- Email: [email protected]
FINITE ELEMENT FREE VIBRATION ANALYSIS OF DOUBLY CURVED LAMINATED COMPOSITE SHELLS
Journal of Sound and Vibration (1996) 191(4), 491–504
FINITE ELEMENT FREE VIBRATION ANALYSIS OF DOUBLY CURVED LAMINATED COMPOSITE SHELLS D. C, J. N. B Department of Civil Engineering, Indian Institute of Technology, Kharagpur-721302, India
P. K. S Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721302, India (Received 2 September 1993, and in final form 8 February 1995) A finite element analysis for the free vibration behaviour of doubly curved shells is presented in which eight-noded curved quadrilateral isoparametric finite elements are used. The first order shear deformation theory for thin and shallow shells is used in the formulation. Results are obtained for comparison with those in the existing literature and to investigate the effects of various composite parameters relevant to doubly curved shells, such as fibre orientations and lamination schemes and several geometrical parameters like aspect ratio, smaller height to greater height ratio (for conoids), thickness to radius ratio (for hyperbolic and elliptic paraboloids), and radii of curvature ratio (for elliptic paraboloids). 7 1996 Academic Press Limited
1. INTRODUCTION
Composite shell structures, are extensively used in aerospace, civil, marine and other engineering applications. In civil engineering construction the conoidal (see Figure 1), hyperbolic paraboloidal (see Figure 2) (among the anticlastic) and the elliptic paraboloidal (see Figure 3) (among the synclastic) shells are used as roofing units to cover large column-free areas. Conoidal shells which are of ruled surfaces provide ease of fabrication and also allow north light to come in and hence are preferred in many situations. The hyperbolic paraboloidal shells are aesthetically appealing although they offer less stiffness than other doubly curved shells. The elliptic paraboloidal shells are both architecturally acceptable and structurally stiff due to their surface geometry. It is obvious that an in depth study of doubly curved composite shell behaviour is needed to exploit the fullest potential of these curved forms. There are several aspects of shell behaviour such as bending, buckling, vibration, impact etc. which are important. The present investigation is, however, restricted only to the free vibration behaviour of doubly curved composite shell structures. During the last decade there has been an active interest among some researchers in the dynamic behaviour of doubly curved composite shells. Chao and Reddy [1] presented numerical results for the non-linear bending, free vibration and transient response of laminated composite spherical shells with different boundary conditions. They used a three-dimensional element incorporating the geometrical non-linearity. Reddy [2] 491 0022–460X/96/140491 + 14 $18.00/0
7 1996 Academic Press Limited
492
. .
Figure 1. Conoidal shell.
proposed exact solutions for the bending and free vibration of moderately thick simply supported spherical shells. The free vibration characteristics of anisotropic laminated doubly curved shells, determined by using nine-noded isoparametric quadratic elements, were presented by Chandrashekhara [3]. He obtained results for spherical shells. Qatu and Leissa [4] investigated the free vibration behaviour of completely free doubly curved laminated composite hyperbolic paraboloidal and spherical shells. They used the Ritz method in conjunction with an algebraic polynomial displacement function. In another paper [5] they used the same approach to analyze free vibrations of cantilevered doubly curved composite shallow shells. The effects of various parameters upon the frequencies of saddle, cylindrical and spherical shells were studied. The non-linear vibration of moderately thick antisymmetric angle-ply shallow spherical shells with simply supported boundaries was studied by Chia and Chia [6]. The solution was formulated in the form of generalized double Fourier series with time dependent coefficients to satisfy the boundary conditions. Sheinman and Reichman [7] studied the
Figure 2. Hyperbolic paraboloidal shell.
493
Figure 3. Elliptic paraboloidal shell.
buckling and vibration of laminated shallow curved panels. The equations in terms of the transverse displacement and Airy stress function were derived and solved by the Ritz method, by using the eigenfunctions of an isotropic beam. Although in the formulation two principal radii of curvature were considered, the authors solved only the problems related to simply supported cylindrical shells. This review of the existing literature clearly reveals that the information available for free vibration behaviour of all common forms of both anticlastic and synclastic shells is far from complete. The present work is therefore aimed at investigating free vibration characteristics of several laminated composite shell forms like conoids, and hyperbolic and elliptic paraboloids by employing the finite element method based on an eight-noded isoparametric doubly curved shell element. 2. GOVERNING EQUATIONS
We consider a doubly curved laminated composite shell element of uniform thickness h and principal radii of curvature Rxx and Ryy (see Figure 4). Each of the thin laminae
Figure 4. Laminated composite doubly curved shell.
. .
494
can be oriented at an arbitrary angle u with reference to the x-axis. The constitutive equations for the shell are given by (Definitions of symbols used in this paper are given in the Appendix) {F} = [D]{e},
(1)
where {F} = {Nx , Ny , Nxy , Mx , My , Mxy , Qx , Qy }T, 0 0 , kx , ky , kxy , gxz , gyz0 }T, {e} = {ex0 , ey0 , gxy
KA11 GA12 GA16 G B [D] =G 11 G GB12 GB16 G0 k0
A12 A22
A16 A26
B11 B12
B12 B22
B16 B26
0 0
A26 B12 B22
A66 B16 B26
B16 D11 D12
B26 D12 D22
B66 D16 D26
0 0 0
B26 0 0
B66 0 0
D16 0 0
D26 0 0
D66 0 0 S44 0 S45
L G 0G G 0G . 0G G 0G S45G S55l 0 0
(2)
The force resultants are expressed as {F} = [Nx , Ny , Nxy , Mx , My , Mxy , Qx , Qy ]T =
$g
h/2
(sx , sy , txy , sx z , sy z , txy z , txz , tyz ) dz
−h/2
%
T
(3)
The elements of the stiffness matrix [D] are defined as n
Aij = s (Qij )k (zk − zk − 1 ),
n
Bij = 12 s (Qij )k (zk2 − zk2 − 1 ),
k=1
k=1
n
Dij = 13 s (Qij )k (zk3 − zk3 − 1 ),
i, j = 1, 2, 6,
k=1
and n
Sij = s Fi Fj (Gij )k (zk − zk − 1 ),
i, j = 4, 5,
(4)
k=1
where Qij are elements of the off-axis elastic constant matrix which is given by [Qij )off = [T1 ]−1 [Qij ]on [T1 ]−T , [Gij ]off = [T2 ]−1 [Qij ]on [T2 ],
&
m2 [T1 ] = n 2 −mn
'
n2 2mn m 2 −2mn , mn m 2 − n 2
i, j = 1, 2, 6,
(5)
i, j = 4, 5,
(6)
[T2 ] =
$
m n
%
−n , m
495
in which m = cos u and n = sin u,
&
'
Q11 Q12 0 [Qij ]on = Q12 Q22 0 , 0 0 Q66 [Qij ]on =
$
Q44 0
%
0 , Q55
i, j = 1, 2, 6,
i, j = 4, 5,
in which Q11 = (1 − n12 n21 )−1E11 , Q12 = (1 − n12 n21 )−1E11 n21 ,
Q22 = (1 − n12 n21 )−1E22 ,
Q66 = G12 ,
Q44 = G13 ,
Q55 = G23 .
