ORNL-2442.txt

 

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CENTRAL

ORNL=-2442

THERMAL STRESS ANALYSIS OF THE ART HEAT

EXCHANGER CHANMNELS AND HEADER PIPES

D. L. Platus

CENTRAL RESEARCH LiBRARY
DOCUMENT COLLECTION

LIBRARY LOAN COPY
DO NOT TRANSFER TO ANOTHER PERSON

If you wish someone else to see this

document, send in name with document
and the library will arrange o loan.

 

   
  

TSTARCH LIBRARY

CULLECTION

2 cy. 774
C-84 - Reactors - Special
AEC RESEARCH AND DEVELOPMENT REPORT

of Aircraft Reactors

M-3679 (20th ed., Rev.)

eatures

3

 

OAK RIDGE NATIONAL LABORATORY

operated by
UNION CARBIDE CORPORATION
for the
U.S5. ATOMIC ENERGY COMMISSION

 
 
:i A
=& ‘&"-';;é?_ D
ST

   

ORNL -2442
C-84 - Reactors =~ Special Features

of Aircraft Reactors
M-3679 (20th ed., Rev.)

This document consists of 38 pages.
Copy 77 of 227 copies. Series A.
Contract No. W~7405-eng-20

REACTOR PROJECTS DIVISION

THERMAL. STRESS ANALYSIS OF THE ART HEAT

EXCHANGER CHANNELS AND HEADER PIPES

D. L. Platus

DATE ISSUED

fiflflp'4£1QEg

   
  

- T
g TENE

M g "TGAK R.lDGE NATIQNAL LABORATORY
i Qak Rldge Tennessee

____ m:,,-“ 1N

 

£

4
-

4 R s

 

 

  

05 fl i

”“rvfi%Wflwwmwwrw%fl,Q; 3 yy5kL 03kL22? ?

 
Are
 

TR AR ' b ogE e T e T,
e . e . K
AL AR oy e
WA K g s SRR A T 5

    

m’c‘)r e i th T TS ST,
~ [, Tyt A Rt AN

i oENX ) a"{?a«,‘é'«t- -"‘?"o,“"- S AT T

e _

NOMENCLATURE

Subscripts on Deflections:

 

Szmbols:

X3¥s2

Xp? Zp

-

H

{

D, I

stényZ
ép, én

deflections due to in-plane bending of channel
deflection due to out-of-plane bending of channel
deflection due to rigid-body rotation of plane of
channel about y-axis

deflections due to relative thermal expansion of channel

deflections due to deformations of header pipe

rectangular coordinates

coordinates of point b

distance from origin to point b

angle between negative x-axis and ob

radius of curve of channel

maximum radius of header pipe

length of header pipe

directions in xz-plane, parallel and normal to plane of
curve, respectively

deflections parallel to x~-, y-, and z- axes, respectively

deflections in plane and normal to plane of curve, respectively

§0, 88, &7 angular deflections with rotation vectors parallel to x-, y-,

S Yp 28y

and z- axes, respectively
angular deflections with rotation vectors parallel to p and n,
respectively

forces parallel to x- and z- axes, respectively

   

. . 4
0 — R e i) iy g g
S T e
B T R SN TR et R

&

T T
NN

 

 

