AD 2000-MB - B5 -03-2009

AD 2000-Merkblatt

Supersedes August 2007 edition; I = Amendments to previous edition

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

ICS 23.020.30 March 2009 edition

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

The AD 2000-Merkblätter contain safety requirements to be met under normal operating conditions. If above-normal loadings are to beexpected 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 ofthis 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 Maschinenund 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 forthis to the publisher:

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

Design ofpressure vessels

Unstayed and stayed flat endsand plates

AD 2000-Merkblatt

B 5

Contents

0 ForewordThe AD 2000 Code can be applied to satisfy the basic safetyrequirements of the Pressure Equipment Directive, princi-pally for the conformity assessment in accordance withmodules “G” and “B + F”.

The AD 2000 Code is structured along the lines of a self-con-tained concept. If other technical rules are used in accord-ance with the state of the art to solve related problems, it isassumed that the overall concept has been taken into ac-count.

The AD 2000 Code can be used as appropriate for othermodules of the Pressure Equipment Directive or for differentsectors of the law. Responsibility for testing is as specifiedin the provisions of the relevant sector of the law.

1 ScopeThe design rules hereafter apply to flat ends and plates andtube bundles of heat exchangers as far as their staying ef-fect is concerned. They are based on the Kirchhoff equa-tions for plates taking into consideration the effect of theboundary conditions and multiple openings approximately.

In addition, the C-factors also include the effect of a Pois-son’s ratio of 0,3.

In the case of materials where the Poisson’s ratio differsconsiderably, and in those cases where the dimensions ex-ceed the limits:

a separate stress and deformation analysis is necessary.

For D, the relevant design diameter shall be substituted.This distinction does not apply to tube plates where mutualbracing is provided by the tubes.

2 General2.1 This AD 2000-Merkblatt shall only be used in conjunc-tion with AD 2000-Merkblatt B 0.

2.2 When using blind flanges to DIN 2527 and blind covers(flat covers of steel) to DIN 28122 the requirements of thisAD 2000-Merkblatt are deemed to be met if non-metallicgaskets (e.g. flat gaskets for flanges with a flat face toDIN 2690 to 2692) are used.

0 Foreword1 Scope2 General3 Symbols and units4 Weakenings

5 Allowances6 Calculation7 LiteratureAppendix 1: Explanatory notes

se c1 c2––

D----------------------------- 0,0087 p

E----4 ; s

D---- 1

3---≤≥

a

BA75E3EC9238321DFDAD9E8AEA77DB4BB3DF8BFFFF8CE3D73A9D38EFA8

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3 Symbols and unitsIn addition to AD 2000-Merkblatt B 0 the following applies:

d1, d2 design diameters mm

lK effective length mm

lw length over which tubeis roll-expanded into tube plate mm

l*w length of connection betweentube and tube plate mm

pi, pu design pressure within oraround the tubes bar

D1, D2, D3, D4 design diameter mm

FA axial force N

FK buckling force N

FR tube force N

t in this context: pitch mm

l slenderness ratio –

4 Weakenings

4.1 Openings in unstayed flat ends and plates

4.1.1 Central openings with diameter d1 can be as shownin Fig. 21 for designs as specified in 6.1 and 6.2 and asshown in Fig. 22 for designs as specified in 6.3 and 6.4.

4.1.2 The required wall thickness of the plate with anopening is determined from formulae (2) to (4) where the C-or C1-factor as specified in Table 1 or Fig. 5 is multiplied bythe opening factor CA or CA1.

4.1.3 The values of CA or CA11 shall be taken from curveA or B depending on whether an opening does not have aconnector (design A in Figs. 21 and 22) or does have aconnector (design B in Figs. 21 and 22). If the diameter ratiodi/dD ≥ 0,8, the flange design rules as specified in AD 2000-Merkblatt B 8 shall be applied.

4.1.4 Eccentric openings can be considered in the sameway as central openings.

4.1.5 For round unstayed plates having an equidirectionaladditional peripheral moment where the ratio se – c1 – c2/dt ≥ 0,1, if there are several cut-outs, the cut-out correctionvalue CA1 can be determined as follows:

(1)

where A is the cross-sectional area of the unpierced plateand AA, is the sum of the cross-sectional areas of the open-ings in a cross section which represents the maximumweakening effect.

4.1.6 For tube plates, the efficiency factors shall be deter-mined in accordance with formulae (17) and (18).

5 AllowancesPlease refer to AD 2000-Merkblatt B 0, Section 9, but notethat, unlike that section, there is no c1 allowance for wallsthicker than 25 mm.

6 Calculation

6.1 Unstayed circular flat ends and plates with noadditional peripheral moment

6.1.1 The required wall thickness s of unstayed circular flatends or plates with no additional peripheral moment is

(2)

with the design factor C and the design diameter D1 toTable 1.

6.2 Unstayed rectangular or elliptical flat ends and plates with no additional peripheral moment

6.2.1 The required wall thickness s of unstayed rectangu-lar or elliptical plates with no additional peripheral momentto Fig. 1 is

(3)

The factor CE from Fig. 2 allows for the special conditionspresented by rectangular or elliptical plates. The C valueshall be taken from Table 1 in accordance with the boundaryconditions relative to the short side.

6.2.2 For covers as shown in Fig. 1 with an additional loadfrom U-bolts, the permissible bolt load acting in the same di-rection as the internal pressure shall be taken into account.Generally it is adequate to replace p by 1,5 p in formula (3).

