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

[first posted: 23 December 1995, revised 10 July 2000, 30 January 2001, and 12 September 2003]

  CorTech Training, Red Wing, MN

Weibull Trending Toolkit (WTT) Courses I, II and III are presented to students of engineering everywhere, at all career stages. They contain elements useful for development of Safety Critical (SC) Systems for many types of containment structures. Many containments are pressure vessels. When a fluid is contained under pressure, it is important that it be designed to leak before it breaks. Corrosion pits can cause leaks. They can also serve as initiation sites for crack growth. When a growing crack penetrates a container wall, the crack growth mechanism must not convert from a slow growth to a fast or catastrophic fracture mode. Spend time searching topics such as safety critical systems, artificial neural networks for SC systems.

INTRODUCTION
This course examines historic data sets containing depths of pits measured after sets of ferrous alloy specimens had been exposed to corrosion environments for extended periods of time. Ferrous alloys require corrosion protection. Depending on the type of alloy and application, the protection can be in the form of zinc galvanizing, cadmium or other plating, or an inhibitive coating or lining system. In critical applications, such as aircraft structures, periodic inspections are performed to ensure protection system integrity. A surficial irregularity such as a carbon or carbide particle, or minute scratch can cause a local spot to be galvanically noble with respect to the base alloy. A corrosion pit can form at the irregularity.

COURSE FOCUS
Corrosion consultant, test engineer, and inspector roles in pipeline testing are central. Main points of lessons and numerical exercises are based on instructor's experience in reviewing and analyzing pit depth data sets. The Weibull Trending Toolkit and QBasic programs are used extensively. Auditors are expected to read lessons and follow instructor's guidelines in repeating numerical exercises.

COURSE OUTLINE

  1. Task 1
    Review G. N. Scott's 1933 analysis of Kay County pipeline maximum pit depth versus exposed area and analyze the data by linear regression.
  2. Task 2
    Analyze the Kay County data
    by means of two Weibull curve fitting relations and compare with Scott's results.
  3. Task 3
    Apply Weibull relations to G. N. Scott's Mt. Auburn and Council Hill data and to Mahoning and Almeda-Lynchburg data obtained by other investigators.
  4. Task 4
    Review G. G. Eldredge's 1957 interpretation of Bessemer steel data obtained by K. H. Logan (1945) and analyze the data by Weibull analyses.
  5. Task 5
    Analyze K. H. Logan's 1945 wrought iron specimen data and compare with Bessemer steel specimen results.
  6. Task 6
    Review P. M. Aziz's 1956 experimental procedures and data for Al alloy specimens immersed in Kingston tap water.
  7. Task 7
    Interpret Aziz's pitting depth frequency distributions by Weibull analysis.
  8. Task 8
    Interpret Aziz's Alcan 55S-T maximum pit depth data for 9 or 10 specimens at each of six exposure times.
  9. Task 9
    Generalize interpretations of Aziz's data to obtain a relation in terms of exposed area, maximum pit depth, and exposure time.

