1

0 0 1

0 0 0 1

x

x

esymmetric

R

T

The generalized displacements of the y-stiffener and the shell are

related by the transformation matrix syi yT where

1

0 1

0 0 1

0 0 0 1

y

y

esymmetric

R

T

These transformations are required due to curvature of x-

stiffener and y-stiffener.

is the appropriate portion of the

114

displacement vector of the shell excluding the displacement component along the other axis. Elasticity matrices are as follows:

11

3

664412/

12/

12/

11/

11/

12/

11/

11

000

06

1

0

0

Sb

bdQQbDbB

bDbDbB

bBbBbA

D

sx

sxsxsxsx

sxsxsx

sxsxsx

sx

22

311

/12

/12

/

12/

664422/

12/

22/

22

000

0

0)(6

1

0

Sb

bdDbDbB

bDbQQbB

bBbBbA

D

sy

sysysysy

sysysy

sysysy

sy

. where,

ijijijij AeeBDD 2/ 2 ; ijijij eABB /

, (13) and Aij, Bij, Dij and Sij are explained in an earlier paper by Sahoo and Chakravorty [29]. Here the shear correction factor is taken as 5/6. The sectional parameters are calculated with respect to the mid-surface of the shell by which the effect of eccentricities of stiffeners is automatically included. The element stiffness matrices are of the following forms.

for X-stiffener: dxBDBK sxsx

T

sxxe ;

for Y-stiffener: dyBDBK sysy

T

syye (14)

The integrals are converted to isoparametric coordinates and are carried out by 2-point Gauss quadrature. Finally, the element stiffness matrix of the stiffened shell is obtained by appropriate matching of the nodes of the stiffener and shell elements through the connectivity matrix and is given as:

yexeshee KKKK . (15)

The element stiffness matrices are assembled to get the global matrices. Element Mass Matrix The element mass matrix for shell is obtained from the integral

dxdyNPNMT

e , (16)

where,

8

1

0000

0000

0000

0000

0000

i

i

i

i

i

i

N

N

N

N

N

N ,

8

1

0000

0000

0000

0000

0000

i

I

I

P

P

P

P ,

in which

np

k

z

z

k

k

dzP1

1

and

np

k

z

z

k

k

dzzI1

1

(17)

Element mass matrix for stiffener element

dxNPNMT

sx for X stiffener

and dyNPNMT

sy for Y stiffener (18)

Here, N is a 3x3 diagonal matrix.

3

1

33

2

12/)..(000

012/.00

00.0

000.

i

sxsxsxsx

sxsx

sxsx

sxsx

dbdb

db

db

db

P

for X-stiffener

3

133

2

12/)..(000

012/.00

00.0

000.

i

sysysysy

sysy

sysy

sysy

dbdb

db

db

db

P

for Y-stiffener The mass matrix of the stiffened shell element is the sum of the matrices of the shell and the stiffeners matched at the appropriate nodes.

yexeshee MMMM . (19)

The element mass matrices are assembled to get the global matrices. 3.3 Modeling the cutout The code developed can take the position and size of cutout as input. The program is capable of generating non uniform finite element mesh all over the shell surface. So the element size is gradually decreased near the cutout margins. One such typical mesh arrangement is shown in Fig. 3. Such finite element mesh is

115

redefined in steps and a particular grid is chosen to obtain the fundamental frequency when the result does not improve by more than one percent on further refining. Convergence of results is ensured in all the problems taken up here.

Fig. 3 Typical 10x10 non-uniform mesh arrangements drawn to scale

3.4 Solution Procedure for Free Vibration Analysis The free vibration analysis involves determination of natural frequencies from the condition

02 MK (20)

This is a generalized eigen value problem and is solved by the subspace iteration algorithm. 3. VALIDATION STUDY The results of Table 1 show that the agreement of present results with the earlier ones is excellent and the correctness of the stiffener formulation is established. Free vibration of corner point supported, simply supported and clamped saddle shells of (0/90)4 lamination with cutouts is also considered. The fundamental frequencies of saddle shell with cutout obtained by the present method agree well with those reported by Chakravorty et al. [12] as evident from Table 2, establishing the correctness of the cutout formulation. Thus it is evident that the finite element model proposed here can successfully analyse vibration problems of stiffened composite saddle shells with cutout which is reflected by close agreement of present results with benchmark ones. The present approach uses the improved first order approximation theory for thin shells [28] considering the radius of cross curvature. For this class of thin shells a shear correction factor of unity is found to yield good results. It is observed that the results remain the same when analysis is repeated with the

commonly used shear correction factor of 12

.

