Use of the cylindrical instability stress for blunt

metal loss defects in linepipe

M. Lawa,*

aInstitute of Materials and Engineering Science, Australian Nuclear Science and Technology Organisation (ANSTO), PMB 1, Menai, NSW 2234, Australia

Received 21 October 2004; revised 14 March 2005; accepted 4 April 2005

Abstract

Assessment of metal loss defects in gas pipelines can be analysed by a number of methods. In analyses with finite element methods a

failure criterion is required. A material property is introduced, the cylindrical instability stress, which determines the plastic collapse of

cylindrical pressure containing vessels. The use of this is extended to cover blunt metal loss defects. Some published finite element studies of

defected vessels are re-analysed using this failure criterion.

The cylindrical instability stress is a more accurate failure criterion for plastic collapse in pipelines and pressure vessels than commonly

used measures such as flow stress, Specified Minimum Yield Stress plus 10 ksi or multiplied by 1.1. It can be used in determining burst

pressure of defected and un-defected pressure vessels and piping.

q 2005 Published by Elsevier Ltd.

Keywords: Analysis; Burst pressure; Metal loss defect; Blunt defect; Plastic collapse; Considere’s construction

1. Introduction

Gas pipelines are a safe and economical method of

transporting energy; this is due to detailed materials

characterisation and stress analysis. Corrosion and metal

loss defects are a potential cause of pipeline failure and their

management is essential if pipeline integrity is to be

ensured. Common methods of assessing integrity of

pipelines with corrosion defects are approximate and

necessarily conservative.

Work at the Battelle Memorial Institute developed

methods of determining the failure behaviour of pipes

containing defects. In pipes of adequate toughness, failure

occurs by plastic collapse. In this work Duffy and McClure

[1] proposed that the critical stress was above the yield

stress of materials because of strain hardening. This stress

was termed the flow stress and was estimated to be the

average of the yield strength (YS) and tensile strength (TS

or UTS).

0308-0161/$ - see front matter q 2005 Published by Elsevier Ltd.

doi:10.1016/j.ijpvp.2005.04.002

* Tel.: C 61 2 9717 9102.

E-mail address: mlx@ansto.gov.au.

Alternate definitions of flow stress have been used.

Instead of the average of YS and TS, 1.1!SMYS (Specified

Minimum Yield Strength) was used in B31.8 [2]. Later work

by Kiefner and Veith [3] led to a less conservative measure

(in lower strength pipe) of SMYS C10 ksi (69 MPa). These

measures of flow stress are compared to SMYS and TS for

grade X70 in Fig. 1. These values depend on the pipe grade

and yield to tensile ratio, while the reported yield to tensile

ratio (typically 0.75–0.93) in turn depends on the method of

measurement of the yield stress [4–7]. Fig. 1 indicates that

while the flow stress is always below the TS, other measures

may exceed the TS.

The flow stress referred to in the Battelle formula is a

measure of the stress where plastic instability occurs and is

dependent on the shape of the stress–strain curve after yield.

Plastic instability (collapse) occurs differently in a pressurized

cylinder to a tensile test. This is described by an extension of

Considere’s construction in Section 2. A failure criterion, the

cylindrical instability stress (CIS), is proposed for cylindrical

pressure vessels. This is the material behaviour in a particular

stress state (restrained pipeline or pressure vessel) which can

be determined from a conventional tensile test.

The use of finite element analysis to analyse defects

potentially offers greater accuracy, but requires a failure

International Journal of Pressure Vessels and Piping 82 (2005) 925–928

www.elsevier.com/locate/ijpvp

X70

300

350

400

450

500

550

600

650

0.8 0.825 0.85 0.875 0.9 0.925Y/T ratio

Str

ess

(MP

a)

TS

SMYSx 1.1

SMYS+ 10 ksi

Flowstress

Fig. 1. Estimates of flow stress for X70 grade material.

M. Law / International Journal of Pressure Vessels and Piping 82 (2005) 925–928926

criterion. Common failure criteria are based on the tensile

strength and the true stress or strain at UTS. Previous studies

have presented detailed finite element analyses and

compared the results to experimental results. These results

are re-analysed with the CIS as a failure criterion.

True strain

uniform

True

str

ess

Pipefailure strain

True stressat UTS

Instabilitystress

Engineeringstress-strain curve

d /d1/2

d /d

Fig. 2. Considere’s construction, stress in system (s) and rate of increase of

strain hardening (ds/d3).

