Thermal shock of a pipe with a partly circumferential surface crack

Engmeermg Fracrure Mechanics Vol. 2X. No. 3. pp 31W317. Printed in Great Britain.

0013-7944/87

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THERMAL SHOCK OF A PIPE WITH A PARTLY CIRCUMFERENTIAL SURFACE CRACK Brown,

Boveri

HANS GREBNER & Cie AC, Mannheim,

F.R.G.

Abstract-Stress intensity factors are calculated at the deepest point of partly circumferential surface cracks in a pipe loaded thermally and by internal pressure. The method of calculation is based on weight functions using the programm package “EASY” developed by Mattheck and Munz. Numerical values of the stress intensity factors are given for various crack depths and crack lengths in a pipe with an inner radius to wall thickness ratio of IO.

1. INTRODUCTION CIRCUMFERENTIAL surface cracks in piping systems-especially at welding joints-may cause a severe influence on the reliability of these pipes. Thermal transients occurring for example during an emergency cooling of the primary loop of a reactor can be the reason for even an instable growth of circumferentially oriented surface cracks. To decide whether unstable crack growth must be expected or not, or to calculate fatigue crack growth it is necessary to have knowledge of the stress intensity factors of the cracks under consideration. Stress intensity factors at the deepest point of circumferential surface cracks as shown in Fig. I may be calculated by means of a weight functions approach proposed by Morawietz, Mattheck and Munz[l]. Meanwhile the weight functions method has shown to be a very effective method for the calculation of stress intensity factors in problems with complex stress distributions. A number of papers by Mattheck and Munz eta/. (e.g. [2-41) demonstrates this very clearly. 2. THE WEIGHT FUNCTIONS As outlined in [l] the stress intensity equation of the weight functions method

K(u) =

factors

zI KR

METHOD

in interest

a

a(x)

2

can be evaluated

using

the basic

(1)

dx,

o

where KR and UR are the stress intensity factor and the crack opening displacement, respectively, of a reference solution with the same crack geometry and an arbitrary other loading. H is a constant and equal to E/( I - u’) for plane strain. KR and UR are derived in [l] by means of finite element calculations, and will not be repeated here.

SX

Fig. 1. Partly

circumferential

surface

crack

in a pipe

310

HANS v(x)

thermal

GREBNER

is the actual stress distribution of the untracked pipe. In this paper stresses and internal pressure, as described in the next section.

3. THE

TEMPERATURE OF THE

a(x)

is due to

AND STRESS DISTRIBUTION UNCRACKED PIPE

The thermal stresses produced in the primary loop of a reactor by an emergency cooling can be calculated conservatively by considering a thermoshock problem with a sudden cooling down of the inner surface of the pipe. The temperature distribution for this problem is gained by the well known solution of the heat conduction equation for problems with radial symmetry. The following initial and boundary conditions are assumed. Initial temperature: T( r,

7

S

RiSrS R,.

0) = T,, = 0;

(2)

Boundary conditions, (a) case I: T( Ri, T> 0) = Ti,

(3)

T>O) = T,,

(4)

T(R,, (b) case 2:

T(Ri,T>O)

= Ti,

(3

= 0.

(6)

aT(r, 4 ar Thereby The completely and 3 for The

r=R,

T denotes the time, and Ri and R, are the solutions of the boundary value problems described in [S] and will not be repeated a pipe with Ri= 50 mm, R, = 55 mm, K = axial stresses in rhe untracked pipe acting 2v

a,,(r,d=-

I-vR2,-R:

Fig. 2. Temperature

Ea

R. rT(r, IRI

inner and outer radius of the pipe. above are found in literature. They are here. Some results are presented in Figs 2 4.219 mm*/% To- Ti= 120 K and T, = 0. as loading of the crack faces are given by

ECY

2vR: T) dr + R;

_ R’

fi -G

distribution through the wall of the pipe for thermoshock 7= 1s; 5, 7+m). 7~0.2~; 3, 7=0.4s;4,

T(r’

case

‘I’

I (1, 7 = 0.05s; 2,

(7)

Thermal

shock

of a pipe

ill

t ITI

-80

50.

Fig. 3. Temperature

55.

