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Limit load solutions for elbows with circumferential through-wall crack under the pressure-induced bending restraint effect
Journal Pre-proof Limit load solutions for elbows with circumferential through-wall crack under the pressure-induced bending restraint effect Seok-Jun Kang, Jun Hyeok Choi, Hoomin Lee, Doo Ho Cho, Jae Boong Choi, Moon Ki Kim PII:
S0308-0161(19)30182-6
DOI:
https://doi.org/10.1016/j.ijpvp.2019.103983
Reference:
IPVP 103983
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 8 May 2019 Revised Date:
30 August 2019
Accepted Date: 7 September 2019
Please cite this article as: Kang S-J, Choi JH, Lee H, Cho DH, Choi JB, Kim MK, Limit load solutions for elbows with circumferential through-wall crack under the pressure-induced bending restraint effect, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/ j.ijpvp.2019.103983. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Limit Load Solutions for Elbows with Circumferential ThroughWall Crack under the Pressure-Induced Bending Restraint Effect
Seok-Jun Kang1), Jun Hyeok Choi2), Hoomin Lee2), Doo Ho Cho3), Jae Boong Choi2), Moon Ki Kim2),* 1)
Institute of Advanced Machinery and Technology, Sungkyunkwan University
2066, Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, Republic of Korea 2)
School of Mechanical Engineering, Sungkyunkwan University
2066, Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, Republic of Korea 3)
Korean Institute of Nuclear Safety,
62 Gwahak-ro, Yuseong-gu, Daejeon 34142. Republic of Korea
* Corresponding author ☎ +82-31-299-4840 (E-Mail) [email protected]
Manuscript for submission to International Journal of Pressure Vessels and Piping Date: Aug 2019
ABSTRACT A typical pipe finite element model is made by assuming there is internal pressure with a single crack located in the middle of the pipe where both ends are constraint-free. In reality, however, both ends are constrained by their connections to components such as the pressure vessel or pump. Furthermore, due to the complexity of the pipe system, it also experiences a self-constraint effect. This is called the pressure-induced bending restraint effect, and is known to affect the fracture mechanics parameters at the crack tip. Some studies exist on the pressure-induced bending restraint effect of straight pipes, but none have focused on elbow pipe and junction between straight pipe and elbow. This study suggests limit load solutions for elbow and their jointing regions with circumferential through-wall cracks under the pressure-induced bending restraint effect. The relevant equations can be used to estimate the maximum withstand load evaluation of a highly ductile pipe, and can also be used to estimate the elasto-plastic fracture mechanics parameters using the reference stress method.
Keywords: Pressure-induced bending restraint effect, Nuclear power plant, Limit load, Leak-before-break, Pipe, Circumferential through-wall crack.
NOMENCLATURE DEGB
Double-ended guillotine break
LBB
Leak-before-break
SRP
Standard Review Plan
PIB
Pressure-induced bending
λ
Non-dimensional design parameter of elbow
Rb
Bending radius of elbow (mm)
rm
Mean radius of pipe cross section (mm)
t
Thickness of pipe (mm)
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L
Length of attached straight pipe (mm)
θ
Half angle of crack (radian)
E
Elastic modulus (GPa)
ν
Poisson’s ratio
Flow stress (MPa)
Limit load of un-cracked straight pipe under free bending condition (MPa)
Limit load of un-cracked elbow under free bending condition (MPa)
Limit load of cracked elbow under free bending condition (MPa)
,
Limit load of cracked elbow under PIB restraint condition (MPa)
PIB correction factor of elbow with extrados crack
PIB correction factor of elbow with intrados crack
1. INTRODUCTION
The early design concept of nuclear power plants assumed a double-ended guillotine break (DEGB) for its failure mode, as shown in Fig. 1(a). This design concept had to consider additional phenomena such as water jets and pipe whip, and thus needed several auxiliary dynamic failure restraining devices, as illustrated in Fig. 1(b) [1-4]. However, besides increasing construction costs, the long maintenance pathway of these additions increased workers’ radioactive exposure, thus raising safety concerns [3,4]. In response, the leak-before-break (LBB) design concept was proposed, which prevents DEGB by detecting pipe leakage. A more practical design can be drawn by removing the redundant jet impingement shield and pipe whip restraint, as shown in Fig. 2 [1-4]. In 1986, the General Design Criteria (Section 4 of Title 10) of the U.S. Code of Federal Regulations pertaining to nuclear power regulation, was amended so that the LBB design concept could be applied. A Nuclear Regulatory Report (NUREG-0800) by the U.S. Nuclear Regulatory Commission (NRC) included an SRP for the review of safety analysis reports for nuclear power plants; Section 3.6.3 of its LWR Edition – Design of Structures, Components, Equipment, International Journal of Pressure Vessels and Piping
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and Systems – includes a procedure to assess the LBB applicability of pipe systems [1]. Implementing the LBB concept requires an accurate calculation of the fracture mechanics parameters at the crack tip. Because nuclear power plant pipes are usually welded in circumferential directions, most research has been conducted on circumferential through-wall cracks. Nuclear power plants have especially complex pipe systems, and the elbows are connected by welds to straight pipes. Therefore, LBB evaluation of cracks in these connection areas must be done, and several studies have reported on circumferential through-wall crack behaviors [2]. However, the preceding studies did not consider the pressure-induced bending (PIB) restraint effect, and thus improvements are still needed [5-8]. When a pipe with a circumferential through-wall crack is under internal pressure, the crack opening phenomena cause the pipe to bend; this is called the PIB effect. Actual pipe behavior is constrained by nearby components, concrete structures, and supports, as illustrated in Fig. 3. The PIB restraint effect is well known to affect the fracture mechanics parameters in pipes [5-13]. This paper investigates changes in fracture mechanics parameters with respect to PIB restraint effects in pipe elbows and their joints with straight pipes, by assuming there is a circumferential throughwall crack. Material behavior is assumed to be elastic-perfectly plastic at flow stress for the limit load analyses. Here the limit load is defined as the minimum load that makes the ligaments completely yield by von Mises criterion. The prediction equations are developed based on simulation results with the minimum root mean square error method. The correction factor of the boundary conditions in this study is suggested as a function of geometrical parameters rm/t, Rb/rm and θ/π. These equations can be used to estimate the maximum withstand load of a highly ductile pipe and can also be used to estimate the elastoplastic fracture mechanics parameters using the reference stress method.
(a) DEGB consequences
(b) Additional structures
Figure 1. Schematic of the DEGB concept [1-4]
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Figure 2. LBB concept and its advantages [1-4]
Figure 3. Pressure-induced bending restraint effect for an elbow
2. LIMIT LOAD SOLUTIONS FOR ELBOWS International Journal of Pressure Vessels and Piping
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Pipe elbows are crucial within entire pipe systems because they redirect the flow into the desired components by effectively using the given limited space. Nuclear power plants have many elbows within their pipe systems, mainly in LBB-designed main coolant loops and pressurizer surge lines, making it necessary to exactly predict the fracture mechanics parameters of elbows with circumferential throughwall cracks. Many researchers [e.g., 14-23] have conducted studies on limit loads, but not all of them considered the PIB effect; they delivered limit load solutions assuming the pipe is in a free bending condition. The following sections describe limit load solutions of an elbow with a PIB restraint effect using finite element analysis. 2.1 Geometry and finite element model A 90° elbow pipe is modeled here for limit load analysis. Fig. 4 depicts the pipe geometry, its related parameters, and the analysis conditions. Straight pipes were connected to both ends of the elbow to control the PIB restraint effect location. The geometric characteristics of the elbow can be defined as a non-dimensional parameter λ as written in Eq. (1).
λ=
R t R /r = r /t r!
(1)
Here, rm and t are the average radius and thickness, respectively, and Rb is the bend radius. To quantify the effect of elbow geometry on the limit load, different λ values were used in the analysis. Its range was limited to 0.1 through 1.2 to consider the geometrical effect, and this range covers the practical geometry of most nuclear plants [20]. Also, by changing the length of the connected straight pipe into 1R, 3R, and 5R, we investigated the PIB restraint effect location. Only circumferential through-wall cracks were considered in this study to observe the PIB restraint effect, which constrains the crack opening rotational degree. We performed multiple analyses with various crack lengths (θ/π = 0, 0.125, 0.250, 0.375, 0.500).