Fig. 1. Unstayed rectangular or elliptical plate placed in frontfrom inside with no additional peripheral moment

6.3 Unstayed circular plates with an additionalperipheral moment

Fig. 3. Unstayed circular plate with an additional periph-eral moment acting in the same sense as thepressure load

CA1A

A AA–-----------------=

s C D1p S⋅10 K------------ c1 c2+ +⋅ ⋅=

s C CE f p S⋅10 K------------ c1 c2+ +⋅ ⋅ ⋅=


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Fig. 4. Unstayed circular plate with an additional periph-eral moment acting against the pressure load

6.3.1 The required wall thickness s of unstayed circularplates with an additional peripheral moment acting in thesame direction is

(4)

The C1 value can be taken from Fig. 5 as a function of theratio dt/dD and the value d. Here, the ratio of the requiredbolt load and the internal pressure is

(5)

with SD generally equal to 1,2 and the gasket characteristick1 to be taken from AD 2000-Merkblatt B 7. C1 = 0,35 if themoment is acting against pressure load.

6.3.2 Plates which are designed in accordance with theabove formulae meet the strength requirements. It is possi-ble however, e.g. in the case of plates made from highstrength steels, and in the case of plates made from nonfer-rous metals or plates of larger diameter, for problems to beencountered in respect of sealing and permissible bolt de-flection which are attributable to excessive angling of theplates. It is recommended therefore, when non-metallicseals and combined seals are used, that the inclination ofthe plate f should be kept to within about 0,5° to 1° [17]. Itwill then sometimes be necessary to make the plate thickerthan would be demanded on the basis of the strength re-quirements.

6.4 Unstayed rectangular or elliptical plate withadditional peripheral moment acting in the same direction

The required wall thickness s of unstayed rectangular or el-liptical plates with additional peripheral moment acting in thesame direction is determined by applying formula (3) of6.2.1 accordingly, with C replaced by C1, to 6.3.1 taken fromFig. 5 and relative to the small side of the plate.

6.5 Circular flat ends and plates with central staying provided by a tube or a plain bar stay

6.5.1 The required wall thickness s of circular flat ends andplates with central staying provided by a tube or a plain barstay is

(6)

with the design factor C2 and the design diameters D1 andd1, to be in accordance with Table 2.

6.5.2 The central stays or stay tubes shall be capable ofwithstanding the axial load (tensile or compressive load) onthem with a safety factor S = 1,5. The axial load is

(7)

with Cz to be taken from Fig. 6 as a function of D1/d1. For-mula (6) does not take into account the effect of the variousthermal expansions of the shell, the tubes and the plate it-self. Where the effect of different thermal expansions has tobe taken into account the design method is to be agreedupon between manufacturer and customer/user.

6.5.3 If the stay is subjected to axial pressure, evidence ofthe buckling strength shall also be provided in accordancewith Euler’s formula. The permissible buckling load is

(8 a)

where SK = 3,0 for the operational condition. In the test con-dition, S'K = 2,2 shall be inserted instead of SK. The length lKshall be determined from Table 3 as a function of l0 depend-ing on the loading case. l0 is the distance between the endpoints of the original run of the stay.

Slenderness ratios

(9 a)

greater than 200 shall be avoided. In equation (9 a) da is theoutside diameter and Di the inside diameter of the staytubes. Equation (8 a) applies only in the range of slender-ness ratios

(9 b)

When the slenderness ratios are smaller, the permissiblebuckling load of stay tubes is

(8 b)

where SK = 3,0 for the operational condition. In the test con-dition, S'K = 2,2 shall be inserted instead of SK.

6.6 Flat plates with stay bolts

6.6.1 The required wall thickness s of flat plates with staybolts where the stays are distributed uniformly over the load-ed area as shown in Fig. 7 is

(10)

The design factor C3 shall be taken from Table 4.

Fig. 7. Uniformly distributed stays

s C1 dDp S⋅10 K------------ c1 c2+ +⋅ ⋅=

d 1 4k1 SD⋅

dD-----------------+=

s C2 D1 d1–( ) p S⋅10 K------------ c1 c2+ +⋅ ⋅=

FA CZ

p D12 p⋅ ⋅

40--------------------------⋅=

FKp2

E I⋅ ⋅

lK2

SK⋅----------------------=

l4 lK⋅

da2 di

2+

---------------------------=

l l0 p EK----≈>

FKKS---- p

da2 di

2–

4---------------------- 1

ll0------ 1 S

SK-------–⎝ ⎠

⎛ ⎞–⋅=

s C3 t12 t2

2+( ) p S⋅

10 K------------⋅ c1 c2+ +⋅=


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Table 3. lK for different load cases

Fig. 8. Irregularly distributed stays

6.6.2 The required wall thickness s of flat plates with staybolts where the stays are distributed irregularly as shown inFig. 8 is

(11)

The design factor C3 shall be taken from Table 4.

Table 4. Design factor for flat plates with stay bolts

6.7 Circular flat plates in heat exchangers

The design shall be in accordance with the requirements of6.7.1 to 6.7.6. The requirements of 6.7.7 shall be noted inany case.

6.7.1 Circular flat plates mutually stayed by the tubes and the shell

Fig. 9. Circular flat plates mutually stayed by the tubes andthe shell

6.7.7.1 The required wall thickness s of circular flat plates(see Fig. 9) mutually stayed by the tubes and the shell is

(12)

where p is the greater of the tube or shell side pressure. Thedesign diameter d2 shall be the diameter of the greatest in-scribed circle in the unpierced part of the plate (see Fig. 10).