2. Considere’s construction

Straining after yielding in the tensile test results in strain

hardening which increases the load bearing capacity of the

material. This rate of strain hardening is the slope of the

stress–strain curve (ds/d3). Up to the UTS the rate of

strength increase is faster than the accompanying reduction

in load bearing capacity due to reduction in cross section.

Between yield and failure, straining is stable since an

increase in strain increases the load bearing capability. It is

also uniform as local strain concentrations increase the local

strength and redistribute the strain.

At the strain where the rate of strain hardening balances

the reduction in load bearing capacity, plastic instability

occurs because further straining reduces the load the

specimen can support. This load defines the UTS and the

strain at maximum load is the uniform elongation. Straining

thereafter is unstable and non-uniform, and necking begins.

A mathematical treatment known as Considere’s construc-

tion [8] shows this occurs when ds/d3Zs. Where the rate of

strength increase due to strain hardening (ds/d3) equals the

stress increase due to load and specimen thinning (s),

necking initiates and the material has reached the UTS.

Pipe failure is also due to the onset of plastic instability.

Once again, the increase in strength of the material with

increasing strain eventually becomes less than the increase

in stress due to increasing load, and plastic instability

occurs. The geometric increase in stress comes from two

causes: reduced cross sectional area (as in the tensile test)

and increasing internal diameter D (unlike the tensile case).

Both these factors raise the hoop stress via sZPD/2t.

Considere’s analysis can be extended to the stress state

found in a pressure vessel [9]. This predicts that for a pipe or

pressure vessel under internal pressure only, the point of

instability is where sZ1/2 ds/d3 (Fig. 2).

It is important to note that the reduced stress and strain

where instability occurs is not a result of the biaxial stress

state, but of the vessel geometry where increased stress

comes from both increased ID and reduced wall thickness.

The stress where instability occurs is referred to as the

cylindrical instability stress (CIS).

3. Comparison with published analyses

Karstensen et al. [10] carried out a single test on a pipe

made from X52 material with a manufactured defect, and a

large deformation finite element model was made of this

defected pipe. Failure was predicted when the von-Mises

stress at the base of the ligament in the FEA model reached

X52 material

0

100

200

300

400

500

600

700

0 0.05 0.1 0.15 0.2True strain

True

str

ess

(MP

a)

dσ/dε

1/2dσ/dε

Area of curve fitting

Fig. 3. True stress–strain curve of material X52 material.

Table 1

Estimated CIS (MPa) from different curve fitting for X52 material

Quadratic 561

Power law 555

Average 558

Table 2

Estimated CIS (MPa) from different curve fitting for X65 material

Quadratic 618

Power law 607

Average 612

M. Law / International Journal of Pressure Vessels and Piping 82 (2005) 925–928 927

the true stress at UTS (referred to as st-uts). This was found

to overestimate the burst pressure by some 3%.

To find the CIS it is necessary to find the slope of the

stress–strain curve. The true stress–plastic true strain was

digitized and the plastic strain converted to total strain.

Using a slope calculated from the data points alone results in

considerable variation in slope. To reduce inaccuracies from

this cause, a limited section of the stress–strain curve

(between strain values of 0.035 and 0.17) was fitted with

power-law and quadratic equations, and the slope calculated

from these (Fig. 3). For clarity, only the slopes from the

power law fit are shown. The CIS calculated from these

methods, and the average, are given in Table 1.

In [10] there is a graph showing the evolution of the von-

Mises stress at the ligament at the base of the flaw with

increasing pressure. Use of the CIS (sCISZ558 MPa, 94.1%

of st-uts) in this graph gave a predicted burst pressure of

666 bar, within 0.3% of the actual burst pressure (661 bar).

X65 material

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2

True strain

Tru

e st

ress

(MP

a) dσ/dε

1/2dσ/dε

Arfitting

/

Area of curve

Fig. 4. True stress–strain curve of material X65 material.

Choi and Goo [11] used the following criteria: failure

occurred when the von-Mises stress in a defect reached 90%

of the true stress at UTS (st-uts) for a rectangular defect, or

80% of st-uts for an elliptical defect.

This work performed seven burst tests using rectangular

machined defects in X65 pipe material, and modeled the

defects with FEA. The best-fit failure criterion based on the

testing was when the von-Mises stress distribution reached

the value of 90% of st-uts. ‘Since the failure mechanism

is controlled by plastic collapse as observed from the test,

the prediction on the basis of su provided a more reasonable

sensitivity on defect geometry’ [10]. Since plastic collapse

is the controlling failure mode, it is more appropriate to use

the CIS as a failure criterion rather than st-uts.