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Figure 4. Schematics of elbow; (a) Case 1, rotational and translational restraint; (b) C ase 2, rotational restraint without radial expansion; (c) Case 3, rotational restraint with r adial expansion
The finite element model used in these analyses is shown in Fig. 5. Regarding symmetric geometry, a half model was used; the boundary conditions and loading conditions are also illustrated. The commercial finite element analysis program ABAQUS was used with a 20-node element (C3D20R). The material was assumed to be elastic-perfectly plastic, which shows perfect plasticity at flow stress after the elastic behavior. Internal pressure was considered and an internal pressure-induced tensile load, as well as half of the internal pressure, were applied to the crack surface. In the limit load analysis, strain hardening effect is negligible because of the assumption of elasticperfectly plastic behaviors of the material. In addition, stress distribution is hardly affected by geometrical nonlinearity [9, 14] within small deformation. As a result, the limit load increases monotonically until the stress of the net section reaches the flow stress. The simulation cases and material properties used in this study are summarized in Table 1 and Table 2, respectively.
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Table 1. Cases of limit load analysis for various elbow sizes λ
rm/t
Rb/rm
L/Rb
θ/π
Crack locatio n
Loading c ondition
0.1~1.2
5, 10, 20
2, 3, 4, 5, 6
1, 3, 5
0, 0.125, 0. 250, 0.375, 0.500
Extrados, Intr ados
Internal pr essure
Table 2. Material properties Young’s modulus, E [GPa] 200
Poisson’s ratio, ν 0.3
Flow stress, σf [MPa] 100
Figure 5. Finite element model of a pipe assembly consisting of straight pipes and an elb ow with a crack on it
In order to verify the FE model used in this study, the free bending simulation results are compared to the FE analysis data and proposed equation data of the previous study [21]. As simulation results in Fig. 6 show good agreement with each other, it can be concluded that our FE model is qualified in terms of mesh quality and loading/boundary conditions. In addition, the PIB restraint condition is applied to this International Journal of Pressure Vessels and Piping
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FE model exclusively.
(a) Limit load equation of extrados crack as function of rm/t and θ/π
(b) Limit load equation of intrados crack as function of θ/π only Figure 6. Model verification for elbow-centered crack cases
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According to previous study [15], the limit load of an uncracked elbow in a free bending condition can be estimated by Eq. (2).
P$%& '(
)*+,-./* P$
=
1 1 + 2 exp6−8 ∙ : /;< =
2 = 1.196;< /A + 1=B.BC − 1, 8 = 0.0013;< /A + 0.307
(2)
Here, POElbow and POStraight refer to the limit load of an uncracked elbow and a straight pipe, respectively, in a free bending condition. Additionally, from other studies [21, 22], the limit load solution of an elbow, including a circumferential through-wall crack with respect to its extrados and intrados, is suggested by Eqs. 3(a) and 3(b).
PG%& '( P$%& '(
r θ r θ ! = min K1.0, 1.0 + L1.3 − 0.06 N OP Q T + L−5.1 + 0.12 N OP Q T W t π t π PG%& '( P$%& '(
θ = min X1.0, 1.5 − 2.4 Q TZ π
(3a)
(3b)
Here, PLElbow is the limit load of an elbow with a circumferential through-wall crack under a free bending condition at extrados or intrados. Various limit load analyses are conducted using the aforementioned FE model. Using both the analyses results and previous studies, a limit load solution considering the PIB restraint effect is suggested in Eq. (4).
%& '( PG,[\]
PG%& '(
= 6^/_, ;< /A, : /;< =
(4)
Here, PL,PIBElbow is the limit load of an elbow with a circumferential through-wall crack considering the International Journal of Pressure Vessels and Piping
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PIB restraint effect. The three considered boundary conditions are illustrated in Fig. 4. Case 1 (Fig. 4(a)) shows that both rotational and translational motions are constrained. Cases 2 and 3 (Figs. 4(b) and 4(c), respectively) include the rotational restraint. The difference is that the pipe in Case 2 cannot expand in the radial direction, but with the pipe in Case 3 this is possible. The results are shown in Fig. 7.
Figure 7. The effects of three different boundary conditions
The analyses indicated that the translational restraint did not affect the limit load of the elbow. We therefore concluded that the PIB restraint effect was mainly due to the rotational displacement restraint. Furthermore, the pipe deformation was restricted to near the crack locally, where the effect of ovalization was negligible. Therefore, in this study, the boundary condition in Case 1 (rotational and translational restraint) was applied since the Case 1 conditions are more similar to a pipe connected to a large mass structure, such as a pressure vessel or a pump. For the cases summarized in Table 1, we performed a parametric study on pipe length, rm/t, and Rb/rm to achieve a more precise solution. The results are shown in Fig. 8. Fig. 8 is the analysis results with different lengths of straight pipe connected to the elbow. As seen in Fig. 8, there is no length effect. Therefore, we
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analysis only for shortest straight pipe case to reduce analysis time
Figure 8. The effects of straight pipe length attached to both elbow ends
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2.3 Limit load solutions for crack positions 2.3.1 Extrados crack Stress distributions resulted from FEA are shown in Fig. 9. Gray area represents a yield area. As seen in this figure, stress distribution is concentrated at ligament near the crack tip in case of free bending condition. However, in case of considering PIB restraint effect, large area of the elbow yields before plastic collapse occurs. Since the PIB restraints interrupt crack opening, more limit load is required to fully plasticize the net section including crack.
(a) Free bending condition
(b) PIB restraint condition
Figure 9. Stress distribution of the extrados elbow-centered crack
Additionally, stress distribution of the net section is displayed in Fig. 10. Crack opening stress is extracted along the y direction from crack tip and it is normalized by flow stress. Stress distributions of free bending condition and PIB restraint condition are compared to each other. Most of ligament is plasticized to the tensile direction at the fracture moment. The difference between two conditions is noticeable after 40 mm point where the compression stress reaches the flow stress. As indicated by blue arrows, when considering PIB restraint effect, the magnitude of bending moment decreases due to International Journal of Pressure Vessels and Piping
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reaction moment caused by the boundary condition. Eventually, the PIB restraint effect suppresses the bending deformation of the pipe resulting in an increase of the limit load
Figure 10. PIB restraint effect on stress distribution of the extrados elbow-centered crack
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Figure 11. The effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the extra dos elbow-centered crack
Fig. 11 presents the results for the extrados crack in elbow center, including the effects of different rm/t values and different curvatures of the pipe. The correction factor for the extrados cracked model is affected by θ/π, rm/t and Rb/rm. As the crack length becames longer, the PIB restraint effect increases the limit load so that the pipe could endure until it fully collapses. It was especially noticeable that the results for the free bending condition are the same as those for the PIB restraint effect at a certain length crack or less. The PIB restraint effect does not affect the models with a crack length of θ/π ≤ 0.25. rm/t and Rb/rm also have no effect on the analysis results in this range. From the FE analysis, the PIB restraint effect is valid for models with θ/π > 0.25. Accordingly, we propose correction factors for the boundary conditions in the crack length range of 0.25 to 0.50. In cases with θ/π = 0.50, where the PIB restraint effect is the most significant, the analysis results are distributed with a deviation of 24.9% due to the rm/t effect. The effects of Rb/rm are 14.2% (rm/t = 5), 12.1 (rm/t = 10), and 11.0% (rm/t = 20), depending on rm/t. The correction factor of the PIB restraint effect for the extrados crack in an elbow center is proposed in Eq. (5), with θ/π, rm/t, and Rb/rm as its variables. The equation is extracted by curve-fitting for representative data and then the θ/π, rm/t, and Rb/rm effects are considered.
6^/_, ;< /A, : /;< = = `
1 1 + 2 ∙ 86^/_ − 0.25=!
^/_ ≤ 0.25 0.25 < ^/_ ≤ 0.50
A = 20.14 − 0.4456r /A=
(5)
B = 0.9 + 0.036R /;< =
2.3.2 Intrados crack When there is an intrados crack, it also restrains the crack opening, which increases the limit load compared to a free bending condition. Stress distributions of the case of intrados crack are displayed in Fig. 12. Similar to the case of extrados crack, plastic zone of a free bending condition is limited near the crack tip. However, the pipe with a PIB restraint condition is ruptured after plastic zone widely propagates along the longitudinal direction. International Journal of Pressure Vessels and Piping
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(a) Free bending condition
(b) PIB restraint condition
Figure 12. Stress distribution of the intrados elbow-centered crack
Stress distribution of the pipe with intrados elbow-centered crack is presented in Fig. 13. Similar to the extrados case, the difference between two conditions, indicated by blue arrows, stands out when the compressive stress reaches the flow stress. Remarkably, the bending effect almost disappears in a PIB restraint condition and every ligament is yielded by membrane stress. As a result, plastic zone propagates as shown in Fig. 12(b). Consequently, the PIB restraint effect hinders the behavior of pipe and increases the limit load, not only in the extrados case, but also in the intrados case.
