ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules
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2010 SECTION VIII, DIVISION 2
4-425
b) The corresponding axial force
x
F applied to the ends of the bellows is given by the following equation.
x
b
FKx= (4.19.30)
4.19.8.3 Lateral deflection
a) When the ends of the bellows are subjected to a lateral deflection
y (see Figure 4.19.5), the equivalent
axial displacement per convolution is given by the following equation where
y
shall be taken positive.
()
3
m
y
Dy
q
NNq x
Δ=
+
(4.19.31)
b) The corresponding lateral force
y
F applied to the ends of the bellows is given by the following equation.
()
2
2
3
2
bm
y
KD y
F
Nq x
=
+
(4.19.32)
c) The corresponding moment
y
M
applied to the ends of the bellows is given by the following equation.
()
2
3
4
bm
y
KD y
M
Nq x
=
+
(4.19.33)
4.19.8.4 Angular rotation
a) When the ends of the bellows are subjected to an angular rotation
θ
(see Figure 4.19.6), the equivalent
axial displacement per convolution is given by the following equation where
θ
, expressed in radians,
shall be taken positive.
2
m
D
q
N
θ
θ
Δ= (4.19.34)
b) The corresponding moment
M
θ
applied to the ends of the bellows is given by the following equation.
2
m
8
b
KD
M
θ
θ
= (4.19.35)
4.19.8.5 Total equivalent axial displacement range per convolution
a) Equivalent axial displacement per convolution – The equivalent axial displacement per convolution, in
extension or compression, is given by the following equations.
exy
()q q q q extended convolution
θ
Δ=Δ+Δ+Δ (4.19.36)
cxy
()q q q q compressed convolution
θ
Δ=Δ−Δ−Δ
(4.19.37)
b) Bellows installed without cold spring – If the bellows is subjected to displacements from the neutral
position
()
00 0
0 , 0 , 0xy
θ
=== to the operating position
(
)
,,
x
y
θ
(see Figure 4.19.7), the
2010 SECTION VIII, DIVISION 2
4-426
equivalent axial displacement per convolution, in extension or compression, for the initial and operating
positions and the total equivalent axial displacement range are given by the following equations.
Initial Position:
e
0.0 ( )q extensionΔ= (4.19.38)
c
0.0 ( )q compressionΔ= (4.19.39)
Operating Position:
exy
()q q q q extension
θ
Δ=Δ+Δ+Δ (4.19.40)
cxy
()q q q q compression
θ
Δ=Δ−Δ−Δ (4.19.41)
Total Equivalent Axial Displacement Range:
max ,
ec
qqqΔ= ⎡Δ Δ ⎤
⎣⎦
(4.19.42)
c) Bellows installed with cold spring – If the bellows is subjected to displacements from an initial position
()
00 0
, , xy
θ
, which is not the neutral position to the operating position
(
)
,,
x
y
θ
(see Figure 4.19.8),
the equivalent axial displacement per convolution, in extension or compression, for the initial and
operating positions and the total equivalent axial displacement range are given by the following
equations.
Initial Position:
e,0 x,0 y,0 ,0
()q q q q extension
θ
Δ=Δ+Δ+Δ (4.19.43)
,0 ,0 ,0 ,0
()
cxy
q q q q compression
θ
Δ=Δ−Δ−Δ
(4.19.44)
Operating Position:
exy
()q q q q extension
θ
Δ=Δ+Δ+Δ (4.19.45)
()
cxy
q q q q compression
θ
Δ=Δ−Δ−Δ (4.19.46)
Total Equivalent Axial Displacement Range:
ec,0 ce,0
max , qqqqq
⎡⎤
Δ= Δ −Δ Δ −Δ
⎣⎦
(4.19.47)
d) Bellows operating between two operating positions – If the bellows is subjected to displacements from
operating position number 1
()
11 1
, , xy
θ
to the operating position number 2
(
)
222
,,xy
θ
(see Figure
4.19.9), the equivalent axial displacement per convolution, in extension or compression, for operating
positions number 1 and 2 and the total equivalent axial displacement range are given by the following
equations. An initial cold spring (initial position 0) has no effect on the results.