Fig. 10. Determination of design diameter d2

6.7.1.2 In the case of tubes roller-expanded into tube plate,adequate safety against pulling out the tubes shall be avail-able. This is the case where the strength in the tube to tubeplate connection given by the tube force FR (see 6.7.1.4) andthe effective area Aw does not exceed the values of Table 5.

The effective area is

(13)

with a maximum of

(14)

Bar-ends unconstrained along the axis

One bar-end clamped, the

other unconstrained along the axis

Bar-ends clamped along

the axis

(Tube or stay between 2 bracing plates)

(Tube or stay between

tube plate and bracing plate)

(Tube or stay between

2 tube plates)

Descrip-tion of the load case

Uncon-strained buckling lengthlK =

l0 0,7 l0 0,5 l0

Design of stay boltsDesign factor

C3

screwed and riveted orscrewed and expanded

0,47

screwed with nuts on both ends 0,44

welded 0,40

s C3

tu1 tu2+

2-------------------- p S⋅

10 K------------ c1 c2+ +⋅ ⋅=

s 0,40 d2p S⋅10 K------------ c1 c2+ +⋅=

Aw da di–( ) lw⋅=

Aw 0,1 da lw⋅ ⋅=


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Table 5. Permissible strength for tubes roller-expanded intothe tube plate

The expanded length lw shall be at least 12 mm and may beconsidered up to and including 40 mm for the determinationof the effective area.

6.7.1.3 In the case of welded tube to tube plate connec-tions to Fig. 11 the welded joints shall be able to sustain thetotal force exerted on the tube. In the shearing cross sectionthe joint thickness shall be at least

(15)

6.7.1.4 The calculation of the tube force FR is to be basedon the loading area AR corresponding to one tube. For fullyperforated tube plates the loading area is designated by theshaded part in Fig.12. In the case of partially perforated tubeplates the portion of the border area shall be taken into ac-count. In the case of border areas in flat ends the area of theend shall be taken into account up to the transition from flatplate to flange. In the case of border areas in flat plates thestrength of the border area may be assumed to be taken upby up to 50 % by the direct adjoining wall of the vessel.

6.7.1.5 If the tubes are subjected to buckling loads, the re-quirements of 6.5.3 shall also be noted. If the buckling loadis higher than the permissible buckling load to formula (8 a),then the required wall thickness s of the plates is

Fig. 11. Thickness of a welded tube to tube plate connection

Fig. 12. Loading area AR

(16)

The design factors C are to be taken from Table 1 andFig. 5. The weakening factor shall be determined as follows

(17)

where

(18)

In formula (18), the index “t” represents the tube parametersand the length of the connection between tube and tubeplate ( in the case of welded tubes;

in the case of roller expanded tubes; in the case of welded and roller-expanded tubes). The max-imum values selected for Et/E and Kt/K as well shallbe 1.

6.7.1.6 Formula (12) does not take into account the effectof the various thermal expansions of the shell, the tubes andthe plate itself. Where the effect of different thermal expan-sions has to be taken into account, the design method is tobe agreed upon between manufacturer and customer/user.

6.7.1.7 Where the pressure in the pipes is more than twicethe pressure around the pipes (i.e. pi > 2 · pu), it shall beproven that the shell can also withstand the axial force aris-ing from p.

6.7.2 Circular fully perforated tube plates for U-tubes

6.7.2.1 The required wall thickness s of circular fully perfo-rated tube plates for U-tubes to Fig. 13 is

(19)

Fig. 13. Circular fully perforated tube plates for U-tubes

Type of roller expanded jointPermissible strength of

roller-expanded jointFR/AW in N/mm2

evenwith groovewith flange

150300400

g 0,4 FR S⋅da K⋅---------------=

s CD1

2 n di2⋅–

v-------------------------------

pi S⋅10 K-------------⋅ c1 c2+ +⋅=

v t d *a–t

---------------=

d *a max †da 2 st

Et

E-----⎝ ⎠⎛ ⎞ Kt

K-----⎝ ⎠⎛ ⎞ lw

*

s-----⎝ ⎠⎜ ⎟⎛ ⎞

⋅ ⋅ ⋅ ⋅– ¢;da

1,2--------

⎩ ⎭⎨ ⎬⎧ ⎫

=

lw*

lw* g da st⋅+=

lw* lw= lw

* g lw+=

lw* s⁄

s C D1pi S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=


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and

(20)

The greater wall thickness obtained from formula (19) or(20) is decisive. Design diameter and design factors for tubeand shell side are to be taken from Table 1 and Fig. 5. Forthe weakening factor the formula (17) and (18) apply.

6.7.2.2 For tubes roller-expanded into the tube plate, therequirements of 6.7.1.2 shall be noted as appropriate with

(21)

6.7.3 Partially or irregularly perforated flat tube plates for U-tubes

6.7.3.1 The required wall thickness s of circular partially orirregularly perforated flat tube plates for U-tubes is

(22)

and

(23)

In accordance with 6.7.2.1, the wall thickness shall be deter-mined using formulae (22) and (23) with the relevant C4 de-sign factors, with the larger wall thickness being thedetermining factor for the design. However, the wall thick-ness shall not fall below the required value for unpiercedplate.

The design factors C4 are to be taken from Fig. 14. For theweakening factor the formulae (17) and (18) apply.

