ad 2000-mb - b13 -02-2010

AD 2000-Merkblatt ICS 23.020.30 February 2010 edition

Design of

pressure vessels Single-ply bellows expansion joints

AD 2000-Merkblatt

B 13

The AD 2000-Merkblätter are prepared by the seven associations listed below who together form the “Arbeitsgemeinschaft Druck-behälter” (AD). The structure and the application of the AD 2000 Code and the procedural guidelines are covered by AD 2000-Merk-blatt G 1.

The AD 2000-Merkblätter contain safety requirements to be met under normal operating conditions. If above-normal loadings are to be expected during the operation of the pressure vessel, this shall be taken into account by meeting special requirements.

If there are any divergences from the requirements of this AD 2000-Merkblatt, it shall be possible to prove that the standard of safety of this Code has been maintained by other means, e.g. by materials testing, tests, stress analysis, operating experience.

Fachverband Dampfkessel-, Behälter- und Rohrleitungsbau e.V. (FDBR), Düsseldorf

Deutsche Gesetzliche Unfallversicherung (DGUV), Berlin Verband der Chemischen Industrie e.V. (VCI), Frankfurt/Main

Verband Deutscher Maschinen- und Anlagenbau e.V. (VDMA), Fachgemeinschaft Verfahrenstechnische Maschinen und Apparate, Frankfurt/Main

Stahlinstitut VDEh, Düsseldorf VGB PowerTech e.V., Essen

Verband der TÜV e.V. (VdTÜV), Berlin The above associations continuously update the AD 2000-Merkblätter in line with technical progress. Please address any proposals for this to the publisher:

Verband der TÜV e.V., Friedrichstraße 136, 10117 Berlin.

Contents

Page

0 Foreword.................................................................................................... 2 1 Scope......................................................................................................... 2 2 General ...................................................................................................... 2 3 Symbols and units ..................................................................................... 3 4 Safety factor............................................................................................... 4 5 Allowances................................................................................................. 4 6 Calculation ................................................................................................. 4 7 Literature.................................................................................................. 10 Appendix 1: Explanatory notes ........................................................................ 35

Supersedes October 2000 edition; | amendments to previous edition

AD 2000-Merkblätter are protected by copyright. The rights of use, particularly of any translation, reproduction, extract of figures, transmission by photomechanical means and storage in data retrieval systems, even of extracts, are reserved to the author. Beuth Verlag has taken all reasonable measures to ensure the accuracy of this translation but regrets that no responsibility can be accepted for any error, omission or inaccuracy. In cases of doubt or dispute, the latest edition of the German text only is valid.

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AD 2000-Merkblatt AD 2000-Merkblatt B 13, 02.2010 edition

0 Foreword

The AD 2000 Code can be applied to satisfy the basic safety requirements of the Pressure Equipment Directive, principally for the conformity assessment in accordance with Modules “G” and “B F”.

The AD 2000 Code is structured along the lines of a self-contained concept. If other technical rules are used in accordance with the state of the art to solve related problems, it is assumed that the overall concept has been taken into account.

The AD 2000 Code can be used as appropriate for other modules of the Pressure Equipment Directive or for different sectors of the law. Responsibility for testing is as specified in the provisions of the relevant sector of the law.

1 Scope

The design principles specified hereafter apply to metallic1) single-ply bellows expansion joints with parallel-sidewall or lyre-shaped convolutions (Figs. 1 and 2), within the following limits:

3 d/h 100

0,1 r/h 0,5

0,018 s/h 0,1

Instructions for the calculation of bellows expansion joints outside the aforementioned scope (multi-ply bellows, geometric parameters) may be taken from [1, 2, 3, 4, 5, 6], for example.

sdb 2

1

Figure 1 — Bellows expansion joints with

parallel-sidewall convolutions

Figure 2 — Bellows expansion joints with lyre-

shaped convolutions (angle 8°)

Figure 3 — Butt weld

Figure 4 — Inserted butt weld Figure 5 — Lap weld

2 General

2.1 This AD 2000-Merkblatt shall only be used in conjunction with AD 2000-Merkblatt B 0.

2.2 This AD 2000-Merkblatt covers bellows expansion joints loaded by pressure and forced movements (axial or lateral movements, angular rotation) and subject to cyclic loading during operation.