Position Number 1:
,1 ,1 ,1 ,1
()
exy
q q q q extension
θ
Δ=Δ+Δ+Δ (4.19.48)
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2010 SECTION VIII, DIVISION 2
4-427
,1 ,1 ,1 ,1
()
cxy
q q q q compression
θ
Δ=Δ−Δ−Δ (4.19.49)
Position Number 2:
,2 ,2 ,2 ,2
()
exy
q q q q extension
θ
Δ=Δ+Δ+Δ (4.19.50)
,2 ,2 ,2 ,2
()
cxy
q q q q compression
θ
Δ=Δ−Δ−Δ
(4.19.51)
Total Equivalent Axial Displacement Range:
,2 ,1 ,2 ,1
max ,
ec ce
qqqqq
⎡⎤
Δ= Δ −Δ Δ −Δ
⎣⎦
(4.19.52)
4.19.9 Fabrication
The following requirements shall be met in the fabrication of expansion joint flexible elements.
a) All welded joints shall comply with the requirements of Part 6.
b) All longitudinal weld seams shall be butt-type full penetration welds; Type 1 welds of paragraph 4.2.
c) Bellows shall be attached to the weld end elements by circumferential butt or full fillet welds as shown in
Figure 4.19.10.
d) Other than the attachment welds, no circumferential welds are permitted in the fabrication of bellows
convolutions.
4.19.10 Examination
The following examinations are required to verify the integrity of expansion joints.
a) All expansion joint flexible elements shall be visually examined for and shall be free of injurious defects,
such as notches, crevices, material buildup or upsetting, weld spatter, etc. which may serve as points of
local stress concentrations. Suspect surface areas shall be further examined by liquid penetrant.
b) All bellows butt-type welds shall be examined 100 % on the inside and outside surfaces by the liquid
penetrant method before forming. This examination shall be repeated after forming to the maximum
extent possible considering the physical and visual access to the weld surfaces after forming. The butt
weld shall be full penetration.
c) The circumferential attachment welds between the bellows and the weld ends shall be examined 100 %
by liquid penetrant.
d) Liquid penetrant examination shall be in accordance with Part 7. However, any linear indication found
by examination shall be considered relevant if the dimension exceeds
0.25
m
t , but not less than 0.25
mm (0.010 in.), where
m
t is the minimum bellows wall thickness before forming.
4.19.11 Pressure Test Requirements
The pressure testing requirements for expansion joints shall be as follows.
a) The completed expansion joint shall be subjected to a pressure test in accordance with paragraph
4.1.6.2 and Part 8. However the designer shall consider the possibility of instability of the bellows due to
internal pressure if the test pressure exceeds the value determined using the following equation. In such
a case, the designer shall redesign the bellows to satisfy the test condition.
[
]
,
1.5max ,
t s sc si
PPP= (4.19.53)
For reinforced and toroidal bellows, use
0
si
P
=
in the above equation.
2010 SECTION VIII, DIVISION 2
4-428
b) The pressure testing of an expansion joint may be performed as a part of the vessel pressure test,
provided the joint is accessible for inspection during pressure testing.
c) Any expansion joint restraining elements (see Figure 4.19.1 Sketch (b) and paragraph 4.19.3.1(c)) shall
also be pressure tested in accordance with paragraph 4.1.6.2 and Part 8 as a part of the initial
expansion joint pressure test, or as a part of the final vessel pressure test after installation of the joint.
4.19.12 Marking and Reports
a) The expansion joint Manufacturer, whether the vessel Manufacturer or a parts Manufacturer, shall have
a valid ASME Code U2 Certificate of Authorization and shall complete the appropriate Data Report in
accordance with Part 2.
b) The Manufacturer responsible for the expansion joint design shall include the following additional data
and statements on the appropriate Data Report:
1) Axial movement (+ and -), associated design life in cycles, and associated loading condition, if
applicable;
2) Spring rate; and
3) That the expansion joint has been constructed to the rules of this paragraph.
c) A parts Manufacturer shall identify the vessel for which the expansion joint is intended on the Partial
Data Report.
d) Markings shall not be stamped on the flexible elements of the expansion joint.
