Large deflections of a shallow elastic conical membrane fixed at the outer edge and loaded by either uniform or hydrostatic pressure are investigated. The overning equations were solved by the method of matched asymptotic expansions and by a finite difference method. Agreement between the two methods was excellent for the small values of the perturbation parameter

Chang, W. H.

Peddieson, J., Jr.

English

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LARGE DEFLECTIONS OF A SHALLOW CONICAL MEMBRANE 
Wen-Hu Chang and John Peddieson, Jr. 
Tennessee Technological University 
SUMMARY 
This work is concerned with large def lect ions of a shallow elastit! conical 
membrane fixed a t  t he  outer edge and loaded by e i t h e r  uniform or hydrostatic 
pressure. 
t o t i c  expansions and by a f ini te-difference method. 
methods w a s  excel lent  f o r  t he  s m a l l  values of t h e  perturbation parameter. 
The governing equations were solved by the  method of matched asymp-. 
Agreement between the two 
INTRODUCTION 
This paper is  concerned with the  moderately la rge  axisymmetric deformation 
of a shallow. e l a s t i c  conical membrane. 
invest igate  the  application of the method of matched asymptotic expansions (see 
Van Dyke, reference 1) t o  the  solut ion of membrane-shell problems involving 
large deflections.  
small loads the  l i n e a r  membrane solut ion is  a good approximation t o  the  actual 
solut ion everywhere except i n  the immediate v i c in i ty  of boundaries. In  these 
regions t h i n  boundary layers  exist: where the var iables  unGergo rapid changes t o  
accommodate themselves t o  the  boundary conditions t h a t  cannot be s a t i s f i e d  by 
the  l i n e a r  membrane solution. In  the  method of matched asymptotic expansions 
separate perturbation expansions me found i n  the  i n t e r i o r  and boundary-layer 
regions and matched i n  an appropriate way t o  insure t h a t  they jo in  smoothly. 
The purpose of t h i s  work is  t o  fur ther  
The success of t h i s  method is based on the  fact t h a t  for 
Bromberg and Stoker ( r e f .  2 )  i n i t i a t e d  t h i s  type of analysis of membrane 
s h e l l s  when they found one term of both the  i n t e r i o r  and boundary-layer expan- 
sions f o r  a uniformly pressurized shallow spherical  she l l .  The next two terms 
i n  the  i n t e r i o r  and boundary-layer expansions were found by Smith, Peddieson, 
and Chung ( re f .  3 )  and used by them t o  invest igate  the  accuracy of f i n i t e -  
difference solut ions of the  same problem. 
layer  expansions f o r  deep membranes of a rb i t r a ry  shape 
Rossettos ( r e f .  4) .  
l i s t e d  i n  reference 4. 
One t e r m  of t he  i n t e r i o r  andboundary- 
has been given by 
This work generalizes the r e s u l t s  given i n  t he  references 
I n  the  present paper’ th ree  terms of the i n t e r i o r  and boundary-layer expan- 
If i s  found tba? complications arise which do 
sions are found for the  case of a shallow conical membrane loaded by e i t h e r  
uniform OF hydrostat ic  pressure. 
not appear i n  the  so lu t ion  of the  corresponding sphere problem. 
method is  modified somewhat t o  account for t h i s .  
sented t o  i l lus t ra te  some of the  in t e re s t ing  features of the solution. 
The solut ion 
Numerical r e s u l t s  are pre- 
5 75 
https://ntrs.nasa.gov/search.jsp?R=19770003342 2020-03-22T12:10:50+00:00Z
GOVERNING EQUATIONS 
Consider a shallow conical membrane (opening upward) with base radius  a,  
thickness h ,  and i n i t i a l  angle @o with the horizontal  made of a l i n e a r l y e l a s t i c  
material with modulus of e l a s t i c i t y  E and Poisson's r a t i o  v. The equations gov- 
erning moderately large axisymmetric def lect ions of such a s t ruc ture  can be 
obtained from t h e  work of Reissner ( re f .  5 ) .  
dimensionless form) 
?he resu l t ing  equations are ( i n  
where ar is  the r a d i a l  coordinate, V0a$/@, is a stress function (Vo being a 
cha rac t e r i s t i c  v e r t i c a l  force r e su l t an t ) ,  VoV is the v e r t i c a l  force r e su l t an t ,  
VoNp/$o i s  the r a d i a l  stress re su l t an t ,  VoN~/$, i s  the  transverse stress resul-  
t a n t ,  E is a load parameter, 
the vept ica l  displacement, $ o ~ 2 B  is  the ro t a t ion ,  and a prime denotes differen- 
t i a t i o n  with respect t o  r. 
is the  horizontal  displacement, a@os2w is 
In the present paper a uniform pressure po and a hydrostat ic  loading 
Yo@oa(l-r) are considered. I t  can be shown by 
librium of the membrane centered on the vertex 
v = r / 2  - j r2 /3  
where j 0 f o r  the uniform pressure and j = 1 
The cha rac t e r i s t i c  v e r t i c a l  force resu l tan t  is  
The load parameter E is defined t o  be 
considering the v e r t i c a l  equi- 
and having radius r t h a t  
f o r  the  hydrostat ic  pressure. 
given by 
In  the  present work it is desired t o  solve equations (1) subject  t o  the 
boundary conditions 
u(1)  = w(1) = 0 (5)  
Special  a t ten t ion  w i l l  be given t o  s i tua t ions  where E << 1. 
