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ICS 27.040 Edition August 1996

Technical Rules for

Steam Boilers (TRD)

Calculation for cyclic loading due to pulsating internal pressure

or combined changes of internal pressure and temperature

TRD 301 Annex 1 Design

The Technical Rules for Steam Boilers (TRD) reflect the present state of safety requirements for the materials, manufac- ture, design, equipment, erection, inspection, and testing as well as the operation of steam boilers. They are prepared and updated according to the most recent technical developments by

Deutscher Dampfkesselausschuß (DDA)

Reference is also made to § 6, Paragraph 2 of the Steam Boiler Decree (DampfkV) (Clause on the equivalence of technical rules and standards within the EC)

The TRD sheets are published on behalf of "Deutscher Dampfkesselausschuß" by

Vereinigung der Technischen Überwachungsvereine e.V., P.O.Box 10 38 34, 45038 Essen

Translated by:

Fachverband Dampfkessel-, Behälter- und Rohrleitungsbau e.V., Düsseldorf

If there is any doubt regarding the interpretation of this sheet, the German wording shall apply.

Preliminary note

The calculation for cyclic loading shall be governed by local peak stresses. For static loading, these are only indirectly ac- counted for by the mean stress and the related efficiency coefficients which take into consideration a limited and partially plastic deformation.

Contents

1 Scope

2 Design values and units

3 Individual stresses

4 Total stress condition

5 Allowable streses and load cycles

6 Superposition of various load cycles

7 Literature

Ira S .y o Z3 .1^ Q. Q.

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1 Scope

1.1 This Annex 1 to TRD 301 shall apply to the contirma- tory calculation of components which were designed in ac- cordance with TRD 301, with respect to cyclic loadings which are due to internal pressure, on the one hand, and due to radial temperature differentials during start-up (heating-up) and shutdown (cooling down), on the other hand, in the areas subjected to maximum loadingi).

It is assumed that on components subjected to pressure and temperature the maximum local peak stresses occur on the inside of the edges of openings or on branches in cylin- drical shells. Additional forces and moments of a significant magnitude shall be calculated separately (design rules in preparation).

The design rules following hereafter will be supplemented inasmuch as new knowledge is gained.

1.2 The calculation shall be based on the actual dimen- sions of the respective plant component which shall be de- termined by confirmatory measurements. If the actual wall thickness is unknown, the probable wall thickness shall be calculated as follows:

If the wall thickness Se is a mean wall thickness, then the design wall thickness shall be taken as Sb = Se. If Se is a minimum wall thickness, then Sb = 1.15 • Se shall be used for seamless cylindrical shells, and Sb = Se + 1 for welded shells made of plate^).

^) Explanations in course of preparation 2) Ttie factor of 1.15 is about half ttie plus tolerance of 25 % for tu-

bes with minimum wall thickness.

To replace the April 1975 edition of TRD 301, Annex 1 and to replace Draft TRD sheet 301, Annex 1, edition Ivlay 1995;

I = changed with respect to revtous edition

This TRD sheet shall only be applied in conjunction with TRD 300 and TRD 301

:üpyright © 1997 by Vulkan-Verlag, P.O.Box 10 39 62, 45039 Essen

Page 2 TRD 301, Annex 1

At the same time, the following shall apply, depending on d or da. whichever is the nominal diameter:

da = d, + 2Sb

di = da-2 Sb

dm = 0.5 (i^ + d\)

1.3 For calculation purposes, the following shall be defined as governing temperature &* during a start-up and shut- down period (load cycle) under consideration;

1Î* = d +0,75(0-0)

= 0,75 iW 0,25 0 (1)

All temperature-dependent values shall be referred to this governing cycle temperature i3* (or the respective load cy- cles).

1.4 Because of the approximately linear dependence of stresses on the governing internal pressure p', the calcula- tion of the allowable temperature differentials may be lim- ited solely to the two points of minimum and maximum pressure of the load cycle under consideration. Intermediate values shall be obtained by linear interpolation.

2 Design values and units

Cf. TRD 301, Section 2.

