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Normalizing the influence of flaw length on failure pressure of straight pipe with wall-thinning
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Nuclear Engineering and Design 238 (2008) 8–15
Normalizing the influence of flaw length on failure pressure of straight pipe with wall-thinning Masayuki Kamaya a,∗ , Tomohisa Suzuki b , Toshiyuki Meshii b a
Institute of Nuclear Safety System, Inc., 64 Sata Mihama-cho, Fukui, Japan b University of Fukui, Bunkyo 3-9-1, Fukui, Japan
Received 21 November 2006; received in revised form 13 May 2007; accepted 3 June 2007
Abstract Burst tests using wall-thinned pipe of carbon steel for high-temperature use were conducted in order to examine the influence of length of wall-thinning on burst pressure. Then, three-dimensional elastic-plastic large deformation finite element analyses (EP-FEA) were performed to accurately predict the burst pressure obtained by the tests. The failure pressure corresponding to the burst pressure in tests was defined as the maximum pressure during the analysis including the instability condition after the peak of pressure. The results showed that the failure pressure obtained by EP-FEA agreed well with the experimental results. Finally, failure pressures of wall-thinned pipes with various sizes, thicknesses, flaw lengths and depths were examined by EP-FEA with the same procedure of analysis as validated in this paper. The results showed that, from the standpoint of influence of flaw length on failure pressure, it is preferable to normalize flaw length by pipe mean radius of the unflawed section R rather than by shell parameter (Rt)0.5 , where t is the thickness of the unflawed section. © 2007 Elsevier B.V. All rights reserved.
1. Introduction Structural integrity of wall-thinned pipes in nuclear components has continuously been an issue of interest to engineers. However, in the boiler and pressure vessel code section XI of the American Society of Mechanical Engineers (ASME) (ASME, 2004), only a Code Case for evaluation of wall-thinned pipe is provided (ASME, 2003). The code mainly prescribes assessment rules for flawed components containing crack-like flaws. The British Standard (BS) also provides an assessment method for general corrosion in pipes and pipelines (BS, 2005), though it is still an Annex. In order to enhance the rules for non-cracklike flaws, the failure strength of flawed components has to be clarified. Especially, strength against internal pressure is important to prevent catastrophic accidents due to burst of flawed pipe (NRC, 2006). A number of experimental studies have been conducted to assess the integrity of pipes containing wall-thinning under internal pressure. Most of these studies focused on pipe line steel using full-scale specimens containing artificial wall-thinning
∗
Corresponding author. Tel.: +81 770379114; fax: +81 770372009. E-mail address: [email protected] (M. Kamaya).
0029-5493/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2007.06.006
in addition to corroded pipe taken from actual lines (Vieth and Kiefner, 1993). Empirical and semi-empirical failure criteria were developed based on the results for fitness-for-service (Turbak and Sims, 1994), and some modifications were made in order to reduce conservativeness included in the criteria (Vieth and Kiefner, 1989). Meanwhile, theoretical and numerical approaches were made based on elastic shell theory (Folias, 1965; Kanninen et al., 1992) and finite element analysis (FEA), respectively. Because of limitations in the number of experiments and experimental conditions, numerical analyses are inevitable to complement experimental results. Especially, the elastic-plastic FEA based on shell modeling (Sims et al., 1992; Stephens and Leis, 1997) and three-dimensional solid modeling (Roy et al., 1997; Chouchaoui et al., 1992; Netto et al., 2005; Oh et al., 2006) seemed give good agreement with experimental results. In the FEA, several analysis procedures (criteria) were used as follows. Limit load (LL) analysis is widely used to assess the strength of components. In LL analysis, an elastic-perfect-plastic stress–strain curve is usually used together with the twice-elastic slope criterion. This approach is easy to perform due to the simple material properties and analysis procedure, and has been used in studies of wall-thinning problems under internal pressure (Kim et al., 2002a, 2004; Choi et al., 2005). Large deforma-
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
tion analysis using work-hardening stress–strain curves is also a practical procedure. By using an actual stress–strain relation of the material, deformation of a pipe due to applied internal pressure can be simulated accurately. The problem with this approach is definition of the failure pressure of the pipe, which corresponds to the burst pressure in experiments. Since, in general, it is difficult to obtain the stress–strain relation above the ultimate strength of material, deformation above the ultimate strength is difficult to simulate. Shim et al. (2004) defined failure pressure as the pressure when stress reached ultimate strength at the deepest portion of wall-thinning. Roy et al. (1997) assumed a stress–strain curve above the ultimate strength. Miyazaki et al. (2002) also extrapolated the stress–strain curve and defined failure by considering the multi-axial stress condition, although they treated failure due to bending load. Thus, there are various approaches to the evaluation of failure strength of wall-thinned pipe. The objective of the current study is to quantify the influence of length of wall-thinning on burst pressure. At the same time, the validity of various FEA procedures is discussed with comparisons with experimental results. The experiments were carried out using pipes of carbon steel for high-temperature use containing artificial wall-thinning of different lengths. In these experiments, in order to compare the burst pressure with that obtained by FEA, the precise thickness of pipe at wall-thinned portion was identified. Three-dimensional elastic-plastic FEA was then performed for wall-thinned pipes. Several analysis procedures were applied and their results were compared with those of experiments. Based on the FEA results, the influences of flaw shape on failure pressure were examined, and discussions were made for how to prescribe the influence of flaw length for assessments of failure pressure of straight pipe with wall-thinning.
Table 1 Chemical composition of test material (wt%) C
Si
Mn
P
S
Fe
0.19
0.19
0.40
0.021
0.02
Bal.
Table 2 Mechanical properties of test material
Mill sheet Circumferential direction
Yield strength (MPa)
Tensile strength (MPa)
Elongation
288 260
447 446
0.4 0.37
positions and mechanical properties of the material are shown in Tables 1 and 2, respectively. The geometry of tested pipe is depicted in Fig. 1. The nominal outer diameter and thickness of the pipes were Do = 107.1 mm and t = 4 mm, respectively. In order to obtain uniform dimensions of diameter and thickness, the outer and inner surfaces of the pipes were machined before the flaws were introduced. The flaw lengths (length in longitudinal direction) were So = 72.5, 50 and 25 mm. Two specimens were prepared for each flaw length. These flaws were machined at the pipe internal. The thickness of the flawed portion was measured using an ultrasonic thickness gage at 34 points in the circumferential direction, while measurements were made every 10 mm in the longitudinal direction including the point z = 0. The thickness at z = 0 position is shown in Fig. 2, and Table 3 shows a summary of measurements together with names of the specimens. The depths of flaws were approximately from d = 1.9–2.2 mm, and were almost uniform in circumferential direction and longitudinal direction except the transition section, which was set at the end of the flawed section. The equivalent flaw length S is defined by the following equation:
2. Experimental procedure Six experiments were carried out using pipes made of carbon steel for high-temperature use (STPT370 in JIS) with artificially introduced wall-thinning (hereafter, flaw). The chemical com-
9
S=
Ao 2.0 mm
(1)
where Ao is the cross-sectional area of the flaw. Therefore, the equivalent flaw size becomes S = So + 22.86/2 mm.