6.7.3.2 In the case of plates with tube lanes (multi-passheat exchangers), in the case of plates whose tubed regiondoes not extend as far as the edge of the plate (e.g. rectan-gular tubed region), or in cases where for the individual di-ameters of concentric tube rows the pitches are unequal,separate calculations shall be performed for each distance l(mean distance of the centre-lines of the tubes for the tuberow under consideration to the centre point of the plate), with

the highest value being the determining factor for the

design. For information on determining l, see Appendix 1.

Individual tubes located outside the tubed region may be ig-nored here.

6.7.3.3 In the case of roller-expanded tubes, the require-ments of 6.7.1.2 shall be noted as appropriate. FR shall bedetermined here according to equation (21).

6.7.4 Circular, flat tube plates in floating head heatexchangers

6.7.4.1 The required wall thickness s of circular flat tubeplates in floating head exchangers to Fig. 15 is

(24)

and

(25)

Fig. 15. Circular, flat tube plates of heat exchangers,where tube bundle is provided with a floating head

The design diameters for the shell and tube side are to betaken from Table 1 and Fig. 3 or 4. The greatest wall thick-ness s obtained from formula (24) or (25) is decisive for thedesign. For the weakening factor v formulae (17) and (18)apply. When calculating the required wall thickness s of thefloating head tube plate, D1 is to be replaced by D2 as shownin Fig. 15 in the formulae (24) and (25).

6.7.4.2 The design factors C5 for the shell and tube sideshall be taken from Fig. 16. The decisive curve for the typeof design results from the boundary conditions. The decisivecurves for additional peripheral moments apply to d = 1,5(see formula (5)). For gaskets having other values for d the

design factor C5 shall be multiplied by . For calculation

the design factor C5 shall not be lower than 0,15.

6.7.4.3 The distance l is the mean distance of centre linesof the outer tubes from the centre point of the plate plus halfthe tube diameter. For explanatory notes, see Appendix 1.Individual tubes located outside the tubed region shall beignored here.

6.7.4.4 A check shall be made moreover on whether theboundary tubes (roughly speaking the two outermost tuberows), together with their joints in the tube plate, are capableof withstanding the buckling and compressive loading

and the tensile load . D2 here re-

lates to the floating tube plate.

If the stress on the boundary tubes is too high, the neces-sary number of load-bearing tubes shall be determined. De-duction of these loadbearing tubes from the existing tubedregion results in a smaller diameter for the tubed region, andthus in a smaller l' and, from Fig. 16, a larger value for C5

which is used to determine the thickness of the tube plate.For the loading of the boundary tubes the following is usedfor calculating the compressive load

s C D1pu S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=

FRdi

2 p pi⋅ ⋅40

-------------------------=

s C4 D1pi S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=

s C4 D1pu S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=

C4

v-------

s C5 D1pi S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=

s C5 D1pu S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=

d1,5--------

pu D22 p

40------⋅ ⋅ pi D2

2 p40------⋅ ⋅


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and for the tensile load

. n is the number of loadbearing

boundary tubes. If for C5 the maximum value in

each case is applicable.

6.7.4.5 The pressure pi is decisive for the buckling load onthe tubes in the inner tube field. The loading area of a tubeshall be assumed to be the loading area given in 6.7.1.4 andextended by the cross-sectional area of the tube.

6.7.5 Circular, flat tube plates of heat exchangers with bellows in the shell

6.7.5.1 The required wall thickness s of circular, flat tubeplates of heat exchangers with bellows in the shell as shownin Figs. 17 and 18 is with

(26)

(27)

The diameter D1 shall be taken from Table 1 and Figs. 3 or 4in accordance with the boundary conditions on the tube side.

The requirements of 6.7.4.3 shall apply as appropriate forthe determination of l in formula (26). The design factor C5shall be taken from Figure 16 in accordance with the bound-ary conditions on the tube side. The requirements of 6.7.4.2shall be noted. Formulae (17) and (18) apply to the efficiency.

6.7.5.2 A check shall be made moreover on whether theboundary tubes (roughly speaking the two outermost tuberows), together with their joints in the tube plate, are capable

of withstanding the tensile load . Here, p shall be

substituted in accordance with formula (26). If pi or pu areunderpressures, the boundary tubes shall also be capableof withstanding the buckling and compressive loading

and .

If this requirement is not satisfied, the number of loadbear-ing boundary tubes shall be increased. Deduction of theseload-bearing tubes from the tubed region results in a smallerradius l' for the tubed region and, from Fig. 16, a larger valuefor C5 for dimensioning the tube plate. The tensile loading ofthe boundary tubes is thus

If pi or pu are negative pressures, the compressive load may

be calculated from and

.

n is the number of load-bearing boundary tubes. If the shell-side pressure is greater than the tube-side pressure

(pu > pi), by forming the total stress , it shall be

proven that the boundary tubes withstand the loading. Thefollowing tangential stress

shall be added to the ten-

sile stress. applies.

6.7.5.3 The pressure pi is decisive for the buckling load onthe tubes in the inner tube field. The loading area of a tubeshall be assumed to be the loading area given in 6.7.1.4 andextended by the cross-sectional area of the tube.