4.19.13 Specification Sheet for Expansion Joints
Specification sheets for expansion joints are provided in paragraph 4.19.17 for Metric and US customary
units.
4.19.14 Nomenclature
A cross sectional metal area of one convolution.
c
A cross sectional metal area of all reinforcing collars for toroidal bellows.
f
A cross sectional metal area of one reinforcing fastener
r
A cross sectional metal area of one bellows reinforcing ring member.
α
offset angle of the U-shape bellows side-wall.
123
,,
B
BB stress and stiffness coefficients used for toroidal bellows.
,,
pfd
CCC
stress coefficients for U-shaped convolutions.
r
C convolution height factor for reinforced bellows.
12
,CC coefficients used to determine the coefficients ,,
pfd
CCC
3
C coefficient used to determine the coefficients
123
,,
B
BB
r
C convolution height factor for reinforced bellows.
wc
C longitudinal weld joint efficiency for tangent collar.
wr
C longitudinal weld joint efficiency for reinforcing member.
b
D inside diameter of bellows convolution and end tangents.
c
D mean diameter of collar.
eq
D equivalent outside diameter.
m
D mean diameter of bellows convolution.
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2010 SECTION VIII, DIVISION 2
4-429
eq
e equivalent wall thickness.
b
E modulus of elasticity of bellows material at design temperature.
c
E modulus of elasticity of collar material at design temperature.
f
E
modulus of elasticity of reinforcing fastener material at design temperature.
r
E modulus of elasticity of reinforcing ring member material at design temperature.
0
E modulus of elasticity of bellows material at room temperature.
x
x
I
moment of inertia of one convolution cross section relative to the axis passing by the center of
gravity and parallel to the axis of the bellows.
b
K bellows axial stiffness.
g
K
fatigue strength reduction factor.
k factor considering the stiffening effect of the attachment weld and the end
convolution on the pressure capacity of the end tangent
c
L bellows collar length.
t
L end tangent length.
f
L effective length of one reinforcing fastener.
w
L distance between toroidal bellows attachment welds.
N number of convolutions.
alw
N allowable number of fatigue cycles.
s
pe
N specified number of fatigue cycles.
n number of plies.
b
ν
Poisson's ratio of bellows material.
P design pressure.
s
c
P allowable internal design pressure to avoid column instability.
s
i
P allowable internal design pressure to avoid in-plane instability.
q
convolution pitch as shown in Figure 4.19.1. If the convolutions show an off-set angle of the
sidewalls in the neutral position, as shown in Figure 4.19.2,
q is defined as the length between
two consecutive convolutions when their sidewalls have been made parallel.
R
ratio of the internal pressure force resisted by the bellows to the internal pressure
force resisted by the reinforcement
r mean radius of toroidal bellows convolution.
ic
r crest convolution radius.
ir
r root convolution radius.
S
allowable stress from Annex 3.A of bellows material at design temperature.
c
S allowable stress from Annex 3.A of collar material at design temperature.
f
S allowable stress from Annex 3.A of reinforcing fastener material at design temperature.
r
S allowable stress from Annex 3.A of reinforcing ring member material at design temperature.
t
S total stress range due to cyclic displacement.
y
S yield strength of the bellows material at the design temperature.
*
y
S effective yield strength of the bellows material at the design temperature.
2010 SECTION VIII, DIVISION 2
4-430
1
S circumferential membrane stress in bellows tangent, due to pressure P .
'
1
S circumferential membrane stress in collar, due to pressure P .
2
S circumferential membrane stress in bellows, due to pressure P .
'
2
S circumferential membrane stress in reinforcing member, due to pressure P .