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Figure 13. PIB restraint effect on stress distribution of the intrados elbow-centered crack
When there is an intrados crack, it also restrains the crack opening, which increases the limit load compared to a free bending condition. Fig. 14 present the results for the intrados crack, including the effects of different rm/t values and different curvatures of the pipe. In the case of the intrados crack, seen in Fig. 14, all parameters except for θ/π had a negligible effect on the limit load.
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Figure 14. The effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the extra dos elbow-centered crack
The same as with the extrados crack, the correction factor increased rapidly after θ/π = 0.25. The PIB restraint effect correction factor of the intrados crack is approximated by Eq. (6).
6^/_= `
1 1 1 1.366^/_ 7 0.25= 1 17.356^/_ 7 0.25=!
^/_ a 0.25 0.25 b ^/_ a 0.50
(6)
The proposed equation indicates a maximum difference of 5.6% from the limit load analysis result at θ/π = 0.50.
3. LIMIT LOAD SOLUTION FOR JUNCTION BETWEEN STRAIGHT PIPE AND ELBOW Nuclear power plant pipe systems consist of various components to effectively transport fluids and International Journal of Pressure Vessels and Piping
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their contained energy. The components are usually welded together and, in general, the weldments are designed to have more structural stiffness than the base material. However, due to the inherent difficulties in the welding mechanism, cracks are usually reported near welds. According to previous studies, LBB evaluation must include weldment evaluation [2]. Other studies [21-23] suggested limit load solutions for both straight pipe and elbow junction, but they all assumed a free bending condition. In this study, using the aforementioned limit load solutions for both straight pipes and elbows, a new limit load solution and correction factor on straight pipe-elbow junction is suggested, considering the PIB restraint effect. 3.1 Geometry and finite element model Here, the crack was assumed to be located in a 90° elbow and straight pipe connection, whose schematics are shown in Fig. 4. For direct comparison and considering the actual geometry of nuclear power plant components, all of the simulation cases are used from Table 1. A circumferential throughwall crack was assumed and positioned at the junction. Fig. 15 presents the FE model we used, which is a half symmetric model for effective calculation. The boundary conditions and loading conditions were same as described in Section 2.
Figure 15. Finite element model of a pipe assembly consisting of a straight pipe and an elbow with a crack on the junction
Similar to the case of elbow-centered crack, our FE model for elbow-straight pipe junction crack is also verified by comparison with the previous study [21]. Fig. 16 shows that the results from the proposed International Journal of Pressure Vessels and Piping
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model here (except for the PIB restraint effect) are well matched with the reference data.
(a) Limit load equation of extrados crack as function of rm/t and θ/π
(b) Limit load equation of intrados crack as function of θ/π only Figure 16. Model verification for elbow-straight pipe junction crack cases
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3.2 Limit load analysis When there is no crack, the limit load can be calculated by Eq. (2). Also, from previous work [23], the limit load of junction through-wall crack in an elbow-straight pipe assembly under a free bending condition is not significantly different from the limit load calculated from Eq. 3(a) and 3(b). For the model where the crack was between a straight pipe and an elbow, the sensitivity evaluations of the boundary condition and the straight pipe length effect were carried out according to the same procedure as in Section 2.2. The three boundary cases in Fig. 4 were considered. The FE analyses were performed with each boundary case by changing the lengths of the straight pipes connected at both ends of the elbow. From the comparison we concluded that there was no significant effect of those variables. The Case 1 boundary condition and the shortest straight pipe for the junction-cracked model was applied to describe the PIB restraint effect. 3.3 Limit load solutions for crack positions 3.3.1 Extrados crack In free bending conditions, the longer the crack length, the larger the deformation (i.e., displacement and rotation angle) of pipe ends due to crack opening. On the other hand, considering the PIB restraint effect, the boundary condition doesn’t allow any deformation of pipe ends by generating reaction force and moment. Therefore, a pipe with longer crack length is expected to yield larger deformation under free bending conditions. Conversely, that means the PIB restraint effect increases in this case in order to compensate for that large deformation. Empirically, the PIB restraint effect should be considered in the range of θ/π ≥ 0.25. It is also noted that rm/t and Rb/rm affect to the relatively long crack. In addition, Fig. 17 shows stress distributions resulted from FEA. Unlike a free bending condition, the plastic zone in a PIB restraint condition propagates laterally prior to plasticizing ligament of the net section including crack
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(a) Free bending condition
(b) PIB restraint condition
Figure 17. Stress distribution of the extrados junction crack
The stress distribution for ligament is presented in Fig. 18. In case of a free bending condition, compressive stress determines the neutral axis far from the crack tip. When the PIB restraint boundary condition is applied, bending behavior of the pipe is limited resulting that the neutral axis is shifted further away from the center line.
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Figure 18. PIB restraint effect on stress distribution of the extrados junction crack
Figure 19. Effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the extrad os junction crack
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The limit load analysis and the results are illustrated in Fig. 19. Fig. 19 is the correction factor distributions for crack length of the extrados case. The data for the model with a crack in the junction between the straight pipe and the elbow are more widely distributed than for the elbow-centered-crack case. This was likely due to the asymmetry between the geometry and its volume on both sides of the crack. Eq. (7), which was derived from the data in Fig. 19, shows the correction factor of the PIB restraint effect for the model where an extrados crack existed in the weldment. The boundary condition effect does not appear for θ/π ≤ 0.25, but rm/t and Rb/rm have a significant effect on the relatively long crack.
6^/_, ;< /A, : /;< = = ` C 20.36 − 0.5056r /A=
1 1 + e ∙ f6^/_ − 0.25=!
^/_ ≤ 0.25 0.25 < ^/_ ≤ 0.50
B 0.78 + 0.056R /;< =
(7)
The PIB restraint correction factor was more sensitive to Rb/rm than in the case where the crack was in the elbow center. We believe that asymmetry at both sides of the crack caused this result. Since the crack existed in the connection between the straight pipe and the elbow, the geometry of one side of the crack was straight pipe, and the other side remained as elbow and straight pipe. Therefore, the effect of Rb/rm was predominantly on the elbow and straight pipe side. This is the factor that increased the Rb/rm effect. In the θ/π = 0.50 cases, the data were scattered with a deviation of 29.2% by the rm/t effect. The effects of Rb/rm were 20.6% (rm/t = 5), 24.5 (rm/t = 10), and 25.3% (rm/t = 20).
3.3.2 Intrados crack FEA results of intrados crack on the junction are illustrated in Fig. 20. As described in the extrados case, collapse region of the pipe is very localized in a free bending condition, while collapse region of the PIB restraint pipe grows into elbow side before the ligament becomes fully plastic.
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(a) Free bending condition
(b) PIB restraint condition
Figure 20. Stress distribution of the intrados junction crack
Fig. 21 displays stress distribution of the intrados junction crack case. Similar to the aforementioned three cases, compressive stress eventually disappears because bending moment is canceled by the PIB restraint effect.
Figure 21. PIB restraint effect on stress distribution of the intrados junction crack International Journal of Pressure Vessels and Piping
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Figure 22. Effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the intrad os junction crack
Fig. 22 is the correction factor distributions for crack length of intrados crack. rm/t and Rb/rm had little effect on the elbow-centered-crack and junction-crack cases. In these limit load analysis results of the elbow, including the intrados crack on the junction, the PIB restraint effect occurred at θ/π ≥ 0.25. Eq. (8) is the correction factor estimating equation. The factors have low sensitivity of rm/t and Rb/rm, and the equation is well matched with a maximum of 4.2% difference.
6^/_= `
1 1 1 0.2556^/_ 7 0.25= 1 23.466^/_ 7 0.25=!
^/_ a 0.25 0.25 b ^/_ a 0.50
(8)
Using Eqs. (2) and (3b), one can calculate the limit load for the cracked elbow with a free bending condition, and then calculate the limit load considering the PIB restraint effect with the correction factor of Eq. (8).
