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International Journal of Pressure Vessels and Piping 120-121 (2014) 80e92

Contents lists avai

International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

Model error assessment of burst capacity models for energy pipelinescontaining surface cracks

Zijian Yan, Shenwei Zhang, Wenxing Zhou*

Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario N6A 5B9, Canada

a r t i c l e i n f o

Article history:Received 4 June 2013Received in revised form21 May 2014Accepted 23 May 2014Available online xxx

Keywords:PipelineSurface crackBurst capacityModel errorProbability distribution

* Corresponding author. Tel.: þ1 519 661 2111x879E-mail address: wzhou@eng.uwo.ca (W. Zhou).

http://dx.doi.org/10.1016/j.ijpvp.2014.05.0070308-0161/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

This paper develops the probabilistic characteristics of the model errors associated with five well-knownburst capacity models/methodologies for pipelines containing longitudinally-oriented external surfacecracks, namely the Battelle and CorLAS™ models as well as the failure assessment diagram (FAD)methodologies recommended in the BS 7910 (2005), API RP579 (2007) and R6 (Rev 4, Amendment 10). Atotal of 112 full-scale burst test data for cracked pipes subjected internal pressure only were collectedfrom the literature. The model error for a given burst capacity model is evaluated based on the ratios ofthe test to predicted burst pressures for the collected data. Analysis results suggest that the CorLAS™model is the most accurate model among the five models considered and the Battelle, BS 7910, API RP579and R6 models are in general conservative; furthermore, the API RP579 and R6 models are markedlymore accurate than the Battelle and BS 7910 models. The results will facilitate the development ofreliability-based structural integrity management of pipelines.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Mechanical defects, such as metal-loss corrosion, gouges andstress corrosion cracking, are major threats to the safety andstructural integrity of oil and gas transmission pipelines.Reliability-based integrity management program has beenincreasingly adopted by pipeline operators to ensure the safeoperation of pipelines [1,2]. Central to this program is to evaluatethe failure probability of the pipeline with respect to various limitstates, such as bursts of pristine pipes, corroded pipes, crackedpipes and pipes containing stress corrosion cracking under internalpressure, and to ensure that the maximum allowable failureprobability is met for a reference length (e.g. 1 km) over a referenceperiod of time (e.g. one year). Therefore, it is of great importance toaccurately evaluate the model errors of the deterministic pipe ca-pacity models and incorporate these model errors in the reliabilityanalysis.

Various models and methodologies are available to predict theburst capacities of pipes containing cracks (i.e. planar defects), e.g.the Battelle model [3e5], CorLAS™ model [6e8] and the failureassessment diagram (FAD) methodologies [9e11]. A number ofexperimental studies have been reported in the context of

31; fax: þ1 519 661 3779.

investigating fracture-based burst capacity models for pipes andvessels containing cracks [e.g. [3e5,12,13]]. For example, Kiefneret al. [3] conducted 140 tests for thin-walled pipes, including 92tests for pipes with through-wall flaws and 48 tests for pipes withpart-through-wall (surface) flaws, for the purpose of developingsemi-empirical equations to predict the ductile failure stress levelsof through-wall and surface flaws. Stoppler et al. [12] employedfour engineering approaches, namely the local collapse loads basedon the flow stress, toughness, plastic instability and ligament stresscriteria, respectively, to predict the burst pressures for 134 pipesand vessels containing longitudinally oriented cracks. Large de-viations were observed between the predicted and test burst ca-pacities, especially for deeply-cracked specimens. Motivated by thestudy in Stoppler et al. [12], Staat [13] collected a total of 293 full-scale tests mostly carried out in Germany and improved the for-mulas for local and global collapse loads of thick-walled pipes andvessels containing cracks. Furthermore, studies involving a limitednumber of burst tests for pipes with external surface cracks havealso been reported in the literature [14e21].

The model errors associated with the burst capacity of pristinepipes and pipes containing metal-loss corrosion defects (i.e. volu-metric defects) have been investigated and reported in the litera-ture [22,23]. However, reports of model errors associated with theburst capacity models for pipes containing cracks (referred to ascracked pipes) are scarce in the literature. The objective of the workreported in this paper was therefore to evaluate the model errors

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e92 81

associated with the burst capacity models for cracked pipes. Wefocused on thin-walled pipes containing longitudinally-orientedexternal surface cracks, which is of direct relevance to the integ-rity management of oil and gas pipelines [24,25]. We consideredfive widely used models/methodologies in this study, namely theBattelle and CorLAS™ models, as well as the FAD methodologiesrecommended in the British Standard 7910 (BS 7910) [9], AmericanPetroleum Institute Recommended Practice 579 (API RP579) [10]and R6 [11]. A total of 112 full-scale burst test data for crackedpipes were collected from the literature. Themodel error for a givenburst capacity model/methodology was evaluated based on theratios of the test to predicted burst pressures for the collected data.

The organization of this paper is as follows: Section 2 presents abrief description of the models and methodologies considered inthis study; Section 3 describes the full-scale test data collected fromthe literature; the analysis results are presented in Section 4, fol-lowed by the conclusions in Section 5. The equations to evaluate theapplied J-integral associatedwith the CorLAS™model and calculatethe parameters involved in the FAD methodologies are given inAppendixes A and B, respectively.