:
IR i M I BN AR e [ R

VRS W Y et .- At e e et T o LT g s
BRI R L R A T R T e

 
F, N forces parallel to p and n, respectively
X

y
M

MffimrjMn mements tangent, radial, and normal to plane of curve at

sz moments parailel to X-, y-, and z- axes,; respectively

fa

MN moments paralilliel to p and n, respectively

equator; respactively

MH resuttant bending moment acting on header pipe

MHX9MHy x= and y= components of MH
69615@2 normal stresses
GNpqn stresses due to ipn-piane bending of channel
OpsC, gtresses due to out-of-plane bending of channel
T shear stress or twisting stress
C a functicn of the cress section used in calculating shear stress:
K a function of the cross section used in calculating torsional
rigidity
Ip moment of inertia of channel cross section about an axis
radia. to the curve
IN moment of inertia of channel cross section about an axis
normal. to plane of curve
ZP section modulus of channel about an axis radial to the curve
ZN section modulus of the channel about an axis normal to the curve
IH moment of inertia of cross section of header pipe
JH polar moment of inertia of cross section of header pipe
v  peisson's ratic
E modusus of siasticity
G shear modulus, G = §?f§$?7)
By B ov==8sp coaefficients in set of linear algebraic equations
a linear coefficient of expansion; angle describing direction of MH
T temperature
 

B R AR A ek S g
" g . B et g -

g W e o SErtD) L
BN e e WD o
Erssy o SRR
> SRR SR Ll T e s g
e a N g

M; ey -
o ,“'mtm ‘fi PN e
ol -'.‘4&;‘,'\1" 3

  

       
 

THERMAL STRESS ANALYSIS OF THE ART HEAT EXCHANGER
CHANNELS AND HEADER PIPES

This report summarizes the study which was made to determine the
stresses; deflections, and the forces and moments acting on the ART
heat exchanger channels and header pipes due to relative thermal expan-

sion between the channels and the pressure shell at full power operation.

Introduction
Figure 1 shows a sketch of a channel and header pipes, and a portion

of the pressure shell to which they are connected. During full power
operation the temperature of the channel will be higher than that of the
pressure shell, and thereby produce relative thermal expansion. The
resulting forces and moments will cause deformation of the channel, the
header pipes, and the thermal sleeves which connect the pipes to the pres-
sure shell, ‘The stresses due to these loads will be transmitted to the
incoming NaK piping.

Figure 2 shows the idealized system used for the analysis. The
channel was treated as a semi-circular-arc curved-beam connected directly
to the header pipes, which were treated as cantilever beams. This analysis
assumes that there is no deformation in the incoming NaK piping, or in the -
thermal sleeves. It is expected that this assumption will yield an ade~
quate initial estimate for the analysis of the channel.

Because of symmetry it was sufficient to consider only one-half of
the channel and one header pipe. The channel was assumed fixed at the
mid-point and the deflections due to the relative thermal expansion were

applied to the system. Elastic theory was assumed for all calculations.

 

g e———————
—, e e T T o o

     

TRl

ek Gt S TR L

  

AR e L o .
St m e rneeierdy At TR

; 3 SRR G i IR N e e
AT SRR AR ‘

. o crob
AR R R R )

1

 

 

 
FIG.{- SKETCH OF CHANNEL, HEAPER PIPES AND ORIENTATION I[N
REACTOR

INLET
NaoK PIPE

|
\‘ I P

.
. LT )
O
2
O
O

.

4 ORNL-LR-DWG. 27494

 

 

 

  
     
    
        
 

 

CHANNEL

 

c
e
= / PRESSURE
< / SHELL
1]
" /
) A % A
SECTION A-A /
]
/ LINER (XI)
SOUTH //
| HEADER ;
— 1
.. THERMAL
SLEEVES

 

OUTLET
‘ NaK PIPE

T

 

 

 

 

 
ORNL-LR-DWG. 27495

 

 

FIG.2— SKETCH BEAM STRUCTURE USED TO
APPROXIMATE CHANNEL AND HEADERS

g

-5

 
“
YN s

Thermal Expansions

 

The total vertical expansion of the channel relative to the
pressure shell was reported in ART Design Memo 8«D-5 as 82 mils.
This was assumed to be distributed equally between the sections
above and below the mid-point, so that 41 mils was the vertical
deflection used in the calculations.

For the radial expansion, the channel was assumed to be at an
average temperature of 1&250F’and the pressure shell at 1240°F,
Taking the radial position of the header pipes to be 19.59 inches,
this gives a radial deflection of 32 mils.

