ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules

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2010 SECTION VIII, DIVISION 2

4-435

Table 4.19.5 – Allowable Number Of Cycles For U-Shaped Unreinforced Bellows

Stress in MPa

Stress Range Criteria Allowable Number Of Cycles

()

448

gobt

KEES MPa≥

()

35850

264

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

257.2 448

gobt

Pa K E E S MPa<<

()

46200

211

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

257.2

gobt

KEES MPa≤

alw

Ncycles=

Stress Range in psi

Stress Range Criteria Allowable Number Of Cycles

()

65000

gobt

KEES psi≥

()

5.2(10)

38300

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

37300 65000

gobt

si K E E S psi<<

()

6.7(10)

30600

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

37300

gobt

KEES psi≤

alw

Ncycles=

Notes:

1) In the above equations,

K is the fatigue strength reduction factor that accounts for geometrical stress

concentration factors due to thickness variations, weld geometries, surface notches, and other surface

or environmental conditions. The range for

K is 1.0 4.0

≤

≤ with its minimum value for smooth

geometrical shapes and its maximum for 90 deg. welded corners and fillet welds. Fatigue strength

reduction factors may be determined from theoretical, experimental, or photoelastic studies. A factor

has already been included in the above equations for

N to account for normal effects of size,

environment, and surface finish. If the expansion bellows does not have circumferential welds and

satisfies all of the design and examination requirements of this paragraph, a

1.0

K = may be used.

2) The allowable number of cycles given in this table includes a reasonable design margin (3 on cycles

and 1.25 on stress) and represents the maximum number of cycles for the operating condition

considered. Therefore an additional design margin should not be applied. An overly conservative

estimate of cycles can necessitate a greater number of convolutions and result in a bellows more prone

to instability.

2010 SECTION VIII, DIVISION 2

4-436

Table 4.19.6 – Stress Calculations And Acceptability Criteria For U-Shaped Reinforced Bellows

Subject To Internal Pressure

Stress Calculation

Bellows Reinforcing Ring Member and Fastener

()

btb

btbcccc

PD nt LEk

nt D nt L E t D L E k

++⎡⎤

⎣⎦

()

ctc

btbcccc

PD L E k

nt D nt L E t D L E k

++⎡⎤

⎣⎦

PD q

⎛⎞

⎜⎟

⎝⎠

()

0.85

wCqP

−

0.85

wCq

⎛⎞

−

⎜⎟

⎝⎠

PD q

Sringmember

⎛⎞

⎜⎟

⎝⎠

PD q

S fastener

⎛⎞

⎜⎟

⎝⎠

1.5

100

0.3

1048 320

CPinMPa

⎛⎞

=−

⎜⎟

⎝⎠

mffrr

AE D

DAEAE

⎛⎞

⎜⎟

⎝⎠

1.5

100

0.3

0.6 320

C P in psig

⎛⎞

=−

⎜⎟

⎝⎠

Acceptance Criteria

Bellows Reinforcing Ring Member and Fastener

SS≤

1 wc c

SCS≤

SS≤

SS KS+≤

2rwr

SCS ringmember≤

2 f

SS fastener≤

Notes:

1. k ,

D ,

and

are evaluated using the equations in Table 4.19.1.

C is evaluated using Table 4.19.2.

R= for integral reinforcing ring members and

for reinforcing ring members joined by

fasteners.

4. In the case of reinforcing members which are made in sections, and joined by fasteners in tension, the

equation for

S assumes that the structure used to retain the fastener does not bend so as to permit the

reinforcing member to expand diametrically. In addition, the end reinforcing members must be restrained

against the longitudinal annular pressure load of the bellows.

5. In the case of equalizing rings, the equation for

S provides only the simple membrane stress and does

not include the bending stress caused by the eccentric fastener location. Elastic analysis and/or actual

tests can determine these stresses.

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2010 SECTION VIII, DIVISION 2

4-437

Table 4.19.7 – Allowable Number Of Cycles For U-Shaped Reinforced Bellows

Stress in MPa

Stress Range Criteria Allowable Number Of Cycles

()

567

gobt

KEES MPa≥

()

45505

334

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

326.1 567

gobt

Pa K E E S MPa<<

()

58600

267.5

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

326.1

gobt

KEES MPa≤

alw

Ncycles=

Stress Range in psi

Stress Range Criteria Allowable Number Of Cycles

()

82200

gobt

KEES psi≥

()

6.6(10)

48500

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

47300 82200

gobt

si K E E S psi<<

()

8.5(10)

38800

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

47300

gobt

KEES psi≤

alw

Ncycles=

Notes:

1) In the above equations,

K is the fatigue strength reduction factor that accounts for geometrical stress

concentration factors due to thickness variations, weld geometries, surface notches, and other surface

or environmental conditions. The range for

K is 1.0 4.0

≤

≤ with its minimum value for smooth

geometrical shapes and its maximum for 90 deg. welded corners and fillet welds. Fatigue strength

reduction factors may be determined from theoretical, experimental, or photoelastic studies. A factor

has already been included in the above equations for

N to account for normal effects of size,

environment, and surface finish. If the expansion bellows does not have circumferential welds and

satisfies all of the design and examination requirements of this paragraph,

1.0

K = may be used.