Fig. 1. Geometry of specimen for burst test.
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Fig. 2. Thickness distribution at z = 0 position (unit: mm): (a) specimen: S1A (So = 72.5 mm); (b) specimen: S1B (So = 72.5 mm); (c) specimen: S2A (So = 50 mm); (d) specimen: S2B (So = 50 mm); (e) specimen: S3A (So = 25 mm); (f) specimen: S3B (So = 25 mm).
In the tests, internal pressure was applied by injecting water into the pipe using a pump after evacuating air in room temperature environment. The pressure when water leaked from the mouth due to fracture was defined as burst pressure, Pf . 3. Experimental results and discussions The burst pressures obtained by the tests were Pf = 17.49 MPa (S1A) and 17.46 MPa (S1B) in the case of So = 72.5 mm,
18.17 MPa (S2A) and 19.50 MPa (S2B) in the case of So = 50 mm and 23.90 MPa (S3A) and 24.16 MPa (S3B) in the case of So = 25 mm, respectively. The burst pressure decreased as flaw length increased. It was also influenced by flaw depth. In the cases of So = 50 and 25 mm, burst pressure was lower in thinner pipe at z = 0 position, although it was almost the same in the case of So = 72.5 mm, in which the averaged thickness was the same as at z = 0 position.
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
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Table 3 Summary of measurement of thickness of flawed portion Flaw length (So ) 72.5 mm S1Aa
50 mm S1Ba
S2Aa
25 mm S2Ba
S3Aa
S3Ba
At z = 0 Number of data Average (mm) Minimum (mm) Maximum (mm)
34 1.92 1.90 1.94
34 1.92 1.88 1.95
34 1.82 1.78 1.86
34 1.97 1.92 2.01
34 1.98 1.96 1.99
34 2.04 2.03 2.05
All data Number of data Average (mm) Minimum (mm) Maximum (mm)
360 1.91 1.87 1.95
360 1.95 1.88 1.99
264 1.81 1.76 1.87
264 2.00 1.92 2.07
170 1.97 1.95 1.99
170 2.07 2.03 2.11
a
Specimen no.
Fig. 3 shows the relationship between flaw length and burst pressure normalized by Po , which is the failure pressure of unflawed pipe calculated by: t 2 Po = √ σf 3 R
(2)
where σ f is the flow stress defined as σ f = 0.5(σ y + σ b ) using the yield strength, σ y (=288 MPa), and ultimate strength, σ b (=447 MPa), of the test material. R denotes the nominal mean radius of the pipe in unflawed sections. The burst pressure decreases as flaw length increases. The specimens after the experiment are shown in Fig. 4. A straight longitudinal crack ran at the center of the flaw, except in specimen S3A. In specimen S3A (So = 25 mm), the crack curved toward the circumferential direction at the end of the wall-thinned region. Large deformation (bulging) occurred in all cases. Through measurements of strain during the tests, it was confirmed that bending stress caused by discontinuity of geometry at the end of pipe (z = 170 mm) was negligibly small. The burst pressures obtained in other studies are also plotted in Fig. 3. In these studies, experiments were conducted using pipeline steel (Kim et al., 2002b; Kiefner et al., 1973) and low alloy steel (Kobayashi et al., 2001), with an artificial flaw, of
Fig. 3. Relationship between failure pressure and flaw.
which depth was in the range from 0.48t to 0.51t. The shape of the flaw was a saw-cut longitudinal notch (Kiefner et al., 1973), U-shape longitudinal groove (Kobayashi et al., 2001) or uniform depth patch (Kim et al., 2002b). These data show similar burst pressures as those in this study, except for the saw-cut flaw. It was pointed out that flaws with a sharp notch resulted in lower burst pressure than blunt flaws (Kiefner, 1969). Especially, in the case of a long flaw, further reduction in burst pressure would be caused by ductile crack initiation (Kiefner et al., 1973; Hahn et al., 1969). The burst pressures obtained in the present study show similar dependency on flaw length with other results. It should be noted that the flaw depth in the present study was not exactly 0.5t; it was approximately 0.52t in specimens S1A and S1B, and 0.55t in S2A. The lines shown in Fig. 3 indicate the assessment curves for failure pressure expressed by the equation: 1 − d/t Pf = Po , (3) 1 − d/(tMt ) where Mt represents the so-called bulging factor. Several expressions of Mt have been proposed (Folias, 1965; Kiefner et al.,
Fig. 4. Specimens after tests: (a) S1A (So = 72.5 mm); (b) S2A (So = 50 mm); (c) S3A (So = 25 mm).
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M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
1973; Hahn et al., 1969; Erdogan and Sih, 1976), and the following two equations are used in Fig. 3. 1.61S 2 Mt = 1 + (4) Rt S4 S2 (5) Mt = 1 + 1.255 − 0.0135 Rt (Rt)2 Eq. (4) was derived from elastic shell theory for a throughwall crack in a cylinder, and was adopted in ASME B31G (ANSI/ASME, 1984), which provides an acceptance criterion for non-planar flaws, as well as the limit load failure criterion for axial cracks under internal pressure in the ASME boiler and pressure vessel code, section XI (ASME, 2004) and fitnessfor-service code of the Japan Society of Mechanical Engineers (JSME) (JSME, 2004). The validity of failure pressure of unflawed pipe, Po , defined by Eq. (2) was shown by burst experiments with an unflawed vessel with end-cap (Kobayashi et al., 2001). The reduction in failure pressure due to non-planar flaws is addressed by the term in parentheses in Eq. (3), the form of which is empirically derived (Kiefner, 1969). In the prediction using these equations, it is assumed that the material will fail in a ductile manner. These curves make conservative evaluation of the experimental results, except in the case of saw-cut flaws. Stephens and Leis (1997) proposed a different type of assessment curve based on FEA, which is expressed by the following equation and shown by the broken line in Fig. 3: t d 2S Pf = σb . (6) 1− 1 − exp −0.157 √ R t R(t − d)
Fig. 5. Geometry of a pipe containing wall-thinning.