Fi and Fj of equation (4) are two factors presently taken as unity when the shell is thin. When the shell is moderately thick the product of Fi and Fj is taken as 5/6, which is commonly used since the evaluation of shear correction factors from exact theory of elasticity is difficult. The strain–displacement relations are as follows: 0 0 [ex , ey , gxy , gxz , gyz ]T = [ex0 , ey0 , gxy , gxz , gyz0 ]T + z[kx , ky , kxy , kxz , kyz ]T,
(7)
where
F ex0 J F 1u/1x − w/Rxx J G ey0 G G G 1v/1y − w/R yy G G G G 0 ggxyh=g1u/1y + 1v/1x − 2w/Rxyh, Ggxz0 G G G a + 1w/1x G G G G fgyz0 j f b + 1w/1y j
F kx J F 1a/1x J G ky G G G 1b/1y G G G G gkxyh=g1a/1y + 1b/1xh. GkxzG G G 0 G G G G fkyzj f 0 j
(8)
3. FINITE ELEMENT FORMULATION
An eight-noded doubly curved isoparametric finite element is used for the present formulation. Five-degrees-of-freedom are considered at each node, which include three displacements u, v, w and two rotations a and b. The element stiffness and mass matrices are derived by using the principle of minimum potential energy. In the isoparametric T 1 2 Non-dimensional fundamental frequencies [v¯ = {rvn2 (1 − n 2 )Rxx }/E ] of diaphragmsupported isotropic hyperbolic and elliptic paraboloids (a/b = 1)
Rxx /Ryy
Leissa and Kadi [9] (exact)
Present FEM
Hyperbolic paraboloid
−1·5 −1·0 −0·5
0·23514 0·03505 0·23886
0·23445 0·03504 0·23881
Elliptic paraboloid
0·5 1·0 1·5
0·55346 0·94808 1·01470
0·55322 0·94796 1·01442
Shell type
h/Rxx = 0·001, a/Rxx = 0·4, n = 0·3 A 6 × 6 mesh is used.
. .
496
T 2 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(Ezz h 2 )}1/2 ] for simply supported laminated composite spherical shells (a/b = 1)
h/Rxx 1 300 1 400 1 500 1 1000
Plate
0°/90° ZXXXXXCXXXXXV Reddy [2] Present FEM 46·002 35·228 28·825 16·706 9·6873
0°/90°/0° ZXXXXXCXXXXXV Reddy [2] Present FEM
45·801 35·126 28·778 16·706 9·6893
47·265 36·971 30·993 20·347 15·183
47·035 36·890 30·963 20·356 15·192
Rxx /Ryy = 1, a/h = 100, E11 /E22 = 25, G12 = G13 = 0·5E22 , G23 = 0·2E22 , n12 = 0·25. A 6 × 6 mesh is used.
formulation the displacements and rotations are expressed in terms of their nodal values by the element shape functions as follows: 8
8
u = s Ni ui ,
8
v = s Ni vi ,
i=1
w = s Ni wi ,
i=1 8
a = s Ni ai ,
i=1 8
b = s Ni b i .
i=1
(9)
i=1
The shape functions are Ni = 14 (1 + jji )(1 + hhi )(jji + hhi − 1)
for i = 1, 2, 3, 4,
Ni = 12 (1 + jji )(1 − h 2 )
for i = 5, 7,
Ni = 12 (1 + hhi )(1 − j 2 )
for i = 6, 8.
(10)
T 3 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported laminated composite shells (a/b = 1)
Laminations 0°/90°/0° (0°/90°)s 0°/45°/0° (0°/45°)s 0°/90° (0°/90°)2 +45°/−45° (+45°/−45°)2
Shell type ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV Hyperbolic Elliptic paraboloid paraboloid ZXXXXXXXXXCXXXXXXXXXV Conoid Rxx /Ryy = −1 Rxx /Ryy = 0·5 Rxx /Ryy = 1·0 Rxx /Ryy = 1·5 66·297 66·797 73·575 75·008 53·864 64·267 53·117 70·058
15·118 15·120 15·335 15·756 9·643 13·956 15·174 19·393
25·345 25·400 33·468 36·634 22·557 24·726 37·985 50·421
For conoid, a/hh = 2·5, hl/hh = 0·15, a/h = 100. For others, h/Rxx = 1/500, a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
30·963 31·050 41·748 45·535 28·778 30·506 49·289 58·940
36·865 36·987 49·705 53·784 35·114 36·537 52·756 61·703
497
The strain–displacement relation is derived as {e} = [B]{d},
(11)
where {d} = [u1 , v1 , w1 , a1 , b1 , . . . , u8 , v8 , a8 , a8 , b8 ]T and 0 L KNi,x 0 −Ni /Rxx 0 G 0 Ni,y −Ni /Ryy 0 0 G GN N −2N /R 0 0 G i,y i,x i xy G G 8 0 0 0 Ni,x 0 G G [B] = s . 0 0 0 Ni,yG i = 1G 0 G0 0 0 Ni,y Ni,xG G G 0 Ni,x Ni 0 G G0 k0 0 Ni,y 0 Ni l
(12)
The element stiffness matrix is then given by [Ke ] =
gg
[B]T[D][B] dx dy,
(13)
[N]T[P][N] dx dy,
(14)
The element mass matrix is obtained as [Me ] =
gg
T 4 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported laminated composite shells having different aspect ratios Shell type
a/b
0·5
1
2
Conoid
0°/90° 0°/90°/0°
22·905 32·931
53·804 66·297
91·704 107·834
Hyperbolic paraboloid (Rxx /Ryy = −1)
0°/90° 0°/90°/0°
9·664 15·059
9·683 15·118
33·027 59·381
Elliptic Paraboloid (Rxx /Ryy = 0·5)
0°/90° 0°/90°/0°
20·744 20·853
22·557 25·345
39·581 63·127
Elliptic paraboloid (Rxx /Ryy = 1)
0°/90° 0°/90°/0°
28·756 30·889
28·778 30·963
42·016 64·612
Elliptic paraboloid (Rxx /Ryy = 1·5)
0°/90° 0°/90°/0°
35·009 36·637
35·114 36·865
44·535 66·194
For conoid a/hh = 2·5, hl/hh = 0·15,, a/h = 100. For others h/Rxx = 1/500, a/h = 100. Mesh: conoid 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
. .
498 where
KNi 0 0 0 0 L G 0 Ni 0 0 0 G 8 [N] = sG 0 0 Ni 0 0 G G G i=1 G 0 0 0 Ni 0 G k 0 0 0 0 Ni l and
KP G0 [P] =G G0 G0 k0
0L 0G 0G G
0 0 0 P 0 0 0 P 0 0 0
0 0
0G Il
I 0
with n
p= s k=1
g
zk
r dz
n
and I = s k=1
zk − 1
g
zk
z 2r dz
(15)
zk − 1
The element stiffness and mass matrices are first evaluated by expressing the integrals in the local natural co-ordinates j and h of the element and then performing numerical integration by using the Gaussian quadrature. The element matrices are then assembled to obtain the global [K] and [M] matrices after appropriate transformation to account for the curved nature of the shell surface [8].
