Stresses at the intersection of two cylindrical shells

Nuclear Engineering and Design

ELSEVIER

Nuclear Engineering and Design 154 (1995) 231 238

Stresses at the intersection of two cylindrical shells M.D. Xue, W. Chen, K.C. Hwang Tsinghua University, Beijing, People's Republic of China Received 31 August 1994

Abstract

The stress analysis based on the theory of a thin shell is carried out for two normally intersecting cylindrical shells with a large diameter ratio. Instead of the Donnell shallow shell equation, the modified Morley equation, which is applicable to po(R/T) 1/2 >> 1, is used for the analysis of the shell with cut-out. The solution in terms of displacement function for the nozzle with a non-planar end is based on the Love equation. The boundary forces and displacements at the intersection are all transformed from Gaussian coordinates (~, fl) on the shell, or Gaussian coordinates ((, 0) on the nozzle into three-dimensional cylindrical coordinates (p, 0, z). Their expressions on the intersecting curve are periodic functions of 0 and expanded in Fourier series. Every harmonics of Fourier coefficients of boundary forces and displacements are obtained by numerical quadrature. The results obtained are in agreement with those from the finite element method and experiments for diD <~0.8.

I.

Introduction

Cylindrical pressure vessels with radially attached nozzles as shown in Fig. 1 are c o m m o n l y used in nuclear engineering pressure vessels and piping. In order to provide a design criterion by analysis, extensive investigations on analytical solution have been conducted since the 1960s. Eringen et al. (1965), Lekerkerker (1972) and Qian et al. (1965) obtained analytical solutions based on the Donnell shallow shell equation for P0 = ro/ R ~< 0.25 and p o ( R / T ) '/2 < 1, where ro and R are radii o f the nozzle and the vessel respectively and T is the thickness o f the vessel. Steele et al. (1986) and K h a t h l a n and Steele (1987) developed an approximate m e t h o d for intersecting cylindrical shells with large P0. However, for the case o f 0029-5493/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved S S D 1 0029-5493(94)00920-1

l

•

Z

t

)

-

\ \

Fig. 1. Two intersecting cylindrical shells.

232

M.D. Xue et al. / Nuclear Engineering and Design 154 (1995) 231 238

= {p2 COS2 0 + [sin-l(p sin 0)]2} 1/2

Az

(l)

fl = sin-1([sin-i(p sin 0)] x {p2 cos 2 0 -]- [sin

I(p

sin 0)] 2} 1/2)

(2)

z-R CR ~'~

¢ = - ro

~.

(3)

and

i

(i~, i/~, i,)v = [B](i~,, io, i: )T

(4)

q~R

x

Here, the elements of [B] are functions of (p, 0) respectively and can be obtained from Eqs. (1) and (2) as shown by Xue et al. (1991). The expressions for the intersecting curve F in the system (~, fl) are

f [ i~ ! \ Oo', " "~y Fig. 2. Three-coordinatesystems.

small Po ( p o ( R / T ) I / 2 ~ 1) the method of Steele and coworkers does not lead to results coincident with those obtained by Eringen et al. (1965), Qian et al. (1965) and Lekerkerker (1972) and the accuracy of his method diminishes with increasing d / D ratio as shown by Mokhtarian and Endicott (1991) and Mershon (1989). The authors of the present paper developed an asymptotic method (Deng, 1991; Xue, 1991) for the solution of cylindrical shells with large openings (Po ~<0.7). In this paper the solution of shells with cut-outs has been improved and extended to the case of shells with attached nozzles. The results obtained by present method are in agreement with those from the finite element method (FEM) and experiments for Po ~< 0.8.

