icoincte., tl TrSS

d aateelinreduch

a. Thprinensnjussu

es t

I

ia

a

e

Sbyield theory generally predicts a lower bound of experimental dataof burst pressure, whereas the von Mises yield theory predicts anupper bound of burst pressure for end-capped pipes or cylindricalpps

fpftFts

lbptffr

t

flow rule as its default plasticity model for isotropic hardeningmaterials. Accordingly, the RIKS method that was built in ABAQUScan only determine a von Misesbased upper bound solution for

i2W

6

Downloaded Frressure vessels, as reviewed in Refs. 2,3. This stimulates theresent work to develop a better solution to predict the burst pres-ure of line pipes.

The application of the finite element analysis FEA to the burstailure prediction of pipelines with or without corrosion defectsotentially offers greater accuracy, but it requires an appropriateailure criterion. Such failure criteria available are often related tohe ultimate tensile stress UTS, but inconsistent with each other.or instance, a pipeline can be considered as a burst failure when

he von Mises equivalent stress on the defect ligament in FEAimulations reaches the true UTS. This failure criterion was uti-ized by Fu and Kirkwood 4 for X46 and X60 pipeline steels andy Karstensen et al. 5 for an X52 pipeline steel in their FEAredictions of burst pressure. However, Choi et al. 6 predictedhe burst pressure for X65 corroded pipelines using a differentailure criterion in the FEA calculations. They assumed that burstailure occurs when the von Mises equivalent stress in a defecteaches 90% of the true UTS for a rectangular defect and 80% ofhe true UTS for an elliptical defect. Therefore, further investiga-

the burst pressure of pipelines, as demonstrated in our recent work7 for a defect-free pipe. Likewise, Lam et al. 8 showed that theFEA results of burst pressure determined using ABAQUS and theRIKS method overestimate the experimental data for thin-wallcylindrical pressure vessels. This indicates that ABAQUS with theRIKS method may determine an unreliable FEA result of burstpressure for end-capped pipes or cylindrical shells.

To effectively predict the actual burst pressure, theoretical andnumerical investigations are carried out for defect-free end-capped pipes in this paper. Based on the proposed average shearstress yield ASSY theory, a new theoretical solution for the burstpressure of pipes is obtained and validated by extensive experi-mental data. Four failure criteria are proposed for the burst pres-sure prediction in the von Misesbased FEA simulations. Thesefailure criteria are the von Mises equivalent stress criterion, themaximum principal stress criterion, the von Mises equivalentstrain criterion, and the maximum tensile strain criterion. Appli-cations of these failure criteria to a numerical analysis for end-capped pipes using ABAQUS and ANSYS are discussed.

A New Multiaxial Yield TheoryIn the plasticity analysis of metallic materials, the classical

Tresca and von Mises theories are commonly used. Many experi-mental investigations have indicated that the test data for initialyielding and postyielding lie between the Tresca and von Misespredictions. To more effectively describe the plastic yield behav-

Contributed by the Pressure Vessel and Piping Division of ASME for publicationn the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received October 6,006; final manuscript received February 22, 2007. Review conducted by G. E. Ottoidera. Paper presented at the 2006 ASME Pressure Vessels and Piping Conference

PVP2006, Vancouver, BC, Canada, July 2327, 2006.

44 / Vol. 129, NOVEMBER 2007 Copyright 2007 by ASME Transactions of the ASMEXian-Kui ZhuMem. ASME

e-mail: zhux@battelle.org

Brian N. LeisMem. ASME

Battelle Memorial Institute,505 King Avenue,

Columbus, OH 43221

TheoretPredictiof PipelTo accurately charanew yield theory, i.eence to the classicarials. Based on the Apipelines is obtaineexponent, and ultimdata for various pipare developed for pelement softwares sassociated flow rulehardening materialsrion, the maximumand the maximum tfailure criteria in copredict the burst pre

