**THE USE OF PROBABILITY THEORY IN
FRACTURE MECHANICS – A CASE STUDY**

**J.
Wannenburg:** University of Pretoria, Pretoria 0002, South Africa;
and Laboratory for Advanced Engineering (Pty) Ltd, P0 Box 30536, Sunnyside,
Pretoria 0132, South Africa

** **

**G.C. Klintworth:**
University of Pretoria, Pretoria 0002, South Africa; and Mecaic (Pty) Ltd, P0
Box *7958, *Hernopsemmer 0046, South Africa

** **

**A. D.
Raath:** University
of Pretoria, Pretoria 0002, South Africa; and Laboratory for Advanced Engineering
(Pty) Ltd, P0 Box 30536, Sunnyside, Pretoria 0132, South Africa

**Keywords:**
Probability theory, Fracture mechanics, Pipeline, Probability of failure, Case
study, Statistics, Non-Destructive Inspection

**ABSTRACT**

A probabilistic approach to fracture mechanics in the
form of a typical case study is described wherein the integrity of a
high-pressure, water pipeline is assessed. An analysis methodology is discussed
incorporating the probability density functions of defect sizes, the statistics
of defect occurrences and the statistical distribution of material properties.
This method enables the analyst to supply a very simple assessment of safety,
based on the probability of failure, (a single number) which may be compared to
accepted industrial standards. (e.g. 10^{-6} for nuclear applications)
It is argued that this method often offers the only way to scientifically and
economically assess the integrity of fracture prone structures.

**INTRODUCTION**

Failure
of a stressed member may occur when an inherent crack propagates in an unstable
manner to cause separation. Such failure is termed fracture failure and has
become increasingly important in modern engineering due to the use of stronger
materials in optimal designs. This is especially true for welded constructions
where the use of stronger materials implies higher operating stresses, which,
together with the fact that the occurrence of welding defects is usually
unavoidable, causes such designs to be prone to fracture failure.

The task of a fracture mechanics analyst is to
provide information (normally a safety assessment) on which a fracture control
plan for a particular component or structure may be based. In the design phase
the fracture control plan involves materials selection, choice of operating
stress levels, manufacturing method and planned inspection and repair
schedules. This responsibility is however often ignored in the design phase
and, when the problem becomes apparent during operation, any viable fracture
control measure becomes extremely costly.

In engineering, a safety
assessment is always required to be both conservative as well as sufficiently
accurate. This aspect is illustrated in Fig. 1. A simplified analysis (to the
left on the horizontal axis) would imply that conservative assumptions are made
wherever uncertainties exist, therefore yielding a conservative assessment of
safety (low on the vertical axis). If the level of safety thus obtained would
imply costly fracture control measures such as replacement or repair, the
option remains to perform a more detailed analysis which could lead to less
conservative results. Performing a more detailed analysis could therefore be
seen as an option for fracture control although the true safety level is not
altered.

It is obviously not known whether a more detailed
analysis would in fact result in an appreciably higher estimated level of
safety since the true safety level is unknown. A simplified analysis should
however always result in a more conservative assessment than a more detailed
analysis. If this is not the case then non-conservative assumptions had been
made, implying an invalid assessment.

It is well accepted that certain parameters involved
in a fracture mechanics analysis are probabilistically distributed variables.
Material properties always exhibit scatter, defect occurrences and sizes are
statistically variable and loading may also be random or pseudo-random. When
required to supply a deterministic assessment of safety, the analyst is forced,
according to the philosophy described above, to assume the lowest values of
various variables.

This often leads to very conservative results. A
deterministic analysis of any welded structure could, in fact, often conclude
that the structure is unsafe since there may often be a possibility that a
welding defect exists which is larger than the critical size. This fact is
usually ignored and ambiguous assumptions are made on a ‘gut feel’ basis to
prove or disprove integrity.

A probabilistic approach, in the opinion of the
present author, therefore often offers the only way with which to incorporate
sufficient detail into an analysis to be able to furnish a realistic assessment
of safety. The purpose of such an assessment is then to supply a single number,
namely, the “probability of failure”,* *to decision-makers. Even if this
number turns out to be very small, it should not be the responsibility of a
fracture mechanics analyst to decide whether it is negligible or not. Such a
decision should be made on a management level where a simple calculation,
namely, the probability of failure multiplied by the consequential cost of
failure, should indicate how money should be spent on further fracture control
measures.

