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Prediction of crack coalescence of steam generator tubes in nuclear power plants
Nuclear Engineering and Design 229 (2004) 175–187
Prediction of crack coalescence of steam generator tubes in nuclear power plants Jeries Abou-Hanna a,∗ , Timothy E. McGreevy b , Saurin Majumdar c a
Department of Mechanical Engineering, Bradley University, Peoria, IL 61625, USA b Oak Ridge National Laboratory, Oak Ridge, TN, USA c Argonne National Laboratory, Argonne, IL, USA
Received 8 April 2003; received in revised form 8 October 2003; accepted 3 November 2003
Abstract Prediction of failure pressures of cracked steam generator tubes of nuclear power plants is an important ingredient in scheduling inspection and repair of tubes. Prediction is usually based on nondestructive evaluation (NDE) of cracks. NDE often reveals two neighboring cracks. If the cracks interact, the tube pressure under which the ligament between the two cracks fails could be much lower than the critical burst pressure of an individual equivalent crack. The ability to accurately predict the ligament failure pressure, called “coalescence pressure,” is important. The failure criterion was established by nonlinear finite element model (FEM) analyses of coalescence of two 100% through-wall collinear cracks. The ligament failure is precipitated by local instability of the ligament under plane strain conditions. As a result of this local instability, the ligament thickness in the radial direction decreases abruptly with pressure. Good correlation of FEM analysis results with experimental data obtained at Argonne National Laboratory’s Energy Technology Division demonstrated that nonlinear FEM analyses are capable of predicting the coalescence pressure accurately for 100% through-wall cracks. This failure criterion and FEA work have been extended to axial cracks of varying ligament width, crack length, and cases where cracks are offset by axial or circumferential ligaments. © 2003 Elsevier B.V. All rights reserved.
1. Background Prediction of crack failure in steam generator tubes (SG tubes) of nuclear power plants is an important ingredient in scheduling inspection and repair of tubes. SG tubes account for more than 50% of the primary pressure boundary surfaces in pressurized water reactor (PWR) nuclear power plants (http://www.et.anl. gov/sections/corrosionmm/research/steamgen.html). These tubes experience corrosive and mechanical degradation during service, eventually resulting in ∗ Corresponding author. Tel.: +1-309-677-2725; fax: +1-309-677-3453. E-mail address: [email protected] (J. Abou-Hanna).
tube cracking. Single and multiple cracks in close proximity are observed. When cracks become large enough, two failure modes can occur: (1) the tubes either burst (fail mechanically), or (2) the crack opening area becomes sufficiently large to consider the leak rate unacceptable. The SG tube is then “plugged” and the SG continues to operate at a reduced efficiency. Eventually, a sufficient number of tubes fail and replacement of the SG is required. Replacement costs range from $100 to $200 million dollars per SG; nearly 70 SGs in 22 US plants were scheduled for replacement by 1998 with additional ones expected. Hence, the ability to predict under what conditions these cracks will fail is essential, both from safety and cost standpoints.
0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2003.11.011
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Prediction of tube failure is usually based on nondestructive testing of cracks. Rupture and leak rates for single crack configurations in SG tubes have been well established (Cochet and Flesch, 1987; Gorman et al., 1995; Hahn et al., 1995/1996; Erdogan, 1976; Majumdar, 1999; Eiber et al., 1971; Alzheimer et al., 1979). However, nondestructive testing may reveal and report two neighboring but apparently independent cracks. The tube pressure under which the ligament (material between the two cracks) fails could be much lower than the critical burst pressure of an individual crack equal in size to the sum of the two cracks and ligament; these conditions have not been well established as most efforts to date have not addressed coalescence. Thus, capabilities to predict the pressure causing failure of the ligament between any two neighboring cracks (coalescence criterion) and the corresponding crack opening area evolution with pressure are essential. Preliminary nonlinear finite element studies of crack coalescence of two in-line cracks (Analysis and Testing of Rupture of Steam Generator Tubing with Flaws, 2001) demonstrated that the finite element model (FEM) nonlinear models are capable of predicting coalescence criteria fairly accurately for 100% through-wall (TW) cracks, as opposed to the flow stress criterion used for single crack case (Lee et al., 2001). The ligament rupture starts when local instability of the ligament occurs under plane strain conditions. As a result of this local instability, the ligament thickness in the radial direction reduces drastically. The thickness/pressure gradient becomes very high and the ligament is no longer capable of resisting the applied tube pressure. A picture of an actual SG tube that failed from coalescence of two cracks is illustrated in Fig. 1; this paper further investigates the coalescence of such cracks.
Fig. 1. Coalescence of cracks from actual SG tube. Table 2 SG tube material properties (Alloy 600) Stress (MPa)
Plastic strain (mm/mm)
300 512 687 840 946 1125 1700
0 0.09274 0.178865 0.2582 0.33177 0.5 1.05
Modulus of elasticity (GPa) Poisson’s ratio
200 0.3
2. Crack coalescence modeling This paper presents results of two collinear cracks offset axially or circumferentially (Type 2 and Type 4, respectively) as outlined in Table 1. The numerical study considered 100% TW cracks, 1.27 mm thick tubes, 22 mm tube diameter, stainless steel (Alloy 600) material with properties specified in Table 2, and a crack tip radius of 0.05 mm. Fig. 2 illustrates the crack configurations. Experiments were conducted on tubes with 70, 80, and 100% TW cracks and compared to the finite element predictions.
Table 1 Analysis matrix Specimen type
No. of notches
Notch length, mm (in.)
Ligament width, mm (in.)
Type Type Type Type Type Type
2 2 2 2 2 2
6.35 (0.25) 12.7 (0.5) 25.4 (1) 6.35 (0.25) 12.7 (0.5) 25.4 (1)
0.254/1.27/2.54 (0.01/0.05/0.1) 0.254/1.27/2.54 (0.01/0.05/0.1) 0.254/1.27/5.08 (0.01/0.05/0.2) 0.254/1.27/2.54 (0.01/0.05/0.1) 0.152/0.254/1.27/2.54 (0.006/0.01/0.05/0.1) 0.152/0.254/1.27/2.54/5.08 (0.006/0.01/0.05/0.1/0.2)
2 2 2 4 4 4
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Notch Length Ligament Width
177
100% Through-wall notch
Type 2 Axial Ligament
Ligament width Notch Length
Type 4 Circumferential Ligament Fig. 2. Types 2 and 4 specimens and crack configurations.
