Analysis of circular pipe bends with flanged ends

Nuclear Engineering and Design 72 (1982) 175-187 North-Holland Publishing Company

175

ANALYSIS OF CIRCULAR PIPE BENDS WITH FLANGED ENDS J.F. W H A T H A M Australian Atomic Energy Commission, Research Establishment, Private Mail Bag, Sutherland, NS IV. 2232, Australia

Received 19 April 1982

Thin shell theory is used to analyse flange-ended pipe bends, of circular cross-section, subjected to internal pressure or end loading, either in-plane or out-of-plane. Wall thicknesses should not exceed 30% of the pipe radius. Graphs of flexibility factors for deflections in each direction are plotted versus pipe bend characteristic and those of a pressure deflection factor are plotted versus bend radius for a wide range of 90 ° flanged bends. Numerical values of flexibility factors are tabulated for 90 ° and 180° flanged bends, the bend radius being two and three times the pipe radius and the wall thickness ranging from 1% to 10% of the pipe radius. Illustrative stress distributions are plotted for a typical 90 ° flanged bend subjected to internal pressure, in-plane bending or shear.

1. Introduction

As stated in a previous paper [1], the primary cooling circuit of the D I D O type research reactor at the Lucas Heights Research Laboratories consists in the main of short runs of piping connected with flange-ended elbows thus minimising the heavy water inventory. The stresses in the pipework from, for example, assembly misalignments or thermal expansion during start-up, depend on the flexibility of the elbows; since little information on this topic has been published, a solution was sought for pipe bends with rigidly constrained ends subjected to various end ioadings. Such solutions would also be applicable to the primary cooling circuits in power reactors where pipe bends occur adjacent to heavy pressure vessel nozzles, the nozzles having the effect of a rigid flange. A flanged pipe bend is shown schematically in fig. l, the pipe cross section being circular and the bend angle q~' arbitrary. Forces Fx, Fy, F~ and moments Mx, My, M z may act on the bend, or the pipe may be pressurised, causing translations 8~, By, 8z and rotations Vx, Yy, "tzAssumptions are that: (i) the pipe wall is thin (t/r<0.3), (ii) normal stresses through the wall are negligible, (iii) normals through the wall remain normal to it and unchanged in length, and (iv) the flanges are infinitely stiff. The stress analysis of an endless curved pipe under pressure or any end loading was given previously [1] 0029-5493/82/0000-0000/$02.75

y

Fy

.

_

,,ZYXz v x ~ M z Mx

6z

Fz

Fig. 1. Flange-ended pipe bend.

and the intention now is to construct a solution for the addition of flanged ends. A number of solutions of the basic equations which represent the effect of self-equilibrating end loads, decaying with distance from the ends, will be added to the earlier solution to enable the constraints imposed by the flanges to be satisfied and yield the complete flanged bend solution.

2. Solution unknowns

For this analysis, the curved pipe is represented by its middle surface, that is, an imaginary surface mid-way

© 1982 N o r t h - H o l l a n d

176

J.F Whatham / Analysis of circularpipe bend~ withflanged ends" not constant but decay with distance ~/r from a flanged end, or, to be more exact, each coefficient is the sum of a series of decaying coefficients. For example: J

(2)

u . : E Gu.ie-<", i~l

where 2 N (in-plane) 2 N - 1 (out-of-plane),

J=

~2/= aa ± ibj = eigenvalue (decay constant), R -+ lU • nj t _-- eigenvector component, U nj = Idn)

Cj = CAj +- iCBj = eigenvector coefficient. Fig. 2. Pipe middle surface.

between the inner and outer surfaces; fig. 2 shows a short segment of such a middle surface with one end flanged. The forces Tn, P, Q and moment Mn per unit length of circumference act on the cross-section, resulting in the displacements u, v, w and a rotation ~. These are the eight unknowns to be determined; they may be grouped non-dimensionally as:

Substituting the complex forms of the eigenvalue, vector and coefficient into eq. (2) and combining the effects of flanges from opposite ends, since flange effects from bending persist to a distance of approximately one pipe circumference along a curved pipe [2], then