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Nuclear power plants have approximately 80 km of pipes per unit. Therefore, maintaining the structural integrity of these pipes is critical to the safe operation of the plants. LBB evaluation is performed at the design stage to ensure structural integrity, and various fracture mechanics parameters are used in this process. However, previous studies have considered a free bending condition for such analysis. In real pipes, the PIB restraint effect is engendered by the supports, concrete structures, and other components. This boundary condition must be considered in an LBB analysis for a precise calculation of fracture mechanics parameters. This study used FE analysis to suggest limit load solutions for elbows and their welding regions with circumferential through-wall cracks under PIB restraint effects. To consider the PIB restraint effect, the rotational movement of the elbow was constrained by applying internal pressure to the pipe, while moments were constrained by boundary conditions. Various FE analyses were performed by changing the elbow curvature, length of connected straight pipe, rm/t, crack length, and crack location. Analysis results revealed that the elbow curvature and straight pipe length had no effect on the limit load, and this was confirmed by the pipe rotational angle. Correction factors of the PIB restraint effect were developed from closed-form equations using the parameters that affected the limit load, which were rm/t, Rb/rm, crack length, and crack location. Although this study can be applied to evaluate the structural integrity of cracks located in the center of an elbow or a junction (between straight pipe and elbow) made of highly ductile austenitic steel, there are still several limitations. First, only an ideal boundary condition is considered by assuming that both ends of a pipe are fully restrained by two large mass components. However, in many cases, the restraint effect is caused by the complex piping system (i.e., combination of straight and elbow pipes) itself. Second, the equations presented here don’t take into account eccentricity of the crown crack. Therefore, these limitations should be carefully considered in practical use.
ACKNOWLEDGMENT This work was supported by the National Research Foundation of Korea [grant number: NRF2019M2D2A1A02057338].
REFERENCES International Journal of Pressure Vessels and Piping
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[1] Cho, D., Development of non-linear fracture mechanics parameters estimation for nonidealized through-wall crack in pipes, Ph.D. dissertation, Sungkyunkwan University, 2013, https://lib.skku.edu/suwon/en/#/search/detail/4133802 . [2] Scott, P., Olson, R., Wilkowski, G., Development of technical basis for leak-before-break evaluation procedures, Nuclear Regulatory Commission, 2002, NUREG/CR-6765-Vol.1, https://www.nrc.gov/reading-rm/doc-collections/nuregs/contract/cr6765/#pub-info . [3] Swamy, S.A., Bhowmick, D.C., Prager, D.E., Tanaka, T., Ogata, Y., Yokota, H., LBB evaluation for a typical Japanese PWR primary loop by using the U.S., NRC approved method, Nuclear Regulatory Commission, 1997, NUREG/CP-0115, https://inis.iaea.org/search/searchsinglerecord.aspx?recordsFor=SingleRecord&RN=28076014 . [4] Swamy, S.A., Mandava, P.R., Bhowmick, D.C., Prager, D.E., LBB considerations for a new plant design, Nuclear Regulatory Commission, 1997, NUREG/CP-0155, https://inis.iaea.org/search/searchsinglerecord.aspx?recordsFor=SingleRecord&RN=28076030 . [5] Miura, N., Evaluation of crack opening behavior for cracked pipes effect of restraint on crack opening, ASME pressure vessels and piping conference 423 (2001) 135-143, http://refhub.elsevier.com/S0029-5493(16)00089-3/sbref0035 . [6] Kim, H.S., Jin, T.E., Kim, H.D., Chang, Y.S., Kim, Y.J., Restraining effect of support plates on the limit loads for circumferential cracks in the steam generator tube, Journal of nuclear engineering and design 238(1) (2008) 135-142, https://doi.org/10.1016/j.nucengdes.2007.06.002 . [7] Kim, Y., Hwang, I.S., Oh, Y.J., Effective applied moment in circumferential through-wall cracked pipes for leak-before-break evaluation considering pipe restraint effects, Journal of nuclear engineering and design 301 (2016) 175-182, https://doi.org/10.1016/j.nucengdes.2016.02.028 . [8] Rahman, S., Ghadiali, N., Wilkowski, G.M., Bonora, N., Effects of off-centered cracks and restraint of induced bending caused by pressure on the crack-opening-area analysis of pipes, Journal of nuclear engineering and design, 167 (1996) 55-67, https://doi.org/10.1016/S00295493(96)01289-7 . [9] Shim, D.J., Plastic limit loads for pipes with part-through surface cracks and its application to failure assessment of pressurized piping, Ph.D. dissertation, Sungkyunkwan University, 2003, https://lib.skku.edu/suwon/en/#/search/detail/1110952 .
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[10] Scott, P., Olson, R., Bockbrader, J., Wilson, M., Gruen, B., Morbitzer, R., Yang, Y., Williams, C., Brust, F., Fredette, L., Ghadiali, N., Wilkowski, G., Rudland, D., Feng, Z., Wolterman, R., The Battelle Integrity of Nuclear Piping (BINP) Program Final Report, Nuclear Regulatory Commission, 2005, NUREG/CR-6837, https://www.nrc.gov/reading-rm/doccollections/nuregs/contract/cr6837/ . [11] Kim, J.W., Evaluation of restraint effect of pressure induced bending on the elastic-plastic crack opening behavior, International journal of pressure vessels and piping 81(4) (2004) 355362, https://doi.org/10.1016/j.ijpvp.2004.02.011. [12] Rahman, S., Brust, F., Ghadiali, N., Krishnaswamy, P., Wilkowski, G., Choi, Y.H., Moberg, F., Brickstad, B., Refinement and evaluation of crack-opening-area analyses for circumferential through-wall cracks in pipes, Nuclear Regulatory Commission, 1995, NUREG/CR-6300, https://doi.org/10.2172/46620 . [13] Ghadiali, N., Wilkowski, G., Rahman, S., Choi, Y.H., Deterministic and probabilistic evaluations for uncertainty in pipe fracture parameters in leak-before-break and in-service flaw evaluations, Nuclear Regulatory Commission, 1996, NUREG/CR-6443, https://doi.org/10.2172/256412 . [14] Miller, A.G., Review of limit loads of structures containing defects, Journal of pressure vessels piping 32 (1988) 197-327, https://doi.org/10.1016/0308-0161(88)90073-7 . [15] Kim, Y.J., Je, J., Oh, C., Han, J., Budden, P.J., Plastic loads for 90° thick-walled elbows under combined pressure and bending, International journal of strain analysis 45 (2009) 115127, https://doi.org/10.1243/03093247JSA568 . [16] Mohan, R., Brust, F., Ghadiali, N., Wilkowski, G., Development of a J-estimation scheme for internal circumferential and axial surface cracks in elbows, Nuclear Regulatory Commission, 1996, NUREG/CR-6445, https://doi.org/10.2172/249297 . [17] Chattopadhyay, J., Nathani, D.K., Dutta, B.K., Kushwaha, H.S., Closed-form collapse moment equations of elbows under combined internal pressure and in-plane bending moment, International journal of pressure vessel technology 122 (2000) 431-436, https://doi.org/10.1115/1.1285988 . [18] Robertson, A.C., Li, H., Mackenzie, D., Plastic collapse of pipe bends under combined internal pressure and in-plane bending, International journal of pressure vessels and piping 82(5) (2005) 407-416, https://doi.org/10.1016/j.ijpvp.2004.09.005 . [19] Yahiaoui, K., Moffat, D.G., Moreton, D.N., Piping elbows with cracks part 1: a parametric International Journal of Pressure Vessels and Piping
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study of the influence of crack size on limit loads due to pressure and opening bending, International journal of strain analysis 35(1) (2000) 35-46, https://doi.org/10.1243/0309324001513991 . [20] Kim, Y., Kim, Y., Song, T., Finite element plastic loads for circumferential cracked pipe bends under in-plane bending, International journal of engineering fracture mechanics 74 (2007) 643-668, https://doi.org/10.1016/j.engfracmech.2006.07.001 . [21] Hong, S.P., Kim, J., Kim, Y., Limit pressures of 90° elbows with circumferential surface cracks, International journal of engineering fracture mechanics 76 (2009) 2202-2216, https://doi.org/10.1016/j.engfracmech.2009.07.005 . [22] Hong, S.P., Effect of Internal Pressure on Plastic Loads of 90° Elbows with Circumferential Cracks under Combined Bending, Ph.D. dissertation, Korea University, 2011, https://library.korea.ac.kr/search/detail/CAT000045696066#.XNI0O-gzaUk. [23] Song, T.K., Kim, Y., Oh, C., Jin T., Kim, J., Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes, International journal of pressure vessels and piping 86 (2009) 495-507, https://doi.org/10.1016/j.ijpvp.2009.03.008 .