1) See explanatory notes in Appendix 1 to this AD 2000-Merkblatt.


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2.3 Expansion joint bellows are not suited to withstand considerable forces and moments. Therefore the design shall ensure, e.g. by guides, internals, supports, that bellows are not subjected to these loads. As torsional moments, however, cannot be avoided in any case, relations have been introduced in this AD 2000-Merkblatt to consider such moments.

2.4 Expansion joint bellows may be connected to adjacent components by circumferential butt welds (Fig. 3). In the case of differing wall thicknesses between bellows and connecting components clauses 2.7 and 2.8 of AD 2000-Merkblatt HP 5/1 shall be taken into account.

Joints other than butt welds, e.g. according to Figs. 4 and 5, are also permitted.

2.5 Where corrosion is expected to occur, a corrosion-resistant material for the bellows shall be agreed between manufacturer and customer/user. It is not suitable to provide a wall thickness allowance for compensation of corrosive attacks on the bellows.

2.6 The distance between the expansion joint welded end and the radius discharge of the expansion joint knuckle shall not be less than the maximum value of the three individual amounts 3 s, 10 mm and ,25,0 sd if the weld is to be disregarded in consideration of the fatigue.

Additionally, the following applies: sdb

2

1

3 Symbols and units

In addition to the specifications of AD 2000-Merkblatt B 0 the following applies:

b length of reinforcing collar mm

c angular working spring rate of a convolution N m/degrees

cw axial working spring rate of a convolution N/mm

Cf design factor for column instability –

Cp design factor for convolution instability –

d mean internal diameter of expansion joint mm

c lateral working spring rate of an expansion joint bellows or an expansion joint consisting of two identical bellows with intermediate pipe section N/mm

di internal diameter of the internal convolution (d s) mm

dm mean diameter of the expansion joint (d h) mm

f1 cyclic strength factor for circumferential joints on the bellows –

f2 characteristic value for partial plastic deformation –

h convolution depth mm

IXX surface moment of inertia of a convolution mm4 l convolution length, measured in neutral position mm

n here: shape factor –

n1 shape factor for application of 1 % proof stress –

n0,2 shape factor for application of 0,2 % proof stress –

r knuckle radius (for different radii at the internal and external knuckle the arithmetical mean applies) mm

s* design wall thickness following deformation mm

seq equivalent wall thickness mm

w axial movement on one side of a convolution, measured from the neutral position mm

w equivalent axial movement of a convolution for the rotation angle mm

w equivalent axial movement of the convolution subject to maximum loading for lateral movement of the expansion joint mm

z number of bellows convolutions –

zl number of bellows convolutions of an expansion joint with two identical bellows and intermediate pipe section –

C3, C4 geometry factor for column instability – E20 modulus of elasticity at 20 °C N/mm2

L1 overall length of bellows (Fig. 6) mm

Lz length of intermediate pipe (Fig. 6) mm

MT torsional moment acting on the bellows Nm

N here: number of cycles –


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Nzul allowable number of cycles –

Ps,c allowable pressure due to column instability bar

Ps,i allowable pressure due to convolution instability bar

R here: design factor –

)(cR w

R(p) design factor for pressure loading –

design factor for axial working spring rate –

R(w) design factor for axial loading –

Sum safety factor for circumferential stress –

Svp safety factor for stress intensity –

here: rotation angle on one side of a convolution, measured from straight position degrees

stress intensity factor for convolution instability –

stress ratio for convolution instability –

lateral movement of one side of an expansion joint with one bellows or two identical bellows and intermediate pipe section, measured from straight position (Fig. 6) mm

