Dynamic plastic response of a thin cylindrical shell, immersed in a potential fluid initially at rest and subjected to internal pressure pulse of arbitrary shape and duration, is examined. The shell is assumed to respond as a rigid-perfectly plastic material while the fluid is taken as inviscid and incompressible. The fluid back pressure is incorporated into the equation of motion of the shell as an added mass term. Since arbitrary pulses can be reduced to equivalent rectangular pulses, the equation of motion is solved only for a rectangular pulse. The influence of the fluid in reducing the final plastic deformation is demonstrated by a numerical example

Krajcinovic, D.

English

NASA Technical Reports Server

FLUID-PLASTICITY OF THIN CYLINDRICAL SHELLS* 
Dusan K r a j  cinovic 
University of I l l i n o i s  a t  Chicago Circle 
M. G. S r in ivasan  and Richard A. Valentin 
Argonne National Laboratory 
SUMMARY 
The paper considers dynamic p l a s t i c  response of a t h i n  c y l i n d r i c a l  s h e l l ,  
immersed i n  a p o t e n t i a l  f l u i d  i n i t i a l l y  a t  rest, subjected t o  i n t e r n a l  
pressure  pu l se  of a r b i t r a r y  shape and duration. 
spond as a r ig id -pe r fec t ly  p l a s t i c  material whi le  t h e  f l u i d  i s  taken as 
inv isc id  and incompressible. The f l u i d  back pressure  i s  incorporated i n t o  
t h e  equation of motion of t h e  s h e l l  as an added m a s s  term. Since a r b i t r a r y  
pulses can be reduced t o  equivalent rec tangular  pu l se s ,  t h e  equation of mo- 
t i o n  i s  solved only f o r  a rec tangular  pulse.  The inf luence  of t h e  f l u i d  i n  
reducing t h e  f i n a l  p l a s t i c  deformation i s  demonstrated by a numerical example. 
The s h e l l  i s  assumed t o  re- 
FORMULATION OF THE PROBLEM 
Consider a r ig id - idea l ly  p l a s t i c ,  thin-walled, c i r c u l a r ,  c y l i n d r i c a l  s h e l l  
of i n f i n i t e  length. The s h e l l  i s  surrounded by a pool of p o t e n t i a l  ( i nv i sc id  
and incompressible) f l u i d  i n f i n i t e l y  extended i n  a l l  d i r e c t i o n s .  The s h e l l  
i s  subjected t o  an i n t e r n a l  pressure  pulse  P ( z , t ) ,  varying both  along t h e  a x i s  
and with t i m e .  P ( z , t )  is  f u r t h e r  assumed t o  be  axisymmetric and symmetric i n  
z with respec t  t o  z = 0 ( f i g .  1 ) .  
This paper  examines t h e  inf luence  of t h e  f l u i d  i n  reducing t h e  p l a s t i c  
( res idua l )  deformation of t h e  s h e l l .  
which p o t e n t i a l  f l u i d  resists t h e  motion of a deforming s o l i d  can be con- 
sidered as an increase  i n  t h e  i n e r t i a  of the  s o l i d .  Therefore i n  order t o  
so lve  t h e  problem it is necessary t o  e s t a b l i s h  t h e  so-called e f f e c t i v e  m a s s  
cons is t ing  of t h e  a c t u a l  m a s s  of t h e  s h e l l  and t h e  added ( v i r t u a l )  m a s s  
r e f l e c t i n g  t h e  f l u i d  r e s i s t ance .  Then t h e  problem i s  reduced t o  t h e  ana lys i s  
of a s h e l l  deforming i n  vacuum. 
t h e  no ta t ion  introduced i n  re ference  1. 
It is known t h a t  t h e  p re s su re  wi th  
For t h e  sake of con t inu i ty ,  w e  w i l l  adopt 
GOVERNING EQUATIONS 
The equation of motion of t h e  s h e l l  is: 
Research performed under t h e  auspices of t h e  U. S. Energy Research and 
Development Administration 
* 
627 
https://ntrs.nasa.gov/search.jsp?R=19770003346 2020-03-22T12:11:01+00:00Z
- -  N av a2M
az 
- P - Pf - - -  pH at 2 R 
where M is t h e  axial bending moment, N t h e  c i rcumferent ia l  (hoop) normal 
force ,  R, H and p t h e  rad ius ,  t h e  w a l l  thickness and t h e  m a s s  dens i ty  of t h e  
s h e l l  r e spec t ive ly ,  V t h e  r a d i a l  ve loc i ty  of t h e  po in t s  on t h e  middle su r face  
of t h e  s h e l l ,  and P(z , t )  and P f ( z , t )  are t h e  ex te rna l ly  applied pressure  pulse  
and t h e  back pressure  of t h e  f l u i d  r e s i s t i n g  t h e  motion of t h e  s h e l l  
respec t ive ly .  
