Environmental Engineering Reference
In-Depth Information
4.2
Methods of determining failure probability
using a binomial distribution
4.2.1 Methods of determining the probability of failure
on the basis of computer program MAVR-1.1
Probabilistic analysis of fracture of a section of a component based on
the methodology proposed by A.A. Tutnov
57
is performed using the computer
program MAVR-1.1
58,59
. In this method it is assumed that:
− the predicted depth distribution of calculated cracks is described by the
density distribution
p
(
a
), which is determined by the distribution of defects
detected by monitoring or by the conditional distribution of the calculated
probability
p
cond
(
a
)
of
cracks and the probability of detection of defects
p
d
(
a
):
pa
()
pa
( )=
cond
,
s
pa
()
[4.9]
∫
p a
()
()
da
cond
d
pa
d
0
and for the predicted distribution of the lengths of the calculated cracks
l
:
p
(
l
) =
p
cond
(
l
);
[4.10]
− fatigue growth of cracks is described by the Paris equation;
− distribution of the probability
k
of finding defects (
k
= 1, 2, 3,...) along
the length of a heat exchanger pipe is described by the Poisson distribution:
[ ]
k
pk
( )
= µ
exp
−µ
/
k
!,
[4.11]
where μ
0
is the mathematical expectation of the number of cracks in the
reference area (e.g. the length of a section of the pipe);
− distribution of critical crack dimensions
0
0
F at
can be
determined on the basis of various criteria, for example, two criteria used
in Ref. 59 for heat exchanger pipes of steam generators:
1) occurrence of stress corrosion cracking;
2) onset of the limiting plastic state (ductile fracture).
In this case, the stress intensity factors for the Paris equation are
calculated using quasi-elastic stresses, but in the conditions of stress
corrosion cracking and formation of a plastic hinge the actual stresses are
taken into account.
Each area of the heat exchanger pipe is divided into
Q
reference areas.
In the presence of
k
cracks in the reference area the probability of failure
of the investigated section
l
of the heat exchanger pipe
P
1
jL
by the
j-
th
failure mechanism for calculation cracks oriented in the direction
L
(
L
=
1 - axial,
L
= 2 - circumferential) is defined as follows:
F at
and
( , )
( , )
cr
cr
1
kQ
Pt
() 1 [1
=−−
Pt j
()] , 1,2,
=
[4.12]
1
jL
1
jL
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