3 Individual Stresses

3.1 Ideal-elastic mechanical hole-edge stresses

3.1.1 The maximum hole-edge stress for cylindrical shells with vertical branches shall be determined by using the fol- lowing relation:

d P* o¡p -a

2 s. (2)

For the stress concentration factor am the following shall apply:

+ f..-a. (3) «m ^ «mO

If om or the stress concentration factor a^o cannot be de- termined either by measuring or by calculation, the following shall be used:

(1) omo = 2.6 for set-through and full penetration welded branches as per Fig. 2, as well as for die- forged branches with conical transition and fillets as per Figs. 2 and 3; in each case without residual gap

Omo = 2.9 for welded-on branches; seating surface adapted or milled plane; root drilled out or ground over, without residual gap, Fig. 4

Deviating herefrom, the stress concentra- '" tion factor ctmo = 2.4 may be used for

dAi £ 50 mm;

dAi/oS < 0.2 and 1.6 < ^ < 2.0

ctmo = 3-2 for flanged-out main bodies with welded branch; root drilled out or ground over, without residua! gap, Fig. 5

Where the requirements for the welded joint are not met. e.g. in the case of a non-machined root of the weld or a residual gap of < 1.5 mm, amo shall be increased by the factor for the influence of the root gap

U- 1

1 - 0.5 I -^ d;

(4)

Root gaps up to approximately 1.5 mm are taken into consideration by this (actor íA. However, the require- ments according to (1) shall always be met for diame- ter ratios of dAi/d > 0.5 and simultaneously

d > 300 mm or

for welded-on stubs or nozzles respectively with dAi > 120 mm, provided the pertinent basic shell consists of a material having a guaranteed minimum yield strength of Os > 355 N/mm2 at 20 "C.

OmO = 3.5 for expanded joints

amo = 5.0 for expanded and welded joints with re- sidual gap on the root side > 1.5 mm

Fig. 1: Reinforcement by means of set-through and full- penetration welded branch

3) For material properties cf. VDE Guidelines 3128. Fig. 2: Die-forged branch

Page 3 TRD 301. Annex 1

3.1.2 For the time being, cylindrical shells with oblique and/or non-radially arranged branches shall be calculated as per para 3.1.1 above.

3.1.3 In this connection, arrays of holes in cylindrical shells shall be treated like single openings as per para 3.1.1 above.

3.1.4 For the time being, cylindrical shells with Y-shaped branches under an opening angle of VA shall be calculated as follows:

Fig. 3: Branch forged from solid material and subsequently bored and turned

2 s. (6)

Here the following shall be used:

a„ = 2.5 + (90-VA)'

10^ (7)

Design a Design b

Fig. 4: Reinforcement by means of welded-on branch

Fig. 5: Reinforcement by means of branch welded to flanged-oul main body

{2} If the stress concentration factor ob has neither been determined by measuring nor by calculation, ob = 2.0 shall be used for all branches.

(3) The out-of-roundness factor AJ for elliptical deviations from circularity as per Fig. 6 shall be

fu = l-S •

1 + 1-v'

U_ 3 ' 100

(5)

but not less than 3.2.

A weld root in the gusseted area shall be machined. If this is impossible, am shall be multiplied by ii = 1.2.

3.2 Ideal-elastic thermal stresses at hole edge

These stresses solely depend on the radial temperature pattern within the vessel wall. For the inner wall, and as- suming a rotation-symmetrical temperature characteristic (also under thermal shock),

Ö:,, - a. 1- V

(^m-^i) (8)

shall be used. Herein, a stress concentration factor of ati = 2 shall be used unless another value can be proved to be correct by calculation.

Assuming a quasi-static nary temperature pattern for an in- sulated outer wail, the temperature differential will become Au = i'>rn - ^\ = const. - A-öoo and, as a function of the rate of temperature change 1/0 can be stated as follows:

Ai3_ - — • 0»

In this case <Pt is a shape factor in accordance with Fig. 7.

(9)

For the purpose of measurement dm can be measured in the centre of the wall, with adequate accuracy.

4 Total stress condition

If no considerable exterior forces and moments exist except for internal pressure, then the total hole edge stress can be determined as follows:

(10)

5 Allowable stresses and load cycle numbers

5.1 stress limits with known load cycle numbers

5.1.1 If only the number n of the start-ups from cold condi- tion is stated which are provided or may be expected, then

Page 4 TRD 301, Annex 1

the incipient crack formation load cycle number ñ > 5 n shall be used as basis, so as to ensure adequate reseives for start-ups from hot condition {Cf. TRD 301, Clause 6.2.1). However, if combined load cycles have to be assumed, consisting of ni start-ups from cold and nz, ns . . start-ups from hot condition (with possible differences between the initial and the finai conditions), then the incipient crack for- mation load cycle numbers/ii, 02- ns.etc, shall be so se- lected that the requirement of equation (25) is complied with.