W e  assume t h a t  the y ie ld  condition i n  t h e  M,N space i s  defined by the  
l imi ted  i n t e r a c t i o n  curve of f i g .  2 .  The implications of t h i s  assumption are 
discussed i n  d e t a i l  by Drucker ( r e f .  2) and Hodge ( r e f .  3). The y i e l d  values 
and N are given by Y 
1 M = - H R P  
Y 4  0 
where 
H 
= 
0 
N = R P  
Y 
( 3 )  
with  ay being t h e  y i e l d  stress. 
It i s  known (see,  f o r  example, r e f .  1) t h a t  four d i f f e r e n t  phases (modes) 
of p l a s t i c  deformation may occur during t h e  motion. 
only one of these  phases which occurs f o r  a l l  poss ib l e  types of loading, 
though t h i s  restricts t h e  magnitude of t h e  loading t o  a c e r t a i n  l i m i t .  
t h e  considered phase t h e  deformation i s  charac te r ized  by a s t a t i o n a r y  p l a s t i c  
hinge circle a t  z = 0 and two moving hinge c i r c l e s  a t  z = -t < ( t ) .  
We w i l l  consider here in  
I n  
It can be shown (see,  f o r  example, Eason and Shield ( r e f .  4 ) )  t h a t  the  
p l a s t i c  regimes (see f i g .  2) are as follows: 
Y N = N  Regime A 
Y N = N  Regime B 
Y 
Y 
z = 0 :  M =  -M 
Y 
z = < :  M = M  
Y 
O < Z < < :  -M < M < M  N = N  Regime AB 
Y Y 3  Y 
Thus, from t h e  normality of t h e  s t r a i n - r a t e  vec tor  t o  t h e  y i e ld  sur face ,  
(4) 
For t h i s  deformation mode t h e  ve loc i ty ,  V(z , t ) ,  i s  the re fo re  a l i n e a r  function 
of 2, i .e . ,  
628 
I n  t h e  above equations and i n  t h e  sequel because of symmetry i t  is enough t o  
consider only ha l f  of t h e  s h e l l  z 2 0. 
DETERMINATION OF THE ADDED MASS 
Before attempting t o  so lve  equation (11, t h e  backpressure P f ( z , t )  should 
be determined as a func t ion  of V(z,t) and i t s  de r iva t ives .  
governing t h e  flow of t h e  p o t e n t i a l  f l u i d  i s  i n  po la r  coordinates 
The equation 
n n 
1 aF 3°F L - a F + - -  + - = O  i n  r > R  2 r ar a z  2 a r  
(7) 
where F ( r , z , t )  is  t h e  f l u i d  v e l o c i t y  p o t e n t i a l .  
As t h e  s h e l l  is  impermeable, t h e  v e l o c i t i e s  of t h e  f l u i d  and t h e  s h e l l  
a t  t h e  po in t s  of contac t  must be i d e n t i c a l ,  i .e . ,  
- -  a F - ~  a t  r = ~  
3 r  
Furthermore, from t h e  r a d i a t i o n  p r i n c i p l e ,  
aF - 3 0 as max(r,z) -+ m aF v - t o  , - + o  y ar az (9) 
Once t h e  f l u i d  v e l o c i t y  p o t e n t i a l - i s  determined from the  Laplace equa- 
t i o n  (7), subjec t  t o  t h e  boundary conditions (eqs. (8) and (9) )  t h e  pressure  
exerted by t h e  f l u i d  on t h e  s h e l l  can be computed from the  Cauchy i n t e g r a l ,  
where p f  is  t h e  mass dens i ty  of t h e  f l u i d .  
The equations (7)  and (10) imply t h e  assumption t h a t  t h e  per turba t ions  
about average va lues  can be neglected. 