Table 1 Natural frequencies (Hz) of centrally stiffened clamped square plate

Mode no.

Mukherjee and Mukhopadhyay [24]

Nayak and Bandyopadhyay[25]

Present method

N8 (FEM)

N9 (FEM)

1 711.8 725.2 725.1 733

a=b=0.2032 m, shell thickness =0.0013716 m, stiffener depth 0.0127 m, stiffener width =0.00635 m, stiffener eccentric at bottom, Material property: E=6.87x1010 N/m2, =0.29, =2823 kg/m3

Table 2: Nondimensional Fundamental Frequencies ( ) for laminated

composite saddle shell with cutout.

a/b=1, a/h=100, a//b/=1, h/Rxx= -h/Ryy=1/300, CS=Corner point supported, SS=Simply supported, CL=Clamped 4. RESULTS AND DISCUSSION In order to study the effect of cutout size and position on the free vibration response additional problems for spherical shells with 0/90/0/90 and +45/-45/+45/-45 lamination and different boundary conditions have been solved. The positions of the cutouts are varied along both of the plan directions of the shell for different practical boundary conditions to study the effect of eccentricity of cutout on the fundamental frequency. 4.1 Free vibration behaviour of shells with concentric cutouts Tables 3 and 4 furnish the results of non-dimensional frequency

( ) of 0/90/0/90 and +45/-45/+45/-45 stiffened spherical shells with cutout. The shells considered are of square plan form (a=b) and the cutouts are also taken to be square in plan (a/=b/). The cutouts placed concentrically on the shell surface. The cutout sizes (i.e. a//a) are varied from 0 to 0.4 and boundary conditions are varied along the four edges. Cutouts are concentric on shell surface. The stiffeners are place along the cutout periphery and extended up to the edge of the shell. The boundary conditions are designated by describing the support clamped or simply supported as C or S taken in an anticlockwise order from the edge x=0. This means a shell with CSCS boundary is clamped along x=0, simply supported along y=0 and clamped along x=a and simply supported along y=b. The material and geometric properties of shells and cutouts are mentioned along with the figures. Effect of cutout size on fundamental frequency From the Tables 3 and 4 it is seen that when a cutout is introduced to a stiffened shell the fundamental frequency increases in all the cases. This increasing trend is noticed for both cross ply and angle ply shells. This initial increase in frequency is due to the fact that with the introduction of cutout, numbers of stiffeners are increase from two to four in the present study. It is evident from Tables 3 and 4 that in all the cases with the introduction of cutout with a//a=0.1 the frequencies increase. But further increase in cutout size, fundamental frequency may increase or decrease. When the cutout size is further increased, but the number and dimensions of the stiffeners do not change, the shell surface undergoes loss of both mass and stiffness. It is

a’/a CS SS CL

Chakravorty et. al.(12)

Present model

Chakravorty et. al.(12)

Present model

Chakravorty et. al.(12)

Present model

0.0 13.485 13.249 14.721 14.686 113.567 112.926 0.1 13.060 13.084 14.350 14.695 97.753 98.041 0.2 12.530 12.449 13.544 13.507 97.599 97.032 0.3 12.016 12.038 12.908 12.882 111.489 111.033

0.4 11.721 11.733 12.560 12.559 110.210 110.20

116

evident from Table 3 and 4 that when the cutout size is increased, fundamental frequency is increased in all cases of cross ply shells. For angle ply shells also with the increase of cutout size fundamental frequency increases upto a//a=0.2, but with further increase of cutout size reverse trend is observed. So for angle ply shells with the increase of cutout size loss of stiffness is more than loss of mass. Hence fundamental frequency decreases except in case of CSCS and corner point supported angle ply shells (a//a=0.2 to 0.3). In such cases, as the cutout grows in size the loss of mass is more significant than loss of stiffness, and hence the frequency increases. As with the introduction of a cutout of a//a=0.2, in shell surface, the frequency increases in most of the cases, this leads to the engineering conclusion that concentric cutouts with stiffened margins may be provided safely on shell surfaces for functional requirements upto a//a=0.2. Effect of boundary conditions The boundary conditions may be arranged in the following order, considering number of boundary constraints. CCCC , CSCC, CSSC, CSCS, SSSS and Corner Point supported. It is seen from Table 3 and 4, that fundamental frequencies of members belonging to same number of boundary constraints may not have close values. So the boundary constraint is not the sole criteria for its performance. The free vibration characteristics mostly depends on the arrangement of boundary constrains rather than their actual number, is evident from the present study. It can be seen from the present study that if the circular edge along x=a is released from clamped to simply supported, the change of frequency is not very significant for cross ply shells and angle ply shells. Again if the two alternate edges are released from clamped to simply supported fundamental frequency do not change to a great extent but whereas, if the two adjacent edges are released, fundamental frequency decreases significantly than that of a clamped shell. This is true for both cross and angle ply shells. For both the type of shells if all the edges are simply supported, frequency values undergo marked decrease. The results indicate that the two alternate edges should preferably be clamped in order to achieve higher frequency values.