3. Finite element model features and outputs Fig. 3a–d shows a typical FEM for the Type 2 crack configuration—(a) the overall model, (b) the ligament region and one notch, (c) the ligament and the notch tip, and finally (d) an end view of the tube illustrat-
ing typical tube deformation under internal pressure. The model consists of full 3-D shell five degree of freedom reduced integration finite elements. Similar models were constructed for Type 4 analysis; however, half symmetry required as opposed to quarter symmetry.
Fig. 3. Typical Type 2 FE mesh and corresponding overall deformation.
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The end conditions, far removed from the crack location, were such that the tube segment was free to expand radially and axially. The tube inner surface was subjected to an increasing pressure, while the longitudinal pressure was captured by applying pressure to a membrane on the far-field free end of the tube segment. Outputs from the finite elements consisted of (1) the average tube thickness in the ligament area, and (2) the displaced coordinates of the crack edge. The ligament thickness was averaged based on thickness of a selected group of finite elements that lie in the observed thinnest portion of the ligament. Results consisted of the crack opening profile, crack opening area, and the average ligament thickness versus tube pressure.
4. Case identification scheme (nomenclature) Each case is identified as Tx-n-c-l. Here, x, n, c and l are variables as follows: x crack type number (2 for Type 2, 4 for Type 4), n number of notches in a crack, c notch length, and l ligament width. The values assigned to the notch length (c) and ligament width (l) represent their dimension in thousandths of inches. The models and their results however are all presented in metric units. For example, case ID T2-2-500-010 means:
T2 means Type 2 (which is the only type presented in this paper) -2 means two (2) notches, -500 means a notch length of 12.7 mm (0.500 in.), -010 means a ligament width of 0.254 mm (0.010 in.).
5. Typical behavior of SG tubes under pressure The instability condition of the ligament is depicted by the drastic reduction in ligament average thickness, as shown in Fig. 4 for several Type 2 cases; all other cases including Type 4 cases behave similarly. The coalescence pressure (Pcoal ) is defined at the point where the rate of change of thickness to the applied pressure becomes significantly steep. This point was defined at the intersection of the two linear thickness profiles as shown; the steep thickness/pressure gradient being indicative of ligament rupture. After coalescence, FEM instability occurs, and the simulation is terminated; this is referred to as the terminal pressure condition. Fig. 5 shows typical crack opening profiles (COP) for Type 2 cracks at terminal pressure conditions (only the top half of one notch or crack is shown); note the location of the ligament. Fig. 6 illustrates the progression in COP as a function of tube pressure for case T2-2-250-100. Fig. 7 illustrates a typical case (T2-2-1000) where an individual crack bursts or ruptures, coalescence does not occur. In this case, the COP evolves symmetrically as a single
1.30
Thickness (mm)
1.20 1.10 1.00
T2-2-250-100
0.90
T2-2-250-050 T2-2-250-010
0.80 0.70 0.60 0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
Pressure (MPa) Fig. 4. Average ligament thickness profile and Pcoal for T2-2-250 cases.
40.00
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179
0.60
Minor Coordinate (mm)
0.50 0.40 0.30 0.20 T2-2-250-100 0.10
Ligament location
T2-2-250-050 T2-2-250-010
0.00 62
63
64
65
66
67
68
69
70
Major Coordinate (mm) Fig. 5. COP at terminal conditions for T2-2-250 cases.
crack would, as opposed to the asymmetric COP of a crack interacting with a neighboring crack as shown in Fig. 6. Figs. 4–7 exemplify all Type 2 cases presented herein. Fig. 8 illustrates a typical deformed mesh for a Type 4 case, and Fig. 9 illustrates the reduction in ligament
thickness with pressure for four Type 4 cases. COP of an entire notch for the same four cases are shown in Fig. 10; the ligament location is left of the given COPs. All trends are similar to that of Type 2; however, due to the crack configuration, the COP for Type 4 cracks are asymmetric, as expected.
0.60 T2-2-250-100
17 MPa
Minor Coordinate (mm)
0.50 22 MPa 0.40
25 MPa 31 MPa
0.30
34 MPa
0.20
Ligament location
0.10
0.00 -2
0
2
4
6
Major Coordinate (mm) Fig. 6. COP vs. pressure for T2-250-100 case.
8
10
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Minor Coordinate (mm)
3.00
T2-2-1000-200 T2-2-1000-050 T2-2-1000-010
2.50 2.00 1.50 1.00
Ligament location
0.50 0.00 -10.00
0.00 10.00 20.00 Major Coordinate (mm)
30.00
Fig. 7. COP at terminal conditions for T2-2-1000 cases.
6. Crack coalescence pressure Table 3 lists the coalescence pressure (Pcoal ) and the normalized coalescence pressure, which was obtained as the ratio of Pcoal to the critical pressure or burst pressure (Pcr ) of a single equivalent crack (SEC). The burst pressure for a single crack is based on (Majumdar, 1999; Eiber et al., 1971; Alzheimer et al., 1979). For small ligaments, Pcoal is less than Pcr . However, larger ligaments strengthen the tube, increasing Pcoal , eventually requiring larger pressures for ligament rupture than that required for a SEC to rupture. Several crack configurations examined in this study never coalesced and are denoted by NF in Table 3;
Fig. 8. Typical Type 4 FEM.
1.20
Thickness (mm)
1.00 0.80 T4-2-500-006 T4-2-500-050 T4-2-500-100 T4-2-500-010
0.60 0.40 0.20 0.00
5.00
10.00
15.00
20.00
25.00
30.00
Pressure (MPa)
Fig. 9. Average ligament thickness profile and Pcoal for T4-2-500 cases.
35.00
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181
-4.0
Minor Coordinate (mm)
-3.0 -2.0 -1.0 0.0 1.0 2.0
T4-2-500-006 T4-2-500-010 T4-2-500-050 T4-2-500-100
3.0 4.0 24
26
28
30
32
34
36
38
40
Major Coordinate (mm)
Fig. 10. COP at terminal conditions for T4-2-500 cases.
