Engineering estimates of crack opening displacement for non-idealized circumferential through-wall cracks in pipe

Engineering Fracture Mechanics 142 (2015) 236–254

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Engineering estimates of crack opening displacement for non-idealized circumferential through-wall cracks in pipe Doo-Ho Cho a, An-Dong Shin a, Nam-Su Huh b,⇑, Hyun-Ik Jeon c a

Korea Institute of Nuclear Safety, 34 Gwahak-ro, Yuseong-gu, Daejeon 305-338, Republic of Korea Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 139-743, Republic of Korea c Korea Laboratory Engineering System, 40 Techno 3-ro, Yuseong-gu, Daejeon 305-509, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 19 February 2014 Received in revised form 30 May 2015 Accepted 1 June 2015 Available online 9 June 2015 Keywords: Crack opening displacement (COD) Enhanced reference stress (ERS) method Finite element (FE) analysis Circumferential non-idealized through-wall crack (Circumferential NiTWC) Optimized reference load (QoR) Plastic influence function (h2)

a b s t r a c t The present paper provides the engineering estimates of non-linear crack opening displacement (COD) for a circumferential non-idealized through-wall crack in a pipe based on both the GE/EPRI and the enhanced reference stress methods. For the GE/EPRI-type solution, the h2 values are newly calibrated based on the elastic–plastic finite element analyses. Also, elastic COD and the reference load solutions are proposed. Then, the proposed engineering estimates are validated against the FE results using actual tensile data of SA312 Type 316 stainless steel. From these comparisons, it is revealed that the enhanced reference stress method provides the best and overall satisfactory results. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Over the last few decades, in accordance with the U.S. Nuclear Regulatory Commission (USNRC) Standard Review Plan (SRP) 3.6.3 [1], the Leak-Before-Break (LBB) concept has been widely used in a nuclear piping design to exclude the dynamic effect due to postulated high-energy pipe ruptures from the design basis of primary nuclear piping systems. The LBB concept can be applied to nuclear piping design on condition that a crack length corresponding to a detectable leak rate determined through a calculation of a crack opening displacement (COD) would remain stable, that is, this detectable leakage crack length is smaller than a critical crack length that can cause an unstable pipe rupture. In this basic premise, a circumferential through-wall crack (TWC) is typically employed considering dominant stress components acting on a nuclear piping. In this context, an accurate estimation of relevant fracture mechanics parameters, i.e., COD and J-integral, for a circumferential TWC is important in a LBB analysis. Recently, in order to calculate these parameters, finite element (FE) analysis has been popularly adapted. Since a LBB analysis or a structural integrity assessment for defective components, however, often involves numerous parametric and sensitivity analyses, FE analyses sometimes has limitation in calculating fracture parameters for an actual structural integrity assessment. In this respect, the engineering methods to estimate a COD and a J-integral are typically required for a pipe with a circumferential TWC [2–4].

⇑ Corresponding author. Tel.: +82 2 970 6317; fax: +82 2 974 8270. E-mail address: [email protected] (N.-S. Huh). http://dx.doi.org/10.1016/j.engfracmech.2015.06.004 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

D.-H. Cho et al. / Engineering Fracture Mechanics 142 (2015) 236–254

237

Nomenclature a1 reference half crack length at inner surface a2 reference half crack length at outer surface h2(n) plastic influence function for dp in the GE/EPRI method h2(n ¼ 1) value of h2 for the elastic case(n = 1) M global bending moment ML global limit bending moment MoR optimized reference bending moment n strain hardening index for Ramberg-Osgood model p internal pressure pL Internal limit pressure poR optimized reference internal pressure Q generalized load (tension, bending moment or internal pressure) QL generalized limit load QoR optimized reference load (General) r perpendicular distance from the inner surface to the outer surface at the center of crack Ri inner radius of a pipe Rm mean radius of a pipe (=(Ri + Ro)/2) Ro outer radius of a pipe t wall thickness of a pipe T tensile load TL limit tensile load ToR optimized reference tensile load VQ shape factor of the elastic COD for idealized TWC under generalized loading a coefficient of Ramberg-Osgood model cQ non-dimensional factor of optimized reference load for non-idealized circumferential TWC under generalized loading de elastic component of crack opening displacement dp plastic component of crack opening displacement eref reference strain at the reference stress rref h1 half crack angle at inner side in the circumferential cracked pipe h2 half crack angle at outer side in the circumferential cracked pipe m Poisson’s ratio rref reference stress ry yield strength u non-dimensional correction factor of elastic crack opening displacement for circumferential non-idealized TWC in pipe

Among several engineering estimates for predicting a COD and a J-integral, most popular ones are the GE/EPRI method [5,6] using the Ramberg–Osgood (R–O) fit to tensile data and the reference stress (RS) method [7] using actual tensile data of material of interest. Firstly, although the GE/EPRI method [5] has been widely used for practical applications, it has known that the results are very sensitive to how to characterize the stress–strain data of material based on the R–O model [8,9]. On the other hand, the RS method can either reduce a sensitivity associated with a material characterization or give a robust result since this method does not require material idealization, and then this method has been utilized in many structural integrity assessment methods, for instance, R6 code [7]. In the RS method, the reference stress is defined by the plastic limit load of the cracked components. Although, the plastic limit loads are widely available even for complex geometry and loading conditions, this method also suffers from inaccuracy associated with the definition of the reference stress. Thus, to improve the accuracy of the RS method, the Enhanced Reference Stress (ERS) method has been proposed by Kim and Budden [10] for a pipe with a circumferential TWC. The underlying idea of the ERS method is introducing new reference load, called optimized reference load based on the limited FE analyses instead of plastic limit load in the typical RS method. The validity of ERS method has been confirmed by comparing with 19 pipe experimental data, extracted from Kishida and Zahoor [11] and the Pipe Fracture Encyclopedia [12], where the results showed the overall excellent agreement [8]. At this point, it should be pointed out that, although, the value of the COD at the outer surface can be different from that at the inner surface of a pipe, the existing solution [10] gives only the value of COD at the mid-thickness. For this reason, when the detectable leak rate is calculated during the crack growth, the initially penetrated crack is assumed as fully penetrated crack (referred to as idealized TWC in Fig. 1) based on average crack angle concept for conservative LBB analysis [13]. However, it has been reported by Shim [14] that the equivalent area method based on the average crack angle would lead to excessively conservative result for a LBB analysis. Besides, this average crack angle concept does not physically simulate a realistic crack shape development. Note that, as shown in Fig. 1, during an actual crack growth resulting from a fatigue or a