Task 1

e-mail from Bob Heidersbach concerning G. N. Scott:
Glenn Thanks to you and your website, I know about Gordon Scott's early work. I am proof reading a book I've written and needed a slightly better reference than the one you have on the problem #1 web page. I found this information, which seems to be from something at the U of Michigan. The URL is as the bottom. Thanks again for remaining a valuable contributor to rusty knowledge. Bob Heidersbach Cape Canaveral, Florida Scott, G. N. The Use and Behavior of Protective Coatings on Underground Pipes. Am. Petroleum Inst. Bull. I, 1928, pp. 78-93. 186. - API Pipe Coating Tests-Progress Report No. 1 - Installation of Test Coatings. Proc. Am. Petroleum Inst. [IV], vol. 12, 1931, pp. 55-72. 187. API Pipe Coating Tests Progress Report No. II - Initial Investigation of Specimens. Proc. Am. Petroleum Inst. [IV], vol. 12, 1931, pp. 72 -107. 188. A Review of Recent Progress in Mitigating Soil Corrosion. Proc. Am. Petroleum Inst. [I], vol. 14, 1933, (Production), pp. 130-142. 189. Report of API Research Associate to the Committee on Corrosion of Pipe Lines. Pt I. Adjustment of Soil-Corrosion Pit-Depth Measurements for Size of Sample. Proc. Am. Petroleum Inst. [iV, vol. 14, 1933 5 pp. 204-211. 190. - -Report of API Research Associate to the Committee on Corrosion of Pipe Lines. Pt. II. A Preliminary Study of the Rate of Pitting of Iron Pipe in Soils. Proc. Am. Petroleum Inst. [IV], vol. 14, 1933, pp. 212-220. 191. API Coating Tests. Progress Report No. IV. Third Inspection of Pipe-Coating Tests on Operating Lines and Second Inspection of Short Coated Specimens. Proc. Am. Petroleum Inst. [IVJ, vol. 15, 1934, pp. 18-33. 192. A Rational Approach to Cathodic Protection Problems. Petroleum Engineer, vol. 12, No. 8, 1940-41, pp. 27-30, No. 9, pp. 59-64kNo.ll, PP. 74-76. 193. An Analysis of Certain Circuits in Cathodic Protection. Proc. Am. Petroleum Inst. [IV, vol. 23, 1942, pp. 36-48. 194. An Aspect of the Pipe-to-Soil Potential and Related Measurements. Gas, vol. 20, Feb. 1944, pp. 30, 34, 37, 38. 195. Somastic Pipe Coating and Cathodic Protection. Proc. 1st Ann. Meeting NACE, 1944, pp. 238-245. 196. Senatoroff, N. K. Fifteen Years Experience in Application of External Corrosion Mitigaticn Methods to a High Pressure Natural Gas Transmission Line. Corrosion, vol. 4, No. 10, 1948, pp. 479-91. 197. Shepard, E. R. Pipe Line Currents and Soil Resistivity as Indicators of Local Corrosive Soil Areas. Nat. Bur. Stds. J. Res., vol. 6, 1931, pp. 683-708, RP 298. 198. ----— Some Factors Involved in Soil Corrosion. Ind. Eng. Chem., vol. 26, No. 7, 1934, pp. 723-32. 199. Shepard, E. R. and Graeser, HI. J. Jr. Design of Anode Systems for Cathodic Protection of Underground and Water Submerged Metallic Structures, Corrosion. vol. 6, No. 11, 1950, pp. 360-375. http://deepblue.lib.umich.edu/bitstream/2027.42/7620/4/ajq2382.0001.001.txt September 19, 2010 Bob Bob Heidersbach Dr. Rust, Inc. Cape Canaveral, Florida Cell phone: 321-626-8125

Review G. N. Scott's 1933 analysis of Kay County pipeline maximum pit depth versus exposed area and analyze the data by linear regression.

I, Glenn E. Bowie, studied G. N. Scott's 1933 analysis in 1975. My results as well as those of four co-authors were published in a research and development report. The reference to Scott's work is given in the following image:


There are 12 pairs of measurements in the Kay County pipeline data set. We need to file them. Use the QBasic icon in your wtt02 Course I program group. Open program pairfile.bas. Run the program, enter Kay as file name and 12 as the number of data pairs. Enter the first pair as   56.16, 39.25    and the remaining 11 pairs as you are prompted to do so. On exiting the program you will have made data file Kay.dat in directory wtt02.

We need to think about the data set before analyzing it by means of a linear regression program. Divide the largest area by the smallest. Find the ratio = 1647. Now divide the deepest maximum pit depth by the shallowest. The ratio = 8.15. Imagine a linear graph with depth on the y-axis and area on the x-axis. Plot the data points in your mind. You see a curve. Can you describe it?

G. N. Scott attached particular significance to the Kay County pipeline example because it represented maximum pit depths over the largest length of pipeline available to him and the line examined traversed "a great variety of soil, topographical and other conditions." See p. 206 in the above reference to Scott's report.