''
2
S membrane stress in fastener, due to pressure
P
.
3
S meridional membrane stress in bellows, due to pressure
P
.
4
S meridional bending stress in bellows, due to pressure
P
.
5
S meridional membrane stress in bellows, due to total equivalent axial
displacement range qΔ .
6
S meridional bending stress in bellows, due to total equivalent axial
displacement range qΔ .
t nominal thickness of one ply.
c
t collar thickness.
p
t thickness of one ply, corrected for thinning during forming.
U usage factor.
w convolution height.
qΔ total equivalent axial displacement range per convolution.
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2010 SECTION VIII, DIVISION 2
4-431
4.19.15 Tables
Table 4.19.1 – Stress Calculations And Acceptability Criteria For U-Shaped Unreinforced Bellows
Subject to Internal Pressure
Stress Calculation
()
()
2
1
2
btb
btbcccc
PD nt LEk
S
nt D nt L E t D L E k
+
=
++⎡⎤
⎣⎦
()
2
'
1
2
ctc
btbcccc
PD L E k
S
nt D nt L E t D L E k
=
++⎡⎤
⎣⎦
()
()
2,
2
mtb
E
pt cc
PqD L D nt
S for end convolutions
AntL tL
++
⎡⎤
⎣⎦
=
++
2,
2
m
I
PqD
S for intermediate convolutions
A
=
3
2
p
Pw
S
nt
=
2
4
2
p
p
PC
w
S
nt
⎛⎞
=
⎜⎟
⎜⎟
⎝⎠
where
min , 1.0
1.5
t
b
L
k
Dt
⎡⎤
⎛⎞
=
⎢⎥
⎜⎟
⎜⎟
⎢⎥
⎝⎠
⎣⎦
b
P
m
D
tt
D
=
2
2
2
p
Aqwnt
π
⎡− ⎤
⎛⎞
=+
⎜⎟
⎢⎥
⎝⎠
⎣⎦
2
cb c
DD ntt=+ +
mb
DDwnt=++
Acceptance Criteria
1
SS≤
'
1
wc c
SCS≤
2,E
SS≤
2,I
SS≤
34
f
SS KS+≤
Notes:
1)
3.0
f
K = for as-formed bellows and 1.5
f
K
=
for annealed bellows.
2)
p
C is evaluated using Tables 4.19.2.
2010 SECTION VIII, DIVISION 2
4-432
Table 4.19.2 – Method To Determine Coefficient
p
C
2
C
1
0.3C
≤
0
α
1
α
2
α
3
α
4
α
5
α
0.2 1.001 -0.448 -1.244 1.932 -0.398 -0.291
0.4 0.999 -0.735 0.106 -0.585 1.787 -1.022
0.6 0.961 -1.146 3.023 -7.488 8.824 -3.634
0.8 0.955 -2.708 7.279 14.212 -104.242 133.333
1.0 0.95 -2.524 10.402 -93.848 423.636 -613.333
1.2 0.95 -2.296 1.63 16.03 -113.939 240
1.4 0.95 -2.477 7.823 -49.394 141.212 -106.667
1.6 0.95 -2.027 -5.264 48.303 -139.394 160
2.0 0.95 -2.073 -3.622 29.136 -49.394 13.333
2.5 0.95 -2.073 -3.622 29.136 -49.394 13.333
3.0 0.95 -2.073 -3.622 29.136 -49.394 13.333
3.5 0.95 -2.073 -3.622 29.136 -49.394 13.333
4.0 0.95 -2.073 -3.622 29.136 -49.394 13.333
2
C
1
0.3C >
0
α
1
α
2
α
3
α
4
α
5
α
0.2 1.001 -0.448 -1.244 1.932 -0.398 -0.291
0.4 0.999 -0.735 0.106 -0.585 1.787 -1.022
0.6 0.961 -1.146 3.023 -7.488 8.824 -3.634