2 ages effective range of total expansion %

vges total equivalent stress range N/mm2

v(w) equivalent stress range due to axial movement, e.g. v(w) 2 v(w) for axial movement w N/mm2

v() equivalent stress range due to angular rotation, e.g. v() 2 v() for rotation N/mm2

v() equivalent stress range due to lateral movement, e.g. v() 2 v() for lateral movement N/mm2

v(p) equivalent stress range due to alternating gauge pressure, e.g. v(p) v(p) for pressure cycles 0 p N/mm2

v(T) equivalent stress range due to cyclic torsional moment, e.g. v(T) 2 v(T) for torsional moment MT N/mm2

um mean circumferential stress N/mm2

v maximum stress intensity N/mm2

v() maximum stress intensity due to angular rotation N/mm2

v(T) maximum stress intensity due to torsional moment N/mm2

v(p) maximum stress intensity due to internal or external gauge pressure N/mm2

v(w) maximum stress intensity due to axial moment N/mm2

Poisson’s ratio –

4 Safety factor

As a deviation from AD 2000-Merkblatt B 0, the specifications of clause 6 apply.

5 Allowances

As a deviation from clause 9 of AD 2000-Merkblatt B 0, allowances are not taken into account.

6 Calculation

For the calculation against internal or external gauge pressure and torsional loading the bellows minimum wall thickness, and for the calculation against axial, lateral or angular movement the bellows maximum wall thickness shall be used. In such cases the minimum or maximum wall thickness shall be used in due consideration of the tolerances specified in the standards for semi-finished products, in accordance with the AD 2000-Merkblätter of the W series, with the wall thickness being reduced or increased by the thickness change due to forming, as appropriate.

In the calculation, the relevant third party shall take into account the manufacturing method used by the particular manufacturer in order to make allowance for any change in wall thickness occurring as a result of the forming process.

Alternatively, the reference value for the design wall thickness following forming given in clause 6.6 can be used.


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6.1 Stresses, spring rates

6.1.1 Loading due to internal or external gauge pressure

Maximum stress intensity:

v(p) R(p) p (1)

6.1.2 Loading due to torsional moment


v(T) sd

M2

3T 104

with MT in Nm (2)

6.1.3 Loading due to axial movement


v(w) 2,4 104 hE

R(w) w (3)

Axial working spring rate:

cw 0,15 104 (d h) E (4) )( wcR

6.1.4 Loading due to angular rotation

The values for loading due to deflection can be determined by means of the axial loadings.

Equivalent axial movement of convolution for the rotation angle:

21015,1

2 hdw (5)


v() 2,1 106 hE

R(w) (d 2 h) (6)

Angular working spring rate:

c 2,2 106 (d 2 h)2 cw (7)

6.1.5 Loading due to lateral movement

As a deviation from equations (3) to (7) applying to a single convolution, equations (8) to (13) apply to the complete expansion joint.

6.1.5.1 Lateral single-bellows expansion joint

Lateral movement of this expansion joint is only possible, if the bellows has at least two convolutions.

Equivalent axial movement of the convolution subject to maximum loading for lateral movement of the expansion joint:

zzlhdw

2

3 (8)


v() 2,4 104 hE

R(w) w (9)

Lateral working spring rate of expansion joint:

zc

zlhdc w

22

2

3

(10)


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6.1.5.2 Lateral expansion joint with two identical bellows (double expansion joint) with unguided intermediate pipe section

Equivalent axial movement of convolution subject to maximum loading for lateral movement of expansion joint:

11

1

z2

1

z

1

z

22

2

3

42

22

zLhd

LL

LL

LL

w

(11)


v() 2,4 104 hE

R(w) w (12)

Lateral working spring rate of expansion joint:

1

w2

1

1

z2

1

z22

2

3

42

2

zc

Lhd

LL

LL

c

(13)

These relations apply to the lateral expansion joint according to Fig. 6 where the expansion joint tie only effects parallel guidance of expansion joint ends, but no guidance of the intermediate pipe section.

6.1.5.3 Lateral expansion joint with two identical bellows and guided intermediate pipe section

The two single bellows shall be calculated in accordance with clause 6.1.4.