576 
STRAIGHTFORWARD SOLUTION 
To begin the so lu t ion  process a straightforward perturbation 
equations (1) is sought for E << 1. 
equations ( l a )  and (lb) t o  y ie ld  
To do t h i s  it is convenient t o  rearrange 
E ~ ( * T '  + q ' / r  - Q/r2) - (1 - ( r ~ / ~ / . ~ ) 2 > / ( 2 r )  = o 
B = (rv/* - L ) / E ~  ( 6 )  
A straightforward perturbation solut ion f o r  E << 1 has the form 
where the  subscr ipt  s indic'ates t he  straightforward solution. Subst i tut ing 
equation ( 7 )  i n t o  equation ( 6 a ) ,  expanding for E << 1, s e t t i n g  the coeff ic ient  
of each power of E equal t o  zero i n  the  usual way, and solving the  r e su l t i ng  
algebvai c e quat ions y ie Ids 
% ( r 2 / 2  - j r 3 / 3 )  + ~ 2 ( r 2 / 2  - j r 3 / 3 > ( 3 r / 2  
$S 
- 8 j r 2 / 3 )  + d + ( r 2 / 2  - j r3 /3 ) (75 r2 /8  
- j (79r3 /2  - 32r4 ) )  + . . ( 8 )  
From equations (1) and (6b) it can then be shown t h a t  
Nrs 'L ( r / 2  - j r 2 / 3 )  + ~ ~ ( r / 2  - j r 2 / 3 ) ( 3 r / 2  
- 8 j r 2 / 3 )  + ~ ~ ( r / 2  - j r2 /3 ) (75 r2 /8  
- j (79 r3 /2  - 32r4 ) )  + . . . 
+ (r - j r 2 )  + ~ ~ ( 9 r ' / 4  - j (22 r3 /3  - 40r4/9) )  
+ C4(75r3/4 - j(915r'/8 - 175r  
f 224r6/3))  + . . . 
0s 
5 
BS - (3 r /2  - 8 j r 2 / 3 )  - E2(57r2/8 
- j (63 r3 /2  - 224r4/9))  + . . . 
5-7 7 
w 'L -3(r - 1) /4  + j 8 ( r 3  - 1)/9 - tz2(19(r3 - 1 ) / 8  S 
- j(63(r4 - 1)/8 - 224(r4  - 1)/45))  + . . . (9)  
where equation (5b) has been used t o  determine the  constants of i n t  
equation (9c). By comparison with the r e s u l t s  given i n  Kraus 
be seen t h a t  t he  first term i n  each series expansion is the 1 
solution. 
(9e) are due t o  the  second term i n  equation (8).  
0(c2) it is necessary t o  f ind  9, t o  O b 4 ) .  
by equation (5a) cannot be s a t i s f i e d  by equation (9d). 
expansion is needed i n  the v i c i n i t y  of r = 1. 
I t  should a l so  be noted t h a t  t he  first terms i n  equations (9d) and 
Thus t o  obtain Bs and ws t o  
Thus a boundary-layer 
The boundary condition represented 
BOUNDARY-LAYER SOLUTION 
There are several ways t o  carry out the  boundary-layer analysis i n  t h i s  
problem. One is t o  work i n  terms of the  o r ig ina l  stress function +. If t h i s  
is  done the  d i f f e r e n t i a l  equation f o r  the first boundary-layer approximation 
turns  out t o  be nonlinear. 
l i n e a r  equation could be obtained i n  the first approximation f o r  a spherical  
membixaae by a method which is equivalent t o  working with a dependent var iable  
which is the  difference between the  actual  and the  l i nea r  stress functions. 
This w a s  t r i e d  i n  the present problem but matching d i f f i c u l t i e s  wereencountered. 
These were due t o  the f a c t  t h a t  equations ( 8 )  and (9)  do not terminate with one 
term f o r  the  cone as the  corresponding straightforward expansions do fo r  a 
sphere. I t  w a s ,  therefore ,  decided t o  use the  difference between the  ac tua l  
stress function and the  s t ra ightfornard stress function as the dependent vari-  
able. 
be zero. 
