nozzle corner cracks

Int. J. Pres. Printed ELSEVIER

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An approximate expression of K,.. for sphere/nozzle corner cracks Xu Yigeng Zhejiang

Silk Engineering

College, 88 Wenyi Road, Hangzhou,

310033, P.R. of China

&

Xu Yizhong Lan Xi Phoenix

Chemical

Company

Ltd., Lan Xi, Zhejiong

321100, P.R. China

(Received 6 January 1995;accepted 9 February 1995)

Based on the flat-nozzle model loaded by uniform biaxial tension, using the O-integral of the weight function method and introducing suitable free-surface correcting factors M,, M,, and a curvature correcting factor M,, an approximate expression of Klmax of sphere/nozzle circular corner cracks has been presented in this paper. The results so calculated are compared with those reported in the literatures and they show good agreement,the discrepancies being lessthan 10%. This showsthat the approximate expressionis effective and can be usedin practice. ”

NOTATION

relative errors distance from a point load at Q to the center of dS equivalent normal stress acting at point Q in UQ crack area ue circumferential stress (70 uniform tension stress 8

pQ

radius of the circular crack 1 area of the crack surface (see Fig. 2) dA, infinitesimal area around point Q dS - infinitesimal portion of the crack front D internal diameter of spherical shell stress intensity factor of Mode I cracks K, 1QQ’ distance from the point load at Q to the point Q’ on the crack front front free-surface correction factor Ml M2 back free-surface correction factor MC curvature correction factor internal pressure of vessel P axial membrane stress of vessel body 9 point on the surface of a crack point on the front of a crack :I r distance from a point load to the circular center R radius of nozzle contour line of the crack area S t thickness of nozzle T thickness of vessel body

1 INTRODUCTION Due to the inconsistency of the local structure, the high concentration of stress and strain and the influence of residual and thermal stresses possibly existing during fabrication, the local stress level in the nozzle corner area may be very high. Sometimes the peak stress umax may reach twice as much as the yield strength u,, of the relevant materials and the strain concentration may be up to six times the yield strain eY. Additionally, various kinds of defects are often formed at the nozzle corners during fabrication which may not be found easily during inspection. These defects can gradually grow during the 273

214

Xu Yigeng, Xu Yizhong

service period until vessel failure occurs. Consequently it is to be expected that any failure of pressure vessels often originates in fatigue crack growth at the nozzle corners. The fracture analysis of nozzle corner cracks is therefore very important. Owing to the complication of loading conditions and constructive geometry in this area, no satisfactory theoretical solutions of this problem have been found as yet. There are many difficulties if we want to solve this problem completely by mathematical means. Through some suitable reduction and approximation, an expression of K,,,, of sphere/nozzle corner cracks is reported in this paper. This approach can be valuable for the approximate prediction of fatigue life and failure analysis of pressure vessels. 2 DERIVATION EXPRESSION

OF K,,,,

APPROXIMATE

This work begins with a simple model - flat plate with a central circular hole loaded by (q :q) biaxial uniform tension as shown in Fig. 1. The radius of the circular hole equals that of the radius of the nozzle. The biaxial uniform tension q equals that of the membrane stress of the spherical pressure vessel loaded by internal pressure p. According to the elastic mechanics analysis result, the stress distribution in 8 = 0 plane is ue =pDl(47)(1+ (wX)2) (1) Since the crack surface is also loaded by internal pressure p, the equivalent circumferential stress in 8 = 0” plane is: ue = pDl(4T)(

1 + (RI.q2)

+p

(2)

According to Ref. 1, for cracks embedded in an infinite body subjected to Mode I loading, the general expression of K, at an arbitrary point Q’ along the crack front is: (3) The meaning of various symbols above is shown in Fig. 2 and eqn (3) is the O-integral of the weight function method. It is obvious that the stress distribution given in eqn (2) does not correspondend to the flat plate/nozzle problem. According to the 3D finite-element analysis in Refs 2 and 3, in the nozzle corner area the stresses along nozzle side and plate side vary sharply and it can be observed that near the inner corner u0 values along the vessel and along the nozzle sides decrease at almost the same speed. Supposing that the stress variation along the plate side is in the form of eqn (2), we can think that the ue values along the nozzle side approximately also vary in the same way. Furthermore for safety reasons we can suppose that the ue values are axially symmetric. In other words, we can think that the effect of the nozzle makes the stress distribution in eqn (2) at the nozzle corner area become an axisymmetric distribution, that is: UQ

