An equation describing the radial displacement in a k layered anisotropic cylinder was obtained. The cylinders are initially unstressed but are subjected to either a time dependent normal stress or a displacement at the external boundaries of the laminate. The solution is obtained by utilizing the Vodicka orthogonalization technique. Numerical examples are given to illustrate the procedure

Gupta, B. P.

Mulholland, G. P.

English

NASA Technical Reports Server

VIBRATIONS AND STRESSES I N  LAYERED ANISOTROPIC CYLINDERS 
G .  P .  Mulholland 
New Mexico S t a t e  University 
B. P. Gupta 
Fluor Engineers and Constructors, Inc. 
SUMMARY 
An equation describing t h e  r a d i a l  displacement i n  a k layered an i so t rop ic  
cy l inder  has been obtained. The cy l inders  a r e  i n i t i a l l y  unstressed but are 
subjected t o  e i t h e r  a time-dependent normal s t r e s s  o r  a displacement a t  t h e  
ex terna l  boundaries of t h e  laminate. The so lu t ion  is  obtained by u t i l i z i n g  
t h e  Vodicka orthogonalization technique. 
i l l u s t r a t e  the  procedure. 
Numerical examples are given t o  
INTRODUCTION 
The problems assoc ia ted  with t h e  v ib ra t ions  of p l a t e s  and s h e l l s  have been 
of concern t o  many i n v e s t i g a t o r s  over t he  years.  Most of these works f o r  a 
s i n g l e  layered homogeneous material a r e  summarized i n  two monographs by Leissa 
( r e f .  1 , 2 )  and the  reader  i s  r e f e r r e d  t h e r e  f o r  f u r t h e r  re ferences ,  Since 
composite mater ia l s  have become popular due t o  t h e i r  mechanical and thermal 
p r o p e r t i e s ,  it has become necessary t o  study t h e i r  behavior t o  determine t h e i r  
unique c h a r a c t e r i s t i c s  before they can be used e f f e c t i v e l y .  
( r e f .  3)  and Dong and Nelson ( r e f .  4)  considered t h e  v ib ra t ion  prablem i n  
laminated p l a t e s  and t h e  re ferences  contained i n  These papers summarize t h e  
work i n  t h i s  a r e a  q u i t e  w e l l .  
cy l inders ,  t h e  book of Ambartsumyan ( r e f .  5) and Hearmon ( r e f .  6) and t h e  
papers of Gula t i  and Essenburg ( r e f .  7 ) ,  Stavsky and Smolash (ref. 8) ,  Cheung 
and Wu ( r e f .  9), and Nelson e t  a l .  (ref. 10) are r ep resen ta t ive .  
Recently Cobble 
For works concerned with an i so t rop ic  and layered 
In t h i s  paper, t h e  r a d i a l  v ib ra t ions  of  a layered an i so t rop ic  cy l inder  
are considered. The cy l inders  are s o l i d l y  joined a t  t h e i r  i n t e r f a c e s ,  are 
i n i t i a l l y  unstressed, and can be subjected t o  e i t h e r  a r b i t r a r y  time-dependent 
normal stresses o r  displacements a t  t he  ex te rna l  boundaries of  t h e  system. 
The so lu t ion  i s  obtained by using a dependent va r i ab le  transformation i n  t h e  
displacement equation thereby obta in ing  a new p a r t i a l  d i f f e r e n t i a l  equation 
with homogeneous ex te rna l  boundary conditions;  t h e  Vodicka or thogonal i ty  
conditions are then appl ied  t o  t h i s  new system t o  ob ta in  t h e  f i n a l  so lu t ion .  