NA denotes cases for which Pcr was unavailable. Ligament size for which normalized Pcoal equals unity is defined as the transitional ligament size. Type 2 crack configurations are observed to have a transitional ligament size of 2.54 mm, whereas those of Type 4 are observed to be only 1.27 mm. Table 3 Crack coalescence pressure and normalized pressure with respect to burst pressure of SEC Type and case
Pcoal (MPa)
Normalized Pcoal
T2-2-250-010 T2-2-250-050 T2-2-250-100 T2-2-500-010 T2-2-500-050 T2-2-500-100 T2-2-1000-010 T2-2-1000-050 T2-2-1000-100 T4-2-250-010 T4-2-250-050 T4-2-250-100 T4-2-500-006 T4-2-500-010 T4-2-500-050 T4-2-500-100 T4-2-1000-006 T4-2-1000-010 T4-2-1000-050 T4-2-1000-100 T4-2-1000-200
19.50 24.00 30.50 6.00 13.25 18.50 0.20 5.20 14.20 17.00 35.00 38.00 7.00 9.00 17.00 27.00 1.80 3.00 7.00 NF NF
0.65 0.80 1.01 0.34 0.75 1.05 0.02 0.54 1.47 0.56 1.16 1.26 0.40 0.51 0.96 1.53 0.19 0.31 0.72 NA NA
The predicted results for cases T2-250 produce higher Pcoal than predicted by Lee et al. (2001), as is illustrated in Fig. 11. Two experimental test results are included in Fig. 11 for ligament widths of 0.254 mm, 100% TW cracks, and temperatures of 282 and 23 ◦ C. The test at room temperature did not cause rupture of the ligament or failure of either individual crack at a pressure of 17.2 MPa; higher pressures could not be applied due to limitations in feasible flow rates since the cracks were 100% TW. However, the test at 282 ◦ C did fail at a pressure of 15.5 MPa. Since the yield strength of Alloy 600 at 282 ◦ C is only 10% less than that at room temperature, the room temperature test can readily be deduced to have been near failure.
Fig. 11. Comparison of predictions for different failure criterion.
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40
T2-2-250M
70%
Coalescence Pressure (MPa)
35
T2-2-250
80% T2-2-250 Experiment
30 80%
T2-2-500
25
T2-2-500 Experiment
20
T2-2-1000
100%- No Rupture 100%, 282C
15 10 5 0
1
0
2
3 4 Ligament Width (mm)
5
6
Fig. 12. Crack coalescence pressure vs. ligament width (Type 2).
The predicted Pcoal of 17.5 MPa agrees well with those observed, while results of Lee et al. are conservative by about a factor of 2. A rupture criterion based on plastic instability of the ligament is clearly seen to be more appropriate than one based on flow stress as used by Lee et al. Fig. 12 illustrates how Pcoal increases with ligament size for various notch lengths for Type 2 cracks. The rate of increase of Pcoal diminishes with the increase in ligament width. T2-2-250M predictions are additional cases where the notch length was 6 mm rather than the nominal 6.35 mm. Additional tests conducted on 70% (T2-2-250) and 80% (T2-2-500) TW cracks are also
shown in Fig. 12, along with the two tests from Fig. 11. Results support the predicted coalescence pressure for 100% TW cracks; Pcoal observed for 70 and 80% cases are consistently higher than that predicted for 100% TW cases, as expected. Modeling efforts for Type 2 partial TW cracks are underway. Ligament size effect on Pcoal for Type 4 cracks is shown in Fig. 13 for various notch lengths. Again, the rate of increase of Pcoal diminishes with the increase in ligament width. Experiment results are also illustrated in Fig. 13 for 6.35 and 12.7 mm notches with various ligament widths. Note that all tests were conducted on 80% TW cracks. Pcoal observed in experiments either
45 T4-2-250
40 Pressure (MPa)
35
T4-2-500
30 T4-2-1000
25 20
T4-2-250 Experiment
15 10
T4-2-500 Experiment
5
All tests 80% TW
0 0
1
2
3
Ligament Width (mm) Fig. 13. Crack coalescence pressure vs. ligament width (Type 4).
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183
40 T2-2-250 Coalescence Pressure (MPa)
35
T2-2-500 T2-2-1000
30
T4-2-250
25
T4-2-500 20
T4-2-1000
15 10 5 0 0
1
3 2 4 Ligament Width (mm)
5
6
Fig. 14. Coalescence pressure: Type 2 vs. Type 4.
Table 4 Crack opening area of Type 2 cases Area (mm2 )
Area-M (mm2 )
SCA (mm2 )
Type and case
Pressure (MPa)
T2-2-250-010
7 17.225 20∗
2.41 3.34 4.25
0.124 1.054 1.964
0.273 2.25 3.89
0.45 0.47 0.50
T2-2-250-050
7 17.225 24∗
2.34 3.06 5.27
0.054 0.774 2.984
0.273 2.25 8
0.20 0.34 0.37
T2-2-250-100
7 17.225 31∗
2.33 2.75 8.3
0.044 0.464 6.014
0.273 2.25 29.25
0.16 0.21 0.21
4.9 5.55 6.5
0.328 0.978 1.928
0.7 1.84 3.63
0.47 0.53 0.53
Normalized COA (Area-M/SCA)
T2-2-500-010
2 4 6∗
T2-2-500-050
7 10 13∗
5.7 7.5 11.4
1.128 2.928 6.828
4.99 15 45.3
0.23 0.20 0.15
T2-2-500-100
7 17.225 18.5∗
5.1 15.7 19.6
0.528 11.128 15.028
4.99 314 610
0.11 0.04 0.02
T2-2-1000-050
2 3 5.2∗
10.7 12.6 20.6
8.16 10.06 18.06
13.60 36.00 508.00
0.60 0.28 0.04
T2-2-1000-200
7 10 14.2∗
15.46 48.46 146.46
6263.00 BR BR
0.00 NA NA
18 51 149
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exceed or are approximately equal to the predictions for 100% TW cases. Further testing and modeling efforts are underway to determine how sensitive Pcoal is as TW cracks vary from 70 to 100%. Currently, the trends and magnitudes of the experimental results appear consistent with the predictions. Fig. 14 compares Pcoal of Types 2 and 4. Pcoal for Type 4 configurations are consistently higher than for Type 2 configurations for all ligaments investigated, with the exception of T2-2-250/T4-2-250 and a 0.254 mm ligament. Type 4 cases are increasingly more resistant to coalescence than Type 2 for larger ligaments.
