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sphere intersecting pressure vessels
NUCLEAR STRUCTURAL ENGINEERING 2 (1965-) 169-180. NORTH-HOLLAND PUBLISHING COMP., AMSTERDAM
THE COLLAPSE INTERSECTING
OF CYLINDERISPHERE PRESSURE VESSELS *
F. ELLYIN and A. N. SHERBOURNE Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada
Received 24 May 1965
A general method is presented for finding lower and upper bounds on the collapse pressure for a plastic-rigid vessel composed of intersecting cylindrical and spherical shells obeying Tresca's yield condition and associated flow rule. Depending upon the ratio (t/d)/(T/D) different types of solutions are generated. The solutions are presented in terms of generalized stresses and may be used in conjunction with any choice of yield surface of the form ~(N~p,No, Mq)) = C 2.
1. INTRODUCTION In r e c e n t y e a r s the p r o b l e m of i n t e r s e c t i n g s h e l l s has b e e n the subject of study by a n u m b e r of d i s tinct groups. T h e s e i n v e s t i g a t i o n s have b e e n s y s t e m a t i c a l l y r e v i e w e d by M e r s h o n [1]. So far, much of this work has been of an e x p e r i m e n t a l n a t u r e (mostly t h r e e d i m e n s i o n a l p h o t o - e l a s t i c studies) and a s u m m a r y of the r e l e v a n t e x p e r i m e n t s is given by Leven [2]. It should be pointed out, however, that t h e r e exist some d i f f e r e n c e s b e t w e e n the r e s u l t s of p h o t o - e l a s t i c studies and steel model t e s t s . N e v e r t h e l e s s , p h o t o - e l a s t i c r e s u l t s may be used as a guide for checking t h e o r e t i c a l a n a l y s e s . A n a l y t i c a l l y , the p r o b l e m has been studied f r o m two different points of view, b a s e d upon the t h e o r i e s of e l a s t i c s h e l l s and p l a s t i c - r i g i d solids. Although some r e s u l t s have been obtained f r o m "two-piece" e l a s t i c shell t h e o r y by W a t e r s [3] and others [1], a s a t i s f a c t o r y solution has yet to be obtained. On the b a s i s of p l a s t i c - r i g i d t h e o r y the solutions proposed by Lind [4] and Gill [5] might be mentioned. The a s s u m p t i o n s and s h o r t c o m i n g s of these t h e o r i e s have been the subject of review by E l l y i n and S h e r b o u r n e [6]. The p r e s e n t i n v e s t i g a t i o n is an attempt to i m p r o v e on existing t h e o r i e s ; some of the r e s u l t s of p r e vious work may be d e r i v e d as p a r t i c u l a r c a s e s of this m o r e g e n e r a l approach. The p r o b l e m has b e e n f o r m u l a t e d and r e s u l t s are obtained in t e r m s of g e n e r a l i z e d s t r e s s e s which may be used in a s s o c i a t i o n with one of the many yield s u r f a c e s of g e n e r a l f o r m ~(N(p,No,Mgo) = C 2. Depending upon the shell geo m e t r y , types of solutions a r e obtained which differ in r e s p e c t of the position of the c e n t r a l hinge at the i n t e r s e c t i o n . F o r design p u r p o s e s , however, a p r o c e d u r e is c l e a r l y e s t a b l i s h e d r e l a t i n g the bounds on the collapse p r e s s u r e to the g e o m e t r i c a l p a r a m e t e r s of the two shells.
2. YIELD CONDITION In g e n e r a l , the l i m i t i n g c o m b i n a t i o n of i n t e r n a l force i n t e n s i t y r e q u i r e d to develop full p l a s t i c i t y in a shell of r e v o l u t i o n is given by an equation of the f o r m : f(N~a,Yo,Mq~,Ms) = C 2 , where C is a constant. The effect of t r a n s v e r s e s h e a r on the yielding i s neglected. A yield h y p e r s u r f a c e for shells of r e v o l u t i o n subject to a x i s y m m e t r i c a l loading has been d e s c r i b e d by Onat and P r a g e r [7]. In * Accepted by A. Sawczuk.
170
F. ELLYIN
and A. N. SHERBOURNE
N--B=n8 No
_~b_ n~or n x No -
A
I
N8 =no) No c~ A ~
.' /
d
No=n~
°rnx
1 "~-~" "~e=m~°r mx
4 ~ : ;
M' = mc~°rmx
D
Fig. 1. Yield surface.
Fig. 2. Yield hexagonal prism.
c y l i n d r i c a l s h e l l s , the c i r c u m f e r e n t i a l bending m o m e n t M 0 does not appear in the equations of e q u i l i b r i u m ; it is an induced or p a s s i v e value d e t e r m i n e d explicitly w h e n N ¢ , N o and M e lie on the yield s u r face. Fig. 1 shows a yield s u r f a c e for a solid c y l i n d r i c a l shell composed of p l a s t i c - r i g i d m a t e r i a l obeying the m a x i m u m s h e a r s t r e s s (Tresca) yield condition and a s s o c i a t e d flow r u l e . It h a s been d e s c r i b e d independently by Hodge [8] and Onat [9]. An a p p r o x i m a t i o n to this s u r f a c e , suggested by D r u c k e r and Shield [10], is shown in fig. 2, c o n s i s t i n g of a somewhat s i m p l e r hexagonal p r i s m which leads to a s u r , i face i n s c r i b e d within the o r i g i n a i yield s u r f a c e when its v e r t i c e s are r e d u c e d by a f a c t o r ~(52 - 1). A l though this yield s u r f a c e is d e r i v e d for thin c y l i n d r i c a l s h e l l s , Drucker and Shield [10, 11] contend that this c o n s t i t u t e s a r e a s o n a b l e a p p r o x i m a t i o n to the yield s u r f a c e for any s y m m e t r i c a l l y loaded shell of revolution.
3. YIELD MECHANISM The a x i s y m m e t r i c i n t e r s e c t i n g shell configuration to be a n a l ysed is shown in fig. 3. When the p r e s s u r e i n t e n s i t y is i n c r e a s e d to a c e r t a i n value, the p l a s t i c - r i g i d shell will begin to deform in different locations depending upon the shell geometry. In the case X where both c y l i n d e r and sphere have a p p r o x i m a t e l y equal m e m b r a n e s t r e n g t h s and no r e i n f o r c e m e n t at the junction, p l a s t i c d e f o r m a t i o n will f i r s t o c c u r in this region. P h o t o - e l a s t i c studies by T a y l o r et al. [12] c o n f i r m that there a r e high c i r c u m f e r e n t i a l s t r e s s e s and m e r i d i o n a l bending m o m e n t s p r e s e n t at the zone of / / i n t e r s e c t i o n . It s e e m s r e a s o n a b l e , t h e r e f o r e , to a s s u m e that the f i r s t hinge c i r c l e will f o r m at the junction, the g e o m e t r y of this r e g i o n being n e i t h e r c y l i n d e r nor sphere but, in fact, a r e g i o n of t r a n s i t i o n between the two shells. The deformation, however, will at f i r s t be s m a l l and the p r e s s u r e i n t e n s i t y may continue to i n c r e a s e until yielding takes place in the m e r i d i o n a l plane at two additional locations. Collapse o c c u r s only when the f o r m a t i o n of the Fig. 3. Geometrical relations. m e c h a n i s m is complete. In c a s e s where the i n t e r s e c t i o n is r e i n f o r c e d (by additional m a t e r i a l , welds, etc.) the yielding in the junction is b i a s e d toward e i t h e r sphere or c y l i n d e r depending upon the r e l a t i v e s t i f f n e s s e s of the two shells as m e a s u r e d by the r e l a t i o n s h i p t / T = n d / D . F o r n > 2, the collapse m e c h a n i s m will c o n s i s t of two hinge c i r c l e s in the s p h e r i c a l shell, one located near the t r a n s i t i o n zone and the other some d i s -
THE COLLAPSE OF CYLINDER/SPHERE INTERSECTING PRESSURE VESSELS
171
tance away; the t h i r d hinge c i r c l e will f o r m in the c y l i n d e r somewhat r e m o v e d f r o m the zone of i n t e r section. C o n v e r s e l y , in the case of n < 2, the collapse m e c h a n i s m will contain two hinge c i r c l e s in the c y l i n d r i c a l shell and one in the spherical shell. F o r a range of s m a l l values n the collapse m e c h a n i s m will be confined a l m o s t e n t i r e l y to the c y l i n d e r and the p r o b l e m may be t r e a t e d as d e s c r i b e d e l s e w h e r e [9]. C o n v e r s e l y , for l a r g e values of n, t h e r e will be a m e c h a n i s m a s s o c i a t e d a l m o s t e n t i r e l y with the sphere. In this paper, c o n s i d e r a t i o n is given to c o m b i n e d m e c h a n i s m s which n o r m a l l y occur in p r e s s u r e v e s s e l s when the shell s t i f f n e s s e s a r e of the s a m e o r d e r . In the a n a l y s i s which follows, the o r d e r at which hinge c i r c l e s f o r m is of no c o n s e quence. R e g a r d l e s s of the i n t e r m e d i a t e hinge, however, t h e r e m u s t be c i r c u m f e r e n t i a l d i s p l a c e m e n t between the hinge c i r c l e s in c y l i n d e r and s p h e r e , accompanied by an outward d i r e c t e d r i g i d body motion of the c y l i n d r i c a l p a r t , in o r d e r to yield a compatible m e c h a n i s m . Such a m e c h a n i s m c o r r e s p o n d s to face I of the yield s u r f a c e s shown in figs. 1 and 2. The n o r m a l i t y condition of the s t r a i n rate vector thus i m p o s e s the condition: ~p and dx = 0,
~
and )
and ~e = u n r e s t r i c t e d
(1)
between the hinge c i r c l e s . At the hinge locations, the s t r e s s state is r e p r e s e n t e d by the parabolic axes of the yield s u r f a c e of fig. 1, or by the edges p a r a l l e l to AB of the yield hexagonal p r i s m of fig. 2. The s t r a i n rate v e c t o r , in t h i s case, has two c o m p o n e n t s which impose the condition: d(p and dx = 0,
)~o and Xx = u n r e s t r i c t e d ,
d~; = u n r e s t r i c t e d .