Table3: Nondimensional Fundamental Frequencies ( ) for laminated composite (0/90/0/90 ) stiffened saddle shell for different sizes of the

central square cutout and different boundary conditions.

Boundary conditions Cutout size (

/a a)

0 0.1 0.2 0.3 0.4

CCCC 93.08 107.18 129.7 135.35 138.56

CSCC 79.68 92.13 108.7 120.34 121.57

CSSC 61.39 72.13 81.88 87.4 91.17

CSCS 84.89 87.82 107.4 116.61 118.1

CSSS 57.39 69.92 76.06 80.13 83.51

SSSS 49.1 63.99 68.27 71.36 73.77

Point supported 27.81 30.62 35.72 41.93 47.84

a/b=1, a/h=100, / /a b =1, h/Rxx=- h/Ryy=1/300; E11/E22 = 25, G23 =

0.2E22, G13 = G12 = 0.5E22, 12 =21 =0.25.

Table4: Nondimensional Fundamental Frequencies ( ) for laminated

composite (+45/-45/+45/-45) stiffened saddle shell for different sizes of the central square cutout and different boundary conditions.

Boundary conditions Cutout size (

/a a)

0 0.1 0.2 0.3 0.4

CCCC 81.34 98.71 103.2 107.23 112.06

CSCC 75.81 91.13 95.02 98.14 101.27

CSSC 68.36 81.17 85.72 89 92.23

CSCS 73 87.56 91.25 94.6 97.53

CSSS 64.37 77.65 81.45 84.33 87.34

SSSS 51.61 67 71.16 74.09 77.28

Point supported 34.28 36.7 42.48 45.71 47.15

a/b=1, a/h=100, / /a b =1, h/Rxx= -h/Ryy=1/300; E11/E22 = 25, G23 =

0.2E22, G13 = G12 = 0.5E22, 12 =21 =0.25. Table 5 and 6 shows the efficiency of a particular clamping option in improving the fundamental frequency of a shell with minimum number of boundary constraints relative to that of a clamped shell. Marks are assigned to each boundary combination in a scale assigning a value of 0 to the frequency of a corner point supported shell and 100 to that of a fully clamped shell. These marks are furnished for cutouts with a//a=0.2 These tables will enable a practicing engineer to realize at a glance the efficiency of a particular boundary condition in improving the frequency of a shell, taking that of clamped shell as the upper limit. Table 5: Clamping options for 0/90/0/90 saddle shells with central cutouts having a//a ratio 0.2.

Number of sides to be clamped

Clamped edges

Improvement of frequencies with respect to point supported shells

Marks indicating the efficiencies of no of restraints

0 Corner Point supported

- 0

0 0 Simply supported no edges clamped (SSSS)

Good improvement

35

Edge along x=0 (CSSS)

Good improvement

43

2 a)Two alternate edges along x=0 and x=a

Marked improvement

76

117

(CSCS) c)Any two edges except the above option (CSSC)

Good improvement

49

3 3 edges (CSCC)

Marked improvement

78

4 All sides (CCCC)

Frequency attains highest value

100

Table 6: Clamping options for +45/-45/+45/-45 saddle shells with central cutouts having a//a ratio 0.2.

Number of sides to be clamped

Clamped edges

Improvement of frequencies with respect to point supported shells

Marks indicating the efficiencies of no of restraints

0 Corner Point supported

- 0

0 0 Simply supported no edges clamped (SSSS)

Good improvement

47

Edge along x=0 (CSSS)

Marked improvement

64

2 a)Two alternate edges along x=0 and x=a (CSCS)

Remarkable improvement

80

c)Any two edges except the above option (CSSC)

Marked improvement

71

3 3 edges (CSCC)