7. Crack opening area Crack opening area is directly related to the leak rate of a cracked tube. Table 4 lists the crack opening area (COA) as a function of pressure for Type 2 cases; Table 5 lists COA for Type 4 cases. In the second column, pressure values with an asterisk indicate coalescence, Pcoal . Area corresponds to the total sum of all notch areas, and Area-M is the total area minus the initial crack area (not shown). Single crack area (SCA), the COA of a single equivalent crack (SEC) arrived at with ANL’s COA mathematical model that assumes an initial crack area of zero (Majumdar, 1999; Eiber
Table 5 Crack opening area for Type 4 cases Area (mm2 )
Area-M (mm2 )
SCA (mm2 )
Type and case
Pressure (MPa)
T4-2-250-010
7 10 14∗
1.48 1.65 2.00
0.21 0.38 0.73
0.27 0.53 1.17
0.77 0.71 0.62
T4-2-250-050
7 17.5 35∗
1.50 3.24 26.90
0.23 1.97 25.63
0.27 2.25 59.43
0.84 0.88 0.43
T4-2-250-100
7 17.5 38∗
0.12 0.83 14.43
0.12 0.83 14.43
0.27 2.25 101.60
0.42 0.37 0.14
T4-2-500-010
4 7 9∗
3.50 5.60 8.83
0.96 3.06 6.29
1.73 4.99 10.37
0.55 0.61 0.61
T4-2-500-050
7 10 17.5∗
4.60 7.60 38.40
2.06 5.06 35.86
4.99 15.00 314.00
0.41 0.34 0.11
T4-2-500-100
7 17.5 27∗
3.90 25.00 81.00
1.36 22.46 78.46
4.99 314.00 BR
0.27 0.07 NA
T4-2-1000-010
1 2 3∗
7.30 10.90 15.90
2.22 5.82 10.82
5.00 13.60 36.00
0.44 0.43 0.30
T4-2-1000-050
3 5 7∗
11.40 22.60 50.00
6.32 17.52 44.92
36.00 389.00 6263.00
0.18 0.05 0.01
T4-2-1000-100
7 10∗
34.00 86.00
28.92 80.92
6263.00 BR
0.00 NA
T4-2-1000-200
7 10 15∗
22.00 74.00 210.00
16.92 68.92 204.92
6263.00 BR BR
0.00 NA NA
Normalized COA (Area-M/SCA)
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185
30 T4-2-250-010 T4-2-250-050
COA, Area-M (mm2)
25
T4-2-250-100 20
T2-2-250-010 T2-2-250-050
15
T2-2-250-100
10
5 0 0
10
20 Pressure (MPa)
30
40
Fig. 15. COA for T2-2-250 and T4-2-250 cracks.
et al., 1971; Alzheimer et al., 1979), is also given. The SEC is assumed to be the sum of both notch lengths and the ligament width for Type 2 cases; for Type 4 cases, the ligament is not considered in the SEC length. In several cases, the pressure is larger than the burst pressure for a SEC and is denoted by BR; correspondingly the normalized area is not applicable (NA). The normalized COA is observed to be a fraction since the ligament effectively stiffens the crack configuration. Fig. 15 illustrates the evolution of the predicted COA as a function of pressure for both T2-2-250 and T4-2-250 cases. Initially, COA increases only slightly with pressure to about 17 MPa; thereafter the COA increases at a much higher rate until Pcoal occurs, where Pcoal varies with ligament width
and crack size. Upon reaching Pcoal , the ligament ruptures, and the rate of change in COA is tremendous. Observation of COA in experiments are consistent with this rate of change of COA as illustrated in Fig. 16; COA was observed to change radically with only a slight change in pressure (31.4–32.5 MPa). Differences in COA for Types 2 and 4 cases are illustrated in Fig. 17. When the ligament is small, Types 2 and 4 COA are approximately equal. Since Type 2 cracks coalesce at lower pressures, Type 4 COA will increase stably rather than bursting as for the Type 2 case as indicated in Fig. 17. While Type 4 configurations were observed to have superior resistance to coalescence, simulations reveal that larger ligament widths result in significantly larger COA for Type 4 ligaments than Type 2, as illustrated in Figs. 15
Fig. 16. Observed increase in COA from 31.4 to 32.5 MPa (T4-2-250-050, 80% TW).
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14
T4-2-500-010 T4-2-500-050
COA, Area-M (mm2)
12
T4-2-500-100 T2-2-500-010
10
T2-2-500-050 8
T2-2-500-100
6 4 2 0 0
5
10
15
20
Pressure (MPa) Fig. 17. COA: Type 2 vs. Type 4.
and 17. Furthermore, the differences in COA increase with increasing pressure.
8. Observations and conclusions Nonlinear finite element modeling of collinear cracks with axial and circumferential ligaments were conducted to predict pressures for which coalescence would occur; several experiments were conducted which support and verify the predicted coalescence pressure (Pcoal ). Pcoal for both Types 2 and 4 crack configurations increases with increasing ligament width; however, the rate of increase of Pcoal diminishes with increasing ligament width. A transitional ligament width was predicted to be 2.54 mm for Type 2 crack configurations, i.e. predicted Pcoal is less than the critical pressure (Pcr ) for a single equivalent crack if ligaments are smaller than 2.54 mm. Coalescence was predicted not to occur when ligaments are larger than the transitional width. Collinear cracks with circumferential ligaments (Type 4) have a transitional ligament width of approximately half that of collinear cracks with axial ligaments (Type 2), 1.27 mm versus 2.54 mm, respectively. Comparing Types 2 and 4 crack configurations the authors have shown that Type 4 configurations are more resistant to coalescence than
Type 2 configurations, for a given ligament width. These results will be critical in assisting engineers with maintenance scheduling for SG tubes of PWRs. In addition to coalescence conditions, crack opening areas were also investigated since failure of SG tubes may occur when excessive leak rates are experienced. Simulations reveal that under internal pressure loading, all axial and circumferential ligaments in this study stiffened the crack opening, resulting in smaller COA than that of a single equivalent crack, and increasing the ligament width further stiffens the crack opening, further reducing the COA. Crack openings for collinear cracks with axial ligaments (Type 2) are smaller than those with circumferential ligaments (Type 4). Hence, while Type 4 crack configurations may exhibit superior resistance to coalescence, they are inferior to Type 2 crack configurations with respect to COA. Numerous Type 4 cracks are observed in a single tube. As such, total leak rates may be unacceptable, possibly requiring a tube to be plugged (removed from service) earlier than expected.