J

CAj

Un= X

j= 1 + CBj +{

CA'j

+ CB;

,~, =- {u/r, w/r, M:, T*},

( .Lc + u'o,s)] ( < s -
(., i,} UnjC -{- UnjS )

( UnjSR t__ UInjc ')

(3) "

where

~2~- ( v / r , ¢ , P * , Q * } ,

c=e

where T*, P*, Q* = ( T, , P, Q ) /Et,

a/Ocosbj~,

M~ = Mn/rEt,

s :

e

a~n s i n 3~'q,

E = Young's modulus.

c' =- e-~,l¢-"~ cos bj(~ ' - n),

Now let

s' = e -a*(n'-n) sin b j ( n ' - r~), T/' = p~', and (l)

CA, CBj = eigenvector coefficients from the end q~= O, CA'j, CBj = eigenvector coefficients from the end 4, = qa'.

where ~ , ~e2 are endless pipe bend solutions from [1] and ~j~, ~j~ are the flange effects.

The ( ± ) sign between the bracketed terms is: (i) in-plane loading, ( + ) for ~:[ coefficients and ( - ) for ~ r and (ii) out-of-plane loading, ( - ) for ~[ coefficients and ( + )

3. Flange effects

for ~r.

~ =~+~,

The distributions of the eight components of the vectors ~[, ~2r around the pipe cross section are given in Appendix A for in-plane and out-of-plane loadings, the non-dimensional stress resultants T~, Pf, Qf, M f being self-equilibrating. The coefficients in Appendix A are

4. Eigenvalue determination The eigenvalues and eigenvectors depend on the pipe bend curvature, diameter, wall thickness and Poisson's

177

J.F. Whatham / Analysis of circular pipe bends with flanged ends

ratio, regardless of bend angle, end conditions and loading apart from knowing whether the loading is in-plane or out-of-plane. They are determined from the homogeneous form of the system equations derived previously [1]; in matrix form:

8N- 1 eigenva[ues ( out-of-prone )

P r

I

!

i

,t~

• 8N+3

i

i

I

N,:4 ,D

F

[L,IN

}-

'

eigenvcflues (in-ptane)

:o, 0

f

[ L 2 ] ( ~ } --~-~ {~, } = 0 ,

(4)

~'.o,

,ID,D"~

~o ,em,D P,O~ j

where L~, L z are presented in table 1. Assuming that the flange effects decay as

.,D•.,o • , o •

• .~ •.DO o o

,f}:ooO(°)l

,o -5

then

t

I

L

i

1

i

t

i

i

i

I

0

-5

i

1

5

Re

L,

(5)

L

o lk ;l

Substituting the expansions from Appendix A and expanding by Fourier analysis gives 8N + 3 equations (in-plane) or 8N-1 equations (out-of-plane). This is an algebraic eigenvalue problem which can be solved by a

Fig. 3. Typical curved pipe eigenvalues.

computer code such as EB06AD [3] for the eigenvalues and eigenvectors. All are complex, as indicated by the typical eigenvalue distribution in fig. 3, where only those

Table 1 ILl], [L2]

[L,] (i) (ii)

uf/r

wgr

sin 0 p sin 0 O

r8 v8

3 30

T7

cos 0 - ~ - r8 p

3

-

80

a2 -- + r6-O O0 802

sin 0 3

(iii)

- (1 + 12 ) 0-~(8 O-~)

--~-0 ( 8 - Y-O2

~

(iv)

0 600

[L2]

vf/r

v 3z (8~)~ i2 302

(ii) (iii) (iv) p=R/r;

p

6+~-~

3

a

y 3 3 2(1 + v)(3+ y) 0 0 ( 6 ~ ) -'t 3 ,~) 2(1+ 2)('3+ ),) gO( "~ 8=l+cosO/p;

y=(t/r)2;

o

0 sin 0

~-a-~(~ )

0_( sin 0 o0 o

3z

)-"~(~)

pr 3+y

cos 0 P

12(1--u2)8 Y sin 0 3

8~-0-~)

