CorTech Training Red Wing MN
Weibull Trending Toolkit (WTT)
Pitting Corrosion Project
Based on work performed by Glenn Bowie during 1974-75

Task 9 is the last of four Tasks about pitting corrosion in aluminum alloys. The first experimental data set analyzed includes effects of time of exposure. I suggest you follow the Tasks meticulously. If you are a student, retiree, or particularly diligent working technician or engineer, try to repeat the steps taken to deduce curve fitting relations, tables, and graphs.

Task 9

Generalize interpretation of Aziz's data to obtain a relation in terms of exposed area, maximum pit depth, and exposure time.

Programs WRISK.EXE and WDESIGN.EXE used in Task 9 were prepared in 1996. They each contain a first screen which is to be ignored. An e-mail address given there is obsolete.

References

• P. M. Aziz, "Application of the Statistical Theory of Extreme Values to the Analysis of Maximum Pit Depth Data for Aluminum", Corrosion, Vol. 12, October 1956, pp.35-46.
• H. P. Godard, W. B. Jepson, M. R. Bothwell, and R. L. Kane, "The Corrosion of Light Metals", John Wiley and Sons, Inc., New York, 1967.

Analysis

• P. M. Aziz's Data
• Plot Grouped Data
• Include H. P. Godard's Industrial Water Pipeline Data
• Create Modeling Program DESIGN
• Model Grouped Aziz Data for Three Week Exposure and Godard Data for 13 Year Exposure
• Industrial Water Pipeline with 50 Year Design Life

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P. M. Aziz's Data

Task 8 Suggestions

• Make data files aziz01.dat through aziz06.dat for the specimens exposed for six exposure times.
• Convert pit depth units from microns to mils.
• Run program wrisk.exe. Study three-parameter Weibull analysis results.

Task 9 Suggestions

• Examine results in Task 8 in pairs. Average Weibull threshold parameter E for exposure times of one week and one month equals 3.49 mils. Average E for exposure times of six months and one year equals 13.0 mils. Average Weibull characteristic value V also increased with exposure time. The Weibull shape parameter K seems to have a random behavior. It is clear that some smoothing or grouping of Aziz's data must be used to generalize shape parameter behavior.
• Group Aziz's data for exposure times of one week and one month into data file task901.dat using MS-DOS Edit. Express the 19 values in units of mils.
• Group Aziz's data for the remaining four exposure times into a set of 39 maximum of depths, mils, in file task902.dat.
• Apply QBasic program SORT.BAS given in the Weibull Trending Toolkit to files task901.dat and task902.dat. SORT does just that, and creates two new files otask901.dat and otask902.dat.
• Apply Weibull Trending Toolkit program WRISK.EXE to files otask901 and otask902. Program sees the leading o's and calls the sorted files TASK901 and TASK902.
• Examine WRISK numerical results for files task901 and task902.

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Numerical Results for 19 Maximum Pit Depths

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Numerical Results for 39 Maximum Pit Depths

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Plot Grouped Data

Use WRISK to plot results for data files task901 and task902 with graph upper limit equal to 30 mils in each case.

Superimpose the two graphs to obtain the following image.

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Include H. P. Godard's Industrial Water Pipeline Data

• Examine H. P. Godard's industrial water pipeline data on p. 64 of the above reference. He measured maximum pit depth in each of 20 three foot long sections of AA 5052 pipe after 13 years exposure.
• The instructor was able to read 16 pit depths from Godard's graph. As a course participant, you may make data file task903.dat to store Godard's data.
• Apply program WRISK to file task903.
• Examine the following numerical results.
• Plot results of analyzing Godard's data. Choose a graph upper limit of 80 mils.

Create Modeling Program DESIGN

Model Grouped Aziz Data for Three Week Exposure and Godard Data for 13 Year Exposure

Industrial Water Pipeline with 50 Year Design Life

Note written in 1995 about using wrisk.exe and risk.bas.

The Weibull Trending Toolkit Program wrisk.exe is 1995
copyrighted property of:
Glenn E. Bowie
2426 Hallquist Ave.
Red Wing, MN 55066
(651) 388-2374
e-mail: glennbowie@hotmail.com

Do not try to use WRISK.EXE to process a file with more than 1024
data values.  WRISK.EXE runs on an IBM compatible PC with Microsoft
Windows.  Make sure you have VBRUN300.DLL in your
C:\>WINDOWS\SYSTEM directory.


The Weibull Trending Toolkit program WRISK.EXE has three curves instead
of two.
blue = Weibull probability density
green = Weibull cumulative probability
yellow = Weibull risk
Weibull Probability Density, p

p = k * ((x - e)/(v - e)^(k - 1) *

Exp(-((x - e)/(v - e))^k/(v - e).