Acknowledgements This work was supported by United States Nuclear Regulatory Commission.
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Prediction of crack coalescence of steam generator tubes in nuclear power plants Jeries Abou-Hanna a,∗ , Timothy E. McGreevy b , Saurin Majumdar c a
Department of Mechanical Engineering, Bradley University, Peoria, IL 61625, USA b Oak Ridge National Laboratory, Oak Ridge, TN, USA c Argonne National Laboratory, Argonne, IL, USA
Received 8 April 2003; received in revised form 8 October 2003; accepted 3 November 2003
Abstract Prediction of failure pressures of cracked steam generator tubes of nuclear power plants is an important ingredient in scheduling inspection and repair of tubes. Prediction is usually based on nondestructive evaluation (NDE) of cracks. NDE often reveals two neighboring cracks. If the cracks interact, the tube pressure under which the ligament between the two cracks fails could be much lower than the critical burst pressure of an individual equivalent crack. The ability to accurately predict the ligament failure pressure, called “coalescence pressure,” is important. The failure criterion was established by nonlinear finite element model (FEM) analyses of coalescence of two 100% through-wall collinear cracks. The ligament failure is precipitated by local instability of the ligament under plane strain conditions. As a result of this local instability, the ligament thickness in the radial direction decreases abruptly with pressure. Good correlation of FEM analysis results with experimental data obtained at Argonne National Laboratory’s Energy Technology Division demonstrated that nonlinear FEM analyses are capable of predicting the coalescence pressure accurately for 100% through-wall cracks. This failure criterion and FEA work have been extended to axial cracks of varying ligament width, crack length, and cases where cracks are offset by axial or circumferential ligaments. © 2003 Elsevier B.V. All rights reserved.
1. Background Prediction of crack failure in steam generator tubes (SG tubes) of nuclear power plants is an important ingredient in scheduling inspection and repair of tubes. SG tubes account for more than 50% of the primary pressure boundary surfaces in pressurized water reactor (PWR) nuclear power plants (http://www.et.anl. gov/sections/corrosionmm/research/steamgen.html). These tubes experience corrosive and mechanical degradation during service, eventually resulting in ∗ Corresponding author. Tel.: +1-309-677-2725; fax: +1-309-677-3453. E-mail address: [email protected] (J. Abou-Hanna).
tube cracking. Single and multiple cracks in close proximity are observed. When cracks become large enough, two failure modes can occur: (1) the tubes either burst (fail mechanically), or (2) the crack opening area becomes sufficiently large to consider the leak rate unacceptable. The SG tube is then “plugged” and the SG continues to operate at a reduced efficiency. Eventually, a sufficient number of tubes fail and replacement of the SG is required. Replacement costs range from $100 to $200 million dollars per SG; nearly 70 SGs in 22 US plants were scheduled for replacement by 1998 with additional ones expected. Hence, the ability to predict under what conditions these cracks will fail is essential, both from safety and cost standpoints.
0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2003.11.011
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Prediction of tube failure is usually based on nondestructive testing of cracks. Rupture and leak rates for single crack configurations in SG tubes have been well established (Cochet and Flesch, 1987; Gorman et al., 1995; Hahn et al., 1995/1996; Erdogan, 1976; Majumdar, 1999; Eiber et al., 1971; Alzheimer et al., 1979). However, nondestructive testing may reveal and report two neighboring but apparently independent cracks. The tube pressure under which the ligament (material between the two cracks) fails could be much lower than the critical burst pressure of an individual crack equal in size to the sum of the two cracks and ligament; these conditions have not been well established as most efforts to date have not addressed coalescence. Thus, capabilities to predict the pressure causing failure of the ligament between any two neighboring cracks (coalescence criterion) and the corresponding crack opening area evolution with pressure are essential. Preliminary nonlinear finite element studies of crack coalescence of two in-line cracks (Analysis and Testing of Rupture of Steam Generator Tubing with Flaws, 2001) demonstrated that the finite element model (FEM) nonlinear models are capable of predicting coalescence criteria fairly accurately for 100% through-wall (TW) cracks, as opposed to the flow stress criterion used for single crack case (Lee et al., 2001). The ligament rupture starts when local instability of the ligament occurs under plane strain conditions. As a result of this local instability, the ligament thickness in the radial direction reduces drastically. The thickness/pressure gradient becomes very high and the ligament is no longer capable of resisting the applied tube pressure. A picture of an actual SG tube that failed from coalescence of two cracks is illustrated in Fig. 1; this paper further investigates the coalescence of such cracks.
Fig. 1. Coalescence of cracks from actual SG tube. Table 2 SG tube material properties (Alloy 600) Stress (MPa)
Plastic strain (mm/mm)
300 512 687 840 946 1125 1700
0 0.09274 0.178865 0.2582 0.33177 0.5 1.05
Modulus of elasticity (GPa) Poisson’s ratio
200 0.3
2. Crack coalescence modeling This paper presents results of two collinear cracks offset axially or circumferentially (Type 2 and Type 4, respectively) as outlined in Table 1. The numerical study considered 100% TW cracks, 1.27 mm thick tubes, 22 mm tube diameter, stainless steel (Alloy 600) material with properties specified in Table 2, and a crack tip radius of 0.05 mm. Fig. 2 illustrates the crack configurations. Experiments were conducted on tubes with 70, 80, and 100% TW cracks and compared to the finite element predictions.
Table 1 Analysis matrix Specimen type
No. of notches
Notch length, mm (in.)
Ligament width, mm (in.)
Type Type Type Type Type Type
2 2 2 2 2 2
6.35 (0.25) 12.7 (0.5) 25.4 (1) 6.35 (0.25) 12.7 (0.5) 25.4 (1)
0.254/1.27/2.54 (0.01/0.05/0.1) 0.254/1.27/2.54 (0.01/0.05/0.1) 0.254/1.27/5.08 (0.01/0.05/0.2) 0.254/1.27/2.54 (0.01/0.05/0.1) 0.152/0.254/1.27/2.54 (0.006/0.01/0.05/0.1) 0.152/0.254/1.27/2.54/5.08 (0.006/0.01/0.05/0.1/0.2)
2 2 2 4 4 4
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Notch Length Ligament Width
177
100% Through-wall notch
Type 2 Axial Ligament
Ligament width Notch Length
Type 4 Circumferential Ligament Fig. 2. Types 2 and 4 specimens and crack configurations.