T 5 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] of simply supported and clamped laminated composite shells (a/b = 1) Lamination
Simply supported
Clamped
Conoid
Shell type
0°/90° 0°/90°/0°
53·864 66·297
106·845 137·721
Hyperbolic paraboloid (Rxx /Ryy = −1)
0°/90° 0°/90°/0°
9·643 15·118
71·265 79·841
Elliptic paraboloid (Rxx /Ryy = 0·5)
0°/90° 0°/90°/0°
22·557 25·345
54·589 56·000
Elliptic paraboloid (Rxx /Ryy = 1)
0°/90° 0°/90°/0°
28·778 30·963
74·149 81·409
Elliptic paraboloid (Rxx /Ryy = 1·5)
0°/90° 0°/90°/0°
35·114 36·865
75·441 103·137
For conoid a/hh = 2·5, hl/hh = 0·15, a/h = 100. For others h/Rxx = 1/00, a/h = 100. Mesh: conoid, 8 × 8, hyperbolic and elliptic paraboloid, 6 × 6.
499
The free vibration analysis involves determination of natural frequencies from the condition =[K] − vn2 [M]= = 0.
(16)
This is a generalized eigenvalue problem and is solved by using the subspace iteration algorithm.
4. NUMERICAL RESULTS AND DISCUSSION
The accuracy of the present finite element formulation was first established by comparing the results with those in the literature. The full shell is discretized to allow for any combination of symmetric and/or antisymmetric modes of vibration along the two directions.
T 6 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported laminated composites shells (a/b = 1) with different hl/hh ratios ( for conoid ) and h/Rxx ratios ( for hyperbolic and elliptic paraboloids) Shell type Conoid
Stacking sequence hl/hh ratio 0·25 0·20 0·15 0·10 0·05 0·00
0°/90° 55·082 54·469 53·864 51·175 45·359 39·437
0°/90°/0° 78·086 72·477 66·297 59·424 51·872 44·111
h/Rxx ratio 1/300 1/400 1/500 1/1000 10
0°/90° 9·550 9·614 9·643 9·682 9·6893
0°/90°/0° 14·981 15·075 15·118 15·175 15·192
Elliptic paraboloid (Rxx /Ryy = 0·5)
1/300 1/400 1/500 1/1000 10
35·099 27·196 22·557 14·079 9·6893
36·884 29·519 25·345 18·283 15·192
Elliptic paraboloid (Rxx /Ryy = 1)
1/500 1/400 1/500 1/1000 10
45·801 35·126 28·778 16·706 9·6893
47·035 36·890 30·963 20·356 15·192
Elliptic paraboloid (Rxx /Ryy = 1·5)
1/300 1/400 1/500 1/1000 10
56·206 43·076 35·114 19·963 9·6893
57·119 44·440 36·865 22·739 15·192
Hyperbolic paraboloid (Rxx /Ryy = −1)
For conoid a/hh = 2·5, a/h = 100. For others a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
. .
500
Non-dimensional fundamental frequencies (v¯ ) of isotropic hyperbolic paraboloids and elliptic paraboloids with different Rxx /Ryy ratios are presented in Table 1 along with exact results obtained by Leissa and Kadi [9]. Table 2 compares the present results with those of Reddy [2] for antisymmetrically and symmetrically laminated spherical shells. In both cases the present results exhibit good agreement with others. Numerical results were then obtained for several composite doubly curved thin and moderately thick shells with varied fibre orientation, lamina stacking sequences, shell aspect ratio, smaller height to greater height ratio (for conoids), thickness to radius ratio (for hyperbolic and elliptic paraboloids) and radii of curvature ratio (for elliptic paraboloid). The shell thickness h is assumed to be the same for all thin shell problems, although the stacking sequence may vary from one problem to the other. The full shells are also discretized for these problems. The lamina properties used are E11 = 25E22 ,
G12 = G13 = 0·5E22 ,
G23 = 0·2E22
and
n12 = 0·25.
Non-dimensional fundamental frequencies (v¯ ) for various thin shell forms (a/b = 1) having simply supported boundaries are presented in Table 3. It is observed that v¯ increases with the increase in the number of layers in the symmetrically laminated configurations. In the case of antisymmetric laminates this increase is quite significant in many cases. The effects of the aspect ratio on v¯ of simply supported composite thin shells are given in Table 4. In all cases v¯ increases with the increase in the aspect ratio (a/b). The v¯ values for symmetrical laminates are observed to be higher than those for antisymmetric laminates for all aspect ratios. The v¯ values of simply supported and clamped antisymmetrically and symmetrically laminated composite thin shells are compared in Table 5. In all cases the expected increase in stiffness with the increase of support constraints is observed. As usual the symmetrically laminated shells are stiffer than the antisymmetrically laminated ones. Table 6 contains the results for simply supported laminated composite thin shells for different hl/hh ratios (for conoids) and h/Rxx ratios (for hyperbolic and elliptic paraboloids). The results indicate that a truncated conoid is stiffer than a full conoid
T 7 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/E22 h 2 )}1/2 ] for simply supported laminated composite moderately thick hyperbolic and elliptic paraboloid shells for different h/Rxx ratios h/Rxx ratio
0°/90°
0°/90°/0°
Hyperbolic paraboloid (Rxx /Ryy = −1)
1/5 1/10 1/20 1/100 1/1000
6·802 8·592 9·126 9·050 9·007
9·714 12·668 13·209 12·606 12·530
Elliptic paraboloid (Rxx /Ryy = 1)
1/5 1/10 1/20 1/100 1/1000
20·956 14·877 11·148 9·116 9·009
21·198 17·634 14·793 12·681 12·531
a/h = 10. Mesh: hyperboloid paraboloid and elliptic paraboloid, 6 × 6.
501
(hl/hh = 0). The frequency increases steadily but at a slower rate at the higher values of hl/hh ratio. It is further observed that for hyperbolic paraboloids the v¯ value increases with the decrease of the h/Rxx ratio but this increase is quite insignificant. It is also noted that this shell form does not provide stiffness additional to that of a flat construction. The reverse trends are observed for elliptic paraboloids where v¯ decreases with the decrease of the h/Rxx ratio, indicating that the shell stiffness increases with the increase in the shell curvature. The values of the non-dimensional fundamental frequencies v¯ for moderately thick hyperbolic and elliptic paraboloid shells are given in Table 7, corresponding to different h/Rxx ratios. For both the shell forms the symmetrically laminated ones give higher frequencies than the antisymmetrically laminated ones. The v¯ value of hyperbolic paraboloids initially increases with decrease of the h/Rxx ratio and then again decreases with decrease of the above ratio. It therefore seems that a suitable h/Rxx ratio can be chosen to achieve maximum vibrational rigidity. For elliptic paraboloids, however, the frequency always decreases with decrease of the h/Rxx ratio. It is further to be noted that for a particular pair of stacking sequence and h/Rxx ratio elliptic paraboloids give a higher frequency than hyperbolic parabolids and this difference decreases as the h/Rxx ratio decreases because both the shell forms approach a plate configuration. Table 8 provides further information on the effects of fibre orientation and number of layers on the fundamental frequencies of antisymmetrically laminated composite thin shells. It is observed that the fibre orientation considerably influences the fundamental
T 8 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported antisymmetric (0/u)n laminated composite shells (a/b = 1) Shell type
u
n=1
n=2
n=3
n=4
n = 10
Conoid
30° 45° 60° 90°
59·711 56·020 53·890 53·864
70·617 72·352 71·321 64·267
71·769 73·659 72·526 65·145
72·290 74·204 73·007 65·552
72·680 74·576 73·293 65·874
Hyperbolic paraboloid (Rxx /Ryy = −1)
30° 45° 60° 90°
13·007 11·697 10·843 9·643
15·597 15·612 15·091 13·956
15·939 16·155 15·679 14·555
16·140 16·425 15·963 14·836
16·287 16·644 16·197 15·071
Elliptic paraboloid (Rxx /Ryy = 0·5)
30° 45° 60° 90°
29·776 35·680 32·264 22·557
32·115 37·696 34·053 24·726
32·503 37·901 34·183 25·073
32·691 37·999 34·243 25·238
32·891 38·012 34·251 25·380
Elliptic paraboloid (Rxx /Ryy = 1)
30° 45° 60° 90°
38·345 42·470 40·493 28·778
40·527 47·161 42·412 30·506
40·850 47·322 42·520 30·787
40·996 47·392 42·567 30·921
41·122 47·402 42·583 31·037
Elliptic paraboloid (Rxx /Ryy = 1·5)
30° 45° 60° 90°
43·881 44·327 46·169 35·114
48·961 56·079 50·518 36·537
49·204 56·212 50·625 36·772
49·303 56·263 50·669 36·884
49·334 56·271 50·708 36·981
For conoid a/hh = 2·5, hl/hh = 0·15, a/h = 100. For others h/Rxx = 1/500, a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
. .