A s  a f u r t h e r  approximation, w e  w i l l  assume t h a t  t h e  func t iona l  r e l a t i o n  
between P f ( z , t )  and 5 i s  not s e n s i t i v e  t o  the  t i m e  dependence of 5 ,  and hence 
5 may be t r e a t e d  as a constant f o r  t h e  determination of t h i s  r e l a t i o n .  Then 
i n  view of equations (6) ,  (7) ai.d (81, w e  may w r i t e  
F ( r , z , t )  = Vo(t) f ( r , z )  (11) 
P f ( Z Y t )  = --Pff(R,Z) dt 
and from equation (10) , 
(12) 
0 
dV 
where -pff(R,z) i s  t h e  added ( v i r t u a l )  mass a r i s i n g  due t o  t h e  r e s i s t a n c e  of 
t h e  f l u i d  being displaced by t h e  s h e l l .  
Subs t i t u t ing  equations (11) and ( 6 ) ,  (with 5 being constant)  i n t o  equa- 
t i ons  (7) t o  (9) ,  i t  follows 
629 
a2f + - - + - =  1 af a2f 0 
2 r ar az  2 
-
ar 
s u b j e c t  t o  
and 
as max(r,z) -t m (15) a f  af - + o  y a r  a Z  - + o  f + O  y 
The d e t a i l s  of the s o l u t i o n  of equat ion  (13) are omit ted h e r e i n  f o r  t h e  
sake of b r e v i t y .  
ducing t h e  F o u r i e r  cos ine  t r ans fo rm.  The argument of t h i s  i n t e g r a l  is 
r a t h e r  complicated and t h e  i n t e g r a t i o n  is performed i n  t h r e e  s t a g e s  us ing  
asymptot ic  formulae and F i l o n ' s  method, s u b j e c t  t o  the r e s t r i c t i o n ,  5 L R 
which i s  subsequent ly  seen  t o  be  n o t  severe. In o r d e r  t o  make t h i s  numerical  
s o l u t i o n  amenable f o r  s u b s t i t u t i o n  i n t o  equat ion  (11, the r e s u l t  is  sub jec t ed  
t o  a series of polynomial r e g r e s s i o n  ana lyses .  
A c losed  form i n t e g r a l  s o l u t i o n  is obta ined  a f t e r  i n t r o -  
F i n a l l y  w e  o b t a i n  
where 
and 
2 3 
go(x) = Bo + B1x + B 2 x + B 3 x 
3 g,(x) = a  + a x + a x 2 + a x  2 3 0 1  
a = .004994512 Bo = .02050149 0 w i t h  
a = -.5420473 El  = 1.664447 
1 
2 
3 
= -.lo58701 B 2  = -1.105309 
a = .1627719 B3 = .4096600 
Note a i  and Bi are dimensionless  cons t an t s  t h a t  do not  depend on t h e  s h e l l /  
f l u i d  parameters .  
PLASTIC DEFORMATION OF THE SHELL 
Equation (1) now becomes, i n  view of equat ions  (2 ) ,  ( 6 ) ,  (12) and (16) ,  
6 30 
I n  equation (20), 5 = z;(t). For a r b i t r a r y  P (z , t )  t h e  above equation may only 
be solved by numericalmethods. A s  a f i r s t  s t e p  i n  simplifying t h e  work, t h e  
approach introduced by Youngdahl ( r e f .  1) w i l l  be used t o  approximate a com- 
plex loadiag func t ion  by (i.e., c o r r e l a t e  i t  to )  a simple rec tangular  pulse.  
Since t h e  standard l i m i t  ana lys i s  of t h e  s h e l l  is independent of any sur- 
rounding medium, t h e  method given by Youngdahl ( r e f .  1) t o  determine t h e  
equivalent rec tangular  pu lse  does not need any modification i n  t h i s  case. 
Thus co r re l a t ed ,  P ( z , t )  can be  expressed as 
P (z , t )  = P 121 5 Le and 0 5 t 5 te e 
P ( z , t )  = 0 IzI > L~ o r  t > te 
where Pe is  t h e  magnitude, te t h e  dura t ion  and 2Le t h e  length  of t he  loaded 
area of t h e  equivalent rec tangular  pu lse  (see r e f .  1 f o r  t h e i r  de r iva t ion ) .  
For p l a s t i c  deformation t o  occur, Pe must be  g r e a t e r  than the  l i m i t  load. 