Method of Analysis

Deflection equations were written to determine the forces and

 

moments acting on the channel at point b, from the thermal expansions

applied between points a and c. The coordinate system is shown in
Fig. 3. Since the channel is free to grow radially, forces were not

applied in the y-direction. The modes of deformation included in-

plane and out-of-plane bending of the channel from flexure and torsion

and deformation of the header pipe by flexure and torsion.

The deflections for point b due to deformation and rotation of the

channel may be expressed by the following equations, in which the sub-

scripts refer to the modes of deflection.

fx = SXP + SxN + éxr
Jz = 8zP + SzN +>cSzr
do = dep, + é‘eN
§g = &gy + 4@,
dr = §rp + Sy

at

wBm o w %

(1)
(2)
(3)
(1)
(5)

 
 

ORNL -LR-DWG. 27496

 

 

FIG.3- COORDINATE SYSTEM SHOWING FORCES AND MOMENTS
ACTING ON CHANNEL Afi@,__il:]\EADER PIPE

=

 
Fo

The relative thermal expansions applied between points a and c
must be equal to the differences in the deflections of point b caused
by deformations of the channel and those caused by deformations of the
header pipe. Hence, the following relations may be written, in which
the subscfipts T and H refer to the appiied relative therm@l expansions

and the deflections due to deformation of the header pipe, réspectivelyl,

Sxp + Sxy + 8%, - 8xy = Iy (6)
Szp + Sz +dx, = - Iz, (7)
Jep + S@N - 86, = 0 (8)
Sgy, + 59 - gy = 0 (9)
§rp +&7y =Sry = O (10)

By expressing the deflections in Eqs (6) through (10) in terms of
the loads acting on the channel at point b, a set of equations results
from which these loads may be determined. Since the rigid body rotation
of the plane of the channel about the y-axis is an unknown in addition
to the five loads ij Fz’ Mk, My and Mz’ an additional equation is re-

quired, and may be written by summing moments about the y-axis.

EMy = M& +Fz -Fx =0 (11)

Deflections From In-Plane Bending of Channel

 

It is seen from Fig. 3; that the force and moment producing in-
plane bending of the channel are P and MN° These may be resolved into

forces and moments parallel to the coordinate axes.

P = F_ sing - F, cos ¢ (12)

My = M_sin ¢ + M, cos @ (13)

1. The applied relative thermal expansions are taken as positive for both
the x and z directions, Note also that the deformation of the header
pipe in the z direction has been neglected.

‘I 1

5
- Lag

The deflections duvue to these loads with respect to the p~ and

. 2
n- axes are given by

2
5
Pr MNI

§ r - == (14)

P I EL; ~ EI

D My

- Pr L

Y, = - BT, © 2 BT, (15)

Resplving these deflections into components along the coordinate axes,

§xp = ~&p cos ¢ (16)
§z, = &p sin ¢ (17_
d’eP = dY_ sin ¢ (18)
§rp = &Y, cos ¢ (19)

Substituting Eqs (12) through (15) into Egs (16) through (18),

5
CSDXP =£f %E(Fx cos § - F, sin @) cos ¢
(20)
+ %TE (MX sin ¢ + M, cos Q) cos ¢
2
c?zP = -iz-fif— (F, cos ¢ - F, sin @) sin ¢
N
(21)

o ,
+-E——I-§ (MX sin ¢ + M, cos @) sin ¢

23 See Refo l’ Part l’ Ppa 7"’80

 
2
cSGP = EI (F, cos g - F_ sin @) sin ¢
(22)
+ g-—;’—N (M, sin @ + M cos @) sin ¢
2
§7p = EIN (F, cos ¢ - F, sin @) cos ¢
(23)
+ g:_%fi (M, sin ¢ + M, cos @) cos ¢

Deflections From Out-of-Plane Bending of Channel
The force and moments producing out-of-plane bending of the channel

are N, MP’ and My. Resolving N and MP along the coordinate axes,
N = F_sin g + F, cos @ (24)
M, = M sin ¢ - M, cos ) (25)