Remarkable improvement

87

4 All sides (CCCC)

Frequency attains highest value

100

Mode shapes The mode shapes corresponding to the fundamental modes of vibration are plotted in Fig.4 and Fig.5 for crossply and angle ply shells respectively. The normalized displacements are drawn with the shell midsurface as the reference for all the support condition and for all the lamination used here. The fundamental mode is clearly a bending mode for all the boundary condition for cross ply and angle ply shell, except corner point supported shell. For corner point supported shells the fundamental mode shapes are complicated. With the introduction of cutout mode shapes remain almost similar. When the size of the cutout is increased from 0.2 to 0.4 the fundamental modes of vibration do not change to an appreciable amount. Effect of eccentricity of cutout position Fundamental frequency The effect of eccentricity of cutout positions on fundamental frequencies, are studied from the results obtained for different locations of a cutout with a//a=0.2. The non-dimensional

coordinates of the cutout center (,

x yx y

a a

) was varied from 0.2 to 0.8 along each directions, so that the distance of a cutout margin from the shell boundary was not less than one tenth of the plan dimension of the shell. The margins of cutouts were stiffened with four stiffeners. The study was carried out for all the seven boundary conditions for both cross ply and angle ply shells. The fundamental frequency of a shell with an eccentric cutout is expressed as a percentage of fundamental frequency of a shell with a concentric cutout. This percentage is denoted by r. In Table 7 and 8 such results are furnished. Table 7: Values of ‘r’ for 0/90/0/90 saddle shells.

Edge condition

y x

0.2 0.3 0.4 0.5 0.6 0.7 0.8

CCCC

0.2

91.20

85.18

88.20

91.75

88.17

85.18

85.03

0.3

84.39

85.22

89.00

93.85

89.00

85.22

84.39

0.4

86.94

88.31

92.41

96.84

92.41

88.31

86.94

0.5

90.16

91.96

96.15

100.00

96.15

91.96

90.16

0.6

86.94

88.31

92.41

96.84

92.41

88.31

86.94

0.7

84.39

85.22

89.00

93.64

89.00

85.22

84.39

0.8

84.76

84.97

88.01

91.77

88.13

85.07

87.34

CSCC

0.2

91.12

90.60

91.90

93.11

91.90

90.60

91.10

0.3

94.86

95.19

97.57

99.66

97.57

95.18

94.86

0.4

95.93

99.74

105.62

109.00

106.03

99.76

95.88

0.5

93.08

93.79

97.03

100.00

97.33

93.89

92.79

0.6

86.93

85.83

87.77

89.93

87.92

85.92

86.50

0.7

82.66

81.28

82.85

84.66

82.94

81.34

82.26

0.8

80.75

79.37

80.75

82.34

80.82

79.43

80.40

CSSC

0.2

67.33

73.00

81.56

92.87

96.24

87.04

78.21

0.3

72.96

79.10

88.04

99.06

101.48

92.50

83.52

0.4

76.94

83.79

93.47

104.13

105.86

97.08

87.64

0.5

73.28

78.60

87.24

100.00

104.47

94.43

83.29

0.6

65.80

70.00

77.99

91.48

98.34

87.23

77.64

0.7

60.39

64.56

72.67

86.24

92.93

81.74

72.47

0.8

57.62

62.10

70.46

83.79

89.52

78.58

69.52

CSCS

0.2

78.77

78.00

78.58

79.36

78.58

78.00

78.77

0. 81.6 80.8 81.9 83.1 81.9 80.8 81.6

118

3 4 5 4 8 4 5 4

0.4

86.58

86.04

87.77

89.54

87.77

86.04

86.58

0.5

92.51

94.34

97.12

100.00

97.09

94.34

92.51

0.6

86.58

86.04

87.77

89.54

87.77

86.04

86.58

0.7

81.64

80.85

81.94

83.18

81.94

80.85

81.64

0.8

78.76

78.00

78.58

79.35

78.58

78.00

78.77

CSSS

0.2

53.39

58.77

68.30

82.91

90.38

78.57

68.38

0.3

58.51

63.66

72.90

87.51

94.99

83.39

73.30

0.4

65.33

70.81

80.02

94.19

100.46

89.57

79.40

0.5

70.35

77.20

87.33

100.00

103.67

93.82

83.51

0.6

65.33

70.81

80.02

94.20

100.46

89.57

79.40

0.7

58.51

63.66

72.89

87.51

94.99

83.39

73.30

0.8

53.31

58.73

68.25

82.84

90.28

78.49

68.28

SSSS

0.2

50.81

60.23

73.00

87.40

73.00

60.25

50.81

0.3

56.41

65.40

77.74

91.59

77.74

65.40

56.41

0.4

63.78

72.87

84.90

96.78

84.90

72.87

63.78

0.5

69.18

79.20

91.26

100.00

91.26

79.20

69.18

0.6

63.78

72.87

84.90

96.78

84.90

72.87

63.78

0.7

56.41

65.40

77.75

91.59

77.74

65.40

56.41

0.8

50.71

60.19

72.95

87.33

72.96

60.19

50.73

CS 0.2

81.33

91.69

101.88

106.75

101.88

91.69

81.33

0.3

87.40

94.82

101.06

103.50

101.04

94.82

87.40

0.4

94.65

98.49

100.45

101.04

100.45

98.49

94.65

0.5

101.51

100.62

100.11

100.00

100.11

100.62

101.51

0.6

94.65

98.46

100.45

101.04

100.45

98.49

94.65

0.