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(a) Surface crack

(b) Initially penetrated crack

(c) Partially penetrated crack (referred to as “non-idealized TWC”)

(d) Fully penetrated crack (referred to as “idealized TWC”)

Fig. 1. Schematics of typical crack growth behavior in pipe (Circumferential direction).

stress corrosion cracking, an internal surface crack develops through the wall thickness of a pipe and may partially form a TWC at its deepest point, which creates a TWC with different crack length on two surfaces, i.e., the inner and outer surfaces of a pipe which is referred to as ‘non-idealized TWC (NiTWC)’ hereafter [15]. On the basis of such a background, to consider more realistic crack growth behavior and improve the accuracy of a LBB analysis, the engineering estimation for a pipe with a NiTWC should be proposed. Recently, for a pipe with a NiTWC, many numerical and experimental works have been conducted to determine the fracture parameters, i.e. elastic stress intensity factors, elastic–plastic J-integrals and CODs, used for predicting crack growth behavior due to fatigue or primary water stress corrosion cracking (PWSCC), instability and leak rate [16–21]. For instance, Huh and Shim [18,19] have proposed the elastic stress intensity factor and COD solutions for a plate and a pipe by performing 3-dimensional (3-D) elastic FE analyses as tabular solutions. Also, Yellowes et al. [20] presented the distributions of J-integral along the crack front based on limit FE analyses for a plate with a NiTWC. Then, Benson et al. [21] have proposed the COD model for axial non-idealized TWCs. However, the engineering estimation methods for calculating the elastic–plastic J-integral and the COD, which are key elements in the LBB analysis, of a pipe with a NiTWC are still lacking. The objective of this paper is to propose new engineering estimates of CODs for a pipe with a NiTWC either based on the GE/EPRI concept or based on the ERS concept. For the GE/EPRI-type estimate, the plastic influence functions (h2 functions) are newly calibrated based on the detailed 3-D elastic–plastic FE analyses. For the ERS-type estimate, the new elastic COD solution as a closed-form expression and the optimized reference load of a pipe with a circumferential NiTWC are proposed. Finally, the confidence of the proposed engineering estimates is gained by comparing the predictions from the present engineering estimates with those from the FE analyses using actual stress–strain data. 2. COD estimations based on the ge/EPRI method 2.1. FE analyses based on the R–O idealization Fig. 2(a) depicts a pipe with a circumferential NiTWC subjected to an axial tension (T), a global bending moment (M) and an internal pressure(p), where Rm, Ri, Ro and t represent the mean radius, inner radius, outer radius and thickness of a pipe, respectively. The r denotes perpendicular distance from the inner surface to the outer surface of a pipe at the center of a circumferentual NiTWC along the thickness. The shape of circumferential NiTWC in a pipe is represented by a half crack angle, i.e., h1 and h2, on the inner and outer surfaces of a pipe, where h1 > h2, which is usually observed during typical crack growth behavior [16,17]. In the present study, to cover the practical wide ranges of crack geometries, sixty cases are considered as summarized in Table 1. In terms of pipe geometries, three different values of Rm/t are employed: Rm/t = 5, 10 and 20, and four values of h1/p are considered: h1/p = 0.125, 0.250, 0.300 and 0.400. Also, five values of h1/h2 are considered to take into account the geometries of the circumferential NiTWC in a pipe as a significant variable in the present work: h1/h2 = 1, 1.5, 2, 3 and 4. Note that h1/h2 = 1 represents an circumferential idealized TWC as demonstrated in Fig. 1. To propose the GE/EPRI-type solution for predicting the CODs of a pipe with a circumferential NiTWC, firstly, the tensile properties for the FE analyses should be characterized in accordance with R–O idealization as follows [22]:



e r r ¼ þa ry eo ry

n ð1Þ

where ry denotes the 0.2% yield strength, eo = ry/E is the reference strain and E is the Young’s modulus, respectively. a and n are the R–O constants. In the present study, the Poisson’s ratio is set to be m = 0.3, and ry = 165 MPa, E = 190 GPa and a = 1 are used. As for strain hardening exponents, three different values of n, n = 1, 3 and 5, are selected because plastic influence functions of the GE/EPRI method are strongly dependent on the strain hardening exponent. Elastic–plastic FE analyses are performed using the general-purpose FE program, ABAQUS [23], where deformation plasticity option within ABAQUS using the R–O constants is invoked. By considering a symmetric condition, only a quarter of a

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T

p

M

(a) Schematics of non-idealized circumferential TWC

(b) Typical FE mesh employed in the present study Fig. 2. Schematics of a pipe with circumferential non-idealized TWC and typical FE mesh employed in the present work.