=@/(473(1

+

(R/r)‘)

p

(4)

The stress distribution in the supposed crack surface is known. We may calculate the stress intensity factor at an arbitrary point in the nozzle corner crack front from eqn (3). However, due to the non-axial symmetry of the quarter circular geometry, the above integral has to be solved by a numerical integral method. To avoid this trouble, we have treated it as follows. First, it can be considered as an infinite body

f I 4 4 -

I I 9 I I Fig. 1. Sketch of model consideredin this paper.

+

Fig. 2. Key to symbols.

An approximate

expression

of KImax for spherelnozzle

containing an embedded crack with a stress acting on the crack surface; is assumed the same as in the case of the quarter circular crack (see Fig. 3). Then the solution of this problem, is obtained which can be fulfilled easily. Then, a front free-surface correction factor M, and a back free-surface correction factor M2 are introduced and a expression for the quarter circular crack in the structure (see Fig. 4) loaded by the same ue is obtained. Obviously, values obtained in this way are constant (KI)Q~ along the crack front. It is reasonable to think that the (K,)Q, value is the approximation of the practical (K,),,, in the crack front and can be used as the K,,,, that is important in engineering. Considering Fig. 3, we obtain

corner cracks

275

uQ

(K,)Q+,

(K,)Q,

dS = 2Eal(a2 - r2)

pa2

(5)

(KdQ’

d?7[@/(47-)(1

=

+ (R/r)* d&i

= ;pm%

[D,(4T)

+ pDI ZTV’&

l&Q.

+ l]

a R’/(R

o

+p]dAQ

+ r)*r

(f-9

Since r/R < r/a < 1, we obtain R’/(R

+ r)’ =

(a2 - r*)-l’*

1 - 1*965(r/R)

+ 2*560(r/R)*

- 2.070(r/R)3

+ 0*738(r/R)4

= (l/a)[l

(7)

- 0*114(r/a)

+ 1*605(r/a)*

- 3*225(r/a)3

+ 4*475(r/a)4]

(8)

Substituting eqns (7), (8) into eqn (6), after integration, we obtain (K,),,

=

P&(1-128 - 0.403(u/R) - 0*294(u/R)3

+ (DIT)[O.572

1.28 x Za,d& = l-28 (10) 2aJLG As to the back free-surface correction factor M2, referring to Ref. 5 and the analysis of Folias, we have M2 = 1 + 0*15[u x (t’ + T2)-1’2]2 (11) After correcting eqn (9) with M, and M2, we obtain the (K&.,,ax of flat plate/nozzle corner cracks. To go from the plate/nozzle solution to a sphere/nozzle, a curvature correction factor M, must be introduced. According to Ref. 5 and the analysis of Folias, we have M, = dl + 1.86 X a*/(DT) (12) Correcting eqn (9) with MI, M, and M,, we finally obtain an approximate (K,),,, expression

+ 0*428(a/R)* + 0~092(u/R)“]}

According to the geometric characteristic of a flat plate nozzle, two free-surface correction factors M, and M2 to Wdmax should be introduced. M, can be represented by the ratio of (KJ,,, along the quarter circular crack front in a quarter infinite body (see Fig. 5) loaded by uniform normal tension stress (T, to the (K,),,, value along the crack front embedded in an infinite body loaded by the same uniform tension stress. Referring to the table given in Ref. 4, we obtain WJmax along the crack front in Fig. 5 as (KJmax = 1.28 X Za,&& therefore M, =

dr

m

Fig. 4. The quarter circular crack and the free-surface correction factors, M, and M,.

(9)

c- -.

w

Fig. 3. Sketch of circular crack.