In 1995, I prepared the following files to accomplish Task 1. As a student of engineering, you have the opportunity to duplicate the results in order to satisfy yourself that you have met self-imposed performance-based training objectives.
  1. Kay.dat --- Measured corrosion pit depths and exposed areas
  2. pairfile.bas --- QBasic routine to prepare a data file
  3. plotdata.bas --- QBasic routine to plot normalized depth versus area
  4. linrgr.bas --- QBasic routine to solve G. N. Scott's equation and plot

In the United States, the most extensive studies of pitting corrosion were performed at the National Bureau of Standards. Congress approved funding for investigation of stray current electrolysis of buried pipelines in vicinities of street railways in 1910. Field and laboratory were carried out in the following eleven years. More general investigations were conducted from 1922 to 1955. By the early 1930's, the following two important empirical corrosion effects were known:

  • at a given time of exposure maximum pit depth depends on the amount of surface area inspected
  • maximum pit depth for a given surface area increases relatively rapidly in the first few years of exposure and increases less rapidly at longer exposure times
QBasic programs linrgr.bas, wybrgr01.bas, and wybrgr02.bas were prepared to help you accomplish performance-based training objectives. The following code fragment is given to remind you to let QBasic know where you have placed data file Kay.dat.

REM  ************************************************
REM *     Written by CorTech Training for WTT Course II     *
REM *     linrgr.bas     *
REM  ************************************************
'
CLS
PRINT "ENTER FILE NAME, EX. Kay";
INPUT FILE$
FILE$ = "C:\wtt02\" + FILE$ + ".DAT"

Task 2

Analyze the Kay County data by means of two Weibull curve fitting relations and compare with Scott's results.


CorTech Training prepared the following files for course participants:
  1. Kay.dat --- Measured corrosion pit depths and exposed areas
  2. wttrgr01.bas --- QBasic routine to solve the first Weibull equation, where maximum pit depth P is regarded as a function of exposed area
  3. wttrgr02.bas --- QBasic routine to solve the second Weibull equation, where exposed area is regarded as a function of maximum pit depth

Task 3

Apply Weibull relations to G. N. Scott's Mt. Auburn and Council Hill data and to Mahoning and Almeda-Lynchburg data obtained by other investigators.

In 1933, G. N. Scott published results of maximum corrosion pit depth analyses, performed while serving as API Research Associate at the US Bureau of Standards. His paper reveals his highly intuitive understanding of extreme value statistics. He was vitally interested in measurement and interpretation of maximum corrosion pit depth as a function of exposed pipeline surface area.

The following image reminds me of many good times spent at Lockheed's Rye Canyon Research and Development Center two decades ago. I had a wonderful opportunity to study distributions of corrosion pit depth data while working at Rye Canyon. I also think of the open pit mines I saw in Minnesota four decades ago. Steel was made from open pit mine oxides of iron. Many of the steel components made from the iron ore now are protected by crusts of rust, or iron oxides with compositions similar to the parent iron ore.

In Northern Minnesota open pit mines, hematite was a direct shipping ore. It was only necessary to scoop up relatively loosely consolidated red earth to ship by rail to the head of Lake Superior and by freighter to hungry open hearths at lower Great Lake ports. It was common for property owners and mining companies to contract for removal of all the high grade ore in the pits. Owners were paid for the amounts removed, even if local geometry of a pit made mining relatively costly. While male miners operated machines, female miners swept remnants of ore into baskets to ensure complete removal. The mine openings were indeed people-made, not just man made.

Scott was involved in measuring pit depths on two of the seven pipelines discussed in his 1933 report. The Mt. Auburn and Council Hill pipelines were 3 years old, the first was 8 in. diameter and the second 10 in. Mt. Auburn pipeline was treated with enamel, and Council Hill was painted thinly. Mt. Auburn was 1024 ft. long, and measurements were made at increments of 1 ft. Council Hill was 992 ft. long, and interval was 1 ft. It would be a very worthwhile project for a student to determine whether Scott's 1024 Mt. Auburn and 992 Council Hill pit depth measurements can be recovered from US Bureau of Standards archives.