Figure 6 — Lateral expansion joint with intermediate pipe section

6.1.6 Design values R

The design values R(p), R(w) and )( wc required for equations (1), (3), (4), (6), (9) and (12) shall be taken from Tables 2 to 25 for rated parameters. Intermediate values shall be subject to linear interpolation. See also clause 6.5.

R

6.2 Calculation against static loading

6.2.1 Gauge pressure and torsion

v(T) vpSn

1,5 v(p) vpSn

K (14)

Here, v(p) shall be determined elastically as specified in clause 6.1.1.

The shape factor n shall be inserted using

n0,2 1,55 2,8 104 K (15)

or n1 1,55, and the safety factor as Svp 1,2.

In addition, the following applies to the mean circumferential stress due to gauge pressure and torsional moment

umv(T))14,1(40

)(

SK

vhrsplhd

(16)

with Sum 1,5


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6.2.2 Verification against instability

The verifications listed in the following apply mainly to expansion joint bellows having more than one convolution.

6.2.2.1 Instability due to internal pressure

Allowable design pressure to overcome column instability:

3

f22

2

cs,* )(

)1(24,3

hs

clzEhdP

(17 a)

Here, cf shall be determined in accordance with Appendix 1, taking into consideration s* as given in clause 6.6.

Allowable design pressure to overcome convolution instability:

ldRAP

m

*e

is, )2(

10 (17 b)

In this equation, is the effective yield strength of the bellows material at design temperature in the formed or annealed state. If no values for are specified in the material standards, the following values shall be used for austenitic steel and other equivalent materials:

*eR

*eR

a) *eR Kd Rp1,0/T for bellows in the formed state (cold work hardening)

b) *eR 0,75 Rp1,0/T for annealed bellows (no cold work hardening)

where

Rp1,0/T is the 1 % yield strength at design temperature

Kd is the consolidation factor

1 5 sd if sd 0,2

Kd 2,0 if sd 0,2

For non-austenitic steel, Rp0,2/T *eR

The degree of deformation sd depends on the deflection component sb and the circumferential component s:

sd 1,04 )( 2b

2 ss

sb ln [1 s/(2 r s)]

The decisive factor for the circumferential component is the deformation method selected:

For hydraulic or similar methods where forming is 100 % directed outwards, starting from the initial cylinder:

s ln [1 2 h/di]

For roll forming methods with 50 % inward forming and 50 % outward forming, starting from the initial cylinder:

s ln [1 h/di]

For half convolutions manufactured by curling or other methods from circular ring plates whose deformation degree is highest at the internal knuckle:

)(

)2()12/(1ln

i sdsrs

with

)421(21 422

and 2

m

p

*3

sh

ldCA

with Cp in accordance with Appendix 1, taking into consideration s* as given in clause 6.6

and

A [( 2)/2 l 2 h] s*


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6.2.2.2 Instability due to external pressure

The instability behaviour in the circumferential direction of expansion joint bellows having several convolutions shall be checked by means of the following analogy consideration.

An external pressure calculation of an equivalent cylinder shall be performed in accordance with AD 2000-Merkblatt B 6, where the external diameter of the internal knuckle of the expansion joint shall be taken to be the design diameter, the overall length of the expansion joint between the subsequent reinforcements shall be taken to be the length and the following value of seq shall be taken to be the equivalent wall thickness:

3 xx2eq 112

lIs (17c)

with

2

3

xx 2,04,048

2* lhllhsI

6.3 Calculation against variable2) loadings

The effective total strain range is calculated as:

2 ages E

f2vges f1 102 (18)

vges is composed of the variable part of the loading due to axial and lateral movement, angular rotation, internal pressure and torsion and shall be determined from the sum of the resulting stress components occurring at the same time.

By approximation vges may be determined by addition of the individual stress intensity components v from:

vges v(p) v(w) v() v(T) v() (19)

Partial plastic deformations are taken in account by the factor f2.

This factor is obtained from:

f2 1 C

2

vges

K

0,1 B (20)

B is the greater value of K

Sn

vp

v(p) and

um

um

SK

Equation (20) applies to:

2vges K

For 2vges K

is f2 1

The C values for axial, angular and lateral movement shall be taken from Table 1.