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Highlights Limit load solutions are developed by proposing pressure induced bending correction factors for elbow pipe system. Several sensitivity evaluations are performed and the stress field of the net section is discussed. Various crack locations are addressed such as intrados or extrados crack in elbow center or straight pipe-elbow junction.
S0308-0161(19)30182-6
DOI:
https://doi.org/10.1016/j.ijpvp.2019.103983
Reference:
IPVP 103983
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 8 May 2019 Revised Date:
30 August 2019
Accepted Date: 7 September 2019
Please cite this article as: Kang S-J, Choi JH, Lee H, Cho DH, Choi JB, Kim MK, Limit load solutions for elbows with circumferential through-wall crack under the pressure-induced bending restraint effect, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/ j.ijpvp.2019.103983. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Limit Load Solutions for Elbows with Circumferential ThroughWall Crack under the Pressure-Induced Bending Restraint Effect
Seok-Jun Kang1), Jun Hyeok Choi2), Hoomin Lee2), Doo Ho Cho3), Jae Boong Choi2), Moon Ki Kim2),* 1)
Institute of Advanced Machinery and Technology, Sungkyunkwan University
2066, Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, Republic of Korea 2)
School of Mechanical Engineering, Sungkyunkwan University
2066, Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, Republic of Korea 3)
Korean Institute of Nuclear Safety,
62 Gwahak-ro, Yuseong-gu, Daejeon 34142. Republic of Korea
* Corresponding author ☎ +82-31-299-4840 (E-Mail) [email protected]
Manuscript for submission to International Journal of Pressure Vessels and Piping Date: Aug 2019
ABSTRACT A typical pipe finite element model is made by assuming there is internal pressure with a single crack located in the middle of the pipe where both ends are constraint-free. In reality, however, both ends are constrained by their connections to components such as the pressure vessel or pump. Furthermore, due to the complexity of the pipe system, it also experiences a self-constraint effect. This is called the pressure-induced bending restraint effect, and is known to affect the fracture mechanics parameters at the crack tip. Some studies exist on the pressure-induced bending restraint effect of straight pipes, but none have focused on elbow pipe and junction between straight pipe and elbow. This study suggests limit load solutions for elbow and their jointing regions with circumferential through-wall cracks under the pressure-induced bending restraint effect. The relevant equations can be used to estimate the maximum withstand load evaluation of a highly ductile pipe, and can also be used to estimate the elasto-plastic fracture mechanics parameters using the reference stress method.
Keywords: Pressure-induced bending restraint effect, Nuclear power plant, Limit load, Leak-before-break, Pipe, Circumferential through-wall crack.
NOMENCLATURE DEGB
Double-ended guillotine break
LBB
Leak-before-break
SRP
Standard Review Plan
PIB
Pressure-induced bending
λ
Non-dimensional design parameter of elbow
Rb
Bending radius of elbow (mm)
rm
Mean radius of pipe cross section (mm)
t
Thickness of pipe (mm)
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L
Length of attached straight pipe (mm)
θ
Half angle of crack (radian)
E
Elastic modulus (GPa)
ν
Poisson’s ratio
Flow stress (MPa)
Limit load of un-cracked straight pipe under free bending condition (MPa)
Limit load of un-cracked elbow under free bending condition (MPa)
Limit load of cracked elbow under free bending condition (MPa)
,
Limit load of cracked elbow under PIB restraint condition (MPa)
PIB correction factor of elbow with extrados crack
PIB correction factor of elbow with intrados crack
1. INTRODUCTION
The early design concept of nuclear power plants assumed a double-ended guillotine break (DEGB) for its failure mode, as shown in Fig. 1(a). This design concept had to consider additional phenomena such as water jets and pipe whip, and thus needed several auxiliary dynamic failure restraining devices, as illustrated in Fig. 1(b) [1-4]. However, besides increasing construction costs, the long maintenance pathway of these additions increased workers’ radioactive exposure, thus raising safety concerns [3,4]. In response, the leak-before-break (LBB) design concept was proposed, which prevents DEGB by detecting pipe leakage. A more practical design can be drawn by removing the redundant jet impingement shield and pipe whip restraint, as shown in Fig. 2 [1-4]. In 1986, the General Design Criteria (Section 4 of Title 10) of the U.S. Code of Federal Regulations pertaining to nuclear power regulation, was amended so that the LBB design concept could be applied. A Nuclear Regulatory Report (NUREG-0800) by the U.S. Nuclear Regulatory Commission (NRC) included an SRP for the review of safety analysis reports for nuclear power plants; Section 3.6.3 of its LWR Edition – Design of Structures, Components, Equipment, International Journal of Pressure Vessels and Piping
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and Systems – includes a procedure to assess the LBB applicability of pipe systems [1]. Implementing the LBB concept requires an accurate calculation of the fracture mechanics parameters at the crack tip. Because nuclear power plant pipes are usually welded in circumferential directions, most research has been conducted on circumferential through-wall cracks. Nuclear power plants have especially complex pipe systems, and the elbows are connected by welds to straight pipes. Therefore, LBB evaluation of cracks in these connection areas must be done, and several studies have reported on circumferential through-wall crack behaviors [2]. However, the preceding studies did not consider the pressure-induced bending (PIB) restraint effect, and thus improvements are still needed [5-8]. When a pipe with a circumferential through-wall crack is under internal pressure, the crack opening phenomena cause the pipe to bend; this is called the PIB effect. Actual pipe behavior is constrained by nearby components, concrete structures, and supports, as illustrated in Fig. 3. The PIB restraint effect is well known to affect the fracture mechanics parameters in pipes [5-13]. This paper investigates changes in fracture mechanics parameters with respect to PIB restraint effects in pipe elbows and their joints with straight pipes, by assuming there is a circumferential throughwall crack. Material behavior is assumed to be elastic-perfectly plastic at flow stress for the limit load analyses. Here the limit load is defined as the minimum load that makes the ligaments completely yield by von Mises criterion. The prediction equations are developed based on simulation results with the minimum root mean square error method. The correction factor of the boundary conditions in this study is suggested as a function of geometrical parameters rm/t, Rb/rm and θ/π. These equations can be used to estimate the maximum withstand load of a highly ductile pipe and can also be used to estimate the elastoplastic fracture mechanics parameters using the reference stress method.
(a) DEGB consequences
(b) Additional structures
Figure 1. Schematic of the DEGB concept [1-4]
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Figure 2. LBB concept and its advantages [1-4]
Figure 3. Pressure-induced bending restraint effect for an elbow
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Pipe elbows are crucial within entire pipe systems because they redirect the flow into the desired components by effectively using the given limited space. Nuclear power plants have many elbows within their pipe systems, mainly in LBB-designed main coolant loops and pressurizer surge lines, making it necessary to exactly predict the fracture mechanics parameters of elbows with circumferential throughwall cracks. Many researchers [e.g., 14-23] have conducted studies on limit loads, but not all of them considered the PIB effect; they delivered limit load solutions assuming the pipe is in a free bending condition. The following sections describe limit load solutions of an elbow with a PIB restraint effect using finite element analysis. 2.1 Geometry and finite element model A 90° elbow pipe is modeled here for limit load analysis. Fig. 4 depicts the pipe geometry, its related parameters, and the analysis conditions. Straight pipes were connected to both ends of the elbow to control the PIB restraint effect location. The geometric characteristics of the elbow can be defined as a non-dimensional parameter λ as written in Eq. (1).
λ=
R t R /r = r /t r!
(1)
Here, rm and t are the average radius and thickness, respectively, and Rb is the bend radius. To quantify the effect of elbow geometry on the limit load, different λ values were used in the analysis. Its range was limited to 0.1 through 1.2 to consider the geometrical effect, and this range covers the practical geometry of most nuclear plants [20]. Also, by changing the length of the connected straight pipe into 1R, 3R, and 5R, we investigated the PIB restraint effect location. Only circumferential through-wall cracks were considered in this study to observe the PIB restraint effect, which constrains the crack opening rotational degree. We performed multiple analyses with various crack lengths (θ/π = 0, 0.125, 0.250, 0.375, 0.500).
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Figure 4. Schematics of elbow; (a) Case 1, rotational and translational restraint; (b) C ase 2, rotational restraint without radial expansion; (c) Case 3, rotational restraint with r adial expansion
The finite element model used in these analyses is shown in Fig. 5. Regarding symmetric geometry, a half model was used; the boundary conditions and loading conditions are also illustrated. The commercial finite element analysis program ABAQUS was used with a 20-node element (C3D20R). The material was assumed to be elastic-perfectly plastic, which shows perfect plasticity at flow stress after the elastic behavior. Internal pressure was considered and an internal pressure-induced tensile load, as well as half of the internal pressure, were applied to the crack surface. In the limit load analysis, strain hardening effect is negligible because of the assumption of elasticperfectly plastic behaviors of the material. In addition, stress distribution is hardly affected by geometrical nonlinearity [9, 14] within small deformation. As a result, the limit load increases monotonically until the stress of the net section reaches the flow stress. The simulation cases and material properties used in this study are summarized in Table 1 and Table 2, respectively.