Using the method described above, the true stress true

strain curve was digitized (Fig. 4), and a section of the curve

fitted with power law and quadratic equations. The half-

slope was used to estimate the CIS. Individual and average

stresses from these two methods are given in Table 2.

The results for the seven burst tests were presented in a table

showing the ratio of predicted failure pressure by FEA to the

actual failure pressure, for a number of failure criteria:

sy(Z453 MPa), sflow(Z572 MPa), 0.8st-uts, 0.9st-uts, st-uts

(Z691 MPa). This is reproduced in Table 3 with the

predicted ratio for the CIS added. The results for the CIS

(sCISZ612 MPa, 88.6% of st-uts) were estimated by interp-

olation. Use of the CIS for this material gave a similar

prediction to the chosen criterion (0.9st-uts), with less over-

estimation of burst strength. The variability in results is

explained by variation in material properties between

individual pipes and the measured properties taken from a

single pipe [6].

Choi and Goo also presented a failure criterion of 80%

st-uts for elliptical defects, but this was based on an invalid

method of comparing the stresses in modeled elliptical

defects with the burst pressures in actual rectangular

defects. As the stresses in the elliptical defects are less

than those seen in rectangular defects, a lower failure

criterion was arrived at.

4. Discussion

Use of the CIS is a rational solution to the problem of

defining the flow stress of a pipe because it takes into

account the actual stress state of a pressurized pipe, and the

strain hardening properties of the material.

The CIS is a unique value defining plastic instability

for a material in an internally pressurised pipe or

Table 3

Ratio of PFEA/Ptest for X65 defected vessels

Pipe Burst pressure

(MPa)

Failure criterion

sy sflow 0.8st-uts sCIS 0.9st-uts st-uts

DA 24.11 0.81 0.98 0.99 0.99 1.01 n/v

DB 21.76 0.66 0.93 0.95 1.02 1.04 1.10

DC 17.15 0.42 0.84 0.86 0.94 0.95 1.05

LA 24.30 0.68 0.94 0.95 0.98 1.00 n/v

LC 19.80 0.61 0.86 0.88 0.96 0.98 1.06

CB 23.42 0.57 0.84 0.86 0.92 0.93 1.00

CC 22.64 0.59 0.85 0.88 0.94 0.95 1.02

Average, std dev 0.62, 0.12 0.89, 0.06 0.91, 0.05 0.96, 0.03 0.98, 0.04 1.05, 0.04

M. Law / International Journal of Pressure Vessels and Piping 82 (2005) 925–928928

pressure vessel that can be gained from a tensile test.

The material stress versus strain curve is required.

Where this is not available the flow stress may be

used. As the value of the CIS was above the flow stress

for both these materials, this will be a conservative

assumption. When the stress state is no longer a simple

pressurised pipe, for instance in the case of bending or

additional thermal loading, the CIS is no longer an

appropriate failure criteria.

The method offinding the failure criterion in the Choi paper

is an empirical procedure that will give good results for any

given material if enough burst tests are performed, but for

another material the criterion will be a different value again.

For example, while the value of 90% st-uts was found for this

material, in the Karstensen et al. [10] paper the value is 94% st-

uts.

There will always be some variation between the

measured material properties and what is found in each

pipe section. This is shown in the variation in the ratios

between the predicted and actual failure pressures for all

criteria in Table 3 [6,8]. Extremely long defects will gain

support in the longest direction and failure may not be

adequately described by this criteria.

5. Conclusions

The use of cylindrical instability stress as a failure

criterion is based on the physical phenomenon of plastic

collapse in a pipe or pressure vessel. The value depends

on the shape of the stress–strain curve and is generally

greater than the flow stress calculated from the average of

YS and TS.

In the two cases analysed, the cylindrical instability stress

was a more accurate failure criterion for plastic collapse in

defected pipelines and pressure vessels than commonly used

measures such as flow stress, SMYS C10 ksi, 1.1 SMYS, etc.

Further cases will need to be analysed to demonstrate the

accuracy of this method.

Acknowledgements

The work reported herein was undertaken as part of

a Research Project of the Cooperative Research Centre

for Welded Structures (CRC-WS). It was jointly

sponsored by the Australian Pipeline Industry Associ-

ation (APIA), many of its member companies, and the

CRC-WS. The Cooperative Research Centre for

Welded Structures was established and is supported

under the Australian Government’s Cooperative

Research Centres Program.

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