Weibull Cumulative Probability, F

F = 1 - Exp(-((x - e)/(v - e))^k)

Let P = Exp(-((x - e)/(v - e))^k)

Weibull Risk

R = k * ((x - e)/(v - e))^(k - 1)/(v - e)

You see that p = R * P, or R = p/P.

The probability density and risk curves are scaled so you can see their
shapes.  When you are ready to study probability density and risk mag
nitudes, you can use QBasic program RISK.BAS as a starting tool.

I like to use to PCs side by side when studying Weibull risk.  A good
starting point is to examine file 6061.DAT using Edit.  There are 102
fatigue lives in the file.  The first two lives are 233 and 258.  Count
down to find the median life in the file equals 400.  Examine the last
entry in the file.  The 102nd value in the file is 560.

Run WRISK and choose file 6061.  Use a graph upper limit = 800.
There are 19 tics on the Y-axis.  Each scale division is worth 40 life
units.  20 x 40 = 800.  From the summary screen,
k = 3.61498
e = 195.1232
v = 420.0841
Median = 398.3941

Type QBasic risk, press enter on your second PC.  RISK asks you for
the Weibull shape parameter k, threshold parameter e, median, and
production run size.  Enter the above values for k, e, and median and
102 for production run size.  RISK tells you characteristic value v =
420.0842.  The value in the summary screen of WRISK is truncated
rather than rounded.  We accept that the two characteristic values
are the same.  RISK tells you the probability of survival at the first
failure is 0.9903.  The cumulative distribution curve, green, is a
probability of failure curve.

P = probability of survival
F = probability of failure
F = 1 - P.

The height of the green curve is 1 - 0.9903 or 0.0097 at the first
failure, according to RISK.  RISK predicts the time to first
failure = 257.6.
Compare with actual first experimental value in 6061.DAT, 233.  RISK
tells us the probability density at time 257.6 equals 0.0005588.
Calculate the risk at first failure:

R = p/P = 0.0005588/0.9903 = 0.0005643.  RISK tells us
R = 0.0005643 at first failure.

Work your way through second failure comparisons at your leisure.
Move to the median.  RISK predicts the median life equals 398.3941,
and the experimental value is 400.  WRISK shows the mean, median,
and mode are close together for file 6061.  RISK tells us the height of
the probability density curve at the median equals 0.00616.  And at time
538.956, the density drops back down to 0.000473.  The difference in
time units between the last failure and the threshold e equals 538.956 -
195.1232 = 343.8328.  Multiply this difference by the density at the
median to obtain 2.1192.  Remember the area of a triangle equals 0.5 x
base x height.  We just estimated the area under the density curve to be
0.5 x 2.1192 = 1.0596.  The area under a probability density curve
equals 1.0.

I used a clear, plastic scale divided in mm to find the height of the
yellow risk curve at the mode and at 538 time units to be 19 and 79 mm.
The height ratio equals 19/79 or 0.24.  RISK predicts the ratio to be
0.0123/0.0487 = 0.25.

Stare at the graph for file 6061.  Use a scale to confirm that the
cumula tive probability is at 0.5 of the maximum height, for this case.
The probability of survival at the median = 0.5.  RISK tells us risk
R = 0.1232696 at the median.  At the median, find R x P = 0.0616348.
The density p at the median is then 0.0616348, as RISK told us already.
It is important to practice looking at the curves when interpreting
numerical results.

Look at the probability density and cumulative probability curves for
file 6061.  Think of the probability density as the rate of change of
the cumulative probability.  The slope of the cumulative probability
curve, the green one, is maximal at the mode.  The probability density
is maximal at the mode.

Glenn E. Bowie
Red Wing, MN
September 1, 1995

Note written in 1995 about using programs wdesign.exe and wrisk.exe.

The Weibull Trending Toolkit Program wdesign.exe is 1995 copyrighted
property of:
Glenn E. Bowie
2426 Hallquist Ave.
Red Wing, MN 55066
(651) 388-2374
e-mail: glennbowie@hotmail.com

It is important to remember that WDESIGN writes a data file and a
results file each time it is used.  When WDESIGN asks you for a file
name, NEVER give it the same name as an experimental file such as
6061, 7075 and so on.  It is a good practice to use the leading
letter D in all file names you give WDESIGN.  Avoid
naming experimental data files with a leading D.

Let me assume you have applied WRISK.EXE to study data file 6061 as
outlined in the above note about WRISK.  Run WDESIGN.  Click the box
"Enter N, k, e and Median".  Be sure to enter a file name.  This
time enter D6061.  Enter the following values:
For N: 102
For k: 3.61498
For e: 195.1232
For Median: 398.3941

Compare the summary results and graph with the summary and graph
you get when you apply WRISK to file 6061.  Keep the median at
398.3941 and decrease k gradually.  At each step, compare with
WRISK and 6061 results.  Finally, choose
N: 102
k: 3.55
e: 194
Median: 398.3941


Apply WRISK to file D6061.  You get the same results given by
WDESIGN for the above parameters.