(2)
4. U P P E R BOUND ANALYSIS An upper bound on the collapse p r e s s u r e can be found by equating the e x t e r n a l rate of doing work to the i n t e r n a l rate of e n e r g y dissipation for a k i n e m a t i c a l l y a d m i s s i b l e p a t t e r n of t h r e e hinge c i r c l e s . If n o r m a l s to the u n d e f o r m e d middle s u r f a c e continue to r e m a i n n o r m a l in the bent configuration, the s h e a r i n g f o r c e Q is not a g e n e r a l i z e d s t r e s s but has the n a t u r e of a r e a c t i o n . Also, since we a r e conc e r n e d with the a x i a l l y s y m m e t r i c d e f o r m a t i o n s of the shell, component V of the velocity in the c i r c u m f e r e n t i a l d i r e c t i o n is z e r o , and we need to c o n s i d e r only the components U and W in the m e r i d i o n a l and radial directions. A velocity field of the f o r m V = C[1 - cos(/3 -(p)] ,
W = -C sin(/~ -~p)
(3)
s a t i s f i e s the c o n s t r a i n t s (1) and (2) for the s p h e r i c a l p a r t of the shell, /3 being the angle which locates the position of the hinge c i r c l e . The p r i n c i p a l s t r a i n s and c u r v a t u r e r a t e s a r e then given by
1 ~dU -
- w) =0 ,
d e = ~1 - ( U c o t e - w) = RsinC
_1
(cos~0 -cos/3) ,
R
d (U~-)=0 ,
)(0
cot~0 ( u
dW
C
(4)
:R-2eot
An e x a m i n a t i o n of the o r d e r s of magnitude of v a r i o u s t e r m s in the e x p r e s s i o n for the rate of work done by the s t r e s s s y s t e m indicates that, when the t h i c k n e s s of shell T is s m a l l c o m p a r e d with the local r a d i u s r o , the r a t e at which work is done by the s t r e s s s y s t e m on the change in c i r c u m f e r e n t i a l c u r v a t u r e i s s m a l l c o m p a r e d with r a t e of work done on the change in c i r c u m f e r e n t i a l s t r a i n . T h e r e f o r e , the c o n t r i b u t i o n of c i r c u m f e r e n t i a l bending m o m e n t M e to the work equation can be neglected. This point has b e e n d i s c u s s e d by D r u c k e r and Shield [10]. The e x t e r n a l and i n t e r n a l work done by the f o r c e s of fig. 4 a r e given by:
v
Iw = 2 ~ r M c ~ + 2 ~ r M k
(v~
+ e
)
+ 2 n R M s O cosB + ~ l v N c t 2 ~ R e N s [ s i n / 3
E w = ~rlvP + nr26p
- s i n a - (/3-a) cos/3] ,
+ n R 3 0 [sin(/3 - a ) c o s a - (/~ - a ) cos/3] P ,
where M c ~< ¼a0t2, N c --< cr0t, M k --< ¼a0t2 or ¼ g o T 2 , M s ~ ¼cr0T2 and N s ~< gOT.
F. ELLYIN and A. N. SHERBOURNE
172
/I
7°
/
Jj
/I
i,
Z
f
I Fig. 4. Force-deformation scheme. Since no e n e r g y c a n be s t o r e d in the p l a s t i c - r i g i d s h e l l , the t o t a l e n e r g y d i s s i p a t i o n m u s t e q u a l the w o r k done by the a p p l i e d load, i.e. E w = Iw .
(5)
The c o n t i n u i t y of d i s p l a c e m e n t at the j u n c t i o n r e q u i r e s that v = OR(cosa
- cosB) ,
5 = 0R(sin~ - sina) .
(6)
I n s e r t i o n of e q s . (6) and (5) w i l l l e a d to an u p p e r bound on the c o l l a p s e p r e s s u r e
[R2p- 2NsR ] sin/3
- It(/+ R cos a)P - ~
(M c + M k) + 2M s - lNc] cos/3 + (/~ - a) [2NsR - R 2 p ] c o s B
+ [Prl -~(Mc+Mk)
- /Nc] c o s a + R [ 2 N s R
-Pr 2 -2Mk] = 0.
(7)
Eq. (7) is a f u n c t i o n of two v a r i a b l e s , ~ and l, w h i c h s t i p u l a t e the hinge l o c a t i o n s in s p h e r i c a l and c y l i n d r i c a l s h e l l r e s p e c t i v e l y . A l i m i t i n g v a l u e o f P is o b t a i n e d f r o m eq. (7) by m i n i m i z i n g with r e s p e c t to t h e s e two v a r i a b l e s to y i e l d the f o l l o w i n g e x p r e s s i o n s : [ 2 ( M c + Mk)]½ l = L (Yc/r) -PJ
r(l + R c o s a ) P and
~ =a +
- ( 2 r / / ) ( M c + M k ) + 2M s - l Y c 2NsR
- R2p
S u b s t i t u t i o n of l and ~ f r o m e q s . (8) in eq. (7) w i l l g i v e the l o w e s t v a l u e of p o s s i b l e u p p e r b o u n d s the c o l l a p s e p r e s s u r e .
(8) for
5. L O W E R BOUND ANALYSIS F o r the c y l i n d r i c a l p a r t of the s h e l l , the e q u a t i o n s of e q u i l i b r i u m , with the n o t a t i o n of fig. 5, a r e g i v e n by:
No ~
+
dMx =P '
dx
=v '
Nx = ½Pr "
(9)
E q s . (9) m a y be i n t e g r a t e d and t h e c o n s t a n t s of i n t e g r a t i o n c a n be found by i n v o k i n g a s u i t a b l e s t r e s s f i e l d N o = N c f o r 0 ~< x --< l, in c o n j u n c t i o n with t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s :
173
T H E C O L L A P S E O F C Y L I N D E R / S P H E R E I N T E R S E C T I N G P R E S S U R E VESSELS
\ Nx - - - ~
cb
----.:- ~.---~j.~-
R
,," % ,"w ~-,'l ~X
No
d~ Fig. 5. Notation for cylindrical part of shell. Q =0, M=Mcwhenx=l
Fig. 6. Notation for spherical part of shell. ,
Q =Q', M=Mkwhenx
=O,
where
IMcl ~<¼~0t2 ,
[Ncl
~< a0 t ,
IMkl --< ¼~0t2 or ¼a0 T2 .
This procedure yields:
o
. ~ __~(~ __})~2 + Q,~ + Mk,
Nx : ~ ,
(lO)
where Q' =
(Mc - M k)
-
,
I = L(Nc/r )
_pj
.
(11)
S i m i l a r l y f o r the s p h e r i c a l p a r t of the s h e l l , the e q u i l i b r i u m e q u a t i o n s with the n o t a t i o n of fig. 6, become Nq~ sinq~ + Qcos~o = ½PR sin~o , N~sin~
+ N Osinq~ + ~d (Q sin~o) = P R s i n ~o
dM¢
dq~ sinq~ + (Mcp - M 0 ) c o s ~ o - QR s i n e = 0 .
(12a) (12b) (12c)
As i n d i c a t e d p r e v i o u s l y [10], the c o n t r i b u t i o n of the s e c o n d t e r m in eq. (12c) to the load c a r r y i n g c a p a c i t y of the s h e l l is s m a l l when c o m p a r e d with the r e m a i n i n g t e r m s a n d m a y , t h e r e f o r e , be o m i t t e d . Eq. (12c) t h u s r e d u c e s to: dM~o d e sinq~ - QR sin(p = 0 .