2. Models and methodologies for predicting burst capacity

2.1. Battelle model

The Battelle model, also known as the log-secant approach orNG-18 Equation, is a semi-empirical model developed at theBattelle Memorial Institute to predict the burst pressure of pipescontaining longitudinally-oriented surface cracks subjected tointernal pressure only [3e5]. The model assumes a rectangularcrack profile in the through-wall thickness direction defined bythe maximum crack depth and length and employs two criteria,namely the flow stress- and fracture toughness-based criteria, todetermine the burst pressure. The flow stress-based criterionaddresses the plastic collapse failure mode, whereas thetoughness-based criterion addresses the fracture failure mode.According to the Battelle model, the burst pressure, Pb1, is givenby

Pb1 ¼ min

8><>:2tsfD

1� at

1� aMt;4tsfpD

1� at

1� aMt

arccos

exp

� pK2

mat

8cs2f

!!9>=>;

(1)

where t and D denote the pipe wall thickness and outside diameter,respectively; sf is the flow stress of pipe steel and equalssy þ 68.95 MPa with sy being the yield strength of the pipe steel; aand 2c denote the crack depth (i.e. in the through pipe wall thick-ness direction) and length (i.e. in the longitudinal direction of thepipeline), respectively; Kmat denotes the fracture toughness of pipesteel in terms of the stress intensity factor, and M is the so-calledFolias factor and calculated by

M ¼

8>>>><>>>>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:6275

ð2cÞ2Dt

� 0:003375ð2cÞ4ðDtÞ2

sð2cÞ2Dt

� 50

3:3þ 0:032ð2cÞ2Dt

ð2cÞ2Dt

>50

(2)

If direct measurement of Kmat is not available, it is suggested tobe evaluated by an empirical equation, i.e. Kmat ¼ (CvE/Ac)0.5, withCv, Ac and E denoting the upper shelf Charpy V-notch (CVN) impactenergy, net cross-sectional area of the Charpy impact specimen (i.e.80 mm2 for full-size and 53.33 mm2 for 2/3-size specimens) andYoung's modulus of steel, respectively.

Note that the two terms in the curly brackets in Eq. (1) are theburst pressures corresponding to the plastic collapse and fracturefailure modes, respectively.

2.2. CorLAS™ model

CorLAS™ is a widely used tool in the pipeline industry to assessthe integrity of cracked pipes [20]. Similar to the Battelle model, theburst capacity model incorporated in this tool also considers twoindependent failure criteria: the flow stress- and toughness-basedcriteria. For simplicity, we refer to the corresponding burst capacitymodel as the CorLAS™ model. If the detailed crack depth profile isavailable, the CorLAS™ model uses the so-called effective areamethod [8] to evaluate the burst capacity (i.e. an iterative proce-dure to find the critical portion of the defect profile that leads to thelowest predicted burst pressure); otherwise, a semi-elliptical crackprofile is assumed in the model, with the length and depth of thesemi-ellipse equal to the crack length and maximum depth,respectively. The latter was considered in this study given that theactual crack profiles are, more often than not, unavailable inpractice.

The burst pressure, Pb2, according to the CorLAS™ model isgiven by

Pb2 ¼ 2tDscrit

0B@ 1� pa

4t1� pa

4tM

1CA (3a)

scrit ¼ minnsf ; sl

o(3b)

where the flow stress sf is defined as (sy þ su)/2 (as opposed tosf ¼ sy þ 68.95MPa in Eq. (1)) with su denoting the tensile strengthof the pipe steel, and sl is the local failure stress at the crackdetermined by the toughness-based criterion. The value of sl isobtained by solving Jc ¼ J, where J is the applied J-integral, i.e. thecracking driving force, and Jc is the fracture toughness of the pipesteel. Detailed formulations to evaluate J [6,8] are given inAppendix A. An empirical equation, i.e. Jc ¼ Cv/Ac, which is equiv-alent to the one used for the Battelle model in terms of Kmat, issuggested to estimate Jc from the CVN impact energy, if more ac-curate information about Jc (e.g. from fracture toughness tests) isunavailable. This empirical equation was adopted in this study.

2.3. Failure assessment diagram (FAD) methodologies

2.3.1. OverviewThe failure assessment diagram (FAD) was proposed by Dowling

and Townley [26] based on fracture mechanics and involves twokey parameters, namely the brittle fracture parameter, Kr, andplastic collapse parameter, Lr. The use of FAD to carry out theintegrity assessment of cracked pipes involves three components,namely evaluating the assessment point (Lr, Kr), establishing theassessment line and checking the relative position of the assess-ment point with respect to the assessment line (or cut-off line) (seeFig. 1). The integrity of a cracked pipe under a given pressure and/orother loading conditions is acceptable if the assessment point fallswithin the region bounded by the assessment and cut-off lines aswell as the two axes, and unacceptable otherwise.