Fig. 17. Circular, flat tube plates of heat exchangers withexpansion bend in shell

Fig. 18. Circular, flat tube plate of heat exchanger withgland in shell

6.7.6 Circular, flat tube plates of heat exchangers with a gland sealing the floating tube plate

6.7.6.1 The required wall thickness s of circular, flat tubeplates of heat exchangers having a gland sealing the float-ing tube plate as shown in Fig. 19 shall be calculated usingformula (27) taking into consideration for the fixed tube plate

(28)

and

(29)

pu 40⁄ 4 l�2 n da2⋅+⋅( ) p⋅ ⋅

pi 40⁄ 4 l�2 n di2⋅+⋅( ) p⋅ ⋅

l�

D1------- 0,1<

p pi pu

D32 4 l2

–

D12

--------------------------⋅+=

s C5 D1p S⋅

10 K v⋅-------------------- c1 c2+ +⋅ ⋅=

p D12 p

40------⋅ ⋅

pi D12 p

40------⋅ ⋅ pu D3

2 4 l2

–( ) p40------⋅ ⋅

pi 4 l�2 n di2⋅+⋅( ) pu D3

2 4 l�2 n da2⋅–⋅–( )⋅+⋅[ ] p 40⁄⋅

pu D32 4 l�2 n da

2⋅–⋅–( ) p 40⁄⋅ ⋅

pi 4 l�2 n di2⋅–⋅( ) p 40⁄⋅ ⋅

s sa st+=

st pi da 2sR–( ) puda–[ ] 20sR⁄=

s KR≤ S⁄

p pi

D12 D4

2–

D12

-------------------------⋅=

p pu

D12 D4

2–

D12

-------------------------⋅=


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and for the floating tube plate

(30)

and

(31)

The maximum value of p obtained from the formulae (28)and (29) and (30) and (31) is decisive for the design of thetube plate. For the floating tube plate C5 shall be replacedby 0,45. For the determination of l in formulae (30) and (31),appropriate reference shall be made to the provisions of6.7.4.3.

6.7.6.2 The assessment of the tensile and compressivestresses on the tubes, and of the buckling resistance, shallbe made through appropriate reference to 6.7.5 for themean load on the tubes.

6.7.7 Circular, flat tube plates with projecting flange rings on heat exchangers

6.7.7.1 The required wall thickness s of circular, flat tubeplates with projecting flange rings as shown in Fig. 20 shallbe calculated for the range given by the design diameter D1according to 6.7.1 to 6.7.6.

Fig. 19. Circular, flat tube plates on heat exchangers with asealing gland at the floating end

Fig. 20. Circular, flat tube plates with projecting flange ring

6.7.7.2 For through tube plates as shown in Fig. 20, theprojecting ring shall be recalculated in cross-section C-C inaccordance with preliminary standard DIN 2505 (October1964).

6.7.7.3 The axial stress in the shell shall be assessed asappropriate in accordance with 6.7.1.7.

6.8 Rectangular flat tube plates in heat exchangers

Rectangular tube plates are dealt with in accordance with6.7.1 to 6.7.6 depending on their shape and taking into ac-count the design factor CE from Fig. 2. This means multiply-ing the value for C by CE in each of the formulae concerned.The values for C are determined in accordance with 6.4 fromthe geometric conditions related to the short side of theplate. In the relevant equations, replace the design diameterD1 by the length of the short side f of the plate.

7 Literature[1] Föppl, A.: Vorlesungen über Techn. Mechanik; Bd. III,

Festigkeitslehre. Teubner Verlag, Berlin (1922).

[2] Timoshenko, S.: Theory of plates and shells. Mc-GRAWHill Book Company, Inc., New York/London (1940).

[3] Filonenko-Boroditsch: Festigkeitslehre, VEB VerlagTechnik, Berlin (1954).

[4] Hampe, E.: Statik rotationssymmetrischer Flächen-tragwerke; Bd. 1. VEB Verlag für Bauwesen, Berlin(1966).

[5] Föpp, L., u. G. Sonntag: Tafeln und Tabellen zur Fes-tigkeitslehre. Oldenbourg-Verlag, München (1951).

[6] Miller, K.A.G.: The Design of Tube Plates in Heat-Ex-changers. Proc. Inst. Mech. Engineers Series B, Vol. 1(1952) pp. 215/31.

[7] Sterr, G.: Berechnungsfragen von Rohrböden imDruckbehälterbau. Verlag Ernst & Sohn, München(1967).

[8] Sterr, G.: Die genaue Ermittlung des C-Wertes für dieam Rande mit einem Schuß verschweißte Kreisvoll-platte unter Berücksichtigung der im Schuß auftreten-den Spannungen. Techn. Überwach. 4 (1963) No. 4,pp. 140/43.

[9] Wellinger, K., u. H. Dietmann: Bestimmung von Form-dehngrenzen. Materialprüfung 4 (1962) No. 2, pp. 41/47.

[10] Siebel, E.: Festigkeitsrechnung bei ungleichförmigerBeanspruchung. Die Technik 1 (1946) No. 6, pp. 265/69.

[11] Hübner, F.-W.: Berechnung der Axialkraft von Ankernund Ankerrohren zur zentralen Verankerung ebenerBöden. Techn. Überwach. 9 (1968) No. 3, pp. 95/97.

[12] Physikhütte, 29. Auflage, pp. 240 ff.

[13] Dietmann, H.: Spannungen in Lochfeldern. Konstruk-tion 18 (1966) Vol. 1, pp. 12/23.

[14] Nadai, A.: Die elastischen Platten. Springer-VerlagHeidelberg, Berlin, New York (1968).

[15] Sterr, G.: Die festigkeitsmäßige Berechnung von Wär-metauschern mit geraden Rohren. Verlag TÜV Bayern,München (1975).