0.8 0.622 1.685 -9.347 18.447 -15.991 5.119
1.0 0.201 2.317 -5.956 7.594 -4.945 1.299
1.2 0.598 -0.99 3.741 -6.453 5.107 -1.527
1.4 0.473 -0.029 -0.015 -0.03 0.016 0.016
1.6 0.477 -0.146 -0.018 0.037 0.097 -0.067
2.0 0.935 -3.613 9.456 -13.228 9.355 -2.613
2.5 1.575 -8.646 24.368 -35.239 25.313 -7.157
3.0 1.464 -7.098 17.875 -23.778 15.953 -4.245
3.5 1.495 -6.904 16.024 -19.6 12.069 -2.944
4.0 2.037 -11.037 28.276 -37.655 25.213 -6.716
Notes:
1)
11
01
2
q
CC
w
=≤≤
2)
22
0.2 4.0
2.2
mp
q
CC
Dt
=≤≤
3)
2345
01121 31 41 51
p
C CCCCC
αα α α α α
=+++++
4) A plot of
p
C versus
1
C and
2
C is shown in Figure 4.19.11
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2010 SECTION VIII, DIVISION 2
4-433
Table 4.19.3 – Method To Determine Coefficient
f
C
2
C
0
α
1
α
2
α
3
α
4
α
5
α
0.2 1.006 2.375 -3.977 8.297 -8.394 3.194
0.4 1.007 1.82 -1.818 2.981 -2.43 0.87
0.6 1.003 1.993 -5.055 12.896 -14.429 5.897
0.8 1.003 1.338 -1.717 1.908 0.02 -0.55
1.0 0.997 0.621 -0.907 2.429 -2.901 1.361
1.2 1 0.112 -1.41 3.483 -3.044 1.013
1.4 1 -0.285 -1.309 3.662 -3.467 1.191
1.6 1.001 -0.494 -1.879 4.959 -4.569 1.543
2.0 1.002 -1.061 -0.715 3.103 -3.016 0.99
2.5 1 -1.31 -0.829 4.116 -4.36 1.55
3.0 0.999 -1.521 -0.039 2.121 -2.215 0.77
3.5 0.998 -1.896 1.839 -2.047 1.852 -0.664
4.0 1 -2.007 1.62 -0.538 -0.261 0.249
Notes:
1)
11
01
2
q
CC
w
=≤≤
2)
22
0.2 4.0
2.2
mp
q
CC
Dt
=≤≤
3)
2345
01121314151f
C CCCCC
αααααα
=+++++
4) A plot of
f
C versus
1
C and
2
C is shown in Figure 4.19.12
2010 SECTION VIII, DIVISION 2
4-434
Table 4.19.4 – Method To Determine Coefficient
d
C
2
C
0
α
1
α
2
α
3
α
4
α
5
α
0.2 1 1.151 1.685 -4.414 4.564 -1.645
0.4 0.999 1.31 0.909 -2.407 2.273 -0.706
0.6 1.003 2.189 -3.192 5.928 -5.576 2.07
0.8 1.005 1.263 5.184 -13.929 13.828 -4.83
1.0 1.001 0.953 3.924 -8.773 10.44 -4.749
1.2 1.002 0.602 2.11 -3.625 5.166 -2.312
1.4 0.998 0.309 1.135 -1.04 1.296 -0.087
1.6 0.999 0.12 0.351 -0.178 0.942 -0.115
2.0 1 -0.133 -0.46 1.596 -1.521 0.877
2.5 1 -0.323 -1.118 3.73 -4.453 2.055
3.0 1 -0.545 -0.42 1.457 -1.561 0.71
3.5 1 -0.704 -0.179 0.946 -1.038 0.474
4.0 1.001 -0.955 0.577 -0.462 0.181 0.08
Notes:
1)
11
01
2
q
CC
w
=≤≤
2)
22
0.2 4.0
2.2
mp
q
CC
Dt
=≤≤
3)
2345
01121314151d
C CCCCC
αααααα
=+++++
4) A plot of
d
C versus
1
C and
2
C is shown in Figure 4.19.13
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