Analytical model for failure bending moment of straight pipes including residual stress and ovality

lnt J Pres Ves & Pipmg 30 (1987) 311-319

Analytical Model for Failure Bending Moment of Straight Pipes Including Residual Stress and Ovality

A n t o m o M o r e n o Gonzfilez Empresanos Agrupados, SA, Magallanes, 3, 28015--Madrid, Spare (Received 10 May 1987, accepted 31 May 1987)

ABSTRACT When a ptpe breaks the fracture starts m the vicinity of a n mlttal or nucleatmg defect, such a defect being usually located near to the weldment Generally, thts means that breakage ts an elastw-plastw process takmg place in areas wtth heterogeneous properttes This comple~ctty is increased by the restdual stress m the weldment, whwh is always dtffieult to evaluate Moreover, the breakage ts usually assoctated with a gross loss o f clrculartO,, mdependent of the pipe dtmenslons In thts paper a four-parameter model is gwen which de,tribes four fundamental aspects o f the pipe section at the moment o f breakage These parameters are (a) the effective stress m the uncracked area as defined by M F Kannmen, (b) the effecttve stress m the cracked area as defined b)' K Havegawa, and two newparameter~ to dewrlbe (c) residual ~tre,w~ and (d) ovahty Addlttonally, a slmphfied approach to dewrtbe cracl~ growth t~ mcluded m the model Thts approach requtres t . o new parameters, namely the mlntmum defect length and the effective defect area at the fatlure instant

INTRODUCTION The detection of cracks, caused by stress corrosion m stainless steel piping of nuclear plants with hght water reactors, has promoted the study of the potential rupture of th~s pxpmg Stainless steel is a material which admits a slgmficant plastic strain before rupture Therefore, the study o f cracks appearing m large diameter piping, with appreciable thicknesses, is a 311 Int J Pres Ves &Ptpmg 0308-0161/87/$03 50 ~ Elsevier Apphed Science Pubhshers Ltd. England, 1987 Printed m Great Britain

312

4 M Gonzalez

complex problem of stress d~strlbunon m three dimensions on the elasncplasnc range In addition, aspects such as the lmtial geometry of any defects and heterogeneltles m the material due to differences between the base material, the support material and the heat-affected material play a slgmficant role The difficulty in describing the indicated effects has led to the development of models with different grades of complexity The most complex model amongst those proposed is that due to Paris i In this model, a tearing instability criterion is used, based on the material resistance curve This method has two drawbacks it reqmres complex 3dlmens~onal calculanon systems on the elasnc plasnc range and a material resistance curve data base, including heterogeneIties On the other hand, empmcal methods are found, such as the one proposed by Belbe 2 for piping without defects, m which the plastic moment is corrected by empirical funcnons related to geometry (3 = R/t) and material (~=So/S~), which permit enveloping the failure moments of several experimental series The advantage of this method Is its simplicity of apphcatlon, although m certain cases it may prove to be very conservative In an earher par~er Kannmen 3 had incorporated the geometry of the defect into the calculanon of the plastic moment of the section, and Wllkowskl 4 presented the safety factors of the model compared with a series of tests The previous model was completed by Hasegawa, 5 introducing a new parameter of the material that permitted d~fferentiatmg between maximum stress in the hgament w~th a crack and maxxmum stress in the material without a crack at the rime of maximum moment The model presented m the present work completes the Hasegawa model with four new parameters The first one permits incorporation of the effect of the residual stresses In the secnon and the second ovahzanon The third and fourth are parameters that model the growth of the defect, one being the minimum length of the defect at the leaking instant and the other its effecnve area

R E S I D U A L STRESS T R E A T M E N T The basic geometry of the secnon with the defect and the stress distribution is described in Fig 1 at the time of maximum moment Figure 2 shows the distribution of relative residual stresses of the type described by Hale et al 6 The areas between the maximum and the residual stress curve show the section load capacity It can be seen that the presence of residual stresses increases the capacity to absorb tension, area A~, whilst it decreases the capacity of absorbing compressions, a r e a A 2 The new admissible stress values, the allowable maximum average stress

Model for fazlure bending moment of ptpes

313

R Pr 1

l

Fig. 1.