The deflections due to these loads with respect to the n, p, and y
5

axes are given by

e Mre 11
Sn = N [EEI +6+_7T )GK] T’Cfi;*fif} ~5- [g ET—-G_K-(Q——

b

3, See Ref. 1, Part 2, pp. 13-16,

. 5
. X
. &

-10-

 

 
1 1y M 1
SV ="1'\I'§'(E"“ 1; %MPI'(ET_ GK) - (EI "“é'fi) (27)

P

 

2 r
N 1 N1l M 1 v 1.1
dty = = [g BT, ~ (2 - é') fi{'] =z @IP "d'K) ty My GIP * E}"K’) (28)

Resolving (Jn and § \}&) into components parallel to the coordinate axes

§xy =dn sin ¢ (29)
‘ §2y =dn cos ¢ (30)
§7y =3y, sin ¢ (31)
foy = - Y, cos ¢ € ‘

Substituting Eqs (24) through (27) into Egs (28) through (32),

Je; ="l:2'2"(Fxsin¢+F cos¢)[ E_];"" (2,.17) ]

_ (33)
i o L
‘ ’g(Mx cos¢=MZ sin @) “'fi%;'”“@fi)"”gMyr (f?[:;-"aj;f)
Gy = 2w, sin g+, con 9) [P b+ (3 - ) | oin 6
I'2 1 1
- % (M_cos ¢ - M, sin ?) (fi'fl;' + ‘fi) sin ¢ (34)
e T 1 Tyl
. +_2—M.V[Q ;E"i; u(?m-z—) fi}Slnfd

P

wl]l=

 
P

am

3T

1‘2 1 1
_—é-(chosgj-MZs:.nQfl) ET1;+§'I€>C°S¢
2

r T 1 my 1
_2_1My‘|:2 E'f;'(e'z)GK]°°S¢

+

I‘2 1 1
Z’_(Fx 51n¢+FZ cos @) -E—I—I:-i--(-fi(—)s:_ngfi

dry
] . 1 1 .
_gr(MXcosgfi-Mz sin @) fi;+°§f(-> sin ¢
T 1 L .
+§My(fifli-;-a{')51n¢

2
r . 1 1
CSQN = - = (Fx sin @ + F, cos @) (fi; + ————GK> cos ¢

+{TTI'(MX cos § - M sin ) EJI‘-I-:--i--é]-i-(-) cos ¢

r 1 1
- 'fé’My (E——IP - Efi) cos ¢

 

Jzy, =r5(FX sin § + F_ cos ¢)\:%E%I-;+ (T - 2) &

(35)

(36)

(37)
 

Deflections From Rigid-Body Rotation of Channel About y-axis ‘
; :

 

- The x- and z- deflections from rigid-body rotation of the channel

about the y-axis are given by

‘ Sz = b 8F_ cos @ (38)
SXI_ = ob 5’¢r sin ¢ -(39)

Deflections From Deformation of Header Pipe

 

Since the loads acting on the channel are transmitted to the
header pipe in the opposite directions, the deflections of the header

pipe may be written in terms of these loads.

3 2
FJ M_{
‘PXH - - E]I-H ( 3 ye) (k0)

 

Jog = i (41)
" ET,
1 FXFQ
: Jf; = ;EI—H(—Q——-M},E) (42)
" vy = 2—-; (43)
H
\ Solution of Deflection Equatipn§

 

Substituting Egs (20)#(£3), (33)-(37), (38), (39), and (40)-(L3)
into Eqs (6)~(11) gives six equations in six unknowns. These can be

written in terms of coefficients 84717 through 8,47 a8 follows:

“ 4, Note that the distance from the origin to b has been denoted by ob
instead of r. ©Since the channel is not c¢ircular, the actual distances
have been used in Eqs (38) and (39) instead of an average radius. The
discrepancy involved here is small since ©b = 24.1 inches and r has been
taken as 21.9. '