Probabilistic fracture mechanics has mainly been used
in the nuclear energy field where it has been recognized that it is
unacceptable to base a safety assessment on unscientific assumptions. In the
present paper, a case study is presented to illustrate the use of probabilistic
fracture mechanics in a non-nuclear application.

**PROBLEM DEFINITION**

Assume
a high-pressure water pipeline with a diameter of 6 m and a wall-thickness of
30mm. The pipe is made up from cans which are welded together with
circumferential manual arc butt welds on site. Each can is made up from four
rings, each ring consisting of three rolled plates welded together with axial
machine welds. The rings are welded together with circumferential machine welds.
The configuration is shown in Fig. 2. Note that the rings are rotated so that
the axial welds do not fall in line.

The pipe is manufactured from a quenched and tempered
steel with a yield strength of 700MPa. The design was based on a simple
yielding criterion without consideration to fracture mechanics. The concern
regarding the integrity of the pipeline developed when welding defects were
found during inspection of the manual arc welds joining the prefabricated cans
of the pipeline. These defects developed due to hydrogen cracking, which
occurred due to insufficient preheating in a damp environment on site, together
with the presence of welding induced residual stresses.

The pipeline is mainly stressed by internal water
pressure caused by a static head. Superimposed on the static head pressure are
pressure fluctuations caused by operational mode changes. Fatigue crack
propagation therefore had to be considered as a potential complicating aspect.
In addition to this, the welds were not stress relieved and welding residual
stresses also have to be considered during the analyses.

The consequence of failure of the pipeline would be
flooding of costly plant as well as possible loss of life. The logic followed
during the integrity assessment of the pipeline is diagrammatically depicted in
Fig. 3.

**NON DESTRUCTIVE INSPECTION (NDI) RESULTS**

As
shown in Fig. 3, the first step was to conduct an extensive NDI exercise, involving
magnetic particle as well as ultrasonic inspection techniques. This was done
after a preliminary analysis had indicated that welding cracks would indeed
have an influence in decreasing the integrity of the pipeline. The aim of the
NDI exercise was to determine the extent of occurrences as well as the sizes of
the welding defects. Without this information, no analyses would have been
possible. In excess of 1000 m of weld were inspected.

The inspection exposed more than 500 defects. All
were situated in the circumferential manual arc welds. A little less than half
of these defects were found to be orientated in the axial direction, as shown
in Fig. 4. The defects were all surface breaking or near surface breaking.

An analysis was conducted according to the procedures stipulated in the
British Standards Institution document PD 6493,^{1} in order to
determine tolerable defect sizes. The results of this analysis are shown in
Fig. 5, depicting a tolerable defect depth versus length graph, any defect
lying below this curve being safe. Based on these results, as well as the fact
that probable fatigue crack growth of the defects was not quantified at that
stage, it was decided to repair all defects found during the inspection. The
repair exercise represented the first excursion down the assessment logic flow
diagram depicted in Fig. 3.

The defects were sized during the inspection and
repair exercise according to their length and depth dimensions. It is proposed^{6
}that a lognormal probability density function could be fitted to defect
size results, or

(1)

with x = defect size related
parameter (x = length/l0 and x = depth/2); = variance; and = standard deviation.

The lognormal distribution function achieved a remarkably good fit to
the size data, as can be observed in Fig. 6. The defect length and depth
dimensions are shown in the upper and lower graphs respectively. The ‘after
fatigue growth’ curves are discussed later.

**MATERIAL PROPERTIES**

As
shown in Fig. 3, the next step was to assess the post-repair integrity of the
pipeline. The PD 6493 method used to establish tolerable defect sizes for the
repair program is inherently very conservative due to reasons beyond the scope
of this paper. It was therefore decided to perform a more detailed analysis to
obtain the best estimate of critical defect sizes. A J-integral based analysis
method, proposed by the British Central Electricity Generation Board (CEBG) in
their document known as R6,^{2 }was subsequently employed.

This necessitated a material testing exercise to
determine the material toughness properties relevant to the R6 method. The
method requires the initiation toughness in terms of J (=J_{IC}) or K
(= K_{IC}),* *as well as the crack growth resistance or J_{r}
-curve, to be quantified (commonly used fracture mechanics notation has been
used). Fracture tests on three-point bend specimens were performed according to
the procedures proposed in R6. These tests were performed on welded as well as
base metal specimens.

The
unloading compliance technique^{7 }was used to construct the J - Da curves. According to this
technique, the stable crack extension is measured by partially unloading the
specimen at various clipgauge displacement levels. The slopes of the unloading
curves are then a measure of the change in compliance of the specimen due to
stable crack growth. A typical test record is shown in Fig. 7.