Distribution

M4

of stresses

in c o m p r e s s i o n a c and m tension at, are defined as a f u n c t i o n o f the material yield stress % as ac = A 2 % < ap

a t = Al%

> %

(1)

It is a s s u m e d that there is internal pressure apt at the u m f o r m tension stress in the section I f the defect has a length 20 and a d e p t h d, so t h a t x = d / t (Fig 1), the average stress, with R being the average r a d m s o f the section, will be n ( 2 / ? - t) 2

(2)

% t = P 8 R t ( n - Ox)

D u e to the presence o f internal pressure, the m a x i m u m allowable stress due to the b e n d i n g m o m e n t ~s m the c o m p r e s s i o n zone O'mc = (9"c -'{'- a p t

(3)

O'mt = 0"! - - a p t

(4)

m the u n c r a c k e d tension z o n e

It ms a s s u m e d that the defect relieves a n y residual stresses. T h e r e f o r e , m a x i m u m stress due to the m o m e n t equals an average stress o f O'mF = O F - - Opt

(5)

where av ~s the average stress p a r a m e t e r defined by H a s e g a w a 5 at the t~me o f m a x i m u m m o m e n t m the zone o f the defect

'

~ 0 . 5 0

-1 Fig. 2

Typical distribution

of residual stresses

A M Gonza&z

314

The external m o m e n t at the plastic collapse o f the section has ~ts neutral pomt m 1 fl - (trc + a,~ [(rt - 0)a t + 0(1 - x)a F - (n - 0x)%t]

(6)

The above includes the Hasegawa 5 and K a n n m e n 3 model as a specific case The total external m o m e n t as regards the central axis of the circular section will be Mext=2tR2[amcSlnfl+amt(Slnfl--slnO)+amV(1--x)sxnO]

(7)

or Mad = 2t/~2[0"c s i n fl + o t ( s m fl - s m 0) + trF(1 - - X) s m 0]

Mse = 2t/~2%t x sm 0

(8) (9)

Alex ' = Mad + Ms~

where Mad lS the m a x i m u m plastic m o m e n t that section can w~thstand, Ms~ is a secondary m o m e n t due to internal pressure and fixed end c o n d m o n If the hypothesis had been free rotation of the non-symmetric section due to internal pressure, Ms~ would have a negatwe sign Therefore, the m a x i m u m apphcable external m o m e n t in the piping, depending on piping end conditions, will be Mext = Mad _+ M,e

(10)

OVALITY T R E A T M E N T Wxlkowskl 4 presents a correction of the failure m o m e n t m terms of the ovahzatlon reached by the sectxon It has been observed during experiments that piping with short cracks with complete penetration becomes deformed as if there were no cracks, i e w~th the greater axis of the section remammg horizontal (Fig 3a) Pxplng with long cracks of complete penetration becomes deformed with the greater axis of the section remaining vertical (Fig 3b) In this paper, an expression slmdar to the one used by Moreno 7 will be used to represent the radius as a function o f o v a h t y ~ The case m Figs 3a and 3b will be represented by R = R(1 -T-~ cos 2q~)

(11)

The neutral hne is obtained in a slmdar way to that in the case of residual stress treatment The total force in the compression zone (see Fig 1) is F1 = 2

fi' amct / 1 + (\ R ddRd p }~2Rd~)=2(Tmclll

(12)

Model for failure bending moment of pipes

-"-

315

1M

(a) i,/

/i

t

\\

(b) Fig. 3. Deformation mode m the tube section (a) short crack, (b) long crack

where 11 =

1 + IkR~-~) Rd~b

(13)

T a k i n g into account that ~ << 1, the following is obtained dR Rd~b

4- 2~ sin 2q5 << 1 (l-T- ~cos2(b)