Table 1 Summary of FE analyses for the present work. Geometry Circumferential TWC

Rm/t 5, 10, 20

h1/p 0.125, 0.25, 0.3, 0.4

h1/h2 1, 1.5, 2, 3, 4

Loading condition

n

Axial tension, Bending moment, Internal pressure

1, 3, 5

pipe is modeled. Fig. 2(b) shows the typical FE meshes where reduced integration, 20-nodes brick elements (element type C3D20R in ABAQUS element library) are used. The numbers of nodes and elements in a typical FE mesh are approximately 80,978 and 9360, respectively. Although, there is some curvature along the crack front that is resulting from the cylindrical

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transformation of a straight crack front in a plate, it has been reported that this technique does not affect the calculation of the results [18,19]. As shown in Fig. 2(b), each crack tip of the FE model is modeled with focused wedge-type elements. Note that, to simulate the crack tip singularity for an elastic–plastic analysis, crack tip nodes of focused element are not constrained together and the mid-nodes are not moved to the quarter points. Also, in order to obtain the values of COD at the various locations along the thickness at the center of the crack, twenty elements are used along the thickness of a pipe. The validation of the mesh sensitivity to the results has been made in the previous study. [24]. As for a loading condition, an axial tension, a global bending moment and an internal pressure are considered. In particular, the internal pressure is applied as a distributed load to the inner surface of a pipe together with a equivalent axial tension to the end of a pipe to simulate the closed-end effect of a pipe. Furthermore, to consider the effect of crack face pressure, 50% of the internal pressure is also applied to the crack face in the present work [25]. 2.2. Proposed GE/EPRI-type COD estimates According to the fundamental concept of the GE/EPRI method, the plastic components of CODs, dnon-idealized , of a pipe with a p circumferential NiTWC using the plastic influence function, i.e. h2, are proposed in the present study as

dnon-idealized p;Q;i

¼ aeo Rm p

non-idealized h2;Q ;i

!n   Rm h1 h1 Q ; n; ; t p h2 Q idealized L

ð2Þ

where subscript ‘i’ means the location at the center of a NiTWC along the thickness, i.e., innermost, mid-thickness, and outermost surface, and Q is the generalized loads (axial tension, global bending moment and internal pressure). Q idealized is generL alized plastic limit load of idealized TWC, which is defined for each loading mode as follows [26].

10

n=3 n=5 Inner Mid. Outer

h2 (n)

8

6

4

2

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

idealized

T/TL

(a) Tension 10

n=3 n=5 Inner Mid. Outer

h2 (n)

8

6

4

2

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

idealized

M/ML

(b) Bending moment Fig. 3. Variations of the h2(n) values obtained from the FE results, with the load magnitude for Rm/t = 10, h1/p = 0.4 and h1/h2 = 2 under (a) axial tension and (b) global bending moment.

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( )  2   h1 h1 þ 3:14 T idealized ¼ 2R t r 2:19  6:27 m y L

p

ð3Þ

p

( Midealized L

¼

4R2m t

)  2   h1 h1 þ1 0:35  1:74

ry

2try pidealized ¼ pffiffiffi L 3Rm

p

( 0:92

 2 h1

p

ð4Þ

p

 0:28

  h1

p

) þ1

ð5Þ non-idealized

As noted in Eq. (2), the plastic influence functions of a pipe with a circumferential NiTWC, h2 , are functions of Rm/t, strain hardening exponent n, reference crack length h1/p, shape of NiTWC h1/h2, and locations along the thickness i, whereas the plastic influence functions of idealized TWC depend only on Rm/t, strain hardening exponent n and reference crack length h1/p. Another notable point is that, in the Eq. (2) for predicting the CODs of circumferential NiTWC, the plastic limit load of , is used as a normalizing load instead of that of NiTWC for the simplicity when calculating the CODs idealized TWC, Q idealized L of NiTWC using Eq. (2) proposed in the present study. non-idealized

The values of h2

of Eq. (2) are calibrated using the present FE results for each strain hardening exponent as follows.

, from the total FE CODs at each location along The plastic CODs are determined by subtracting elastic components, dnon-idealized e the thickness as follows. non-idealized non-idealized dp;Q;i ¼ dnon-idealized  de;Q;i FE;Q ;i

non-idealized

non-idealized de;Q;i ¼ aeo Rm ph2;Q ;i



ð6Þ

Rm h1 h1 ; n ¼ 1; ; t p h2



Q

! ð7Þ

Q idealized L

The elastic FE analyses with n = 1 provide the elastic values of CODs of a pipe with a circumferential NiTWC. In principle, the calibrated values of h2 strongly depend on the magnitude of the applied load, thus the h2 values for a NiTWC by using the present FE results are determined as asymptotic value at sufficiently larger loads to obtain fully plastic components of CODs, as given in Fig. 3, according to the fundamental premise of the GE/EPRI method as shown in Eq. (2) [8,9,21]. To prove the accuracy of the procedure employed in the present study to calibrate the plastic influence functions of the CODs, the plastic influence functions for an idealized TWC (h1/h2 = 1) from the procedure employed in the present study are compared with those from the existing solution [5] in Table 2. As summarized in Table 2, the values of h2 for idealized TWC obtained from the procedure employed in the present study are in good agreement with those from the existing solution. Fig. 4 shows typical variations of h2 for NiTWC along the thickness, and the resulting values of h2 at the three important locations along the thickness, i.e., inner surface, mid-thickness and outer surface, are tabulated in Tables 3–5. For brevity, although only results for a pipe under axial tension with Rm/t = 5 are given, the general trend of overall cases is identical. As shown in Fig. 4, as crack shape becomes sharper, i.e. as the values of h1/h2 increase, the values of plastic influence functions rapidly decrease, which leads to smaller CODs. Finally, the predictions of CODs of a pipe with a circumferential NiTWC at each location along the thickness can be made by adding its elastic component to plastic component using the plastic influence functions proposed in the present study as follows.