Scott reported maximum corrosion pit depths for 11 exposed areas on each of the two pipelines. For example, he reported a maximum pit depth of 27.59 mils in the smallest of the 11 exposed areas, 2.258 sq. ft. for the Mt. Auburn case. If we assume the pipe had an internal diameter of 8 in. and a wall thickness of 0.125 in., then the outer diameter was 8.25 in. In that event, a length of pipe equal to 12 in. would have an external surface exposed area of 2.258 sq. ft. This interpretation does not make sense. The wall thickness seems insufficient. It is more likely that the wall thickness was greater than 0.3 in. Then 8 in. must have been the nominal innner diameter of a pipe with an actual inner diameter of 8.25 in. Scott's reported exposed areas must be internal surface areas. His sequences of exposed areas are binomial. Raise 2 to the power 10 and multiply by 2.258 to obtain 2312.192. The largest exposed area for the Mt. Auburn case was 2312.192 sq. ft., according to Scott. Maximum pit depth measured in 1024 ft. was 78.00 mils.

Course II includes QBasic programs linrgr.bas, wttrgr01.bas, and wttrgr02.bas. These programs were prepared to help you accomplish performance-based training objectives. The following code fragment is given to remind you to let QBasic know where you have placed data file Auburn.dat.

REM  ************************************************
REM *     Written by CorTech Training for WTT Course II     *
REM *     linrgr.bas     *
REM  ************************************************
'
CLS
PRINT "ENTER FILE NAME, EX. AUBURN";
INPUT FILE$
FILE$ = "C:\wtt02\" + FILE$ + ".DAT"

CorTech Training prepared the following files for course participants:
  1. auburn.dat, council.dat, mahoning.dat, almeda.dat --- Measured corrosion pit depths and exposed areas
  2. wttrgr01.bas --- QBasic routine to solve the first Weibull equation, where maximum pit depth Px is regarded as a function of exposed area
  3. wttrgr02.bas --- QBasic routine to solve the second Weibull equation, where exposed area is regarded as a function of maximum pit depth

In Task 3, course participants explore applications of two, two-parameter Weibull relations to four pipelines. The following normalized graphs summarize one set of results.

I urge serious students of material property behavior to reproduce the numerical and graphical results summarized in Task 4.

Task 4

Review G. G. Eldrege's 1957 interpretation of Bessemer steel data obtained by K. H. Logan(1945) and analyze the data by Weibull analyses.

In 1945, K. H. Logan presented results of tests performed on two sizes of Bessemer steel and wrought iron specimens buried in 38 types of soils for 12 years. Ref: K. H. Logan, Underground Corrosion, Circular of the Bureau of Standards C450, Bureau of Standards, US Department of Commerce, 1945. On 11/29/95, the author of this course received an e-mail message from Didier Crusset, ANDRA, France, concerning Circular 579, dated 1957, by M. Romanoff with the same title as Logan's Circular C450. In his message, Didier stated he is interested in clay soils with steel specimens and in analysis of general and pitting corrosion.

In March 1957, three papers were presented at a corrosion engineers' conference on application of extreme value statistics to interpretation of pit depth data. Papers by G. G. Eldredge and P. M. Aziz were published subsequently. A paper by G. N. Scott was presented but not published. It is probable that the three authors were influenced by a report by E. J. Gumbel: Statistical Theory of Extreme Values and Some Practical Applications, US Bureau of Standards, Applied Mathematics Series No. 33, 1954.

G. G. Eldredge applied Gumbel's double exponential function to interpretation of K. H. Logan's data. Eldredge used the equation

F(Px) = Exp(- Exp( - alpha(Px - mu))),

where Px is maximum pit depth in a specimen and alpha and mu are parameters of the distribution.

After Gumbel, Eldredge approximated F(Px) by means of the order statistic i /(N + 1), where i is the rank order of a specimen with maximum pit depth Px(i) and N equals the number of soil types, 38. Eldredge found values of alpha and mu by graphical techniques.

For this course there are two Bessemer steel specimen data files, bess66.dat and bess126.dat, each containing 38 maximum pit depths rank ordered from smallest to largest. File bess66.dat contains Logan's data for 66 sq.in. Bessemer steel specimens and bess126.dat the 126 sq. in. specimen data. Program gumrgr.bas was written for those course participants who actually perform suggested exercises. The program can help you determine parameters alpha and mu for the two data sets. The following two graphs show how well Gumbel's double exponential distribution applies to interpretation of Logan's data.