To predict the burst pressure of a cracked pipe based on FAD is toevaluate the pressure that causes the assessment point to fall on theassessment or cut-off line. Note that the FAD-based burst pressureprediction can account for the interaction between the plasticcollapse and fracture failure modes, whereas the Battelle and

Fig. 1. Illustration of the FAD model.

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e9282

CorLAS™ models consider these two failure modes simultaneouslybut independently [20]. Furthermore, the use of FAD-based modelstends to result in a conservative assessment of the integrity ofcracked pipes because FAD is developed for failure avoidance ratherthan failure prediction [20,25]. The FAD methodologies as pre-sented in BS 7910 (2005) [9], API RP579 (2007) [10] and R6 (Revi-sion 4, Amendment 10) [11] are described and analyzed in thisstudy. It is noted that BS 7910 was updated at the end of 2013, afterthe completion of the present study; however, it is believed that theupdate has a minimal impact on the analysis results presented inthis paper.

2.3.2. British Standard 7910 (BS 7910)Three FAD-based assessment levels are included in BS 7910,

namely Levels 1, 2 and 3 [9]. Level 1 is the most simplistic assess-ment method that is in general applicable to conditions withlimited information of material properties or applied stress; Level 2is the normal assessment level, and Level 3 is the most sophisti-cated and applicable to ductile materials exhibiting stable tearing.Furthermore, two FADs for Level 2 are given in BS 7910, namelyLevels 2A and 2B. Level 2B is more accurate than Level 2A becausethe former incorporates the stress-strain curve of the materialwhereas the latter is independent of the stress-strain curve of thematerial. Only Level 2A was considered in this study based on theconsideration that the stressestain curve of the material may notalways be readily available.

The assessment line of Level 2A in BS 7910 defines Kr as thefollowing function of Lr:

Kr ¼(�

1� 0:14L2r��

0:3þ 0:7exp��0:65L6r

��0< Lr < Lmax

r

0 Lr > Lmaxr

(5)

where Lmaxr denotes the cut-off line (see Fig.1) and equals (syþ su)/

(2sy).The assessment point is evaluated by

Lr ¼ srefsy

(6a)

Kr ¼ KI

K(6b)

mat

where sref denotes the applied reference stress, and KI and Kmat

denote the applied stress intensity factor (crack driving force) andfracture toughness of the material, respectively. The equations toevaluate sref and KI are given in Appendix B. It must be emphasizedthat the equations summarized in Appendix B are only applicable tothin-walled pipes that are subjected to internal pressure only andcontain longitudinally-oriented external surface cracks with asemi-elliptical depth profile. An empirical equation is suggested inBS 7910 (see Clauses J.2.1 and J.2.4 of BS 7910) to evaluate Kmat andgiven by

Kmat ¼ min

(�12

ffiffiffiffiffiCv

p� 20

��25t

�0:25þ 20; 0:54Cv þ 55

)

(7)

Note that the units of Kmat, Cv and t in Eq. (7) areMPa(m)0.5, Jouleand mm, respectively.

2.3.3. American Petroleum Institute Recommended Practice 579(API RP579)

API RP579 provides guidance for carrying out fitness-for-service(FFS) assessment of pressurized equipment with flaws or damagebased on the FAD method [10]. API RP579 includes three levels ofassessment, i.e. Levels 1, 2 and 3. Level 1 Assessment providesconservative screening criteria based on the minimum amount ofinformation of cracks, pipes and operation and design conditions,e.g. depth and length of crack, pipe wall thickness and operatingtemperature. Level 2 Assessment requires the detailed informationon material properties, loading conditions and stress state at thelocation of the flaw, and provides a better estimate of the structuralintegrity of a component than Level 1. Level 3 Assessment is themost accurate because it incorporates more detailed information ofthe material and crack, e.g. the material-dependent FAD, J-R curveof the material and growth of cracks [10]. In this study, Level 2specified in API RP579 was considered, which is the same as Level2A of BS 7910 except that different equations are used to evaluatesref, Kmat and KI. The equations for calculating sref and KI are sum-marized in Appendix B. Note that these equations are only appli-cable to cracks satisfying the following conditions: 1) 0 � a/t � 0.8;2) 0 < a/c � 2.0, and 3) D/t � 4. In the absence of accurate infor-mation about the fracture toughness, Kmat can be evaluated from Cvusing the Rollfe-Novak-Barsom empirical equation [10,30]:

Kmat ¼ sy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:64

�Cvsy

� 0:01�s

(8)

where Kmat, Cv and sy are in MPa(m)0.5, Joule and MPa, respectively.

2.3.4. R6R6 [11] provides a procedure to evaluate the structural integrity

of structures with defects based on the FAD method and includesthree options, namely Options 1, 2 and 3. Option 1 only requires theyield and tensile strengths as the input for material properties;Option 2 requires the stressestrain relationship of thematerial, andOption 3 requires results of detailed elasticeplastic analysis of thedefective component. R6 also provides an approximate FAD modelfor Option 2, referred to as Option 2A in this study. Option 2A re-quires the yield and tensile strengths as well as Young’s modulus asthe input for material properties, and differentiates continuous

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e92 83

yielding materials (i.e. materials without a yield plateau) fromdiscontinuous yielding materials (i.e. materials with a yieldplateau). We considered Option 1 and Option 2A in this study.