Table 2 Comparisons of plastic influence function, h2, for plastic COD between FE result and existing solution for pipe with circumferential idealized TWC under axial tension (h1/h2 = 1). Rm/t

h1/p

Axial tension, n = 3

Bending moment, n = 5

Zahoor [5]

FE

Diff. (%)

Zahoor [5]

FE

Diff. (%)

5

0.125 0.250 0.300 0.400

0.533 0.859 0.932 0.986

0.514 0.851 0.914 0.944

3.528 0.965 1.976 4.214

0.629 0.783 0.808 0.811

0.612 0.754 0.763 0.743

2.703 3.753 5.541 8.409

10

0.125 0.250 0.300 0.400

0.629 1.134 1.207 1.282

0.616 1.158 1.265 1.306

2.080 2.140 4.819 1.838

0.802 1.096 1.102 1.108

0.732 1.022 1.061 1.047

8.678 6.793 3.691 5.524

20

0.125 0.250 0.300 0.400

0.797 1.550 1.660 1.747

0.789 1.624 1.776 1.808

0.968 4.777 6.980 3.496

1.081 1.547 1.554 1.525

1.020 1.520 1.531 1.496

5.643 1.745 1.505 1.898

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1.2

θ1/θ2

1.0

1.5

2.0

3.0

1.2

4.0

Non-idealized TWC

1.0

1.5

2.0

3.0

4.0

0.9

0.6

Inner

Mid.

Outer

r/t =0.0

r/t =0.5

r/t =1.0

h2 (n)

h2 (n)

0.9

0.3

0.6

0.3

0.0 0.0

0.2

1.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

0.4

0.6

0.8

r/t

r/t

(a) θ1/π = 0.125

(b) θ1/π = 0.250

θ1/θ2

1.0

1.5

2.0

3.0

1.2

4.0

Non-idealized TWC

θ1/θ2

1.0

1.5

2.0

3.0

1.0

4.0

Non-idealized TWC

0.9

0.9

h2 (n)

h2 (n)

θ1/θ2

Non-idealized TWC

0.6

0.6

0.3

0.3

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.0

0.2

0.4

0.6

0.8

r/t

r/t

(c) θ1/π = 0.300

(d) θ1/π = 0.400

1.0

Fig. 4. The distributions of h2 values at the center of crack along the thickness of circumferential non-idealized through-wall cracked pipe under axial tension for Rm/t = 5 and n = 5.

Table 3 Values of the plastic influence functions, h2(n), at inner surface point of pipe with circumferential non-idealized TWC under axial tension. Rm/t

h1/p

h1/h2

5

0.125

1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0

0.250

0.300

0.400

non-idealized dnon-idealized ¼ de;Q þ dnon-idealized total;Q ;i ;i p;Q ;i

n 1

3

5

0.375 0.307 0.275 0.245 0.230 0.789 0.516 0.402 0.314 0.279 0.991 0.594 0.436 0.320 0.275 1.439 0.726 0.472 0.302 0.243

0.438 0.311 0.258 0.212 0.191 0.736 0.387 0.263 0.174 0.140 0.806 0.379 0.240 0.146 0.112 0.861 0.295 0.161 0.083 0.058

0.418 0.264 0.206 0.157 0.135 0.541 0.229 0.139 0.082 0.061 0.540 0.185 0.102 0.054 0.039 0.511 0.096 0.041 0.018 0.011

ð8Þ

The proposed GE/EPRI-type estimates can be applied for 5 6 Rm =t 6 20; 0:125 6 h1 =p 6 0:400; 1 6 h1 =h2 6 4, and 1 6 n 6 5, and the plastic influence functions can be interpolated to predict the values of CODs of a pipe with a circumferential NiTWC of interest.

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D.-H. Cho et al. / Engineering Fracture Mechanics 142 (2015) 236–254 Table 4 Values of the plastic influence functions, h2(n), at mid-thickness point of pipe with circumferential non-idealized TWC under axial tension. Rm/t

h1/p

h1/h2

5

0.125

1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0

0.250

0.300

0.400

n 1

3

5

0.451 0.333 0.275 0.217 0.189 0.950 0.612 0.456 0.323 0.264 1.171 0.709 0.507 0.342 0.272 1.638 0.854 0.557 0.341 0.258

0.518 0.356 0.282 0.212 0.177 0.856 0.460 0.313 0.202 0.157 0.919 0.445 0.284 0.171 0.129 0.949 0.336 0.188 0.098 0.069

0.479 0.297 0.224 0.159 0.128 0.608 0.264 0.161 0.093 0.068 0.599 0.211 0.118 0.062 0.044 0.552 0.107 0.047 0.020 0.013

Table 5 Values of the plastic influence functions, h2(n), at outer surface point of pipe with circumferential non-idealized TWC under axial tension. Rm/t

h1/p

h1/h2

5

0.125

1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0

0.250

0.300

0.400

n 1

3

5

0.526 0.367 0.286 0.205 0.164 1.096 0.703 0.512 0.340 0.263 1.334 0.815 0.576 0.370 0.282 1.819 0.973 0.638 0.382 0.281

0.596 0.403 0.312 0.223 0.178 0.965 0.527 0.359 0.229 0.175 1.020 0.505 0.325 0.196 0.146 1.026 0.373 0.212 0.112 0.079

0.537 0.329 0.244 0.166 0.129 0.670 0.295 0.181 0.104 0.075 0.652 0.234 0.132 0.070 0.049 0.589 0.116 0.052 0.023 0.015

3. COD estimations based on the ERS method 3.1. Proposed ERS-type COD estimates For a pipe with a circumferential NiTWC, the COD estimates based on the reference stress concept can be expressed as non-idealized dtotal;Q ;i non-idealized de;Q;i

¼

Eeref

rref

þ

1 2



rref ry

2

rref

ð9Þ

Eeref

where the rref is reference stress, and eref is true reference strain at r = rref. The reference stress is defined as

rref ¼

Q Q non-idealized oR

ry ¼

T non-idealized T oR

ry ¼

M non-idealized M oR

ry ¼

p ry pnon-idealized oR

where Q non-idealized is the optimized reference load for each loading mode in reference stress based approach. oR

ð10Þ

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Table 6 Coefficients of correction factor (uQ,i) for the elastic COD in Eq. (13). Type

Coeffi.