~, . 8 0

3+~;~a0

( 1 - v2)8

v 02 ( 82

q,f

sin 0

(i)

0

--UQr

6(1 +,,) 8

BO

3+3,

-6

0

2(1+ p)(3+ y) (6 ) 3, ~0 -~0 2(I + p)(3+ y) (8 )

3 + y O0 sin0 3 8 P 3 + y O0 ( 8 )

u=Poisson'sratio.

p

~-~(8 )

cos 0 + v8

0 0

cos 0 P

178

J.F. Whatharn / Analysis of circular pipe bends with flanged ends"

eigenvalues with negative real parts represent decaying end effects; with their conjugates, these total 4 N for in-plane and 4 N - 2 for out-of-plane loading. Hence J equals 2 N or 2 N - 1 in eqs. (2) and (3) while each eigenvalue has 8N + 3 or 8 N - 1 complex eigenvector components corresponding to the coefficients in Appendix A.

described by four dimensionless variables ~, x, k and/3 which depend on the state of strain of the cross-section: for a built-in end or a rigid flange all are zero. These variables are related to the displacements by 1 3u

~=r~+~ K--

5. S p e c i f i c

IN- P L A N E DEFLECTION

-,

1 au

1 ~2w

r ~O

r 302,

1 3v

3~

r ~0

30'

problems

The eigenvector coefficients in eq. (3) are derived from the bend angle, end conditions and loading. The five flanged bends in fig. 4, designated by the computer program developed for their analysis, will be considered for the following loading conditions: P R E S E F - internal pressure, B E N D E F - in-plane moment, S H E R E F - any in-plane f o r c e / m o m e n t combination other than B E N D E F , C O I L E F - out-of-plane shear and twist as on a coil spring, T U R N E F - any out-of-plane f o r c e / m o m e n t combination other than C O I L E F . According to the thin shell theory of Novozhilov [4], the distortion of a pipe cross section under load can be

1 32v =

-

w

-

-

-

-

r 002

t-~b.

(6)

When c, ~, ~ and /3 are zero, so must be the displacements u, w, v, ~p and aw/O~l, the last by eqs. (4), apart from components of those displacements which represent rigid body motion. For example, in in-plane loading, these displacement components are w / r = w 1 cos 0, u/r=

v / r = v o + v I cos 0,

-w I sin0,

~b=v 1 c o s 0 ,

1 0w

r ~)~1-- ( % / P - - v j ) c o s O,

where the coefficients w z, v 0 and v~ are displacements of the pipe cross section in the directions shown in fig. 5, the superscripts referring to the ~ values at each end. The eigenvector coefficients are now derived from the requirement that, at the pipe ends, ~ f -['-~e = ff f ~- Ke ~---~kf-'~'-~ke= /3 f "~ Be = 0 ,

PRESEF

BENDEF

C = f" - t t 2

OUT-OF-PLANE COILEF

(7)

SHERE F

DEFLECTION TURNEF

where c f, ~¢f, ~f,/3r represent the flange effect and c c, KL he, /3e the endless pipe solution [1]. The value of each coefficient of the expansions for u, w, v, q~ in Appendix A at the pipe ends is equated to the negative value of that coefficient from the endless pipe solution, except that v 0, u 0 are omitted and the combinations (u I + Wm) and (v I - q q ) are treated as single coefficients. The resulting 8N equations (in-plane) or 8 N - 4 equations (out-of-plane) are then solved for the eigenvector coefficients CA j, CBj, CAj and CBf. PRESEF, B E N D E F and C O I L E F are special cases since, without end effects, the stresses are independent of position around the bend. Thus the coefficients, CAj, CBj equal the coefficients CAj, CBf and only half the equations are required. 6. Flange

Fig. 4. Flanged bend loading conditions.

displacement

The solutions thus obtained displace both flanges as rigid bodies, the movements being given by single coef-

179

J . F W h a t h a m / Analysis o f circular pipe bends with flanged ends

¢

ficients from the total solution as shown in fig. 5. By stipulating that v 0 = v I = 0 when 0 = 0 ' / 2 in P R E S E F and BENDEF,