[mm1

distribution through the wall of the pipe for thermoshock 7=0.4s; 3, T= 1.6s;4, 7=3.2s; 5, 7-+-m).

ease 2 (I, 7 = 0.05 s; 2.

r 5.

Fig. 4. Axial stresses

due to thermoshock

=zz t

case

1 and internal

mm

pressure

(notations

as in Fig.

2)

N rnme2

0. 50.

Fig. 5. Axial stresses

due to thermoshock

55.

case 2 and internal

mm

pressure

(notations

as in Fig. 3).

312

HANS GREBNER

where a is the coefficient of thermal expansion, pi is the internal pressure and T(r, T) is the temperature distribution described above. Figures 4 and 5 show axial stresses for different times, calculated with E = 200,000 Nmmb2, v = 0.3, CY= 1.2 x lo-” K-’ and pi = 200 bar, including the temperature distributions for the two different boundary conditions cases 1 and 2. Furthermore, for the evaluation of stress intensity factors there were also considered axial stresses resulting from the thermoshock case 1 alone, i.e. with pi = 0. This loading case is referred to as case 3 in the following sections. 4. CALCULATION

OF STRESS INTENSITY

FACTORS

Now introducing uZzzas a(x) in eq. (1) stress intensity factors can be calculated for various time-steps and different crack depths and lengths. According to [I] the crack depth to wall thickness ratio was varied between a/t = 0.2 and a/t = 0.8 and four crack lengths were considered, given by values of the crack angle cp of 10, 20, 30 and 40“. The evaluation of eq. (1) was done numerically using a computer programm which is part of the weight functions programm package “EASY”, developed by Mattheck and Munz[6]. Figures 6-16 present results of the stress intensity factor calculations.

1

t

“it I

0

0.5

1.

Fig. 6. Stress intensity factors as functions of crack depth for thermoshock case I and internal pressure and time-step T = 0.05 s (1, cp= IO”; 2, cp= 20”; 3, @= 30’; 4, v = 40”).

K t

0.

0.5

1

Fig. 7. Stress intensity factors as functions of crack depth for thermoshock case 1 and internal pressure at time-step 7 = 0.4 s (notations as in Fig. 6).

Thermal

shock

313

of a pipe

1600.

Fig. 8. Stress intensity

factors

as functions of crack depth for thermoshock at r- ~0 (notations as in Fig. 6).

K

case

I and internal pressure

Nmm-'

’

L

2LOO.

T 0.

0.5

1.

*

5

Fig. 9. Stress intensity factors as functions of time for thermoshock case I and internal pressure (1, cp = 10” and a/r = 0.2; 2, cp = 20” and a/t = 0.4; 3, (p = 30” and a/r = 0.6; 4, cp = 40” and a/r = 0.8).

I

K

N

mm-*2

2100.

:7 L

1604

3

2

800.

1

A

O/t 1

0. Fig.

10. Stress intensity factors pressure at timestep

I

0.5

1

-

1.

as functions of crack depth for thermoshock case 2 and r = 1.6 s (1, cp = IO”; 2, cp = 20”; 3, cp = 30”; 4, cp = 40’).

internal

314

HANS

K

N mm-

GREBNER

%

I

J-a/t

0.h

Fig.

11. Stress

intensity

factors

as functions

pressure

at ~+m

of crack depth for (notations

thermoshock

as in Fig.

case 2 and internal

10).

Figures 6-8 show stress intensity factors as functions of the crack depth for thermoshock case 1 including pi = 200 bar for T = 0.05 s, T = 0.4 s and T-+ ~0, respectively. Figure 9 gives the time dependence of the stress intensity factors for case 1 and 4 selected cracks, namely a/t = 0.2 and cp = lo”, a/t = 0.4 and cp = 20”, a/t = 0.6 and cp = 30” as well as a/t = 0.8 and cp = 40”. Figures 10 and 11 present stress intensity factors as functions of the crack depth for thermoshock case 2 including pi = 200 bar for T = 1.6 s and T+W, while Figs 12 and 13 show corresponding results for case 3 (thermoshock case 1 without internal pressure) for T = 0.05 s and T = 0.4 s respectively. Figures I4 and 15 show normalized stress intensity factors (or geometry functions) F(a/t), which are defined as usual by K = moJn-a F(a/t) with a, = 105 Nmm-* in our case. Figure 14 gives a comparison

K

500.

of F(a/t)

as a function

of the crack

depth

for cracks

42

L-Nmm-

(8)

t

250

0.