Keywords: von Mispipeline

ntroductionAn accurate prediction of the burst pressure of pipelines is very

mportant in the engineering design and integrity assessment of oilnd gas transmission pipelines. Experimental results showed thatnalytical, numerical, and empirical predictions available are gen-rally inconsistent and inaccurate, and have limited applications.tewart and Klever 1 pointed out that the theoretical solutions ofurst pressure depend on the yield theory adopted. The Trescaom: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20al and Numericalns of Burst Pressurees

rize plastic yield behavior of metals in multiaxial stress states, ahe average shear stress yield (ASSY) theory, is proposed in refer-esca and von Mises yield theories for isotropic hardening mate-Y theory, a theoretical solution for predicting the burst pressure ofs a function of pipe diameter, wall thickness, material hardeningtensile strength. This solution is then validated by experimentale steels. According to the ASSY yield theory, four failure criteria

icting the burst pressure of pipes by the use of commercial finiteas ABAQUS and ANSYS, where the von Mises yield theory and the

re adopted as the classical metal plasticity model for isotropicese failure criteria include the von Mises equivalent stress crite-cipal stress criterion, the von Mises equivalent strain criterion,

ile strain criterion. Applications demonstrate that the proposednction with the ABAQUS or ANSYS numerical analysis can effectivelyre of end-capped line pipes. DOI: 10.1115/1.2767352heory, Tresca theory, ASSY theory, ABAQUS, ANSYS, burst pressure,

tions are needed to develop a consistent and valid failure criterionfor the burst pressure prediction of line pipes using the FEA cal-culations.

As is well known, the commercial finite element softwareABAQUS 22 provides a modified RIKS method for a plastic in-stability analysis, which can be used to determine global failureloads for engineering structures. Most commercial FEA packages,including ABAQUS and ANSYS 23, adopt the classical metal plas-ticity model, i.e., the von Mises yield theory and the associated15 Terms of Use: http://asme.org/terms

iyrda

a

M=

msa

Ftm

a

s

y

e

wl

a

EwMA

ndess

J

Downloaded Fror, the present authors 9 recently developed a new multiaxialield theory, i.e., the ASSY theory for isotropic hardening mate-ials. In the following sections, the ASSY theory is first intro-uced, and its validations for both initial and subsequent yieldingsre then demonstrated with extensive experimental results.

Average Shear Stress Yield Theory. An average shear stressA is defined as the average of the maximum shear stress maxnd the von Mises equivalent shear stress M. Since the vonises shear stress is related to the octahedral shear stress by M3/2oct, the average shear stress is a weighted average of theaximum shear stress and the octahedral shear stress. It is as-

umed that plastic yielding will occur if the average shear stress ofmaterial reaches a critical value, namely,

A =12max +32oct = Ac 1

or convenience, this yield theory is referred to as the ASSYheory hereafter. In reference to the uniaxial tension test, the maxi-

um shear stress and the octahedral shear stress at initial yieldingre 0 /2 and 20 /3, respectively. Therefore, the critical averagehear stress at yielding is Ac= 2+30 /43, where 0 is theield strength of the material in tension.

In the principal stress space 1 ,2 ,3 with an assumption of123, from Eq. 1, the ASSY equivalent stress A can bexpressed as

A =1

2 + 33T + 2M 2

here the Tresca equivalent stress T and the von Mises equiva-ent stress M are defined, respectively, by

T = 2max = 1 3 3nd

M =32

oct =12 1 22 + 2 32 + 3 12 4quation 2 indicates that the ASSY equivalent stress is aeighted average of the Tresca equivalent stress and the vonises equivalent stress. For the simple tension, from Eq. 2, theSSY yield theory can be simplified as A=0.

Fig. 1 Three yield theoretical loci aLode 10, Ros and Eichinger 11, LMarin and Hu 14, and Maxey 15

ournal of Pressure Vessel Technologyom: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20Experimental Validation for Initial Yielding. Under the planestress condition 3=0 and in the principal stress plane 1 ,2,three yield loci for the Tresca, von Mises, and ASSY theories arecompared with extensive experimental data for tube or pipe speci-mens in biaxial stress states, as plotted in Fig. 1. In this figure, theexperimental data for initial plastic yielding are taken from thewell-known classical experiments by Lode 10, Ros and Eich-inger 11, Lessels and MacGregor 12, Davis 13, and Marinand Hu 14 for different structural steels and the mill experimentby Maxey 15 for X52 and X60 pipeline steels. The comparisonsindicate that the ASSY yield locus lies between the Tresca andvon Mises yield loci and provides the best correlation to the av-erage experimental data for all ductile metals considered.