As mentioned earlier, fracture toughness data will
always exhibit scatter. In order to quantify the distribution of fracture
toughness data, a sufficient number of specimens need to be tested to represent
an adequate statistical sample. Initiation fracture toughness data may be
described using a Weibull probability density function.^{3}

In
the present case, a limited number of specimens were available. A conservative
engineering approach therefore had to be assumed by choosing the lowerbound
toughness values obtained in the tests. These results are listed below:

Weld metal: J_{0.2} = 46 KJ m^{2} dJ/da = 70
MPa

Base metal: J_{0.2} = 119 KJ m^{2} dJ/da = 216 MPa

Initiation is assumed to commence
after 0.2% apparent crack extension.

The
J_{r} - curve is approximated^{2 }in R6, by:

(2)

**CRITICAL DEFECT SIZE ANALYSIS**

The CEGB R6 method is a descendant
of the Two Criteria method. These criteria are LEFM and plastic collapse and
are represented on a Failure Assessment Diagram (FAD) as a normalized crack
driving force, K_{r} = K/K_{IC} on the vertical axis with a
brittle fracture limit at K_{r} = 1(K = K_{IC}) and a
normalized stress on the horizontal axis with a plastic collapse failure limit
at L_{r} = 1 (s
= s_{yield}).
Assessment points falling inside the diagram (shown in Fig. 8) are considered
safe. In R6 this concept is refined by using EPFM to interpolate between the
two extremes, as shown by the dashed line in Fig. 8. This curve then represents
the R6 assessment line.

The philosophy behind this assessment line and how
the R6 method relates to the J-integral approach in elastic-plastic fracture
mechanics is described in the following paragraphs.

In the linear elastic regime, the stress intensity
factor, K,* *is used as a measure for the crack driving force and the
failure criterion is simply:

K
³ K_{IC }(3)

K* *is a function of
geometry, stress and crack size, or

(4)

with Y = geometry correction
factor; s = applied
stress; and a = crack size.

Equation
(3) may also be written in terms of fracture energy parameters, or:

G
> G_{IC} (5)

with

(5a)

and

(5b)

The expression for G may be
written in more general terms (using ):

(6)

It
has been shown^{4 }that eqn (6) is also valid for non-linear elastic
material behaviour (¹ s/E) In the non-linear case, the
geometry factor Y* *would be different to Y in the linear case and is
therefore given a new name:

(7)

It has also been established^{4
}that G_{non-linear}** **is equivalent to the value of the
well-known J-integral, or

(8)

The failure criterion then
becomes:

J ³
J_{IC} (9)

with

(9a)

Equation (9) may then be
manipulated by dividing with G (linear), inverting and taking the square root,
or

(10)

As
was mentioned earlier, the vertical axis of the R6 FAD represents a normalized
crack driving force:

(11)

and from eqns (5a) and (9a) it is
clear that

(12)

Evaluation of the left-hand side
of eqn (10) therefore only requires the linear elastic crack driving force (K)*
*and the fracture toughness (K_{IC})* *to be known. The
right-hand side of eqn. (10) is denoted K_{r}^{f }in R6,
therefore establishing the R6 failure criterion:

K_{r}
³ K_{r}^{f} (13)

K_{r}^{f} as a
function of a normalized applied stress parameter (L_{r} = s/s_{yield})
then represents the R6 failure assessment line. Evaluation of K_{r}^{f}
requires the elastic-plastic crack driving force (J) to be known. Using eqns.
(5a) and (8) K_{r}^{f} may then be written as:

(14)

K_{r}^{f}
therefore a function of geometry factors, the material stress-strain
relationship ( = f(s)),**
**as well as the applied stress. Finite element analysis results for
different geometries and different materials have shown

K_{r}^{f}
= (1 – 0.14 L_{r}^{2})(0.3 + 0.7 exp(-0.65 L_{r}^{6})) (15)

with L_{r}** **=
applied stress/collapse stress; and K_{r}^{f} = critical value
for K_{r}.

Residual stresses are
accounted for by adding the crack driving force due to residual stresses (K_{s})*
*to the crack driving force due to externally applied stresses (K_{p}).*
*Residual stresses are assumed not to contribute towards plastic collapse
and therefore have no influence on the L_{r}** **coordinate. A
plasticity correction factor (r)
is then added to the K_{r} coordinate to take into account plasticity
at the crack tip caused by residual stresses:

K_{r}
= (K_{s} + K_{p}) / K_{IC} + r (16)

Having broadly described the principles behind R6, it can be
applied to the present case study.