The assessment line corresponding to Option 1 is defined as [11]

Kr ¼8<:�1þ 0:5L2r

��0:5�0:3þ 0:7exp

��0:6L6r

��0< Lr < Lmax

r

0 Lr � Lmaxr

(9)

where Lr is evaluated based on the limit load solution (as opposedto the reference stress solution adopted by both BS 7910 and APIRP579), i.e. Lr¼ p/pL; p denotes the internal pressure; pL denotes thelimit pressure leading to plastic collapse of the pipe, andLmaxr ¼ (sy þ su)/(2sy). Note that Eq. (9) is applicable to thecontinuous and discontinuous yielding materials if Lr < 1 but onlyapplicable to the continuous yielding material if Lr � 1. It is sug-gested in R6 [11] that Option 2 can be employed for the discon-tinuous yielding material with Lr � 1 [11].

The FAD corresponding to Option 2A is defined as [11]

Kr ¼

8>>>>>><>>>>>>:

�1þ 0:5L2r

��0:5�0:3þ 0:7exp

��mL6r

��0< Lr � 1

f ð1ÞLN�12Nr 1< Lr < Lmax

r

0 Lr � Lmaxr

(10a)

for the continuous yielding material, and

Kr ¼ f ðLrÞ ¼

8>>>>>>>>><>>>>>>>>>:

�1þ 0:5L2r

��0:50< Lr <1

ðlþ 1=2lÞ0:5 Lr ¼ 1

f ð1ÞLN�12Nr 1< Lr < Lmax

r

0 Lr � Lmaxr

(10b)

for the discontinuous yielding material, where

m ¼ min�0:001E

�sy;0:6

(10c)

N ¼ 0:3�1� sy

su

�(10d)

l ¼ 1þ EDεsy

(10e)

Dε ¼ 0:0375�1� sy

1000

�(10f)

Table 1Ranges of the geometric properties of cracks and pipes and the material properties.

a (mm) 2c (mm) D (mm) t (mm)

Upper bound 16.8 850 1422.4 21.7Lower bound 1.0 20.0 88.9 4.0

The evaluation of Kr is obtained through Eq. (6b), whereasthe equations for calculating KI are detailed in Appendix B. Assuggested in R6 [11,31], the fracture toughness Kmat is evaluatedfrom the same empirical equation adopted in API RP579, i.e. Eq.(8). Two solutions are provided in R6 to calculate the limitpressure pL, namely the global and local solutions, both ofwhich were considered in this study. The global solution is ayield criterion-dependent solution; therefore, we furtherconsidered the von Mises and Tresca yield criteria [32] toevaluate the global solution. The corresponding equations forevaluating pL are summarized in Appendix B. Note that theequations provided in R6 to calculate KI are only applicable topipes with 10 � D/t � 22 (see Table IV.3.4.3.1 in R6) [11]. Forpipes with D/t > 22, it is suggested in R6 that the equationsadopted by API RP579 as summarized in Appendix B be used tocalculate KI. Because all of the test data collected in this studyhave the D/t ratios greater than 22, the same equations wereused to calculate KI in API RP579 and R6.

For brevity, the three FAD methodologies described above arereferred to as BS 7910, API RP579 and R6 models, respectively, inthe following sections.

3. Full scale burst test data

To evaluate the model errors associated with the burst capacitymodels described in Section 2, a database of 112 full scale burst testdata for cracked pipes was established according to the criteriadescribed as follows:

(1) The specimens are thin-walled pipes, i.e. D/t � 20 [32],subjected to internal pressure only.

(2) Each specimen contains an external longitudinally orientedpart-through-wall flaw/crack with a rectangular or semi-elliptical profile. Both machined flaws and real cracks areconsidered.

(3) Information about the specimen (e.g. the geometric prop-erties of the pipe and crack as well as material properties ofsteel) is sufficient such that it is applicable to at least one ofthe burst capacity models considered. For example, a totalof 48 test data were reported by Kiefner et al. [3], but only36 test data were included in the database because theCharpy V-notch impact energy for the other 12 tests isunavailable.

Based on the above-described criteria and information providedin the literature, a database containing 112 test data was estab-lished. The ranges of the geometric properties of cracks and pipes aswell as the material properties for these data are summarized inTable 1.

Only subsets of the test data were used to evaluate the modelerrors associated with the CorLAS™, BS 7910, API RP579 and R6models because of the limited applicability associated with boththe models (e.g. the a/t limit for the API RP579 model) and test data(e.g. test data for which tensile strengths are unknown inadmissibleto the four models). The data points that were used to evaluate themodel errors associated with the five models are summarized inTable 2, where the number in the brackets represents the total

sy (MPa) su (MPa) Cv (J) a/t a/2c

1096 1179 261 0.95 0.226246 463 15.2 0.18 0.002

Table 2Summary of test data applicable to individual burst models.