(13)
E q s . (12a), (I'2b) t o g e t h e r with eq. (13) c a n now be i n t e g r a t e d ; the c o n s t a n t s of i n t e g r a t i o n a r e d e t e r m i n e d by i n v o k i n g the s t r e s s d i s t r i b u t i o n NO = N s f o r a --< ¢ ~< 8 along with b o u n d a r y c o n d i t i o n s when q~ = 8, Q = 0, Mq~ = M s ,
when q~ = a, Q = Q", Mq~ = M k.
(14)
T h i s f i n a l l y l e a d s to: V = (½PR -Ys)(q~ - 8) ,
Mq~ = ~ZR (~/-" I P R -gs)(q~ - 8 ) 2 + M s ,
Nq~ = ½PR - Q cot~o ,
where Q, :
1
(M s _ M k )(N s__SPR
,
V
Ms-Mk 1½ /3 = a + L2R(Ns _ ~PR)J "
(15)
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F. ELLYIN and A. N. SHERBOURNE
6. EQUILIBRIUM OF FORCES AT THE JUNCTION Fig. 7 shows the e q u i l i b r i u m of s h e a r and m e m b r a n e f o r c e s from which the following i d e n t i t i e s r e s u l t : Q" c o s a + N a s i n a = ½ P r ,
N a c o s a - Q" s i n a = Q' ,
(17)
leading to
(18)
Q " = ½ P r c o s a - Q' s i n a .
Substituting for Q' and Q" from eqs. (11) and ~1~) r e s p e c t i v e l y in eq. (18) and noting that s i n a = r / R , c o s a = [1 - ( r / R ) Z ] ~ , yields
Fig. 7. 1
(Ms-Mk)(Ns-½PR
( M c - M k)
~ + ,~
= ½Pr[1 - (r/R)2] ~ .
-
(19)
Once the s t a t i c a l l y a d m i s s i b l e s t r e s s field and shell g e o m e t r y a r e specified, the above e x p r e s s i o n will yield a lower bound on the collapse p r e s s u r e . An a p p r o p r i a t e statically admissible, s t r e s s field between the hinge c i r c l e s will c o r r e s p o n d to face I of either yield loci. F o r the c y l i n d r i c a l p a r t of the shell lying outside the hinge location, i.e. for x > l, a s t a t i c a l l y a d m i s s i b l e s t r e s s field which s a t i s f i e s both end conditions and e q u i l i b r i u m equations for a rigid, fixed ended c l o s u r e , may be adequately d e s c r i b e d by the following: Mx =Mc ,
N 0 = Pr ,
N~
-~ ± 2 Pr °
F o r a free end at distance b, the s t r e s s s y s t e m b e c o m e s : Mx = Mc
(b+ x - 2l)(b (b - / ) 2
- x)
'
2Mcr NO = P r + (b - / ) 2 '
Nx = ½Pr .
S i m i l a r l y for the s p h e r i c a l part of the p r e s s u r e v e s s e l , the equations M¢ = M s ,
N(p = ½ P R ,
N e = l~p R
forq~ >/3
constitute a s t a t i c a l l y a d m i s s i b l e s t r e s s field which s a t i s f i e s the boundary conditions and nowhere v i o lates the yield surface.
7. RESULTS F o r the p u r p o s e s of d e r i v i n g typical n u m e r i c a l r e s u l t s f r o m the foregoing a n a l y s i s , the p r i s m of fig. 2 is chosen as the yield condition. On substituting for i n t e r n a l f o r c e s c o r r e s p o n d i n g to this yield s u r f a c e in eqs. (11), (16)and ( 1 9 ) r e s p e c t i v e l y and n o n - d i m e n s i o n a l i z i n g , the following r e s u l t s a r e obtained: Lower bound (a) n < 2
(20)
Is
- ~'l/J
+ 2~
(1
- D2 j p~ ;
THE COLLAPSE OF CYLINDER/SPHERE INTERSECTING PRESSURE VESSELS
175
(b) n > 2 t=2
[ ( 1 - 2 d T ~s~| T~I]
[~-(I+~) (D t
'
/3
=
sln
~
+
11-~Z~]
(21)
'
~
s)]½ D [ . T
(1-p~,]½
[1
d2] ½
where p~ = P/(aot/r), p[ = P/(2~oT/R), and t, T, d, D are thickness and diameter of cylindrical and spherical shell respectively. A similar substitution of internal f o r c e s corresponding to this yield surface in eqs. (8) and (7) r e spectively leads to non-dimensional equations of the form: Upper bound l __d2_½ c t d TT l t / =[ d/t l $ . -1 d D+ l - D 2 j } P u - 7 ~ + ~ t - - 2 7 ~ (22) 7 L2(l_pC)j ' /3=sin ~+ T D c '
2 T - ~ Pu
pC - 2 T d ] u
[
sin/3
d2 l[ 2 ~I t + D~
-
~-(I
c 2"½7 d J Pu - [t + tT Td - 2~//_fcos/31t + (/3 - sin- I ~d) - D~)
,dt[1
lt dt - lt - 2[
- D2 j +
i2Td I}- _pC]cos~ u [ d - Pu -
l
=0;
(b) n > 2 d I
1
T2\
7
dl2lt ,
/3 = sin-1 d +D D
[D +
1 - pzs
1
[Pu - 1] sin/3 -
~
+ (1 _D~d 2"~1)] ps
+ DtD
4It TIDal (1 + ~2) + 2D - t T _I ~ l c to sI/ 3 +
u-21Z) 7
~
-1~
D2 j +/5
t [/3_sin_ld
-D2PU-
(l _puSS)cos l3
=0.
(23)
The eqs. (20) to (23) w e r e p r o g r a m m e d on a computer and the results for various shell g e o m e t r i e s are plotted in figs. 8 to 12. T h e s e results may be useful for design purposes. For purposes of direct comparison, upper and lower bounds are plotted in figs. 13 and 14 for typical shell p a r a m e t e r s . It can be shown that for n = 2 and t = T, the collapse p r e s s u r e as given by the curves of figs. 10 and 11 will be identical r e g a r d l e s s of whether the central hinge f o r m s in sphere or cylinder. This implies an identity between eqs. (20) and (21) and between (22) and (23).
8. CONCLUSIONS (i) The computations of lower bound based on an inscribed and upper bound upon a c i r c u m s c r i b e d hexagonal yield surface of fig. 2 will differ considerably. Since a t h r e e - q u a r t e r size p r i s m lies within the actual yield surface o v e r an extended range, the actual value of collapse p r e s s u r e deduced f r o m the exact yield surface will almoBt always lie between 3/4p and p. It seems reasonable, t h e r e f o r e , to assume 7/8p as being sufficiently close to the actual value. This argument brings upper and lower bounds, plotted in figs. 8 to 14, even c l o s e r together. For design purpose, however, the value p = 7/16(Pl+pu) will be quite reasonable. This theory seems to be in good agreement with the experimental results reported by Cloud [13]. (ii) The method d e s c r i b e d herein, calculates bounds on the collapse p r e s s u r e in t e r m s of generalized s t r e s s e s . The value of generalized s t r e s s M k is known once the g e o m e t r y of p r e s s u r e vessel is speci-
F. E L L Y I N and A. N. SHERBOURNE
176
LOI
(a)
I
G$
Q8
0J
~2 j
~
I
0.7
0~ C PJ 03
o~1
04
C)AI
03
o~1
n.t/--.~,l.5 VD
o~1
I
I
I
I
I
I
I
2
3
4
5
6
7
8
O/d
_o d
Fig. 8.
(a)
Q8
oJ
0.7 c Pu
0.015
0.~
o.~
0.01~ / 0.01/ 0.4
G4
n- ~ m--m
'2
'
'6
_o
.
-I.75
03
.
0.2
,o
d
I
I
4
I
6
I
_o d
Fig. 9.
8
I
I0
I
12
THE COLLAPSE OF CYLINDER/SPHERE INTERSECTING PRESSURE VESSELS
(a)
(b)
177
T GO] 0.025 0.02: 0.015
c PU or pS u
o~ c of
o~
n -
n "t/-d t
'=
I
t/._~. Z.O
T/D t
~
-~,
4
6
8
I0
i2
J4
16
D
D
¥
7 Fig. 10.
(b)
(a)
•
T
O
0.0
o,o 0.8
°
S
S
0.(
Pl
05
Pl or ¢
PU or C Pu
0.4
V., " " T/~ =2.0
t>T
n - t/d = 2.0
T/o
t>T
O5
O.
0.4~
0
18
i 8
D
D d
Fig. 11.