The plane s t r a i n  s i t u a t i o n  i s  considered f o r  t h i s  ana lys i s .  
To i l l u s t r a t e  t h e  e f f i c i e n t  and s t ra ight - forward  manner i n  which so lu t ions  
can be obtained with t h i s  method, numerical examples are given f o r  a two-layered 
459 
https://ntrs.nasa.gov/search.jsp?R=19770003331 2020-03-22T12:12:05+00:00Z
composite. 
tangential stress, and axial stress components at two interior positions. 
Results are presented for the displacement, normal stress, 
SYMBOLS 
t 
v 
constants, (eq. (1)) 
constants, (eq. (2)) 
Young’s modulus for r, 8,  and z directions, dyne/cm 
function of time, (eq. (5) )  
function o f  displacement and time, (eq. (10)) 
Bessel function of first kind of order D 
function of  r, (eq. (4)) 
constants, (eq. (23)) 
radial coordinate, cm 
time, seconds 
radial displacement, cm 
function of time, (eq. (11)) 
weighting function, (eq. (17)) 
eigenfunct ion 
Bessel function of the second kind of order D 
eigenvalues, l/sec 
constant 
Poisson’s ratio 
2 
i 
i 
normal stress in r, 6, and z directions, dyne/cm 
functions of time, (eq. (2)) 
functions of r ,  (eq. (9)) 
460 
PROBLEM 
The p a r t i a l  d i f f e r e n t i a l  equation descr ib ing  t h e  displacement u f o r  t h e  i i t h  l aye r  of a multi layered c y l i n d r i c a l  composite whose material p rope r t i e s  are 
constant f o r  each l aye r  is given by 
( r , t >  (1) 
i a 2~ - ( r , t )  + - - (r,t) - - u i ( r , t )  = - -
ar 
a 2u 
r2 B~~ a t 2  
i 1 i  Di2 1 au 
2 r ar 
where 
31iv13i 
32iv23i 
1 -v  
1-v 
E2 i 
Eli 
Di2 = -
32iv23i 1-v 2 Eli B = -  
’i ‘i i 
The boundary and i n i t i a l  conditions associated with equation (1) a r e :  
e )  u.  ( r ,o)  = 0 
1 
46 1 
where 
(1-v 32iv23i) 
- Eli - -  
‘ill A~ 
+ v  Eli (‘21i 31iv23i) ‘i12 A~ = -  
The boundary and initial conditions given by equation ( 2 )  assume that either 
the radial stresses o r  displacements are known at the external boundaries 
and that the radial stresses and displacements are continuous at the interfaces. 
To obtain homogeneous external boundary conditions, let 
3 
I 
u. (r,t) = U. (r,t) + g L. .(r)F.(t) ( 3 )  
1 1 j-1 1~ J 
where 
F.(t) = $.(t), j=1,2 
7. J 
and 
D . 2  
L.. (r) = 0 v~L.. (r) - - -i 
1 J  r2 1J 
(4) 
(5) 
For a cylinder with r 
following form for i = 1: 
= 0 (solid cylinder) and D1~l, Eq. (4) and ( 6 )  take the 1 
and 
-A D lj 1 L (r) = V2L - - D12 
Ij r2 1j r2 
The functions L (r) satisfy the following boundary conditions: ij 
462 
i+l j (ri+l) c) L.  .(r ) = L ij i+l 
Substitution of  equation ( 3 )  into equations (1) and (2) yields the following 
partial differential equation with homogeneous external boundary conditions: 
a2ui 
(r,t> (8) 
aui Di2 1 
U.(r,t) = -- (r,t) + Hij i 1 
. a2u 
(r,t) + - - (r,t) - -
r2 1 Bi2 at2 r ar ar2 
with 
463 
aui 2 
f) - ( r ,o )  = - C L . .  (r) F. ’ (O)  = q2(r) at j=l 13 J 
and where 
2 
.C  L . .  (r)F.”(t) H. .(r,t) = - 1 
13 ~~2 J=1 13 J 
SOLUTION: Ui(r,t) 
The problem has now been sufficiently simplified so that a series 
solution for Ui(r,t) can be assumed where the orthogonality conditions developed 