7

87.4

0

94.8

2

101.

04

103.

50

101.

04

94.8

2

93.0

0

0.

8

81.3

0

91.6

6

101.

82

106.

69

101.

79

91.6

3

81.2

7

a/b=1, a/h=100, / /a b =1, h/Rxx=- h/Ryy=1/300; E11/E22 = 25, G23 = 0.2E22,

G13 = G12 = 0.5E22, 12 =21=0.25. Table8: Values of ‘r’ for +45/-45/+45/-45 saddle shells.

Edge condition

y x

0.2 0.3 0.4 0.5 0.6 0.7 0.8

CCCC 0.2

62.14

67.01

74.35

81.73

74.25

66.95

62.10

0.3

66.84

72.12

79.94

87.32

79.80

72.03

66.79

0.4

73.89

79.66

87.87

94.77

87.69

79.53

73.81

0.5

80.79

86.62

94.42

100.00

94.42

86.62

80.79

0.6

73.81

79.53

87.69

94.77

87.87

79.66

73.89

0.7

66.79

72.03

79.80

87.32

79.94

72.12

66.84

0.8

62.10

66.94

74.23

81.72

74.34

67.00

62.13

CSCC 0.2

66.99

72.23

79.93

86.96

79.79

72.14

66.93

0.3

72.28

77.96

86.19

93.34

86.02

77.86

72.22

0.4

79.85

85.92

94.21

100.31

94.02 8.92

79.73

0.5

80.52

86.75

94.69

100.00

94.61

86.31

79.75

0.6

72.23

78.15

86.18

92.38

86.09

77.79

71.77

0.7

66.63

72.04

79.59

85.89

79.49

71.76

66.29

0.8

63.09

68.09

75.23

81.47

75.12

67.87

62.86

CSSC 0.2

67.62

71.93

78.69

88.09

87.51

79.16

73.02

0.3

73.11

77.87

85.09

94.56

93.95

85.32

78.76

0.4

79.35

84.54

92.11

100.94

101.35

93.36

86.30

0.5

76.87

82.51

90.76

100.00

101.20

93.33

85.17

0.6

69.88

75.13

82.93

92.58

93.30

84.62

76.75

0.7

64.93

69.54

76.54

85.90

86.63

78.14

70.92

0.8

61.76

65.78

72.13

81.23

82.09

73.92

67.34

CSCS 0.2

65.00

70.15

77.46

83.55

77.68

70.47

65.30

0.3

68.71

74.33

82.18

88.50

82.47

74.77

69.17

0.4

74.35

80.57

88.92

95.05

89.24

81.16

75.05

0.5

81.86

87.95

95.39

100.00

95.39

87.95

81.90

0.6

75.05

81.16

89.24

95.05

88.92

80.57

74.35

0.7

69.17

74.77

82.47

88.50

82.18

74.33

68.71

0.8

65.29

70.44

77.63

83.47

77.42

70.13

64.99

CSSS 0. 62.0 66.7 74.0 83.9 85.3 77.4 70.7

119

2 7 0 6 9 9 2 7

0.3

66.19

71.52

79.36

89.37

90.44

82.20

75.17

0.4

71.50

77.58

86.11

95.97

97.07

89.02

81.61

0.5

75.65

82.34

91.09

100.00

101.52

94.65

86.92

0.6

70.51

76.70

85.24

95.03

96.11

87.64

79.66

0.7

64.81

70.30

78.26

88.19

89.43

80.91

73.46

0.8

60.37

65.02

72.49

82.53

84.37

76.26

69.33

SSSS 0.2

56.00

66.46

78.09

86.57

77.50

66.16

56.07

0.3

60.91

71.33

83.09

91.39

82.70

71.14

61.06

0.4

66.81

77.63

89.45

96.92

89.15

77.50

67.02

0.5

70.52

81.69

93.41

100.00

93.41

81.69

70.52

0.6

67.03

77.50

89.15

96.92

89.45

77.63

66.81

0.7

61.06

71.14

82.70

91.39

83.09

71.33

60.91

0.8

55.96

66.09

77.46

86.54

78.08

66.41

55.89

CS 0.2

59.06

66.85

75.42

81.90

75.35

66.74

59.09

0.3

63.54

70.88

79.61

86.39

79.54

70.90

63.63

0.4

69.44

76.32

85.12

92.28

85.15

76.44

69.37

0.5

76.67

83.22

91.74

100.00

91.74

83.22

76.67

0.6

69.40

76.44

85.17

92.28