502
T 9 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported shells (a/b = 1) with (0/u/0/u) and (0/u/u/0) lamination schemes Shell type
u
0/u/0/u
0/u/u/0
Conoid
0° 30° 45° 60° 90°
59·085 70·617 72·352 71·321 64·267
59·085 72·984 75·008 73·985 66·797
Hyperbolic paraboloid (Rxx /Ryy = −1)
0° 30° 45° 60° 90°
15·124 15·597 15·612 15·091 13·956
15·124 15·614 15·756 15·580 15·116
Elliptic paraboloid (Rxx /Ryy = 0·5)
0° 30° 45° 60° 90°
22·965 32·115 37·696 34·053 24·726
22·965 32·160 36·634 33·149 25·400
Elliptic paraboloid (Rxx /Ryy = 1)
0° 30° 45° 60° 90°
27·481 40·527 47·161 42·412 30·506
27·481 40·169 45·535 41·233 31·050
Elliptic paraboloid (Rxx /Ryy = 1·5)
0° 30° 45° 60° 90°
32·296 48·961 56·079 50·518 36·537
32·296 48·103 53·784 49·060 36·987
For conoid a/hh = 2·5, hl/hh = 0·15, a/h = 100. For other h/Rxx = 1/500, a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
frequency. The stiffness is found to increase with the increase in the number of layers. However, the increase is insignificant when the number of antisymmetric units (0/u) exceeds four. Table 9 contain the v¯ values of laminated composite thin shells with four-layered antisymmetric and symmetric stacking sequences for different fibre orientations. It is observed that in all the cases v¯ increases as u increases from 0° to 45° and decreases as u further increases beyond 45° to 90°. Thus u = 45° appears to provide the maximum dynamic stiffness for the assumed four-layered antisymmetrically and symmetrically laminated composite shells. From the results presented in Tables 3–8 it is evident that the fundamental frequency and hence the vibrational rigidity of a thin elliptic paraboloid shell always increases with an increase in the Rxx /Ryy ratio. 5. CONCLUSIONS
A finite element analysis has been carried out to study the free vibration characteristics of doubly curved laminated thin and moderately thick composite shells. The fundamental
503
frequency is observed to increase with the increase in the number of layers for both antisymmetric and symmetric laminations in the case of thin shells. It is further noticed that such an increase for antisymmetric laminations is insignificant when the number of antisymmetric units (0/u) exceeds four. The dynamic stiffness of thin shells also increases with the increase of aspect ratio and increase in boundary constraints. It is also inferred that a thin truncated conoid is preferred to a full one from the vibrational rigidity point of view. The results also indicate that thin hyperbolic paraboloids, plates and elliptic paraboloids provide stiffness in an ascending order of magnitude. Moreover, for a thin shell the fibre orientation u = 45° gives the highest natural frequency for four-layered antisymmetric (0/u/0/u) and symmetric (0/u/u/0) laminations. Further, the fundamental frequency of a thin elliptic paraboloid increases with the increase of the Rxx /Ryy ratio. For moderately thick hyperbolic and elliptic paraboloid shells the latter offers greater stiffness for a particular lamination and h/Rxx ratio. However, this difference decreases for low values of the h/Rxx ratio. For a hyperbolic paraboloid an optimum h/Rxx ratio may be chosen for obtaining the highest fundamental frequency while for an elliptic paraboloid h/Rxx must be kept as high as possible in view of the other design criteria to achieve maximum dynamic stiffness.
REFERENCES 1. W. C. C and J. N. R 1984 International Journal for Numerical Methods in Engineering 20, 1991–2007. Analysis of laminated composite shells using a degenerated 3-D element. 2. J. N. R 1984 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 110, 794–809. Exact solutions of moderately thick laminated shells. 3. K. C 1989 Computers and Structures 33, 435–440. Free vibrations of anisotropic laminated doubly curved shells. 4. M. S. Q and A. W. L 1991 Journal of Sound and Vibration 151, 9–29. Free vibrations of completely free doubly curved laminated composite shallow shells. 5. M. S. Q and A. W. L 1991 Composite Structures 17, 227–256. Natural frequencies for cantilevered doubly-curved laminated composite shallow shells. 6. C. Y. C and D. S. C 1992 Computers and Structures 44, 797–805. Nonlinear vibration of moderately thick antisymmetric angle-ply shallow spherical shells. 7. I S and Y R 1992 International Journal of Solids and Structures 29, 1329–1338. A study of buckling and vibration of laminated shallow curved panels. 8. R. D. C 1981 Concepts and Applications of Finite Element Analysis. New York: John Wiley. 9. A. W. L and A. S. K 1971 Journal of Sound and Vibration 116, 173–187. Curvature effects on shallow shell vibrations.