L
i
i
i
IO
z2
14
16
18
178
F. ELLYIN
and A N. SHERBOURNE
(b)
(a3 0.9
0.025 ~
/
0.9
gT"
g. O.a
$
PU OY
e= t/d-
3.0
T/D
O~
t/d= 3,0
n - .r/D
;
'
,'
,
•
,',
,~,
,'
D d
O
Fig. 12.
\
LO
0.9
O.
pS
t
pS o8
/ / 1 t
/
/
/
11
DT-"= 0.055
n-3
0.~
/" I "
~
I~s
~
~. t ' ~ " \
I / \\
t ~.,~.//
TD= 0.02
n-5.0
\ 0.~ ~ pS 0.8
~ I ~
~
-
\ 0.{ -~ = 0.02
pS
,02
0.7
\ \
1"
/
/
/
------
i
;
~,
n- 4
~
,"
,,
UPPER BOUND LOWER BOUND
,,
,~
G( 05
UPPER BOUND ----~LOWER BOUND I
D d
Fig. 13.
Fig. 14.
n = 6,0
THE C O L L A P S E OF C Y L I N D E R / S P H E R E I N T E R S E C T I N G P R E S S U R E VESSELS
179
fled. Using his ap p r o a c h , it is p o s s i b l e to choose any i n t e r a c t i o n c u r v e f o r m e r i d i o n a l m o m e n t M(p and m e m b r a n e f o r c e N ~ . C l o s e r bounds can be obtained by substituting in eq. (7) and (14) the v a l u e s of moment: M c = ½(~0t2 [Pr(1 - ½Pr)] ,
M s = ½~0T2 [PR (1 - ½PR)] ,
which c o r r e s p o n d to face I of the yield s u r f a c e of fig. 1. (iii) The e f f e c t s of c h a n g e s in g e o m e t r y and s t r a i n h a r d e n i n g a r e not c o n s i d e r e d in this a n a l y s i s . T h e s e f a c t o r s , h o w e v e r , tend to i n c r e a s e the load c a r r y i n g c a p a c i t y of the p r e s s u r e v e s s e l . The m a t e r i a l is a s s u m e d to be p l a s t i c - r i g i d and e l a s t i c s t r a i n s a r e , t h e r e f o r e , neglected. T h i s condition i m p l i e s that the s h e l l s be m o d e r a t e l y thick. F u r t h e r m o r e , s i n c e the yield s u r f a c e s a r e obtained on the a s s u m p ti on s of the thin s h e l l t h e o r y , the t h i c k n e s s to d i a m e t e r r a t i o s of the s h e l l s m u s t be r e s t r i c t e d . R e a s o n i ably a c c u r a t e r e s u l t s m a y be obtained within t h e s e l i m i t a t i o n s when ~0 --< T / D --< ~ . M o r e o v e r , f o r the a s s u m e d f a i l u r e mode and t h r e e - d i m e n s i o n a l y i el d s u r f a c e , the d i a m e t e r r a t i o s should a l s o be r e s t r i c t e d to the r a n g e 2 --< D i d ~< 15. (iv) When the s p h e r i c a l s h e l l is c o n s i d e r a b l y stiff and n < 2, the c o l l a p s e p r e s s u r e is g r e a t e r than that f o r an infinite c y l i n d e r . C o n v e r s e l y , f o r a r i g i d c y l i n d e r and n > 2, the c o l l a p s e p r e s s u r e is g r e a t e r than that of an u n p i e r c e d s p h e r e . S i m i l a r r e s u l t s w e r e o b s e r v e d by Hodge [14] in the c a s e of a cutout in a s p h e r i c a l cap. (v) If the p r e s s u r e v e s s e l is r e q u i r e d to withstand g r e a t e r p r e s s u r e than that g i v en by the above t h e o r y , then s o m e s o r t of r e i n f o r c e m e n t is evidently r e q u i r e d . This r e i n f o r c e m e n t m u s t be p o s i t i o n e d b e tw e en the two o u te r hinge c i r c l e s whose l o c a t i o n s a r e d e t e r m i n e d f r o m eq. (8) and (11). The r e q u i r e d t h i c k n e s s of r e i n f o r c e m e n t s may be c a l c u l a t e d u s i n g eq. (19).
9. NOTATION b l M6 M~o , M s
= = = =
length of the c y l i n d r i c a l s h e l l hinge l o c a t i o n in c y l i n d r i c a l s h e l l m e a s u r e d f r o m the junction c i r c u m f e r e n t i a l bending m o m e n t p e r unit width of sh el l m e r i d i o n a l bending m o m e n t p e r unit width of s p h e r i c a l sh el l and its value on the yield s u r face Mx, M c = m e r i d i o n a l bending m o m e n t p e r unit width of c y l i n d r i c a l s h e l l and its v al u e on the yield s u r face = value of m e r i d i o n a l bending m o m e n t p e r unit width of c e n t r a l hinge c i r c l e Mk
t/d
n =
N0, Ns Nc N} P P Q Q', Q" R t T U, W
?0 ~6
T/D
= c i r c u m f e r e n t i a l m e m b r a n e f o r c e p e r unit width of s p h e r i c a l sh el l and its value on the yield surface = value of c i r c u m f e r e n t i a l m e m b r a n e f o r c e p e r unit width of c y l i n d r i c a l sh el l on yield s u r f a c e = m e r i d i o n a l m e m b r a n e f o r c e p e r unit width of s p h e r i c a l sh el l = m e r i d i o n a l m e m b r a n e f o r c e p e r unit width of c y l i n d r i c a l sh el l = internal pressure = dimensionless pressure = s h e a r f o r c e p e r unit width = s h e a r f o r c e p e r unit width in c y l i n d r i c a l and s p h e r i c a l sh el l at the junction = m e a n r a d i u s of c y l i n d r i c a l s h e l l = m e a n r a d i u s of s p h e r i c a l s h e l l = t h i c k n e s s of the c y l i n d r i c a l s h e l l = t h i c k n e s s of the s p h e r i c a l s h e ll = m e r i d i o n a l and r a d i a l v e l o c i t i e s in the s p h e r i c a l s h e l l = angle subtended by c y l i n d r i c a l s h e l l at c e n t e r of s p h e r i c a l sh el l = l o c a t i o n of the hinge c i r c l e in the s p h e r i c a l sh el l = y i el d s t r e s s in s i m p l e t e n s i o n = m e r i d i o n a l r a t e of s t r a i n in s p h e r i c a l sh el l = c i r c u m f e r e n t i a l r a t e of s t r a i n in s p h e r i c a l sh el l
180 ex = meridional rate ~ = meridional rate X0 = circumferential Xx = meridional rate Other symbols are defined
F. ELLYIN and A N. SHERBOURNE of s t r a i n i n c y l i n d r i c a l s h e l l of c u r v a t u r e i n s p h e r i c a l s h e l l r a t e of c u r v a t u r e in s p h e r i c a l s h e l l of c u r v a t u r e in c y l i n d r i c a l . s h e l l a s t h e y a p p e a r in t h e p a p e r .
ACKNOWLEDGEMENT T h e r e s u l t s p r e s e n t e c l i n t h i s p a p e r w e r e o b t a i n e d in t h e c o u r s e of a r e s e a r c h p r o g r a m m e s p o n s o r e d , i n p a r t , b y t h e S u b c o m m i t t e e o n R e i n f o r c e d O p e n i n g s of t h e P r e s s u r e V e s s e l R e s e a r c h C o u n c i l , G r a n t No. C - 5 1 9 . T h a n k s a r e d u e to M r . S. K e s h a v a u f o r h i s a s s i s t a n c e i n c o m p u t e r p r o g r a m m i n g .