[16] Hütte I, 28. Auflage, pp. 940 ff.

[17] Schwaigerer, S.: Festigkeitsberechnung im Dampf-kessel-, Behälter- und Rohrleitungsbau; 4. Auflage(1983), Springer Verlag Berlin, Heidelberg, New York.

p pi

D42 4l2–

D12

------------------------⋅=

p pu

D42 4l2–

D12

------------------------⋅=


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Table 1. Design factors for unstayed circular flat ends and plates without additional peripheral moment

Type of flat end design(principle only)

ConditionsDesign factor

C

a) flanged flat end 1. knuckle radius:

and r ≥ 1,3 s

2. cylindrical part: h ≥ 3,5 s

0,30

b) forged or pressed flat end 1. knuckle radius:

, however

at least 8 mm

2. cylindrical part: h ≥ s

0,35

c) flat plat welded into the shell from both sides plate thickness: s ≤ 3 s1s > 3 s1

0,35

0,40

d) bolted flat plat with full face gasket D1 ≥ Di 0,35

e) flat plate with stress-relief groove1)

s1 = effective wall thickness of the cylindrical portion of the head at the junction with the cylindrical shell

1. residual wall thickness at the groove:

,

but not less than 5 mmand with Da > 1,2 D1 sR ≤ 0,77 s1

2. groove radius:r ≥ 0,2 s, but not less than 5 mmand

3. only killed steels shall be used. If plate material is used, the plate shall show no evidence of laminations in the weld zone over a width of at least 3 s1

2).4. This type of design is not suitable with-

out detailed verification, where high stress concentrations can occur.

0,40

Dar

min

up to500 30over 500 up to 1400 35over 1400up to 1600 40over 1600 up to 1900 45over 1900 50

r s3---≥

sRp10------

D1

2------- r–⎝ ⎠⎛ ⎞ 1,3 S

K--------------≥


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f) plate welded to the shell with welds at both sides of the latter plate wall thickness: s ≤ 3 s1s > 3 s1

only killed steels shall be used. If plate material is used, the plate shall show no evidence of laminations in the weld zone over a width of a least 3 s1

2).

0,400,45

g) simply supported flat plate residual plate thickness at the gasket circle or at the grooves:

sR ≥ 0,7 s

0,40

h) flat plate welded into the shell from one side only plate thickness: s ≤ 3 s1s > 3 s1

0,450,50

i) partial penetration welded flat plate plate wall thickness: s ≤ 3 s1s > 3 s1

Conditions for a:a ≥ 0,5 s and a ≥ 1,4 s1

Df /D1 ≥ 0,7

0,450,50

k) flat plate placed in front from the external side 1. residual plate thickness at the gasket circle:sR ≥ 0,7 s

2. non-metallic gasket:D1 ≤ 500 mm

1,25

l) flat plate placed in front from inside Residual plate thickness at gasket circle:sR ≥ 0,7 s

0,45

1) Stress-relief grooves with different cross-sections may be more effective in reducing stresses and are acceptable if appropriate evidence is produced.2) As a rule, this the case when the inspections are carried out as specified in Stahl-Eisen-Lieferbedingungen DIN EN 10160, quality class E 3. This can

be stipulated at the time when the order is placed.



C


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Table 2. Design factors for circular flat ends and plates with central stay



C2

flanged flat enda) with inward flange

orb) with protruding stay

1. knuckle radius:

and r ≥ 1,3 s

2. cylindrical part: h ≥ 3,5 s

0,25

c) flat plate welded into the shell from both sides withprotruding stay

plate thickness:s ≤ 3 s1s > 3 s1

0,30

0,35

d) flat plate welded into the shell from one side only withprotruding stay

plate thickness:s ≤ 3 s1s > 3 s1

0,400,45

Dar

min

up to 500 30over 500 up to 1400 35over 1400 up to 1600 40over 1600 up to 1900 45over 1900 50


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Fig. 2. Design factor CE for rectangular or elliptical flat plates

Rectangular plates Elliptical plates

f = short side of the rectangular platee = long side of the rectangular plate

f = short side of the elliptical platee = long side of the elliptical plate

A1 = 1,58914600A2 = – 0,23934990A3 = – 0,33517980A4 = 0,08521176

A1 = 1,48914600A2 = – 0,23934990A3 = – 0,33517980A4 = 0,08521176


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Fig. 5. Design factor C1 of plates with additional peripheral moment in the same direction(note: approximation functions in preparation)


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Fig. 6. Design factor CZ for central stays

Central stays

D1 and d1 = design diameters according to Table 2 from AD-Merkblatt B 5 A1 = 0,4092950A2 = – 0,1073072 · 10–1

A3 = 0,1128268 · 10–1

A4 = – 0,1518604 · 10–2

A5 = 0,9880992 · 10–4

A6 = – 0,3485928 · 10–5

A7 = 0,6391361 · 10–7

A8 = – 0,4773844 · 10–9


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Fig. 14. Design factor C4 for tube plates with U-tubes (note: approximation functions in preparation)


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Fig. 16. Design factor C5 for tube plates of heat exchangers with floating head