Weibull Trending Toolkit

Three executable programs in the Weibull Trending Toolkit find three parameters of the Weibull distribution by the method of moments. A program matches sample and Weibull mean, standard deviation, and skewness in the process of finding three Weibull parameters k, e, and v. The Weibull distribution for maximum pit depths Px may be expressed as

F(Px) = 1 - Exp(-((Px - e)/(v - e))^k), where

parameter e is the Weibull threshold value, v is the characteristic value, and k is the Weibull exponent. In present form, the Weibull Trending Toolkit applies only to cases where exponent k is greater than 1.0. When k is greater than 1.0, the distribution has a most frequent value. The most frequent value is called the mode. Program wtutor.exe was applied to data files bess66 and bess126 to obtain results shown below. Pit depths are in units of mils. Specimen areas were 66 sq. in. and 126 sq.in.


Course participants may use Weibull modal maximum pit depth values and the two specimen areas to write data file bessemer.dat. Two pairs of values may be analyzed by means of Task 1 program linrgr.bas to estimate parameters a and b in Scott's power law equation.


CorTech Training prepared the following files for participants:
  1. bess66.dat, bess126.dat, and bessemer.dat
  2. gumrgr.bas --- QBasic routine to find Gumbel distribution parameters
  3. wtt01.exe --- executable program to find three parameters of the Weibull distribution
Every trainer takes her or his own course. I took this course by preparing it. The first time was in 1975, the second in 1996. It takes a lot of detailed, individual effort to extract meanings from historic pit depth data sets. There is satisfaction in finding simple relations which can model the data. By understanding past material corrosion behavior we can better predict future behavior.

Task 5

Analyze K. H. Logan's 1945 wrought iron specimen data and compare with Bessemer steel specimen results.

In 1945, K. H. Logan presented results of tests performed on two sizes of Bessemer steel and wrought iron specimens buried in 38 types of soils for 12 years. Ref: K. H. Logan, Underground Corrosion, Circular of the Bureau of Standards C450, Bureau of Standards, US Department of Commerce, 1945.

were presented at a corrosion engineers' conference on application of extreme value statistics to interpretation of pit depth data. Papers by G. G. Eldredge and P. M. Aziz were published subsequently. A paper by G. N. Scott was presented but not published. It is probable that the three authors were influenced by a report by E. J. Gumbel: Statistical Theory of Extreme Values and Some Practical Applications, US Bureau of Standards, Applied Mathematics Series No. 33, 1954.

Eldredge applied Gumbel's double exponential function to interpretation of K. H. Logan's data. Eldredge used the equation F(Px) = Exp(-(-alpha(Px - mu))), where Px is maximum pit depth in a specimen and alpha and mu are parameters of the distribution.
After Gumbel, Eldredge approximated F(Px) by means of the order statistic i/(N + 1), where i is the rank order of a specimen with maximum pit depth Px(i) and N equals the number of soil types, 38. Eldredge found values of alpha and mu by graphical techniques.

For this course there are two wrought iron specimen data files, wfe66.dat and wfe126.dat, each containing 38 maximum pit depths rank ordered from smallest to largest. File wfe66.dat contains Logan's data for 66 sq.in. wrought iron specimens and wfe126.dat the 126 sq. in. specimen data. Program gumrgr.bas was written for course participants to determine parameters alpha and mu for the two data sets. The following two graphs show how well Gumbel's double exponential distribution applies to interpretation of Logan's data.


Weibull Trending Toolkit

Three executable programs in the Weibull Trending Toolkit find three parameters of the Weibull distribution by the method of moments. A program matches sample and Weibull mean, standard deviation, and skewness in the process of finding three Weibull parameters k, e, and v. The Weibull distribution for maximum pit depths Px may be expressed as

F(Px) = 1 - Exp(-((Px - e)/(v - e))^k), where

parameter e is the Weibull threshold value, v is the characteristic value, and k is the Weibull exponent. In present form, the Weibull Trending Toolkit applies only to cases where exponent k is greater than 1.0. When k is greater than 1.0, the distribution has a most frequent value. The most frequent value is called the mode. Program wtutor.exe was applied to data files wfe66 and wfe126 to obtain results shown below. Pit depths are in units of mils. Specimen areas were 66 sq. in. and 126 sq.in.