The maximum externally applied stress in the pipeline
is caused by a maximum internal pressure peak of 3.8 MPa:

s_{hoop}
= Pr/t (17)

= 3.8 x 3/0.03

= 380 MPa

A simplified version of the
well known Newman & Raju solution for elliptical surface cracks subjected
to uniform tensile loading^{9 }was used to calculate K_{p}:

(18)

with a = defect depth; c =
defect length/2; and t = wall-thickness.

A
parabolic through-thickness residual stress profile, as shown in Fig. 9, was
assumed. A weight function technique^{10 }was used to calculate K_{s}.

An initiation
analysis was performed to determine the critical crack dimensions at which
initiation will take place. The results of this analysis are shown in Fig. 10
for an axial defect in weld metal, giving the relationship between defect depth
and length for the conditions of initiation and stable growth instability
(discussed later) respectively.

The
influence of the residual stress profile on the initiation condition can
clearly be seen.

A stable crack growth analysis was subsequently
performed to determine the critical defect dimensions at which instability
would occur. A locus of assessment points was obtained for different initial
crack depths and postulated crack growth increments. Instability would occur
when a locus falls in its entirety outside the FAD and touches at one point
only, or

K_{r}(a
+ Da) = K_{r}^{f} (19)

(20)

This
procedure is illustrated in Fig. 11 which depicts a portion of the FAD shown in
Fig. 8. The assessment line in Fig. 11 is described by eqn (15).* *The
crack depth is incrementally increased until the initiation condition is
reached (a_{i} in Fig. 11). A crack growth increment is then added to
this initial depth and a new assessment coordinate is calculated. Due to the
rising fracture toughness, the curve drops below the assessment line, implying
arrest. The initial depth is then incrementally increased and the procedure
repeated. The critical initial crack depth (a_{cr})** **is found
when the locus of stable crack growth assessment coordinates does not intersect
the assessment line.

The result of the stable crack growth analysis is
shown in Fig. 10 for an axial defect in weld metal.

The
critical crack size analyses discussed above were based on the most severe
loading conditions expected to occur during the life of the pipeline. The
fatigue crack growth of defects due to variable loading during the operational
life were however not taken into account.

A fatigue crack growth analysis was subsequently
performed. Material tests were performed to determine the fatigue crack growth
properties of the weld metal based on the Paris crack growth rate expression:

(21)

with DK in MPaÖm
and da/dN* *in m/cycle.

It
was assumed that the growth rate would double in water:

(22)

A fatigue crack growth analysis was performed for
each defect found during the inspection exercise to determine estimated
end-of-life size probability density functions for both length and depth. The
analyses were performed by calculating the incremental crack growth in both the
length and depth directions for each sequential cycle and then updating the
aspect ratio before continuing to the next cycle. The sequence of cycles and
stress ranges for each cycle expected during the life of the pipeline were
accurately simulated.

Typical crack growth curves for defect depth and
length are shown in Fig. 12. The end-of-life PDFs for length and depths are
shown in Fig. 6.

**Figure 12:**
Fatigue crack growth curves. (a) Length; (b) depth.

It
became obvious that the integrity of the pipeline could not be
deterministically demonstrated since there was always a possibility that a
critical defect had been missed during the inspection and repair exercise. It
was therefore necessary to quantify this risk.

The greatest risk involved axially oriented defects
which are situated in circumferential welds but which are aligned with axial
welds. Axially orientated defects not aligned with axial welds would propagate
into the very much tougher base metal and would subsequently arrest.
Circumferentially orientated defects are subjected to a fraction of the hoop
stress and would not become critical.

The probability that, if such a defect exists and was
missed during the inspection and repair program, it would be critical before or
at the end of the design life of the pipeline was quantified as follows:

P_{d}
= P(size > critical size)

(23)

with
l = length; lc = critical length; d* *= depth; and dc = critical depth.

The stable crack growth critical size curve shown in
Fig. 10, establishes the relationship between critical depth and critical
length: lc = f(dc).

The above function was approximated by a straight
line fit of the actual curve with a vertical cut-off at l = 62 mm and a horizontal
cut-off at d *= *2.5 mm. When a surface defect fails it must be
re-categorized as a through-thickness crack and should then be re-analyzed. The
critical length of a through-thickness crack was calculated as 50 mm. This
implies that, in this case, any surface defect that becomes critical would
propagate in the thickness direction until it breaks the surface and would then
further propagate in an unstable manner in the length direction to split open
the pipe. The approximated function is shown in Fig. 13.