Number of applicable data points Flaw (crack) profile Source

Battelle CoLAS™ BS 7910 API RP579 R6

(112) (103) (109) (82) (82)

3 3 3 3 3 S Cravero and Ruggieri, 20062 2 2 2 2 S Garwood et al., 19814 4 4 4 4 S Hosseini et al., 201015 15 15 12 12 S Keller et al., 198736 33 33 27 27 R Kiefner et al., 19732 2 2 2 2 R Mannucci et al., 20006 e 6 6 6 S Rana and Rawls, 20078 8 8 5 5 S Rothwell and Coote, 200929 29 29 14 14 S Staat, 20047 7 7 7 7 S Wang and Smith, 1988

Fig. 2. Comparison of the predicted and test burst pressures based on the Battelle model.

Fig. 3. Comparison of the predicted and test burst pressures based on the CorLAS™ model.

Fig. 4. Comparison of the predicted and test burst capacities based on the BS 7910 model.

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e92 85

Fig. 5. Comparison of the predicted and test burst pressures based on the API RP579 model.

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e9286

number of data points applicable to a particular model, and thesymbols S and R represent the semi-elliptical and rectangular crackprofiles, respectively.

4. Model error assessment

We calculated the burst pressures using the five models for thecorresponding test data sets summarized in Table 2. It should beemphasized that the actual (as opposed to nominal) geometric andmaterial properties of the pipe specimens were employed in thecalculation. For example, the actual yield strengths rather than thespecified minimum yield strengths (SMYS) of the specimens wereused in the calculation. We also applied a profile transformation tothe test data based on the equivalent-area criterion such that thetransformed crack profile is consistent with the assumptions in theburst models. For example, a semi-elliptical crack profile wasconverted to a rectangular crack profile with an equivalent lengthbased on the assumption that both profiles have the same area anddepth, when the Battelle model was employed to predict the burstpressure.

A comparison of the predicted and test burst pressures isdepicted in Fig. 2 through 6 for the Battelle, CorLAS™, BS 7910, APIRP579 and R6 models, respectively, where the straight line is the1:1 line, i.e. the predicted burst pressure equal to the test burstpressure. Fig. 2 suggests that the Battelle model is in general con-servative because the majority of the data points are below the 1:1line. Note that the 36 test data reported by Kiefner et al. [3] arecloser to the 1:1 line than the other test data. This makes sense asthese 36 test data are the majority of the data set used to developthe Battelle model [3]. Fig. 3 suggests that the CorLAS™model is ofhigh accuracy because the data spread over a narrow region alongthe 1:1 line.

Note that a safety factor of 1.2 is applied to the referencestress equation in the BS 7910 model (see Appendix B). Wetherefore considered two BS 7910 models with respect to thesafety factor, denoted by BS 7910-I and BS 7910-II, respectively.The former includes the safety factor in the reference stresscalculation whereas the latter ignores the safety factor (i.e.assuming the safety factor to be unity) in the calculation. Thevalue of Kmat was evaluated from Eq. (7) for BS 7910-I and BS

7910-II. We further considered one additional model, denoted byBS 7910-III, which is the same as BS 7910-II except that Kmat isevaluated as Kmat ¼ (CvE/Ac)0.5, which is the same as that used inthe Battelle and CorLAS™ models. These models allow us toinvestigate the impact of the safety factor and empirical equationfor evaluating Kmat on the statistics of the model error associatedwith the BS 7910 model. The results corresponding to these threemodels are plotted in Fig. 4(a) through (c), respectively, each ofwhich also includes a small window showing a comparison ofthe predictions corresponding to the three models. A comparisonbetween Fig. 4 and Figs. 2, 3, 5 and 6 indicates that the BS 7910model is the most conservative among the five models consid-ered because almost all of the data points shown in Fig. 4 arebelow the 1:1 line, with many of them markedly below the 1:1line.

As described in Section 2.3.4, a total of six R6 models wereconsidered, denoted by R6-I through R6-VI, respectively, as sum-marized in Table 3. Because the stress-strain curves of the testspecimens included in the databasewere not reported, we assumedthe pipe steel to be the continuous yieldingmaterial in applying theR6 models. This assumption can be justified from the fact that themechanical work involved in the pipe manufacturing processtypically leads to a roundhouse stress-strain curve as opposed to astress-strain curve with a yield plateau [33]. A comparison of thepredicted and test burst pressures corresponding to the six modelssummarized in Table 3 is depicted in Fig. 6(a) through (f), respec-tively. Fig. 6 indicates that there is no significant difference betweenthemodels considered. The trend reflected in Fig. 6 is similar to thatobserved from Fig. 5.

The test-to-predicted ratio corresponding to each of theapplicable experimental data was calculated for a given burstcapacity model. The samples of the test-to-predicted ratio for agiven burst capacity model were then plotted on various prob-ability papers to identify the best-fit probability distribution forthe corresponding model error based on the Kolmogor-oveSmirnov test [27]. The best-fit distributions as well as thecorresponding means and coefficients of variation (COV) for themodel errors associated with all the models considered aresummarized in Table 4. Note that the cumulative distributionfunction (CDF) of a Frechet distributed (i.e. the Type II extreme

Fig. 6. Comparison of the predicted and test burst pressures based on the R6 models.