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Table 1. Cases of limit load analysis for various elbow sizes λ
rm/t
Rb/rm
L/Rb
θ/π
Crack locatio n
Loading c ondition
0.1~1.2
5, 10, 20
2, 3, 4, 5, 6
1, 3, 5
0, 0.125, 0. 250, 0.375, 0.500
Extrados, Intr ados
Internal pr essure
Table 2. Material properties Young’s modulus, E [GPa] 200
Poisson’s ratio, ν 0.3
Flow stress, σf [MPa] 100
Figure 5. Finite element model of a pipe assembly consisting of straight pipes and an elb ow with a crack on it
In order to verify the FE model used in this study, the free bending simulation results are compared to the FE analysis data and proposed equation data of the previous study [21]. As simulation results in Fig. 6 show good agreement with each other, it can be concluded that our FE model is qualified in terms of mesh quality and loading/boundary conditions. In addition, the PIB restraint condition is applied to this International Journal of Pressure Vessels and Piping
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FE model exclusively.
(a) Limit load equation of extrados crack as function of rm/t and θ/π
(b) Limit load equation of intrados crack as function of θ/π only Figure 6. Model verification for elbow-centered crack cases
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According to previous study [15], the limit load of an uncracked elbow in a free bending condition can be estimated by Eq. (2).
P$%& '(
)*+,-./* P$
=
1 1 + 2 exp6−8 ∙ : /;< =
2 = 1.196;< /A + 1=B.BC − 1, 8 = 0.0013;< /A + 0.307
(2)
Here, POElbow and POStraight refer to the limit load of an uncracked elbow and a straight pipe, respectively, in a free bending condition. Additionally, from other studies [21, 22], the limit load solution of an elbow, including a circumferential through-wall crack with respect to its extrados and intrados, is suggested by Eqs. 3(a) and 3(b).
PG%& '( P$%& '(
r θ r θ ! = min K1.0, 1.0 + L1.3 − 0.06 N OP Q T + L−5.1 + 0.12 N OP Q T W t π t π PG%& '( P$%& '(
θ = min X1.0, 1.5 − 2.4 Q TZ π
(3a)
(3b)
Here, PLElbow is the limit load of an elbow with a circumferential through-wall crack under a free bending condition at extrados or intrados. Various limit load analyses are conducted using the aforementioned FE model. Using both the analyses results and previous studies, a limit load solution considering the PIB restraint effect is suggested in Eq. (4).
%& '( PG,[\]
PG%& '(
= 6^/_, ;< /A, : /;< =
(4)
Here, PL,PIBElbow is the limit load of an elbow with a circumferential through-wall crack considering the International Journal of Pressure Vessels and Piping
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PIB restraint effect. The three considered boundary conditions are illustrated in Fig. 4. Case 1 (Fig. 4(a)) shows that both rotational and translational motions are constrained. Cases 2 and 3 (Figs. 4(b) and 4(c), respectively) include the rotational restraint. The difference is that the pipe in Case 2 cannot expand in the radial direction, but with the pipe in Case 3 this is possible. The results are shown in Fig. 7.
Figure 7. The effects of three different boundary conditions
The analyses indicated that the translational restraint did not affect the limit load of the elbow. We therefore concluded that the PIB restraint effect was mainly due to the rotational displacement restraint. Furthermore, the pipe deformation was restricted to near the crack locally, where the effect of ovalization was negligible. Therefore, in this study, the boundary condition in Case 1 (rotational and translational restraint) was applied since the Case 1 conditions are more similar to a pipe connected to a large mass structure, such as a pressure vessel or a pump. For the cases summarized in Table 1, we performed a parametric study on pipe length, rm/t, and Rb/rm to achieve a more precise solution. The results are shown in Fig. 8. Fig. 8 is the analysis results with different lengths of straight pipe connected to the elbow. As seen in Fig. 8, there is no length effect. Therefore, we
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analysis only for shortest straight pipe case to reduce analysis time
Figure 8. The effects of straight pipe length attached to both elbow ends
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2.3 Limit load solutions for crack positions 2.3.1 Extrados crack Stress distributions resulted from FEA are shown in Fig. 9. Gray area represents a yield area. As seen in this figure, stress distribution is concentrated at ligament near the crack tip in case of free bending condition. However, in case of considering PIB restraint effect, large area of the elbow yields before plastic collapse occurs. Since the PIB restraints interrupt crack opening, more limit load is required to fully plasticize the net section including crack.
(a) Free bending condition
(b) PIB restraint condition
Figure 9. Stress distribution of the extrados elbow-centered crack
Additionally, stress distribution of the net section is displayed in Fig. 10. Crack opening stress is extracted along the y direction from crack tip and it is normalized by flow stress. Stress distributions of free bending condition and PIB restraint condition are compared to each other. Most of ligament is plasticized to the tensile direction at the fracture moment. The difference between two conditions is noticeable after 40 mm point where the compression stress reaches the flow stress. As indicated by blue arrows, when considering PIB restraint effect, the magnitude of bending moment decreases due to International Journal of Pressure Vessels and Piping
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reaction moment caused by the boundary condition. Eventually, the PIB restraint effect suppresses the bending deformation of the pipe resulting in an increase of the limit load
Figure 10. PIB restraint effect on stress distribution of the extrados elbow-centered crack
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Figure 11. The effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the extra dos elbow-centered crack
Fig. 11 presents the results for the extrados crack in elbow center, including the effects of different rm/t values and different curvatures of the pipe. The correction factor for the extrados cracked model is affected by θ/π, rm/t and Rb/rm. As the crack length becames longer, the PIB restraint effect increases the limit load so that the pipe could endure until it fully collapses. It was especially noticeable that the results for the free bending condition are the same as those for the PIB restraint effect at a certain length crack or less. The PIB restraint effect does not affect the models with a crack length of θ/π ≤ 0.25. rm/t and Rb/rm also have no effect on the analysis results in this range. From the FE analysis, the PIB restraint effect is valid for models with θ/π > 0.25. Accordingly, we propose correction factors for the boundary conditions in the crack length range of 0.25 to 0.50. In cases with θ/π = 0.50, where the PIB restraint effect is the most significant, the analysis results are distributed with a deviation of 24.9% due to the rm/t effect. The effects of Rb/rm are 14.2% (rm/t = 5), 12.1 (rm/t = 10), and 11.0% (rm/t = 20), depending on rm/t. The correction factor of the PIB restraint effect for the extrados crack in an elbow center is proposed in Eq. (5), with θ/π, rm/t, and Rb/rm as its variables. The equation is extracted by curve-fitting for representative data and then the θ/π, rm/t, and Rb/rm effects are considered.
6^/_, ;< /A, : /;< = = `
1 1 + 2 ∙ 86^/_ − 0.25=!
^/_ ≤ 0.25 0.25 < ^/_ ≤ 0.50
A = 20.14 − 0.4456r /A=
(5)
B = 0.9 + 0.036R /;< =
2.3.2 Intrados crack When there is an intrados crack, it also restrains the crack opening, which increases the limit load compared to a free bending condition. Stress distributions of the case of intrados crack are displayed in Fig. 12. Similar to the case of extrados crack, plastic zone of a free bending condition is limited near the crack tip. However, the pipe with a PIB restraint condition is ruptured after plastic zone widely propagates along the longitudinal direction. International Journal of Pressure Vessels and Piping
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(a) Free bending condition
(b) PIB restraint condition
Figure 12. Stress distribution of the intrados elbow-centered crack
Stress distribution of the pipe with intrados elbow-centered crack is presented in Fig. 13. Similar to the extrados case, the difference between two conditions, indicated by blue arrows, stands out when the compressive stress reaches the flow stress. Remarkably, the bending effect almost disappears in a PIB restraint condition and every ligament is yielded by membrane stress. As a result, plastic zone propagates as shown in Fig. 12(b). Consequently, the PIB restraint effect hinders the behavior of pipe and increases the limit load, not only in the extrados case, but also in the intrados case.