In accordance with Clause 5,1.2 the reduced alternating stress ranges shall be determined for the load cycle num- bers Hi, 02, etc.

5.1.2 For unnotched bars, and in accordance with Fig. 8, the allowable alternating stress ranges of 2oa shall be de- termined as a function of the incipient crack formation load cycle number n. For components, these ranges shall be re- duced because of surface influences. This shall be done by using the correction factor fs as per Table 1 and results in

Aa* = ^ f.

[11:

This alternating stress range Ao* shall be corrected as indi- cated in Fig. 7.

(1 ) In the elastic case of Aa' < 2 ho.2/f a correc- tion is required in order to take account of the influence of the maximum possible mean stress. The result upon using the Gerber parabola is the allowable reduced al- ternating stress range of the ideal-elastic stresses in the form of

AO; = 2 Or ''0.2/tí' ''E

Ao*

2 '^02/-d' _^

Ao* Aa*

(12)

(2) In the 0 v e r e I a s t i c case of Ao* > 2 ao2/it* it must be considered that, in fact, the calculation must be made with a higher strain than that in the idealised elastic case. Then the following applies:

AOi - 1/2 • f^o.s/ö* • ^° ' (13)

5.1.3 After the allowable reduced alternating stress ranges Aoi have been determined for the individual load cycles, the allowable maximum stresses a, and the allowable minimum stresses oi shall be determined, by means of which a, shall be limited during start-up and shutdown and finally the al- lowable temperature differentials are to be calculated.

As the stress limits for the individual cycles can be influ- enced by the selection of the start-up and shutdown rate, they may be freely selected within a certain range. In this case, it is recommended to determine o¡ as follows, using a factor Y-

y > 0 is the absolute relation between the allowable thermal stress at the beginning of shutdown and the allowable thermal stresses at the beginning of start-up.

If during start-up not only the temperature, but also the pressure is reduced, and the component is not regularly cooled by an additionally feeded colder fluid, and unless otherwise agreed between boiler manufacturer, purchaser and inspection body, y - 0 may be used in the calculation. Then, in this case ój = á,p and Oi ^ G¡p - AG,.

Otherwise the maximum is

o¡ = a,p + A 0| (15)

5.1.4 For parts in contact with water which are manufac- tured from non-austenitic steels, special care shall be taken to preserve the mangetite protection layer. Therefore, the stress limits for these parts are additionally restricted as follows:

> Oip^ - 600 N/mm2

â| < öjp^ + 200 N/mm2

(16)

[17)

Table 1: Correction factor fe to take into account the influ- ence of the surface as a function of the yield strength

Os in N/mm2 h <355 1.0

> 355 to 600 1.2

>600 1.4

5.1.5 Equations (14) to (17) provide the allowable maxi- mum and minimum stresses for the individual load cycles. Therefrom, and using equations (8) and (10), the allowable temperature differentials upon start-up and shutdown can be calculated as follows:

Start-up:

Ai3 = i5^-0¡ 1- V

ßu ^.

Shutdown:

AÛ = ^^ - oi 1- V

Öi - Oi,

O; - O;,

[18)

(19)

5.2 Allowable number of load cycles for given stresses

5.2.1 If for a load cycle the stress limits a, and a, are known, then the incipient crack formation load cycle number n for this cycle can be determined as follows:

Based on the existing alternating stress range

AO: = Ó: - Ó, (20)

a,= — .a. (14) there is

(1 ) for the over-elastic case of Aoi < 2 • GO 2/,r

Page 5 TRD 301, Annex 1

•SH¿4?> M Bmmmmm W'.f WmMM^i^&^MM"Ae\¡

!^iLü"[!iLi!DiHííljiJllj!:!lJil!Íihi^?^

0.5x10"^

Flg. 6: Out-of-roundness factor fu

Page 6 TRD 301, Annex 1

-0.48

-0.45 ^^^

-0.35,

-0.32

é, -0.40 ^ggíPS^;

Diameter ratio íJ^=-T-

Ftg. 7: Shape factors for cylindrical shells

2aa - ACf: • h —3- AG;

0,2/û*

(2) for the elastic case of AGí < 2 oo.2/ô'

2 0a - Aa¡ f^ • (2Öp)'

(2 0B)^-(2GO2/,,.-AOj)^

(21 ;

(22)

The results as per equations (23) and (24) are conservative, because the maximum thermal stress and the maximum mechanical stress are added without considering an addi- tional phase shift. For more accurate calculations, the ac- tual time phasing must be taken into account.