2. Three coordinate systems and the geometric description of the intersection curve Three coordinate systems as shown in Fig. 2 are used. The polar coordinate system (~, fl) and Cartesian ((, 0) on the developed surfaces of the cylindrical shell and nozzle respectively are taken as Gaussian coordinates; cylindrical coordinates (p, 0, z) are taken as the global system. The accurate transformations are expressed as follows:

c~t~= {po2 cos 2 0

+

[sin-l(po sin 0)]2} I/2

(5)

fir = sin-1([sin-1@o sin 0)] x {po 2

cos z 0 + [sin-l(po sin 0)]2} -1/2)

(6)

and in the system ((, O) are 1

g r ( O ) = ~ (1 - po2 sin 2 0) I/2

(7)

Po Let (iv, it, i,) be triad with it being the unit vector tangential to the boundary F, and iv being the unit vector normal to F in the tangent plane of the shell. We have ip x i n

it = lip ; i, I

iv = it × i,

(8)

Substituting Eq. (4) (set p = Po) into Eq. (8), we get cos ~9 = iv " i~ = it " i/~

~r (1

(

-- P° 2

1

sin4 0)1/2 cos 2 0 -}- --posin 0

x [sin-l(po sin O)](1 - po2 sin 2 0)1/2] / sin O = iv' ils = - i t ' i~ Po cos 0 ccr (1 - po2 sin 4 0) 1/2 x ( - s i n 0(1 - po2 sin 2 0)1/2 k

(9)

233

M.D. X u e et al. / Nuclear Engineering and Design 154 (1995) 2 3 1 - 2 3 8

,

+ - - sin-l(p0 sin 0) P0

)

(10)

F o r the case symmetrical about fl = 0, re/2, the solution of Eq. (12) is

The triads o f unit vectors on F in the surface o f the nozzle are shown in Fig. 3:

x =

it (t) = it

where

i. m = i/,

iv(t) = it (t) X in(t)

(1 1)

the Donnell shallow shell equation, modified Morley equation, which is p o ( R / T ) ~/~ .> 1 with an error o f the (Xue, 1991), is used for the solution shells with cut-out:

((V2 + ½)2 - 4fl2i ~@2)Z = 0

(12)

where

i

)

(15)

[ J . ( ( - i)l'/2/tc~)H. (r/a) m= 0 F,.,, = ~[J2 . . . . . (( - i) I"2/tc~) (16) l [ + J 2...... ((-i)'12~u~)]H,,(~/~x) m > 0

3. The solution for shells with cut-outs Instead o f the following applicable to order O ( T / R ) o f cylindrical

( - 1 ) " C , F , . . cos(2m n=0m=0

4]A2

and i = ( - 1 ) 1/2. u, is the normal displacement o f the shell, ~b is the Airy stress function and 4/t ~ = [12(1 - v2)] 1/~ R T

(14) ][z

J . and /4,, are nth-order Bessel and Hankel functions respectively and C n = C m q- iC.2 /I =

(17)

~1 ' 2~ 1,'2 [ ~ - - 1/2 I

(18)

The components of forces, moments, displacements and rotations in the shell are all expressed through the partial derivatives o f Z with respect to ~ and fl, as shown by Deng et al. (1991) and Xue et al. (1991). The general forces Tv, S~, Q~ and M~ and the displacements, u~, u/i, u. and 7v, on the boundary o f shell can be expressed by Z~'/)= [(?~+/~Z/ (~i8flJ)]]~,/3~ and cos ~9, sin ~9, so that they are periodic functions of 0. Take Mv as an example: M~ = M~ cos 2 ,9 + M/~ sin 2 9 + 2M~/¢ sin ~9 cos ,9 (19) where M~ = 12(1 - v 2) + I-" / M~ r ,/: r

In

_ I.~

il v . o

V (2m)2F,.,,

/i i

x

,

cos(2mflr )

g2

rF

II

i

; /

r ~,:

'02F,..

v 0F,....

~, Im ~ + - -

-C,2 m:

o

V (2m)2F,."

cos( 2mfl r

~2

)

~F 2 i~

~y Fig. 3, The directions of unit vector boundary forces and displacements on F.

n=O

1=

I

m=O

(2o) M/~ and M~z: can be expressed in the same way as

M . D . X u e et al. / Nuclear Engineering a n d Design 154 (1995) 2 3 1 - 2 3 8

234

M~. Substituting Eqs. (9), (10) and (20) into Eq. (19), we get

M~ =

~. C.,rn.tV(~r (Po, 0), fir (Po,

1o

0))

(21)

i

n=0/=l

where m.tV(l = 1, 2) are periodic functions of 0. The decompositions of the boundary force vector F and boundary displacement under u in the global coordinate system (p, 0, z) are