(14)

Therefore, the expression below the square root sign in the integral 11 can be expanded as ? x 4 #~2 ? (~, -• 2) x2 (1 4- x) v/x = 1 4- -~

(15)

"

Applying the expression to the square root in 11 and ignoring the higherorder terms, the following is obtained for I 1 I1=/~

(1 ~ c~cos 2~b)d4~ = / ~ ~b T-2 sm 24~ do

(16)

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Calculating the integral for the other two zones and balancing forces gives the equation of fl 2(0"m¢ + O'mt)t/~(fl + ~/2 sin 2fl) = 2O'mtt/~(rc -- 0) + 2amy(1 -- .~)tRO - ( -T-~/2) sin 20[-2trmt t/2 + 20"mF(1 - x)t/~]

(17)

where trm¢, am, and amy have the meanings indicated in eqns (3)-(5) The external moment around the central axis of the tube section due to the compression zone is Mmc=

2amct

1+ ~ )

Rcosq~Rdq5 = 20"mJI 2

(18)

By applying the expansion o f e q n (15) and ignoring higher-order terms in ~, the following is obtained

12 =

R2ff(1

-T-2~ cos 2qS)cos ~bdq5 = ]~2[sln t~ ~- ~(sln ~b + {sin 3~b)]~o (19)

Calculating the integral for the other two zones and adding moments, the expression for the total external moment is obtained Mex t = 2t/~216mc sin fl + O'mt(Sln fl -- sin 0) + (1

~7)O'mFsin 0] -T-2tRZ~[amc(Sln [3 + ½ sin 33) + amt(Sln/3 - sin 0 + ½sin 3/3 - ½sin 30) + (1 - V)amv(Sln 0 + I sm 30)] 120) -

-

which, with the help of eqns (3)-(5), can be written in a similar wa~ to eqn (10), and with the same notation

CRACK GROWTH TREATMENT One of the reasons for lack of conservatism in certain cases, using the Kanninen formulae, is that they ignore the crack growth process This can lead to a calculated failure moment higher than the actual one This results from the assumption of a greater load absorbing area than that actually existing after the crack has grown One way of correcting this loss of conservaUsm is to define an effecnve defect area at the failure instant Let us consider the defect shown in Fig 4 of depth d The correction assumes that the effecnve area of the defect at the failure instant is the shaded area, instead of the mitml area of the defect The approach also assumes that there is no circumferential growth of the defect until it reaches complete penetranon This approach is applicable to defects with large OR/t fracnons, but not to small ones The approach should be completed with an

Modelfor fadure bendingmoment of ptpes

317

A

..x

..o Fig. 4

L

i

Definmon of effective defect areas

experimental Omm parameter, which gives the minimum circumferential development of any defect at the instant when complete penetration is reached All defects with a development of less than 0mm wall be treated as If they were defects of size 0mm For greater defects, it will be assumed that they do not experience orcumferentlal development before reaching complete penetration It is assumed that the effective defect has an elliptical front The elhpse is defined as passing through points A and B of Fig 4 of coordinates (Rex, n/2), (R,n q- d, n / 2 - 0) The equation of this ellipse is

a2b 2 p2 = b 2 c o s 2 q~ q- a 2 sin 2 q~

(21)

and with the condition to pass through the indicated points, b = Rex

a2 =

2

Rex(R'" + d)2 sln2 0 R2x-- (R m + d) 2 c o s 2 0

(22)

Upon integration of the area, the effective area of the defect, shaded area of Fig 4, is

Seff=ab[2-tan-l(bCOtO)l-R2,n 0

(23)

The effectwe depth of the defect will be [-(Rln -~- def) 2 - - R2n]O

=

Sef f

(24)

To a first approximation, de¢- Serf 2R,nO

(25)

