Free vibrations of multilayered thick composite shells

Computers & Structures Vol. 28, No. 6, pp. 717-722, 1988 Printed in Great Britain.

0045-7949/88 $3.00+0.00 Pergamon Press plc

FREE VIBRATIONS OF MULTILAYERED THICK COMPOSITE SHELLS R. RAMESH KUMAR~" and Y. V. K. SADASIVAR A O ~ tStructural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum-22, India :~Stage Engineering and Analysis Division, Liquid Propellant Systems Unit, Valiamala, Trivandrum, India

(Received 8 April 1987) Abstract--Free vibration characteristics of a multi layered graphite/epoxy circular cylindrical shell are presented using an eight-noded isoparametric quadrilateral shear flexible shell element. The effects of rotary inertia are considered in the formulation of the mass matrix. Variations of length to radius and radius to thickness ratios of the shell, and different fibre orientations such as (0/90°), (0/90/0/90°), (0/90°)s and (0/90/0/90°)s with respect to the frequency parameter, K~, are studied. The influence of the ratio of major to minor Young's moduli on frequency parameter is also shown. Comparison of numerical results with those available in the literature shows a good agreement.

NOTATION

hk i IJ I ksh K~, L Ni m n

thickness of kth layer ith node determinant of Jacobian matrix transverse shear correction factor normalised frequency length of the shell shape function number of half-waves in the x-direction number of half-waves in the circumferential direction ri total number of layers in the laminate R mean radius of the shell t thickness of the shell V3 thickness vector eigenvector 2 eigenvalue ~o angular frequency, rad/sec # direction cosines a x, ay, oz direct stresses rxy inplane shear stress rxz, ryz transverse shear stresses uc, vc, wc displacements at the mid-surface x~, Yc, zc mid-surface co-ordinates Z L2/Rt

l. I N T R O D U C T I O N

With the rapid development in the use of fibre reinforced composite plate and shell type structures in the aerospace industry, the need for an accurate assessment of its behaviour under static and dynamic situations still persists. Considering the dynamic behaviour of composite shells, it is observed that a few studies have been reported in the literature[I-4]. These studies have been based on either a Donnel type theory or a refined Kirchhoff-Love type theory of motion. However, in the case of multilayered composite shells where the degree of anisotropy between each adjacent layer becomes large, one has to seek 717

new approaches. Sadasiva Rao and Raju [5, 6] formulated an analytical solution for the free vibration of two-, four- and eight-layered composite cylindrical shells neglecting [5] and considering transverse shear and rotary inertia. It has been found that the effects of transverse shear and rotary inertia are predominant in the case of two-layered shells and should be included for R / t <~20. Recently Sheinman and Grief [7] presented numerical solutions for laminated thin shells of revolution, neglecting transverse shear effects based on Hamilton's variational principle. In the present work free vibration characteristics of multilayered graphite/epoxy shells of varying L / R and R / t ratios are presented using a degenerated eight-noded isoparametric quadrilateral shear flexible shell element [8, 9].

2. FINITE ELEMENT F O R M U L A T I O N

A typical degenerated eight-noded isoparametric shell element is shown in Fig. l(a). The nodal degrees of freedom are three displacements u, v, w along the global Cartesian coordinates x, y, z and two rotations and fl of the thickness vectors V2 and V~ respectively [Fig. l(b)]. As the detailed form of element description and derivation of the element stiffness matrix are available in the literature [8, 10], only a brief presentation of finite element formulation for free vibration analysis of a multilayered composite shell is made in this work. The element geometry in terms of mid-surface nodal coordinates is defined as

=

N~ y~ +N~-r~,. iffi I

lZ~J

2

(1)

718

R. RAMESHKUMARand Y. V. K. SADASlV^RAO

~ . ~ f ~i

~ ~i 7 z,tO[

/

(a)

zl

v,

=- Y,1)i

X~U

Fig. I. Details of element. The total displacements at any point due to translation and rotation are given by

N,

=

J=J

,(Lu],~

v~ +

/~ .