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e92 87

Table 3Summary of the six scenarios considered for the R6 model.

Model Option of FAD Solution of pL Yield criterion

R6-I 1 Local e

R6-II Global von MisesR6-III Global TrescaR6-IV 2A Local e

R6-V Global von MisesR6-VI Global Tresca

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e9288

value distribution) random variable X, Fx(x), is given byFx(x) ¼ exp(�(x/s)�x)I(x) [28], where x (x > 0) and s (s > 0) denotethe shape and scale parameters, respectively, and I(x) is an in-dicator function and equal unity if x > 0 and zero otherwise.Given x and s, the mean and variance of X equal sG(1�1/x) forx > 1 and s2(G(�2/x)�( G(1�1/x))2) for x > 2, respectively, withG($) denoting the Gamma function [28]; therefore, given themean and variance of X, x and s can be obtained by solving themean and variance equations simultaneously. The values of x ands for a given Frechet distribution shown in Table 4 were alsocalculated and are shown in the same table. Fig. 7(a) through (e)depict the test-to-predicted ratios plotted on the probabilitypapers corresponding to the best-fit distributions for fiverepresentative models summarized in Table 4. For the Battelleand CorLAS™ models, the portions of the data points for whichthe predicted burst pressures are governed by the local fractureand plastic collapse criteria, respectively, are shown in Table 4.For the FAD approaches, the portions of the data points for whichthe predicted burst pressures are governed by the assessmentline and the cut-off line, respectively, are also shown in Table 4.The results indicate that the majority of the data points aregoverned by the local fracture criterion for the Battelle andCorLAS™ models, and by the assessment line for the FADapproaches.

The results shown in Table 4 suggest that the CorLAS™ modelis the most accurate among the five models considered becausethe mean and COV of the corresponding model error are 0.96 and22.8%, respectively e the former is closest to unity than those ofthe model errors for all the other models, and the latter is onlyslightly higher than those of the model errors for the API RP579and R6 models. On the other hand, the Battelle, BS 7910, APIRP579 and R6 models are in general conservative. In particular,BS 7910-I and BS 7910-II tend to result in extremely conservativeand highly variable predictions. Furthermore, the API RP579 and

Table 4Basic statistics of model errors.

Model Number of applicable data Mea

Battelle 112 (109)a 1.40CorLAS™ 103 (73)a 0.96BS 7910-I 109 (106)b 3.87BS 7910-II 109 (109)b 3.73BS 7910-III 109 (72)b 1.63API RP579 82 (77)b 1.35R6-I 82 (76)b 1.54R6-II 82 (77)b 1.30R6-III 82 (77)b 1.39R6-IV 82 (80)b 1.52R6-V 82 (82)b 1.29R6-VI 82 (81)b 1.39

Note:a The number in the brackets indicates the number of data points for which the predib The number in the brackets indicates the number of data points for which the predicte

R6 models are markedly more accurate than the Battelle and BS7910 models given that the COVs of the model errors for theformer two models (ranging from 19.3 to 22.3%) are much lowerthan those of the model errors for the latter two models (rangingfrom 43.1 to 101%).

A comparison of the three BS 7910 models suggests that themodel error is highly sensitive to the empirical equation forevaluating Kmat but insensitive to the safety factor of 1.2 appliedin calculating sref. The large model errors associated with BS7910-I and BS 7910-II are mainly due to the very conservativeempirical equation (i.e. Eq. (7)) used to evaluate Kmat.

A comparison of the model errors associated with the R6models suggests that the difference between Option 1 andOption 2A for the continuous yielding material is negligible ifboth options employ the same solution for the limit pressurepL. For example, if pL is evaluated based on the local solution(i.e. R6-I and R6-IV), the mean (COV) of the model error is 1.54(22.3%) for Option 1 and 1.52 (21.2%) for Option 2A. On theother hand, the local solution for pL on average leads to moreconservative predictions of the burst pressure than the globalsolution for pL for both Option 1 and Option 2A. This is ex-pected because the global solution addresses the overallcollapse of the defective structure assuming the elastic-perfectly plastic behavior, whereas the local solution ad-dresses the yielding of a limited area surrounding the flaw onthe structure so that the local limit pressure is in general lessthan or equal to the global limit pressure [11]. Furthermore, forthe global solution, the Tresca yield criterion leads to margin-ally more conservative predictions of the burst pressure thanthe von Mises yield criterion. This is also expected given thatthe Tresca- and von Mises criterion-based solutions tend toprovide the lower and upper bounds of the burst pressure forpristine thin-walled pipes [29]. Finally, the model errors for theR6 models are similar to that of the API RP579 model, which is

n COV (%) Fitted distribution

43.1 Frechet (s ¼ 1.14, x ¼ 3.96)22.8 Normal96.7 Frechet (s ¼ 2.64, x ¼ 2.56)

101.0 Frechet (s ¼ 2.52, x ¼ 2.52)57.8 Frechet (s ¼ 1.25, x ¼ 3.28)19.8 Lognormal22.3 Lognormal22.119.821.221.819.3

cted burst pressures are governed by local fracture as opposed to plastic collapse.d burst pressures are governed by the assessment line as opposed to the cut-off line.