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A M Gonzalez

N U M E R I C A L V A R I A T I O N I N T R O D U C E D BY T H E N E W MODEL We will consider the case of a pipe with the following characteristics /~ = 190mm, t = 2 6 m m , 2 0 = 132 °, % = 4 5 k g m m -2 If a complete penetration defect and no internal pressure are assumed, the consideration of residual stresses as those described in Fig 2 means a reduction o f the plastlfiCatlon m o m e n t in the range of 6% over the initial one obtained without considering residual stresses If an Internal pressure of 1 kg mm - 2 is considered, the secondary m o m e n t means a variation in the external m o m e n t In the range of 10% with the sign depending on the b o u n d a r y conditions Ovahzation involves a variation in the external m o m e n t in the range of 1 35e which, for small ovahzatlons of 5%, means a variation of 7% The accumulation of these corrections entails a variation either side of the estimate of over 20% If we assume that the defect has a penetration of 0 65 and a length of 171 ~, apphcatlon of expression (25) lmphes an effective size of 0 84 at the leak m o m e n t This means, by application of eqn (8), that the correction due to crack growth reduces the calculated failure moment, with the initial size, by a factor of 0 7, very close to the value estimated by Wllkowskl 4 for a similar case which is of the order of 0 68, which suggests that the estimate of effective defect sizes suitably corrects the lack of conservatism in the K a n n l n e n 3 and Hasegawa 5 expressions CONCLUSIONS The model presented in this paper for the plastlfication m o m e n t in piping with defects has the following features • It introduces a new parameter to evaluate the effect of residual stresses at the failure moment • A first-order correction is introduced in ovahty The formula used allows corrections of a higher order to be considered only by maintaining them in the expansion • The model maintains the secondary m o m e n t due to internal pressure, allowing consideration of different b o u n d a r y conditions to be accepted • Ignoring the previous effects could lead to unrealistic plastlfiCatlon moments As the above-mentioned effects may have an alternative sign, an accumulation with the same sign of all the effects could mean differences of up to 26%, plus or minus

Model for fadure bending moment of ptpes

319

• Apart from the two parameters indicated, the new model includes two new parameters relating to the instant of failure, namely the m l m m u m defect length Oraln and the effective area of the defect (Serf) The introduction o f these parameters allows coherent correcUon o f the lack of conservatism of the present models for small length or small penetration defects, which do not include defect growth • As all these parameters have an analytical formula, the model is suitable for introducing into a small computer With the new parameters introduced, the model includes six parameters which allow a wide parametric sweep

REFERENCES 1 Pans, P C, Tada, H, Zahoor, A and Ernst, H, The theory ofmstabthty of the tearing mode ofelastw-plastw crack growth, ASTM STP 668 (1979), p 5 2 Belke, L, A stmple approach for fadure bending moments of straight pipes, Nuclear Engng Deszgn, 77 (1984), pp 1-5 3 Kannmen, M F, Broek, D, Hahn, G T, Marshall, C W, Ryblckl, E F and Wdkowskl, G M, Towards an elastic-plastic fracture mechanics predlctwe capabdlty for reactor piping, Nuclear Engng Destgn, 48 (1978), p 17 4 Wdkowskl, G M, Margins of safety based on circumferential cracked depth using the net section collapse analysis, CSNI/RCC Meeting on 'Leak before break', NUREG-CP-0051, Monterey, Cahforma, September 1983 5 Hasegawa, K, Shlmlza, T, Sakata, S and Shlda, S, Leakage and breakage estimation based on a net stress approach for stamless steel p~pes w~th orcumferentlal cracks, Nuclear Engng Design, 81 (1984), pp 285-90 6 Hale, D A, Heald, J D, Horn, R M, Jewett, C W, Kass, J N, Mehta, H S, Ranganath, S and Sarma, S R The growth and stabthty of stress corroston cracks tn large diameter BWR ptpmg, EPRI-NP-2472, General Electric Company, San Jos6, Cahforma, 1982 7 Moreno, A, Collapse of nearly orcular thin walled tubes, Nuclear Engng Design, 58 (1980), p 1