In LL analyses, the elastic-perfect-plastic stress–strain relation was assumed. Various failure stresses have been used in the evaluation of failure strength of wall-thinned pipes, such as σ y + 10 ksi (Kiefner et al., 1973; Kiefner and Vieth, 1989), σ y /0.9 (API, 2000) and 1.1σ y (ANSI/ASME, 1984), σ b (Leis and Stephens, 1997b) and 0.5(σ y + σ b ), which is used in the current codes of ASME (ASME, 2004) and JSME (JSME, 2004). In order to investigate the validity of these definitions, three kinds of yield strengths in the elastic-perfect-plastic stress–strain relation were used: 0.5(σ y + σ b ), σ y + 10 ksi and σ b . The twice-elastic slope criterion was adopted to obtain the failure pressure. FEA using a work-hardening stress–strain relation obtained from the test material was also performed. Hereafter, in this study, this type of analysis is referred to as large deformation (LD) analysis. The stress–strain relation was obtained from a specimen cut from the test material for the circumferential direction. Fig. 7 shows the obtained stress–strain curve together with
This criterion exhibits relatively good agreement with the experimental results. In this equation, however, no end-capped (multi-axial stress) effect in the Mises criterion is considered, while this effect is considered in Eq. (2), and ultimate strength of the material is used for the material constant. 4. Procedure of finite element analysis The three-dimensional elastic-plastic FEA was performed using the general purpose program ABAQUS, Version 6.5 (ABAQUS Inc., 2005). The geometry of a pipe with wallthinning (flaw) is schematically shown in Fig. 5. The half-length of the flaw is expressed by So with an edge length of Se . Pipe length was set to L = 5Do , which is enough to ignore the boundary effect at the end of the specimen. The pipe was modeled by 8-node solid elements as shown in Fig. 6. At the bottom of the flaw, the model was divided by 4 elements in the thickness direction and 20 elements in the longitudinal direction, while element width in the circumferential direction is 5◦ . There are 17,280 elements and 24,384 nodes are included in the model shown in Fig. 6, which corresponds to the test of So = 50 mm. It was confirmed that well converged failure pressure were obtained with this mesh size. Axial tensile load was applied at the end of the pipe in order to simulate the end-capped condition.
Fig. 6. Finite element mesh for pipe with wall-thinning: (a) whole view; (b) magnified cross-sectional view of flawed portion.
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
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to geometrical symmetries, only one quarter of the full pipe was modeled with displacement boundary conditions on the planes of symmetries. 5. Analysis results and discussions 5.1. Prediction of experimental results
Fig. 7. Stress–strain relation used in analyses.
a simplified one, which is expressed by a bi-linear relation and is denoted as the “modeled curve”. To simulate large inelastic deformation during the tests, the non-linear geometry option with ABAQUS (the NLGEOM option) was invoked for the analysis procedure together with the modified Riks method, which allows analysis under static instability after plastic collapse. Two kinds of failure pressure were obtained by different definitions. The pressure when the Mises equivalent stress reaches the upper limit of the true stress–strain curve, σ bc , at any element is one of them and is denoted Pf(mi) , while the other failure pressure is defined as the maximum pressure during the analysis including the instability condition after the peak of the pressure and is denoted Pf(max) . In the analyses, internal pressure was increased until the instability condition. The magnitude of increase of pressure was controlled not to exceed 0.06 MPa in the plastic region. The typical step size in pressure just before the failure was 0.001 MPa. At first, analyses were performed in order to simulate the experiments. In the analysis, the transition section Se was set to 22.86 mm following the geometry of the test specimen. The thickness at the wall-thinned portion was determined by interpolating the measured thickness. Then, in order to examine the influence of the pipe and flaw geometry, the analyses were performed for various pipe and flaw configurations with uniform flaw depth and edge length of Se = 0.2So . In these analyses, due
The FEA results simulating those of the experiments are shown in Table 4 together with the experimental results. The failure pressures obtained by LL analyses give conservative results, except those obtained using σ b as the yield strength. The evaluated failure pressures are less than 30% of those obtained by the experiments in some cases. With use of σ b , results are less conservative, although, in general, σ b is not used as the yield strength in LL analyses. The safety margin changes with the flaw length. On the other hand, the failure pressures obtained by LD analysis agree well with the experimental results. Especially, the error in Pf(max) was less than 5% in all cases when the actual stress–strain relation was used. The Pf(max) seems better than Pf(mi) , whereas the difference between the two failure pressures is small. Even when the simplified bi-linear stress–strain relation was used, error in the predicted failure pressure was 11.15% at maximum. 5.2. Influence of pipe and flaw geometry Fig. 8 shows the relationship between failure pressure and flaw size obtained by FEA with a constant flaw depth of d = 0.5t together with the experimental results (Kim et al., 2002b; Kobayashi et al., 2001), which are the same data as shown in Fig. 3. The failure pressures obtained by LD analyses (Pf(max) ) using the actual stress–strain relations and LL analyses using Sf as the yield strength are also shown in the figure. The FEA results obtained by LD analyses exhibited good agreement with the experimental results including those obtained in other papers. This supports the validity of the analyses. On the other hand, LL analyses show poor correlation with the experimental results. This implies that LL analyses are not effective in simulating experiments, and that it is reasonable to estimate failure pressure conservatively.
Table 4 Comparison of failure pressure obtained by experiments and finite element analyses (unit: MPa) Specimen
Experiment
LL analysis
LD analysis
Yield strength
Stress–strain relation
0.5 (σ y + σ b )
S1A S1B S2A S2B S3A S3B
17.49 17.46 18.17 19.50 23.90 24.16
14.78 (15.51) 14.88 (14.79) 13.90 (23.48) 15.29 (21.58) 15.77 (34.02) 16.42 (32.02)
σ y + 10 ksi
13.77 (21.28) 13.86 (20.6) 12.95 (28.71) 14.25 (26.92) 14.69 (38.53) 15.31 (36.65)
σb
18.67 (−6.75) 18.80 (−7.68) 17.57 (3.32) 19.32 (0.9) 19.93 (16.63) 20.76 (14.09)
Experimental material
Modeled curve
Pf(mi)
Pf(max)
Pf(mi)
Pf(max)
17.77 (−1.58) 17.98 (−2.97) 17.51 (3.66) 19.60 (−0.54) 22.66 (5.18) 23.46 (2.91)
17.90 (−2.33) 18.14 (−3.92) 18.03 (0.79) 19.77 (−1.36) 22.90 (4.16) 23.72 (1.81)
17.13 (2.06) 17.38 (0.48) 16.97 (6.58) 18.52 (5.01) 21.24 (11.15) 21.92 (9.29)
17.22 (1.52) 17.41 (0.27) 17.33 (4.6) 18.97 (2.69) 21.84 (8.6) 22.58 (6.55)
The parentheses in the table indicate difference, which is defined by (experiment − analysis)/(experiment) × 100.