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Figure 13. PIB restraint effect on stress distribution of the intrados elbow-centered crack
When there is an intrados crack, it also restrains the crack opening, which increases the limit load compared to a free bending condition. Fig. 14 present the results for the intrados crack, including the effects of different rm/t values and different curvatures of the pipe. In the case of the intrados crack, seen in Fig. 14, all parameters except for θ/π had a negligible effect on the limit load.
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Figure 14. The effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the extra dos elbow-centered crack
The same as with the extrados crack, the correction factor increased rapidly after θ/π = 0.25. The PIB restraint effect correction factor of the intrados crack is approximated by Eq. (6).
6^/_= `
1 1 1 1.366^/_ 7 0.25= 1 17.356^/_ 7 0.25=!
^/_ a 0.25 0.25 b ^/_ a 0.50
(6)
The proposed equation indicates a maximum difference of 5.6% from the limit load analysis result at θ/π = 0.50.
3. LIMIT LOAD SOLUTION FOR JUNCTION BETWEEN STRAIGHT PIPE AND ELBOW Nuclear power plant pipe systems consist of various components to effectively transport fluids and International Journal of Pressure Vessels and Piping
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their contained energy. The components are usually welded together and, in general, the weldments are designed to have more structural stiffness than the base material. However, due to the inherent difficulties in the welding mechanism, cracks are usually reported near welds. According to previous studies, LBB evaluation must include weldment evaluation [2]. Other studies [21-23] suggested limit load solutions for both straight pipe and elbow junction, but they all assumed a free bending condition. In this study, using the aforementioned limit load solutions for both straight pipes and elbows, a new limit load solution and correction factor on straight pipe-elbow junction is suggested, considering the PIB restraint effect. 3.1 Geometry and finite element model Here, the crack was assumed to be located in a 90° elbow and straight pipe connection, whose schematics are shown in Fig. 4. For direct comparison and considering the actual geometry of nuclear power plant components, all of the simulation cases are used from Table 1. A circumferential throughwall crack was assumed and positioned at the junction. Fig. 15 presents the FE model we used, which is a half symmetric model for effective calculation. The boundary conditions and loading conditions were same as described in Section 2.
Figure 15. Finite element model of a pipe assembly consisting of a straight pipe and an elbow with a crack on the junction
Similar to the case of elbow-centered crack, our FE model for elbow-straight pipe junction crack is also verified by comparison with the previous study [21]. Fig. 16 shows that the results from the proposed International Journal of Pressure Vessels and Piping
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model here (except for the PIB restraint effect) are well matched with the reference data.
(a) Limit load equation of extrados crack as function of rm/t and θ/π
(b) Limit load equation of intrados crack as function of θ/π only Figure 16. Model verification for elbow-straight pipe junction crack cases
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3.2 Limit load analysis When there is no crack, the limit load can be calculated by Eq. (2). Also, from previous work [23], the limit load of junction through-wall crack in an elbow-straight pipe assembly under a free bending condition is not significantly different from the limit load calculated from Eq. 3(a) and 3(b). For the model where the crack was between a straight pipe and an elbow, the sensitivity evaluations of the boundary condition and the straight pipe length effect were carried out according to the same procedure as in Section 2.2. The three boundary cases in Fig. 4 were considered. The FE analyses were performed with each boundary case by changing the lengths of the straight pipes connected at both ends of the elbow. From the comparison we concluded that there was no significant effect of those variables. The Case 1 boundary condition and the shortest straight pipe for the junction-cracked model was applied to describe the PIB restraint effect. 3.3 Limit load solutions for crack positions 3.3.1 Extrados crack In free bending conditions, the longer the crack length, the larger the deformation (i.e., displacement and rotation angle) of pipe ends due to crack opening. On the other hand, considering the PIB restraint effect, the boundary condition doesn’t allow any deformation of pipe ends by generating reaction force and moment. Therefore, a pipe with longer crack length is expected to yield larger deformation under free bending conditions. Conversely, that means the PIB restraint effect increases in this case in order to compensate for that large deformation. Empirically, the PIB restraint effect should be considered in the range of θ/π ≥ 0.25. It is also noted that rm/t and Rb/rm affect to the relatively long crack. In addition, Fig. 17 shows stress distributions resulted from FEA. Unlike a free bending condition, the plastic zone in a PIB restraint condition propagates laterally prior to plasticizing ligament of the net section including crack
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(a) Free bending condition
(b) PIB restraint condition
Figure 17. Stress distribution of the extrados junction crack
The stress distribution for ligament is presented in Fig. 18. In case of a free bending condition, compressive stress determines the neutral axis far from the crack tip. When the PIB restraint boundary condition is applied, bending behavior of the pipe is limited resulting that the neutral axis is shifted further away from the center line.
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Figure 18. PIB restraint effect on stress distribution of the extrados junction crack
Figure 19. Effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the extrad os junction crack
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The limit load analysis and the results are illustrated in Fig. 19. Fig. 19 is the correction factor distributions for crack length of the extrados case. The data for the model with a crack in the junction between the straight pipe and the elbow are more widely distributed than for the elbow-centered-crack case. This was likely due to the asymmetry between the geometry and its volume on both sides of the crack. Eq. (7), which was derived from the data in Fig. 19, shows the correction factor of the PIB restraint effect for the model where an extrados crack existed in the weldment. The boundary condition effect does not appear for θ/π ≤ 0.25, but rm/t and Rb/rm have a significant effect on the relatively long crack.
6^/_, ;< /A, : /;< = = ` C 20.36 − 0.5056r /A=
1 1 + e ∙ f6^/_ − 0.25=!
^/_ ≤ 0.25 0.25 < ^/_ ≤ 0.50
B 0.78 + 0.056R /;< =
(7)
The PIB restraint correction factor was more sensitive to Rb/rm than in the case where the crack was in the elbow center. We believe that asymmetry at both sides of the crack caused this result. Since the crack existed in the connection between the straight pipe and the elbow, the geometry of one side of the crack was straight pipe, and the other side remained as elbow and straight pipe. Therefore, the effect of Rb/rm was predominantly on the elbow and straight pipe side. This is the factor that increased the Rb/rm effect. In the θ/π = 0.50 cases, the data were scattered with a deviation of 29.2% by the rm/t effect. The effects of Rb/rm were 20.6% (rm/t = 5), 24.5 (rm/t = 10), and 25.3% (rm/t = 20).
3.3.2 Intrados crack FEA results of intrados crack on the junction are illustrated in Fig. 20. As described in the extrados case, collapse region of the pipe is very localized in a free bending condition, while collapse region of the PIB restraint pipe grows into elbow side before the ligament becomes fully plastic.
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(a) Free bending condition
(b) PIB restraint condition
Figure 20. Stress distribution of the intrados junction crack
Fig. 21 displays stress distribution of the intrados junction crack case. Similar to the aforementioned three cases, compressive stress eventually disappears because bending moment is canceled by the PIB restraint effect.
Figure 21. PIB restraint effect on stress distribution of the intrados junction crack International Journal of Pressure Vessels and Piping
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Figure 22. Effects of θ/π, rm/t, and Rb/rm on the PIB correction factor, f, of the intrad os junction crack
Fig. 22 is the correction factor distributions for crack length of intrados crack. rm/t and Rb/rm had little effect on the elbow-centered-crack and junction-crack cases. In these limit load analysis results of the elbow, including the intrados crack on the junction, the PIB restraint effect occurred at θ/π ≥ 0.25. Eq. (8) is the correction factor estimating equation. The factors have low sensitivity of rm/t and Rb/rm, and the equation is well matched with a maximum of 4.2% difference.
6^/_= `
1 1 1 0.2556^/_ 7 0.25= 1 23.466^/_ 7 0.25=!
^/_ a 0.25 0.25 b ^/_ a 0.50
(8)
Using Eqs. (2) and (3b), one can calculate the limit load for the cracked elbow with a free bending condition, and then calculate the limit load considering the PIB restraint effect with the correction factor of Eq. (8).