Experimental Validation for Subsequent Yielding. To ana-lyze the subsequent plastic yielding of multiaxial stress states forisotropic hardening materials, an equivalent strain must be definedin accordance with its corresponding equivalent stress for a yieldtheory. According to the definition of the average shear stress, anASSY equivalent strain is defined as the average of the Trescaequivalent strain and the von Mises equivalent strain,

A =12T + M =

1212max + 12oct 5

where the maximum shear strain max and the octahedral shearstrain oct are expressed, respectively, by

max = 1 3 6and

oct =231 22 + 2 32 + 3 12 7

Marin and Hu 14 developed useful experimental data of biax-ial stress-strain relations for a mild SAE1020 steel. All test speci-mens were tubes with a wall thickness of 1.27 mm and an insidediameter of 25.40 mm, and loaded by the internal pressure and anaxial tension for different constant stress ratios. The experimentalstress-strain curves up to the maximum load were reported in Fig.1 of their paper in both axial and hoop directions for five loadingcases of S2/S1=0.0, 0.5, 1.0, 2.0, and , where S1 is the hoopstress and S2 is the axial stress. From these experimental data, themaximum shear stress max and the maximum shear strain max

initial yielding experimental data ofels and MacGregor 12, Davis 13,

NOVEMBER 2007, Vol. 129 / 64515 Terms of Use: http://asme.org/terms

dFsFeassdvt

relations. Therefore, it is concluded that the ASSY yield theorycan effectively characterize the initial and subsequent plasticyielding at the multiaxial stress states for isotropic ductile mate-

Flmts

6

Downloaded Fruring loading are calculated from Eqs. 3 and 6 and plotted inig. 2a; the octahedral shear stress oct and the octahedral sheartrain oct are calculated from Eqs. 4 and 7 and displayed inig. 2b; and the equivalent average shear stress A and thequivalent average strain A are obtained from Eqs. 1 and 5nd shown in Fig. 2c. These figures show that a the maximumhear stress-strain curve or the Tresca yield theory can only rea-onably correlate the biaxial stress-strain relations, b the octahe-ral shear stress-strain curve or the von Mises yield theory pro-ides a better correlation for the biaxial stress-strain relations, andc the equivalent shear stress-strain curve or the ASSY yieldheory provides the best correlation for the biaxial stress-strain

ig. 2 Experimental results of biaxial plastic stress-strain re-ations for five loading cases: a maximum shear stress versus

aximum shear strain; b octahedral shear stress versus oc-ahedral shear strain; c equivalent average shear stress ver-us equivalent average strain

46 / Vol. 129, NOVEMBER 2007om: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20rials.

Theoretical Solution of Pipeline Burst PressureASSY-Based Burst Pressure. Consider a long thin-wall pipe or

pipeline with end caps and subjected to internal pressure. For purepower-law hardening materials, using the Tresca, von Mises, andASSY yield theories, Zhu and Leis 7,9,16 obtained three differ-ent theoretical solutions for the burst pressure Pb of the pipe,which are expressed in the general form of

Pb = C2 n+14t0

D0UTS 8

where t0 is the original thickness, D0 is the original average di-ameter of the pipe, C is a yield theory-dependent constant withvalues as

C = 1 for the Tresca theory2/3 for the von Mises theory1/2 + 1/3 for the ASSY theory 9

UTS is the engineering UTS, and n is the strain hardening expo-nent and usually ranges from 0 to 0.3 for most pipeline steels. Ourrecent work 17 indicated that for a given yield-to-tensile stressY /T ratio, n can be solved from