Weibull Numerical and Graphical Results


In Task 4, Weibull modal maximum pit depth values and the two specimen areas were used to write data file bessemer.dat. Two pairs of values were analyzed by means of Task 1 program linrgr.bas to estimate parameters a and b in Scott's power law equation. As can be seen above, the Weibull modal estimate for 66 sq. in. wrought iron specimens is 51.81 mils and for 126 sq. in. specimens the estimate is 44.50 mils. Scott did not anticipate a decrease in modal depth with increase in specimen area. However, the Gumbel analyses for wrought iron specimens yielded different modal estimates. The parameter mu in Gumbel's equation is an estimate of modal value. For 66 sq. in. specimens the Gumbel mode equals 50.29696 mils and the 126 sq. in. specimen mode equals 61.06899 mils. MS-DOS editor Edit was used to write file wrought.dat and QBasic was used to run program linrgr.bas to obtain the following result.


Modeling Weibull Behavior

All four Weibull probability density distributions are positively skewed. The distributions have relatively long tails toward deeper maximum pit depths. The Weibull exponent k for 126 sq. in. wrought iron specimens seems to be significantly lower than the exponents for the other three cases. There is a possibility that surficial material properties of 126 sq. in. wrought iron specimens were different than those of the 66 sq. in. specimens. Seemingly exceptional numerical results do not necessarily imply that experimental results are not in order.

The average of three Weibull exponents equals 2.035. The average of four threshold values equals 20.37 mils and the average of four characteristic values equals 70.37 mils. All of Logan's maximum pit depth measurements for two iron based alloys and two specimen sizes may be modelled by means of the Weibull cumulative distribution

F(Px) = 1.0 - Exp(-((Px - 20)/(70 - 50)^2)

If, for example, it is desired to model the upper control limit at 100 x F(Px) = 98.865% then

UCL = 20 + 50(-LOG(1 - 0.99865))^(1/2) = 149 mils.

A more conservative model could be chosen by letting k = 2, v = 70, and e = 0. In that event the model is a Rayleigh distribution

F(Px) = 1.0 - Exp(-(Px/70)^2)

For this model, 99.865% of the maximum pit depths would be predicted to be less 180 mils.


Comparison Of Weibull And Gumbel Results For Bessemer Steel And Wrought Iron Specimens

In his 1957 review of K. H. Logan's 1945 data, G. G. Eldredge decided the effect of difference in specimen size could be accounted for on extreme value probability graphs by shifting a linear fitting relation by an amount equal to the logarithm of the specimen area ratio. He concluded that differences in maximum pit depths for the Bessemer steel specimens could be accounted for in terms of an area effect, but pit depth differences for the wrought iron specimens could not. WTT Course I participants are encouraged to discuss the Bessemer steel and wrought iron specimen results. Task 6 is the first 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 6

Review P. M. Aziz's 1956 experimental procedures and data for Al alloy specimens immersed in Kingston tap water

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.

Aziz described experimental procedure on pp. 38-39:

"Strings of 2S-O alloy sheet, ten panels to string, were prepared by threading 5 inch x 2 inch coupons on glass hangers through 3/8 inch holes centered 3/4 inch from the top of the panels. These were vapor degreased using trichloroethylene, etched for two minutes in 85 percent phosphoric acid at 70 deg. C and washed for half an hour in cold running water; they were then immersed in a 300 gallon tank of Kingston tap water, held at 25 deg. C. The water flowed continuously and slowly and changed every 24 hours.

The immersion periods ranged from one week to one year.

At the end of each time period a string was removed from the tank, cleaned of corrosion products and the depth of all the pits on both sides of each of the ten panels measured by the calibrated microscope technique. These data were grouped over the pit depth ranges 0-99, 100-199 microns etc., and frequency curves were constructed for the various time periods."