APPENDIX: NOTATION
a, b E E11 , E22 FEM G12 , G13 , G23 h hh hl kx , ky , kxy kxz , kyz
7
Mx , My , Mxy
length and width of shell Young’s modulus of isotropic shell Young’s modulii of a lamina along and transverse to the fibres finite element method shear modulii of a lamina shell thickness greater height conoid smaller height of conoid curvatures of shell internal moment resultants per unit length
504 Ni Ni,x , Ni,y Nx , Ny , Nxy Qx , Qy Rxx , Ryy , Rxy u, v, w ui , vi , wi x, y, z X, Y, Z zk − 1 , zk a, b a i , bi 0 ex0 , ey0 , gxy 0 0 gxz , gyz u n n12 , n21 j, h v¯ vn
. . shape functions at node i derivatives of shapes functions with respect to local x- and y-axes, respectively internal in-plane force resultants per unit length transverse shear resultants per unit length radii of curvatures of shell displacements of the mid-plane along x-, y- and z-axes displacements at a node i along the x-, y- and z-axes local co-ordinate axes global co-ordinate axes bottom and top distances of a lamina from the mid-plane rotations of theshell element along x- and y-axes rotations at node i along x- and y-axes in-plane strains of the mid-plane transverse strains of the mid-plane fibre orientation Poisson’s ratio of isotropic shell Poisson’s ratios of an orthotropic lamina local natural coordinates of an element non-dimensional natural frequency natural frequency
FINITE ELEMENT FREE VIBRATION ANALYSIS OF DOUBLY CURVED LAMINATED COMPOSITE SHELLS D. C, J. N. B Department of Civil Engineering, Indian Institute of Technology, Kharagpur-721302, India
P. K. S Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721302, India (Received 2 September 1993, and in final form 8 February 1995) A finite element analysis for the free vibration behaviour of doubly curved shells is presented in which eight-noded curved quadrilateral isoparametric finite elements are used. The first order shear deformation theory for thin and shallow shells is used in the formulation. Results are obtained for comparison with those in the existing literature and to investigate the effects of various composite parameters relevant to doubly curved shells, such as fibre orientations and lamination schemes and several geometrical parameters like aspect ratio, smaller height to greater height ratio (for conoids), thickness to radius ratio (for hyperbolic and elliptic paraboloids), and radii of curvature ratio (for elliptic paraboloids). 7 1996 Academic Press Limited
1. INTRODUCTION
Composite shell structures, are extensively used in aerospace, civil, marine and other engineering applications. In civil engineering construction the conoidal (see Figure 1), hyperbolic paraboloidal (see Figure 2) (among the anticlastic) and the elliptic paraboloidal (see Figure 3) (among the synclastic) shells are used as roofing units to cover large column-free areas. Conoidal shells which are of ruled surfaces provide ease of fabrication and also allow north light to come in and hence are preferred in many situations. The hyperbolic paraboloidal shells are aesthetically appealing although they offer less stiffness than other doubly curved shells. The elliptic paraboloidal shells are both architecturally acceptable and structurally stiff due to their surface geometry. It is obvious that an in depth study of doubly curved composite shell behaviour is needed to exploit the fullest potential of these curved forms. There are several aspects of shell behaviour such as bending, buckling, vibration, impact etc. which are important. The present investigation is, however, restricted only to the free vibration behaviour of doubly curved composite shell structures. During the last decade there has been an active interest among some researchers in the dynamic behaviour of doubly curved composite shells. Chao and Reddy [1] presented numerical results for the non-linear bending, free vibration and transient response of laminated composite spherical shells with different boundary conditions. They used a three-dimensional element incorporating the geometrical non-linearity. Reddy [2] 491 0022–460X/96/140491 + 14 $18.00/0
7 1996 Academic Press Limited
492
. .
Figure 1. Conoidal shell.
proposed exact solutions for the bending and free vibration of moderately thick simply supported spherical shells. The free vibration characteristics of anisotropic laminated doubly curved shells, determined by using nine-noded isoparametric quadratic elements, were presented by Chandrashekhara [3]. He obtained results for spherical shells. Qatu and Leissa [4] investigated the free vibration behaviour of completely free doubly curved laminated composite hyperbolic paraboloidal and spherical shells. They used the Ritz method in conjunction with an algebraic polynomial displacement function. In another paper [5] they used the same approach to analyze free vibrations of cantilevered doubly curved composite shallow shells. The effects of various parameters upon the frequencies of saddle, cylindrical and spherical shells were studied. The non-linear vibration of moderately thick antisymmetric angle-ply shallow spherical shells with simply supported boundaries was studied by Chia and Chia [6]. The solution was formulated in the form of generalized double Fourier series with time dependent coefficients to satisfy the boundary conditions. Sheinman and Reichman [7] studied the
Figure 2. Hyperbolic paraboloidal shell.
493
Figure 3. Elliptic paraboloidal shell.
buckling and vibration of laminated shallow curved panels. The equations in terms of the transverse displacement and Airy stress function were derived and solved by the Ritz method, by using the eigenfunctions of an isotropic beam. Although in the formulation two principal radii of curvature were considered, the authors solved only the problems related to simply supported cylindrical shells. This review of the existing literature clearly reveals that the information available for free vibration behaviour of all common forms of both anticlastic and synclastic shells is far from complete. The present work is therefore aimed at investigating free vibration characteristics of several laminated composite shell forms like conoids, and hyperbolic and elliptic paraboloids by employing the finite element method based on an eight-noded isoparametric doubly curved shell element. 2. GOVERNING EQUATIONS
We consider a doubly curved laminated composite shell element of uniform thickness h and principal radii of curvature Rxx and Ryy (see Figure 4). Each of the thin laminae
Figure 4. Laminated composite doubly curved shell.
. .
494
can be oriented at an arbitrary angle u with reference to the x-axis. The constitutive equations for the shell are given by (Definitions of symbols used in this paper are given in the Appendix) {F} = [D]{e},
(1)
where {F} = {Nx , Ny , Nxy , Mx , My , Mxy , Qx , Qy }T, 0 0 , kx , ky , kxy , gxz , gyz0 }T, {e} = {ex0 , ey0 , gxy
KA11 GA12 GA16 G B [D] =G 11 G GB12 GB16 G0 k0
A12 A22
A16 A26
B11 B12
B12 B22
B16 B26
0 0
A26 B12 B22
A66 B16 B26
B16 D11 D12
B26 D12 D22
B66 D16 D26
0 0 0
B26 0 0
B66 0 0
D16 0 0
D26 0 0
D66 0 0 S44 0 S45
L G 0G G 0G . 0G G 0G S45G S55l 0 0
(2)
The force resultants are expressed as {F} = [Nx , Ny , Nxy , Mx , My , Mxy , Qx , Qy ]T =
$g
h/2
(sx , sy , txy , sx z , sy z , txy z , txz , tyz ) dz
−h/2
%
T
(3)
The elements of the stiffness matrix [D] are defined as n
Aij = s (Qij )k (zk − zk − 1 ),
n
Bij = 12 s (Qij )k (zk2 − zk2 − 1 ),
k=1
k=1
n
Dij = 13 s (Qij )k (zk3 − zk3 − 1 ),
i, j = 1, 2, 6,
k=1
and n
Sij = s Fi Fj (Gij )k (zk − zk − 1 ),
i, j = 4, 5,
(4)
k=1
where Qij are elements of the off-axis elastic constant matrix which is given by [Qij )off = [T1 ]−1 [Qij ]on [T1 ]−T , [Gij ]off = [T2 ]−1 [Qij ]on [T2 ],
&
m2 [T1 ] = n 2 −mn
'
n2 2mn m 2 −2mn , mn m 2 − n 2
i, j = 1, 2, 6,
(5)
i, j = 4, 5,
(6)
[T2 ] =
$
m n
%
−n , m
495
in which m = cos u and n = sin u,
&
'
Q11 Q12 0 [Qij ]on = Q12 Q22 0 , 0 0 Q66 [Qij ]on =
$
Q44 0
%
0 , Q55
i, j = 1, 2, 6,
i, j = 4, 5,
in which Q11 = (1 − n12 n21 )−1E11 , Q12 = (1 − n12 n21 )−1E11 n21 ,
Q22 = (1 − n12 n21 )−1E22 ,
Q66 = G12 ,
Q44 = G13 ,
Q55 = G23 .