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M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
Fig. 8. Relationship between the failure pressure and flaw length obtained by FEA together with experimental results.
As shown in Figs. 3 and 8, failure pressure is dependent on flaw length. The failure pressure is near Po if the flaw length is small, and decreases as flaw length becomes larger. When the flaw length is large enough, the failure pressure should be equivalent to that of pipe of thickness (t–d). Thus, there are two extremes of failure pressure of a pipe with wall-thinning. Leis and Stephens (1997a) proposed a concept in the following form: d Pf = 1 − f (geometry). t Po
Fig. 9. Relationship between the failure pressure and flaw length for various pipe geometry: (a) flaw length is normalized by (Rt)0.5 ; (b) flaw length is normalized by R.
(9)
Eq. (6) is based on the concept of Eq. (9) and yields a relatively good correlation with the experimental results as shown in Fig. 3, although the definition of Po is different from that in Eq. (2). f (geometry) is a transition function connecting the two extremes, and is related to pipe and flaw geometry. In order to examine the function f, failure pressures were evaluated for various pipe configurations, which were possible combinations of diameters of 100A, 200A, 300A, 400A and 500A and thicknesses of schedule 40 and 80 with different flaw depths of d/t = 0.1, 0.25, 0.5 and 0.75. Fig. 9 shows the failure pressure, Pf(max) , evaluated by LD analysis. The failure pressure is normalized by the value obtained for unflawed pipe. Therefore, normalized failure pressure is unity at S = 0. In most previous studies, as expressed by Eqs. (4)–(6), the influence of flaw length was normalized by the shell parameter (Rt)0.5 or (R(t − d))0.5 , as in the current fitness-for-service criteria, such as the API (API, 2000), ASME (ASME, 2003; ANSI/ASME, 1984) and BS (BS, 2005). However, as shown in Fig. 9(a), there is scattering in failure pressure when the flaw length is normalized by (Rt)0.5 , although the magnitude of scattering is limited compared to the degree of dependency on flaw depth and length. On the other hand, as shown in Fig. 9(b), normalization by R yields a good correlation between the failure pressure and flaw size. Sims et al. (1992) and Turbak and Sims (1994) also normalized flaw length by pipe diameter to show the change in failure pressure obtained by elastic-plastic FEA. It was pointed
out that the parameter Mt is independent of thickness for thin cylinder (Hahn et al., 1969). Furthermore, as shown in Fig. 4, fracture of flawed pipes is accompanied by large plastic deformation, suggesting that elastic theory might not be valid for such large deformations. The results of analyses indicate that, in the assessment criterion for fitness-for-service, normalization by R is better for representation of the influence of flaw length. This supports the proposal made by Sims and colleague. The transition function f can be obtained by interpolating the curves shown in Fig. 9(b). 6. Conclusions Burst tests using wall-thinned pipe of carbon steel for high-temperature use were conducted in order to examine the influence of the length of wall-thinning on failure pressure. Then, three-dimensional elastic-plastic FEA was performed to predict the burst pressure in the tests and investigate the influence of flaw shape on failure pressure. The following conclusions were obtained: (a) The burst pressure of wall-thinned pipe in tests decreased as flaw length increased, and was accurately predicted by large deformation FEA using the work-hardening stress–strain relation and precise thickness of the pipe at wall-thinned portion.
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
(b) LL analysis with an elastic-perfect-plastic stress–strain relation yielded conservative results when the yield strength was set to 0.5(σ y + σ b ) and σ y + 10 ksi. (c) In normalizing the influence of flaw length on failure pressure, it is preferable to normalize flaw length by pipe mean radius of the unflawed section rather than by shell parameter (Rt)0.5 . References ABAQUS Inc., 2005. ABAQUS/Standard User’s Manual Ver. 6.5. ABAQUS Inc., USA. ANSI/ASME, 1984. Manual for Determining the Remaining Strength of Corroded Pipelines, ANSI/ASME B31G. 2000. Fitness-For-Service API579. American Petroleum Institute. ASME, 2003. Requirements for Analytical Evaluation of Pipe Wall Thinning (Cases of ASME Boiler and Pressure Vessel Code N-597-2), ASME, New York, USA. ASME, 2004. Boiler and Pressure Vessel Code Section XI, ASME, New York, USA. BS, 2005. Guide to Methods for Assessing the Acceptability of Flaws in Metallic Structures BS7910: 2005, BSI. Choi, J.B., Lee, S.M., Huh, N.S., Chang, Y.S., Kim, Y.J., Choi, Y.H., 2005. Limit load analyses for wall-thinned elbow with different bend angles subjected to internal pressure. In: The 6th International Symposium on the Structural Integrity of Nuclear Components, pp. 98–109. Chouchaoui, B.A., Pick, R.J., Yost, D.B., 1992. Burst pressure predictions of line pipe containing single corrosion pits using the finite element method. OMAE V-A, 203–210. Erdogan, F., Sih, G.C., 1976. Ductile fracture theories for pressurized pipes and containers. Int. J. Pressure Vessel Piping 4, 253–283. Folias, E.S., 1965. An axial crack in a pressurized cylindrical shell. Int. J. Fract. Mech. 1, 104–113. Hahn, G.T., Sarrate, M., Rosenfield, A.R., 1969. Criteria for crack extension in cylindrical pressure vessel. Int. J. Fract. Mech. 5, 187–210. JSME, 2004. Codes for Nuclear Power Generation Facilities: Rules of Fitnessfor-Service for Nuclear Power Plants, Tokyo, Japan. Kanninen, M.F., Pagalthivarthi, K.V., Popelar, C.H., 1992. A theoretical analysis for the residual strength of corroded gas and oil transmission pipelines. In: Chaker, V. (Ed.), Corrosion Forms and Control for Infrastructure, ASTM Special Technical Publication 1137, 183–198. Kiefner, J.F., 1969. Fracture initiation, Fourth Symposium on Line Pipe Research, PRCI Catalog No. L30075, G1-G36. Kiefner, J.F., Maxey, W.A., Eiber, R.J., Duffy, A.R., 1973. Failure stress levels of flaws in pressurized cylinders. ASTM STP 536, 461–481.
