Axisymmetric dynamic response of spherical and cylindrical shell

Lin, C.-W.

Stadler, W.

Wang, J. T.-S.

English

NASA Technical Reports Server

1967 13000 
AXISYM'XIC DYNAMIC RESPONSE OF 
SPJERICAL ANG CYLINDRICAL SH!2LIS 
P r o j e c t  B - 91 0 
NASA Grant NsG-571 
James Ting-Shun Wang 
Chi-Wen L,in 
Wolfram Stadlei: 
Prepared f o r  
American Ast ronaut ica l  Soc ie ty  
Southeastern Symposim on M i . s s i l z s  and 
Aerospace Vehicles  Sciences 
December 5-7,  1366 Huntsv i l le ,  :ilabana 
School o f  Engineering ?IecE.~u:-cs 
Georgia I n s t i t u t e  of  Technology 
A t l a n t a ,  Georgia 30332 
I 
(.ACCESSION NUMBER) (THRUI 
(NASA CR OR TMX OR AD NUMBER) (CATEGORY) 
GPO PRICE $ 
CFSTI PRICE(S) S 
Microfiche (MF) .A3 i 
ff 663 July 65 
https://ntrs.nasa.gov/search.jsp?R=19670003671 2020-03-16T17:19:55+00:00Z
Axisymw t r ic  Dynamic Response of Spher i ca l  
and Cyl indr ica l  S h e l l s  
James Ting-Shun Wang 
Chi-Wen Lin 
Wolfram S t ad le r  
Georgia I n s t i t u t e  o f  Technology 
Abstract  
The mode-acceleration or Williams method i s  useL t o  c - t a i n  the axisymme- 
t r i c  response,  due t o  a n  a r b i t r a r y  time-dependent loading, of a complete 
sphe r i ca l  s h e l l ,  and of hemispherical  s h e l l s  w i th  ro l le r -h inged ,  rol ler-clamped 
and completely f ixed  edge. Numerical exampLs, where the  s h e l l s  are subjected 
t o  simple time-dependent loads,  are presented; The same method i s  used i n  the  
s tudy of a c y l i n d r i c a l  s h e l l  wi th  clamped-edge boundary condi t ions.  A numerical 
example of a c y l i n d r i c a l  s h e l l  subjected t o  a l i n e a r l y  decaying load,  uniformly 
d i s t r i b u t e d  over the shell, i s  included. The r e s u l t s  here  compare favorably 
with so lu t ions  obtained by means of a modal a n a l y s i s  i n  a previous inves t iga -  
t i o n  of t he  problem by the authors .  
a 
b 
d 
D '  
E 
g 
h 
L C  
R 
t 
ucY 
v, w 
x s  Q 
P 
P 
L i s t  of Symbols 
Radius of sphere 
Loading parameter 
Durat ion of loading 
P l a t e  R i g i d i t y  
Modulus of E l a s t i c i t y  
Grav i t a t iona l  acce le ra t ion  
She 11 t h i c  h e  s s 
S h e l l  l eng th  
Loading components 
Non-dimensional normal loading 
Radius of Cyl inder  
Time 
General displacement components 
Meridional and normal displacements 
Coordinate s 
Mass d e n s i t y  of s h e l l  
Poisson 's  R a t i o  
Frequencies  
Frequency 
Acknowledgment 
The research was supported by NASA Research Grant  No. NsG-571. 
-. 
. .  
I 
General Discussion 
The genera l  equations of motion of a s h e l l ,  when damping is neglected,  may 
be written as a system of three l i n e a r ,  coupled d i f f e r e n t i a l  equat ions  of t h e  
f orm 
where [La81 is a mat r ix  d i f f e r e n t i a l  ope ra to r ,  and where' {ua) and' {pa) are t h e  
displacement and load column v e c t o r s  respec t ive ly .  Natura l ly ,  s u i t a b l e  boundary 
' and i n i t i a l  condi t ions  must be included. , 
Assume t h e  displacement vec to r ,  according to W i l l i a m s '  method, in t h e  form 
. -  
where ha} and' {u ' 1 are so l&ions  of a j  . ,  . ,  
and 
. .  
. .  
r e s p e c t i v e l y .  Here, t h e  U, represent  t h e  q u a s i - s t a t i c  respons , and t h -  U a  
The necessary  coupling equations f o r  t h e  eva lua t ion  of t h e  $ (t> are obta ine  