(2)

i=t

(wcJ

Following the plane stress assumption, the stress-strain relationship in the x,y, z directions is obtained as

As the elasticity matrix [D] is different from layer to layer and not a continuous function of ~, the sum of the elasticity matrix of an element through thickness is obtained by using the thickness concept [11]. This is achieved by suitably modifying the variable to ~k in any kth layer such that ~k varies from -- I to + 1 in that layer. The relationship between ~ and (k is given by

(= -1+ {a}

=

[Dl{e},

1t -h~(l -

(7)

.=

(3)

d( = (hJt ) d~k.

(8)

where

{~} = [B]{~}

Thus eqn (6) takes the form

and [K] = ~= l a -1 a -I f ~ l [B° + ~B*]r[D]

[D] = [T]rIQ]IT].

(4) hk x [Bo+~B*]IJ[- 7 d~ dq d~k.

[D], [B] and [T] are the usual matrices relating stress-strain, strain-displacement and global-local transformation of coordinates, [Q] is given by

The mass matrix of the element with the above layers is given by

Qn = E l/(I -- vl2v21 ) Q22 =

(9)

l

E2/(I -- v12v21 )

= %E2/(I

-

[MI =

%%)

[Nlr[Pl kffilJ-I

1

t

Q4, = GI2 Q55 =

× [ N I I J [ ~ d~ d~/d~k

k,h G23

Q~ = 631 ksh,

(10)

(5) Table 1. Convergence study

where: ksh is the transverse shear correction factor which is taken as 5/6; E, G and v are the material constants; 1 denotes the direction parallel to the fibre, and 2 and 3 the other two transverse directions. The element stiffness matrix [K] is given by [K] = ~ [Bo + (B*Ir[D][Bo + ~B*] dv.

j,

(6)

First mode (m = 1, n = 2) mesh size

Natural frequency in cycles/sec, four layered shell (0/90°),

4x4 8x4 16 × 4

0.6661 0.5791 0.5668

E I/E2=40,

Z = 1000, R / t

G,2/E2=0.5, = I00.

G23/E2=0.416,

%=0.25,

Free vibrations of multilayered thick composite shells

719

Table 2. Natural frequencies of a clamped steel circular cylindrical shell (cycles/sec) Mode m n 1 1 2 2 3 3 3

1 2 2 3 2 3 4

Present result

Flugge's shell equation [13]

FEM [13]

Leissa [12]

3449 2269 4962 3688 5960 3709 5095

3427 1918 3905 2538 5844 4054 2921

3422 1920 3730 2541 5292 3875 2917

---4365 -4350 4921, 5072

R / t = 300, L / R = 4, t = 0.01 in., p = 0.7333 x 10 -3 lb secZ/in4.

8

=

Z

In t h e case o f t h e m a s s m a t r i x full i n t e g r a t i o n (3 x 3) is a d o p t e d .

iffil

x

N

O

O

/Z~l

O

N

O

N ~1~21

O

O

N

F o r the case o f free v i b r a t i o n , p r o b l e m is given by

N t

t

([K] - t o [ n ] ) ~

(11)

N ~1~22

the eigenvalue

= 2~.

(13)

S i m u l t a n e o u s iterative t e c h n i q u e is followed for the e x t r a c t i o n o f t h e eigenvalues.

N t

3. N U M E R I C A L R E S U L T S A N D D I S C U S S I O N

[p] = p,[I].

(12)

Pk is t h e m a s s d e n s i t y p e r u n i t t h i c k n e s s f o r t h e k t h layer. T h e e v a l u a t i o n o f t h e stiffness m a t r i x is c a r r i e d o u t n u m e r i c a l l y f o l l o w i n g t h e 3 x 3 G a u s s rule for ax, try a n d zxy while t h e 2 x 2 i n t e g r a t i o n is for zyz a n d Z~x.