0

Wit

w o:w

,

v

v 0:-v,,

a n d with u o : w 1 = 0 when 0 = 0 ' / 2

uOo

wO=-w;,

in C O I L E F ,

vO:v,,

It is now necessary to restore the flange at 0 = 0 ' to its original position (fig. 1); the total displacements of the flange at 0 = 0 for in-plane deflection ( P R E S E F , B E N D E F a n d S H E R E F ) are then

IN - P L A N E

8 x / r = v 0 - v~ cos 0 ' - w~ sin 8y/r:w°--w~cosO'

0' - v'l P ( I -- COS0 ' ) ,

+v~sinO'--v'lpsinO

',

(8)

= - v ° + v',,

or, for out-of-plane deflection ( C O I L E F a n d T U R N E F ) , 8Jr=

FwIO t

w ° - w~ - v ~ p s i n 0 ' -

Ty = v ° - v'~ s i n 0 ' - u~ sin 0 ' ,

(9)

"G = - u 0 ° + u ~ c o s 0 ' - v ' 1 sin 0 ' . The c o m p o n e n t s of the overall flexibility matrices for curved pipes, which relate relative end displacements a n d rotations to applied forces a n d m o m e n t s , m a y be written as c o m p o n e n t s derived from elastic line theory, suitably corrected by dimensionless flexibility factors

OUT-OF- PLANE

Fig. 5. Flange displacements.

Table 2 Deflection matrices

In-plane

F~R2/E1

ryR2/E1

MzR/E1

8~,/R = ~y/R =

f x x ( e / + B --2S) fyx(l - C - D )

fxy(1 -- C - D ) fyyA

f x z ( S - ep') f y z ( C - I)

v, =

f, As

-

¢3

f,y(c

-

1)

f,,¢'

Out-of-plane

8:/R = yy = •x =

u~p(1 - cos0'),

F, R 2 / E I

MxR/EI

M~R/EI

f-Fr4 + p(ep' + B - 2 S ) f-~y:D + P ( l - C - D ) f-~-~4 + P ( B -- S )

zf~-yD+ P(1 - C - D )

~x,4 + P ( B - S ) ~:,D - PD ~xxA + P B

P = 1 + v, S =sin ok', A =(ok'-- S C ) / 2 , D = S S / 2 I = ~rr3t, C : c o s ~', B = ( ~ ' + S C ) / 2

f'~yyB + P A ~ D -- P D

e.

~a

2+

I

i.--

I

l

7---

I'--+--Tlll

I,,j

,

J

.

i

\

i

_ l l l , i

z~

x

z~

'<9

z~

~

z~

XA

\

\~,.~'b--____.--s \ \ o - o ~. f _ _ S

[ ~ r w - i

L+O I

7

I

i

i

,

,

tT+J~ + _zU

+o-o:~

T--T+

LO.O -

ITI'~-

L I

I

I

~

'.

q

xg

I I

I

~

I

x~

1

"i l l ' I " i "I 'I

PO

I

I

I

'I

I

D~±SI~3J_DV~tVHO aN3g LO'O

~J

L

,

~

\

5

L'O

~,3~

~

~3J.u __+~=

xj

LO-O

5

=x 9

E =zA

L

"spuaq pa~u~ U o06 SJOla~j fil!l!q!xaU au~ld-u I "9 "~!~I

5

5

10

1

2

3

4

i

i r

1 1 \

Fz I

)-01

Y~

I

I

I

4

I

I

[11[

~: [fY2+P]P2/2~"r ~ :z

0.02

/

01

~

I

\ \

I

i

i I

i i i

i

I i iiJi

P:l+v

v:o3

1

hp:RRt,['-2'

i

Fig. 7. Out-of-plane flexibility factors 90 ° flanged bends.