YT

= eynn 10a

where e=2.7183 and y is the engineering yield strain. For theyield stress defined at the 0.5% total strain, y=0.005. For theyield stress defined at the 0.2% offset strain, y=0.002+Y /E, withE being the elastic modulus. Alternatively, n can be approximatedby

n = 0.239T/Y 10.596 for the yeild stress defined

at the 0.5 % total strain0.224T/Y 10.604 for the yield stress defined

at the 0.2 % offset strain10b

At the burst pressure, the equivalent stress, hoop stress, equiva-lent strain, and hoop strain for the three yield theories were ob-tained as

eqb = C2 n

UTS 11

b = 2C2 n+1UTS 12

eqb =nC2

13

b =n

214

where UTS is the true UTS and the correlation between the trueUTS and the engineering UTS is UTS=enUTS . Note that the truestrain at the UTS in the tension test is similar to the strain hard-ening exponent for power-law hardening materials. It is interest-ing that for all three yield theories, the hoop strain of the pipe atthe plastic collapse has the same value of n /2, which is indepen-dent of the yield theory. Therefore, Eq. 14 could be used as amaximum tensile strain criterion to monitor the maximum pres-sure in the hydrostatic testing of a pipeline.

Transactions of the ASME15 Terms of Use: http://asme.org/terms

pceftbtmntt

btissatn

F

ptttvef

ctel.

J

Downloaded FrFull-Scale Experimental Validation. Extensive full-scale ex-erimental data for the burst pressure of defect-free pipes wereollected and compared with theoretical solutions in Fig. 3. Thexperimental results involve 40 burst tests by Kiefner et al. 18or pipeline steels with grades ranging from grade B to X65, 12ests by Amano et al. 19 for X65 and X70 pipeline steels, 3 testsy Maxey 20 for an X70 pipeline steel, and 1 test by Netto et al.21 for an AISI 1020 mild steel. Table 1 compares the threeheoretical solutions with the average experimental data of nor-

alized burst pressure for two typical strain hardening exponents:=0.080.004 and n=0.120.004. The percentages of errors of

he three predictions with respect to the test data are given in theable.

Both Fig. 3 and Table 1 demonstrate the excellent agreementetween the proposed ASSY solution and the average experimen-al data of burst pressure for all pipeline steels considered, includ-ng low and high strength grades. It also shows that the von Misesolution serves as an upper bound and that the Tresca solutionerves as a lower bound of experimental data. Therefore, it isnticipated that the ASSY yield theory can be used as an effectiveool to predict the actual burst pressure of line pipes in the engi-eering design and integrity assessment of pipelines.

ailure Criteria for Burst Pressure PredictionAs demonstrated above, the ASSY-based solution is an effective

rediction of the burst pressure of pipelines. The following sec-ions investigate failure criteria for the ASSY burst predictionhrough the use of the FEA calculations and the von Mises yieldheory for isotropic hardening materials. Four important fieldariablesvon Mises equivalent stress, hoop stress, von Misesquivalent strain, and hoop strainare discussed to develop theailure criteria for the burst prediction.

Fig. 3 Comparisons of ASSY-predidata of Kiefner et al. 18, Amano et afor different pipeline steels

Table 1 Comparison of theoretical soluti

Hardeningexponent n Test data

Tresca

Solution Error

0.080.004 1.0333 0.9461 8.44%0.120.004 0.9869 0.9202 6.76%

ournal of Pressure Vessel Technologyom: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20Elastic and Plastic Deformation Behaviors. For the end-capped pipe, the equivalent stress, hoop stress, equivalent strain,and hoop strain can be generally expressed for the three yieldtheories 9,16 as

eq =1C =

D02t0C

Pd 15

eq = C =C2

ln d 16

where C is defined in Eq. 9 and d= D / t / D0 / t0 denotes anormalized dimension ratio, with D being the current average di-ameter and t being the current wall thickness of the pipe. For theelastic deformation, the pipe geometry change is small, and thusd1. From Eq. 15 and the elastic Hooks law, one obtains thestress-load and strain-load relations,

eq

UTS=

1C

PP0

17

eq =UTS

CEPP0

18

where E is the Youngs modulus and P0 is a reference load definedas P0=2t0UTS /D0.