3.1 Shell modelling In t h e p r e s e n t study, a full c i r c u l a r cylindrical shell is m o d e l l e d w i t h eight e l e m e n t s a l o n g its c i r c u m ferences (at a n i n t e r v a l o f 45 ° p e r e l e m e n t ) a n d f o u r e l e m e n t s a l o n g its length. T h i s c o n c l u s i o n is a r r i v e d at f r o m a c o n v e r g e n c e s t u d y o f t h e f u n d a m e n t a l

Table 3. Normalisedfrequency, K~formultilayeredcircular cylindrical shell

R/t

m

Mode n

2 (0/90 °)

4 (90/90/0/90 ° )

4 (0/90°~

8 (0/90/0/90°)s

5

1 1 2

1 0 3

2.178 3.234 4.095

2.265 3.259 4.096 4.981 5.542 6.335

2.298 3.253 4.096 4.635 5.542 6.350

1

2

--

2 2

1 2

4.940 4.815

2.329 3.246 4.096 4.234 5.544 6.227

10

1 I 1 2 2

1 2 0 2 I

4.490 4.881 6.596 8.312 9.685

4.625 5.740 6.609 9.656 10.280

4.588 6.104 6.616 9.406 9.840

4.607 6.151 6.613 9.557 10.083

20

I 1 I 2 2 2

2 I 0 2 3 1

8.841 9.395 13.350 15.751 16.972 19.374

9.158 9.909 13.357 16.815 16.976 21.056

9.238 10.219 13.360 16.378 16.972 19.552

9.246 10.066 13.360 16.611 16.976 19.712

50

1 1 1 2 2 I 2 2

I 2 0 0 2 3 1 3

23.203 23.406 -33.621 38.953 46.192 48.743 54.323

23.314 23.708 33.628 -39.586 47.399 49.084 55.713

23.308 23.848 33.630 -39.361 48.196 48.788 55.371

23.312 23.779 33.628 -39.586 47.786 49.176 55.917

L/R = 4

E I/E 2

=

40, GI2/E2 = 0.5, G23/E2 = 0.416, vl2 = 0.25.

720

R. RAMF.Srl K u M ~ and Y. V. K. SADASlVA RAO Table 4. Normaliscd frequency, K~, for multilayered circularcylindricalshell L/R = 2

R/t 5

10

Mode m n 1

1

1

2

1

0

1 2 2

3 3 2

L/R = 8

Mode m n

(0/90 °)

(0/90/0/90 °)

(0/90 °)

(0/90/0/90 °)

1.133 1.477 1.611 2.055 2.500 2.578

1.265 1.514 1.614 2.575 2.835 2.867

l 1 2 2 3 1 2

l 0 l 0 1 2 2

5.813 6.569 9.275 12.063 13.603 14.523 18.231

6.041 6.651 10.004 12.136 14.960 15.799 19.589

I

1

1 1 2 2

2 3 3 2

1.818 2.275 3.052 4.172 4.305

2.392 3.096 3.606 4.207 5.153

1 2 1 2 3 3

1 1 2 2 1 2

8.491 18.918 19.793 21.035 27.752 28.463

8.717 19.570 20.277 24.575 28.906 31.889

20

1 1 1 1 2 2

2 1 3 0 2 3

3.510 4.606 6.222 6.672 7.456 8.054

3.711 4.697 6.267 6.676 8.531 9.109

l l l 2 2 3

l 0 2 ! 2 2

17.376 26.705 26.763 38.425 40.465 54.692

17.594 26=793 30.276 38.979 42.594 57.040

50

1 1 I 2 2 2

2 1 3 0 2 3

8.764 11.643 12.561 16.804 17.484 18.569

8.889 11.714 12.885 16.804 17.579 19.096

1 1 2 2 3

1 2 1 2 1

44.007 64.807 97.197 100.794 114.771

44.280 66.329 97.653 102.044 143.555

E j / E 2 = 40, G,2/E 2 = 0.5, G23/E2

=

0.416, v,2 = 0.25.

frequency o f a cylindrical shell taking four, eight a n d 16 elements a l o n g the circumferential direction as s h o w n in Table 1.

E 2 = 0.75 × 106 psi v]2 = 0.25

3.2 Numerical e x a m p l e (isotropic shell) In o r d e r to validate the present finite element analysis, the free v i b r a t i o n o f a clamped steel circular cylindrical shell for which analytical [12, 13] a n d numerical [13] results are available in the literature is studied. A c o m p a r i s o n o f the results with the available ones presented in Table 2 indicates a good a g r e e m e n t for b o t h f u n d a m e n t a l a n d higher frequencies, thus proving the usability o f the element.