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Nuclear Engineering and Design 238 (2008) 8–15
Normalizing the influence of flaw length on failure pressure of straight pipe with wall-thinning Masayuki Kamaya a,∗ , Tomohisa Suzuki b , Toshiyuki Meshii b a
Institute of Nuclear Safety System, Inc., 64 Sata Mihama-cho, Fukui, Japan b University of Fukui, Bunkyo 3-9-1, Fukui, Japan
Received 21 November 2006; received in revised form 13 May 2007; accepted 3 June 2007
Abstract Burst tests using wall-thinned pipe of carbon steel for high-temperature use were conducted in order to examine the influence of length of wall-thinning on burst pressure. Then, three-dimensional elastic-plastic large deformation finite element analyses (EP-FEA) were performed to accurately predict the burst pressure obtained by the tests. The failure pressure corresponding to the burst pressure in tests was defined as the maximum pressure during the analysis including the instability condition after the peak of pressure. The results showed that the failure pressure obtained by EP-FEA agreed well with the experimental results. Finally, failure pressures of wall-thinned pipes with various sizes, thicknesses, flaw lengths and depths were examined by EP-FEA with the same procedure of analysis as validated in this paper. The results showed that, from the standpoint of influence of flaw length on failure pressure, it is preferable to normalize flaw length by pipe mean radius of the unflawed section R rather than by shell parameter (Rt)0.5 , where t is the thickness of the unflawed section. © 2007 Elsevier B.V. All rights reserved.
1. Introduction Structural integrity of wall-thinned pipes in nuclear components has continuously been an issue of interest to engineers. However, in the boiler and pressure vessel code section XI of the American Society of Mechanical Engineers (ASME) (ASME, 2004), only a Code Case for evaluation of wall-thinned pipe is provided (ASME, 2003). The code mainly prescribes assessment rules for flawed components containing crack-like flaws. The British Standard (BS) also provides an assessment method for general corrosion in pipes and pipelines (BS, 2005), though it is still an Annex. In order to enhance the rules for non-cracklike flaws, the failure strength of flawed components has to be clarified. Especially, strength against internal pressure is important to prevent catastrophic accidents due to burst of flawed pipe (NRC, 2006). A number of experimental studies have been conducted to assess the integrity of pipes containing wall-thinning under internal pressure. Most of these studies focused on pipe line steel using full-scale specimens containing artificial wall-thinning
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in addition to corroded pipe taken from actual lines (Vieth and Kiefner, 1993). Empirical and semi-empirical failure criteria were developed based on the results for fitness-for-service (Turbak and Sims, 1994), and some modifications were made in order to reduce conservativeness included in the criteria (Vieth and Kiefner, 1989). Meanwhile, theoretical and numerical approaches were made based on elastic shell theory (Folias, 1965; Kanninen et al., 1992) and finite element analysis (FEA), respectively. Because of limitations in the number of experiments and experimental conditions, numerical analyses are inevitable to complement experimental results. Especially, the elastic-plastic FEA based on shell modeling (Sims et al., 1992; Stephens and Leis, 1997) and three-dimensional solid modeling (Roy et al., 1997; Chouchaoui et al., 1992; Netto et al., 2005; Oh et al., 2006) seemed give good agreement with experimental results. In the FEA, several analysis procedures (criteria) were used as follows. Limit load (LL) analysis is widely used to assess the strength of components. In LL analysis, an elastic-perfect-plastic stress–strain curve is usually used together with the twice-elastic slope criterion. This approach is easy to perform due to the simple material properties and analysis procedure, and has been used in studies of wall-thinning problems under internal pressure (Kim et al., 2002a, 2004; Choi et al., 2005). Large deforma-
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
tion analysis using work-hardening stress–strain curves is also a practical procedure. By using an actual stress–strain relation of the material, deformation of a pipe due to applied internal pressure can be simulated accurately. The problem with this approach is definition of the failure pressure of the pipe, which corresponds to the burst pressure in experiments. Since, in general, it is difficult to obtain the stress–strain relation above the ultimate strength of material, deformation above the ultimate strength is difficult to simulate. Shim et al. (2004) defined failure pressure as the pressure when stress reached ultimate strength at the deepest portion of wall-thinning. Roy et al. (1997) assumed a stress–strain curve above the ultimate strength. Miyazaki et al. (2002) also extrapolated the stress–strain curve and defined failure by considering the multi-axial stress condition, although they treated failure due to bending load. Thus, there are various approaches to the evaluation of failure strength of wall-thinned pipe. The objective of the current study is to quantify the influence of length of wall-thinning on burst pressure. At the same time, the validity of various FEA procedures is discussed with comparisons with experimental results. The experiments were carried out using pipes of carbon steel for high-temperature use containing artificial wall-thinning of different lengths. In these experiments, in order to compare the burst pressure with that obtained by FEA, the precise thickness of pipe at wall-thinned portion was identified. Three-dimensional elastic-plastic FEA was then performed for wall-thinned pipes. Several analysis procedures were applied and their results were compared with those of experiments. Based on the FEA results, the influences of flaw shape on failure pressure were examined, and discussions were made for how to prescribe the influence of flaw length for assessments of failure pressure of straight pipe with wall-thinning.
Table 1 Chemical composition of test material (wt%) C
Si
Mn
P
S
Fe
0.19
0.19
0.40
0.021
0.02
Bal.
Table 2 Mechanical properties of test material
Mill sheet Circumferential direction
Yield strength (MPa)
Tensile strength (MPa)
Elongation
288 260
447 446
0.4 0.37
positions and mechanical properties of the material are shown in Tables 1 and 2, respectively. The geometry of tested pipe is depicted in Fig. 1. The nominal outer diameter and thickness of the pipes were Do = 107.1 mm and t = 4 mm, respectively. In order to obtain uniform dimensions of diameter and thickness, the outer and inner surfaces of the pipes were machined before the flaws were introduced. The flaw lengths (length in longitudinal direction) were So = 72.5, 50 and 25 mm. Two specimens were prepared for each flaw length. These flaws were machined at the pipe internal. The thickness of the flawed portion was measured using an ultrasonic thickness gage at 34 points in the circumferential direction, while measurements were made every 10 mm in the longitudinal direction including the point z = 0. The thickness at z = 0 position is shown in Fig. 2, and Table 3 shows a summary of measurements together with names of the specimens. The depths of flaws were approximately from d = 1.9–2.2 mm, and were almost uniform in circumferential direction and longitudinal direction except the transition section, which was set at the end of the flawed section. The equivalent flaw length S is defined by the following equation:
2. Experimental procedure Six experiments were carried out using pipes made of carbon steel for high-temperature use (STPT370 in JIS) with artificially introduced wall-thinning (hereafter, flaw). The chemical com-
9
S=
Ao 2.0 mm
(1)
where Ao is the cross-sectional area of the flaw. Therefore, the equivalent flaw size becomes S = So + 22.86/2 mm.