F = r,,iv + Svit - Qvi. = rpi,, + Foio + F.i.~

(22)

u = u~,i~ + u/~i/~ + u.i. = u,,ip + uoio + u:i~

(23)

--This paper --Steele = F.E.M o ORNL-1

0 0

I

iI /

2<

By use of the relations (4), (9) and (10) for triad transformation, Fp, Fo, F=, Mv, up, Uo, Un and Yv can be all expanded in Fourier series of 0 as follows:

pR[-3 F,, =

+ ~

-~

+ cos(20) + 4p02 sin 4 0] 4(1 - - p o 2 sin 4 0) 1/2

~ ~ C, Jk,/'(po)COS(2kO)

~

-~

(24)

F.=pRposin20(~-p°2sin2~) 1 / 2 o po2 sin 4 2

(26)

k=On=Ol=l

~ C.,fk.,"(po)

cos(2k0)

(27)

k=On=Ol=l

[3po(1 - v)-- po(1

+ v) cos(20)] 4ET pR2

+ ~'. ~ 2 C.,uk./'(po)COS(2kO)

40

50

(ram)

C.tfk.lO(po) sin(2k0)

(25)

~

2

Uk./'(po)COS(2kO)

(31)

4 £~/2 m,t v cos(2k0) dO

= -

(32)

7~

(28)

k=On=Ol=l

where m,t v comes from Eq. (21).

pR2po 4E----T(1 + v) sin(20)

4. The solution for nozzles with curved ends

C.t Uk.fl(po) sin(2k0)

(29)

The displacement function for closed cylindrical shells by Timoshenko and Woinowsky-Krieger (1959) is used for the solution of the Love equation. ¢ satisfies the following equation:

(30)

V8~9 + 42t4 c34~ = o

k=ln=Ol=l

v)pR 2 (1 2ET

(2 --

uz =

30

Here the first terms of Eqs. (24)-(31) come from membrane solutions, which can also be expanded in Fourier series. The Fourier coefficients fk,~ and Uk,/ in Eqs. (24)-(31) are obtained by numerical integration, such as in Eq. (27):

fk,i "

2

Uo=

20

k=On=Ol=l

+ ~', ~ E C.,fk./(po)C°S(2kO)

up =

10

2

7v = ~

Mv =

0

Fig. 4. Comparison of stresses cr/~/~ro at the line 0 = 0° in the outer surface of model ORNL-I (~o = PR/T).

sin(20)[-1 +po2(2 sin 2 O + sin 4 0 - 2 p o 2 sin 6 0)1+ ~ ~ 2 4 [ ( 1 - - p o 2 sin 4 0)311/2 k=ln=Ol=l 2

=

-10

VESSEL NOZZLE DISTANCE FROM INTERSECTION

k=On=Ol~l

Fo pR

40

-- p0 2 sin 2

0) 1/2

C, lUk,/(po) k=On=Ol=l

cos(2k0)

(33)

M.D. Xue et al. / Nuclear Engineering and Design

,

(1995) 231 238

235

I

-- Steele ~ F.E.M. ~ ORNL-1

I

154

I

r

<

i/ t

I

i

-50

-40

-30

,

I

-20 -10 VESSEL

10

20 30 NOZZLE

40

50

DISTANCE FROM INTERSECTION (mm) Fig. 5. Comparison of stresses G'~/~roat the line 0 = 90° in the outer surface of model ORNL-1 (ao = PR/T). Fig. 7. FEM mesh of model with Po = 0.8. where ,i t = [3(1 - v2)] 1/4

(34)

The c o m p o n e n t s of displacement are expressed through ~ as follows:

a.) -

c~3O &021~¢

c~3O v --

(~¢3

(35c)

tT, m = V4@

(35a)

The solution of Eq. (33) is expanded in Fourier series of 0 as shown by T i m o s h e n k o and W o i n o w s k y - K r i e g e r (1959):

/~o(t)= _ _ ( - ~ 3 + ( 2 - t - V") ~ 6330 '~ )

(35b)

~b = Do~gol (¢) + Do2go2(g ) 4

+ L

~ Dk,gkt(¢) cos(2kO)

(36)

k=l/=l

16

I

i

At 3 2 r a m from Ihejunction At Ihe junction

14

N

12

[

I I [ This Ix'lper Experiment O ....