Gl2 = 0.375 x 106 psi G23/E2 = 0.4 16. z,w

6

3.3 M u l t i l a y e r e d circular cylindrical shells Circular cylindrical shells with two, four a n d eight layers are analysed using the present element with fibre o r i e n t a t i o n s o f (0/90°), (0/90/0/90°), (0/90°)s a n d (0/90/0/90°)s, a n d results are presented in Tables 3 a n d 4 a n d Figs 2-5. The R / t a n d L / R ratios taken for the study are as follows:

Z W 0

R / t = 5, 10, 20 a n d 50 L / R = 2, 4 and 8

I

t = unity.

E l = 30 × 106psi

-0.13

ksh -0.65 0

The material properties for the graphite/epoxy laminate considered in this work are given by

RAdU (WITHOUT TRANSVERSE EFFECTS) G~/Ez [ 2

I 3

I 4

Tt

Fig. 2. Comparison of frequency parameter, Ko, for twolayered circular cylindrical shell with modal number (m = 1, R / t = 20, L / R = 2).

48

64!

44

56

Free vibrations of multilayered thick composite shells 52

721

60

.52 40 4 LAYERS~{019010190)

48 L/R "e

36

44 40

>:

u Z

0 z

28

36 SECOND MODE

32 0

o

28

FIRST MODE r~

~_ ~o

(/)

24 - 4 LAYERS - -

(o/oo/o/9o)

20 16

Z 12

2 LAYERS-k ~

-'LL/R - 4

(o/9o) I \ + ~

12

\

8

8

L/R -2 4 4

5 o

5

L tO

zio

20

Fig. 3. Variation of normalised frequency, K=, based on first mode with R/t and L/R ratios for four-layered shells.

The ratio of Young's moduli, E~/E2, is varied from 1, 10, 20 to 40 and fundamental frequencies are determined for L/R = 2 and 4. 3.4 Normalised frequency, K~ The free vibration characteristics of multi-layered shells are presented based on the non-dimensioned frequency parameter K~ [5] which is defined as

Po2L4 DII~ 4 ,

30

40

50

R/t

R/t

K~-

I0

~:)

(14)

Fig. 4. Variation of normalised frequency, K,o, based on second mode with R/t and L/R ratios for two- and fourlayered shells.

(8 x 4) chosen for the numerical study (Table 1) in order to reduce the computational efforts. The variation of normalised frequency, Ko, for four-layered (0/90/0/90 °) shells based on first mode (m = 1, n = 1 or 2) with different values of R/t and L/R ratio is given in Fig. 3. It is found that K,o increases linearly with R/t for the range of values considered. As anticipated such a linear variation is quite rapid at higher L/R ratios. As the coupling between bending and extension is weak, the frequencies of two-layered (0/90 °) and four-layered (0/90/0/90 °) shells are very closely spaced, which is in

where

Dl] --

(1 + F )

2

F = E2/E,.

Ql]t3/12

12

y,

(]5) /

Figure 2 compares the variation of K~(m = l) for two-layered shells with modal n, obtained from the present numerical approach following the full and selective integration technique (as mentioned earlier) and the analytical method of Sadasiva Rao with [6] and without [5] transverse shear effects and rotary inertia for L/R = 2. It is seen that for n equal to 1 and 2 that there is a good agreement for K,o with the analytical results for both the integration techniques followed. The large discrepancy of the results when n > 2 could be attributed to the medium mesh size CAS

28,~"

=,

o0

~0

G23/EZ: 0,416 ~t2 = 0,25

O/

+

i

I

i

I

I

i

5

I0

I~

20

25

30

35

I

40

45

El/E2~

Fig. 5. Variation of frequency parameter, K,o, with E I/E2 (R/t = I0, L/R = 2 and 4).