REFERENCES 1. J . L . Mershon, PVRC r e s e a r c h on r e i n f o r c e m e n t of openings in p r e s s u r e v e s s e l s , Welding R e s e a r c h Council Bull. No. 77 (May 1962). 2. M.M. Leven, Photoelastic determination of the s t r e s s e s in reinforced openings in p r e s s u r e v e s s e l s , Westinghouse R e s e a r c h Report 64-9D7-514-R1 (October 1964). 3. E.O. Waters, S t r e s s e s n e a r cylindrical outlets in a s p h e r i c a l vessel, Welding R e s e a r c h Council Bull. No. 96 (May 1964). 4. N.C. Lind, P l a s t i c analysis of radial outlets from s p h e r i c a l p r e s s u r e v e s s e l s , ASME p a p e r No. 63, Pet. 2, P e t r o l e u m Mech. Eng. Conf., September 1963. 5. S.S. Gill, The limit p r e s s u r e for a flush cylindrical nozzle in s p h e r i c a l p r e s s u r e vessel, Int. J. Mech. Sci. 6 (1964). 6. F. Ellyin and A. N. Sherbourne, Limit analysis of a x i s y m m e t r i c i n t e r s e c t i n g shells of revolution, Nucl. Struct. Eng. 2 (1965).86. 7. E . T . Onat and W. P r a g e r , Limit analysis of shells of revolution, p a r t s I and II, Proc. Kon. Nederl. Akad. Wetensch. 57 (1954). 8. P.G. Hodge Jr., Rigid-plastic analysis of s y m m e t r i c a l l y loaded cylindrical shells, Trans. ASME 76 (1954). 9. E . T . Onat, The plastic collapse of cylindrical shells under axially s y m m e t r i c a l loading, Quart. Appl. Math. 13 (1955). 10. D.C. Drucker and R. T. Shield, Limit analysis of s y m m e t r i c a l l y loaded thin shells of revolution, J. Appl. Mech. 26 (1959). 11. R.T. Shield and D. C. Drucker, Limit s t r e n g t h of thin walled p r e s s u r e v e s s e l s with an ASME standard t o r i s p h e r ical head, 3rd U.S. Natl. Congr. of Applied Mechanics, June 1958. 12. C.E. Taylor, N.C. Lind and J. W. Schweiker, A t h r e e - d i m e n s i o n a l photoelastic study of s t r e s s e s around r e i n forced outlets in p r e s s u r e v e s s e l s , ASME p a p e r No. 58-A-148 (1958). 13. R. L. Cloud, The limit p r e s s u r e of radial nozzles in s p h e r i c a l shells, Nucl. Struct. Eng. 1 (1965) 403. 14. P . G . Hodge J r . , Rigid-plastic analysis of s p h e r i c a l caps with cutouts, Int. J. Mech. Sci. 6 (1964).
THE COLLAPSE INTERSECTING
OF CYLINDERISPHERE PRESSURE VESSELS *
F. ELLYIN and A. N. SHERBOURNE Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada
Received 24 May 1965
A general method is presented for finding lower and upper bounds on the collapse pressure for a plastic-rigid vessel composed of intersecting cylindrical and spherical shells obeying Tresca's yield condition and associated flow rule. Depending upon the ratio (t/d)/(T/D) different types of solutions are generated. The solutions are presented in terms of generalized stresses and may be used in conjunction with any choice of yield surface of the form ~(N~p,No, Mq)) = C 2.
1. INTRODUCTION In r e c e n t y e a r s the p r o b l e m of i n t e r s e c t i n g s h e l l s has b e e n the subject of study by a n u m b e r of d i s tinct groups. T h e s e i n v e s t i g a t i o n s have b e e n s y s t e m a t i c a l l y r e v i e w e d by M e r s h o n [1]. So far, much of this work has been of an e x p e r i m e n t a l n a t u r e (mostly t h r e e d i m e n s i o n a l p h o t o - e l a s t i c studies) and a s u m m a r y of the r e l e v a n t e x p e r i m e n t s is given by Leven [2]. It should be pointed out, however, that t h e r e exist some d i f f e r e n c e s b e t w e e n the r e s u l t s of p h o t o - e l a s t i c studies and steel model t e s t s . N e v e r t h e l e s s , p h o t o - e l a s t i c r e s u l t s may be used as a guide for checking t h e o r e t i c a l a n a l y s e s . A n a l y t i c a l l y , the p r o b l e m has been studied f r o m two different points of view, b a s e d upon the t h e o r i e s of e l a s t i c s h e l l s and p l a s t i c - r i g i d solids. Although some r e s u l t s have been obtained f r o m "two-piece" e l a s t i c shell t h e o r y by W a t e r s [3] and others [1], a s a t i s f a c t o r y solution has yet to be obtained. On the b a s i s of p l a s t i c - r i g i d t h e o r y the solutions proposed by Lind [4] and Gill [5] might be mentioned. The a s s u m p t i o n s and s h o r t c o m i n g s of these t h e o r i e s have been the subject of review by E l l y i n and S h e r b o u r n e [6]. The p r e s e n t i n v e s t i g a t i o n is an attempt to i m p r o v e on existing t h e o r i e s ; some of the r e s u l t s of p r e vious work may be d e r i v e d as p a r t i c u l a r c a s e s of this m o r e g e n e r a l approach. The p r o b l e m has b e e n f o r m u l a t e d and r e s u l t s are obtained in t e r m s of g e n e r a l i z e d s t r e s s e s which may be used in a s s o c i a t i o n with one of the many yield s u r f a c e s of g e n e r a l f o r m ~(N(p,No,Mgo) = C 2. Depending upon the shell geo m e t r y , types of solutions a r e obtained which differ in r e s p e c t of the position of the c e n t r a l hinge at the i n t e r s e c t i o n . F o r design p u r p o s e s , however, a p r o c e d u r e is c l e a r l y e s t a b l i s h e d r e l a t i n g the bounds on the collapse p r e s s u r e to the g e o m e t r i c a l p a r a m e t e r s of the two shells.
2. YIELD CONDITION In g e n e r a l , the l i m i t i n g c o m b i n a t i o n of i n t e r n a l force i n t e n s i t y r e q u i r e d to develop full p l a s t i c i t y in a shell of r e v o l u t i o n is given by an equation of the f o r m : f(N~a,Yo,Mq~,Ms) = C 2 , where C is a constant. The effect of t r a n s v e r s e s h e a r on the yielding i s neglected. A yield h y p e r s u r f a c e for shells of r e v o l u t i o n subject to a x i s y m m e t r i c a l loading has been d e s c r i b e d by Onat and P r a g e r [7]. In * Accepted by A. Sawczuk.
170
F. ELLYIN
and A. N. SHERBOURNE
N--B=n8 No
_~b_ n~or n x No -
A
I
N8 =no) No c~ A ~
.' /
d
No=n~
°rnx
1 "~-~" "~e=m~°r mx
4 ~ : ;
M' = mc~°rmx
D
Fig. 1. Yield surface.
Fig. 2. Yield hexagonal prism.
c y l i n d r i c a l s h e l l s , the c i r c u m f e r e n t i a l bending m o m e n t M 0 does not appear in the equations of e q u i l i b r i u m ; it is an induced or p a s s i v e value d e t e r m i n e d explicitly w h e n N ¢ , N o and M e lie on the yield s u r face. Fig. 1 shows a yield s u r f a c e for a solid c y l i n d r i c a l shell composed of p l a s t i c - r i g i d m a t e r i a l obeying the m a x i m u m s h e a r s t r e s s (Tresca) yield condition and a s s o c i a t e d flow r u l e . It h a s been d e s c r i b e d independently by Hodge [8] and Onat [9]. An a p p r o x i m a t i o n to this s u r f a c e , suggested by D r u c k e r and Shield [10], is shown in fig. 2, c o n s i s t i n g of a somewhat s i m p l e r hexagonal p r i s m which leads to a s u r , i face i n s c r i b e d within the o r i g i n a i yield s u r f a c e when its v e r t i c e s are r e d u c e d by a f a c t o r ~(52 - 1). A l though this yield s u r f a c e is d e r i v e d for thin c y l i n d r i c a l s h e l l s , Drucker and Shield [10, 11] contend that this c o n s t i t u t e s a r e a s o n a b l e a p p r o x i m a t i o n to the yield s u r f a c e for any s y m m e t r i c a l l y loaded shell of revolution.