For the plastic deformation that is characterized by the power-law stress-strain relation, from Eqs. 15 and 16, the stress-loadand strain-load relations can be written as

eq

UTS=

dC

PP0

19

d burst pressure and experimental19, Maxey 20, and Netto et al. 21

and normalized experimental data Pb /P0

von Mises ASSY

Solution Error Solution Error

1.1051 6.95% 1.02530.77%

1.0810 9.54% 1.0003 1.35%

NOVEMBER 2007, Vol. 129 / 647ons15 Terms of Use: http://asme.org/terms

wTfte

rrl

daepflos

cMv

bEb

tri

F

ti

re

6

Downloaded Freq =n

e dC

1/n PP01/n

20

here e=2.7183. When the total deformation is small, d1, Eq.19 reduces to Eq. 17, but Eq. 20 does not reduce to Eq. 18.his indicates that the stress-load relation in Eq. 19 is satisfied

or both elastic and plastic deformations, but the strain-load rela-ion in Eq. 20 holds only for the plastic deformation. The abovequations are the basis for the following analysis.

Critical D / t Ratio. From Eqs. 16 and 20, one obtains aelationship between the internal pressure P and the dimensionatio D / t or d during loading for the three yield theories as fol-ows:

PP0

=

2dC2

n+1 en

ln dn 21Figure 4 shows the variation of the internal pressure with the

imension ratio d for the von Mises, ASSY, and Tresca theoriesccording to Eq. 21 for a strain hardening exponent n=0.15. It isvident from this figure that when d=e0.151.162, the appliedressure reaches the peak value, and the pipeline burst is initiatedor all three yield theories. Again, the Tresca theory predicts theowest burst pressure, the von Mises theory predicts the highestne, and the ASSY theory predicts an intermediate value. Ashown in Fig. 4, the von Misesbased dimension ratio reaches aritical value dc

M1.044 at the ASSY burst pressure. If the vonises theory is used and its dimension ratio reaches the critical

alue, a critical von Mises load that will be equal to the ASSYurst pressure can be determined, i.e., Pc

M= Pb

A at dM =dcM

. Fromqs. 8, 9, and 21, the critical von Mises dimension ratio cane solved from the following equation:

1dcM e

nln dcMn = 2 + 34

n+1

22

From the above equation, the variation of dcM with n is illus-

rated as the square symbols in Fig. 5. Using the least squaresegression to fit the calculated points, an explicit expression of dc

M

s obtained as

dcM = 1 + 0.882n1.585 23or n=0.15, one yields the critical Mises dimension ratio dc

M

1.044 from Eq. 23, which is the same as that graphically de-ermined in Fig. 4. Apparently, the critical Mises dimension ratios less than the burst value of the Mises dimension ratio.

Fig. 4 Variation of internal pressuthree yield theories

48 / Vol. 129, NOVEMBER 2007om: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20von Mises Equivalent Stress Criterion. From Eqs. 19 and21, the equivalent stress for the three yield theories is generallyexpressed as a function of d,

eq

UTS= Cn e2n ln d

n

24

For a given pressure, d is determined from Eq. 21, the equiva-lent stress is determined from Eq. 24, and the equivalent stressversus the internal pressure is thus established and shown in Fig.6 for n=0.15. In this figure, UTS denotes the engineering UTS. Itis seen that for the von Mises, ASSY, and Tresca yield theories,the corresponding burst pressures are different significantly, butthe three equivalent stresses are very close to each other at theburst pressure. From Eq. 11, the von Mises and ASSY equivalentstresses at the burst failure are eq

Mb=1.070UTS and eqA b

=1.059UTS , respectively, for n=0.15. At the ASSY burst pres-sure, we can determine a critical von Mises equivalent stress. It isassumed here that if the von Mises equivalent stress reaches itscritical value during loading, a pipeline burst will be initiated andthe corresponding critical load will be the ASSY burst pressure,i.e., Pc

M= Pb

A at eqM

= eqMc. From Eqs. 2224, the critical von

Mises equivalent stress is obtained as

with dimension ratio at n=0.15 for

Fig. 5 Variation of critical von Mises dimension ratio dc withstrain hardening exponent n