722

R. RA~mSHKUr~ARand Y. V. K. SADASlVARAO

conformity with the observations made in an earlier study [5]. Figure 4 presents a similar variation for the second mode of vibration of the shell. The behaviour of the shell is similar to that of first mode except for a marginal variation in K~ between the two- and four-layered configurations. For long shells, a parabolic variation of the frequency is observed as against a linear variation in the case of short shells, particularly in the moderately thick range. Table 3 presents the frequency parameter, K~, for the first six fundamental modes for two-, four- [(i) (0/90/0/90 °) and (ii) (0/90°)s] and eight- (0/90/0/90°)~ layered shells with L / R = 4 and R / t = 5, 10, 20 and 50. It is observed that for the lowest frequency the maximum deviation in the numerical values of Ko for all the layers is about 5%, which increases to about 30% for R / t = 5, 6% for R / t = l0 and 20 and then reduces to 3% for R / t = 50 when compared to a corresponding two-layered shell. F o r the range of R / t values considered, K~ for the four-layered shell with fibre orientation (0/90/0/90 °) is quite close to that of eight-layered one (0/90/0/90°)s when compared to a (0/90°).~ shell. Table 4 gives the above information for two- and four-layered shells with L / R = 2 and 8. The results indicate that for a short and thick shell, the frequency parameter varies by about 11% for fourlayered configurations when compared to the twolayered one for higher modes. However, the actual frequency variation for the above case is more than the corresponding one for a long shell. Figure 5 presents the variation Ko with moduli ratio, E~/E2, for L / R = 2 and 4. No variation in the frequency parameter is observed for lower values of Ez/E2 for two- and four-layered shells. However, a marginal difference can be observed between these values for El/E2 > 5. 4. CONCLUSION The free vibration characteristics of simply supported multilayered cross-ply cylindrical shells are studied using the finite element method. An eightnoded isoparametric shear flexible shell element employing a selective integration scheme is used in

the analysis. Comparison of the present results with those available in the literature shows a good agreement. The higher frequencies are observed to be effected due to the presence of transverse shear terms. Although the frequency variation is linear in the case of short shells, a parabolic variation in the moderately thick ranges of long shells is reported. The layering effects are found to be negligible at lower moduli ratios and significant at higher values. REFERENCES

1. S. A. Ambartsumyan, Theory of anisotropic shells. NASA Rep. TTF-118 (1964). 2. S. B. Dong, Analysis of laminated shells of revolution. J. Engng Mech. Die., ASCE EM6, 135-155 (1966). 3. C. W. Bert, J. L. Baker and D. M. Egle, Free vibration of multilayer anisotropic cylindrical shells. J. Comp. Mat. 3, 480 0969). 4. J. B. Greenberg and Y. Stavsky, Vibrations of laminated filament-wound cylindrical shells. AIAA Jnl 19, I055-1062 (1981). 5. Y. V. K. Sadasiva Rao and P. C. Raju, Vibration of Anti-symmetric Laminated Cylindrical Shells, Chapter 3, Developments in Composite Materials--1. Applied Science, Barking, U.K. (1977). 6. Y.V.K. Sadasiva Rao, Vibrations of layered shells with transverse shear and rotary inertia effects. J. Sound Vibr. 86, 147-150 (1983). 7. I. Sheinman and S. Greif, Dynamic analysis of laminated shells of revolution. J. Comp. Mat. 18, 199-296 0984). 8. R. R. Kumar, K. Rajaiah, R. M. Belkune and T. Kant, A finite element method for evaluating stresses around cutouts in shells. In Proc. International Conference in Finite Elements in Computational Mechanics, 2, at Indian Institute of Technology, Bombay, India, 2-6 December 1985 (Edited by Tarum Kant), pp. 955-965. Pergamon Press, Oxford. 9. T. Kant and R. R. Kumar, Equivalent and degenerate concepts in finite element analysis of laminated composite plates (communicated). I0, R. D. Cook, Concepts and Applications of Finite Element Analysis. John Wiley, New York (1974). 11, S. C. Panda and R. Natarajan, Finite element analysis of laminated composite plate. Int. J. Numer. Meth. Engng 14, 69-79 (1979). 12. A. W. Leissa, Vibration of shells. NASA Rep. SP-288 (1973). 13, C. S. Desai and J. F. Abel, Introduction to the Finite Element Method. East-West PUT, New Delhi (1973).