Fi and Fj of equation (4) are two factors presently taken as unity when the shell is thin. When the shell is moderately thick the product of Fi and Fj is taken as 5/6, which is commonly used since the evaluation of shear correction factors from exact theory of elasticity is difficult. The strain–displacement relations are as follows: 0 0 [ex , ey , gxy , gxz , gyz ]T = [ex0 , ey0 , gxy , gxz , gyz0 ]T + z[kx , ky , kxy , kxz , kyz ]T,
(7)
where
F ex0 J F 1u/1x − w/Rxx J G ey0 G G G 1v/1y − w/R yy G G G G 0 ggxyh=g1u/1y + 1v/1x − 2w/Rxyh, Ggxz0 G G G a + 1w/1x G G G G fgyz0 j f b + 1w/1y j
F kx J F 1a/1x J G ky G G G 1b/1y G G G G gkxyh=g1a/1y + 1b/1xh. GkxzG G G 0 G G G G fkyzj f 0 j
(8)
3. FINITE ELEMENT FORMULATION
An eight-noded doubly curved isoparametric finite element is used for the present formulation. Five-degrees-of-freedom are considered at each node, which include three displacements u, v, w and two rotations a and b. The element stiffness and mass matrices are derived by using the principle of minimum potential energy. In the isoparametric T 1 2 Non-dimensional fundamental frequencies [v¯ = {rvn2 (1 − n 2 )Rxx }/E ] of diaphragmsupported isotropic hyperbolic and elliptic paraboloids (a/b = 1)
Rxx /Ryy
Leissa and Kadi [9] (exact)
Present FEM
Hyperbolic paraboloid
−1·5 −1·0 −0·5
0·23514 0·03505 0·23886
0·23445 0·03504 0·23881
Elliptic paraboloid
0·5 1·0 1·5
0·55346 0·94808 1·01470
0·55322 0·94796 1·01442
Shell type
h/Rxx = 0·001, a/Rxx = 0·4, n = 0·3 A 6 × 6 mesh is used.
. .
496
T 2 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(Ezz h 2 )}1/2 ] for simply supported laminated composite spherical shells (a/b = 1)
h/Rxx 1 300 1 400 1 500 1 1000
Plate
0°/90° ZXXXXXCXXXXXV Reddy [2] Present FEM 46·002 35·228 28·825 16·706 9·6873
0°/90°/0° ZXXXXXCXXXXXV Reddy [2] Present FEM
45·801 35·126 28·778 16·706 9·6893
47·265 36·971 30·993 20·347 15·183
47·035 36·890 30·963 20·356 15·192
Rxx /Ryy = 1, a/h = 100, E11 /E22 = 25, G12 = G13 = 0·5E22 , G23 = 0·2E22 , n12 = 0·25. A 6 × 6 mesh is used.
formulation the displacements and rotations are expressed in terms of their nodal values by the element shape functions as follows: 8
8
u = s Ni ui ,
8
v = s Ni vi ,
i=1
w = s Ni wi ,
i=1 8
a = s Ni ai ,
i=1 8
b = s Ni b i .
i=1
(9)
i=1
The shape functions are Ni = 14 (1 + jji )(1 + hhi )(jji + hhi − 1)
for i = 1, 2, 3, 4,
Ni = 12 (1 + jji )(1 − h 2 )
for i = 5, 7,
Ni = 12 (1 + hhi )(1 − j 2 )
for i = 6, 8.
(10)
T 3 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported laminated composite shells (a/b = 1)
Laminations 0°/90°/0° (0°/90°)s 0°/45°/0° (0°/45°)s 0°/90° (0°/90°)2 +45°/−45° (+45°/−45°)2
Shell type ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV Hyperbolic Elliptic paraboloid paraboloid ZXXXXXXXXXCXXXXXXXXXV Conoid Rxx /Ryy = −1 Rxx /Ryy = 0·5 Rxx /Ryy = 1·0 Rxx /Ryy = 1·5 66·297 66·797 73·575 75·008 53·864 64·267 53·117 70·058
15·118 15·120 15·335 15·756 9·643 13·956 15·174 19·393
25·345 25·400 33·468 36·634 22·557 24·726 37·985 50·421
For conoid, a/hh = 2·5, hl/hh = 0·15, a/h = 100. For others, h/Rxx = 1/500, a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
30·963 31·050 41·748 45·535 28·778 30·506 49·289 58·940
36·865 36·987 49·705 53·784 35·114 36·537 52·756 61·703
497
The strain–displacement relation is derived as {e} = [B]{d},
(11)
where {d} = [u1 , v1 , w1 , a1 , b1 , . . . , u8 , v8 , a8 , a8 , b8 ]T and 0 L KNi,x 0 −Ni /Rxx 0 G 0 Ni,y −Ni /Ryy 0 0 G GN N −2N /R 0 0 G i,y i,x i xy G G 8 0 0 0 Ni,x 0 G G [B] = s . 0 0 0 Ni,yG i = 1G 0 G0 0 0 Ni,y Ni,xG G G 0 Ni,x Ni 0 G G0 k0 0 Ni,y 0 Ni l
(12)
The element stiffness matrix is then given by [Ke ] =
gg
[B]T[D][B] dx dy,
(13)
[N]T[P][N] dx dy,
(14)
The element mass matrix is obtained as [Me ] =
gg
T 4 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported laminated composite shells having different aspect ratios Shell type
a/b
0·5
1
2
Conoid
0°/90° 0°/90°/0°
22·905 32·931
53·804 66·297
91·704 107·834
Hyperbolic paraboloid (Rxx /Ryy = −1)
0°/90° 0°/90°/0°
9·664 15·059
9·683 15·118
33·027 59·381
Elliptic Paraboloid (Rxx /Ryy = 0·5)
0°/90° 0°/90°/0°
20·744 20·853
22·557 25·345
39·581 63·127
Elliptic paraboloid (Rxx /Ryy = 1)
0°/90° 0°/90°/0°
28·756 30·889
28·778 30·963
42·016 64·612
Elliptic paraboloid (Rxx /Ryy = 1·5)
0°/90° 0°/90°/0°
35·009 36·637
35·114 36·865
44·535 66·194
For conoid a/hh = 2·5, hl/hh = 0·15,, a/h = 100. For others h/Rxx = 1/500, a/h = 100. Mesh: conoid 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
. .
498 where
KNi 0 0 0 0 L G 0 Ni 0 0 0 G 8 [N] = sG 0 0 Ni 0 0 G G G i=1 G 0 0 0 Ni 0 G k 0 0 0 0 Ni l and
KP G0 [P] =G G0 G0 k0
0L 0G 0G G
0 0 0 P 0 0 0 P 0 0 0
0 0
0G Il
I 0
with n
p= s k=1
g
zk
r dz
n
and I = s k=1
zk − 1
g
zk
z 2r dz
(15)
zk − 1
The element stiffness and mass matrices are first evaluated by expressing the integrals in the local natural co-ordinates j and h of the element and then performing numerical integration by using the Gaussian quadrature. The element matrices are then assembled to obtain the global [K] and [M] matrices after appropriate transformation to account for the curved nature of the shell surface [8].
T 5 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] of simply supported and clamped laminated composite shells (a/b = 1) Lamination
Simply supported
Clamped
Conoid
Shell type
0°/90° 0°/90°/0°
53·864 66·297
106·845 137·721
Hyperbolic paraboloid (Rxx /Ryy = −1)
0°/90° 0°/90°/0°
9·643 15·118
71·265 79·841
Elliptic paraboloid (Rxx /Ryy = 0·5)
0°/90° 0°/90°/0°
22·557 25·345
54·589 56·000
Elliptic paraboloid (Rxx /Ryy = 1)
0°/90° 0°/90°/0°
28·778 30·963
74·149 81·409
Elliptic paraboloid (Rxx /Ryy = 1·5)
0°/90° 0°/90°/0°
35·114 36·865
75·441 103·137
For conoid a/hh = 2·5, hl/hh = 0·15, a/h = 100. For others h/Rxx = 1/00, a/h = 100. Mesh: conoid, 8 × 8, hyperbolic and elliptic paraboloid, 6 × 6.