by s u b s t i t u t i n g  (2) i n t o  equation (1) and s impl i fy ing  by meaAs of equations (3) 
and (4). 
are t h e  normal modes of free v i b r a t i o n  wi th  corresponding eigenfrequencies  
The r e s u l t a n t  coupled equat ion is 
Mu1 t i p l i c a  i o n  of both s i d e s  of equation (5) by t h e  row vec to r  Lu, J and use of 
9: 
t h e  o r thogona l i ty  cond i t ion  [l] F %.e., 
. .  
3 ' 2  1.1 uaiuijdS - 6 1.1 uai dS - Sa-1 '3 sa=l 
. . . .  . .  
* 
All Greek i n d i c e s  range from 1 . t o  3, and t h e  La t in  i n d i c e s  from 1 t o  unless 
s p e c i f i e d  otherwise. 
. , .. 4 2 
.' 
.. 
j 
results in the following expression for the 41 
. .  
.. 3 f t -  - ' L U  u dS S a=l aj a 
s a=1 'j 
2 
3 
$ 1 t12 as 
where S denotes integration over the middle surface of the shell. 
a 
. The Axisymmetricallv Loaded Cy l ind r i ca l  S h e l l  
Under cons idera t ion  i s  a c y l i n d r i c a l  s h e l l  w i th  an a x i s m e t r i c  dynamic 
loading, normal t o  the sur face  of  the s h e l l .  The s h e l l  is of  f i n i t e  length  L 
wi th  mean r a d i u s  R and has  clamped ends. For  t h i s  problem the system of 
equat ions  
i n  non-dimensional Eorm, reduces t o  the s i n g l e  equat ion  f o r  the r a d i a l  d i s -  
placement w [4] a 
where 
To s impl i fy  c a l c u l a t i o n s . t h e  loading i s  taken t o  be of t he  form 
PO Q(5, 7) - - E (l-T)e 
The boundary and i n i t i a l  condi t ions  a r e  
For  t h e  so lu t ion  of  Eq. ( 8 ) ,  w i s  assumed i n  the  form 
W(S,T) = W(5,T) + 1 (pi(T)Wi(5). 
A 
. As discussed previous ly  i s  the  q u a s i - s t a t i c  p a r t  o f  the so lu t ion  and 
isfies the d i f f e r e n t i a l  equat ion  
a w  4 - a3Po -+ alw = - -(1-~) E a% 
a long  w i t h  the boundary condi t ions  (11). Hence, 
n 
sat- 
.. 
(13) 
4 
I 
I 
I where, 
~ . 
, a 
with 
' the q u a n t i t i e s  CY, $, y and 6 a l l  being evaluated a t  5 = 1. 
The w are t he  normal modes of f r e e  v i b r a t i o n  and as such s a t i s f y  
I - i' 
I 
t 
I 
4 d w  i 4  - -  = 0, (15) 
dS4 'i wi c 
I 
f sub jec t  t o  
i 
w i t h  
c 4 2 Ai = a p i  -al. 
. .  
(16) 
These e igenfunct ions  are. 
A 
where 
, 
I 
I '  The func t ions  w, are orthogonal on the bas ic  i n t e r v a l  ( 0 , l )  and they may be 
normalized by clioosing 
i .' 
I 
0 hence 
5 
1 
.. 
. where 6 is the Kronecker del ta .  
i j  
To evaluate the func t ions  cp Eq.  (12) i s  s u b s t i t u t e d  i n  E q .  (8), and 
after s i m p l i f i c a t i o n  by means of 4i3) and (15) the r e s u l t  is 
primes denote d i f f e r e n t i a t i o n  wi th  respec t  t o  5 .  
t i m  is 
The r e s u l t  o f  the evalua- ' 
I , I n  view of t he  o r thogona l i ty  of the wi and the fact that 
f o r  t h i s  problem, t h e  cpi are given by 
wi th  . 
Ai - - s1 &5,0) wi<5) dSo B, = - s1 wi(S> dS 
0 .I. "i 0 
due t o  the i n i t i a l  condi t ions  (11). "he eva lua t ion  of  these  i n t e g r a l s  is 
f a c i l i t a t e d  g r e a t l y  i f  E q s .  (13) and (15) are used and the i d e n t i t y  
. I  
due t o  the fact t h a t ,  i n  the present  case  
I The express ion  K(Ai) is given by 
With these e x p r e s s i m s  i n  mind w may f i n a l l y  be  w r i t t e n  8s 
6 
. 
w h r e  ( 5 , T )  i s  given by (14). 