3. YIELD MECHANISM The a x i s y m m e t r i c i n t e r s e c t i n g shell configuration to be a n a l ysed is shown in fig. 3. When the p r e s s u r e i n t e n s i t y is i n c r e a s e d to a c e r t a i n value, the p l a s t i c - r i g i d shell will begin to deform in different locations depending upon the shell geometry. In the case X where both c y l i n d e r and sphere have a p p r o x i m a t e l y equal m e m b r a n e s t r e n g t h s and no r e i n f o r c e m e n t at the junction, p l a s t i c d e f o r m a t i o n will f i r s t o c c u r in this region. P h o t o - e l a s t i c studies by T a y l o r et al. [12] c o n f i r m that there a r e high c i r c u m f e r e n t i a l s t r e s s e s and m e r i d i o n a l bending m o m e n t s p r e s e n t at the zone of / / i n t e r s e c t i o n . It s e e m s r e a s o n a b l e , t h e r e f o r e , to a s s u m e that the f i r s t hinge c i r c l e will f o r m at the junction, the g e o m e t r y of this r e g i o n being n e i t h e r c y l i n d e r nor sphere but, in fact, a r e g i o n of t r a n s i t i o n between the two shells. The deformation, however, will at f i r s t be s m a l l and the p r e s s u r e i n t e n s i t y may continue to i n c r e a s e until yielding takes place in the m e r i d i o n a l plane at two additional locations. Collapse o c c u r s only when the f o r m a t i o n of the Fig. 3. Geometrical relations. m e c h a n i s m is complete. In c a s e s where the i n t e r s e c t i o n is r e i n f o r c e d (by additional m a t e r i a l , welds, etc.) the yielding in the junction is b i a s e d toward e i t h e r sphere or c y l i n d e r depending upon the r e l a t i v e s t i f f n e s s e s of the two shells as m e a s u r e d by the r e l a t i o n s h i p t / T = n d / D . F o r n > 2, the collapse m e c h a n i s m will c o n s i s t of two hinge c i r c l e s in the s p h e r i c a l shell, one located near the t r a n s i t i o n zone and the other some d i s -
THE COLLAPSE OF CYLINDER/SPHERE INTERSECTING PRESSURE VESSELS
171
tance away; the t h i r d hinge c i r c l e will f o r m in the c y l i n d e r somewhat r e m o v e d f r o m the zone of i n t e r section. C o n v e r s e l y , in the case of n < 2, the collapse m e c h a n i s m will contain two hinge c i r c l e s in the c y l i n d r i c a l shell and one in the spherical shell. F o r a range of s m a l l values n the collapse m e c h a n i s m will be confined a l m o s t e n t i r e l y to the c y l i n d e r and the p r o b l e m may be t r e a t e d as d e s c r i b e d e l s e w h e r e [9]. C o n v e r s e l y , for l a r g e values of n, t h e r e will be a m e c h a n i s m a s s o c i a t e d a l m o s t e n t i r e l y with the sphere. In this paper, c o n s i d e r a t i o n is given to c o m b i n e d m e c h a n i s m s which n o r m a l l y occur in p r e s s u r e v e s s e l s when the shell s t i f f n e s s e s a r e of the s a m e o r d e r . In the a n a l y s i s which follows, the o r d e r at which hinge c i r c l e s f o r m is of no c o n s e quence. R e g a r d l e s s of the i n t e r m e d i a t e hinge, however, t h e r e m u s t be c i r c u m f e r e n t i a l d i s p l a c e m e n t between the hinge c i r c l e s in c y l i n d e r and s p h e r e , accompanied by an outward d i r e c t e d r i g i d body motion of the c y l i n d r i c a l p a r t , in o r d e r to yield a compatible m e c h a n i s m . Such a m e c h a n i s m c o r r e s p o n d s to face I of the yield s u r f a c e s shown in figs. 1 and 2. The n o r m a l i t y condition of the s t r a i n rate vector thus i m p o s e s the condition: ~p and dx = 0,
~
and )
and ~e = u n r e s t r i c t e d
(1)
between the hinge c i r c l e s . At the hinge locations, the s t r e s s state is r e p r e s e n t e d by the parabolic axes of the yield s u r f a c e of fig. 1, or by the edges p a r a l l e l to AB of the yield hexagonal p r i s m of fig. 2. The s t r a i n rate v e c t o r , in t h i s case, has two c o m p o n e n t s which impose the condition: d(p and dx = 0,
)~o and Xx = u n r e s t r i c t e d ,
d~; = u n r e s t r i c t e d .
(2)
4. U P P E R BOUND ANALYSIS An upper bound on the collapse p r e s s u r e can be found by equating the e x t e r n a l rate of doing work to the i n t e r n a l rate of e n e r g y dissipation for a k i n e m a t i c a l l y a d m i s s i b l e p a t t e r n of t h r e e hinge c i r c l e s . If n o r m a l s to the u n d e f o r m e d middle s u r f a c e continue to r e m a i n n o r m a l in the bent configuration, the s h e a r i n g f o r c e Q is not a g e n e r a l i z e d s t r e s s but has the n a t u r e of a r e a c t i o n . Also, since we a r e conc e r n e d with the a x i a l l y s y m m e t r i c d e f o r m a t i o n s of the shell, component V of the velocity in the c i r c u m f e r e n t i a l d i r e c t i o n is z e r o , and we need to c o n s i d e r only the components U and W in the m e r i d i o n a l and radial directions. A velocity field of the f o r m V = C[1 - cos(/3 -(p)] ,
W = -C sin(/~ -~p)
(3)
s a t i s f i e s the c o n s t r a i n t s (1) and (2) for the s p h e r i c a l p a r t of the shell, /3 being the angle which locates the position of the hinge c i r c l e . The p r i n c i p a l s t r a i n s and c u r v a t u r e r a t e s a r e then given by
1 ~dU -
- w) =0 ,
d e = ~1 - ( U c o t e - w) = RsinC
_1
(cos~0 -cos/3) ,
R
d (U~-)=0 ,
)(0
cot~0 ( u
dW
C
(4)
:R-2eot
An e x a m i n a t i o n of the o r d e r s of magnitude of v a r i o u s t e r m s in the e x p r e s s i o n for the rate of work done by the s t r e s s s y s t e m indicates that, when the t h i c k n e s s of shell T is s m a l l c o m p a r e d with the local r a d i u s r o , the r a t e at which work is done by the s t r e s s s y s t e m on the change in c i r c u m f e r e n t i a l c u r v a t u r e i s s m a l l c o m p a r e d with r a t e of work done on the change in c i r c u m f e r e n t i a l s t r a i n . T h e r e f o r e , the c o n t r i b u t i o n of c i r c u m f e r e n t i a l bending m o m e n t M e to the work equation can be neglected. This point has b e e n d i s c u s s e d by D r u c k e r and Shield [10]. The e x t e r n a l and i n t e r n a l work done by the f o r c e s of fig. 4 a r e given by:
v
Iw = 2 ~ r M c ~ + 2 ~ r M k
(v~
+ e
)
+ 2 n R M s O cosB + ~ l v N c t 2 ~ R e N s [ s i n / 3
E w = ~rlvP + nr26p
- s i n a - (/3-a) cos/3] ,
+ n R 3 0 [sin(/3 - a ) c o s a - (/~ - a ) cos/3] P ,
where M c ~< ¼a0t2, N c --< cr0t, M k --< ¼a0t2 or ¼ g o T 2 , M s ~ ¼cr0T2 and N s ~< gOT.
F. ELLYIN and A. N. SHERBOURNE
172
/I
7°
/
Jj
/I
i,
Z
f
I Fig. 4. Force-deformation scheme. Since no e n e r g y c a n be s t o r e d in the p l a s t i c - r i g i d s h e l l , the t o t a l e n e r g y d i s s i p a t i o n m u s t e q u a l the w o r k done by the a p p l i e d load, i.e. E w = Iw .
(5)
The c o n t i n u i t y of d i s p l a c e m e n t at the j u n c t i o n r e q u i r e s that v = OR(cosa
- cosB) ,
5 = 0R(sin~ - sina) .
(6)
I n s e r t i o n of e q s . (6) and (5) w i l l l e a d to an u p p e r bound on the c o l l a p s e p r e s s u r e
[R2p- 2NsR ] sin/3
- It(/+ R cos a)P - ~
(M c + M k) + 2M s - lNc] cos/3 + (/~ - a) [2NsR - R 2 p ] c o s B
+ [Prl -~(Mc+Mk)
- /Nc] c o s a + R [ 2 N s R
-Pr 2 -2Mk] = 0.
(7)
Eq. (7) is a f u n c t i o n of two v a r i a b l e s , ~ and l, w h i c h s t i p u l a t e the hinge l o c a t i o n s in s p h e r i c a l and c y l i n d r i c a l s h e l l r e s p e c t i v e l y . A l i m i t i n g v a l u e o f P is o b t a i n e d f r o m eq. (7) by m i n i m i z i n g with r e s p e c t to t h e s e two v a r i a b l e s to y i e l d the f o l l o w i n g e x p r e s s i o n s : [ 2 ( M c + Mk)]½ l = L (Yc/r) -PJ
r(l + R c o s a ) P and
~ =a +
- ( 2 r / / ) ( M c + M k ) + 2M s - l Y c 2NsR
- R2p
S u b s t i t u t i o n of l and ~ f r o m e q s . (8) in eq. (7) w i l l g i v e the l o w e s t v a l u e of p o s s i b l e u p p e r b o u n d s the c o l l a p s e p r e s s u r e .
(8) for
5. L O W E R BOUND ANALYSIS F o r the c y l i n d r i c a l p a r t of the s h e l l , the e q u a t i o n s of e q u i l i b r i u m , with the n o t a t i o n of fig. 5, a r e g i v e n by:
No ~
+
dMx =P '
dx
=v '
Nx = ½Pr "
(9)
E q s . (9) m a y be i n t e g r a t e d and t h e c o n s t a n t s of i n t e g r a t i o n c a n be found by i n v o k i n g a s u i t a b l e s t r e s s f i e l d N o = N c f o r 0 ~< x --< l, in c o n j u n c t i o n with t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s :
173
T H E C O L L A P S E O F C Y L I N D E R / S P H E R E I N T E R S E C T I N G P R E S S U R E VESSELS
\ Nx - - - ~
cb
----.:- ~.---~j.~-
R
,," % ,"w ~-,'l ~X
No
d~ Fig. 5. Notation for cylindrical part of shell. Q =0, M=Mcwhenx=l
Fig. 6. Notation for spherical part of shell. ,
Q =Q', M=Mkwhenx
=O,
where
IMcl ~<¼~0t2 ,
[Ncl
~< a0 t ,
IMkl --< ¼~0t2 or ¼a0 T2 .