499
The free vibration analysis involves determination of natural frequencies from the condition =[K] − vn2 [M]= = 0.
(16)
This is a generalized eigenvalue problem and is solved by using the subspace iteration algorithm.
4. NUMERICAL RESULTS AND DISCUSSION
The accuracy of the present finite element formulation was first established by comparing the results with those in the literature. The full shell is discretized to allow for any combination of symmetric and/or antisymmetric modes of vibration along the two directions.
T 6 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported laminated composites shells (a/b = 1) with different hl/hh ratios ( for conoid ) and h/Rxx ratios ( for hyperbolic and elliptic paraboloids) Shell type Conoid
Stacking sequence hl/hh ratio 0·25 0·20 0·15 0·10 0·05 0·00
0°/90° 55·082 54·469 53·864 51·175 45·359 39·437
0°/90°/0° 78·086 72·477 66·297 59·424 51·872 44·111
h/Rxx ratio 1/300 1/400 1/500 1/1000 10
0°/90° 9·550 9·614 9·643 9·682 9·6893
0°/90°/0° 14·981 15·075 15·118 15·175 15·192
Elliptic paraboloid (Rxx /Ryy = 0·5)
1/300 1/400 1/500 1/1000 10
35·099 27·196 22·557 14·079 9·6893
36·884 29·519 25·345 18·283 15·192
Elliptic paraboloid (Rxx /Ryy = 1)
1/500 1/400 1/500 1/1000 10
45·801 35·126 28·778 16·706 9·6893
47·035 36·890 30·963 20·356 15·192
Elliptic paraboloid (Rxx /Ryy = 1·5)
1/300 1/400 1/500 1/1000 10
56·206 43·076 35·114 19·963 9·6893
57·119 44·440 36·865 22·739 15·192
Hyperbolic paraboloid (Rxx /Ryy = −1)
For conoid a/hh = 2·5, a/h = 100. For others a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
. .
500
Non-dimensional fundamental frequencies (v¯ ) of isotropic hyperbolic paraboloids and elliptic paraboloids with different Rxx /Ryy ratios are presented in Table 1 along with exact results obtained by Leissa and Kadi [9]. Table 2 compares the present results with those of Reddy [2] for antisymmetrically and symmetrically laminated spherical shells. In both cases the present results exhibit good agreement with others. Numerical results were then obtained for several composite doubly curved thin and moderately thick shells with varied fibre orientation, lamina stacking sequences, shell aspect ratio, smaller height to greater height ratio (for conoids), thickness to radius ratio (for hyperbolic and elliptic paraboloids) and radii of curvature ratio (for elliptic paraboloid). The shell thickness h is assumed to be the same for all thin shell problems, although the stacking sequence may vary from one problem to the other. The full shells are also discretized for these problems. The lamina properties used are E11 = 25E22 ,
G12 = G13 = 0·5E22 ,
G23 = 0·2E22
and
n12 = 0·25.
Non-dimensional fundamental frequencies (v¯ ) for various thin shell forms (a/b = 1) having simply supported boundaries are presented in Table 3. It is observed that v¯ increases with the increase in the number of layers in the symmetrically laminated configurations. In the case of antisymmetric laminates this increase is quite significant in many cases. The effects of the aspect ratio on v¯ of simply supported composite thin shells are given in Table 4. In all cases v¯ increases with the increase in the aspect ratio (a/b). The v¯ values for symmetrical laminates are observed to be higher than those for antisymmetric laminates for all aspect ratios. The v¯ values of simply supported and clamped antisymmetrically and symmetrically laminated composite thin shells are compared in Table 5. In all cases the expected increase in stiffness with the increase of support constraints is observed. As usual the symmetrically laminated shells are stiffer than the antisymmetrically laminated ones. Table 6 contains the results for simply supported laminated composite thin shells for different hl/hh ratios (for conoids) and h/Rxx ratios (for hyperbolic and elliptic paraboloids). The results indicate that a truncated conoid is stiffer than a full conoid
T 7 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/E22 h 2 )}1/2 ] for simply supported laminated composite moderately thick hyperbolic and elliptic paraboloid shells for different h/Rxx ratios h/Rxx ratio
0°/90°
0°/90°/0°
Hyperbolic paraboloid (Rxx /Ryy = −1)
1/5 1/10 1/20 1/100 1/1000
6·802 8·592 9·126 9·050 9·007
9·714 12·668 13·209 12·606 12·530
Elliptic paraboloid (Rxx /Ryy = 1)
1/5 1/10 1/20 1/100 1/1000
20·956 14·877 11·148 9·116 9·009
21·198 17·634 14·793 12·681 12·531
a/h = 10. Mesh: hyperboloid paraboloid and elliptic paraboloid, 6 × 6.
501
(hl/hh = 0). The frequency increases steadily but at a slower rate at the higher values of hl/hh ratio. It is further observed that for hyperbolic paraboloids the v¯ value increases with the decrease of the h/Rxx ratio but this increase is quite insignificant. It is also noted that this shell form does not provide stiffness additional to that of a flat construction. The reverse trends are observed for elliptic paraboloids where v¯ decreases with the decrease of the h/Rxx ratio, indicating that the shell stiffness increases with the increase in the shell curvature. The values of the non-dimensional fundamental frequencies v¯ for moderately thick hyperbolic and elliptic paraboloid shells are given in Table 7, corresponding to different h/Rxx ratios. For both the shell forms the symmetrically laminated ones give higher frequencies than the antisymmetrically laminated ones. The v¯ value of hyperbolic paraboloids initially increases with decrease of the h/Rxx ratio and then again decreases with decrease of the above ratio. It therefore seems that a suitable h/Rxx ratio can be chosen to achieve maximum vibrational rigidity. For elliptic paraboloids, however, the frequency always decreases with decrease of the h/Rxx ratio. It is further to be noted that for a particular pair of stacking sequence and h/Rxx ratio elliptic paraboloids give a higher frequency than hyperbolic parabolids and this difference decreases as the h/Rxx ratio decreases because both the shell forms approach a plate configuration. Table 8 provides further information on the effects of fibre orientation and number of layers on the fundamental frequencies of antisymmetrically laminated composite thin shells. It is observed that the fibre orientation considerably influences the fundamental
T 8 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported antisymmetric (0/u)n laminated composite shells (a/b = 1) Shell type
u
n=1
n=2
n=3
n=4
n = 10
Conoid
30° 45° 60° 90°
59·711 56·020 53·890 53·864
70·617 72·352 71·321 64·267
71·769 73·659 72·526 65·145
72·290 74·204 73·007 65·552
72·680 74·576 73·293 65·874
Hyperbolic paraboloid (Rxx /Ryy = −1)
30° 45° 60° 90°
13·007 11·697 10·843 9·643
15·597 15·612 15·091 13·956
15·939 16·155 15·679 14·555
16·140 16·425 15·963 14·836
16·287 16·644 16·197 15·071
Elliptic paraboloid (Rxx /Ryy = 0·5)
30° 45° 60° 90°
29·776 35·680 32·264 22·557
32·115 37·696 34·053 24·726
32·503 37·901 34·183 25·073
32·691 37·999 34·243 25·238
32·891 38·012 34·251 25·380
Elliptic paraboloid (Rxx /Ryy = 1)
30° 45° 60° 90°
38·345 42·470 40·493 28·778
40·527 47·161 42·412 30·506
40·850 47·322 42·520 30·787
40·996 47·392 42·567 30·921
41·122 47·402 42·583 31·037
Elliptic paraboloid (Rxx /Ryy = 1·5)
30° 45° 60° 90°
43·881 44·327 46·169 35·114
48·961 56·079 50·518 36·537
49·204 56·212 50·625 36·772
49·303 56·263 50·669 36·884
49·334 56·271 50·708 36·981
For conoid a/hh = 2·5, hl/hh = 0·15, a/h = 100. For others h/Rxx = 1/500, a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
. .