Fig. 1. Geometry of specimen for burst test.
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Fig. 2. Thickness distribution at z = 0 position (unit: mm): (a) specimen: S1A (So = 72.5 mm); (b) specimen: S1B (So = 72.5 mm); (c) specimen: S2A (So = 50 mm); (d) specimen: S2B (So = 50 mm); (e) specimen: S3A (So = 25 mm); (f) specimen: S3B (So = 25 mm).
In the tests, internal pressure was applied by injecting water into the pipe using a pump after evacuating air in room temperature environment. The pressure when water leaked from the mouth due to fracture was defined as burst pressure, Pf . 3. Experimental results and discussions The burst pressures obtained by the tests were Pf = 17.49 MPa (S1A) and 17.46 MPa (S1B) in the case of So = 72.5 mm,
18.17 MPa (S2A) and 19.50 MPa (S2B) in the case of So = 50 mm and 23.90 MPa (S3A) and 24.16 MPa (S3B) in the case of So = 25 mm, respectively. The burst pressure decreased as flaw length increased. It was also influenced by flaw depth. In the cases of So = 50 and 25 mm, burst pressure was lower in thinner pipe at z = 0 position, although it was almost the same in the case of So = 72.5 mm, in which the averaged thickness was the same as at z = 0 position.
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
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Table 3 Summary of measurement of thickness of flawed portion Flaw length (So ) 72.5 mm S1Aa
50 mm S1Ba
S2Aa
25 mm S2Ba
S3Aa
S3Ba
At z = 0 Number of data Average (mm) Minimum (mm) Maximum (mm)
34 1.92 1.90 1.94
34 1.92 1.88 1.95
34 1.82 1.78 1.86
34 1.97 1.92 2.01
34 1.98 1.96 1.99
34 2.04 2.03 2.05
All data Number of data Average (mm) Minimum (mm) Maximum (mm)
360 1.91 1.87 1.95
360 1.95 1.88 1.99
264 1.81 1.76 1.87
264 2.00 1.92 2.07
170 1.97 1.95 1.99
170 2.07 2.03 2.11
a
Specimen no.
Fig. 3 shows the relationship between flaw length and burst pressure normalized by Po , which is the failure pressure of unflawed pipe calculated by: t 2 Po = √ σf 3 R
(2)
where σ f is the flow stress defined as σ f = 0.5(σ y + σ b ) using the yield strength, σ y (=288 MPa), and ultimate strength, σ b (=447 MPa), of the test material. R denotes the nominal mean radius of the pipe in unflawed sections. The burst pressure decreases as flaw length increases. The specimens after the experiment are shown in Fig. 4. A straight longitudinal crack ran at the center of the flaw, except in specimen S3A. In specimen S3A (So = 25 mm), the crack curved toward the circumferential direction at the end of the wall-thinned region. Large deformation (bulging) occurred in all cases. Through measurements of strain during the tests, it was confirmed that bending stress caused by discontinuity of geometry at the end of pipe (z = 170 mm) was negligibly small. The burst pressures obtained in other studies are also plotted in Fig. 3. In these studies, experiments were conducted using pipeline steel (Kim et al., 2002b; Kiefner et al., 1973) and low alloy steel (Kobayashi et al., 2001), with an artificial flaw, of
Fig. 3. Relationship between failure pressure and flaw.
which depth was in the range from 0.48t to 0.51t. The shape of the flaw was a saw-cut longitudinal notch (Kiefner et al., 1973), U-shape longitudinal groove (Kobayashi et al., 2001) or uniform depth patch (Kim et al., 2002b). These data show similar burst pressures as those in this study, except for the saw-cut flaw. It was pointed out that flaws with a sharp notch resulted in lower burst pressure than blunt flaws (Kiefner, 1969). Especially, in the case of a long flaw, further reduction in burst pressure would be caused by ductile crack initiation (Kiefner et al., 1973; Hahn et al., 1969). The burst pressures obtained in the present study show similar dependency on flaw length with other results. It should be noted that the flaw depth in the present study was not exactly 0.5t; it was approximately 0.52t in specimens S1A and S1B, and 0.55t in S2A. The lines shown in Fig. 3 indicate the assessment curves for failure pressure expressed by the equation: 1 − d/t Pf = Po , (3) 1 − d/(tMt ) where Mt represents the so-called bulging factor. Several expressions of Mt have been proposed (Folias, 1965; Kiefner et al.,
Fig. 4. Specimens after tests: (a) S1A (So = 72.5 mm); (b) S2A (So = 50 mm); (c) S3A (So = 25 mm).
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M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
1973; Hahn et al., 1969; Erdogan and Sih, 1976), and the following two equations are used in Fig. 3. 1.61S 2 Mt = 1 + (4) Rt S4 S2 (5) Mt = 1 + 1.255 − 0.0135 Rt (Rt)2 Eq. (4) was derived from elastic shell theory for a throughwall crack in a cylinder, and was adopted in ASME B31G (ANSI/ASME, 1984), which provides an acceptance criterion for non-planar flaws, as well as the limit load failure criterion for axial cracks under internal pressure in the ASME boiler and pressure vessel code, section XI (ASME, 2004) and fitnessfor-service code of the Japan Society of Mechanical Engineers (JSME) (JSME, 2004). The validity of failure pressure of unflawed pipe, Po , defined by Eq. (2) was shown by burst experiments with an unflawed vessel with end-cap (Kobayashi et al., 2001). The reduction in failure pressure due to non-planar flaws is addressed by the term in parentheses in Eq. (3), the form of which is empirically derived (Kiefner, 1969). In the prediction using these equations, it is assumed that the material will fail in a ductile manner. These curves make conservative evaluation of the experimental results, except in the case of saw-cut flaws. Stephens and Leis (1997) proposed a different type of assessment curve based on FEA, which is expressed by the following equation and shown by the broken line in Fig. 3: t d 2S Pf = σb . (6) 1− 1 − exp −0.157 √ R t R(t − d)
Fig. 5. Geometry of a pipe containing wall-thinning.