Design factor C5 for tube plates with additional peripheral moment to Fig. 3 or 4

l = mean distance from plate mid-point of the tube row under consideration

D1 = design diameter

dt = pitch circle diameter

dD = mean gasket diameter


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A11 = – 0,236012836 · 10+1; A12 = 0,545217668 · 10+1; A13 = – 0,311489659 · 10+1; A14 = 0,308300374A15 = 0,168134309A21 = 0,101396274 · 10+1; A22 = 0,109609483 · 10+1; A23 = 0,162822737 · 10+1; A24 = – 0,345692712 · 10+1

A25 = 0,126083679 · 10+1

A31 = – 0,316517682 · 10+2; A32 = 0,412763296 · 10+2; A33 = – 0,369557657 · 10+2; A34 = 0,248045141 · 10+2

A35 = – 0,727898100 · 10+1

A41 = 0,472852891 · 10+2; A42 = – 0,522484275 · 10+1; A43 = – 0,334202904 · 10+1; A44 = – 0,426049735 · 10+2

A45 = 0,236709739 · 10+2

A51 = – 0,821294529 · 10+2; A52 = 0,122221210 · 10+3; A53 = – 0,167734885 · 10+3; A54 = 0,166614761 · 10+3

A55 = – 0,574166821 · 10+2

Design factor C5 for tube plates without additional peripheral moment and where C = 0,45 to Table 1

l = mean distance between centres of the tubes of row considered from the plate midpoint


A11 = – 0,236012836 · 10+1; A12 = 0,545217668 · 10+1; A13 = – 0,311489659 · 10+1; A14= 0,308300374A15 = 0,168134309A21 = 0,101396274 · 10+1; A22 = 0,109609483 · 10+1; A23 = 0,162822737 · 10+1; A24 = – 0,345692712 · 10+1

A25= 0,126083679 · 10+1

A31 = – 0,316517682 · 10+2; A32 = 0,412763296 · 10+2; A33 = – 0,369557657 · 10+2; A34 = 0,248045141 · 10+2

A35 = – 0,727898100 · 10+1

A41 = 0,472852891 · 10+2; A42 = – 0,522484275 · 10+1; A43 = – 0,334202904 · 10+1; A44 = – 0,426049735 · 10+2

A45= 0,236709739 · 10+2

A51 = – 0,821294529 · 10+2; A52 = 0,122221210 · 10+3; A53 = – 0,167734885 · 10+3; A54 = 0,166614761 · 10+3

A55 = – 0,574166821 · 10+2




A1 = 0,399827021A2 = 0,870316825A3 = – 0,547933931 · 10+1

A4 = 0,622283882 · 10+1

A5 = 0,747769988 · 10+1

A6 = – 0,208753919 · 10+2




A1 = 0,350103983A2 = 0,426355908 · 10–2

A3 = – 0,153280871A4 = – 0,474043872 · 10+1

A5 = 0,109862460 · 10+2

A6 = – 0,103370105 · 10+2


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Fig. 21. Opening factor CA for flat ends and plates without additional peripheral moment

Type A Type B

d = inside diameter of openingD1 = design diameterf = short side of elliptical end

d = inside diameter of openingD1 = design diameterf = short side of elliptical end

A1 = 0,99903420A2 = 1,98062600A3 = – 9,01855400A4 = 18,63283000A5 = – 19,49759000A6 = 7,61256800

A1 = 1,00100344A2 = 0,94428468A3 = – 4,31210200A4 = 8,38943500A5 = – 9,20628384A6 = 3,69494196


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Fig. 22. Opening factor CA1 for plates with additional peripheral moment

Type Ad = inside diameter of openingdt = pitch circle diameterdD = mean gasket diameterf = short side of an elliptical end

A11 = 0,78361000;A21 = – 6,17657500;A31 = 55,15520000;A41 = – 102,76280000;A51 = 17,63476000;A61 = 76,13799000;

A12 = 0,57648980;A22 = 25,97413000;A32 = – 187,50120000;A42 = 385,65620000;A52 = – 218,65220000;A62 = 99,25291000;

A13 = – 0,50133500;A23 = – 20,20477000;A33 = 151,22980000;A43 = – 328,17740000;A53 = 223,86580000;A63 = 46,20896000;

A14 = 0,14374330A24 = 5,25115300A34 = – 40,46585000A44 = 92,13028000A54 = – 71,60025000A64 = – 3,45883000–


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Type B

d = inside diameter of openingdt = pitch circle diameterdD = mean gasket diameterf = short side of an elliptical end

A11 = 1,00748900;A21 = 3,20803500;A31 = – 13,19182000;A41 = 30,58818000;A51 = – 43,36178000;A61 = 42,25349000;

A12 = – 0,02409278;A22 = – 1,09148900;A32 = 10,65100000;A42 = – 44,89968000;A52 = 79,56794000;A62 = – 92,64466000;

A13 = 0,02144546;A23 = 1,55382700;A33 = – 13,27656000;A43 = 47,62793000;A53 = – 71,67355000;A63 = 74,76717000;

A14 = – 0,004895828A24 = – 0,423889000A34 = 3,525713000A44 = – 11,935440000A54 = 16,794650000A64 = – 17,856930000


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Explanatory notes to AD 2000-Merkblatt B 5

To 1

The question of differentiating numerically, for plate design,between thick plates and membranes is one that has not yetbeen finally settled in the literature. In the present case, inkeeping with the data of Kantorowitsch [1], a wall-thickness/diameter-ratio of 1:3 has been adopted as a limit to distin-guish thick plates. The lower limit for defining extremely thinplates was fixed such that, in the exact design of plates com-pared with design in accordance with AD-2000-Merkblatt B 5,a maximum error of 5 % can occur. This error results for theworst case of the simply supported plate with a ratio of platedeflection to plate thickness of 0,5.