Transactions of the ASME15 Terms of Use: http://asme.org/terms

Af

FTeFb

encs

J

Downloaded FreqMc = 31 + 0.882n1.5852 + 343

n+1

UTS 25

s shown in Fig. 7, Eq. 25 can be well approximated by theollowing quadratic polynomial:

eqMc = 0.797n2 0.417n + 0.932UTS 26

rom this equation, one obtains eqMc=0.887UTS for n=0.15.

his critical von Mises equivalent stress is less than the von Misesquivalent stress at the burst failure of about 17%, as illustrated inig. 6. If the true UTS is used, as shown in Fig. 7, Eq. 26ecomes

eqMc = 1.259n2 1.308n + 0.932UTS 27

Our companion work 16 has indicated that the ratio of the truequivalent stress to the true UTS is equal to the ratio of the nomi-al equivalent stress to the engineering UTS. Therefore, Eq. 27an be used to define a critical nominal von Mises equivalenttress in reference to the engineering UTS.

Fig. 6 Variation of normalized equivn=0.15 for three yield theories

Fig. 7 Variation of critical von Misexponent

ournal of Pressure Vessel Technologyom: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20Maximum Principal Stress Criterion. For a pressurized pipe-line, the hoop stress is the maximum principal stress. From Eqs.15 and 21, the hoop stress for the three yield theories is ex-pressed as a function of d,

UTS= Cn+1 e2n ln d

n

28

From Eqs. 21 and 28, the variations of the hoop stress withthe internal pressure during loading are determined and shown inFig. 8 for n=0.15. It is observed that the three hoop stress versusinternal pressure curves are linear and identical to each other forloading up to 80% of P0, where the elastic deformation domi-nates. From Eqs. 15 and 17, one has the linear relationship /UTS = P / P0 for all yield theories. However, the hoop stressesat the burst failure are significantly different for the three theories.From Eq. 12, the von Mises and ASSY hoop stresses at the burstare

M b=1.236UTS and A b=1.141UTS , respectively, for n

=0.15. At the ASSY burst pressure, one can determine a critical

nt stress versus normalized load at

equivalent stress versus hardening

NOVEMBER 2007, Vol. 129 / 649esale15 Terms of Use: http://asme.org/terms

vsw

ba

a

hc

tieb

br

MMw

b

Tm

lcv

c

=

c

p s

6

Downloaded Fron Mises hoop stress. It is assumed that if the von Mises hooptress reaches its critical value during loading, a pipeline burstill occur and the corresponding critical load will be the ASSYurst pressure, i.e., Pc

M= Pb

A at M

= M c. From Eqs. 22, 23,

nd 28, the critical von Mises hoop stress is determined and hasn equation similar to Eq. 25. Moreover,

M c= 2/3eMcolds. From Eq. 26, an approximate explicit equation for theritical von Mises hoop stress is obtained as

M c = 0.920n2 0.481n + 1.076UTS 29

Using this equation, the critical von Mises hoop stress is ob-ained as

M c=1.025UTS for n=0.15. This critical hoop stresss about 83% of the von Mises burst hoop stress. The big differ-nce is clearly shown in Fig. 8. If the true UTS is used, Eq. 29ecomes

M c = 1.454n2 1.510n + 1.076UTS 30

Due to the similar reason for the use of Eq. 27, Eq. 30 cane used to define a critical nominal von Mises hoop stress ineference to the engineering UTS.

von Mises Equivalent Strain Criterion. Similar to the vonises equivalent stress criterion, one can assume that if the vonises equivalent strain reaches its critical value, a pipeline burstill occur and the corresponding critical load will be the ASSYurst pressure, i.e., Pc

M= Pb

A at eqM

= eqMc. From Eqs. 16 and

23, the critical von Mises equivalent strain is obtained as

eqMc =

13

ln1 + 0.882n1.585 31

hrough curve fitting, the above equation can be further approxi-ated as

eqMc = 0.443n1.524 32

Maximum Tensile Strain Criterion. For a pressurized pipe-ine, the maximum tensile strain is the hoop strain. Similarly, onean assume that if the von Mises hoop strain reaches its criticalalue, a pipeline burst will be initiated and the correspondingritical load will be equal to the ASSY burst pressure, i.e., Pc