Yx:4

5

10

1

yy= 4 3 2

6z=

i

lO _-- F~= [f~+P(3-~-)lPS/4~E'~

1

0"01

m =

I

I

J

I

I

I

I

I II

llIKI

My

6z

__

I

=

I

"yy

O1

Fz

__

My

I

I

01 BEND CHARACTERISTIC

I

h

I

I

I

H i l l

I

I

I I

1

1 % ~

I I

I

I

~

I

I i I II

Mx

YY My

Fz

i

i

t

I

I

for 90 ° bends

My

"Yx

Yx

Mx

6z

I

Mx

Yy

ll~l

Yx Mx

I

I

,,(r

I

I

"la

ll:

i:l

182

J.F. Whatham / Analysis of circular pipe bends with flanged ends"

Table 3 Flexibility factors 90 ° flanged bends (f~v.v= ~f~x,f.-~ are endless pipe flexibility factors [1], ~, =0.3)

t / r = 0.01 R / r =2 ~.,. f,,. f: ~. f~., f...~ f.:

0.02 3

2

0.05 3

2

0.1 3

2

3

f,~

2.66 2.79 2.37 4.16 3.09 2.63 3.07 2.73 2.30 2.32 2.95

2.90 2.83 2.98 3.22 3.25 3.08 3.19 3.02 3.43 2.84 3.71

2.61 2.73 2.30 4.09 3.01 2.56 3.01 2.67 2.21 2.25 2.86

2.80 2.74 2.86 3.13 3.12 2.95 3.07 2.90 3.26 2.72 3.54

2.43 2,56 2,10 3.92 2.78 2.34 2.82 2.49 1.95 2.07 2.61

2.51 2.47 2.51 2.90 2.76 2.60 2.76 2.59 2.79 2.41 3.08

2.16 2.31 1.78 3.66 2.44 2.02 2.54 2.21 1.56 1.79 2.21

2.14 2.12 2.07 2.60 2.31 2.15 2.35 2.18 2.19 2.00 2.48

f e_.

82.58

55.07

41.27

27.53

16.49

11.02

8.21

5.43

f~x~fT:

which then account for b o t h the pipe curvature a n d the e n d effects. Such flexibility factors, symbolised by f, are defined in table 2. N o t e that b o t h matrices are symmetrical a b o u t the leading diagonal, thus bearing out the reciprocal theorem, a n d that for out-of-plane loading the elastic line deforms by b o t h b e n d i n g and torsion. F r o m the analysis, a n d using eqs. (8) a n d (9), these flexibility factors have been calculated a n d are plotted in figs. 6 a n d 7 versus b e n d characteristic h for 90 ° flanged end pipe elbows; all are symmetrical a b o u t the leading diagonal. Numerical values of f are given in tables 3 a n d 4 for 90 ° a n d 180 ° b e n d s with curvature radii two a n d three times the pipe radius. In the case of 180 ° bends, it was f o u n d that

3'x/Mv = 0, and "~v/F: = [A - 2(1 + u ) ] R 2 / E I , where A is a small discrepancy owing to the exclusion of tension a n d shear from the elastic line deflections; for b e n d angles of 179 ° to 180 ° the following equation was adopted:

A pressure deflection factor fp' plotted in fig. 8 shows the small effect that flanges have o n b e n d deflection in the case of 90 ° bends, unless the b e n d radius ratio R / r is less than 2.

Table 4 Flexibility factors 180° flanged bends (f~v = f~v, f.-x -- f~z, xf-~x= f~.-, f~.=0, ~,=0.3)

t / r =0.01

fv v fxy f~x fz~

f~yv fe

0.02

0.05

0.1

R/r=2

3

2

3

2

3

2

3

15.61 16.47 11.12 15.07 19.24 19.31 -0.54 3.23

28.40 27.62 16.66 22.40 31.47 31.48 --0.23 5.48

12.32 12.86 8.94 I 1.86 14.94 15.01 --0.54 3.05

17.01 16.60 10.61 13.74 18.68 18.70 -0.24 4.52

7.75 7.86 5.86 7,41 8.98 9.05 --0.57 2.60

8.12 7.92 5.62 6.83 8.70 8.72 -0.26 3.23

4.98 4.84 3.93 4.71 5.43 5.51 --0.61 2.09

4.48 4.31 3.40 3,89 4.62 4.65 -0.27 2.33

82.58

55.07

41.27

27.53

16.49

11.02

8.21

5.43

183

J.F. Whatham / Analysis of circular pipe bends with flanged ends ~'=9o °

t

0"8

¢': ~ ~°~ p R :?