Point

j=1

j=2

j=3

j=4

j=5

j=6

uT,i

cj

Inner Mid. Outer Inner Mid. Outer

0.726 0.868 1.044 0.398 0.737 1.042

0.322 0.560 1.172 7.277 6.189 5.139

0.055 0.170 0.035 4.941 3.533 1.941

4.161 1.095 0.445 7.603 4.774 4.616

0.953 0.631 1.426 0.903 0.500 1.010

1.622 1.003 3.120 22.867 18.650 16.800

Inner Mid. Outer Inner Mid. Outer

0.754 0.874 1.006 0.482 0.759 0.976

0.109 0.614 1.622 7.528 6.304 5.065

0.149 0.123 0.133 4.657 3.512 2.345

3.071 1.033 0.622 6.130 4.777 5.574

1.137 0.325 1.141 1.169 0.858 0.242

0.863 1.298 3.327 22.433 19.183 17.283

Inner Mid. Outer Inner Mid. Outer

0.728 0.867 1.044 0.400 0.732 1.042

0.308 0.563 1.179 7.297 6.235 5.143

0.059 0.175 0.035 4.928 3.581 1.948

4.128 1.097 0.439 7.575 4.715 4.601

0.953 0.640 1.428 0.886 0.603 0.992

1.638 1.013 3.137 22.917 18.850 16.800

dj

uM,i

cj

dj

up,i

cj

dj

When calculating the CODs of a NiTWC according to the Eqs. (9) and (10), two key elements are the elastic COD at each location along the thickness and the definition of reference load. In this context, the closed-form expression of elastic COD and the new reference load of a pipe with a circumferential NiTWC are proposed in the present study, which is described in the following section. 3.2. New elastic COD and reference load for a circumferential NiTWC in a pipe Based on the present FE results, the elastic COD values are directly obtained from the FE displacement results at the center of the crack along the thickness, and a correction factor, u, which is simply the ratio of the COD for NiTWC to that for idealized TWC, is newly introduced in the present study to quantify the effect of NiTWC on elastic COD values.

dnon-idealized ¼ didealized  uQ;i ¼ e;Q;i e;Q

4r1 a1 V Q  uQ ;i E

ð11Þ

where subscript ‘Q’ denotes the generalized load, i.e., tension, global bending moment and internal pressure, and the reference half crack length at inner surface, a1, is Rmh1. VQ is the shape factor of a circumferential idealized TWC, which can be found in Ref. [5,6]. r1 denotes the remote nominal stress acting on a pipe as follows.

1

r ¼

8 T > > < 2pRm t

for tension

M pR2m t

for bending

> > : pRm

ð12Þ

for pressure

2t

The closed-form expression of uQ for calculating elastic CODs of a circumferential NiTWC is proposed in the present study using the present elastic results based on the numerical regression as follows.

uQ ;i ¼ C Q ;i

 DQ ;i h1 h2

C Q ;i ¼ c1 þ c2

ð13Þ



       2  2 t h1 h1 t h1 t þ c3 þ c4 þ c5 þ c6 Rm Rm p p Rm p

 DQ ;i ¼ d1 þ d2

       2  2 t h1 h1 t h1 t þ d3 þ d4 þ d5 þ d6 Rm Rm p p Rm p

where cj and dj (j = 1  6) mean the coefficient of closed-form expression, and they are functions of Rm/t and h1/p. The resulting values of these coefficients are provided in Table 6 and are also compared with the FE results and Ref. [19] in Fig. 5. As shown in Fig. 5, the elastic CODs of NiTWC using the proposed closed-form expression agree well with the FE results and Ref. [19] within maximum difference of 5% regardless of loading mode, Rm/t, h1/h2 and h1/p. In terms of reference load of a NiTWC in Eq. (10), following expression for reference load is proposed in the present study based on the author’s FE plastic limit loads of a pipe with a circumferential NiTWC performed in Ref. [27].

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1.5

θ1/π 0.125

Inner

0.9

0.250

0.9

0.6

0.6

0.3

0.3

0.0

0.0 1

2

3

1

3

(a) Rm/t=5, Inner

(b) Rm/t=5, Outer

Inner

1.5

0.250

FE result Eq.(13) Ref.[19]

φp,i

0.6

0.3

0.3

0.0 3

0.250

0.9

0.6

2

θ1/π 0.125

4

FE result Eq.(13) Ref.[19]

Outer

1.2

0.9

1

2

θ1/θ2

θ1/π 0.125

1.2

4

θ1/θ2

1.5

φp,i

θ1/π 0.125 FE result Eq.(13) Ref.[19]

Outer

1.2

φp,i

φp,i

1.2

1.5

0.250

FE result Eq.(13) Ref.[19]

0.0

4

1

2

3

θ1/θ2

θ1/θ2

(c) Rm/t=10, Inner

(d) Rm/t=10, Outer

4

Fig. 5. Comparisons of correction factors for elastic COD, up,i, from Eq. (13) with those from FE results [19] for a pipe with a circumferential non-idealized TWC under internal pressure.

800

Stress (MPa)

600

400

Test data Fit A(Entire curve) Fit B(~5%) Fit C(0.1%~0.8 εu)

200

0 0.0

0.1

0.2

0.3

Strain Fig. 6. Stress–strain curve and three different R–O fitting results for SA312 TP316 steel (288 °C).