Table 1 — C values

C (values in parantheses apply to deflection, i.e., angular and lateral movement)

Bellows without circumferential welds in highly loaded zone Material group Bellows with circumferential

welds in highly loaded zone cold work hardened hot formed or normalized

Austenite 0,127 (0,101) 0,105 (0,086) 0,085 (0,067)

Ferrite 0,155 (0,127) 0,155 (0,127) 0,133 (0,109)

2) The variable stress portions are identified with .


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In the case of combined loading vges f2 is calculated as follows:

)2()v()v()v()v(v(w)

)v()v(v(T)v(p)

2(w)v(w))v()v(v(w)

v(w)v(T)v(p)2vges

f

ff

(21)

f2(w) and f2() are the respective factors f2 with C values for axial and angular movement according to equation (20).

The effective total strain range at the point of maximum loading shall be determined in connection with Tables 2 to 13 and f1 1,0.

For bellows with circumferential welds at the internal knuckle the fatigue strength of the circumferential weld shall also be assessed. In this connection, the effective total strain range at the circumferential weld (internal knuckle) shall be determined using the design factors from Tables 14 to 25 and the factor f1 2 which takes into account the lower cyclic strength of the weld.

The greater value of 2 ages shall be decisive for the determination of the number of cycles to failure.

At elevated temperatures the effective total strain range 2 ages shall be multiplied with E20/E.

The expected number of cycles to failure (until onset of leakage) is obtained from:

N

45,3

ages2

10

for 500 N 106 (22)

The allowable number of stress cycles is obtained from:

Nzul LSN

with SL 2,0 (23)

Prerequisite for SL 2,0 is that, by representative component service life tests in due consideration of materials and manufacturing processes, it is proved that the number of cycles to failure N according to equation (22) is obtained for at least 95 % of the expansion joints. Otherwise, SL 5,0 shall be inserted in equation (23).

In the case of non-uniform loading (variable amplitudes) the usage factor of the individual load cycles shall be accumulated according to the linear damage rule.

1i

i

NnD (24)

where ni is the number of load cycles of the respective load cycles and Ni, the allowable number.

6.4 Elevated temperatures

Within the temperature range of the creep rupture stress values3) the shape factor will be smaller than that at room temperature in the case of equal limit loads. For temperatures 500 °C the shape factor n 1,28 shall be inserted in equation (14) as well as for the determination of B (see clause 6.3). Values for temperatures between 350 °C and 500 °C shall be subject to linear interpolation between n according to equation (15) and 1,28.

6.5 Design values

The numerical values for R(p), R(w) and shall be taken from Tables 2 to 13 or 14 to 25. Intermediate values shall be subject to linear interpolation

)( wcR4).

6.6 Design wall thickness following forming

If the wall thickness of the expansion joint following forming is not known, the following design value can be used for the verification of the internal pressure:

m

i*ddss

3) See explanatory note in Appendix 1.

4) The interpolation equations in Appendix 1 to this AD 2000-Merkblatt can be used.


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7 Literature

[1] Friedrich, W.: Festigkeitsberechnung einwandiger Balgkompensatoren. (Strength calculation of single-ply bellows expansion joints). Technisch-wissenschaftliche Berichte der Staatlichen Materialprüfungsanstalt an der Universität Stuttgart; (1973) Vol. 73-01.

[2] Wellinger, K.; Dietmann, H.: Festigkeitsberechnung von Wellrohrkompensatoren. (Strength calculation of corrugated-tube compensators). Technisch-wissenschaftliche Berichte der Staatlichen Materialprüfungsanstalt an der Tech-nischen Hochschule Stuttgart; (1964) Vol. 64-01.

[3] Anderson, W. F.: Analysis of Stresses in Bellows NAA-SR-4527; USAEC, October 1964.