by Vodicka (ref. 11) can be utilized. Let 
W 
U. (r,t) = C um(t)Xim(r) 
r . <  r < r  i = 1,2,3, ..., k,t - > 0 
(11) 1 m= 1 
1 -  - i+ly 
where the function u (t) is to be determined from the initial conditions and 
the functions Xim (r)mare eigenfunctions of the eigenvalue problem 
Bi2 Di2 
r2 
Xim(r) + cxm2Xim(r) = o r dr 
with 
= o  dXkm Xkm(rk+l) b, ‘kll dr (‘k+l) + ‘k12 rk+l 
464 
The solution of equation (12) is 
The eigenvalues, a are found by substituting equations (14) o r  (15) into the 
boundary conditions, (eq. (13)). The 2k linear homogeneous equations that 
result from this substitution are then solved for the constants A 
(ref. 12). 
m y  
and Bim im 
The orthogonality condition for the eigenfunctions is 
where 
wi2 = Cill/Bi2 = Pi (17) 
The functions L . . ( r )  and H..(r,t) will satisfy Dirichlet’s conditions so 
13 13 they can be expanded in an infinite series of the eigenfunctions 
(18) 
co 
L.  .(r) = C k .X. (r) ,  j = 1,2 1 3  m=l m j  i m  
and 
where 
i51 Pi (r) Xim(r) dry j = 1,2 (20) 
k 1 = -  
mj Nm 
and 
1 r 
465 
Substituting equations (ll), (18), and (19) into equation (S), we obtain the 
following relationship: 
The initial conditions asssociated with equation (28) are obtained in the 
following manner : 
co co 
Ui(r,o) = mgl um(o) Xim(r) = $,(r) = - C R .F . (o)  Xim(r) m = l  mj J 
and 
Thus 
a) um(o) = 2 C R F.(o)  = p, 
2 
2 
j=l m j  j 
b) urn' (0) = -jzl Rmj F j  ' (0) = % 
The solution of equation (22) subject to the initial conditions (eq. (23))  is 
2 R  
(24) 
qm 
a m m j=1 a j um(t) = - sin a t + p, cos a t - .C F."(t) * sin (amt) m m 
where the symbol * denotes convolution. Substitution of equation (24) into 
equation (11) and that result into equation (3) gives the desired relationship 
for the radial displacement of the composite cylinders: 
2 2 
u. (r,t) = C L.. (r) F .  (t) + mfl um(t) Xim(r) (3) 
1 j=l IJ J 
where the functions Lij (r), F j  (t), Xjm(r) and u (t) are given by equations ( 4 ) ,  
(5), (14) o r  (lS), and (24),  respectively. m 
466 
STRESS 
The stress i n  t h e  ith sec t ion  of t h e  composite i s  given by 
V E  23 3 vie 
i z  ir E2 . 
EXAMPLE 
Consider a two-layered composite with t h e  following p rope r t i e s :  
Layer 1 Layer 2 
p1 = 1.73 gm/cm3 3 p 2  = 1.75 gm/cm 
= 0.11 V = v  = 0.14 
= 0.18 = 0.16 V 
= 0.1 V = v  = 0.22 
122  132 
212 - v312 
232 322 
131 
311 
32 1 
= v  v121 
- = v  211 
231 
V 
= v  V 
= 6.6 x l o5  newton/cm2 2 = 7.93 x 105 newton/cm2 
- = 1.14 x lo6  newton/cm2 EZ2 - E32 = 8.76 x 105 newton/cm* E21 - E31 
The above p rope r t i e s  are t y p i c a l  of some of  t he  more common g raph i t e s  (ATJ and 
CHQ) ( r e f .  13).  Assume f u r t h e r  t h a t  t h e r e  i s  a normal stress applied a t  t h e  
ou te r  boundary of t h e  cy l inder .  
$ 2 ( t )  = 6895 s i n  ( l o t )  N/cm2 
and t h e  phys ica l  dimensions are 
r = 0; r = 2.54 cm; r = 5.08 cm 1 2 3 
467 
Following the procedures outlined in the text, the radial displacement and the 
radial and tangential stresses within the composite are obtained; values at 
two positions are shown in Figures 1 through 3. 
SUMMARY 
A closed-form solution for the radial displacement in layered orthotropic 