Q non-idealized ¼ cQ Q non-idealized oR L

ð14Þ

where QL are the generalized plastic limit loads for each loading mode. Based on the force and moment equilibrium and the thin-wall assumption for a NiTWC, the plastic limit loads of a NiTWC have been proposed as follows [27].

   h1 þ h2 2 1 cos h2  cos h1 T non-idealized ¼ 2 p R t r 1   sin m y L 2p p 2ðh1  h2 Þ

ð15Þ

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0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.06

0.04

δ /a1

δ /a1

0.06

Inner

0.02

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.6

non-idealized

T/ToR

1.0

1.2

1.4

1.0

1.2

1.4

1.0

1.2

1.4

T/ToR 0.08

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.06

Mid.

0.06

0.04

δ /a1

δ /a1

0.8 non-idealized

0.08

0.02

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Mid.

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.6

0.8 non-idealized

T/ToR

non-idealized

T/ToR 0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.06

Outer

0.06

0.04

δ /a1

δ /a1

Inner

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.04

0.02

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.0

non-idealized

T/ToR

(a) θ1/θ2=1.5

0.2

0.4

0.6

0.8 non-idealized

T/ToR

(b) θ1/θ2=3

Fig. 7. Comparisons of normalized COD values obtained from FE results with the proposed solutions based on the ERS method and the GE/EPRI method along the thickness for Rm/t = 5 and h1/p = 0.125 under tension.

MLnon-idealized ¼

pnon-idealized ¼ L

   R2m t ry cos h1  cos h2 p 1  2 sin  ðh1 þ h2 Þ 2 h1  h2 2 4

   2t ry h1 þ h2 2 1 cos h2  cos h1 1  sin Rm 2p p 2ðh1  h2 Þ

ð16Þ

ð17Þ

By comparing the FE plastic limit loads in Ref. [27] with the results from Eqs. (15)–(17), the non-dimensional factor, cQ, for each loading mode has been suggested as a function of h1/p [27].

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0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.06

δ /a1

δ /a1

0.06

Inner

0.04

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.8

1.0

1.2

1.4

1.0

1.2

1.4

1.0

1.2

1.4

non-idealized

0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Mid.

0.06

δ /a1

0.06

δ /a1

0.6

M/MoR

non-idealized

M/MoR

0.04

0.02

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Mid.

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

non-idealized

M/MoR

0.6

0.8 non-idealized

M/MoR

0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.06

δ /a1

0.06

δ /a1

Inner

0.04

0.02

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

non-idealized

M/MoR

(a) θ1/θ2=1.5

0.00 0.0

0.2

0.4

0.6

0.8 non-idealized

M/MoR

(b) θ1/θ2=3

Fig. 8. Comparisons of normalized COD values obtained from FE results with the proposed solutions based on the ERS method and the GE/EPRI method along the thickness for Rm/t = 5 and h1/p = 0.125 under bending moment.

!  2   2 h1 h1 þ 0:9365 cT ¼ pffiffiffi 0:2143 þ 0:5072 p p 3

ð18Þ

!  2   2 h1 h1 p ffiffiffi þ 0:8622 0:8590 cM ¼ þ 0:1124 p p 3

ð19Þ

 2 h1

cp ¼ 0:7973

p

!   h1 þ 0:5487 þ 0:9349

p

ð20Þ

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0.08 Inner

0.06

δ /a1

δ /a1

0.06

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.00 0.0

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

non-idealized

0.06

δ /a1

δ /a1

Mid.

0.02

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

1.4

1.0

1.2

1.4

1.0

1.2

1.4

Mid.

0.02

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

p/poR

0.6

0.8 non-idealized

p/poR

0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.06

δ /a1

δ /a1

1.2

0.04

non-idealized

0.04

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.04

0.02

0.02

0.00 0.0

1.0

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.06

0.8

p/poR

0.08

0.00 0.0

0.6

non-idealized

p/poR

0.06

Inner

0.2

0.4

0.6

0.8

1.0

non-idealized

p/poR

(a) θ1/π =0.125

1.2

1.4

0.00 0.0

0.2

0.4

0.6

0.8 non-idealized

p/poR

(b) θ1/π =0.250

Fig. 9. Comparisons of normalized COD values obtained from FE results with the proposed solutions based on the ERS method and the GE/EPRI method along the thickness for Rm/t = 5 and h1/h2 = 1.5 under internal pressure.

The expressions of non-dimensional factor of cQ are suggested based on the numerical regression and it was in good agreement with FE results within maximum difference of 10% [27]. Finally, using Eq. (9) together with expressions for elastic COD and reference load proposed in the present study, the CODs of a pipe with a circumferential NiTWC can be calculated at each location along the thickness.

4. Elastic–plastic FE validations 4.1. FE analysis To validate the proposed engineering estimates either based on the GE/EPRI method or based on the ERS method, additional elastic–plastic FE analyses are carried out for a pipe with a circumferential NiTWC. Elastic–plastic FE analyses using

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0.08 Inner

0.06

δ /a1

δ /a1

0.06

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.00 0.0

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.06

δ /a1

δ /a1

Mid.

0.02

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

1.4

1.0

1.2

1.4

1.0

1.2

1.4

Mid.