86.16

76.32

69.44

0.7

63.63

70.93

79.54

86.39

79.61

70.88

63.54

0.8

59.09

66.76

75.38

81.87

75.35

66.83

59.06

a/b=1, a/h=100, / /a b =1, h/Rxx= -h/Ryy=1/300; E11/E22 = 25, G23 = 0.2E22,

G13 = G12 = 0.5E22, 12 =21 =0.25.

It can be seen that eccentricity of the cutout along the length of the shell towards the edges makes it more flexible. It is also seen that almost all the cases r value is maximum in and around

0.5x and 0.5y

. It is noticed that towards the clamped edge r value is greater than that of the simply supported edge when only one edge is simply supported. When two adjacent edges are simply supported no such unified trend is observed. Moreover, when three edges are simply supported, the simply supported edge opposite to the clamped one shows more frequency value than that of the clamped edge. For corner point supported shells the maximum fundamental frequency always occurs along the diagonal of the shell.

Table 9 and 10 provide the maximum values of r together with the position of the cutout. These tables also show the rectangular zones within which r is always greater than or equal to 90. It is to be noted that at some other points r values may have similar values, but only the zone rectangular in plan has been identified. These tables indicate the maximum eccentricity of a cutout which can be permitted if the fundamental frequency of a concentrically punctured shell is not to reduce a drastic amount. So these tables will help practicing engineers Table 9: Maximum values of r with corresponding coordinates of cutout

centres and zones where r≥90 and r≥95 for 0/90/0/90 saddle shells. Boundary Condition

Maximum values of r

Co-ordinate of cutout centre

Area in which the value of r≥90

CCCC

100.00 (0.5,0.5) 0.4 ≤ x ≤0.6,

0.4≤y

≤0.6. CSCC

109.00 (0.5, 0.4) 0.2≤ x ≤0.8,

0.2≤y

≤0.5. CSSC

105.86 (0.6, 0.4) 0.5≤ x ≤0.6,

0.2≤y

≤0.6. CSCS

100.00 (0.5, 0.5) 0.2≤ x ≤0.8,

y=0.5.

CSSS

103.67 (0.6, 0.5) 0.5≤ x ≤0.6,

0.4≤y

≤.0.6. SSSS

100.00 (0.5, 0.5) x =0.5, 0.3≤

y≤0.7.

CS

106.75 (0.5, 0.2) 0.3≤ x ≤0.7,

0.2≤y

≤0.8.

a/b=1, a/h=100, / /a b =1, h/Rxx=- h/Ryy=1/300; E11/E22 = 25, G23 = 0.2E22,

G13 = G12 = 0.5E22, 12 =21 =0.25. Table 10: Maximum values of r with corresponding coordinates of cutout centres and zones where r≥90 and r≥95 for +45/-45/+45/-45 saddle shells.

Boundary Condition

Maximum values of r

Co-ordinate of cutout centre

Area in which the value of r≥90

120

CCCC

100.00 (0.5, 0.5) x =0.5,

0.4≤y

≤0.6. CSCC

100.31 (0.5, 0.4) 0.4≤ x ≤0.6,

0.4≤y

≤0.5. CSSC

101.35 (0.6, 0.4) 0.5≤ x ≤0.6,

0.3≤y

≤0.6. CSCS

100.00 (0.5, 0.5) x =0.5,

0.4≤y

≤0.6. CSSS

101.52 (0.6, 0.5) 0.4≤ x ≤0.6,

0.5≤y

≤0.6. SSSS

100.00 (0.5, 0.5) x =0.5,

0.3≤y

≤0.7. CS

100.00 (0.5, 0.5) x =0.5,

0.4≤y

≤0.6

0.4≤ x ≤0.6,

y=0.5

a/b=1, a/h=100, / /a b =1, h/Rxx= -h/Ryy=1/300; E11/E22 = 25, G23 = 0.2E22,

G13 = G12 = 0.5E22, 12 =21 =0.25.