,

i

10

where gkl is product o f sine-cosine and exponential function of (. The general solutions of the forces, m o m e n t s , displacements and rotations in the nozzle are expressed in terms of @~i'J)(;r (0), 0), where

LL

o

t.O

8 7

(?0i 8¢ /

6

(37)

The b o u n d a r y force vector F (t) in the nozzle is d e c o m p o s e d in the global system as follows:

4 2

F(t) = _ Tvmivm + Sv(t)it(t) q- Qv
10

20

30

40

50

60

70

80

90

= F,,
(38)

0 Fig. 6. Comparison of stresses a t/% in the neighborhood of the intersection of model (ORNL-I).

Similar to the a b o v e discussion for shell, b o u n d a r y forces F, m, F0 m, F__m, m o m e n t Mv "),

M.D. Xue et al. / Nuclear Engineering and Design 154 (1995) 231-238

236

20

T

F

,

,

~ - !

F."'=

:

P;o(1 _ p2 sin 2 0)~/2(1 _ p2 sin 4 0)-1.,2

15

4

+ ~ E D*,~, zc°s(2kO)

10

(41)

k=0/=l

5

i

M v (t) =

. . . .

(m);

o

0

uP")

4 ~ ~ k=0/=l

(2

v)pro 2

-

2Et

-5

l

--F.E.M

-15 0

(42)

4

+ ~" ~ Okt~kfl'cos(2kO) (43) k=0/=l

4

\

--This paper l

-10

Oklfk, m cos(2k0)

uo") = ~ ~ Dkt~k,° sin(2k0)

(44)

k=l /=1 u ( t ) _ (1 - - 2 v ) p r o 2 Cr ( 0 )

10

20

30

40

50

60

70

80

2Et

90

4

0

+ u0 + ~

~ Dk, akS cos(2k0)

(45)

k=l/=l

Fig. 8. C o m p a r i s o n o f S C F for a shell with cut-out (Po = 0.8).

4

7v~t)=

4 ~ Z k=01=l

(39)

Oklfk' p COS(2kO)

4

5. Continuity conditions The following continuity conditions should be satisfied for general forces and displacements in the shell and the nozzle on the intersecting curve:

Fo (t) = --prop o sin 0 cos 0(1 - pg sin 4 0) 1/2 ~

+ ~ 2 Dk,fk, ° sin(2k0)

(46)

k=0l=l

displacements Up (t), UO(t), Uz (t) and rotation ~dv(t) a r e all expanded in Fourier series of 0 as follows: Fp (t)=

~ Dkt~t" cos(2k0)

(40)

k=l/=l

F,, =

-Fp

Mv =

M v (t)

Up ~---Up (t)

(t)

F,, = --Fo (t>

Fz

=

--Fz (t) (47)

lg0 ~ b/0 (t)

Uz ~ ~/ (t)

~v z - - ~v(t)

(48) 3.5

-~

3

2.5

- -Erigen o T a y l o r a n d Lind o Mershon

i i 2

0

0.2

0.4

0.6

0.8

Fig. 9. C o m p a r i s o n o f S C F for the m o d e l s with p o ( R / T ) 112

,~ 1, t / T = Po.

Substituting Eqs. (24)-(31), truncated after the terms of either k = K or n = 2K, and Eqs. (39)(46), truncated after k = K, into Eqs. (46) and (47), the number of equations for all harmonic is 8K + 4 i. The 2K + 1 complex constants Ck~ and 4K + 2 real constants Dk~ are obtained by solving these equations. Then follows 2" by Eq. (7) for the shell and ~ by Eq. (26) for the nozzle.

i The zeroth h a r m o n i c term o f F_ = - F c (t) s h o u l d be satisfied a u t o m a t i c a l l y . The zeroth h a r m o n i c t e r m o f u: = u~ ~°, which d e n o t e s stress-free displacement, can be omitted.