In LL analyses, the elastic-perfect-plastic stress–strain relation was assumed. Various failure stresses have been used in the evaluation of failure strength of wall-thinned pipes, such as σ y + 10 ksi (Kiefner et al., 1973; Kiefner and Vieth, 1989), σ y /0.9 (API, 2000) and 1.1σ y (ANSI/ASME, 1984), σ b (Leis and Stephens, 1997b) and 0.5(σ y + σ b ), which is used in the current codes of ASME (ASME, 2004) and JSME (JSME, 2004). In order to investigate the validity of these definitions, three kinds of yield strengths in the elastic-perfect-plastic stress–strain relation were used: 0.5(σ y + σ b ), σ y + 10 ksi and σ b . The twice-elastic slope criterion was adopted to obtain the failure pressure. FEA using a work-hardening stress–strain relation obtained from the test material was also performed. Hereafter, in this study, this type of analysis is referred to as large deformation (LD) analysis. The stress–strain relation was obtained from a specimen cut from the test material for the circumferential direction. Fig. 7 shows the obtained stress–strain curve together with
This criterion exhibits relatively good agreement with the experimental results. In this equation, however, no end-capped (multi-axial stress) effect in the Mises criterion is considered, while this effect is considered in Eq. (2), and ultimate strength of the material is used for the material constant. 4. Procedure of finite element analysis The three-dimensional elastic-plastic FEA was performed using the general purpose program ABAQUS, Version 6.5 (ABAQUS Inc., 2005). The geometry of a pipe with wallthinning (flaw) is schematically shown in Fig. 5. The half-length of the flaw is expressed by So with an edge length of Se . Pipe length was set to L = 5Do , which is enough to ignore the boundary effect at the end of the specimen. The pipe was modeled by 8-node solid elements as shown in Fig. 6. At the bottom of the flaw, the model was divided by 4 elements in the thickness direction and 20 elements in the longitudinal direction, while element width in the circumferential direction is 5◦ . There are 17,280 elements and 24,384 nodes are included in the model shown in Fig. 6, which corresponds to the test of So = 50 mm. It was confirmed that well converged failure pressure were obtained with this mesh size. Axial tensile load was applied at the end of the pipe in order to simulate the end-capped condition.
Fig. 6. Finite element mesh for pipe with wall-thinning: (a) whole view; (b) magnified cross-sectional view of flawed portion.
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
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to geometrical symmetries, only one quarter of the full pipe was modeled with displacement boundary conditions on the planes of symmetries. 5. Analysis results and discussions 5.1. Prediction of experimental results
Fig. 7. Stress–strain relation used in analyses.
a simplified one, which is expressed by a bi-linear relation and is denoted as the “modeled curve”. To simulate large inelastic deformation during the tests, the non-linear geometry option with ABAQUS (the NLGEOM option) was invoked for the analysis procedure together with the modified Riks method, which allows analysis under static instability after plastic collapse. Two kinds of failure pressure were obtained by different definitions. The pressure when the Mises equivalent stress reaches the upper limit of the true stress–strain curve, σ bc , at any element is one of them and is denoted Pf(mi) , while the other failure pressure is defined as the maximum pressure during the analysis including the instability condition after the peak of the pressure and is denoted Pf(max) . In the analyses, internal pressure was increased until the instability condition. The magnitude of increase of pressure was controlled not to exceed 0.06 MPa in the plastic region. The typical step size in pressure just before the failure was 0.001 MPa. At first, analyses were performed in order to simulate the experiments. In the analysis, the transition section Se was set to 22.86 mm following the geometry of the test specimen. The thickness at the wall-thinned portion was determined by interpolating the measured thickness. Then, in order to examine the influence of the pipe and flaw geometry, the analyses were performed for various pipe and flaw configurations with uniform flaw depth and edge length of Se = 0.2So . In these analyses, due
The FEA results simulating those of the experiments are shown in Table 4 together with the experimental results. The failure pressures obtained by LL analyses give conservative results, except those obtained using σ b as the yield strength. The evaluated failure pressures are less than 30% of those obtained by the experiments in some cases. With use of σ b , results are less conservative, although, in general, σ b is not used as the yield strength in LL analyses. The safety margin changes with the flaw length. On the other hand, the failure pressures obtained by LD analysis agree well with the experimental results. Especially, the error in Pf(max) was less than 5% in all cases when the actual stress–strain relation was used. The Pf(max) seems better than Pf(mi) , whereas the difference between the two failure pressures is small. Even when the simplified bi-linear stress–strain relation was used, error in the predicted failure pressure was 11.15% at maximum. 5.2. Influence of pipe and flaw geometry Fig. 8 shows the relationship between failure pressure and flaw size obtained by FEA with a constant flaw depth of d = 0.5t together with the experimental results (Kim et al., 2002b; Kobayashi et al., 2001), which are the same data as shown in Fig. 3. The failure pressures obtained by LD analyses (Pf(max) ) using the actual stress–strain relations and LL analyses using Sf as the yield strength are also shown in the figure. The FEA results obtained by LD analyses exhibited good agreement with the experimental results including those obtained in other papers. This supports the validity of the analyses. On the other hand, LL analyses show poor correlation with the experimental results. This implies that LL analyses are not effective in simulating experiments, and that it is reasonable to estimate failure pressure conservatively.
Table 4 Comparison of failure pressure obtained by experiments and finite element analyses (unit: MPa) Specimen
Experiment
LL analysis
LD analysis
Yield strength
Stress–strain relation
0.5 (σ y + σ b )
S1A S1B S2A S2B S3A S3B
17.49 17.46 18.17 19.50 23.90 24.16
14.78 (15.51) 14.88 (14.79) 13.90 (23.48) 15.29 (21.58) 15.77 (34.02) 16.42 (32.02)
σ y + 10 ksi
13.77 (21.28) 13.86 (20.6) 12.95 (28.71) 14.25 (26.92) 14.69 (38.53) 15.31 (36.65)
σb
18.67 (−6.75) 18.80 (−7.68) 17.57 (3.32) 19.32 (0.9) 19.93 (16.63) 20.76 (14.09)
Experimental material
Modeled curve
Pf(mi)
Pf(max)
Pf(mi)
Pf(max)
17.77 (−1.58) 17.98 (−2.97) 17.51 (3.66) 19.60 (−0.54) 22.66 (5.18) 23.46 (2.91)
17.90 (−2.33) 18.14 (−3.92) 18.03 (0.79) 19.77 (−1.36) 22.90 (4.16) 23.72 (1.81)
17.13 (2.06) 17.38 (0.48) 16.97 (6.58) 18.52 (5.01) 21.24 (11.15) 21.92 (9.29)
17.22 (1.52) 17.41 (0.27) 17.33 (4.6) 18.97 (2.69) 21.84 (8.6) 22.58 (6.55)
The parentheses in the table indicate difference, which is defined by (experiment − analysis)/(experiment) × 100.