o.31~.0~\\\ I' \\\

,, ¥, E~ P: P- -fiT'

/-,1\

~ 0.1~.g o~

~

C-

/

3

4

6

8

v-o.~

Unttongcd bznds

tl ~/// 2

~=o.os

S~-'T

o.6~- \\ t k o.s\__ r =o.o~ l'-,-Tr~V" o.4~- \ \ ' X x~

-o,,L ,-<\

R

T=2.s

10

20

30

40

~'3f <45°

60

BEND RADIUS RATK) (p)

~0.5

Fig. 8. Pressure deflection factor 90 ° flanged bends.

Fig. 9. Transverse stress from internal pressure.

7. Pipe stresses Pipe outer surface stresses were derived by

oeo = Te/t + 6Me/t 2 , ~'=90°

onn = Tn/t + 6Mn/t 2 ,

----

~ -0.05

s~: ~..~ p

where To=vT, + Et an ), r ( --~-I-w

Me = pMn + yEt ( on ~ - a8

T " 2.5

(lO)

~2W)

~02

'

( t / r ) 2.

Illustrative distributions of outer surface stress are plotted in figs. 9 and 10 for pressurisation, and in figs. 11 to 16 for in-plane end loadings of moment and shear force on a typical flange-ended pipe elbow; the only stress distributions to be experimentally verified were those for the in-plane moment [5]. Stress levels away from the ends are reduced by flanges, and although high longitudinal stresses were calculated at the sharp root of each flange, these stresses would be reduced in practice by a fillet radius. Flanges had negligible effect on the stresses overall in pressurised pipe bends since the flange effect decayed to

Nt';°gg ~ -1.0

Fig. 10. Longitudinal stress from internal pressure.

3) • 0.3

J.F. Whatham /' A na(vsis of circular pipe bends with flanged ends

184

~'~9o'

~ =2.5 ~-o.os v-0-) t

M~¢

/

k-)

L

~

, r:

"

$2-- ~;/" ]1" I'Zl

"

---~'r~ '~o" L~INTRAOOS _/;] t

7 ,I~,

6

:.

M

'1

S, . OM-I]-r rZt

t-¢

"

"

"

-_

l

~

I //

~ '

End

'~''~°°'

Fh, / ~/-

2/ i

47 /', ,!t%dL4.4~

4/

..":7/~,l/ ...V/ ~A

aC

2~LANGE10'

20

uJ

30

40"

~

IN TRAOO5

,

Without

End

tI"RA00S

-2

tn

~Lb

10'

20°

30'

Fig. 11. Transverse stress from in-plane moment.

40* ~Ex TRXOO'S -4

Fig. 12. Longitudinal stress from in-plane moment,

¢

.9o0

/111 ~

/

8

-

-

.s

"

oo5.o

'

~

2

g

F~0

15°

300

450

600

Fig. 13, Transverse stress from in-plane shear without end effects.

75°

--~-f-?'~°

INTI~ADOS

J.F. Whatham / Analysis of circularpipe bends with flanged ends

185

¢

•

r

-

-

-

,

--

,

,

,

180 ° I N T I ~ s

135°

4

o M

O

~ 15 °

30 °

. 45 °

60 ~

• 75 °

0 ° EXTRADOS 90 °

llg -4

Fig. 14. Longitudinal stress from in-plane shear without end effects.

F

•" - - - - -

•

•

~ -', I

18oo INTRAEX3S

r

t

g

•

t 30°

~5 o

I~ -2

Fig. 15. Transverse stress from in-plane shear.

t 6oo

t 7so

i ` / :)o E X T R A D O S 9 o° -2

186

J.F. Whatham / Analysis of circular pipe bends with flanged ends

F

I

= 900

ff

:

2"5

-~ : 0 - 0 5

~)

=

0-3

0"2 IT F2t S2 : ~ i

R

i

F Without •

end e f f e c t s i i

180 ° INTRADOS

o O ),

J 15 0

----T--300

r4so

I

6o o

-XTRADOS

7~o -2

u'l

_j Fig. 16. Longitudinal stress from in-plane shear.

zero within a distance of one pipe radius from the flange, leaving virtually the membrane stress solution.