Mode shapes The mode shapes corresponding to the fundamental modes of vibration are plotted in Figs. 6-9 for crossply and angle ply shell of CCCC and CSCC shells for different eccentric position of the cutout. All the mode shapes are bending mode. It is found that for different position of cutout mode shapes are somewhat similar to one another, only the crest and trough position changes.

Conclusion The following conclusions may be drawn from the present study: 1. As this approach produces results in close agreement with those of the benchmark problems the finite element code used here is suitable for analyzing free vibration problems of stiffened spherical panels with cutouts. 2. The arrangement of boundary constraints along the four edges is far more important than their actual number so far the free vibration is concerned. 3. If cross ply fully clamped shell is released for any functional reason, then opposite edges must be released to avoid excessive loss in frequency. 4. The relative free vibration performances of shells for different combinations of edge conditions along the four sides, are expected to be very useful in decision-making for practicing engineers.

5. The information regarding the behaviour of stiffened spherical shell with eccentric cutouts for a wide spectrum of eccentricity and boundary conditions for cross ply and angle ply shells may also be used as design aids for structural engineers. 6. From this study we get the specific zones within which the cutout centre may be moved so that the loss of frequency is less than 10% with respect to a shell with a central cutout. That will help an engineer to make a decision regarding the eccentricity of the cutout centre that he can allow. NOTATIONS a ,b length and width of shell in plan

/ /,a b length and width of cutout in plan

bst width of stiffener in general bsx, bsy width of X and Y stiffeners respectively dst depth of stiffener in general dsx, dsy depth of X and Y stiffeners respectively {de} element displacement E11, E22 elastic moduli G12, G13, G23 shear moduli of a lamina with respect to 1, 2 and 3 axes of fibre h shell thickness Mx, My moment resultants Mxy torsion resultant np number of plies in a laminate N1-N8 shape functions Nx,, Ny inplane force resultants Nxy inplane shear resultant Qx, Qy transverse shear resultant Ry ,Rx radii of curvature of shell u, v, w translational degrees of freedom x, y, z local co-ordinate axes X, Y, Z global co-ordinate axes zk distance of bottom of the kth ply from mid-surface of a laminate , rotational degrees of freedom x, y inplane strain component xy ,xz, yz shearing strain components 12, 21 Poisson’s ratios , , isoparametric co-ordinates density of material x, y inplane stress components xy, xz, yz shearing stress components natural frequency

non-dimensional natural frequency

2/1222

2 / hEa

121

Fig. 4: First mode shapes of laminated composite (0/90/0/90) stiffened saddle shell for different sizes of the central square cutout and boundary conditions.

Fig. 5: First mode shapes of laminated composite (+45/-45/+45/-45) stiffened saddle shell for different sizes of the central square cutout and boundary conditions.

122

Fig. 6: First mode shapes of laminated composite (0/90/0/90) stiffened saddle shell for different positions of the square cutout with CCCC boundary condition.

Fig. 7: First mode shapes of laminated composite (0/90/0/90) stiffened saddle shell for different positions of the square cutout with CSCC boundary condition.

123

Fig. 8: First mode shapes of laminated composite (+45/-45/+45/-45) stiffened saddle shell for different positions of the square cutout with CCCC boundary condition.

Fig. 9: First mode shapes of laminated composite (+45/-45/+45/-45) stiffened saddle shell for different positions of the square cutout with CSCC boundary condition.

124

References 1. R. K. Kapania, A review on the analysis of laminated shells, J.

Pressure Vessel Technol., vol.111(2), 1989, pp.88-96. 2. A. K. Noor, W. S. Burton, Assessment of computational

models for multi-layered composite shells, Appl. Mech. Rev., vol.43 1990, pp.67-97.

3. J. N. Reddy, Finite element modeling of layered, anisotropic composite plates and shells: review of recent research. Shock and Vibration Digest, vol.13(12) 1981, pp. 3-12.

4. W. C. Chao, J. N. Reddy, Analysis of laminated composite shells using a degenerated 3-d element, Int. J. Numer. Methods in Engg., vol.20, 1984, pp.1991-2007.

5. N. Reddy, K. Chandrashekhara, Geometrically nonlinear transient analysis of laminated doubly curved shells, Int. J. Nonlinear Mech., vol.20(2) 1985,pp.79-90.

6. C. C. Chao, T. P. Tung, Step pressure and blast response of clamped orthotropic hemispherical shells, Int. J. Impact Engg., vol.8(3), 1989, pp.191-207.

7. M. S. Qatu, Natural frequencies for cantilevered doubly-curved laminated composite shallow shells, Composite Structures, vol.17, 1991, pp.227-255.