A .;an

-1%-

 
 

a  F o +ta, F o+ 813 M+ aq) My tagg M +a ¢r =£XT
ayy Fo+ a5, Fz+a25 M, + a5 My+a25 M, + asg ¢r=..QS’ZT
F + =

a5l F‘X+a32 Z-&--:a.35 Mx+8'51+ M.Y a55 MZ 0
23,51Fx+a.52 Fz+sa.55Mx+a5lL My+a55 MZ = 0
a6l FX-+EJ.62 FZ +a61+ My = 0
7
where, 3 3
- 5[Tr 1 (3 )g_ .2 o
a1 T ¥ FT *’(T“QGK Sln¢+gEI cos P + iy
P N H
a ~r51T—4]-'—+3T"-2~]-=—sin¢c§°¢-’gisin¢cos¢
12 - 5 EI; "\I GK ~ EI,
2 2
r 1 1 r .
a1335‘f°"‘2=’(§"j; +§)sin¢cos¢+fi-&-51n¢cos¢
20y 1 A1 >
r
%1, T 7|2 "ET“‘(E’E)GR']Sln¢'2EI
P H
2 2
r 1 1 2 r 2
8.15m-2- ifi*;+61-351n¢+fi§008

5
l o =
1 7t [E fi“*@g’.g) afi] sin ¢ cos ¢ - | BT sin ¢ cos ¢

37 1 > p

8y, =T [E-E—I—+(-h—-2 'G"'I'{'] c052¢+g%f1;_-sin¢
25

8ol

25

%06

31

52

23

5.5,4.

35

a1

B

ahB

i

f

i

° (_l__ !
2 EIP

d

_ . r (...1_-_._
2 EIP'

 

kg
mmy

2
1 2 r . 2
_GK> cos“f - E——IN51H ¢

o\l
- (2 - 2) GK]COS

2
. r :
-fi)sn.ngflcosgfl-fifl;sn.ngfcos{é

2
+'G]'I-T) sin¢cos¢+%—§ sin @ cos ¢

2
1 2 2
+ EK) cos ¢ - %‘T"" sin ¢

1 .
+(-}f) s:.n¢cos¢+gfi-§-—1; sin @ cos ¢

| 2
Ty 1 . fl
- ( “'é")'é‘fi]sm@l“zEzH

-(2-%1) (—:}Lfi] cos ¢

a—é—) tos ¢
ALY

8,5!4_

55

861

6o

8.6,4.

]

]

i

i

]

amw LY

1 1 4
i (EI ") ot

Moy H

H

T,

( 1
ET,

1
I

rd

1

d

(

r (..___l
2 EIP

gr (E__ + l’“) sin?p + T - cosld + -

1
Ip

P

- ék) sin ¢

2 2
r 1 . 2 r 2
5 (-—-—-——E + C?K) sin“g + ETI\; cos“¢
2

2
1 .
5 (ET" + Ef) sin @ cos @ - %T§ sin @ cos ¢

1 1 .
7t 5%) sin @ cos ¢ +’gf E§E sin @ cos ¢

P
7%%) sin ¢

GK

=16~
 

Results
Equations (44) were solved with the aid of an IBM-650., The
numerical data and values of the coefficients are given in Appendix

A. The following results were obtained:

¥

F, = 505.9 Ibs.
FZ = = L|-2607 le s
Moo= - 5462 in-1bs.
(45)
My = 391 in-lbs,
Moo= - 6620 in-lbs.
¢r = - %,L68x107° radian

Calculation of Deflections

The y~deflection at b relative to point a can be calculated from

>

the loads producing in-plane bending”.
2

dy = pr’ (]21 ) 1) el (46)

 

EEIN EIN

Substituting P and M from Egs (12) and (13) into Eq (46),

N

r5 TE
&y = - EETE (Fx cos §§ =~ F, sin @) - (g - %) §T§ (M.X sin @ + MZ cos @) (L47)

5. See Ref. 1, pp. 7-8.

S SN

~17-
Using the values (L45) for the forces and moments in Eq (L47),

8y = - 0.0373 in.