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Fig. 8. Relationship between the failure pressure and flaw length obtained by FEA together with experimental results.
As shown in Figs. 3 and 8, failure pressure is dependent on flaw length. The failure pressure is near Po if the flaw length is small, and decreases as flaw length becomes larger. When the flaw length is large enough, the failure pressure should be equivalent to that of pipe of thickness (t–d). Thus, there are two extremes of failure pressure of a pipe with wall-thinning. Leis and Stephens (1997a) proposed a concept in the following form: d Pf = 1 − f (geometry). t Po
Fig. 9. Relationship between the failure pressure and flaw length for various pipe geometry: (a) flaw length is normalized by (Rt)0.5 ; (b) flaw length is normalized by R.
(9)
Eq. (6) is based on the concept of Eq. (9) and yields a relatively good correlation with the experimental results as shown in Fig. 3, although the definition of Po is different from that in Eq. (2). f (geometry) is a transition function connecting the two extremes, and is related to pipe and flaw geometry. In order to examine the function f, failure pressures were evaluated for various pipe configurations, which were possible combinations of diameters of 100A, 200A, 300A, 400A and 500A and thicknesses of schedule 40 and 80 with different flaw depths of d/t = 0.1, 0.25, 0.5 and 0.75. Fig. 9 shows the failure pressure, Pf(max) , evaluated by LD analysis. The failure pressure is normalized by the value obtained for unflawed pipe. Therefore, normalized failure pressure is unity at S = 0. In most previous studies, as expressed by Eqs. (4)–(6), the influence of flaw length was normalized by the shell parameter (Rt)0.5 or (R(t − d))0.5 , as in the current fitness-for-service criteria, such as the API (API, 2000), ASME (ASME, 2003; ANSI/ASME, 1984) and BS (BS, 2005). However, as shown in Fig. 9(a), there is scattering in failure pressure when the flaw length is normalized by (Rt)0.5 , although the magnitude of scattering is limited compared to the degree of dependency on flaw depth and length. On the other hand, as shown in Fig. 9(b), normalization by R yields a good correlation between the failure pressure and flaw size. Sims et al. (1992) and Turbak and Sims (1994) also normalized flaw length by pipe diameter to show the change in failure pressure obtained by elastic-plastic FEA. It was pointed
out that the parameter Mt is independent of thickness for thin cylinder (Hahn et al., 1969). Furthermore, as shown in Fig. 4, fracture of flawed pipes is accompanied by large plastic deformation, suggesting that elastic theory might not be valid for such large deformations. The results of analyses indicate that, in the assessment criterion for fitness-for-service, normalization by R is better for representation of the influence of flaw length. This supports the proposal made by Sims and colleague. The transition function f can be obtained by interpolating the curves shown in Fig. 9(b). 6. Conclusions Burst tests using wall-thinned pipe of carbon steel for high-temperature use were conducted in order to examine the influence of the length of wall-thinning on failure pressure. Then, three-dimensional elastic-plastic FEA was performed to predict the burst pressure in the tests and investigate the influence of flaw shape on failure pressure. The following conclusions were obtained: (a) The burst pressure of wall-thinned pipe in tests decreased as flaw length increased, and was accurately predicted by large deformation FEA using the work-hardening stress–strain relation and precise thickness of the pipe at wall-thinned portion.
M. Kamaya et al. / Nuclear Engineering and Design 238 (2008) 8–15
(b) LL analysis with an elastic-perfect-plastic stress–strain relation yielded conservative results when the yield strength was set to 0.5(σ y + σ b ) and σ y + 10 ksi. (c) In normalizing the influence of flaw length on failure pressure, it is preferable to normalize flaw length by pipe mean radius of the unflawed section rather than by shell parameter (Rt)0.5 . References ABAQUS Inc., 2005. ABAQUS/Standard User’s Manual Ver. 6.5. ABAQUS Inc., USA. ANSI/ASME, 1984. Manual for Determining the Remaining Strength of Corroded Pipelines, ANSI/ASME B31G. 2000. Fitness-For-Service API579. American Petroleum Institute. ASME, 2003. Requirements for Analytical Evaluation of Pipe Wall Thinning (Cases of ASME Boiler and Pressure Vessel Code N-597-2), ASME, New York, USA. ASME, 2004. Boiler and Pressure Vessel Code Section XI, ASME, New York, USA. BS, 2005. Guide to Methods for Assessing the Acceptability of Flaws in Metallic Structures BS7910: 2005, BSI. Choi, J.B., Lee, S.M., Huh, N.S., Chang, Y.S., Kim, Y.J., Choi, Y.H., 2005. Limit load analyses for wall-thinned elbow with different bend angles subjected to internal pressure. In: The 6th International Symposium on the Structural Integrity of Nuclear Components, pp. 98–109. Chouchaoui, B.A., Pick, R.J., Yost, D.B., 1992. Burst pressure predictions of line pipe containing single corrosion pits using the finite element method. OMAE V-A, 203–210. Erdogan, F., Sih, G.C., 1976. Ductile fracture theories for pressurized pipes and containers. Int. J. Pressure Vessel Piping 4, 253–283. Folias, E.S., 1965. An axial crack in a pressurized cylindrical shell. Int. J. Fract. Mech. 1, 104–113. Hahn, G.T., Sarrate, M., Rosenfield, A.R., 1969. Criteria for crack extension in cylindrical pressure vessel. Int. J. Fract. Mech. 5, 187–210. JSME, 2004. Codes for Nuclear Power Generation Facilities: Rules of Fitnessfor-Service for Nuclear Power Plants, Tokyo, Japan. Kanninen, M.F., Pagalthivarthi, K.V., Popelar, C.H., 1992. A theoretical analysis for the residual strength of corroded gas and oil transmission pipelines. In: Chaker, V. (Ed.), Corrosion Forms and Control for Infrastructure, ASTM Special Technical Publication 1137, 183–198. Kiefner, J.F., 1969. Fracture initiation, Fourth Symposium on Line Pipe Research, PRCI Catalog No. L30075, G1-G36. Kiefner, J.F., Maxey, W.A., Eiber, R.J., Duffy, A.R., 1973. Failure stress levels of flaws in pressurized cylinders. ASTM STP 536, 461–481.
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