Section 7: Confidence Bounds on Component Reliability
COMPONENT RELIABILITY CONFIDENCE BOUNDS
Confidence bounds on component reliability are calculated via the bootstrap technique
utilizing Monte-Carlo simulation (2,000 simulations are used in the example below). Once
the reliability has been estimated 2,000 times the results are sorted and the desired
confidence level is determined. The following is a general overview of the process:
• The biased Weibull modulus (m) and the biased Weibull characteristic strength (sq)
parameters are estimated from the sectored flexure bar failure data (SFB).
• Monte Carlo simulation is used to generate failure data points using the SFB
parameter estimates. In this case 20 data points were generated to match the number
of SFB data.
• Estimate the biased Weibull parameters (m and sq) for each failure data set generated
by the Monte Carlo procedure.
• Repeat the above two steps 2,000 times (or more depending on computational
resources). Save the pairs of parameter estimates (the failure data sets are
discarded).
• Develop the kAA vs mA relationship for the SFB geometry (for all 2,000 sets).
• Develop the kAA vs mA relationship for the GB geometry (for all 2,000 sets).
• Solve for the Weibull Scale Parameter (for all 2,000 sets).
• Solve for the Probability of Failure (for all 2,000 sets)
• Sort the Probability of Failures
• Pick the Confidence level (90%)
Section 7: Confidence Bounds on Component Reliability
An initial failure data set must be available along with a finite element analysis for both the test
specimen and the component. Flexure bar failure data was used for the failure set along with
the finite element analysis of the sectored flexure bar. Here the fractographic analysis
determined that all the specimens had failed due to surface related defects
.
The dimensions of the test specimen are sufficient if the specimen is a standard tensile
specimen or a standard bend bar specimen. Otherwise, a finite element model of the test
specimen is required. For this specific task the finite element model of the sectored flexure bar
(SFB) created under an earlier contract was utilized. Relative to the SFB test specimen, the
following is a list of specific steps associated with that particular specimen:
a) Estimate the Weibull Parameters for the sectored flexure bar. For this specific task the
SN47 specimen data at 21ºC was used. Here the Weibull modulus is
(mA)SFB = 12.01
and the Weibull characteristic strength is
(sqA)SFB = 642.5
Section 7: Confidence Bounds on Component Reliability
b) Next a random number between 0 and 1 is generated, i.e.,
c) Calculate a failure stress associated with the random number from (b) and the Weibull
parameters from (a) using the expression
σi
  ln  Ri  


 σ  
 θA SFB 
=
1/  m A SFB
d) Repeat (b and c) N times, where N is the number of sectored flexure data values in the
original data set. For this particular data set N=20.
TABLE 2 – Generated Data Set
i
si
1
Eq Above
2
Eq Above
…
…
N
Eq Above
Section 7: Confidence Bounds on Component Reliability
e) Repeat steps (b) through (e) as many times as is computational feasible (as noted earlier
the effort being reported on here used 2,000 repetitions). The simulated sectored flex bar
Weibull modulus and characteristic strength, i.e., the (mA)SFB and (sqA)SFB columns of
Table 3 are thus generated.
f) Estimate Weibull parameters for the data set generated in the previous steps using
maximum likelihood estimators (MLE).
g) Establish the relationship between the simulated sectored flex bar Weibull moduli and the
effective area, i.e., (mA)SFB and (kA)SFB. Use the maximum and minimum values of
(mA)SFB from Table 3 as the bounds on (mA)SFB. Use the SFB finite element model to
calculate (kA)SFB as a function of (mA)SFB for a few key values of (mA)SFB, then use a linear
fit for the remaining values. Complete the (kA)SFB column in Table 3. In future work the
linear fit would be refined with a spline.
TABLE 3 – Bootstrap
J
(mA)SFB
(sqA)SFB
(kA)SFB
(kA)Barrel
(s0A)Material
PfBarrel
1
(mA)1
(sqA)1
(kA)1
(kA)1
(s0A)1
Pf1
2
(mA)2
(sqA)2
(kA)2
(kA)2
(s0A)2
Pf2
…
…
…
…
…
…
…
BS
(mA)BS
(sqA)BS
(kA)BS
(kA)BS
(s0A)BS
PfBS
Section 7: Confidence Bounds on Component Reliability
i) Calculate the material characteristic strength or material scale parameter, (s0A)Material,
as a function of (mA)SFB, (sqA)SFB, and (kA)SFB using the following expression:
σ0A  j