cylinders has been obtained. 
computer which enables one t o  calculate natural frequencies, displacements and 
stresses quite easily. 
directly by hand o r  a numerical integration subroutine can be written to perform 
the calculations. 
The solution can be programmed on a modern 
The functions hj and Nm can either be integrated 
REFERENCES 
1. Leissa, A. W., Vibration of Plates, Scientific and Technical Information 
Office, NASA, Washington, D. C., 1969, NASA SP-160. 
2. Leissa, A. W., Vibration of Shells, Scientific and Technical Information 
Office, NASA, Washington, D. C., 1973, NASA SP-288. 
3. Cobble, M. H., "Dynamic Vibrations and Stresses in Composite Elastic Plates," 
Journal of the Acoustical Society of  America, Vol. 46, No. 5, pp. 1969, 
pp. 1175-1179. 
4. Dong, S. B. and Nelson, R. B., "On Natural Vibrations and Waves in 
Laminated Orthotropic Plates," ASME Paper No. 72-APM-14. 
5. Ambartsumyan, S. A., Theory of  Anisotropic Plates, Technomic Publishing Co., 
Stamford, Conn. 
/ 
6. Hearmon, R. F. S . ,  Applied Anisotropic Elasticity, Oxford University Press, 
London, 1961. 
7. Gulati, S. T. and Essenburg, F., "Effects of Anisotropy in Axisymmetric 
Cylindrical Shells," Journal of Applied Mechanics, Vol. 34, No. 3, Trans., 
ASME, Vol. 89, Series E . ,  1967, pp. 659-666. 
8. Stavsky, Y. and Smolash, I., "Thermoelasticity of Heterogeneous Orthotropic 
Cylindrical ShellsYT' International Journal of Solids and Structures, Vol. 
6, 1970, pp. 1211-1231. 
9. Cheung, Y. K., and Wu, C. I., "Free Vibrations of Thick, Layered Cylinders 
Having Finite Length With Various Boundary Conditions," Journal of Sound 
and Vibrations, Vol. 24, No. 2, 1972, pp. 189-200. 
468 
10. Nelson, R. B., Dong, S .  B. and Kalro, R. D., "Vibrations and Waves in 
Laminated OrthotEopic Circular Cylinders , I t  Journal of Sound and 
Vibration, Vol. 18, No. 3, 1971, pp 429-444. 
11. Vodicka, B., "Warmeleitung in geschichteten Kugel and Zylinderkorpern 
Schweiz. Arch. Vol. 10, 1950, pp. 297-304. 
12. Mulholland, G. P. and Cobble, M. H., "Diffusion Through 
International Journal of Heat and Mass Transfer, Vol. 
pp. 147-159 
13. Industrial Graphite Engineering Handbook, Union Carbide 
York, 1970. 
Composite Media , I '  
15, 1972, 
Corporation, New 
469 
(cm) (inch) 
254 100 
190.5 75 
127.0 50 
n 
M 
0 63.5 25 
rl 
x 
7 
v 
U a 
4 
4. 
Q -63.5 -25’ 
U 
d 
-127.0 -50 
-190..5 -75 
-254 -100 
Figure 1.- Radial displacement of composite. 
INNER LAYER 
(r=1.27  cm) 
4 70 
2 
100 
(Newton/cm ) 
103 
68.95 
5 1 . 7  
3 4 . 5  
17.2 
v) 
v) w cz 
f- 
4 
d 
H 
n 
d 
-17.2 
-34.2 
-51.7 
-68.95 
(KSI 1 
100 
75 - 
50  - 
2s - 
0 
-25 - 
-50 
-75- 
-100- 
- INNER LAYER 
(r=1.27 c m )  
0 1.0 2.0 3.0 4.0 5 . 0  
TIME (SECONDS) 
Figure 2 . -  Radial stress compared to external excitation. 
471 
xiox 
68.95 
51 .7  
34.5 
17.2 
v) 
v) 
W e: 
b 
v) 
J 
f- z w 
W z 
4 0  
H 
5 
-17 .2  
-34.5 
-51.7 
-64 
V 
2 (Newton/cm ) 
Figure 3 . -  Tangential stresses within composite. 
INNER LAYER 
(r=1.27 cm) 
472 


Vibrations and stresses in layered anisotropic cylinders

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server