Fig. 7. Test-to-predicted ratios plotted on the probability papers.

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e92 89

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e9290

attributed to that both models employ the same approaches toevaluate KI and Kmat.

5. Conclusion

We developed the probabilistic model errors associated withfive burst capacity models that are commonly used to assess thestructural integrity of energy pipelines containing externallongitudinally-oriented part-through-wall cracks. The burst ca-pacity models considered in this study include the Battelle andCorLAS™ models, as well as the FAD methodologies specified inBS 7910 (2005), API RP579 (2007) and R6 (Rev 4, Amendment10).

Three variations of the BS 7910 Level 2A FAD were considered,denoted as BS 7910-I, II and III, respectively. BS 7910-I includes thesafety factor of 1.2 in the reference stress calculation whereas BS7910-II and III ignore the safety factor in the reference stresscalculation. Furthermore, BS 7910-I and II employ Eq. (7) to eval-uate Kmat from the CVN impact energy whereas BS 7910-III employsthe empirical equation Kmat ¼ (CvE/Ac)0.5 to evaluate Kmat. Weconsidered six R6 models, denoted as R6-I through VI. R6-I throughR6-III are based on the Option 1 FAD and differ by the solution toevaluate pL, i.e. the local solution, global solution with the vonMises yield criterion and global solution with the Tresca yield cri-terion. R6-IV through VI are the same as R6-I through III, respec-tively, except that the former three models are based on thesimplified Option 2 FAD. Finally, the Level 2 FAD in API RP579 wasconsidered.

We established a database of 112 full-scale burst tests of pipescontaining longitudinally-oriented external surface cracks based oninformation in the literature. The probabilistic characteristics of themodel error, including the best-fit distribution, mean value andCOV, for a given burst model were evaluated by analyzing the test-to-predicted ratios for the applicable test data included in thedatabase. The actual as opposed to the nominal geometric andmaterial properties of the pipe specimens were employed in theburst model.

The analysis results suggest that the CorLAS™ model is themost accurate model among the five models considered: thecorresponding model error can be characterized by a normaldistribution with the mean and COV equal to 0.96 and 22.8%,respectively. The Battelle, BS 7910, API RP579 and R6 models ingeneral result in conservative predictions of the burst pressure.But the API RP579 and R6 models are markedly more accuratethan the Battelle and BS 7910 models. The conservatism in theempirical equation used to evaluate the fracture toughness Kmatis the main reason for the highly conservative nature of the BS7910 model. The model error for the API RP579 model, with thecorresponding mean value and COV equal to 1.35 and 19.8%respectively, are similar to those for the R6 models, with themean values and COVs ranging from 1.29 to 1.54 and from 19.3 to22.3%, respectively. Furthermore, the R6 models employing thelocal solution for the limit pressure lead to more conservativepredictions of the burst pressure than the R6 models employingthe global solution for the limit pressure, and for the R6 modelsemploying the global solution for the limit pressure, the Trescayield criterion leads to marginally more conservative predictionsof the burst pressure than the von Mises yield criterion. Theprobabilistic model errors reported in this study will facilitatethe development and application of the reliability-based pipe-line integrity management program with respect to surfacecracks.

Acknowledgments

The authors gratefully acknowledge the financial supportprovided by the Natural Sciences and Engineering ResearchCouncil of Canada and TransCanada Ltd. through the Collabo-rative Research and Development (CRD) program (CRDPJ405856-10). The constructive comments of the anonymousreviewer are appreciated.

Appendix A. Calculation of J in the CorLAS™ model

The J-integral is calculated as follows [6,8]

J ¼ QsfFsfa

ps2lE

þ f3ðnÞεpsl!

(A.1)

where Qsf is the flaw shape factor and calculated by

Qsf ¼ 1:6260� 1:4795� a2c

�� 6:3428

� a2c

�2� 10:2610

� a2c

�3(A.2)

Fsf is the free surface factor and calculated by

Fsf ¼

8>><>>:

�2tpa

�tan�pa2t

��0:50<

at� 0:95

2:918 0:95<at� 1:0

(A.3)

εp is the plastic strain corresponding to sl and evaluated from apower-law stressestrain relationship:

εp ¼ Ks1=nl (A.4a)

with K denoting the strength coefficient evaluated by

K ¼�0:005� sy

E

�.s1=ny (A.4b)

and n denoting the strain hardening exponent estimated from thefollowing empirical equation:

n ¼ �0:00546þ 0:556�sy

su

�� 0:547

�sy

su

�2(A.4c)

f3(n) is the Shih and Hutchinson factor and given by

f3ðnÞ ¼"3:85

�1n

�0:5

ð1� nÞ þ pn

#ð1þ nÞ (A.5)

Appendix B. Calculations of sref, pL and KI in the BS 7910 (2005), API RP579 (2007) and R6 (Rev 4, Amendment 10) models