This condition is  expressed by (see  r e f .  1) 
For t h e  deformation t o  take  p lace  i n  t h e  assumed phase, t h e  following 
boundary conditions must be  s a t i s f i e d  
M = -M _ -  a M - ~  a t  Z = O  
Y Y  az 
M = M  - =  aM 0 a t  z = z ;  Y Y  a Z  
Further,  t h e  condition t h a t  t h e  bending moment does not exceed a t  t h e  
hinge c i r c l e  a t  z = 5 implies 
f i < o  2 a t  z = s  (24) 
az 
I n  add i t ion  t h e  condition t h a t  t h e  bending moment cannot be less than -Mu a t  
t h e  hinge circle a t  z = 0 implies 
a Z "  
For t h e  i n t e r v a l  0 5 t 5 te, a t r ia l  so lu t ion  as i n  t h e  case of a s h e l l  
leforming i n  vacuum is  assumed. This is taken i n  t h e  form 
O 5 t s t e  
2nd Vo(t) = - K1 t 
PH 
There z 1  and K1 are constants.  
md in t eg ra t ing  t w i c e  sub jec t  t o  t h e  boundary conditions (23) y i e l d s  i n  t h e  
Subs t i t u t ing  equation (26) i n t o  equation (20) 
631 
end two equations f o r  z l  and K1.  These two can be reduced to :  
and 
6pH(PeLez1 - PeLe 2 - Porn) 
- 
K1 z: {PH - P f  Rgl (:)} 
where 
The i n e q u a l i t i e s  ( 2 4 )  and (25) can be w r i t t e n  i n  the  form 
2 
e e  and 4PeLez1 - Pezi - 3P L - 3P0RH < 0 
From equations (17), (18) and (19) i t  i s  e a s i l y  v e r i f i e d  t h a t  i nequa l i ty  
(30) is  always s a t i s f i e d .  I f  i nequa l i ty  (31) is not s a t i s f i e d  motion cannot 
start  i n  t h e  phase assumed. 
t h e  bounding value. 
equations (27) and (31) should be solved simultaneously. This is done 
numerically. 
e q u a l i t i e s  (22) and (31) and hence g ives  rise t o  deformation i n  t h e  assumed 
mode. For a Pe belonging t o  t h i s  range, t h e  non-linear equation (27) may be 
solved numerically. Also it may be v e r i f i e d  t h a t  t h e  r e s t r i c t i o n  z1  < R is 
always s a t i s f i e d  i f  Le < R. 
v a l i d  f o r  Le < R and Pe s a t i s f y i n g  i n e q u a l i t i e s  (22) and (31). 
When t h e  inequa l i ty  becomes an equa l i ty ,  Pe takes  
To determine t h e  bounding value of Pey t h e  non-linear 
Figure 3 shows t h e  range of va lues  of Pe t h a t  s a t i s f i e s  in- 
Thus, t h e  s o l u t i o n  discussed i n  t h i s  paper is 
For t > te, t h e r e  is  no i n t e r n a l  pressure .  Le t t ing  P ( z , t )  = 0 i n  equa- 
t i o n  (20), i n t e g r a t i n g  i t  t w i c e  with respec t  t o  z and s u b s t i t u t i n g  t h e  r e s u l t  
i n t o  t h e  boundary conditions (23), w e  arrive a t  t h e  following equations 
Equations (32) and (33) c o n s t i t u t e  a system of non-linear f i r s t  order d i f f e r -  
e n t i a l  equations f o r  Vo and 5 ,  t h e  i n i t i a l  conditions being given by 
632 
K t  l e  Vo(te) = pH ( 3 5 )  
The above d i f f e r e n t i a l  equations are v a l i d  only f o r  t 
defined by 
I t I t e f )  where tf is 
V0(tf) = 0 ( 3 6 )  
W e  w i l l  denote 
5f E 5 ( t f >  ( 3 7 )  
From equations ( 3 2 )  and ( 3 3 ) ,  we can express V as, 
0 
From (36) and ( 3 7 ) ,  w e  see t h a t  G f  can be obtained from t h e  equation, 
9 
3 m  + 5; 
- = o  6RH ( 3 9 )  
Equation ( 3 9 )  can be solved numerically t o  ob ta in  S f .  It is  noted t h a t  
cf depends only on t h e  s h e l l  parameters H and R and t h e  dens i ty  r a t i o  p f / P .  
Since 5f is t h e  quant i ty  t h a t  i s  known and not t f ,  t h e  equations (32) and 
( 3 3 )  are now reformulated wi th  5 being t h e  independent v a r i a b l e  and t ( C )  and 
V (<) being t h e  dependent var iab les .  Thus, 
0 
The new i n i t i a l  conditions are 
Equations ( 4 0 )  and (41) are solved numerically. F ina l ly ,  t h e  maximum 
p l a s t i c  deformation U ( t )  can be obtained as, 
0 
( 4 4 )  
633  
o r  (45) 
The i n t e g r a l  i n  equation (45) can be numerically evaluated a f t e r  t h e  numeri- 
cal so lu t ions  Vo(<) and t ( < )  have been obtained. 