[4] EN 13445-3, May 2002 edition, including Amendment A4 (July 2005)

[5] Standards of EJMA, Edition 2005

[6] DIN EN 14917, 2004 edition (draft)


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Table 2 — Design values (stress intensity at the point of maximum loading of the convolution and axial working spring rate)


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Table 14 — Design values (stress intensity in the middle of internal knuckle of convolution)


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Appendix 1: Explanatory notes

Explanatory notes to AD 2000-Merkblatt B 13

This edition of AD 2000-Merkblatt B 13 is based on the dissertation “Festigkeitsberechnung einwandiger Balgkompensa-toren (Strength calculation of single-ply bellows expansion joints)” by Dr. Friedrich at the MPA (Institute for the Testing of Materials) in Stuttgart. The results determined empirically on the basis of 150 expansion joints are supported by a numerical calculation procedure (transfer matrices). With the aid of the finite element method the correctness of the transfer matrices method is confirmed.

The mathematical formulation of the method is based on a toroidal shell element.

where:

z is the membrane stress b is the bending stress is the shear stress v is the displacement in the direction of the shell normal w is the axial displacement of one bellows convolution (displacement in meridional direction)

is the torsion of the shell normal

The notation of these displacement and stress quantities refers to the calculation model of half the bellows convolution shown above. The designations and quantities used here are not directly related to the designations and quantities used in the Merkblatt itself.

On the basis of this element, a system of linear differential equations can be formulated with the aid of equilibrium conditions, the geometrical relationship between strains and displacements as well as Hooke’s law. This system is principally soluble, however, not as an integral solution but as an approximate solution, with the differential equation being transformed into a difference equation. The system of linear difference equations is solved with the aid of a matrix calculation. The method of transfer matrices is subject to the type of rotation and thin-walledness of the component. These

conditions are met by expansion joints. It was ascertained that at a ratio 42 s

srno influence on the accuracy of

calculation can be noticed.

At each point (step i 0 to n) the stresses z, b and are determined for both strain*) and internal pressure*). The maximum value in the shape is to be determined. With the aid of the maximum distortion energy theory, the individual stresses are combined to form a stress intensity v(p), and v(w) respectively.

Because of the way in which they can be represented, the four variables d, r, s and h of an expansion joint shall be

reduced to three. This is achieved by dividing d, r, s by h as follows: .and,hs

hr

hd

The results were evaluated with the

following values:

h 50 mm

E20 2,1 105 N/mm2

0,3

The convolution depth h was selected to be h 50 mm, since most of the investigated expansion joints were constructed with this convolution depth. Here the linear connection between external load and stress distribution of geometrically similar components was utilized. Here it can be noticed that all geometrical parameters refer to the centre of the shell.

*) Loadings other than those mentioned in clause 2.2 of the AD 2000-Merkblatt will not be taken into account for the calculation. This procedure is not applicable, particularly in respect of lateral deflection.


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The design values R(p), R(w) and refer to the following: )( wcR

hr

hs

hdR ,,(p) stress intensity at 1 bar gauge pressure

hr

hs

hdR ,,(w) stress intensity at 1 mm axial displacement

hr

hs

hdR c ,,)( w

axial load per mm of mean circumference in the case of 1 mm axial displacement.

Explanation of equation (3)

Since the design value R(w) refers to a convolution depth h 50 mm and a modulus of elasticity of 210 000 N/mm2, the exact equation is written as follows:

wREh

(w)v(w) 000210

50

The term 50/210 000 was reduced to a factor 2,38 104 2,4 104.


Since the design value refers to a modulus of elasticity of 210 000 N/mm2, the exact equation is written as follows: )( wcR

cw (d h) 000210

E )( wcR

where (d h) is the mean circumference of the bellows convolution. The term /210 000 was reduced to the factor 0,15 104.


The relationship between rotation angle , in degrees, and an axial displacement w is

hdw

2

/360

The term 360/ approximates to 1,15 102.


Equation (6) is determined by insertion of

hdw 2360

from equation (3).

hdR

hE

2360

104,2 (w)4

v

The term 2,4 10-4 /360 gives the factor 2,1 10-6.