3. Finite element model features and outputs Fig. 3a–d shows a typical FEM for the Type 2 crack configuration—(a) the overall model, (b) the ligament region and one notch, (c) the ligament and the notch tip, and finally (d) an end view of the tube illustrat-
ing typical tube deformation under internal pressure. The model consists of full 3-D shell five degree of freedom reduced integration finite elements. Similar models were constructed for Type 4 analysis; however, half symmetry required as opposed to quarter symmetry.
Fig. 3. Typical Type 2 FE mesh and corresponding overall deformation.
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The end conditions, far removed from the crack location, were such that the tube segment was free to expand radially and axially. The tube inner surface was subjected to an increasing pressure, while the longitudinal pressure was captured by applying pressure to a membrane on the far-field free end of the tube segment. Outputs from the finite elements consisted of (1) the average tube thickness in the ligament area, and (2) the displaced coordinates of the crack edge. The ligament thickness was averaged based on thickness of a selected group of finite elements that lie in the observed thinnest portion of the ligament. Results consisted of the crack opening profile, crack opening area, and the average ligament thickness versus tube pressure.
4. Case identification scheme (nomenclature) Each case is identified as Tx-n-c-l. Here, x, n, c and l are variables as follows: x crack type number (2 for Type 2, 4 for Type 4), n number of notches in a crack, c notch length, and l ligament width. The values assigned to the notch length (c) and ligament width (l) represent their dimension in thousandths of inches. The models and their results however are all presented in metric units. For example, case ID T2-2-500-010 means:
T2 means Type 2 (which is the only type presented in this paper) -2 means two (2) notches, -500 means a notch length of 12.7 mm (0.500 in.), -010 means a ligament width of 0.254 mm (0.010 in.).
5. Typical behavior of SG tubes under pressure The instability condition of the ligament is depicted by the drastic reduction in ligament average thickness, as shown in Fig. 4 for several Type 2 cases; all other cases including Type 4 cases behave similarly. The coalescence pressure (Pcoal ) is defined at the point where the rate of change of thickness to the applied pressure becomes significantly steep. This point was defined at the intersection of the two linear thickness profiles as shown; the steep thickness/pressure gradient being indicative of ligament rupture. After coalescence, FEM instability occurs, and the simulation is terminated; this is referred to as the terminal pressure condition. Fig. 5 shows typical crack opening profiles (COP) for Type 2 cracks at terminal pressure conditions (only the top half of one notch or crack is shown); note the location of the ligament. Fig. 6 illustrates the progression in COP as a function of tube pressure for case T2-2-250-100. Fig. 7 illustrates a typical case (T2-2-1000) where an individual crack bursts or ruptures, coalescence does not occur. In this case, the COP evolves symmetrically as a single
1.30
Thickness (mm)
1.20 1.10 1.00
T2-2-250-100
0.90
T2-2-250-050 T2-2-250-010
0.80 0.70 0.60 0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
Pressure (MPa) Fig. 4. Average ligament thickness profile and Pcoal for T2-2-250 cases.
40.00
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179
0.60
Minor Coordinate (mm)
0.50 0.40 0.30 0.20 T2-2-250-100 0.10
Ligament location
T2-2-250-050 T2-2-250-010
0.00 62
63
64
65
66
67
68
69
70
Major Coordinate (mm) Fig. 5. COP at terminal conditions for T2-2-250 cases.
crack would, as opposed to the asymmetric COP of a crack interacting with a neighboring crack as shown in Fig. 6. Figs. 4–7 exemplify all Type 2 cases presented herein. Fig. 8 illustrates a typical deformed mesh for a Type 4 case, and Fig. 9 illustrates the reduction in ligament
thickness with pressure for four Type 4 cases. COP of an entire notch for the same four cases are shown in Fig. 10; the ligament location is left of the given COPs. All trends are similar to that of Type 2; however, due to the crack configuration, the COP for Type 4 cracks are asymmetric, as expected.
0.60 T2-2-250-100
17 MPa
Minor Coordinate (mm)
0.50 22 MPa 0.40
25 MPa 31 MPa
0.30
34 MPa
0.20
Ligament location
0.10
0.00 -2
0
2
4
6
Major Coordinate (mm) Fig. 6. COP vs. pressure for T2-250-100 case.
8
10
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Minor Coordinate (mm)
3.00
T2-2-1000-200 T2-2-1000-050 T2-2-1000-010
2.50 2.00 1.50 1.00
Ligament location
0.50 0.00 -10.00
0.00 10.00 20.00 Major Coordinate (mm)
30.00
Fig. 7. COP at terminal conditions for T2-2-1000 cases.
6. Crack coalescence pressure Table 3 lists the coalescence pressure (Pcoal ) and the normalized coalescence pressure, which was obtained as the ratio of Pcoal to the critical pressure or burst pressure (Pcr ) of a single equivalent crack (SEC). The burst pressure for a single crack is based on (Majumdar, 1999; Eiber et al., 1971; Alzheimer et al., 1979). For small ligaments, Pcoal is less than Pcr . However, larger ligaments strengthen the tube, increasing Pcoal , eventually requiring larger pressures for ligament rupture than that required for a SEC to rupture. Several crack configurations examined in this study never coalesced and are denoted by NF in Table 3;
Fig. 8. Typical Type 4 FEM.
1.20
Thickness (mm)
1.00 0.80 T4-2-500-006 T4-2-500-050 T4-2-500-100 T4-2-500-010
0.60 0.40 0.20 0.00
5.00
10.00
15.00
20.00
25.00
30.00
Pressure (MPa)
Fig. 9. Average ligament thickness profile and Pcoal for T4-2-500 cases.
35.00
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181
-4.0
Minor Coordinate (mm)
-3.0 -2.0 -1.0 0.0 1.0 2.0
T4-2-500-006 T4-2-500-010 T4-2-500-050 T4-2-500-100
3.0 4.0 24
26
28
30
32
34
36
38
40
Major Coordinate (mm)
Fig. 10. COP at terminal conditions for T4-2-500 cases.