BS 7910 API RP579 R6

sref or pL sref ¼ 1:2MspD2t

where

Ms ¼ 1�½a=ðtMTÞ�1�ða=tÞ

MT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1:6

�c2Rit

�s

Ri ¼ D�2t2

sref ¼gPbþ gPbð Þ2þ9 MsPm 1�að Þ2ð Þ2

� 0:53 1�að Þ2

where

g ¼ 1� 20�

a2c

�0:75a3

a ¼ a=t1þðt=cÞ

Pb ¼ pR2o

R2o�R2

i

"tRi� 3

2

�tRi

�2

þ 95

�tRi

�3#

Ro ¼ D2

Pm ¼ pRit

Ms ¼ 11�a

tþa=tð1=MtðlaÞÞ

MtðlaÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:02þ0:4411l2aþ0:006124l4a

1:0þ0:02642l2aþ1:533�10�6l4a

rla ¼ 1:818cffiffiffiffiffiffi

Riap

Global Solution:PLsy

¼g

1

Mexln�

1þ h

1þ h� ah

�þ lnð1þ h� ahÞ

�

þ24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ h� 1

2

�1� 1

Mex

�ah

�2

þ ðahÞ24

1� 1

M2ex

!vuut

��1þ h� 1

2

�1� 1

Mex

�ah

�35Local Solution:

PLsy

¼ 1h2c þ1

h2c lnð1þ hÞ þ lnð1þ h� ahÞ

�where

Mex ¼ 1þ 1:4 ah

ð1þhÞf2

!0:5

h2c ¼ 1�a

Mex�1

a ¼ at , h ¼ t

Ri, f ¼ a

c, and

g ¼�

1 Tresca based solution2=

ffiffiffi3

pvon Mises based solution

KI KI ¼ MMmpD2t

� � ffiffiffiffiffiffipa

p

M ¼ 1�½a=ðtMTÞ�1�ða=tÞ

MT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3:2

�c22Rt

�s

R ¼ D�t2

Mm ¼ ½M1 þM2ða=tÞ2 þM3ða=tÞ4�gfq=FM1 ¼

�1:13� 0:09 a=ð cÞ 0 � a=2c � 0:5

c=ð aÞ0:5 1þ 0:04 c=ð a½ Þ� 0:5< a=2c � 1:0

M2 ¼�0:89½0:2þ ða=cÞ� � 0:54 0 � a=2c � 0:5

0:2ðc=aÞ4 0:5< a=2c � 1:0

M3 ¼

8><>:

0:5� 1½0:65þ ða=cÞ� þ 14½1� ða=cÞ�24 a=2c � 0:5

�0:11ðc=aÞ4 0:5< a=2c � 1:0

g ¼

8>><>>:

1þ0:1þ 0:35

�at

�2�ð1� sinqÞ2 a=2c � 0:5

1þ0:1þ 0:35

�ca

��at

�2�ð1� sinqÞ2 0:5< a=2c � 1:0

fq ¼(½ða=cÞ2cos2qþ sin2

q�0:25 0 � a=2c � 0:5½ða=cÞ2sin2

qþ cos2q�0:25 0:5< a=2c � 1:0

F ¼(

½1þ 1:464ða=cÞ1:65�0:5 0 � a=2c � 0:5½1þ 1:464ðc=aÞ1:65�0:25 0:5< a=2c � 1:0

q ¼8<:

p

2at the deepest point

0 at flaw surface

KI ¼ pR2i

R2o�R2

i2G0 þ 2G1

aRo

� �þ 3G2

aRo

� �2 þ 4G3aRo

� �3 þ 5G4aRo

� �4 � ffiffiffiffiffiffiffiffiffiffiffiffipa=Q

pRo ¼ D

2

G0 ¼ A0;0 þ A1;0bþ A2;0b2 þ A3;0b

3 þ A4;0b4 þ A5;0b

5 þ A6;0b6

G1 ¼ A0;1 þ A1;1bþ A2;1b2 þ A3;1b

3 þ A4;1b4 þ A5;1b

5 þ A6;1b6

b ¼ 1Ai,j (i ¼ 0, 1, …, 6; j ¼ 0, 1) are given in Table C.13 of Annex C in API RP579.

G2 ¼ffiffiffiffiffi2Q

pp

�1615 þ 1

3M1 þ 16105M2 þ 1

12M3

�

G3 ¼ffiffiffiffiffi2Q

pp

�3235 þ 1

4M1 þ 32315M2 þ 1

20M3

�

G4 ¼ffiffiffiffiffi2Q

pp

�256315 þ 1

5M1 þ 2563465M2 þ 1

30M3

�M1 ¼ 2pffiffiffiffiffi

2Qp ð3G1 � G0Þ � 24

5

M2 ¼ 3M3 ¼ 6pffiffiffiffiffi

2Qp ðG0 � 2G1Þ þ 8

5

Q ¼(1þ 1:464ða=cÞ1:65 a=c � 1:01þ 1:464ðc=aÞ1:65 a=c>1:0

Same as KI in API RP579

Z.Yanet

al./International

Journalof

PressureVessels

andPiping

120-121(2014)

80e92

91

Z. Yan et al. / International Journal of Pressure Vessels and Piping 120-121 (2014) 80e9292

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