UO

Fig. 5. The quarter circular crack in a quarter infinite body loaded by a uniform tensile stress. c

Xu Yigeng, Xu Yizhong

276

along the quarter circular crack front in the sphere/nozzle corner area loaded by the internal pressure p (Kr)max = 1*28[1 + 0.15a2/(t2 -t P)] x ql + 1.86a2/(DT)

p&

X (1.128 + (D/T)

x [O-572 - 0.403(a/R) + 0.428(a/R)* - 0*294(~/R)~ + 0.092@ /R )“I}

(13)

3 THE VALIDITY OF APPROXIMATE EXPRESSION (13) To show the validity of the approximate eqn (13) obtained above, it is necessary to compare the results calculated by eqn (13) with those reported in the literatures and in the photo-elastic experiments that have been done in our laboratory. 3.1 The validity of eqn (13) when sphere/nozzle is transformed into flat plate/nozzle structures The fracture analysis of sphere/nozzle structures is based on the flat plate/nozzle structures through the introduciton of a curvature correction factor M,. The accuracy of the expression for the sphere/nozzle depends on the accuracy of the expression for the flat plate/nozzle. When D + 03, then eqn (13) is transformed into: (K,),,,ax= l-28&

[l + 0.15u2/(t2 + T*)]

x (1.128~ + pD/(4T)

structures are loaded by biaxial uniform tension, the above expression can be simplified into: (KJmax = 1.280,& [l + 0.15u2/(t2 + T*)] x [2.228.- 1.612(u/R) + 1.712(u/R)* - 1*176(~/R)~ + 0.368(~/R)~] (14) Since most (K,) distribution results reported in the literature are flat plate/nozzle loaded by uniaxial uniform tension, according to a similar procedure we obtain the (KI)max of flat plate/nozzle corner cracks loaded by uniaxial uniform tension as follows: (KJmax = 1.28a,& [l + 0.15u2/(t2 + T*)] x [3.335 - 5*465(u/R) + 8*232(~/R)~ - 7.001(~/R)~ + 2.491(&?)4] (15) 3.2 Comparison with photo-elastic experimental results We conducted experimental research of three flat plate/nozzle by the three-dimensional photoelastic frozen method given in Ref. 6. Figure 6 shows the figure of the model. The relative errors comparing these experimental results with those calculated by eqn (15) are shown in Table 1. 3.3 Comparison with the finite-element

results

Broekhoven and van de Ruijtenbeek”’ have calculated the (KJmaX of plate/nozzle models uniaxially tensioned to the level of (T, = 1 N/mm* using different finite element techniques for the same model. The relative errors comparing the UQmax calculated by eqn (15) with those in Refs 7 and 8 are given in Table 2. 3.4 The validity of eqn (W) compared with sphere/nozzle structures Reference 9 obtained the K, distribution along the sphere/nozzle corner circular crack fronts by

x [2.228 - 1.612(u/R) + 1.712(u/R)* - 1*176(~/R)~ + 0.368(u/R)4]} where CT,,= pDl(4T). (T, represents the biaxial tension stress loaded on the flat plate/nozzle structures. Since internal pressure p no longer exists on the crack surface when flat plate/nozzle

I

I

Fig. 6. Model for photo-elastic tests.

An approximate Table

Model 1 2 3

a (mm)

alR

6 10 15

0.3 0.5 0.75

expression

1. Comparison

of KImax for sphere/nozzle

of calculated

corner cracks

values with photo-elastic

277

results

a2/(t2 + T’) (2) 0.0521 0.0521 0.0521

0.125 0.347 0.781

1.193 1.373 1.552

1.346 1.587 1.630

-11-4 -13-5 -4.8

-1.4 -3.5 +5.2

S2is the relative error consideringthe correction of Poissonratio.