5.2.3 For parts in contact with water, o¡ and o¡ shall not exceed the limits stipulated in clause 5.1.4 which, under certain conditions, can be achieved by limiting the ideal- elastic thermal stresses.

Using the value of 2aa, the incipient crack formation load cycle number n is derived from Fig. 8.

Therefrom, the allowable number of load cycles results, for

start-ups from cold only, as n = —.

For combined load cycles the allowable load cycle numbers ni, n2, n3, ... shall be so selected that equation (25) is satis- fied.

6 Superposition of different load cycles

A given combination of load cycles with irregularly fluctuat- ing stresses, regarding magnitude and frequency, is subdi- vided into load cycles with an identical or almost identical alternating stress range (such as, for example, start-ups from cold, or from hot condition), and evaluation is made on the basis of the linear damage rule.

According to this rule, the usage factor D of all load cycles shall be limited to

5.2.2 For preliminary calculations, the stress limits of ôj and Oi, based on the temperature change rates of v,ti and v-dz, assuming quasi-stationary conditions, can be assumed as follows:

D = ^ + ^ + ñr> ño

1 (25)

with a safety factor against usage So > 2.

G; = a m P —

à. =a.

2 s.

2s.

+ a.

-t-a.

ßLö •E^

1- V

ßLü ^. 1- V

<S>,

O,

'flt

'ö2

(23) 7 Literature

(24)

Makinejad, N.: Dynamische Lochrandspannungen und zu- lässige Temperaturdifferenzen in rotationssymmetrisch be- lasteten zylindrischen Bauteilen größerer Dampferzeuger. VGB Kraftwerkstechnik Vol. 54 (1974) No. 3, pp. 186-194.

Page 7 TRD 301, Annex 1

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Page 8 TRD 301, Annex 1

Form for calculating the allowable number of load cycles at specified temperature differentials or temperature change rates, respectively

a) Design and calculation data

1 Type and (nominal) dimensions of vessel

2 Material seamless longitudinally welded

3 Design wall tfiickness (without thermal stress: Sb = Sv)

Sb = (measured or average wall thickness) Sb = X 1.15 (seamless and minimum wall

thickness) Sb = + 1 (longitudinally welded and minimum

wall thickness)

St mm

4 Inside diameter (for outside diameter d\^ da-2 • Sb) a mm

5 Maximum diameter of opening dA\ mm

6 Out-of-roundness U %

7 Opening angle for Y-shaped branches VA "

8 Working pressure (gauge) P4 N/mm2

9 Minimum cycle pressure (for start-up from cold p = 0) P N/mm2

10 Maximum cycle pressure P N/mm2

11 Minimum cycle temperature Û "C

12 Maximum cycie temperature Û °C

13 Governing cycle temperature •Ô* = 0.75 T3 + 0.25 Ü Ô' °C

14 Start-up rate at the beginning of start up, calculated for quasi- stationary condition

(at p) (positive) Vû, K/min

15 Temperature difference at the beginning of start-up

(at p) (negative) AÖ1 K

16 Shutdown rate at the beginning of shutdown, calculated for quasi- stationary condition

(at p) (for 7=0 becomes va^ = 0, otherwise negative)

Vf,2 K/min

17 Temperature difference at the beginning of the shutdown

(at p) (for 7=0 becomes Ai32 = 0, othenwise positive)

Ai32 K

18 Modulus of elasticity (atr) Eö N/mm2

19 Minimum proof stress at elevated temperature

(at r) ao.2/d N/mm2

20 Differential thermal expansion coefficient

(at r) ßu^ 1/K

21 Thermal diffusivity (at Û*) aö mm2/min

22 Minimum tensile strength (at room temperature) ÓB N/mm2

23 Factor fa h=^.0 (if 5s < 355 N/mm2) Í3 = 1.2 (355 < as < 600 N/mm2) Í3=1.4(ifas >600N/mm2)

••• ^ -

24 Theoretical stress concentration factor for membrane stresses

amo = 2.6 (forged/set/through) Omo = 2.9 (welded on without gap) ttmo = 3.2 (flanged)