Please do not apply WDESIGN for N > 1024.

Please supply design threshold e values that are sufficiently
large to keep resulting threshold values in the summary screen
positive.

Please be aware there is room for research.

Please be aware as you practice using WDESIGN in relation to your
database that you are thinking about possible application of
WDESIGN concepts for preparation of production run specifications.
___________________________________________________________________
Design Practice

At the DOS prompt, type QBasic risk.  Run RISK.BAS.  Enter shape
parameter k equal to 1, threshold parameter 0 and median 300.
See the characteristic value v = 432.8085.  Enter run size 500.
It is important to see that risk h(t) is constant from the first
to the last failure. Here h(t) = 0.002310491.  Now find 1/v.
For k = 1 and e = 0, h(t) = 1/v.  The failure rate is constant.

Many reliability and risk analysts limit their work to the special
case where the risk or failure rate is constant.  Programs
WTT01.EXE, WRISK.EXE, and WDESIGN.EXE do not apply to the special
case where k = 1.  In order to show you the difference between the
constant risk case and one where k is slightly greater than 1.0,
the following exercise was constructed.

At the DOS prompt for directory wtt02, load Windows.
C:\wtt02\win.  Hopefully, you see the Program Manager high-
lighted.  Click File, then click Run.  In the Run Window, enter
on the command line: c:\wtt02\wrisk.exe.  Click OK on the WRISK
sign-on screen.  Click "Match Sample Statistics".  Enter file
name 6061.  Click "Plot Control Chart" and enter graph upper
limit 800.  You see the graph.  Notice a portion of the summary
screen is visible below the graph.  Hopefully, you also can see
the Program or File Manager word File. Click File.  Click Run.  Enter
on the command line: c:\wtt02\wdesign.exe.  Click OK on the WDESIGN
sign-on screen.  Click "Enter N, K, E, And Median".  Enter file
name delight.  For N, 500.  For K, 0.95.  For E, 27.0.
For Median, 300.

Notice k = 1.032018, and e = 0.4963179.  If you ever analyze material
property data and obtain a k value as close to 1.0 as this, with e
not negative, consider sending me a copy of your data.  See
UCL = 2743.9.
Click "Plot Control Chart" and enter graph upper limit 3000.  Examine
the yellow RISK curve.  It is a delight.  I believe risk curves such
as this one can represent many data sets which are now represented
by a constant failure rate approach.

In the lower left corner of the screen, click the small rectangular
portion of WRISK summary screen.  Click "Match Sample Statistics".
Enter file name delight.  WDESIGN generated file delight.dat.  WRISK
has now analyzed the file and obtained familiar results.  Click
"Plot Control Chart", and enter 3000.  The graph is again a delight.

Click the small rectangular portion of WDESIGN summary screen in the
lower right corner of your display.  Enter the following:
design11
500
1.1
25
300
Study the summary values.  Since UCL = 2042.8, choose graph upper
limit 2500.  Notice the risk curve.  Click anywhere on the portion
of WDESIGN summary screen visible below the graph.
Enter:
design12
500
1.2
25
300
Study the summary values.  Choose graph upper limit 2000.
Click WDESIGN summary screen at the bottom, enter:
design13
500
1.3
25
300
Choose graph upper limit 2000.  Proceed to generate file design14 with
k = 1.4, design15 with k= 1.5, and so on until you have generated file
design19 with k = 1.9.  Choose graph upper limit 1000.  It is time to
generate a particular case.
Enter file name drayly, with N = 500, k = 1.928, e = 7.9 and median
= 300.  Notice k = 1.999 and e = 0.01767976.  Choose upper limit
1000.  Describe the risk curve.
In the special case where k = 2.0 and e = 0, the Weibull distribution
becomes a Rayleigh distribution.  For a Rayleigh distribution, the risk
is a straight line.
Generate files design20, design21 and so on until you have generated
file design40.  Examine the graphs with upper limit 1000 in each case.
Watch changes in risk.
Click the portion of WRISK summary screen at lower left and review
files delight, design11 through design19, drayly, and design20 through
design40.
Examine skewness aand kurtosis values during your review.  Decide for
yourself what range of k yields probability density curves which might
be considered to be symmetrical or pseudo-normal.


Glenn E. Bowie
Red Wing, MN
September 1, 1995

Copyright Glenn E. Bowie©, CorTech Training, Red Wing, MN 1996, 2000, 2001, 2003. All rights reserved.