From the formula for the deflection of the simply supportedplate.

(1)

where this gives: (2)

(3)

Substituting the numerical value for n and dividing byse – c1 – c2 gives:

(4)

Resolution using (se – c1 – c2)/D finally gives:

(5)

To 6.5.2 and 6.7.1.6

Allowance can be made for thermal stresses occurring in ac-cordance with AD 2000-Merkblatt S 3/0.

To 6.5.3

In order to allow buckling cases encountered in practice tobe embraced in a more discriminating manner, various loadcases have been defined for buckling. In the present case,this is done in such a way that, as the buckling length lK, amultiple of the existing bar length can be substituted accord-ing to the load case.

Literature

[1] Kantorowitsch, S. B.: Die Festigkeit der Apparate undMaschinen für die chemische Industrie. VEB-VerlagTechnik, Berlin (1955).

To 6.7.3.2, 6.7.4.3 and 6.7.5.1

Full tubing is understood to mean the usual arrangement of heat exchanger tubes in the tube plate that allows no furtherregular expansion of the hole pattern within the restraint limits.

Figure 1 Examples of a fully-tubed tube plate

Appendix 1 to AD 2000-Merkblatt B 5

w p R4⋅10 64 N⋅ ⋅-------------------------- 5 n+

1 n+-------------⋅=

NE se c1 c2––( )3⋅

12 1 n2–( )---------------------------------------------=

w p D4⋅

E se c1 c2––( )3⋅--------------------------------------------- 12 1 n2–( ) 5 n+( )⋅

10 64 16 1 n+( )⋅ ⋅ ⋅---------------------------------------------------⋅=

wse c1 c2––----------------------------- 0,5 p D4⋅

E se c1 c2––( )4⋅--------------------------------------------- 0,00435⋅= =

se c1 c2––

D----------------------------- 0,0087 p

E----4≥


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Partial tubing generally follows a section of the surface. l is determined according to the requirements in Table 1.

Type of tubing Symbol

A) Fully-tubed tube plate

l is determined via the arithmetic mean of the mean distance of centre lines of the outer tubes from the centre point of the plate (l1) plus half the tube diameter.

B) Segment of circleCalculation of l

*)

*) Radian measure conversion

a 2 l12 l2

2–⋅=

a 2 arcsin a

2 l1⋅------------⎝ ⎠⎜ ⎟⎛ ⎞ 180

p------------⎝ ⎠⎜ ⎟⎛ ⎞

⋅ ⋅=

b p l1a

180----------⋅ ⋅=

Ab l1⋅ a l2⋅–( )

2-----------------------------------=

l Ap----

da

2------+=


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*) Radian measure conversion

C) Rectangle/arcCalculation of l

D) Sector of circleCalculation of l

Type of tubing Symbol

a1 2 l12 l2

2–⋅=

a2 2 l12

l32

–⋅=

a1 arccos a

1

2 l1⋅------------⎝ ⎠⎜ ⎟⎛ ⎞ 180

p------------⎝ ⎠⎜ ⎟⎛ ⎞

*)⋅=

a2 arccos a

2

2 l1⋅------------⎝ ⎠⎜ ⎟⎛ ⎞ 180

p------------⎝ ⎠⎜ ⎟⎛ ⎞

*)⋅=

b1 2p l1a1

180----------⋅ ⋅=

b2 2p l1a2

180----------⋅ ⋅=

Aa2 l3 a1 l2 l1 b1 b2+( )⋅+⋅+⋅( )

2---------------------------------------------------------------------------------=

l Ap----

da

2------+=

A p l12 a

360----------⋅ ⋅=

l Ap----

da

2------+=


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To 6.7.4.4 and 6.7.5.2

The number of loadbearing boundary tubes n and reduced tube field diameter l' depends on the requirements in Table 2.The calculation procedure is shown in Figure 2.

Step/explanatory note Symbol/comment

A) Specification for the loadbearing boundary tubes using the example of a fully-tubed tube plate

For this, the two outer tube rows are used, n is determined

B) Calculation procedure If the tube plate and boundary tubes are adequately dimensioned, this can be the optimum case. If the number of boundary tubes is insufficient, the procedure is continued as described in step C).

C) Increasing the number of loadbearing boundary tubes

(1) The number of boundary tubes used instep A) are now regarded as n1.

(2) The number of loadbearing boundary tubes should be increased regularly by n2, i.e. inconcentric rows to the centre. For the followingcalculation procedure, this results in a totalnumber of loadbearing boundary tubes ofn = n1 + n2.

(3) The centres of the penultimate innerboundary tube extension row determine themodified (smaller) tube field diameter l'.


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Figure 2 Calculation flow chart

D) Calculation procedure If the tube plate and boundary tubes are adequately dimensioned, this can be the optimum case. If the number of boundary tubes is insufficient, the procedure is continued as described in step E).

E) Increasing the number of loadbearing boundary tubes

Continue procedure as described in C) until adequate dimensioning of the tube plate and boundary tubes is attained.

Step/explanatory note Symbol/comment


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AD 2000-Merkblatt

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AD 2000-Merkblatt

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Publisher: Source of supply:

Beuth Verlag GmbH10772 BerlinTel. 030/26 01-22 60Fax 030/26 01-12 [email protected]

Verband der TÜV e.V.

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


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AD 2000-MB - B5 -03-2009