The following generally applies

Mc

where M 0,125 (d 2 h)2 wc

With

180

the following is obtained

)2 (4401

w2

chdM

w23

w2 2102,22

4401chdchdMc

[Nmm/°]


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or expressed in [Nmm/°]

c 2,2 106 (d 2 h)2 cw


KnKnKSn

1,2

1,5

1,2

5 1, vp

v(p)

with Svp 1,2 and S 1,5. The effective shape factor then becomes 1,251,2

1,5 when n 1,5.


The shape factor n depends on the shape and the material. The dependency on shape is expressed by the factor 1,55 which is increased compared to the case of a beam (with n 1,5) (see Figure 33, page 44 of dissertation). The dependency on the material — herein particularly high-strength materials are meant — is expressed in the second portion of the equation. With an increase in yield strength the shape factor decreases for equal plastic strain. The equation applies up to a temperature of 350 °C; if this value is exceeded, clause 6.4 shall be taken into account.


The mean circumferential stress according to the area method is

pvA

A

pum

) ( 4 )2 ( 2 2 )2 2 ( p hdrrdrrrhdA

srhsrrhA

2444

2842

p

vsrhhdr

244

4um with p in bar

2

um N/mm14,140

pvhrs

hdl

Regarding the determination of cf in (17 a) and cp in (17 b), the following applies:

hlC

23

*2,2 m4

sdlC

Explanation of equation (17 a):

Cf 0 1 C3 2 C32 3 C3

3 4 C34 5 C3

5


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Polynomial coefficients i for the determination of Cf

4C 0 1 2 3 4 5

0,2 1,006 2,375 3,977 8,297 8,394 3,194

0,4 1,007 1,820 1,818 2,981 2,430 0,870

0,6 1,003 1,993 5,055 12,896 14,429 5,897

0,8 1,003 1,338 1,717 1,908 0,020 0,550

1 0,997 0,621 0,907 2,429 2,901 1,361

1,2 1,00 0,112 1,410 3,483 3,044 1,013

1,4 1,00 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 1,002 1,061 0,715 3,103 3,016 0,990

2,5 1,00 1,310 0,829 4,116 4,360 1,555

3 0,999 1,521 0,039 2,121 2,215 0,770

3,5 0,998 1,896 1,839 2,047 1,852 0,664

4 1,00 2,007 1,620 0,538 0,261 0,249

Explanation of equation (17 b):

535

434

333

232310p CCCCCC

Polynomial coefficients i for the determination of Cp for C3 0,3

4C 0 1 2 3 4 5

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 0,95 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 0,95 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 0,95 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 0,95 2,073 3,622 29,136 49,394 13,333


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Polynomial coefficients i for the determination of Cp for C3 0,3

4C 0 1 2 3 4 5

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 0,201 2,317 5,956 7,594 4,945 1,299

1,2 0,598 0,990 3,741 6,453 5,107 1,527

1,4 0,473 0,029 0,015 0,030 0,016 0,016

1,6 0,477 0,146 0,018 0,037 0,097 0,067

2 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 1,464 7,098 17,875 23,778 15,953 4,245

3,5 1,495 6,904 16,024 19,600 12,069 2,944

4 2,037 11,037 28,276 37,655 25,213 6,716


This equation is Hooke’s law where partial plastic deformation is taken into account by the factor f2 and the weld by the factor f1. The significance of the effective total strain range 2 ages is shown by Fig. 41 of the dissertation. lt can be measured directly.

Here, variable stress components mean the start-up and shutdown conditions, e.g. from 0 bar gauge pressure to the maximum allowable working gauge pressure. Pressure fluctuations (as in AD 2000-Merkblatt S 1 or S 2) may be taken into account by equation (24). A further condition is that only stress components occurring at the same time can be included in the calculation.

The addition of stresses (equation 19) is justified since tests have shown that at the location of maximum loading (internal knuckle) the stresses from pressure and axial displacement or deflection act in the same direction and therefore are combined. If one of the components p, w or is constant, the relevant stress portion is equal to zero.

Strain curve for elastic loading in circumferential (subscript u) and meridional (subscript m) direction under internal pressure (subscript p) and axial compression (subscript w).