M

PbA at

M=

M c. From M c= 3/2eMc and Eq. 32, the

ritical von Mises hoop strain can be approximated as

M c = 0.384n1.524 33

Fig. 8 Variation of normalized hoo=0.15 for the three yield criteria

50 / Vol. 129, NOVEMBER 2007om: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20Theoretically, the four failure criteria proposed above can beequally used to determine the burst pressure of pipes using theFEA calculations and the von Mises yield theory. Actually, ourexperience indicated that the two strain criteria are not as efficientas the two stress criteria because a small load increase can cause alarge plastic strain increase near the pipe burst. Therefore, onlyapplications of the two stress failure criteria to the numerical burstprediction are demonstrated in the next sections.

Numerical Prediction of Pipe Burst PressureFinite Element Calculations and Results by ABAQUS. De-

tailed FEA calculations were performed using the commercialpackage ABAQUS Standard 22 for a defect-free pipeline with anoutside diameter of 762 mm and a thickness of 17.53 mm, i.e.,D / t=43.5. Due to the negligible axial strain, the long pipe wassimplified as a plane strain problem note that this plane strainmodel is equivalent to the axisymmetric model for the pipeline.Only one quarter of the circular section was modeled because ofsymmetry. The uniform FEA mesh has four elements in thicknessand 90 elements in circumferential direction, which lead to 360elements and 1269 nodes in total. The eight-node quadratic para-metric element with reduced integration was used in the numericalsimulation. The applied load was internal pressure only, and thesymmetric displacement boundary conditions were employed inthe FEA model. The elastic-plastic finite strain formulation andthe modified RIKS method built in ABAQUS have been adopted inthe FEA simulation. Since ABAQUS adopts the classical metal plas-ticity model, i.e., the von Mises yield theory and the associatedflow rule, as its default plasticity model, all calculated results arethe von Misesbased numerical solutions.

The material considered is an X65 pipeline steel. Experimentaldata of true stresstrue plastic strain curve for the X65 is shown inFig. 9, where the input data of material properties used in the FEAcalculations are also marked. The yield stress defined at the 0.5%total strain is 508 MPa, the UTS is 645 MPa, and thus Y /T=0.788. From Eq. 10a, the strain hardening exponent is esti-mated as n=0.113, and the measured value is n=0.112. In theFEA calculation, the elastic modulus E=207 GPa and the Pois-sons ratio =0.3. From Eq. 8, the von Mises solution and theASSY solution for the burst pressure of this pipe are determinedas Pb

M=32.96 MPa and Pb

A=30.51 MPa, respectively.

Figure 10 shows the variation of the von Mises equivalentstress with internal pressure obtained from the FEA calculations

tress versus normalized load at n

Transactions of the ASME15 Terms of Use: http://asme.org/terms

atafriAa

o

=

c

t=

3mrcpf

Xd

FEA codes ANSYS 23 and LS-DYNA, respectively. Note that thesetwo FEA codes adopted the von Mises yield theory and the asso-ciated flow rule as the classical metal plasticity model. The cylin-

Fp

es

J

Downloaded Frnd the theoretical formula in Eq. 24, respectively. It is foundhat an excellent agreement exists between the numerical resultsnd theoretical solutions for a full-range loading from elastic de-ormation to plastic instability. At plastic instability, the numericalesult of the von Mises pressure is 32.84 MPa, which is almostdentical to its theoretical value of 32.96 MPa. This indicates thatBAQUS can well predict the von Misesbased stress-load relationnd the burst pressure for defect-free pipelines. From Eq. 26,ne has the critical von Mises equivalent stress eq

Mc0.895UTS =577.13 MPa for this X65 pipeline steel. When thisritical von Mises equivalent stress is reached in the FEA simula-ion, a corresponding critical pressure is obtained as Pc

M

30.85 MPa, which is nearly equal to the ASSY burst pressure of0.51 MPa. Similarly, the same critical pressure can be deter-ined by the use of the maximum principal stress failure crite-

ion. Therefore, it is concluded that the proposed stress failureriteria can be effectively used to determine the burst pressure ofipes in the FEA simulations by ABAQUS, and the burst pressureor the X65 pipeline is 30.85 MPa.