2) The allowable number of cycles given in this table includes a reasonable design margin (3 on cycles

and 1.25 on stress) and represents the maximum number of cycles for the operating condition

considered. Therefore an additional design margin should not be applied. An overly conservative

estimate of cycles can necessitate a greater number of convolutions and result in a bellows more prone

to instability.

2010 SECTION VIII, DIVISION 2

4-438

Table 4.19.8 – Stress Calculations And Acceptability Criteria For Toroidal Bellows Subject To

Internal Pressure

Stress Calculation

()

bwb

bwbccc

PD nt LE

nt D nt L E D E A

++⎡⎤

⎣⎦

()

cwc

bwbccc

PD L E

nt D nt L E D E A

++⎡⎤

⎣⎦

nt D r

⎛⎞

−

⎜⎟

−

⎝⎠

where

cb c

DD ntt=+ +

Acceptance Criteria

SS≤

1cwc

SCS≤

SS≤

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2010 SECTION VIII, DIVISION 2

4-439

Table 4.19.9 – Stress And Axial Stiffness Coefficients For Toroidal Bellows

0 1.0 1.0 1.0

1 1.1 1.0 1.1

2 1.4 1.0 1.3

3 2.0 1.0 1.5

4 2.8 1.0 1.9

5 3.6 1.0 2.3

6 4.6 1.1 2.8

7 5.7 1.2 3.3

8 6.8 1.4 3.8

9 8.0 1.5 4.4

10 9.2 1.6 4.9

11 10.6 1.7 5.4

12 12.0 1.8 5.9

13 13.2 2.0 6.4

14 14.7 2.1 6.9

15 16.0 2.2 7.4

16 17.4 2.3 7.9

17 18.9 2.4 8.5

18 20.3 2.6 9.0

19 21.9 2.7 9.5

20 23.3 2.8 10.0

Notes: Equations for

and

are shown below.

33 3

2364

33 3 3

1.00404 0.028725 0.18961 0.00058626

1 0.14069 0.0052319 0.00029867 6.2088(10)

−

++−

+− + −

CC C

CC C C

2 3

33 3

2 3

33 3

1.0 5

0.049198 0.77774 0.13013 0.080371

1 2.81257 0.63815 0.0006405

BforC

CC C

BforC

CC C

=≤

−−+

−++

23 4

33 3 3

2354

33 3 3

0.99916 0.091665 0.040635 0.0038483 0.00013392

1 0.1527 0.013446 0.00062724 1.4374(10)

−

−+− +

−+ − +

CC C C

where,

6.61

2010 SECTION VIII, DIVISION 2

4-440

Table 4.19.10 – Allowable Number Of Cycles For Toroidal Bellows

Stress in MPa

Stress Range Criteria Allowable Number Of Cycles

()

448

gobt

KEES MPa≥

()

35850

264

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

257.2 448

gobt

Pa K E E S MPa<<

()

46200

211

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

257.2

gobt

KEES MPa≤

alw

Ncycles=

Stress Range in psi

Stress Range Criteria Allowable Number Of Cycles

()

65000

gobt

KEES psi≥

()

5.2(10)

38300

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

37300 65000

gobt

si K E E S psi<<

()

6.7(10)

30600

alw

gobt

KEES

⎛⎞

⎜⎟

−

⎝⎠

()

37300

gobt

KEES psi≤

alw

Ncycles=

Notes:

1) In the above equations,

K is the fatigue strength reduction factor that accounts for geometrical stress

concentration factors due to thickness variations, weld geometries, surface notches, and other surface

or environmental conditions. The range for

K is 1.0 4.0

≤

≤ with its minimum value for smooth

geometrical shapes and its maximum for 90 deg. welded corners and fillet welds. Fatigue strength

reduction factors may be determined from theoretical, experimental, or photoelastic studies. A factor

has already been included in the above equations for

N to account for normal effects of size,

environment, and surface finish. If the expansion bellows does not have circumferential welds and

satisfies all of the design and examination requirements of this paragraph,

1.0

may be used.

2) The allowable number of cycles given in this table includes a reasonable design margin (3 on cycles

and 1.25 on stress) and represents the maximum number of cycles for the operating condition

considered. Therefore an additional design margin should not be applied. An overly conservative

estimate of cycles can necessitate a greater number of convolutions and result in a bellows more prone

to instability.

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2010 SECTION VIII, DIVISION 2

4-441

4.19.16 Figures

View A-A

(a) Unreinforced Bellows

Equalizing Ring

End Equalizing Ring

(b) Reinforced Bellows

Reinforcing Rings

Convolutions

Collar

Figure 4.19.1 – Typical Bellows Expansion Joints

2010 SECTION VIII, DIVISION 2

4-442

Figure 4.19.2 – Possible Convolution Profile in Neutral Position

Figure 4.19.3 – Dimensions to Determine

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2010 SECTION VIII, DIVISION 2

4-443

-X

Point of Application of External Forces and Moments

"N" Conv.

Figure 4.19.4 – Bellows Subject to an Axial Displacement

-x

Point of Application of External Forces and Moments

Figure 4.19.5 – Bellows Subject to a Lateral Displacement

2010 SECTION VIII, DIVISION 2

4-444

"N" Conv.

q+ q

q- q

Figure 4.19.6 – Bellows Subjected to an Angular Rotation

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