Bromberg and Stoker ( r e f .  2 )  discovered t h a t  a 
This guarantees t h a t  the  outer expansion f o r  t h i s  dependent var iable  w i l l  
Thus it is necessary t o  f ind only the inner expansion. 
Subst i tut ing 
@ = 9, + $b (10) 
(where t h e  subscr ipt  b denotes the boundary-layer solut ion)  i n t o  equation 
(61, defining the  boundary-layer variables F and 6 by the equations 
= EF, r = 1 - €6, 
expanding F as 
and carrying out the usual perturbation analysis  y ie lds  
.. 
F,, - S2Fo = 0 
and two other  equations governing F1 and F where 2 
% S = (6/(3 - 2 j ) )  
578 
13 = Bs + Bb/E, u = u + l”b, w = ws + Wb (15) 
Now expand as follows 
” A b o  + E%1 + E2%2 + * . (16) 
where Ab is  any one of the  boundary-layer variables.  
(101, (11) , (121, (15 , and (16) i n t o  equations (1) expanding fo r  E << 1, and 
equating the  coef f ic ien ts  of l i k e  powers of 
Subst i tut ing equations 
E t o  zero one obtains 
= -s2F 
0 ’  ’bo 0 
= -F - 
Nrbo - Fo’ Nebo 
and two s i m i l a r  sets of equations r e l a t ing  Abl and A 
s i m i l a r  procedure applied t o  equation (5a) leads t o  boundary condition 
t o  Fo ,  F1, and F,. A 
F,(o) = 1 - j - ( 1 / 2  - j /3)v (18) 
and boundary conditions f o r  I?,(O) and ?,(O). 
To i l l u s t r a t e  the  solut ion procedure the first approximation w i l l  now be 
carr ied out i n  de t a i l .  The solut ion of equation (13) is  eas i ly  seen t o  be 
F, = c 1 exp(S5) + c2 exp(-SS) (19) 
Since the  outer expansion has been forced t o  vanish because of equation (10) 
the  matching process (see Van Dyke, reference 1) i s  equivalent i n  t h i s  case t o  
a statement t h a t  pos i t ive  exponential terms must vanish. Thus 
c = o  1 
Subst i tut ing equations ( 1 9 )  ani! (20) i n t o  equation (18) yie lds  
Thus 
Fo = -(1 - j -(l/2 - j/3)v)exp(-Sf)/S (22) 
Subst i tut ing equation (22) i n t o  equations (17) one obtains 
Nrbo = -(I - j - ( U p  - j / 3 ) v ) e x p ( - ~ < ) / ~  
Neb0 - (1 - j - (1 /2  - j / 3 ) v ) e x p ( - ~ ~ )  - 
579 
ub0 = -(1 - j - (1 /2  - j/3)v)exp(-S<) 
w = -(1 - j - (1/2 - j / 3 ) v ) ( l  - exp(-SS)) (23) 
The r e s u l t s  f o r  higher approximations are found i n  a similar way but  the  calcu- 
l a t ions  are qui te  lengthy. 
bo 
For t he  sake of brevi ty  t h i s  work is omitted. 
To f ind the  complete so lu t ion  t h e  boundary-layer expansions must be added 
t o  the  corresponding straightforward expansions. 
these express: ~ o n s  are 
The f irst  approximations t o  
$, = r 2 / 2  - jr.3/3 - ~ ( 1  - j - (1/2 - j / 3 ) v ) e x p ( - ~ ( 1  - r ) / E ) / S  
= r / 2  - j r2 /3  - ~ ( 1  - j - (1/2 - j / 3 ) v ) e x p ( - ~ ( 1  - r ) / E ) / S  N r o  
N~~ = r - j r 2  t (1 - j - (1/2 - j / 3 ) v ) e x p ( - ~ ( l  - r ) / E )  
B O  = (1 - j - (1/2 - j /3)v)S exp(-S(1 - r ) / E ) / E  
= (1 - v/2)r2 - j ( 1  - v/3)r3 - (1 - j u 
0 
- (1/2 - j/3)v)exp(-S(1 - T I / & )  
w = 3 ( 1  - r 2 ) / 4  - 8 j ( l  - r3)/9 - (1 - j 
0 
- (1/2 - - j / 3 ) v ) ( ~  - exp(-S(l - r ) / E ) )  (24) 
In wri t ing equations (24) the  boundary-layer solut ion w a s  t rea ted  as the  funda- 
mental expansion. A l l  terms i n  the straightforwardexpansion with magnitude 
equal t o  or grea ter  than the  first term i n  the boundary-layer expansion were 
added t o  t h i s  t e r m  t o  form the  first approximation. 
t o  obtain the second and t h i r d  approximations. 