Equation
(15)* *was then evaluated as follows:

(24)

The above
expression was solved numerically and yielded the following result:

P_{d} = 2.2 x 10^{-2}

The
probability of failure of the pipeline was then calculated as follows:

P_{f}
= 1 – (1 – P_{d})^{n} (25)

* *

with
n = number of defects remaining in the pipeline after the inspection program.

The number of defects remaining in the pipeline was
estimated as follows. Approximately 230 axially orientated defects were found
during the inspection. All these defects were repaired. Typical detection
probabilities cited in the literature^{3,6 }range from 0.6 (poor
inspection) to 0.95 (good inspection). The probability would also be a function
of defect size.

A detection probability of 0.95 was assumed in the
present case, independent of crack size. This implied that a probable number of
11.9 defects had been missed. Of these, only a fraction would be aligned with
axial welds. This fraction was quantified as the ratio of six times the width
of an axial weld (there are six axial welds connected to each circumferential
weld), to the circumference of the pipe, or fraction = 6 x 20/(2p x 3000) = 0.007.

The probability of failure was then calculated:

P_{f}
= 1 – (1 – 2.2x10^{-2})^{0.007x11.9}

= 1.87 x 10^{-3}

**DISCUSSION**

The result of the
probabilistic analysis indicated that the probability of failure was
unacceptable. A probability of failure of 5 X 10^{-4} has been proposed
as a maximum tolerable value.^{3 }In nuclear applications a value of 10^{-6}
is usually required. In the present ease, a probable cost of failure was
calculated:

Probable cost of failure = 1.87 x
10^{-3} x $500 000 000

= $1 000 000

Analyses
were thereafter performed to determine which fracture control option would
yield the best results (lower the probability of failure in the most
cost-effective way). One of the greatest advantages of the probabilistic
approach is that the sensitivity of the risk of failure (which is the single
most important result of any failure analysis) to any influencing parameter may
easily be quantified.

The final
solution was to embed the pipeline in concrete, thereby lowering the stresses
to such an extent that the probability of failure became acceptable.

The interaction between
fracture mechanics analyses and NDI results has traditionally been addressed by
assuming a critical defect size based on the NDI techniques’ size limitations.
The fact however remains that detection probabilities are operator dependent
and that there will always be a possibility that a defect larger than this size
would be missed. It is senseless to base a detailed and costly analysis on such
a ‘gut-feel’ baseline assumption.

Probabilistic fracture mechanics therefore often
offers the only scientific way to assess the integrity of fracture prone
structures.

It seems
appropriate that research effort is expended on this subject. Some results on
statistical distributions of fracture initiation toughness data have been
published, very few however on J_{r}** **data. The size dependence
of NDI detection probabilities is also a subject that requires attention.

1. British
Standards Institution, Guidance on some methods for the derivation of
acceptance levels for defects in fusion welded joints. PD 6493, 1980.

2. Milne, I.,
Ainsworth, R. A., Dowling, A. R. & Steward, A. T., Assessment of the
integrity of structures containing defects. R/H/R6-Rev.3, 1986.

3. Mudge, P.
J. & Williams, S., Statistical aspects of defect sizing using ultrasonics,
1986.

4. Broek, D.,
The Practical Use of Fracture Mechanics.* *Kluwer Academic Publishers,
1989, pp.90-1.

5. Takashima,
H., Mimaki, T. & Hagiwara, Y., Reliability analysis of spherical tank by
Monte Carlo simulation. In The Mechanism of Fracture,* *ed. V. S. Goel.
America Society of Metals, 1986, pp.537-46.

6. Carlsson, J.,
Probabilistic fracture mechanics. In Advances in Elasto-Plastic Fracture
Mechanics,* *ed. L. H. Larsson. Applied Science Publishers, 1980,
pp.418-19.

7. Faucher, B. &
Tyson, W. R., A comparison crack-mouth opening and load-line displacement for
J-integral evaluation using bend specimens. Elastic-plastic fracture test
methods. ASTM STP, 856* *(1983).

8. Turner, C. E.,
Methods for post-yield fracture safety assessment. In Post-yield Fracture
Mechanics,* *Elsevier Science Publishers, 1984, pp.326-8.

9. Newman, J. C.
&* *Raju, J. S. Eng. Frac. Mech., 15 (1981) 185-92.

10. Mattheck, C.,
Munz, D. & Stamm, H., Stress intensity factor for semi-elliptical surface
cracks loaded by stress gradients. Eng. Frac. Mech.,18(3) (1983) 633-41.