This result indicates that the channel will expand 37 mils towards Shell V,
at the equator; in addition to the free relative thermal expansion reported
in ART Design Memo 8-D-5. |

Since there are no forces on the channel in the y-direction, the y-
deflection at b, relative to the actual coordinate system, is zero.

The x-deflection at b can be calculated by summing the in-plane and
cut-of-plane x-deflections for the channel, or from the deflection of the

header pipe. Taking the latter, and using Eq (40),

1 [F 2 u g
x = - o= X .= - 0.00299 in.
BT 3 5

 

The angular deflections at b can be calculated from the deformation
cf the header pipe, using Eqs (41), (42), and (43).

M
x -3
Se - &1 1.785x10"° rad.
F f° ,
1 _
8¢) = *E-"f-}-I' —— - M‘YO = 5.657x10 rad.
M {
&y = - % = 2.813x107° rad.
GdH

Stresses in Channel at Header

 

The stresses in the channel at point b (Fig. 2) can be calculated
from the moments, M. My, and My. From Eqs (12), (13), (24), and (25),
with the values from (L495),

-18-
P = F sing¢ - F_ cos g - 708.5 1bs.

]

My = M_sin g+ M, cos @ - 8563 1in-lbs,

N = F_sin¢g + F, cos @ = 16.22 1bs.

i
=

572 in-le -

il

MP . sin ¢ - Mx cos‘¢

The maximum bending stresses due to MN and MP are calculated as

follows:
- +- --M-E- e + 3
op - + 83 psi
P
Oy = - E§ = 8234 psi

The maximum shear stress due to My may be estimated by an
6
approximate method , This stress occurs at or near the inside corner

of the channel. (See Fig. 4)

MyC
: Trax = )
wWhere
K = a function of the cross section of the chanmnel used in

calculating the torsional rigidity7

i

0.09295 inF

___._._E:;ZD {1 + [0,118 1n (1 - %) - 0.238 -21-); tan h %-g} (49)
1+ 5

Q
i

 

See Ref. 2, pp. 170-181,

7. The evaluation of K is given in Appendix B.

 

 
w

where
D = diameter of the largest circle inscribed in the cross .
section = 0.45 in.
r = radius of curvature of the boundary at the point .
(negative when Eq 49 is used) = - 0.110 in,
A = area of the section = 2.582 in?
¢ = angle through which a tangent to the boundary rotates

in turning or traveling around the reentrant portion,
measured in radians = TI/2
For these values,
¢ = 0.8612 in.

Using Eq (48) with the above values of C and K,

T = 3623 psi

max

Stresses in Channel at Equator

 

The moments tangent, radial, and normal to the curve at the equator

may be expressed from equilibrium conditions, (see Figs. % and 5).

M, = MP + Nr = 217.2 in-lbs. -

=
It

- M = - 391 in-1bs.

=
i

o= - MN +lPr'= - 6954 in-lbs.

The maximum bending stresses due to Mr and Mh given by

o =t = * 57 psi

- 6687 psi

Q
i
|

li

 

 
ORNL -LR-DWG. 27497

FIG.4-CROSS-SECTION OF CHANNEL

ORNL -LR-DWG. 27498

M

FIG.5-SKETCH SHOWING PRINCIPAL MOMENTS
) ON CHANNEL AT EQUATOR
g TS Nl TR
a. L "'3 .lfi‘i: i

The meximum shear stress due to M, can be calculated by Eq (48).

t
Using the same C/K ratio as before,

tfiax = 2013 psi .

Stresses in Header Pipe
Bending stresses are produced in the header pipe from Fx’ MX and
Mya Referring to Fig. 3 the moment in the y-direction at the fixed

end of the pipe due to Fx and My is given by

MHy = M -F f = - 385% in-1bs.