4. CONCLUSION International Journal of Pressure Vessels and Piping
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Nuclear power plants have approximately 80 km of pipes per unit. Therefore, maintaining the structural integrity of these pipes is critical to the safe operation of the plants. LBB evaluation is performed at the design stage to ensure structural integrity, and various fracture mechanics parameters are used in this process. However, previous studies have considered a free bending condition for such analysis. In real pipes, the PIB restraint effect is engendered by the supports, concrete structures, and other components. This boundary condition must be considered in an LBB analysis for a precise calculation of fracture mechanics parameters. This study used FE analysis to suggest limit load solutions for elbows and their welding regions with circumferential through-wall cracks under PIB restraint effects. To consider the PIB restraint effect, the rotational movement of the elbow was constrained by applying internal pressure to the pipe, while moments were constrained by boundary conditions. Various FE analyses were performed by changing the elbow curvature, length of connected straight pipe, rm/t, crack length, and crack location. Analysis results revealed that the elbow curvature and straight pipe length had no effect on the limit load, and this was confirmed by the pipe rotational angle. Correction factors of the PIB restraint effect were developed from closed-form equations using the parameters that affected the limit load, which were rm/t, Rb/rm, crack length, and crack location. Although this study can be applied to evaluate the structural integrity of cracks located in the center of an elbow or a junction (between straight pipe and elbow) made of highly ductile austenitic steel, there are still several limitations. First, only an ideal boundary condition is considered by assuming that both ends of a pipe are fully restrained by two large mass components. However, in many cases, the restraint effect is caused by the complex piping system (i.e., combination of straight and elbow pipes) itself. Second, the equations presented here don’t take into account eccentricity of the crown crack. Therefore, these limitations should be carefully considered in practical use.
ACKNOWLEDGMENT This work was supported by the National Research Foundation of Korea [grant number: NRF2019M2D2A1A02057338].
REFERENCES International Journal of Pressure Vessels and Piping
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[1] Cho, D., Development of non-linear fracture mechanics parameters estimation for nonidealized through-wall crack in pipes, Ph.D. dissertation, Sungkyunkwan University, 2013, https://lib.skku.edu/suwon/en/#/search/detail/4133802 . [2] Scott, P., Olson, R., Wilkowski, G., Development of technical basis for leak-before-break evaluation procedures, Nuclear Regulatory Commission, 2002, NUREG/CR-6765-Vol.1, https://www.nrc.gov/reading-rm/doc-collections/nuregs/contract/cr6765/#pub-info . [3] Swamy, S.A., Bhowmick, D.C., Prager, D.E., Tanaka, T., Ogata, Y., Yokota, H., LBB evaluation for a typical Japanese PWR primary loop by using the U.S., NRC approved method, Nuclear Regulatory Commission, 1997, NUREG/CP-0115, https://inis.iaea.org/search/searchsinglerecord.aspx?recordsFor=SingleRecord&RN=28076014 . [4] Swamy, S.A., Mandava, P.R., Bhowmick, D.C., Prager, D.E., LBB considerations for a new plant design, Nuclear Regulatory Commission, 1997, NUREG/CP-0155, https://inis.iaea.org/search/searchsinglerecord.aspx?recordsFor=SingleRecord&RN=28076030 . [5] Miura, N., Evaluation of crack opening behavior for cracked pipes effect of restraint on crack opening, ASME pressure vessels and piping conference 423 (2001) 135-143, http://refhub.elsevier.com/S0029-5493(16)00089-3/sbref0035 . [6] Kim, H.S., Jin, T.E., Kim, H.D., Chang, Y.S., Kim, Y.J., Restraining effect of support plates on the limit loads for circumferential cracks in the steam generator tube, Journal of nuclear engineering and design 238(1) (2008) 135-142, https://doi.org/10.1016/j.nucengdes.2007.06.002 . [7] Kim, Y., Hwang, I.S., Oh, Y.J., Effective applied moment in circumferential through-wall cracked pipes for leak-before-break evaluation considering pipe restraint effects, Journal of nuclear engineering and design 301 (2016) 175-182, https://doi.org/10.1016/j.nucengdes.2016.02.028 . [8] Rahman, S., Ghadiali, N., Wilkowski, G.M., Bonora, N., Effects of off-centered cracks and restraint of induced bending caused by pressure on the crack-opening-area analysis of pipes, Journal of nuclear engineering and design, 167 (1996) 55-67, https://doi.org/10.1016/S00295493(96)01289-7 . [9] Shim, D.J., Plastic limit loads for pipes with part-through surface cracks and its application to failure assessment of pressurized piping, Ph.D. dissertation, Sungkyunkwan University, 2003, https://lib.skku.edu/suwon/en/#/search/detail/1110952 .
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[10] Scott, P., Olson, R., Bockbrader, J., Wilson, M., Gruen, B., Morbitzer, R., Yang, Y., Williams, C., Brust, F., Fredette, L., Ghadiali, N., Wilkowski, G., Rudland, D., Feng, Z., Wolterman, R., The Battelle Integrity of Nuclear Piping (BINP) Program Final Report, Nuclear Regulatory Commission, 2005, NUREG/CR-6837, https://www.nrc.gov/reading-rm/doccollections/nuregs/contract/cr6837/ . [11] Kim, J.W., Evaluation of restraint effect of pressure induced bending on the elastic-plastic crack opening behavior, International journal of pressure vessels and piping 81(4) (2004) 355362, https://doi.org/10.1016/j.ijpvp.2004.02.011. [12] Rahman, S., Brust, F., Ghadiali, N., Krishnaswamy, P., Wilkowski, G., Choi, Y.H., Moberg, F., Brickstad, B., Refinement and evaluation of crack-opening-area analyses for circumferential through-wall cracks in pipes, Nuclear Regulatory Commission, 1995, NUREG/CR-6300, https://doi.org/10.2172/46620 . [13] Ghadiali, N., Wilkowski, G., Rahman, S., Choi, Y.H., Deterministic and probabilistic evaluations for uncertainty in pipe fracture parameters in leak-before-break and in-service flaw evaluations, Nuclear Regulatory Commission, 1996, NUREG/CR-6443, https://doi.org/10.2172/256412 . [14] Miller, A.G., Review of limit loads of structures containing defects, Journal of pressure vessels piping 32 (1988) 197-327, https://doi.org/10.1016/0308-0161(88)90073-7 . [15] Kim, Y.J., Je, J., Oh, C., Han, J., Budden, P.J., Plastic loads for 90° thick-walled elbows under combined pressure and bending, International journal of strain analysis 45 (2009) 115127, https://doi.org/10.1243/03093247JSA568 . [16] Mohan, R., Brust, F., Ghadiali, N., Wilkowski, G., Development of a J-estimation scheme for internal circumferential and axial surface cracks in elbows, Nuclear Regulatory Commission, 1996, NUREG/CR-6445, https://doi.org/10.2172/249297 . [17] Chattopadhyay, J., Nathani, D.K., Dutta, B.K., Kushwaha, H.S., Closed-form collapse moment equations of elbows under combined internal pressure and in-plane bending moment, International journal of pressure vessel technology 122 (2000) 431-436, https://doi.org/10.1115/1.1285988 . [18] Robertson, A.C., Li, H., Mackenzie, D., Plastic collapse of pipe bends under combined internal pressure and in-plane bending, International journal of pressure vessels and piping 82(5) (2005) 407-416, https://doi.org/10.1016/j.ijpvp.2004.09.005 . [19] Yahiaoui, K., Moffat, D.G., Moreton, D.N., Piping elbows with cracks part 1: a parametric International Journal of Pressure Vessels and Piping
28
study of the influence of crack size on limit loads due to pressure and opening bending, International journal of strain analysis 35(1) (2000) 35-46, https://doi.org/10.1243/0309324001513991 . [20] Kim, Y., Kim, Y., Song, T., Finite element plastic loads for circumferential cracked pipe bends under in-plane bending, International journal of engineering fracture mechanics 74 (2007) 643-668, https://doi.org/10.1016/j.engfracmech.2006.07.001 . [21] Hong, S.P., Kim, J., Kim, Y., Limit pressures of 90° elbows with circumferential surface cracks, International journal of engineering fracture mechanics 76 (2009) 2202-2216, https://doi.org/10.1016/j.engfracmech.2009.07.005 . [22] Hong, S.P., Effect of Internal Pressure on Plastic Loads of 90° Elbows with Circumferential Cracks under Combined Bending, Ph.D. dissertation, Korea University, 2011, https://library.korea.ac.kr/search/detail/CAT000045696066#.XNI0O-gzaUk. [23] Song, T.K., Kim, Y., Oh, C., Jin T., Kim, J., Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes, International journal of pressure vessels and piping 86 (2009) 495-507, https://doi.org/10.1016/j.ijpvp.2009.03.008 .
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Highlights Limit load solutions are developed by proposing pressure induced bending correction factors for elbow pipe system. Several sensitivity evaluations are performed and the stress field of the net section is discussed. Various crack locations are addressed such as intrados or extrados crack in elbow center or straight pipe-elbow junction.