Aziz interpreted his frequency distribution curves as having J-shaped and bell-shaped portions. Aziz estimated modal pit depths for the bell-shaped portions and compared modes a as a function of exposure time. He found the mode apparently stabilized after exposure for two months. He noted the maximum pit depth in a data set for exposure times greater than two months continued to increase even though the mode had stabilized. On p. 45 of his paper he concluded:

  1. When freshly etched aluminum is immersed in an aggressive water a relatively large number of pits initiate and develop to the stage where they become visible.
  2. Within two weeks more than half of the pits that start stifle themselves and become inactive. A pit depth distribution curve for this group takes the form of the letter J with many very shallow pits and a few that go somewhat deeper.
  3. The remaining pits continue to propagate for about two months. A pit depth distribution curve for this group has the familiar bell shape. During this period all pits propagate at about the same rate and the "bell" type curve retains its shape and moves in the direction of increasing depth.
  4. After about two months the majority of these pits become stifled and only a small number of the deeper ones continue to propagate. Thus the bell shaped curve remains stationary but develops a tail that lengthens with time.

It is probable that P. M. Aziz was influenced by a report by E. J. Gumbel: Statistical Theory of Extreme Values and Some Practical Applications, US Bureau of Standards, Applied Mathematics Series No. 33, 1954.

Aziz applied Gumbel's double exponential function to interpret the aluminum specimen data. He used the equation
F(Px) = Exp(-(-alpha(Px - mu))),
where P is maximum pit depth in a specimen and alpha and mu are parameters of the distribution.

After Gumbel, Aziz approximated F(Px) by means of the order statistic i/(N + 1), where i is the rank order of a specimen with maximum pit depth Px(i) and N equals the number of number of specimens in a string, that is ten specimens. He found values of alpha and mu by graphical techniques.

Those course participants who learn by doing are encouraged to discuss Aziz's description of shallow pits which stifle themselves and deeper pits which continue to grow in a stable manner. The sketches at page top were included to assist discussion.


Data Obtained By P. M. Aziz, 1956


Task 7 is the second 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 7

Interpret Aziz's pit depth frequency distributions by Weibull analysis.

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.

  • Download the completely free self-extracting Weibull Trending Toolkit wtt01.exe from an archive.
  • Use QBasic program makefile.bas to make data file aziz01.dat for the specimens exposed for one week. Convert pit depth units from microns to mils. Run program wrisk.exe, which is included it archive wtt02.exe.

  • Study three-parameter Weibull analysis results.

  • With the one year data in mind, select a graph upper limit of 20 mils. Study the resulting Weibull curves.

Task 8 is the third 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 8

Interpret Aziz's Alcan 55S-T maximum pit depth data for either 9 or 10 specimens at each of six exposure times.

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.


  • 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 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

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.


Numerical Results for 19 Maximum Pit Depths


Numerical Results for 39 Maximum Pit Depths


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.


  • 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
    • Notice the Weibull shape parameter K values for files task901, task902, and task903 have much less variation than the K values for data files studied in Task 8.
    • Average the three K values found here to obtain K = 2.75439.
    • After some thought, decide to use E and V values for files task901 and task903 to model time dependence. Let E = A * T + B and V = C * T + D, where T is exposure time in years and A, B, C, and D are constants.
    • The Weibull Trending Toolkit includes a program called WDESIGN. It uses an order statistic to estimate N probabilities. A user supplies trial Weibull exponent K, threshold E and an estimate of the median random variable. The program estimates N values of the variate and finds Weibull K, E, and V to match design sample and Weibull mean, standard deviation, and skewness, values.
    • The instructor used WDESIGN as the basis for creating new program DESIGN.
    • DESIGN has an exponent K which is independent of exposure time, and parameters E and V which are linearly dependent on time exposure. When a user runs DESIGN, she/he is prompted to input a file name, number of measurements N being modeled, and exposure time.
    • DESIGN uses an order statistic base on N, the constant K, and calculated E and V to generate N design sample maximum pit depths.
    • DESIGN matches sample and Weibull mean, standard deviation, and skewness values to find K, E, and V for the N design measurements.

    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
    
    
    nul no images

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