502
T 9 Non-dimensional fundamental frequencies [v¯ = vn a 2{r/(E22 h 2 )}1/2 ] for simply supported shells (a/b = 1) with (0/u/0/u) and (0/u/u/0) lamination schemes Shell type
u
0/u/0/u
0/u/u/0
Conoid
0° 30° 45° 60° 90°
59·085 70·617 72·352 71·321 64·267
59·085 72·984 75·008 73·985 66·797
Hyperbolic paraboloid (Rxx /Ryy = −1)
0° 30° 45° 60° 90°
15·124 15·597 15·612 15·091 13·956
15·124 15·614 15·756 15·580 15·116
Elliptic paraboloid (Rxx /Ryy = 0·5)
0° 30° 45° 60° 90°
22·965 32·115 37·696 34·053 24·726
22·965 32·160 36·634 33·149 25·400
Elliptic paraboloid (Rxx /Ryy = 1)
0° 30° 45° 60° 90°
27·481 40·527 47·161 42·412 30·506
27·481 40·169 45·535 41·233 31·050
Elliptic paraboloid (Rxx /Ryy = 1·5)
0° 30° 45° 60° 90°
32·296 48·961 56·079 50·518 36·537
32·296 48·103 53·784 49·060 36·987
For conoid a/hh = 2·5, hl/hh = 0·15, a/h = 100. For other h/Rxx = 1/500, a/h = 100. Mesh: conoid, 8 × 8; hyperbolic and elliptic paraboloid, 6 × 6.
frequency. The stiffness is found to increase with the increase in the number of layers. However, the increase is insignificant when the number of antisymmetric units (0/u) exceeds four. Table 9 contain the v¯ values of laminated composite thin shells with four-layered antisymmetric and symmetric stacking sequences for different fibre orientations. It is observed that in all the cases v¯ increases as u increases from 0° to 45° and decreases as u further increases beyond 45° to 90°. Thus u = 45° appears to provide the maximum dynamic stiffness for the assumed four-layered antisymmetrically and symmetrically laminated composite shells. From the results presented in Tables 3–8 it is evident that the fundamental frequency and hence the vibrational rigidity of a thin elliptic paraboloid shell always increases with an increase in the Rxx /Ryy ratio. 5. CONCLUSIONS
A finite element analysis has been carried out to study the free vibration characteristics of doubly curved laminated thin and moderately thick composite shells. The fundamental
503
frequency is observed to increase with the increase in the number of layers for both antisymmetric and symmetric laminations in the case of thin shells. It is further noticed that such an increase for antisymmetric laminations is insignificant when the number of antisymmetric units (0/u) exceeds four. The dynamic stiffness of thin shells also increases with the increase of aspect ratio and increase in boundary constraints. It is also inferred that a thin truncated conoid is preferred to a full one from the vibrational rigidity point of view. The results also indicate that thin hyperbolic paraboloids, plates and elliptic paraboloids provide stiffness in an ascending order of magnitude. Moreover, for a thin shell the fibre orientation u = 45° gives the highest natural frequency for four-layered antisymmetric (0/u/0/u) and symmetric (0/u/u/0) laminations. Further, the fundamental frequency of a thin elliptic paraboloid increases with the increase of the Rxx /Ryy ratio. For moderately thick hyperbolic and elliptic paraboloid shells the latter offers greater stiffness for a particular lamination and h/Rxx ratio. However, this difference decreases for low values of the h/Rxx ratio. For a hyperbolic paraboloid an optimum h/Rxx ratio may be chosen for obtaining the highest fundamental frequency while for an elliptic paraboloid h/Rxx must be kept as high as possible in view of the other design criteria to achieve maximum dynamic stiffness.
REFERENCES 1. W. C. C and J. N. R 1984 International Journal for Numerical Methods in Engineering 20, 1991–2007. Analysis of laminated composite shells using a degenerated 3-D element. 2. J. N. R 1984 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 110, 794–809. Exact solutions of moderately thick laminated shells. 3. K. C 1989 Computers and Structures 33, 435–440. Free vibrations of anisotropic laminated doubly curved shells. 4. M. S. Q and A. W. L 1991 Journal of Sound and Vibration 151, 9–29. Free vibrations of completely free doubly curved laminated composite shallow shells. 5. M. S. Q and A. W. L 1991 Composite Structures 17, 227–256. Natural frequencies for cantilevered doubly-curved laminated composite shallow shells. 6. C. Y. C and D. S. C 1992 Computers and Structures 44, 797–805. Nonlinear vibration of moderately thick antisymmetric angle-ply shallow spherical shells. 7. I S and Y R 1992 International Journal of Solids and Structures 29, 1329–1338. A study of buckling and vibration of laminated shallow curved panels. 8. R. D. C 1981 Concepts and Applications of Finite Element Analysis. New York: John Wiley. 9. A. W. L and A. S. K 1971 Journal of Sound and Vibration 116, 173–187. Curvature effects on shallow shell vibrations.
APPENDIX: NOTATION
a, b E E11 , E22 FEM G12 , G13 , G23 h hh hl kx , ky , kxy kxz , kyz
7
Mx , My , Mxy
length and width of shell Young’s modulus of isotropic shell Young’s modulii of a lamina along and transverse to the fibres finite element method shear modulii of a lamina shell thickness greater height conoid smaller height of conoid curvatures of shell internal moment resultants per unit length
504 Ni Ni,x , Ni,y Nx , Ny , Nxy Qx , Qy Rxx , Ryy , Rxy u, v, w ui , vi , wi x, y, z X, Y, Z zk − 1 , zk a, b a i , bi 0 ex0 , ey0 , gxy 0 0 gxz , gyz u n n12 , n21 j, h v¯ vn
. . shape functions at node i derivatives of shapes functions with respect to local x- and y-axes, respectively internal in-plane force resultants per unit length transverse shear resultants per unit length radii of curvatures of shell displacements of the mid-plane along x-, y- and z-axes displacements at a node i along the x-, y- and z-axes local co-ordinate axes global co-ordinate axes bottom and top distances of a lamina from the mid-plane rotations of theshell element along x- and y-axes rotations at node i along x- and y-axes in-plane strains of the mid-plane transverse strains of the mid-plane fibre orientation Poisson’s ratio of isotropic shell Poisson’s ratios of an orthotropic lamina local natural coordinates of an element non-dimensional natural frequency natural frequency