0.02

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

non-idealized

0.6

0.8 non-idealized

T/ToR

0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.06

δ /a1

δ /a1

1.2

0.04

T/ToR

0.04

0.02

0.00 0.0

1.0

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.06

0.8

T/ToR

0.08

0.00 0.0

0.6

non-idealized

non-idealized

T/ToR

0.06

Inner

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.04

0.02

0.2

0.4

0.6

0.8

1.0

1.2

non-idealized

T/ToR

(a) θ1/π =0.125 and θ1/θ2=1.5

1.4

0.00 0.0

0.2

0.4

0.6

0.8 non-idealized

T/ToR

(b) θ1/π =0.250 and θ1/θ2=3.0

Fig. 10. Comparisons of normalized COD values obtained from FE results with the proposed solutions based on the ERS method and the GE/EPRI method along the thickness for Rm/t = 20 under tension.

actual stress–strain data are conducted using incremental plasticity within ABAQUS, where experimental uni-axial tensile data of SA312 Type 316 steel at the temperature of 288 °C [26] is employed. In terms of geometries of a pipe, two different values of Rm/t, Rm/t = 5 and 20, are considered. As for a reference crack length at the inner surface, two different crack lengths, h1/p = 0.125 and 0.250, are selected, and two different values of h1/h2, h1/h2 = 1.5 and 3, are considered for each reference crack length. The predictions from the GE/EPRI-type and the ERS-type estimates proposed in the present study are compared with the FE results using actual stress-stain data at the three important locations along the thickness, i.e. inner surface, mid-thickness and outer surface. When we predict the CODs using the proposed GE/EPRI-type estimate, in order to investigate the sensitivity of the GE/EPRI method associated with the R–O idealization, three different R–O fits are made for various fitting ranges using the ROFIT program [12]: ‘Fit A’ using the entire true stress–strain data up to the ultimate tensile strength (a = 8.42 and

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0.08 Inner

0.06

δ /a1

δ /a1

0.06

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.00 0.0

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.02

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.06

δ /a1

δ /a1

Mid.

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

1.4

1.0

1.2

1.4

1.0

1.2

1.4

Mid.

0.04

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

non-idealized

M/MoR

0.6

0.8 non-idealized

M/MoR 0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.06

δ /a1

δ /a1

1.2

0.02

0.02

0.04

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.04

0.02

0.02

0.00 0.0

1.0

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.06

0.8

M/MoR

0.08

0.00 0.0

0.6

non-idealized

non-idealized

M/MoR

0.06

Inner

0.2

0.4

0.6

0.8

1.0

1.2

non-idealized

M/MoR

(a) θ1/π =0.125 and θ1/θ2=1.5

1.4

0.00 0.0

0.2

0.4

0.6

0.8 non-idealized

M/MoR

(b) θ1/π =0.250 and θ1/θ2=3.0

Fig. 11. Comparisons of normalized COD values obtained from FE results with the proposed solutions based on the ERS method and the GE/EPRI method along the thickness for Rm/t = 20 under bending moment.

n = 2.92), ‘Fit B’ using only initial portion of strain up to 5% strain (a = 5.76 and n = 4.11), and ‘Fit C’ using the strain ranging from 0.1% to 0.8eu,t (a = 6.26 and n = 3.46) where eu,t denotes the true ultimate strain. The actual stress–strain data together with the R–O fitting results are shown in Fig. 6.

4.2. Results Figs. 7–12 compare FE COD results using actual stress–strain data (FE(Test data)) with predictions from the proposed GE/EPRI-type and ERS-type estimates at the three important locations, i.e., inner and outer surfaces and mid-thickness, for each loading mode. In these figures, the total COD is normalized with respect to reference crack length at the inner surface. As mentioned previously, the predictions based on the GE/EPRI-type estimate are made using three different R–O results to investigate the sensitivity of the GE/EPRI-type estimate associated with R–O idealization. Thus, ‘GE/EPRI(Fit A)’,

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0.08

0.08 Inner

0.06

δ /a1

δ /a1

0.06

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

0.04

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.6

Mid.

0.06

δ /a1

δ /a1

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.04

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

1.0

1.2

1.4

1.0

1.2

1.4

Mid.

0.04

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

0.6

non-idealized

p/poR

0.8 non-idealized

p/poR 0.08

0.08 GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.06

δ /a1

δ /a1

1.4

0.02

0.02

0.04

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

Outer

0.04

0.02

0.02

0.00 0.0

1.2

0.08

0.08

0.06

1.0

p/poR

p/poR

0.00 0.0

0.8 non-idealized

non-idealized

0.06

Inner

0.02

0.02

0.00 0.0

GE/EPRI(Fit A) GE/EPRI(Fit B) GE/EPRI(Fit C) ERS FE(Test data)

0.2

0.4

0.6

0.8 non-idealized

p/poR

(a) θ1/θ2=1.5

1.0

1.2

1.4

0.00 0.0

0.2

0.4

0.6

0.8 non-idealized

p/poR

(b) θ1/θ2=3.0

Fig. 12. Comparisons of normalized COD values obtained from FE results with the proposed solutions based on the ERS method and the GE/EPRI method along the thickness for Rm/t = 20 and h1/p = 0.250 under internal pressure.

‘GE/EPRI(Fit B)’ and ‘GE/EPRI(Fit C)’ in these figures mean the predictions based on the GE/EPRI-type estimate using R–O constants from Fit A, Fit B and Fit C, respectively. ‘ERS’ denotes the prediction based on the ERS-type estimate. As expected, the predictions based on the GE/EPRI-type estimate are quite sensitive to the R–O constants, and are always very larger than the present FE results using actual stress–strain data regardless of geometries, loading mode and R–O fitting ranges considered in the present study. For a conservative LBB analysis, the COD should be under-estimated for a given load, however, the GE/EPRI-type estimate gives non-conservative COD results for a pipe with a circumferential NiTWC. Among three predictions using the GE/EPRI-type estimate, the prediction using ‘Fit B’ give at least the better results. On the other hand, the ERS-type estimate gives excellent results at overall locations along the thickness comparing with the present FE results using actual stress–strain data. More importantly, it provides not only accurate but also robust and conservative COD estimates for the cases considered in the present study. Although, the ERS-type estimate sometimes produces slightly non-conservative results, but it provides still very accurate results up to high load level. At this point, one

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δ /a1

0.06

0.08

Idealized TWC with average crack angle ERS(Inner) ERS(Mid. thickness) ERS(Outer)