σ θ A  j kA1/j m

A j
j) Calculate the gun barrel probability of failure, PfBarrel, given mA, s0A, (kA)Barrel, and the
smax for the gun barrel (from CARES) using the following expression:
P 
f
j
=
1


exp kA j


m A  j
 σ max 


 σ 0A 
j 





k) The use of this formula allows the Pf column in Table 3 to be populated without the
repeated execution of CARES. This saves on computational resources.
l) Sort by Pf in order to establish percentiles and then choose bounds based on the
percentiles.
WeibPar calculates the confidence bounds on component reliability via integration with
CARES. As WeibPar stores and manipulates the data from Table 3 above, CARES is called to
perform the probability of failure calculations on the finite element model.
Section 7: Confidence Bounds on Component Reliability
The following figures depict the results of the bootstrapping procedure outlined above. A
finite element model of the SFB test specimen is shown below.
Section 7: Confidence Bounds on Component Reliability
The next figure shows a plot of effective area as a function of the Weibull modulus with
values from both the previous contract work (as blue diamonds) along with those
generated through the bootstrap routines (green asterisks). The linear fit between the key
values is shown as the orange line.
Sectored Flexure Bar
Effective Area as a function of Weibull Modulus
Original Report values
SFB
Linear Fit
290
Effective Area
240
190
140
90
4
9
14
19
24
Weibull Modulus
29
34
39
Section 7: Confidence Bounds on Component Reliability
Figure 8 shows the gun barrel effective volume as a function of the Weibull modulus.
Ceram ic Gun Barrel (120 m m )
Very Fine Mesh
SN47
120,000
Effective Area
100,000
80,000
60,000
40,000
20,000
-
10
20
30
Weibull Modulus
40
50
Section 7: Confidence Bounds on Component Reliability
Ceramic Gun Barrel (120 mm) - SN47
Reliability Confidence Bounds
1.0E+00
Probability of Failure
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
0
20
Num ber of Specim ens
We are 90% certain that the actual component reliability falls
between these two points. The open blue circle is the mean
probability of failure.
The next figure depicts the
bounds on component
reliability for the data set
used to generate the
Weibull parameters. The
blue diamond is the
component probability of
failure given the biased
maximum likelihood
Weibull parameters
calculated from the room
temperature specimen
data. For this example an
outer pressure of 255 MPa
was applied to the gun
barrel. The extreme two
data points (red circles)
are the 5% and 95%
confidence bounds.
Section 7: Confidence Bounds on Component Reliability
Ceramic Gun Barrel (120 mm) - SN47
Effect the Number of Specimens have on the
Reliability Confidence Bounds
1.0E+00
Probability of Failure
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
0
10
20
30
40
50
60
70
80
90
100
Num ber of Specim ens
For comparison sake in this hypothetical example we
will assume that even though we are increasing number
of specimens, the Weibull parameters remain the same.
110
The question may be
proposed, what effect does
the limited information
(only 20 failure stresses in
this case) have on the
bounds on reliability? To
answer this question the
bootstrap technique was
rerun for a variety of
specimen data set sizes.
Please note that these are
hypothetical results in the
sense that if we really broke
20 more specimens to
increase our specimen data
set to 40 then it is unlikely
that the recalculated Weibull
parameters would be the
same as for the original 20
data points.
Section 7: Confidence Bounds on Component Reliability