This procedure yields:
o
. ~ __~(~ __})~2 + Q,~ + Mk,
Nx : ~ ,
(lO)
where Q' =
(Mc - M k)
-
,
I = L(Nc/r )
_pj
.
(11)
S i m i l a r l y f o r the s p h e r i c a l p a r t of the s h e l l , the e q u i l i b r i u m e q u a t i o n s with the n o t a t i o n of fig. 6, become Nq~ sinq~ + Qcos~o = ½PR sin~o , N~sin~
+ N Osinq~ + ~d (Q sin~o) = P R s i n ~o
dM¢
dq~ sinq~ + (Mcp - M 0 ) c o s ~ o - QR s i n e = 0 .
(12a) (12b) (12c)
As i n d i c a t e d p r e v i o u s l y [10], the c o n t r i b u t i o n of the s e c o n d t e r m in eq. (12c) to the load c a r r y i n g c a p a c i t y of the s h e l l is s m a l l when c o m p a r e d with the r e m a i n i n g t e r m s a n d m a y , t h e r e f o r e , be o m i t t e d . Eq. (12c) t h u s r e d u c e s to: dM~o d e sinq~ - QR sin(p = 0 .
(13)
E q s . (12a), (I'2b) t o g e t h e r with eq. (13) c a n now be i n t e g r a t e d ; the c o n s t a n t s of i n t e g r a t i o n a r e d e t e r m i n e d by i n v o k i n g the s t r e s s d i s t r i b u t i o n NO = N s f o r a --< ¢ ~< 8 along with b o u n d a r y c o n d i t i o n s when q~ = 8, Q = 0, Mq~ = M s ,
when q~ = a, Q = Q", Mq~ = M k.
(14)
T h i s f i n a l l y l e a d s to: V = (½PR -Ys)(q~ - 8) ,
Mq~ = ~ZR (~/-" I P R -gs)(q~ - 8 ) 2 + M s ,
Nq~ = ½PR - Q cot~o ,
where Q, :
1
(M s _ M k )(N s__SPR
,
V
Ms-Mk 1½ /3 = a + L2R(Ns _ ~PR)J "
(15)
174
F. ELLYIN and A. N. SHERBOURNE
6. EQUILIBRIUM OF FORCES AT THE JUNCTION Fig. 7 shows the e q u i l i b r i u m of s h e a r and m e m b r a n e f o r c e s from which the following i d e n t i t i e s r e s u l t : Q" c o s a + N a s i n a = ½ P r ,
N a c o s a - Q" s i n a = Q' ,
(17)
leading to
(18)
Q " = ½ P r c o s a - Q' s i n a .
Substituting for Q' and Q" from eqs. (11) and ~1~) r e s p e c t i v e l y in eq. (18) and noting that s i n a = r / R , c o s a = [1 - ( r / R ) Z ] ~ , yields
Fig. 7. 1
(Ms-Mk)(Ns-½PR
( M c - M k)
~ + ,~
= ½Pr[1 - (r/R)2] ~ .
-
(19)
Once the s t a t i c a l l y a d m i s s i b l e s t r e s s field and shell g e o m e t r y a r e specified, the above e x p r e s s i o n will yield a lower bound on the collapse p r e s s u r e . An a p p r o p r i a t e statically admissible, s t r e s s field between the hinge c i r c l e s will c o r r e s p o n d to face I of either yield loci. F o r the c y l i n d r i c a l p a r t of the shell lying outside the hinge location, i.e. for x > l, a s t a t i c a l l y a d m i s s i b l e s t r e s s field which s a t i s f i e s both end conditions and e q u i l i b r i u m equations for a rigid, fixed ended c l o s u r e , may be adequately d e s c r i b e d by the following: Mx =Mc ,
N 0 = Pr ,
N~
-~ ± 2 Pr °
F o r a free end at distance b, the s t r e s s s y s t e m b e c o m e s : Mx = Mc
(b+ x - 2l)(b (b - / ) 2
- x)
'
2Mcr NO = P r + (b - / ) 2 '
Nx = ½Pr .
S i m i l a r l y for the s p h e r i c a l part of the p r e s s u r e v e s s e l , the equations M¢ = M s ,
N(p = ½ P R ,
N e = l~p R
forq~ >/3
constitute a s t a t i c a l l y a d m i s s i b l e s t r e s s field which s a t i s f i e s the boundary conditions and nowhere v i o lates the yield surface.
7. RESULTS F o r the p u r p o s e s of d e r i v i n g typical n u m e r i c a l r e s u l t s f r o m the foregoing a n a l y s i s , the p r i s m of fig. 2 is chosen as the yield condition. On substituting for i n t e r n a l f o r c e s c o r r e s p o n d i n g to this yield s u r f a c e in eqs. (11), (16)and ( 1 9 ) r e s p e c t i v e l y and n o n - d i m e n s i o n a l i z i n g , the following r e s u l t s a r e obtained: Lower bound (a) n < 2
(20)
Is
- ~'l/J
+ 2~
(1
- D2 j p~ ;
THE COLLAPSE OF CYLINDER/SPHERE INTERSECTING PRESSURE VESSELS
175
(b) n > 2 t=2
[ ( 1 - 2 d T ~s~| T~I]
[~-(I+~) (D t
'
/3
=
sln
~
+
11-~Z~]
(21)
'
~
s)]½ D [ . T
(1-p~,]½
[1
d2] ½
where p~ = P/(aot/r), p[ = P/(2~oT/R), and t, T, d, D are thickness and diameter of cylindrical and spherical shell respectively. A similar substitution of internal f o r c e s corresponding to this yield surface in eqs. (8) and (7) r e spectively leads to non-dimensional equations of the form: Upper bound l __d2_½ c t d TT l t / =[ d/t l $ . -1 d D+ l - D 2 j } P u - 7 ~ + ~ t - - 2 7 ~ (22) 7 L2(l_pC)j ' /3=sin ~+ T D c '
2 T - ~ Pu
pC - 2 T d ] u
[
sin/3
d2 l[ 2 ~I t + D~
-
~-(I
c 2"½7 d J Pu - [t + tT Td - 2~//_fcos/31t + (/3 - sin- I ~d) - D~)
,dt[1
lt dt - lt - 2[
- D2 j +
i2Td I}- _pC]cos~ u [ d - Pu -
l
=0;
(b) n > 2 d I
1
T2\
7
dl2lt ,
/3 = sin-1 d +D D
[D +
1 - pzs
1
[Pu - 1] sin/3 -
~
+ (1 _D~d 2"~1)] ps
+ DtD
4It TIDal (1 + ~2) + 2D - t T _I ~ l c to sI/ 3 +
u-21Z) 7
~
-1~
D2 j +/5
t [/3_sin_ld
-D2PU-
(l _puSS)cos l3
=0.
(23)
The eqs. (20) to (23) w e r e p r o g r a m m e d on a computer and the results for various shell g e o m e t r i e s are plotted in figs. 8 to 12. T h e s e results may be useful for design purposes. For purposes of direct comparison, upper and lower bounds are plotted in figs. 13 and 14 for typical shell p a r a m e t e r s . It can be shown that for n = 2 and t = T, the collapse p r e s s u r e as given by the curves of figs. 10 and 11 will be identical r e g a r d l e s s of whether the central hinge f o r m s in sphere or cylinder. This implies an identity between eqs. (20) and (21) and between (22) and (23).
8. CONCLUSIONS (i) The computations of lower bound based on an inscribed and upper bound upon a c i r c u m s c r i b e d hexagonal yield surface of fig. 2 will differ considerably. Since a t h r e e - q u a r t e r size p r i s m lies within the actual yield surface o v e r an extended range, the actual value of collapse p r e s s u r e deduced f r o m the exact yield surface will almoBt always lie between 3/4p and p. It seems reasonable, t h e r e f o r e , to assume 7/8p as being sufficiently close to the actual value. This argument brings upper and lower bounds, plotted in figs. 8 to 14, even c l o s e r together. For design purpose, however, the value p = 7/16(Pl+pu) will be quite reasonable. This theory seems to be in good agreement with the experimental results reported by Cloud [13]. (ii) The method d e s c r i b e d herein, calculates bounds on the collapse p r e s s u r e in t e r m s of generalized s t r e s s e s . The value of generalized s t r e s s M k is known once the g e o m e t r y of p r e s s u r e vessel is speci-
F. E L L Y I N and A. N. SHERBOURNE
176
LOI
(a)
I
G$
Q8
0J
~2 j
~
I
0.7
0~ C PJ 03
o~1
04
C)AI
03
o~1
n.t/--.~,l.5 VD
o~1
I
I
I
I
I
I
I
2
3
4
5
6
7
8
O/d
_o d
Fig. 8.