To provide a numerical cornparism wi th  the r e s u l t s  obtained i n  [ 4 ] ,  the  
response was ca lcu la ted  a t  5 = h, with 
2 
I, = 10.0 in ;  R = 5.0 i n ;  p = 0.3; E = 30 x 1 0  6 ps i ;  
h = 0.05 i n ;  d = 20 msec; p = 0.0007298 l b  - sec 2 4  / i n  . 
For convenience p = 1 ps i .  The r e s u l t s  are shown i n  ig .  2. The value of  
It should be noted 
tkhaf thc  extrema of the  response are somwhat 'greater here than obtained i n  
[ 4 ] .  Xn [4], terns l i k e  cosh x - cos  x,  r e s u l t e d  i n  computorial  round-off 
e r r o r  for small values of  X. 
F the q u a s i - s t a t i c  0 so lu t ion  a t  T = 0, namely, 1.67 x 10- , corresponds exac t ly  
' t o  the  r e s u l t s  obtained by using the  membrane theory. 
I 
R e  Axisvmmetrically Loaded Spher ica l  S h e l l s  
The equat ions o f  motion f o r  the  a x i s y m t r i c  response of sphe r i ca l  
.. 
s:iells, used by th? au thors  i n  a previous study [ 4 ] ,  are: 
where 
. 35e meaning o f  t he  g e m e t r i c  symbols may be in fe r r ed  from'Fig. 1. Equations 
( 2 3 )  and ( 2 4 )  were dezived i n  accordance with the e l a s t i c  bending theqry of 
. t h i n  shells. The r e l a t e d  e l a s t i c i t y  equat ions are given i n  [ 4 ] .  As implied 
i n  t h e  pre l iminary  d iscuss ion  the  so lu t ions  are assumed i n  the  form 
I 
J 
Tne functions $ and r ep resen t  the  quasi  s t a t i c  p a r t  of t he  so lu t ion .  
7 
4 
'. 
L. . .  
.. 
For the  ccmplete spher ica l  and the  hemi-spherical  s h e l l  w i th  r o l l e r -  
clanped o r  rol ler-5inged edge, the  adn i s s ib l e  so lu t ions  are given i n  terms 
o f  Legendre polynomials as 
where A = 1, 3, 5 ... corresponds t o  rhe ro l le r -h inged  edge and n = 0, 2, 4 
corresponds t o  t he  roller-clamped edge. j n o m l  modes of r'ree v i b r a t i o n  of the s h e l l .  Details of  t h e i r  Jons t ruc t ion  
xrzy be found i n  [4]. of  the j t h  mode i s  obtained 
The func t ions  JI and w.  are the  
The n a t u r a l  frequency ha 
*om the  expression j 
tix? choice of sign corresponding to tk  des i r ed  frequency on the l e f t  s i d e  
04 .;ha equation. Alj  and A am ,iven by 2 j  
r-t- i n e  subsc r ip t s  P zr?d 2 correspond res;eatfvely tt the upper and iower branch 
f requencies  of a c m p l e t e  sphe r i ca l  s h z l l .  When j i s  an  even in t ege r ,  equa- 
t i o n  (27) y i e l d s  the eigenfrequencies  o f  a hemispherical  s h e l l  wi th  a r o l l e r -  
clamped edge, v l e n  j i s  odd, the  freqzencies  correspond t o  a hemispherical  
s h e l l  with ro l le r -h inged  edge. I n  ehch case the  r e spec t ive  e i g e n f m c t i o n s  
are t h e  LegerrGre 3olynomials P (x) 
. 
J 
..When t h e  Gcr-ding theory of shel ls  i s  Qsed i n  obta in ing  Zhe q u a s i - s t a t i c  
pa r t  of the  so lu t ion ,  no advantags i s  r e a l i z e d  i n  applying the. p re sen t  method 
OVBP the usuril mode displacement mtFisdp ud.ess t he  hadfz;g cicz; be expressed 
i n  the fom 
- . 0- .-.GWP~BT, in mmy casss the  membrane.tZ2eory of s h e l l s  may be used t o  ob ta in  JI 
and w for  an L . > i t r a r y  loading. For a completely f ixed  hemispherical  s h e l l ,  
.essuzir;g a aep&-ia"u:e loading funct ion 
. .  
a , ( x , t )  = q,W m), (31) 
the use of the wnrbrane theory , . inc luding  bending effects, r e s u l t s  in quasi-  