&itß -

25 Factor it

f4= ^.0 (unmachined root)

k — '4

1-0.5

/4 = 1.2 Y-br£ inch ( unmachined)

Page 9 TRO 301, Annex 1

b) Calculating procedure

26 dm = 0.5 (da + d) dm mm

27 uo= 1 + 2-f uo -

28 w °-^^ w

mm^ K N ßL. • ^,.

29 ^ 1 (u^ - 1) (3 u¡ - 1) - 4 u^ In u^

' " 8 (u¡ - 1} (u^ - 1)2

Obtain from diagram or compute! Of -

30 V - ^••' V 1/min

31 f In \ 1 "^1 \ D -* U AJ(P4) -

'u (P4 ) 1-^

1 + 0.455 ÍP4 (d f 100

m

i^b j

32 «m(P4) = «mo''4 +2-.'u(P.) Of «m =

am >

10^ J 3.2

•^A am(p4) -

33 Ojp4 N/mm2

34

1

^b J L/ m) -

1 + 0.455 'P1 3 100

35 «m(P) = «mû'^4+2 i,(p) or «m =

am > 3.2

•u «mÍP) -

36 óip=cx,(p)-P'|f (Tip N/mm2

37

/ dm] «b J u

íiÍP) -

1 + 0.455 ÍPI ¡dm' Ub>

3 100

38 a^(p)^a^o-^4+2-/,(p) or

am >

2.5 + r

3.2

•u am(p) -

39 ó,=ajp).p.^ üip N/mm2

40 (quasi-stationary case!) Wúl N/mm^

41 'i>2 w V W {quasi-stationary case!) CT¡i52 N/mm2

42 a\ = Ojp + Oifl, *) Oi N/mm2

43 Oj = Oip + OiiÍ2 ') (at the beginning of shutdown and Y = 0 become o,-,^^ = 0)

N/mm2

Page 10 TRD 301, Annex 1

44 AOi = CTi - (Tj Aö( N/mm2

45

"^a ACT / ^""^

2 0a N/mm2 .03 -ACT, Í3 2.5^^^^ ifAai>2CT0.2/fl

o „ A„ f (205)" if AOi > 2 00^0

(25Br-(25o.2/ö-Ao^f

46 Incipient crack formation load cycles n for 2 Oa at 1"^' ñ -

47

only start-up from cold: n < —

Allowable number of cycles Combined cycle loading: select m n -

48 Si = Gip4 -600 Si N/mm2

49 S2 = öip, + 200 S2 N/mm2

50 For parts wetted by water, the additional requirements are:

Si < Ci S2>Ôi

See also notes to clause 5.2.2 of Annex 1.

.•iJH?, .' •••••îift^-.v •^!,'.--îjr.:-: „.^-í^V.'-flI^flí'í^:- ,-.í;Ír. •J^v^Tí'^'^í

J

Page 11 TRD 301, Annex 1

Form for calculating the allowable temperature differentials and temperature change rates at specified number of load cycles

a) Design and calculation data

1 Type and (nominal) dimensions of vessel

2 Material seamless longitudinally welded

3 Design wall thickness Sb = (measured or minimum wall thickness Sb = X 1,15 (seamless or minimum wall thickness Sb = + 1 (seamless or minimum wall thickness

Sb mm

4 Inside diameter (for outside diameter d\ = = da - 2 • Sb) oi mm

5 Maximum diameter of opening C^Ai mm

6 Angle of opening for Y shaped branches VA °

7 Working pressure (gauge) P4 N/mm2

8 Minimum cycle pressure (for start-up from cold p = 0) P N/mm2

9 Maximum cycle pressure P N/mm2

10 Minimum cycle temperature •& °C

11 Maximum cycle temperature % °C

12 Specified number of load cycles (for start-up from cold n > 2000) n -

13 Incipient crack formation number of load cycles

only start-up from cold combined loading: sele

5 • n ct n¡

A) < 0.5

h -

14 Governing cycle temperature iV = 0,75 i3 + 0.25 i^ xV X

15 Modulus of elasticity (at xV) E,. N/mm2

16 (Minimum) proof stress at elevated temperature (at xr)