Finite Element Calculations and Results by ANSYS. Recently,ue et al. 24 performed detailed FEA calculations for static andynamic burst analyses of a cylindrical shell using the commercial

ig. 9 True stress versus true plastic strain curve for the X65ipeline steel

Fig. 10 Variation of normalized von Misfor an X65 steel

ournal of Pressure Vessel Technologyom: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 06/10/20drical shell is a thin-wall end-capped pipe with a mean diameterof 606 mm, a wall thickness of 6 mm, and a length of 2400 mm.The material of the cylinder is a low carbon steel, Q235-A. Theyield stress defined at the 0.2% offset strain is 339.4 MPa, theUTS is 472 MPa, and Y /T=0.719. From Eq. 10b, the strainhardening exponent for this material is estimated as n=0.127.From Eq. 8, the theoretical result of the von Mises burst pressurefor this cylinder is determined as 10.06 MPa, and the ASSY burstpressure for this cylinder is 9.31 MPa. The FEA result of the burstpressure for this pipe determined in Ref. 24 is 10.0 MPa in thestatic analysis using ANSYS and 9.6 MPa in the dynamic analysisusing LS-DYNA. Apparently, these two numerical predictions coin-cide with the theoretical result of the von Mises burst pressure.Accordingly, both ANSYS and LS-DYNA can numerically determinethe von Mises burst pressure. Similar to ABAQUS, when the vonMises equivalent stress criterion or the maximum principal stresscriterion developed previously is used in the FEA simulations, theactual burst pressure for this cylinder can be similar to the ASSYburst pressure of 9.31 MPa.

Concluding RemarksThis paper investigated the theoretical and numerical predic-

tions of the accurate burst pressure of pipes or pipelines. Since theTresca yield theory predicts a lower bound of burst pressure andthe von Mises yield theory provides an upper bound of burst pres-sure for pipelines, a new multiaxial yield theory, i.e., the ASSYtheory, was developed for isotropic hardening materials so as toimprove the prediction of burst pressure. The comparisons withclassical experimental data showed that the ASSY yield theorycan much better correlate the stress-strain relations for both initialyielding and subsequent yielding than the Tresca or von Misesyield theory can. A theoretical solution of burst pressure based onthe ASSY yield theory was obtained as a function of pipe diam-eter, wall thickness, material hardening exponent, and ultimatetensile strength for defect-free pipelines. This solution was vali-dated by extensive experimental data of burst pressure for differ-ent pipeline sizes and grades. Therefore, the proposed ASSY-based solution for the burst pressure can be considered as aneffective prediction of burst pressure for pressurized pipes andpipelines.

equivalent stress with normalized load

NOVEMBER 2007, Vol. 129 / 65115 Terms of Use: http://asme.org/terms

Since the commercial finite element codes, ABAQUS and ANSYS,adopt the von Mises yield criterion and the associated flow rule asthe default plasticity model for isotropic hardening metals, onlythe von Misesbased burst pressure of pipes can be determinedusing these FEA codes, as shown in the examples. To effectivelypredict the actual or ASSY burst pressure using these FEA codes,four burst failure criteria: the von Mises equivalent stress crite-rion, the maximum principal stress criterion, the von Misesequivalent strain criterion, and the maximum tensile strain crite-rion were developed in reference to the UTS and the strain hard-ening exponent. The applications showed that the proposed failurecriteria are simple and effective, and can be used to determine theactual burst pressure of pipelines using ABAQUS or ANSYS. It isexpected that the proposed ASSY theory and the four failure cri-teria for burst prediction can provide reliable and useful results ofburst pressure for pipes, pipelines, and cylindrical pressure ves-sels.

AcknowledgmentThe support of the U.S. Department of Transportation through

the Broad-Agency Announcement funding is gratefully acknowl-edged. One of the authors X.K.Z. sincerely thanks ProfessorOtto Widera at Marquette University for his useful discussions.

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