The same method w a s  used 
RESULTS AND DISCUSSION 
Numerical r e s u l t s  were computed f o r  t he  first, second, and t h i r d  approxi- 
mations t o  the var iables  $, N p ,  N e ,  8 ,  u, and w. These calculat ions were made 
f o r  a var ie ty  of values of the  load parameter E and Poisson's r a t i o  v. To 
evaluate the  accuracy of t h e  perturbation method, selected cases were compared 
with numerical solut ions t o  equation (6a) obtained by the  f ini te-difference 
method discussed by Smith, Peddieson, and Chung ( r e f .  3 ) .  It  w a s  found t h a t  
the  t h i r d  approximation t o  the  perturbation solut ion agreed with the  f i n i t e -  
difference r e s u l t s  up t o  E = 0.1. 
values of 
s t e p  s i z e  must be used near t he  edge and the  optimum arrangement of s t ep  s i z e s  
can only be approached by t r i a l  and error .  
the present work are much eas ie r  t o  use for E << 1. 
It should be pointed out t h a t  f o r  s m a l l  
E , ~  the  numerical method is d i f f i c u l t  t o  apply because a var iable  
The exp l i c i t  formulas obtained i n  
580 
To i l l u s t r a t e  t he  behavior of the  solut ion some of the  computed r e s u l t s  
are shorwn i n  f igures  I. -- 4. For the  sake of brev i ty ,  da ta  are presented 
fo r  only the  r a d i a l  stress resu l tan t  Nr, the  transverse stress re su l t an t  Ne, 
and the v e r t i c a l  def lect ion w. The s o l i d  l i nes  represent the  three-term per- 
turbat ion solut ion while the dashed l i n e s  represent t he  l i n e a r  membrane solu- 
t ion.  
from the perturbation solut ion.  
The l i nea r  membrane solut ion is shown only when it d i f f e r s  s ign i f icant ly  
Figures 1 and 2 present r e s u l t s  for uniform pressurizat ion (j = 0) .  
Figure 1 shows 
while Nr does not exhib i t  boundary-layer behavior. A s  E increases the  bound- 
a r y  layers  become wider f o r  a l l  variables.  
Figures 3 and 4 contain r e s u l t s  €or hydrostat ic  loading ( j  = 1). The para- 
metric t rends i l l u s t r a t e d  by these r e s u l t s  are  iden t i ca l  t o  those discussed 
above but t he  behavior of t he  solut ion variables is  more complicated. 
r e s u l t s  i l l u s t r a t e  t he  u t i l i t y  of the perturbation method. 
t i ons  of t h i s  type can be represented numerically only if extreme care is  used. 
t h a t  t h i n  boundary layers  e x i s t  f o r  Ne and w f o r  E = 0.01 
This is i l l u s t r a t e d  by f igure  2, 
These 
Complicated func- 
Results were also computed f o r  several  other values of v .  It w a s  found 
t h a t  the  qua l i ta t ive  behavior of the solut ion is not  s ign i f icant ly  influenced 
by t h i s  parameter. 
CONCLUSION 
In  t h i s  paper, the  ro ta t iona l ly  symmetric moderately large deformation of 
a l i nea r ly  e l a s t i c  shallow conical membrane subjected t o  e i t h e r  uniform or 
hydrostatic pressure w a s  investigated.  A s ingle  d i f f e r e n t i a l  equation having 
a stress function as dependent var iable  w a s  solved by the  method of matched 
asymptotic expansions. The accuracy of the  solut ion was ve r i f i ed  by compari- 
son with a f ini te-difference numerical solut ion of t h e  governing equation for 
the  stress function. Selected r e s u l t s  were presented graphically t o  i l l u s t r a t e  
in te res t ing  features  of t he  solut ions.  
581 
REFERENCES 
1. Van Dyke, M.: Perturbation Methods i n  Fluid Mechanics. Academic Press, 
1964, Chapter 5. 
2. Bromberg, E.; and Stoker, J. J.: Nonlinear Theory of Curved Elastic Sheet. 
Quarterly of Applied Mathematics, Vol, 3, 1950, pp. 27-52. 
3. Smith, Dallas G.; Peddieson, John, Jr.; and Chung, Chin-Jer: F in i t e  Deflec- 
t i on  of a Shallow Spherical Membrane. A I M  Journal, Vol. 11, 1973, 
pp. 736-738. 
4. Rossettos, John N.: Nonlinear Membrane Solutions f o r  Symmetrically Loaded 
Deep Membranes of Revolution. NASA TN D-3297, 1966. 
5. Reissner, E.: Rotationally Symmetric Problems i n  the Theory o f  Thin Elastic 
Shel ls .  Proceedings of the  Third U. S. National Congress of Applied 
Mechanics, 1958, pp. 51-69. 
6. Kraus, H.: Thin Elastic Shells.  John Wiley & Sons, Inc., 1967, Chapter 4. 
582 
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Large deflections of shallow conical membrane

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