(a)
Q8
oJ
0.7 c Pu
0.015
0.~
o.~
0.01~ / 0.01/ 0.4
G4
n- ~ m--m
'2
'
'6
_o
.
-I.75
03
.
0.2
,o
d
I
I
4
I
6
I
_o d
Fig. 9.
8
I
I0
I
12
THE COLLAPSE OF CYLINDER/SPHERE INTERSECTING PRESSURE VESSELS
(a)
(b)
177
T GO] 0.025 0.02: 0.015
c PU or pS u
o~ c of
o~
n -
n "t/-d t
'=
I
t/._~. Z.O
T/D t
~
-~,
4
6
8
I0
i2
J4
16
D
D
¥
7 Fig. 10.
(b)
(a)
•
T
O
0.0
o,o 0.8
°
S
S
0.(
Pl
05
Pl or ¢
PU or C Pu
0.4
V., " " T/~ =2.0
t>T
n - t/d = 2.0
T/o
t>T
O5
O.
0.4~
0
18
i 8
D
D d
Fig. 11.
L
i
i
i
IO
z2
14
16
18
178
F. ELLYIN
and A N. SHERBOURNE
(b)
(a3 0.9
0.025 ~
/
0.9
gT"
g. O.a
$
PU OY
e= t/d-
3.0
T/D
O~
t/d= 3,0
n - .r/D
;
'
,'
,
•
,',
,~,
,'
D d
O
Fig. 12.
\
LO
0.9
O.
pS
t
pS o8
/ / 1 t
/
/
/
11
DT-"= 0.055
n-3
0.~
/" I "
~
I~s
~
~. t ' ~ " \
I / \\
t ~.,~.//
TD= 0.02
n-5.0
\ 0.~ ~ pS 0.8
~ I ~
~
-
\ 0.{ -~ = 0.02
pS
,02
0.7
\ \
1"
/
/
/
------
i
;
~,
n- 4
~
,"
,,
UPPER BOUND LOWER BOUND
,,
,~
G( 05
UPPER BOUND ----~LOWER BOUND I
D d
Fig. 13.
Fig. 14.
n = 6,0
THE C O L L A P S E OF C Y L I N D E R / S P H E R E I N T E R S E C T I N G P R E S S U R E VESSELS
179
fled. Using his ap p r o a c h , it is p o s s i b l e to choose any i n t e r a c t i o n c u r v e f o r m e r i d i o n a l m o m e n t M(p and m e m b r a n e f o r c e N ~ . C l o s e r bounds can be obtained by substituting in eq. (7) and (14) the v a l u e s of moment: M c = ½(~0t2 [Pr(1 - ½Pr)] ,
M s = ½~0T2 [PR (1 - ½PR)] ,
which c o r r e s p o n d to face I of the yield s u r f a c e of fig. 1. (iii) The e f f e c t s of c h a n g e s in g e o m e t r y and s t r a i n h a r d e n i n g a r e not c o n s i d e r e d in this a n a l y s i s . T h e s e f a c t o r s , h o w e v e r , tend to i n c r e a s e the load c a r r y i n g c a p a c i t y of the p r e s s u r e v e s s e l . The m a t e r i a l is a s s u m e d to be p l a s t i c - r i g i d and e l a s t i c s t r a i n s a r e , t h e r e f o r e , neglected. T h i s condition i m p l i e s that the s h e l l s be m o d e r a t e l y thick. F u r t h e r m o r e , s i n c e the yield s u r f a c e s a r e obtained on the a s s u m p ti on s of the thin s h e l l t h e o r y , the t h i c k n e s s to d i a m e t e r r a t i o s of the s h e l l s m u s t be r e s t r i c t e d . R e a s o n i ably a c c u r a t e r e s u l t s m a y be obtained within t h e s e l i m i t a t i o n s when ~0 --< T / D --< ~ . M o r e o v e r , f o r the a s s u m e d f a i l u r e mode and t h r e e - d i m e n s i o n a l y i el d s u r f a c e , the d i a m e t e r r a t i o s should a l s o be r e s t r i c t e d to the r a n g e 2 --< D i d ~< 15. (iv) When the s p h e r i c a l s h e l l is c o n s i d e r a b l y stiff and n < 2, the c o l l a p s e p r e s s u r e is g r e a t e r than that f o r an infinite c y l i n d e r . C o n v e r s e l y , f o r a r i g i d c y l i n d e r and n > 2, the c o l l a p s e p r e s s u r e is g r e a t e r than that of an u n p i e r c e d s p h e r e . S i m i l a r r e s u l t s w e r e o b s e r v e d by Hodge [14] in the c a s e of a cutout in a s p h e r i c a l cap. (v) If the p r e s s u r e v e s s e l is r e q u i r e d to withstand g r e a t e r p r e s s u r e than that g i v en by the above t h e o r y , then s o m e s o r t of r e i n f o r c e m e n t is evidently r e q u i r e d . This r e i n f o r c e m e n t m u s t be p o s i t i o n e d b e tw e en the two o u te r hinge c i r c l e s whose l o c a t i o n s a r e d e t e r m i n e d f r o m eq. (8) and (11). The r e q u i r e d t h i c k n e s s of r e i n f o r c e m e n t s may be c a l c u l a t e d u s i n g eq. (19).
9. NOTATION b l M6 M~o , M s
= = = =
length of the c y l i n d r i c a l s h e l l hinge l o c a t i o n in c y l i n d r i c a l s h e l l m e a s u r e d f r o m the junction c i r c u m f e r e n t i a l bending m o m e n t p e r unit width of sh el l m e r i d i o n a l bending m o m e n t p e r unit width of s p h e r i c a l sh el l and its value on the yield s u r face Mx, M c = m e r i d i o n a l bending m o m e n t p e r unit width of c y l i n d r i c a l s h e l l and its v al u e on the yield s u r face = value of m e r i d i o n a l bending m o m e n t p e r unit width of c e n t r a l hinge c i r c l e Mk
t/d
n =
N0, Ns Nc N} P P Q Q', Q" R t T U, W
?0 ~6
T/D
= c i r c u m f e r e n t i a l m e m b r a n e f o r c e p e r unit width of s p h e r i c a l sh el l and its value on the yield surface = value of c i r c u m f e r e n t i a l m e m b r a n e f o r c e p e r unit width of c y l i n d r i c a l sh el l on yield s u r f a c e = m e r i d i o n a l m e m b r a n e f o r c e p e r unit width of s p h e r i c a l sh el l = m e r i d i o n a l m e m b r a n e f o r c e p e r unit width of c y l i n d r i c a l sh el l = internal pressure = dimensionless pressure = s h e a r f o r c e p e r unit width = s h e a r f o r c e p e r unit width in c y l i n d r i c a l and s p h e r i c a l sh el l at the junction = m e a n r a d i u s of c y l i n d r i c a l s h e l l = m e a n r a d i u s of s p h e r i c a l s h e l l = t h i c k n e s s of the c y l i n d r i c a l s h e l l = t h i c k n e s s of the s p h e r i c a l s h e ll = m e r i d i o n a l and r a d i a l v e l o c i t i e s in the s p h e r i c a l s h e l l = angle subtended by c y l i n d r i c a l s h e l l at c e n t e r of s p h e r i c a l sh el l = l o c a t i o n of the hinge c i r c l e in the s p h e r i c a l sh el l = y i el d s t r e s s in s i m p l e t e n s i o n = m e r i d i o n a l r a t e of s t r a i n in s p h e r i c a l sh el l = c i r c u m f e r e n t i a l r a t e of s t r a i n in s p h e r i c a l sh el l
180 ex = meridional rate ~ = meridional rate X0 = circumferential Xx = meridional rate Other symbols are defined
F. ELLYIN and A N. SHERBOURNE of s t r a i n i n c y l i n d r i c a l s h e l l of c u r v a t u r e i n s p h e r i c a l s h e l l r a t e of c u r v a t u r e in s p h e r i c a l s h e l l of c u r v a t u r e in c y l i n d r i c a l . s h e l l a s t h e y a p p e a r in t h e p a p e r .
ACKNOWLEDGEMENT T h e r e s u l t s p r e s e n t e c l i n t h i s p a p e r w e r e o b t a i n e d in t h e c o u r s e of a r e s e a r c h p r o g r a m m e s p o n s o r e d , i n p a r t , b y t h e S u b c o m m i t t e e o n R e i n f o r c e d O p e n i n g s of t h e P r e s s u r e V e s s e l R e s e a r c h C o u n c i l , G r a n t No. C - 5 1 9 . T h a n k s a r e d u e to M r . S. K e s h a v a u f o r h i s a s s i s t a n c e i n c o m p u t e r p r o g r a m m i n g .
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