The moment in the x-direction at the fixed end 1is
MHx = Mx = - 5)4-62 in~1bs,

Taking the vector sum of these moments gives a maximum bending

¥ -'r'-.z‘;\z'

moment,

MH = - 6684 in-1bs.

The maximum bending stress due to this moment is given by
— H .
o, =1 — = % 6282 psi .

H

The maximum shear stress due to Mz is given by

Mer
T = — = - 3111 psi .

max JH

 

-22- - R, -
ey B

 
i Bk
oo %
b . :

Thermal stresses are produced in the header pipe from radial
and axial temperature gradients, in addition to those caused by relative
thermal expansion of the channel. Figure 6 shows the north and south
headers and pipes with the approximate temperatures of the surrounding
fluids.

The axial temperature gradients should be small except in the
regions close to the headers which are subjected to cross-flow from the
fuel. Because of the azimuthal variation in heat transfer coefficients
and the complicated geometry of these regions, it is possible that
thermal cycling of these small sections of pipe could occur. Although
these temperature fluctuations would be difficult to calculate, small
thermal shields welded to the headers might be warranted.,

Because the layer of fuel surrounding the header pipes over most
of their lengths is very thin, an estimate of the radial temperature
profiles may be calculated using simple conduction with semi-infinite
slab geometry. With the temperatures indicated in Fig. 6, these cal-
culations give temperature differences across the walls of 230F and 52?F
for the north and south header pipes, respectively. The stresses in

the outside walls due to these gradients can be calculated by

QEAT
=TT 43)
% X L-v (

giving - 4574 psi for the north header pipe, and 6176 psi for the south.
The bending and twisting stresses may now be combined with the

stresses due to the radial temperature gradient in the header pipe in

order to get the maximum normal and shear stresses, Figure 7 shows

the fixed end of the header pipe, the direction of the resultant bending

moment, and the bending stresses due to MH' Since the twisting stresses

23

 
 

+ 1I500°F NaK

ORNL -LR-DWG. 27499

FIG.6 ~SKETCH SHOWING NORTH AND SOUTH HEADER
PIPES APPROXIMATE FLUID TEMPERATURES

TENSION

MAXIMUM STRESSES
FOR SOUTH HEADER |

PIPE

X

 

N —"

2

COMPRESSON

>

 

ORNL -LR-DWG. 27500

MAXIMUM STRESSES
FOR NORTH HEADER
PIPE

FIG.7- FIXED END OF HEADER PIPE SHOWING BENDING
STRESSES AND ELEMENTS: AT2AWHICH MAXIMUM
STRESSES OCCUR

-2l

 

 
and temperatuniafggfifnt stresses are uniformly distributed around the
header pipe, the direction of MH determines the location of the maximum
stresses in the pipe. For the north header pipe, the temperature
gradient stresses are compressive at the outside wall, so the maximum
stresses occur where the bending stress is a maximum in compression,
This element is at the outside wall, 900 clockwise from the direction

of - MH and an angle @ counterclockwise frbm the positive y-axis, where

For the south header, the temperature gradient stresses are tensile
at the outer wall, so the maximum stresses occur where the bending
stress is a maximum in tension. This point is 180° from the point of
maximum stress in the north header pipe, or an angle @ counter clock-
wise from the negative y-axis, when the south header pipe is oriented
as in Fig. 7 (eg the z-axis is directed away from the equator)..