static s o l u t i o n s  of the f o w  
. .  . . .~ . .  
- .  
1 .  . . . .  . .. . . 
I .  
R 
8 
Y 
' where 
Z 5  eigenfreqwncies R 
expre s si o= 
as i n  [4], aze given by the  roots of the fol lowing 
j' 
.. 
P'he soPution f o r  a cor.ipfete spher ica l  she l l ,  a hemispheric69 she l l  w i th  
zoLler-clamped edge or a hemispherical she l l  w i th  ro l le r -h inged  edge, sub- 
fected t o  the sarz type of loading as given by Eq. (301, may be derived by 
sa5s t iku tPng Eq. (26) i n t o  equat ions  of the form of Eq. (3), in conjunct ion 
wi2h X q s .  (23) end (24). The resul: of this s u b s t i t u t i o n  i s  
9 
' o$) = - f 
A 
uhre 
1 - n 2 (n-i-1) -(l-p,)n(n+l) , 
(38) I ' 
With Sn = w = P (x), the equstior, iar  Chs coupling fenr~ceiopi R . becomes 
3 a n  
(39) 
s e  
d t  de: 
T"ne f ina l  solutions for  a shell, which is considered to be at res2 i n l r i a l l y ,  
Gnzd has a loading 
20 provide a q u a n i i t a t i v e  eva lua t ion ,  L U T ~ K ~ C ~ P  examples f o r  the response of  
a hemispherical  shell at the  a w x ,  xli ih a ro l le r -c lanped  edge, and wi th  
qo = 1, b = 900 and 0.25, a = 300 in, 
E -3 2 4  30 x 10 6 p s i ,  p, = 0.33, 21lCi p = 0.7298 x PO Ib-sec /in , 
tze presented i n  F igures  (3),  (41,  and (51, 
1. When an exac t  s o l u t i o n  can be obcalned for the q * a s f - s t z L l c  ?art, the  
U i l i i a m s  method has a d e f i n i t e  advmzzge over the usua l  modal ana lys i s ,  i n  . 
t h a t  the  q u a s i - s t a t i c  p a r t  provides ~n i m z d i a t e  average response,  and fu r -  
chermore the spread i n  the ind iv idua l  nodes i s  reduced. 
I 
l 
! 
2. 
by us ing  bending t k o r y ,  i 2  s u f f i c e s  In  many cases *Lo use t>e menbrsne solu-  
cion, fncluding bending co r rec t ions  a t  rhe boundarfes. 
When t h e  q u a s i - s t a t i c  p a r t  of thle s o l ~ t i o a  cannot be ob~afi.i,ed e x a c t l y  
d *  -2 
obtained by means 04 t h e  mod21 ana lys i s  [&I3 tems of t he  form.eos;l x - 
cos x, and s inh  x - s i n  x, resz l zed  L;? smaller values of the  response,  than 
those obtained by r eans  of t h e  W i l i i z a s  r ~ t 2 o d ,  s ince ,  for  small x, fkese 
terns are c lose  t o  zero. A l a rge  parr or" chis error i s  el iminated by ob- 
tsining the quasi-static so lu t ion  in i t s  exac t  form. 
13 fhs nm.erPea2 evaluatiozl 05 L-2 response of a c y l l n d r i c c l  s h e l l ,  as I 
, 
I 
! , 
I 
i 
I . '  Cl3 
f2] 
CS] 
H. ~ r a ~ s  and A ,  Kalnins,  fs~r23s<e;p.~ Vibra t ion  of ~ P R   asti tic Shel l s ,1r  
1 J. A c o ~ s ~ .  SOC. Am. 38, 994-PGG2 (1965)- j 
J. Sheng, I f  Tine Tiespoillse of a Thin Cyl ind r i ca l  S h e l l  t o  a Transient  
J. B. Wilkfnson, 9 r a n s i e n t  Iiesponse of  Thin E la s t i c  Shel l s , "  J, ~ c o u s t .  
Surface Loading," AUA J. 3, 70E-7C9 (1965). 
Soc.  Am. 39, S95-898 (1965), 
Jaaes T. S. !-l.~a~g, K~lfrarn Stadlez ,  Chi-Wen Lin, "The Axisymnetric 
Response of Cyl inckica l  and Eea is2her ica l  S h e l l s  t o  Time-Dependent 
- -  J*-- * 
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* C4] 
r n T A @ A  f iw ,c73  ! i ~ h f ; ] -  
& U & U A L L ~ ; ,  A>-<U-* --. ----, - L 
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Axisymmetric Dynamic Response of Spherical and Cylindrical Shells

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