NA denotes cases for which Pcr was unavailable. Ligament size for which normalized Pcoal equals unity is defined as the transitional ligament size. Type 2 crack configurations are observed to have a transitional ligament size of 2.54 mm, whereas those of Type 4 are observed to be only 1.27 mm. Table 3 Crack coalescence pressure and normalized pressure with respect to burst pressure of SEC Type and case
Pcoal (MPa)
Normalized Pcoal
T2-2-250-010 T2-2-250-050 T2-2-250-100 T2-2-500-010 T2-2-500-050 T2-2-500-100 T2-2-1000-010 T2-2-1000-050 T2-2-1000-100 T4-2-250-010 T4-2-250-050 T4-2-250-100 T4-2-500-006 T4-2-500-010 T4-2-500-050 T4-2-500-100 T4-2-1000-006 T4-2-1000-010 T4-2-1000-050 T4-2-1000-100 T4-2-1000-200
19.50 24.00 30.50 6.00 13.25 18.50 0.20 5.20 14.20 17.00 35.00 38.00 7.00 9.00 17.00 27.00 1.80 3.00 7.00 NF NF
0.65 0.80 1.01 0.34 0.75 1.05 0.02 0.54 1.47 0.56 1.16 1.26 0.40 0.51 0.96 1.53 0.19 0.31 0.72 NA NA
The predicted results for cases T2-250 produce higher Pcoal than predicted by Lee et al. (2001), as is illustrated in Fig. 11. Two experimental test results are included in Fig. 11 for ligament widths of 0.254 mm, 100% TW cracks, and temperatures of 282 and 23 ◦ C. The test at room temperature did not cause rupture of the ligament or failure of either individual crack at a pressure of 17.2 MPa; higher pressures could not be applied due to limitations in feasible flow rates since the cracks were 100% TW. However, the test at 282 ◦ C did fail at a pressure of 15.5 MPa. Since the yield strength of Alloy 600 at 282 ◦ C is only 10% less than that at room temperature, the room temperature test can readily be deduced to have been near failure.
Fig. 11. Comparison of predictions for different failure criterion.
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40
T2-2-250M
70%
Coalescence Pressure (MPa)
35
T2-2-250
80% T2-2-250 Experiment
30 80%
T2-2-500
25
T2-2-500 Experiment
20
T2-2-1000
100%- No Rupture 100%, 282C
15 10 5 0
1
0
2
3 4 Ligament Width (mm)
5
6
Fig. 12. Crack coalescence pressure vs. ligament width (Type 2).
The predicted Pcoal of 17.5 MPa agrees well with those observed, while results of Lee et al. are conservative by about a factor of 2. A rupture criterion based on plastic instability of the ligament is clearly seen to be more appropriate than one based on flow stress as used by Lee et al. Fig. 12 illustrates how Pcoal increases with ligament size for various notch lengths for Type 2 cracks. The rate of increase of Pcoal diminishes with the increase in ligament width. T2-2-250M predictions are additional cases where the notch length was 6 mm rather than the nominal 6.35 mm. Additional tests conducted on 70% (T2-2-250) and 80% (T2-2-500) TW cracks are also
shown in Fig. 12, along with the two tests from Fig. 11. Results support the predicted coalescence pressure for 100% TW cracks; Pcoal observed for 70 and 80% cases are consistently higher than that predicted for 100% TW cases, as expected. Modeling efforts for Type 2 partial TW cracks are underway. Ligament size effect on Pcoal for Type 4 cracks is shown in Fig. 13 for various notch lengths. Again, the rate of increase of Pcoal diminishes with the increase in ligament width. Experiment results are also illustrated in Fig. 13 for 6.35 and 12.7 mm notches with various ligament widths. Note that all tests were conducted on 80% TW cracks. Pcoal observed in experiments either
45 T4-2-250
40 Pressure (MPa)
35
T4-2-500
30 T4-2-1000
25 20
T4-2-250 Experiment
15 10
T4-2-500 Experiment
5
All tests 80% TW
0 0
1
2
3
Ligament Width (mm) Fig. 13. Crack coalescence pressure vs. ligament width (Type 4).
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183
40 T2-2-250 Coalescence Pressure (MPa)
35
T2-2-500 T2-2-1000
30
T4-2-250
25
T4-2-500 20
T4-2-1000
15 10 5 0 0
1
3 2 4 Ligament Width (mm)
5
6
Fig. 14. Coalescence pressure: Type 2 vs. Type 4.
Table 4 Crack opening area of Type 2 cases Area (mm2 )
Area-M (mm2 )
SCA (mm2 )
Type and case
Pressure (MPa)
T2-2-250-010
7 17.225 20∗
2.41 3.34 4.25
0.124 1.054 1.964
0.273 2.25 3.89
0.45 0.47 0.50
T2-2-250-050
7 17.225 24∗
2.34 3.06 5.27
0.054 0.774 2.984
0.273 2.25 8
0.20 0.34 0.37
T2-2-250-100
7 17.225 31∗
2.33 2.75 8.3
0.044 0.464 6.014
0.273 2.25 29.25
0.16 0.21 0.21
4.9 5.55 6.5
0.328 0.978 1.928
0.7 1.84 3.63
0.47 0.53 0.53
Normalized COA (Area-M/SCA)
T2-2-500-010
2 4 6∗
T2-2-500-050
7 10 13∗
5.7 7.5 11.4
1.128 2.928 6.828
4.99 15 45.3
0.23 0.20 0.15
T2-2-500-100
7 17.225 18.5∗
5.1 15.7 19.6
0.528 11.128 15.028
4.99 314 610
0.11 0.04 0.02
T2-2-1000-050
2 3 5.2∗
10.7 12.6 20.6
8.16 10.06 18.06
13.60 36.00 508.00
0.60 0.28 0.04
T2-2-1000-200
7 10 14.2∗
15.46 48.46 146.46
6263.00 BR BR
0.00 NA NA
18 51 149
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exceed or are approximately equal to the predictions for 100% TW cases. Further testing and modeling efforts are underway to determine how sensitive Pcoal is as TW cracks vary from 70 to 100%. Currently, the trends and magnitudes of the experimental results appear consistent with the predictions. Fig. 14 compares Pcoal of Types 2 and 4. Pcoal for Type 4 configurations are consistently higher than for Type 2 configurations for all ligaments investigated, with the exception of T2-2-250/T4-2-250 and a 0.254 mm ligament. Type 4 cases are increasingly more resistant to coalescence than Type 2 for larger ligaments.