The maximum stresses can be calculated for the north and south
header pipes from the combined stresses acting on the elements in the
outer walls of the header pipes, as shown in Fig. 7. The stresses
acting on these elements are shown in Fig. 8. The maximum normal and

shear stresses are given by

+ F (4

 

o, - 0O
Qjmax: (12 2) *'”L2 (45)
ORNL -LR-DWG. 27501

 

 

 

 

 

T,= -11,258
__l_, T ==3
G, = —~4574
‘_T_‘
5, = 12,860
7=~ 31
G, =6176

 

 

 

FIG.8 ~ELEMENTS UNDER MAXIMUM STRESS IN
OUTSIDE WALL OF (a) NORTH HEADER PIPE;

(b) SOUTH HEADER PIPE (STRESSES IN PSi)

. v

-26-~
For the north header pipe,

o
max

T,
max

i

For the south header pipe,

Q
i

max

S
§

max

- 12,480 psi

4565 psi.

14,080 psi

4565 psi.

 
References

D. L. Platus and B. L. Greenstreet, "Deflection Equations for

Various Loadings of Circular-Arc Curved Beams", CF 57-4-G6,

R. J. Roark, Formulas for Stresses and Strain, Third Edition,
1954, pp 170-181.

 

S. Timoshenko and J. N. Goodier, Theory of Elasticity, Second
Edition, 1951, p. 287.

 

S. Timoshenko, Strength of Materials, Part II, 1941, pp 270-271.

28~
L e b r
T “;; 3
- _i‘ -“ -

Appendix A

 

Numerical Data
. 6 .
X, = - 19.59 in. G = 5.8x10° psi
Z, = 14.08 in. IP = 20.72 inh
r = 21.9 in. Iy = 2.46 J‘.n)+
ob = 24,125 in. K = 0.09295uinu
/ = 7.50 én. Ip = 1,53 inu 2-1/2 inch
E = 15x10° psi Iy = 3,06 in Sch 40 Pipe
Y, = 0.3 SXT = 0.032 in.
QS‘ZT = 0.0ll-l ine
Numerical Values of Coefficients Used in Egs (b&4)
a;; = 2558°7x10-§ 8y, = 3212.6x10'6
a = 3012.9x10" " a,. = u693.5x10‘6
12 6 PP 6
a = - 206,19x10 a., = = 299,76x10"
13 6 23 6
a = - 112,53x10 a, = - 154,86x10
14 6 ol _6
815 = 161.18x10 8,5 = 206,19x10
ag =  14.079 s = 19.59
8 = - 206.19x10"6
31 _6
a = - 299,88x10
52 -6
a = 21.173%x10
33 6
a = 16.55%x10
3l 6
g5 = - 15.787x10
— Lt

-29-

 
81
Blyo
auB
nhn
8..)4_5
%46

261

B¢o

g6l

- 112.55x10'6 a
_6 o1
- 154,88x10 8o
6 2
16-55X10 8.55
52.u6hx10': 8c),
- 11.896x10 8
1.0000
14.08
19.59
1.0000

il

Il

161.18x10"

6

206.17x10'6

- lh.787x10-6
- ll.896x10-6
ll.986)n:.10_6

 

 
 

vt
i s

Appendix B

Evaluation of Torsional Rigidity Factor, K

For a narrow rectangular beam of length b and width c, K can be

approximated using the membrane analogy to give

K = %bc2 (B1)

9

Similarly, for a narrow trapezoidal section”,

1 2 2
K = i§-bl(cl + C2)(Cl + c2) (B2)

. For a cross=-section built up of narrow elements, K can be
approximated by summing the K's for the individual elementslo.
Thus, for the channel (Fig. 4), by summing two trapezoidal sections

and one rectangle,

Zvc” + b (c) + )+ c5)  (B3)

8. See Ref. 4, p. 270.
9. See Ref. 4, p. 271.
r lO. See Refo 5, po 2870

M\

-%1-

 
Using Eq (B3) with -
b = 5.00 ino Cl = O-lO ino -
c = 001_25 in. C2 = 0050 in.
bl = 3,45 in,

0.09295 inh

-~
i

 

 
 

-

W00~ vin B~ po =

- -

o

i
O
c..u::«:m_*uzzb:bmwnmzww;‘g—.?pr‘cr_.:npd?dpcpummn:>mhjuu

81,
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