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The factor f2 takes into account partial plastic deformation. It is only determined if the loading vges has exceeded twice the yield strength — i.e. the elastic portion. This is taken into account in the equation in the second term by subtracting 2. The third term of the equation 0,1 B takes into account a progressive distortion (incremental plastic strain or incremental collapse) in the meridional direction. The letter C in the second term indicates that the factor f2 is only determined in the case of axial displacement or deflection (see table).

Graphical representation of equation (20)

where

2 (e)ages is the effective strain range, applied for the failure of bellows subject to cyclic loading

2 (f)ages is the fictitious strain range according to the theory of elasticity

The relationship between 2 (f)ages and 2 (e)

ages can be represented graphically as follows:

Due to the scatter range of C values determined in the tests, an average value of 0,1 was selected for the third term of equation (20).

Where axial displacement and deflection are combined, equation (21) is used. Here the stress component v(p) has been proportionally added to the stress components v(w) and v() respectively.

The factor f1 is explained on page 54 of the dissertation. It takes into account the surface and the structure of the weld. For design reasons, the weld is located in the middle of the external knuckle and/or the internal knuckle. The tests revealed that the centre of the internal knuckle is subject to the greater loading. For this reason, Tables 14 to 25 were established which contain design values for this location. This makes a separate examination of this location possible. Tables 2 to 13 contain the design values for the location of maximum loading of the shape. Where a circumferential weld is provided both locations (location of maximum loading and location of weld) shall be examined. The higher value of 2 ages is decisive.


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This equation represents the equation for the straight line in the log-log diagram (see Figures 70 and 71 of the dissertation). Thus a relationship between the number of load cycles N and the effective total strain range 2 ages can be established.

A figure below this number of load cycles is outside the experience covered by the literature. The aim of the lower limit for the number of load cycles is to

limit the reduction in stiffness from the deflection with regard to stability,

ensure that equation (11) that strictly speaking only applies to the elastic behaviour of the bellows also applies in the elasto-plastic range.

Explanatory note to clause 6.2.2, 2nd sentence

The bellows of an angular or lateral expansion joint is subject to deflection due to angular rotation or lateral movement which under the simultaneous influence of internal pressure may lead to a lateral offset of the bellows along with excessive deformations.

Explanatory note to clause 6.4

The shape factor n 1,28 at elevated temperatures was determined by means of tests as can be found on page 65 of the dissertation.

For the calculation procedure indicated here, only results of individual short-term tests are available for the elevated temperature range. lt is intended to gain experience with these values. Additional surveillance measures are required.

Materials

For the tests, expansions joints made from the following materials were used:

MRSt 34-2 1.0108

H I 1.0345

H II 1.0425

16 CrMo 4 1.7242

X 10 CrNiTi 18 9 1.4541

The C values given in Table 1 refer to these materials.

Interpolation equations

The numerical values for R(p), R(w) und )( wc are taken from Tables 2 to 13 and 14 to 25 respectively. Intermediate values shall be subject to linear interpolation in accordance with the following model:

R

hdu

hru

hro

hdo

hru

hro

hsu u

uuR uuoR

hsu o

uuR ouoR

hso u

ouR uooR

hso o

ouR oooR

hs

hs

hs

hs

hs

uo

u

hr

hr

hr

hr

hr

uo

u

hd

hd

hd

hd

hd

uo

u


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hsRRRR u

uuuou

uuu

uu

hsRRRR u

uouoo

uuo

uo

hsRRRR o

uuoou

ouu

ou

hsRRRR o

uoooo

ouo

oo

hrRRRR u

uuo

uu

u

hrRRRR o

uoo

ou

o

hdRRRR uou


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Publisher: Source of supply:

Verband der TÜV e.V.

E-Mail: [email protected] http://www.vdtuev.de

Beuth Verlag GmbH 10772 Berlin Tel. 030 / 26 01-22 60 Fax 030 / 26 01-12 60 [email protected] www.beuth.de


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ad 2000-mb - b13 -02-2010

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