7. Crack opening area Crack opening area is directly related to the leak rate of a cracked tube. Table 4 lists the crack opening area (COA) as a function of pressure for Type 2 cases; Table 5 lists COA for Type 4 cases. In the second column, pressure values with an asterisk indicate coalescence, Pcoal . Area corresponds to the total sum of all notch areas, and Area-M is the total area minus the initial crack area (not shown). Single crack area (SCA), the COA of a single equivalent crack (SEC) arrived at with ANL’s COA mathematical model that assumes an initial crack area of zero (Majumdar, 1999; Eiber
Table 5 Crack opening area for Type 4 cases Area (mm2 )
Area-M (mm2 )
SCA (mm2 )
Type and case
Pressure (MPa)
T4-2-250-010
7 10 14∗
1.48 1.65 2.00
0.21 0.38 0.73
0.27 0.53 1.17
0.77 0.71 0.62
T4-2-250-050
7 17.5 35∗
1.50 3.24 26.90
0.23 1.97 25.63
0.27 2.25 59.43
0.84 0.88 0.43
T4-2-250-100
7 17.5 38∗
0.12 0.83 14.43
0.12 0.83 14.43
0.27 2.25 101.60
0.42 0.37 0.14
T4-2-500-010
4 7 9∗
3.50 5.60 8.83
0.96 3.06 6.29
1.73 4.99 10.37
0.55 0.61 0.61
T4-2-500-050
7 10 17.5∗
4.60 7.60 38.40
2.06 5.06 35.86
4.99 15.00 314.00
0.41 0.34 0.11
T4-2-500-100
7 17.5 27∗
3.90 25.00 81.00
1.36 22.46 78.46
4.99 314.00 BR
0.27 0.07 NA
T4-2-1000-010
1 2 3∗
7.30 10.90 15.90
2.22 5.82 10.82
5.00 13.60 36.00
0.44 0.43 0.30
T4-2-1000-050
3 5 7∗
11.40 22.60 50.00
6.32 17.52 44.92
36.00 389.00 6263.00
0.18 0.05 0.01
T4-2-1000-100
7 10∗
34.00 86.00
28.92 80.92
6263.00 BR
0.00 NA
T4-2-1000-200
7 10 15∗
22.00 74.00 210.00
16.92 68.92 204.92
6263.00 BR BR
0.00 NA NA
Normalized COA (Area-M/SCA)
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185
30 T4-2-250-010 T4-2-250-050
COA, Area-M (mm2)
25
T4-2-250-100 20
T2-2-250-010 T2-2-250-050
15
T2-2-250-100
10
5 0 0
10
20 Pressure (MPa)
30
40
Fig. 15. COA for T2-2-250 and T4-2-250 cracks.
et al., 1971; Alzheimer et al., 1979), is also given. The SEC is assumed to be the sum of both notch lengths and the ligament width for Type 2 cases; for Type 4 cases, the ligament is not considered in the SEC length. In several cases, the pressure is larger than the burst pressure for a SEC and is denoted by BR; correspondingly the normalized area is not applicable (NA). The normalized COA is observed to be a fraction since the ligament effectively stiffens the crack configuration. Fig. 15 illustrates the evolution of the predicted COA as a function of pressure for both T2-2-250 and T4-2-250 cases. Initially, COA increases only slightly with pressure to about 17 MPa; thereafter the COA increases at a much higher rate until Pcoal occurs, where Pcoal varies with ligament width
and crack size. Upon reaching Pcoal , the ligament ruptures, and the rate of change in COA is tremendous. Observation of COA in experiments are consistent with this rate of change of COA as illustrated in Fig. 16; COA was observed to change radically with only a slight change in pressure (31.4–32.5 MPa). Differences in COA for Types 2 and 4 cases are illustrated in Fig. 17. When the ligament is small, Types 2 and 4 COA are approximately equal. Since Type 2 cracks coalesce at lower pressures, Type 4 COA will increase stably rather than bursting as for the Type 2 case as indicated in Fig. 17. While Type 4 configurations were observed to have superior resistance to coalescence, simulations reveal that larger ligament widths result in significantly larger COA for Type 4 ligaments than Type 2, as illustrated in Figs. 15
Fig. 16. Observed increase in COA from 31.4 to 32.5 MPa (T4-2-250-050, 80% TW).
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14
T4-2-500-010 T4-2-500-050
COA, Area-M (mm2)
12
T4-2-500-100 T2-2-500-010
10
T2-2-500-050 8
T2-2-500-100
6 4 2 0 0
5
10
15
20
Pressure (MPa) Fig. 17. COA: Type 2 vs. Type 4.
and 17. Furthermore, the differences in COA increase with increasing pressure.
8. Observations and conclusions Nonlinear finite element modeling of collinear cracks with axial and circumferential ligaments were conducted to predict pressures for which coalescence would occur; several experiments were conducted which support and verify the predicted coalescence pressure (Pcoal ). Pcoal for both Types 2 and 4 crack configurations increases with increasing ligament width; however, the rate of increase of Pcoal diminishes with increasing ligament width. A transitional ligament width was predicted to be 2.54 mm for Type 2 crack configurations, i.e. predicted Pcoal is less than the critical pressure (Pcr ) for a single equivalent crack if ligaments are smaller than 2.54 mm. Coalescence was predicted not to occur when ligaments are larger than the transitional width. Collinear cracks with circumferential ligaments (Type 4) have a transitional ligament width of approximately half that of collinear cracks with axial ligaments (Type 2), 1.27 mm versus 2.54 mm, respectively. Comparing Types 2 and 4 crack configurations the authors have shown that Type 4 configurations are more resistant to coalescence than
Type 2 configurations, for a given ligament width. These results will be critical in assisting engineers with maintenance scheduling for SG tubes of PWRs. In addition to coalescence conditions, crack opening areas were also investigated since failure of SG tubes may occur when excessive leak rates are experienced. Simulations reveal that under internal pressure loading, all axial and circumferential ligaments in this study stiffened the crack opening, resulting in smaller COA than that of a single equivalent crack, and increasing the ligament width further stiffens the crack opening, further reducing the COA. Crack openings for collinear cracks with axial ligaments (Type 2) are smaller than those with circumferential ligaments (Type 4). Hence, while Type 4 crack configurations may exhibit superior resistance to coalescence, they are inferior to Type 2 crack configurations with respect to COA. Numerous Type 4 cracks are observed in a single tube. As such, total leak rates may be unacceptable, possibly requiring a tube to be plugged (removed from service) earlier than expected.
Acknowledgements This work was supported by United States Nuclear Regulatory Commission.
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