3D photo-elastic frozen method. They expressed the stress intensity factor K, with K = a1aG. The internal diameter of the sphere is D = 164 mm, t = T = 7.5 mm, d = 19 mm. The relative errors compared the results of eqn (13) with those in Ref. 9 and are shown in Table 3.

for Model 1 is very high. This is caused by the difference of nozzle corner geometry between the literature models and the models used in this paper. The inner corner area of the model in this paper has no circular transition and is the complete quarter circle. However, the inner corner area of the model in Ref. 7 and Ref. 8 has circular transition and the practical crack is less than that of quarter circle. When the radius of the corner cracks is not big, for example a = 5 mm in Model 1, the weakness of circular transition to the quarter circular area can not be neglected. Therefore, the fact that the results of Ref. 7 and Ref. 8 are less than those of this paper is reasonable. With the increase of the crack radius, the effect of circular transition becomes less and less, and the difference between the results of Ref. 7, Ref. 8 and eqn (15) becomes less and less. As to Model 4, the fact that the relative errors between Ref. 7, Ref. 8 and eqn (15) are +12*8% and -11.8% respectively, which in the author’s opinion demonstrates the validity of eqn (15). Through the discussion above, we can conclude that the approximate expression eqn (13) of WJmax about the sphere/nozzle corner cracks is effective and can meet the needs of engineering uses to sufficient accuracy. Therefore the acquirement of eqn (13) may be valuable for the workers in this field.

4 ANALYSIS ABOUT THE RELATIVE ERROR AND CONCLUSIONS Through Table 1 and Table 3, we found that photo-elastic experimental results are always higher than those calculated by eqn (13) and eqn (15). The reason is that photo-elastic materials are incompressible below their frozen temperature. Their Poisson ratio p = 0.5, while the Poisson ratio of steel is p = 0.3, thus, the model constructed with photo-elastic materials has more plastic constraint on the corner cracks than that of steel vessels and leads to the higher experimental K, value. According to reports in some literature, the photoelastic results may be 10% higher. Therefore, taking into consideration Poisson ratio correction, the relative errors &, comparing the results calculated by this paper with those experimental results may be in practice -l-4%, -3*5%, 5*2%, -7*5%, 7*9%, 2*1%, 1.4%. From Table 2, we find that the relative error

Table

Model 1 2 3 4

2. Comparison

of results from the approximate

a (mm)

alR

5.00 8.28 13.95 22.60

0.174 0.287 0.484 0.785

approach

with those from finite-element

statistics

(;;)

a2 -t’ + T2

(K,L,x eqn (15)

(Kdmax Ref. 7

s, (%)

(KJmax Ref. 8

62 (%)

1 1 1 1

0.0312 0.0856 0.243 0.638

7.468 8.564 9,376 11.182

5.7 8.2 9.8 12.8

-31 -4.4 4.3 12.6

6.4 8.5 9.6 10

-16.7 0.7 2.3 -11.8

6, is the relative error comparedwith Ref. 7. 6, is the relative error compared with Ref. 8.

Xu Yigeng, Xu Yizhong

278

Table 3. Comparison of the eqn (W) results with those from photo-elastic tests Model

1 2 3 4

all?

3.241 3.412 3.587 5.193

0.341 0.359 0.378 0.547

a2 t2 + T2

eqn (13)

0.093 o-103 0.114 0.240

1.557 1.544 1.538 1.500

WJmax

l-887 1.882 1.670 1.642

-17.5 -17.9 -7.9 -8.6

-75 -7.9 2.1 1.4

S, is the relative error consideringPoissonratio correction.

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Mechanics, Publishing House of Shanghai Transport University, 1987(in Chinese). 6. Xu Yigeng & Cai Shutao, Chinese J. of Applied Mechanics, 4 (1992), 124-127 (in Chinese). 7. Broekhoven, M. J. G. & van de Ruijtenbeek, M. G., Trans. 3rd Int. Conf Techn., 1975,G4/7.

on Structural

8. Broekhoven, M. J. G., Symposium

on

Fracture

Mech.

in Reactor

Proc. of the 9th National Mechanics, ASTM, 1976,

535-558. 9. Gu Shaode et al., The 6th Chinese Conf. on Structural Mech. in Reactor Techn., Beijing, 1990, 216-221 (in Chinese).