8. Available computer programs A computer program package BENDPAC, written in F O R T R A N IV and ASSEMBLER for an IBM 3031 computer, is available from either the Australian Atomic Energy Commission or the National Energy Software Center, Argonne National Laboratory. The programs contained therein, five of which were referred to in section 5, are designed to calculate the following: (a) Stresses and deflections in pipe bends terminated by flanges, infinitely long tangent pipes or short equal length flange-ended tangents when under pressure (PRESEF) or pure in-plane bending (BENDEF). (b) Stresses in flange-ended pipe bends from in-plane end loading other than pure bending (SHEREF), coil spring type out-of-plane loading (COILEF) or any other type of out-of-plane loading (TURNEF). (c) Flexibility matrices for flange-ended pipe bends for any in-plane end loading (FLEXIN) or out-of-plane end loading (FLEXOT).

9. Conclusion A method has been described for analysing the stresses in flange-ended pipe bends subjected to internal pressure or any type of end loading, either in-plane or out-of-plane, using equations published previously [I]. Flexibility factors are plotted versus pipe bend characteristic R t / r 2 for a wide range of bend radii and wall thicknesses. Numerical values of flexibility factors are given for flanged pipe elbows and U bends in which the bend radius is two and three times the pipe radius. Illustrative distributions of outer surface stress are presented for a flange-ended elbow under pressure or with end loadings of in-plane moment or shear force. In each case, high longitudinal stresses are calculated at the sharp root of the flange but in practice these stresses would be reduced by a fillet radius.

Acknowledgement The author acknowledges the advice and encouragement of Professor J.J. Thompson of the School of Nuclear Engineering, University of New South Wales.

J.F. Whatham / Analysis of circular pipe bends with flanged ends

References [ 1] J.F. Whatham, Thin shell equations for circular pipe bends, Nucl. Engrg. Des. 65 (1981) 77. [2] J.F. Whatham and J.J. Thompson, The bending and pressurising of pipe bends with flanged tangents, Nucl. Engrg. Des. 54 (1979) 17.

[3] M.J. Hopper, Harwell subroutine library report AERER7477 (1973). [4] V.V. Novozhilov, Thin shell theory (Wolters-Noordhoff, Groningen, The Netherlands, 1970). [5] J.F. Whatham, In-plane bending of flanged pipe elbows, Trans. Inst. Eng. (Aust.) CE 21 (2) (1979) 80.

Appendix A: Series expansions for the flange effect unknowns In-plane a

Out-of-plane a N

N

u l s i n 0 + ~] u. sinnO

ut/r =

ur/r=-Uo

--UlCOS0--

wf/r=

wIsin0+ ~ w. sinnO

n=2 N

N

wf/r=wo + w l cosO+ ~ w. cosnO n=2 N n=2

r.'=

M.'=

M~sin0+ ~ M. sinn0 n=2

N

-M, coso- E rocosnO N

2 UnCOSn8 n=2

n=2 N

M, cos0+ E M. cosn0

N

--MIsin0-- E T, sinnO

n=2

N

n~2

I

vf/r=v O+v IcOso+

E °ncosn o n=2 N

~r=~o+ ~t+cos0+ ~ %cosn0

vf/r

v t s i n 0 + ~ v~sinn0 n=2 N

~f=

qqsin0+ ~ q,.sinn0 n=2

n=2 N

Pf=

Q t s i n 0 + ~ P. sinnO

Pf:

+

Q1 c O s 0 +

E

N

Qn cOsnO

Qr=

n=2

Q l s i n 0 + ~ Q. sinnO

n=2

8N + 3 coefficients

n=2

8N -- 1 coefficients

a Coefficients are different for in-plane and out-of-plane expansions.

N

-O~cos0- ~ /~.cosn0

n=2 N Qf=Qo

187