0.06

δ /a1

0.08

0.04

0.02

0.00 0.0

Idealized TWC with average crack angle ERS(Inner) ERS(Mid. thickness) ERS(Outer)

0.04

0.02

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.0

1.4

0.2

0.4

non-idealized

0.08

Idealized TWC with average crack angle ERS(Inner) ERS(Mid. thickness) ERS(Outer)

0.06

0.04

0.02

0.00 0.0

1.0

1.2

1.4

(b) θ1/π=0.125 and θ1/θ2=3

δ /a1

δ /a1

0.06

0.8

T/ToR

(a) θ1/π=0.125 and θ1/θ2=1.5 0.08

0.6

non-idealized

T/ToR

Idealized TWC with average crack angle ERS(Inner) ERS(Mid. thickness) ERS(Outer)

0.04

0.02

0.2

0.4

0.6

0.8

1.0

1.2

non-idealized

T/ToR

(c) θ1/π=0.250 and θ1/θ2=1.5

1.4

0.00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

non-idealized

T/ToR

(d) θ1/π=0.250 and θ1/θ2=3

Fig. 13. Comparisons of normalized COD values of circumferential non-idealized through-wall crack with those of idealized through-wall crack with average crack angle for Rm/t = 5 under tension based on the ERS method.

important point is worth noting when we calculate the CODs of a pipe with a circumferential NiTWC along the thickness using the ERS-type estimate. The COD values at arbitrary points along the thickness can be simply estimated in the ERS-type estimate by using the elastic value of COD (in Eq. (13)) relevant to the particular location of interest. For instance, the COD value at the outer surface point can be easily obtained by entering the elastic COD value for the outer surface point on Eq. (9). Moreover, the ERS-type estimate does not require any idealization of material’s tensile data, thus it can provide more robust predictions than GE/EPRI-type estimate as found in Figs. 7–12.

5. Discussion In order to provide technical basis of importance of engineering estimates for a NiTWC, the CODs of NiTWC are compared with those of idealized TWC, where a NiTWC is idealized into idealized TWC by averaging crack angles of both inner and outer surfaces. Figs. 13 and 14 show the results for two different reference crack angle (h1/p = 0.125 and 0.250) with two different h1/h2, where axial tension and bending moment are considered. These comparisons are made based on the ERS-type estimate. Note that the CODs of idealized TWCs with average crack angles are only available at the mid-thickness of a pipe since the existing elastic CODs for an idealized TWC can only give CODs at the mid-thickness, whereas the CODs of NiTWCs are calculated at the three important locations along the thickness as provided in Figs. 13 and 14. As shown in these figures, the predictions using idealized TWC with average crack angle (=(h1 + h2)/2) provides larger and non-conservative COD values than those using NiTWC, and differences between results based on idealized TWC with average crack angle and those based on NiTWC increase as h1/p and h1/h2 increase. Thus, it can be concluded that for the accurate estimates of CODs considering actual crack shape development, the estimates based on a NiTWC might be more desirable. Although the engineering estimates of CODs of a circumferential NiTWC are proposed in the present study, the reliable J-integral estimates are also desirable to assess the instability of a pipe with a circumferential NiTWC for LBB analysis. Thus, the authors are now in the process of developing the engineering estimates of elastic–plastic J-integral of a NiTWC.

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δ /a1

0.06

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0.06

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Fig. 14. Comparisons of normalized COD values of circumferential non-idealized through-wall crack with those of idealized through-wall crack with average crack angle for Rm/t = 20 under bending moment based on the ERS method.

6. Concluding remarks In the present study, the engineering estimates of CODs either based on the GE/EPRI method or based on the ERS method for a pipe with a circumferential NiTWC are newly suggested based on the systematic 3-D elastic and elastic–plastic FE analyses. For the GE/EPRI-type estimate, the plastic influence functions, h2, are calibrated based on the FE results using R–O materials. For the ERS-type estimate, the closed-form expression of elastic COD at three important locations along the thickness and reference loads for a NiTWC are suggested. Finally, the predictions based on the proposed engineering estimates of CODs for NiTWC are compared with elastic–plastic FE results using actual stress–strain data, from which it is revealed that the predictions based on the GE/EPRI-type estimate are very sensitive to R–O fitting ranges, and the accuracy is generally poor. On the contrary, the ERS-type estimate gives very accurate and robust results, and more importantly the results from the ERS-type estimate are conservative than FE results using actual stress–strain data. Moreover, the importance of COD estimates of NiTWC is noted by comparing CODs from NiTWC with those from idealized TWC with average crack angle. The engineering estimation schemes of CODs for a NiTWC proposed in the present study could be applied to evaluate the CODs or the leak rates considering more realistic crack shape development. References [1] USNRC, Standard Review Plan for the Review of Safety Analysis Reports for Nuclear Power Plants: LWR Edition. USNRC, Section 3.6.3; NUREG-0800, 1987. [2] USNRC, Special Meeting on Leak Before Break in Reactor Piping and Vessels, NUREG/CP-0155, 1997. [3] Wilowski G, Schmidt R, Scott P, Olson R, Marschall C, Kramer G, et al. International Piping Integrity Research Group (IPIRG) Program – Final report, NUREG/CR-6233, USNRC, 1997. [4] Hopper A, Wilowski G, Scott P, Olson R, Rudland D, Lilinski T, et al. The Second International Piping Integrity Research Group(IPIRG-2) Program – Final report, NUREG/CR-6452, USNRC, 1997. [5] Zahoor A. Ductile fracture handbook. Novetech Corp 1991. [6] Kumar V, German MD, Andrews WR, deLorenzi HG, Mowbray DF. Advanced in elastic-plastic fracture analysis. EPRI NP-3607 1984.

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