Fig.

12. Stress

intensity

0.5

1.

factors as functions of crack depth for thermoshock case 1 without pressure at timestep 7 = 0.05 s (notations as in Fig. 10).

internal

with

Thermal

K 1

Nmm

13. Stress

intensity

factors pressure

315

of a pipe

-92

0.

Fig.

shock

as functions at timestep

1

0.5

of crack depth for thermoshock case 7 = 0.4 s (notations as in Fig. IO).

Fig. 14. Normalized stress intensity factors as functions of crack thermoshock case 1 with (dashed lines) and without (solid lines) cp = 40”).

q = 10” and cp = 40” at the time r = 0.05 s for pressure, while Fig. IS shows the corresponding r+a. Figure 16 presents stress intensity factors cp = 40” and r+w. Besides the results of the shown for a pure pressure loading of the pipe

5. DISCUSSION

1 without

internal

depth at timestep r = 0.05 s for internal pressure (1, cp = lo”, 2,

thermoshock case 1 with and without results for all three cases considered

internal and for

as functions of the crack depth for cracks with three cases considered in this paper values are with pi = 200 bar and without thermal stresses.

OF THE

RESULTS

Though the temperature and stress distributions of the two thermoshock cases differ rather ‘strongly for greater time values, the behaviour of the stress intensity factors is qualitatively similar. The maximum values for case 2 however are about 30% higher than those of case 1. Due to the fact that in case 3 the cracks are extending from a region of tensile stress to a region with pressure With increasing crack depth, the stress intensity factors in this case partly show a steep decrease with increasing crack depth. As Fig. 16 shows the stress intensity factors evaluated for the thermoshock loading are up to a factor of six higher than the corresponding values for pressure loading only.

HANS

GREBNER

F 9.

6.

3.

“4 _

i

1.

Fig. 15. Normalized stress intensity factors as functions of crack depth at T+ m for thermoshock case 1 with (dashed lines) and without (solid lines) internal pressure as well as for thermoshock case 2 with internal pressure (dot-dashed lines; 1, cp = 10”; 2, cp = 40”).

0.

0.5

1.

Fig. 16. Stress intensity factors as functions of crack depth at ~+m for cp = 40” (1, internal pressure only; 2, thermoshock case 1 without internal pressure; 3, thermoshock case 1 with internal pressure; 4, thermoshock case 2 with internal pressure).

REFERENCES [l]P.

Morawietz, C. Mattheck and D. Munz, Stress intensity factor of partly cirumferential surface cracks at the inside of a pipe loaded arbitrarily. Trans. 8th Int. Conf. on Structural Mechanics in Reactor Technology, Brussels, Belgium (1985). Stress intensity factor for semi-elliptical surface cracks loaded by [2] C. Mattheck, D. Munz and H. Stamm, stress-gradients Engng Fracture Mech. 18, 633-641 (1983).

Thermal shock of a pipe

317

[3] C. Mattheck, P. Morawietz and D. Munz, Stress intensity factor at the surface and at the deepest point of a semi-elliptical surface crack in plates under stress gradients. Inr. J. Fracrure 23, 201-212 (1983). [4] C. Mattheck, P. Morawietz, D. Munz and H. Stamm, Comparison of different methods for the determination of stress intensity factors of cracks in pipes with stress gradients. _l. Press. Vess. Tech&. 106, 209-212 (1984). [5] H. Grebner and U. Strathmeier, Stress intensity factors for circumferential semi-elliptical surface cracks in a pipe under thermal loading. Engng Fracture Mech. 22, l-7 (1985). [6] C. Mattheck and D. Munz, Programmpaket “EASY”. Kernforschungszentrum Karlsruhe, IRB/ZSM. Universitlt Karlsruhe, Institut fiir Zuverllssigkeit und Schadenskunde. (Received

4 March

1987)