M.D. Xue et al. / Nuclear Engineering and Design 154 (1995) 231-238

237

Table 1 Comparison of stress concentration factor values Number

Reference

D/T

d/D

t/T

Test

This paper

1- l

Decock (1975) Mershon (1966) Decock (1975) Mershon (1966) Decock (1975) Findlay (1970) Findlay (1970) Mershon (1966) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Mershon (1966) Corum et al. (1974) Riley (1965) Mershon (1966) Mershon (1966) Taylor and Lind (1966) Findlay et al. (1970) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Decock (1973) Findlay et al. (1970) Decock (1975) Decock (1975) Decock (1975) Decock (1975) Taylor and Lind (1966) Decock (1975)

77.8 14.7 11.61 13.0 26.9 26.9 17.9 98.0 81.4 31.4 31.4 31.4 29.0 15.2 11.6 11.6 13.4 100.0 230.0 62.8 84.4 13.06 21.00 31.40 19.00 31.40 70.00 13.22 10.30 31.40 31.40 15.20 15.20 13.02 31.40

0.131 0.193 0.282 0.289 0.298 0.312 0.310 0.343 0.436 0.411 0.456 0.436 0.448 0.424 0.498 0.416 0.501 0.500 0.500 0.524 0.524 0.566 0.550 0.566 0.650 0.646 0.622 0.716 0.760 0.707 0.748 0.730 0.773 0.80 0.817

0.448 0.752 0.920 0.495 0.615 1.00 0.400 0.343 2.17 1.10 1.59 2.20 1.70 0.55 1.53 1.30 0.50 0.50 0.50 1.00 0.890 1.38 1.82 1.52 0.37 1.64 2.17 0.504 1.50 2.20 3.20 1.10 1.60 0.80 1.66

2.18 2.22 2.74 3.28 3.00 2.60 3.40 5.70 2.36 2.56 2.20 1.91 2.50 3.27 3.24 2.86 3.80 5.4 4.0 3.15 3.86 2.68 2.70 2.61 5.08 2.68 2.49 5.30 3.50 2.16 1.70 3.44 3.10 4.10 3.06

2.65 2.38 2.40 3.04 2.84 2.40 3.39 5.14 1.86 2.46 2.09 1.71 2.34 3.32 3.93 2.32 3.93 5.3 3.7 2.98 3.38 2.57 2.09 2.26 5.40 2.38 2.49 4.85 3.10 2.10 1.68 3.47 2.91 4.51 2.67

1-2 2-1 2-2 2-3 3-1 3-2 3-3 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 6-1 6-2 6-2 7-1 7-2 7-3 7-4 7-5 7-6 8-1 8-2

6.

Verification

A s u b s t a n t i a l a m o u n t o f e x p e r i m e n t a l a n d calc u l a t e d results w a s listed a n d r e v i e w e d b y D e c o c k (1975), M e r s h o n (1966, 1989) a n d M o k h t a r i a n a n d E n d i c o t t (1991). T h e m e t h o d p r e s e n t e d is verified in d e t a i l by test d a t a o f O R N L - 1 a n d F E M results f o r a m o d e l w i t h p a r a m e t e r s D / T = 40, P0 = 0,8 a n d t / T = O. T h e stress c o n c e n t r a tion factor (SCF) values calculated by presented m e t h o d are c h e c k e d w i t h a n u m b e r o f test results at t h e i n n e r c o r n e r o f the line 0 = 0 ° as well.

6.1. Comparison o f stress fields Jbr O R N L - 1 ( D / T = I00; t / T = Po = 0.5) T h e c o m p a r i s o n o f e x p e r i m e n t a l results b y C o r u m et al. (1974) w i t h t h o s e in t h e p r e s e n t p a p e r , t h o s e o f Steele et al. (1986) a n d by t h o s e o b t a i n e d by the F E M is s h o w n in Figs. 4 - 6 . C o n s i d e r i n g a c e r t a i n a m o u n t o f s c a t t e r in the d a t a f o r 0 = 0 ° a n d 0 = 180 ° ( a b o u t 10% f o r m a x i m u m stress) a n d in t h e d a t a f o r 0 = 90 ° a n d 0 = 270 ° ( a b o u t 2 0 % f o r m a x i m u m stress), t h e v a l u e s s h o w n in Figs. 4 a n d 5 r e p r e s e n t a v e r a g e s . T h e p r e s e n t