8. K. M. Liew, C. W. Lim, vibration of perforated doubly-curved shallow shells with rounded corners, Int. J of Solids and Structures, vol.31(11), 1994, pp.1519-1536.

9. K. M. Liew, C. W. Lim, Vibration behaviour of doubly-curved shallow shells of curvilinear planform., ASCE Journal of Engineering Mechanics, vol.121(12), 1995, pp.1277-1283.

10. D. Chakravorty, P. K. Sinha, J. N. Bandyopadhyay, Free vibration analysis of point supported laminated composite doubly curved shells- a finite element approach, Computers and Structures, vol.54(2), 1995,pp.191-207.

11. D. Chakravorty, P. K. Sinha, J. N. Bandyopadhyay, Finite element free vibration analysis of doubly curved laminated composite shells, J. Sound and Vibration, vol.191(4), 1996, pp.491-504.

12. D. Chakravorty, P. K. Sinha, J. N. Bandyopadhyay, Applications of FEM on free and forced vibration of laminated shells. ASCE Journal of Engineering Mechanics vol.124(1),1998,pp. 1-8.

13. D. K. Shin, Large amplitude free vibration behavior of doubly curved shallow open shells with simply supported edges, Computers and Structures, vol.62(1), 1997, pp.35-49.

14. D. Y. Tan, Free vibration analysis of shells of revolution, J. of Sound and vibration, vol.213(1), 1998, pp.15-33.

15. T. Kant, S. Kumar, U. P. Singh, Shell dynamics with three dimensional finite elements, Computers and Structures, vol.50(1),1994, pp.135-146.

16. M. Sathyamoorthy, Nonlinear vibrations of moderately thick orthotropic shallow spherical shells, Computers and Structures, vol.57(1), 1995, pp.59-65.

17. B. P. Gautham, N. Ganesan, Free vibration characteristics of isotropic and laminated orthotropic spherical caps, J. of Sound and Vibration, vol.204(1), 1997,pp.17-40.

18. C. Y. Chia, D. S. Chia, Nonlinear vibration of moderately thick anti-symmetric angle-ply shallow spherical shell, Computers and Structures, vol.44(4), 1997, pp.797-805.

19. B. Sivasubramonian, A. M. Kulkarni, G. Venkateswara Rao, A. Krishnan, Free vibration of curved panels with cutouts, Journal of Sound and Vibration, vol.200(2),1997, pp.227-234.

20. M. S. Qatu, R. W. Sullivan, W. Wang, Recent research advances on the dynamic analysis of composite shells:2000-2009, Composite Structures, vol.93,2010, pp.14-31.

21. X. Wang, G. Lu, S. R. Guillow, Stress wave propagation in orthotropic laminated thick-walled spherical shells, Int. J. Solids struct., vol.39,2002, pp.3027-4037.

22. J. Lellep, H. Hein, Optimization of clamped plastic shallow shells subjected to initial impulsive loading, Engg. Optimization, vol.34, 2002, pp.545-556.

23. H.L. Dai, X. Wang, Stress wave propagation in laminated piezoelectric spherical shells under thermal shock and electric excitation, Eur J Mech A/Solids, vol.24, 2005, pp.263-276.

24. M. Ganapathi, Dynamic stability characteristics of functionally graded materials shallow spherical shells, Composite Structures, vol.79, 2007, pp.338-343.

25. T. Park, S. Y. Lee, Parametric instability osf delaminated composite spherical shells subjected to in-plane pulsating forces. Composite Structures, vol.91, 2009, pp.196-204.

26. V. V. Vasiliev, R. M. Jones, and L. L. Man, Mechanics of Composite Structures, Taylor and Francis, USA, 1993.

27. M. S. Qatu, Vibration of Laminated Shells and Plates, Elsevier, UK, 2004.

28. A. Dey, J.N. Bandyopadhyay, P.K. Sinha, Finite element analysis of laminated composite paraboloid of revolution shells. Computers and Structures vol.44 (3),1992, pp.675-682.

29. S. Sahoo, D. Chakravorty, Finite element vibration characteristics of composite hypar shallow shells with various edge supports. Journal of Vibration and Control vol.11(10),2005, pp.1291-1309.

30. A. Mukherjee, M. Mukhopadhyay, Finite element free vibration of eccentrically stiffened plates. Computers and Structures vol.30,1988;pp.1303-1317.

31. A.N. Nayak, J.N. Bandyopadhyay, On the free vibration of stiffened shallow shells. Journal of Sound and Vibration vol.255(2),2002, pp.357-382.