Ô0.2/Î) N/mm2

17 Differential thermal expansion coefficient (at i3")

ßu 1/K

18 Thermal diffusivity (at Ú-) ai) mm2/min

19 (Minimum tensile strength (at room temperature) OB N/mm2

20 Ratio Y (cf. clause 5.1.3) Y -

21 Factor /b fe = 1.0 (if Os < 355 N/mm2) fe = 1.2 (355 < Os < 600 N/mm2> fa = 1.4 (if Gs > 600 N/mm2)

h -

22 Allowable alternating stress range (2 oa for ñ at xV) 2aa N/mm2

23 Out-of-roundness U % 24 Theoretical stress concentration

factor for membrane stresses amo = 2.6 forged/set-through amo = 2.9 welded-on without gap amo = 3.2 flanged

ClmO -

25 Factor U

f4 = 1.0 (ma

f4=

1-0.5

f4 = 1.2 Y-br

:;hined root)

(unmachined)

anch (unmachined)

U -

Page 12TRD301, Annex 1

b) Calculation procedure

26 drr, = 0.5 (da + a,) dm mm

27

28

29

30

üo=1 +2 -^ = d.

W = 0.35

ßLfl-^ö

^1 (üg - 1) (3 ug - 1) 4 üg In ÜQ

8 {Uo^ - 1) . (üo - 1}' Obtain from diagram or compute!

V = ^^ *f s^

uo

W

a>t

mm^K

1/min

31 . 2 0,

Í. AGZUI N/mm2

32

Aa¡ = ^2 • CTo.2/0 • ^^ ' (for the over-elastic case) Ao* > 2 ÓQ2/O)

AG, - 2 Or Ao*

+ .1-H Ao'

"B O ^0.2/It Ao* O, 'B

(for the elastic case)

33 ^u(P4)-1-5 K'^b

1 - 0.455 V^b ;

Ü 100

ii(p4)

AOi N/mm2

34 am(P4) = «mO-^4+2./,(p4) or a^ - 2.5 +

ID'

2 \

-^

ttm > 3.2

35 ^'ip. =«m(P4)P4' 2s.

am(p4)

Oipa N/mm2

36

37

38

40

/"w \

f.(p) = 15- /• = \

1 -H 0.455 v^«.

U

^w A^ 100

«m(P)= «mO ^ +2/u(P) or a„ = 2.5-H (90-VA)'

10^

am > 3.2

ô^p -a^(p).p 2 s.

/^ \

39 i,(p)-1.5 f •^L \

1 + 0.455 l^oj

U

(M ^^ 100

», =»6

«m(P)-«mO-^4 +2 /-.(p) or

r. - rf.

a„ - 2.5 + (90-n'A)' 10^

am > 3.2

41 Oin =an-(p)p- ip ""m" 2S.

/u(p)

am{p)

^u(p)

Oip N/mm2

am(p)

Oip N/mm2

Page 13 TRD 301, Annex 1

42 S -5 + ^'P Aa¡

Si N/mm2 ^1 "ip ' 1 + y

43 S2 = OifM - 600 S2 N/mm2

44 6\ = S^ 5i = S2 Oi= Si

(not wetted by water) " if S2 > Si if Si > S2

• (if wetted by water) Oi N/mm2

45 S3 = ACT¡ + Ói S3 N/mm2

46 SA = aip4 + 200 S4 N/mm2

47 ai = S3 ai = S4 6i=S3

(not wetted by water) " if SA < S3 if S3 < S4 -

• (if wetted by water) Oi N/mm2

48 AO1 = W • (ói - 5ip) A•ö^ K

49 Ai^r = W- im- Sip) A^r K

50 AT32 = W • (Si - áip) A1Î2 K

51 Ai32' = W • (ai - 5ip) Ai^- K

52 Vù, = V • A\h *) v^ K/min

53 v,i, = V • Atír *) V^r K/min

54 W'^ = V • AÔ2 *) v^ K/min

55 VtV = V • Ai32' *) V^^: K/min

2.2 of Annex 1 See also notes to para 5

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Fig. 1 : Allowable temperature differences Fig. 2: Allowable temperature change rate, calculated for quasi-stationary condition

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