For t h e  s p e c i a l  case i n  which z 1  coincides with < f ,  t h e  s o l u t i o n  f o r  
t > te discussed above i s  not  v a l i d .  
wi th  (35) y i e l d ,  
For t h i s  case, equations (32) and (33) 
< = z  - 1 - <f 
where @(<) is defined i n  equation (32). Note Q , ( z l )  < 0. From equation (47) 
w e  see 
t f  - t e  (1 pHi:zl)} 
From equations (47) and (44) w e  can show t h a t  
F ina l ly  w e  have, f o r  t h e  maximum p l a s t i c  deformation i n  t h i s  s p e c i a l  case 
0 
NUMERICAL EXAMPLE 
For a s h e l l  with H/R = 1/36, Le/R = 1 / 4  surrounded by a f l u i d  of 
p f / p  = 1/10, t h e  complete numerical s o l u t i o n  i s  determined f o r  t h e  admissible 
range of loads Pe. 
g ive  rise t o  motion i n  t h e  assumed phase is  between: 1.33 and 2.97. The 
same range f o r  a s h e l l  i n  vacuum is 1.33 t o  2.19. Figure 4 shows the  f i n a l  
maximum p l a s t i c  deformation, Uo( t f ) ,  (non-dimensionalized as pHUo/Pot$ ) as a 
func t ion  of Pe/Po. 
s h e l l  deforming i n  vacuum is a l s o  shown i n  t h e  same f igure .  
A s  is  seen from f i g u r e  3, t h e  range of Pe/Po t h a t  w i l l  
For t h e  sake of comparison t h e  corresponding curve f o r  a 
I n  t h e  numerical methods used, non-linear a lgebra ic  equations such as 
(27) and (39) w e r e  solved by Newton's i t e r a t i o n  method and t h e  system of 
634 
d i f f e r e n t i a l  equat ions  (40) ,  (41) by a method us ing  automatic  s t e p  change 
( r e f .  5).  
REFERENCES 
1. Youngdahl, C. K.: Cor re l a t ing  t h e  Dynamic P l a s t i c  Deformation of a 
Circular C y l i n d r i c a l  S h e l l  Loaded by a n  Ax ia l ly  Varying Pressure .  
Report  ANL-7738, Argonne Na t iona l  Laboratory,  1970. 
2. Drucker, D. C. :  L i m i t  Analysis  of C y l i n d r i c a l  S h e l l s  Under Axially- 
Symmetric Loading. Proc. of the F i r s t  Midwest Conf. on S o l i d  Mech., 
1953, pp. 158-163. 
3. Hodge, P. G.: L i m i t  Analysis  of Ro ta t iona l ly  Symmetric P l a t e s  and S h e l l s .  
Pren t ice-Hal l  Inc., Englewood C l i f f s ,  N.J., ch. 3, 1963. 
4. Eason, G.;  and Sh ie ld ,  R. T.: Dynamic Loading of Rig id-Plas t ic  
C y l i n d r i c a l  S h e l l s .  J. Mech. Phys. S o l i d s ,  v o l .  4 ,  1956, pp. 53-71. 
5. Chr is t ianson ,  J.: Numerical So lu t ion  of Ordinary Simultaneous D i f f e r e n t i a l  
Equat ions of 1st Order us ing  a Method of Automatic S tep  Change. 
Numer’ische Mathematik, vo l .  14 ,  no. 4, 1970, pp. 317-324. 
635 
F i g u r e  1.- C i r c u l a r  c y l i n d r i c a l  s h e l l  
immersed i n  f l u i d  and loaded by 
i n t e r n a l  p r e s s u r e  pulse .  
N 
3 Y  B 
D C 
Figure  2.- Yield cond i t ion .  
636 
IO 
9 
8 
7 
6 
LL IN VACUUM) 
MOTION NOT IN 
ASSUMED PHASE Pe /Po 5 
4 
3 
2 
I 
0 0.6 1.2 1.8 2.4 3 .O 
Le &H
Figure  3 . -  Range of p u l s e  i n t e n s i t y  
i n i t i a t i n g  mot ion - in  assumed phase. 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0. I
0 
1.5 I .7 I .9 2.1 
'e '0 
Figure  4.- Maximum p l a s t i c  deformation 
as a f u n c t i o n  of p u l s e  i n t e n s i t y .  
637 


Fluid-plasticity of thin cylindrical shells

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