References:

1. The New Weibull Handbook, 4th ed. This text by the author should be in your university library. If it is not, ask your university librarian to write to me to request a copy that I will donate. For an overview, read chapter one. The material in this paper is covered in Chapters 2-4.
2. A demo of the SuperSMITH Software may be downloaded free from either Paul Barringer or Wes Fulton, who produces the software. Both Paul and Wes are good friends and associates in the Weibull analysis research.

A "Panic" Problem: A student from one of our Weibull Workshops called me that he was flying to see me at my home in the Palm Beaches with a problem that required my immediate attention. Heavy trucks all over the country were suddenly failing their steering. As I was on the way to the airport to lecture in Kansas, I had to say no. He asked if he could meet me in the motel that night and I agreed as this was a problem requiring immediate action. Safety, high losses, and adverse "pr" had corporate management concerned. There were 10,010 units in-service. His management wanted the following questions answered as soon as possible:

1. How many more failures will there be? When?
2. What will this problem cost the company?
3. What corrective action can be taken?

The Weibull Plot: To answer these questions the first step is to make a Weibull plot. In this first part of this paper I will skip the how-to-do-it explanation to show the results of the analysis. Later, I will provide step-by-step instruction on how to so the analysis. The data is simulated as the real data was proprietary but it is very similar.

Here is the Weibull plot. The horizontal scale is the aging parameter that produces failures, in this case, mileage. It is a logarithmic scale. The vertical scale shows the cumulative probability of failure. Walloddi Weibull gave a name for the relationship between these two scales. He called the coordinates "B lives." For example B1 life in this plot is 18,000 miles, i.e., we expect one per cent of the trucks to fail the steering by 18,000 miles or less. Ten per cent will fail by 35,000 miles, the B10 life. As truck life is often quoted in millions of miles, a B1 life of 18,000 miles for this failure mode is unacceptable. Some corrective action must be taken to reduce the number of future failures. The world of engineering is below B5 life, with B1 or B0.1 being of much more interest. Fortunately, Weibull probability paper is expanded in the lower area, magnifying the engineering area of interest.

The ten points plotted are failures. The trucks that have not had this failure are called suspensions or censored units. Although their mileage is not plotted it influences the plot positions of the failure points and the nominal line. The goodness of fit is related to how close the points follow a straight line. This is true for all probability papers, Weibull, normal, log normal, etc. Your eyeball will be good for estimating the goodness of fit when you have some experience. In the mean time we can use the correlation coefficient square called the coefficient of determination, as a measure of goodness of fit. In this case it is 0.975, very close to 1.0 which is a perfect fit. Approximately 97.5% of the variation in the data is explained by the Weibull fit.

The slope beta of 3.26 indicates a wearout failure mode. Slopes less than one indicate infant mortality, often quality problems or inadequate environmental stress screening. Slopes near one indicate the exponential distribution is appropriate and that the failures are independent of age. Old units have the same reliability as new units. Slopes greater than one indicate wearout. For example ball bearings have a beta of two, v-belts indicate 2.5, low cycle fatigue cracking ranges from 2.5 to 4.0, stress corrosion usually is quite steep, 4-6, etc. In other words beta tells us something about the physics of failure and is most helpful in determining root cause analysis. In this case the steering links are wearing out prematurely.

Eta is called the characteristic life and is the same as B63.2 life. That is, 63.2% of the units will fail at the time the units age to eta. It is related to the mean-time-to-failure (MTTF) and for all wearout failures eta is slightly greater than MTTF.

Failure Forecast: In the next 12 months we predict 576 failures with an upper 90% bound of 872 failures based on 2000 miles per month usage and no replacement or renewal of the failed part. [With replacement of the failed part with the an identical part, the forecast would be higher because the replacements will fail.] The next failure is expected in six days. Here is a plot of our failure forecast, future months on the horizontal scale and cumulative failures on the vertical scale:

Note that we have added an upper one-sided predictive bound to indicate to management just how bad things may get, worst case conditions.

What will this problem cost? First year loss is predicted to be just over a million dollars but might be as high as $1.6M.

What is the best corrective action? Engineering says the root cause is the recent addition of power steering kits overstressing the links. It will take 2 years to redesign, test and produce 10,000 stronger links for retrofit.

An immediate forced retrofit would be too painful and embarrassing as we do not have enough spare steering links to do it. Many trucks would be down for months. Opportunistic maintenance may be the answer. When the truck is down for other reasons we would change the links if the mileage was sufficiently high. This is called opportunistic maintenance.

The optimal replacement interval for minimum cost is at 22,000 miles, based on the planned replacement cost of $95 and the unplanned failure cost of $1850.

A convenience retrofit at the 20,000 mile oil change would be close to optimal, less painful for our customers, and it would give us time to produce new links and solve the problem.

Solution: Retrofit at 20,000 miles.

Risk Reduction: 12 months from 577 to 187 failures, 24 months from 3470 to 397 failures.

2 Year Cost Analysis: No correction action = 3470 x $1850= $6.4M
20K replace = 397 x $1850 + (10010 - 397) x $95= $1.6M..($4.8M Cost Avoidance)
The sooner the new links are available the sooner the problem is fixed. 3 shift overtime is recommended.

Liabilities? Engineering predicts that the links fail at low speed, usually in reverse in a turn. So far this has been true. No one has been injured to date. Our liability estimates are based on these assumed low cost failures. If the assumption is wrong and the links fail at high speed, liabilities would be increased. Installation of the redesigned replacement links will end the problem. The redesign should be initiated with priority and 3 shift overtime. There is some indication that only one type of power steering kit produces the failures. The power steering vendor and NHSTA should be contacted for a coordinated response to this problem.

How Did I Do The Analysis? Here we will show how the analysis was generated and provide comments and explanation. It is assumed that the reader has the demo SuperSMITH software. All references are to The New Weibull Handbook, 4th edition. Questions are welcome via email.

References: In The New Weibull Handbook (NWH) read Chapter 4 for all the mathematics and case studies. Download the demo SuperSMITH software.

Analysis From WinSMITH Weibull (WSW), click on the folder icon or the file menu to bring in the data file "18WHEELR." You will find the file in the "Smithdat" folder provided with your download of the demo software. Move the pointer to the right and you will see the Weibull plot.

Select the Abernethy Risk Icon. Set Usage at 2000 miles per month. No renewal. Select the green check to see the table of results.

Check the Present Risk Quantity to determine if there is a batch problem. As the present risk, 13.75, is close to the observed number of failures, 10, there is little or no indication of a batch problem. [To confirm "no batch", redo the Abernethy risk with confidence set to lower 90% one-sided. If the present risk is still above the observed number of failures, we are 90% confident that there is a batch problem.] If there is a batch problem, you have the wrong data set and may over estimate the future failures, the Abernethy Risk. In this case the priority is to identify the batch, the correct data set. Then repeat the Abernethy Risk with the correct data set. If this is impossible, there is statistical help. The ratio of the observed number of failures to the Now Risk is approximately the proportion of the data set that constitutes the batch. This is an approximation because it assumes the batch units are randomly distributed in the data set and they are more likely clustered together. If we enter this proportion in Option Z of the risk menu it will approximate the correct risk.

[See Handbook Section 8.11. for batch problems]

To see a plot of the risk click on the bottom icon that shows the little plot or save the table as a WinSMITH file to be viewed in WSV later using the icon above the plot icon.

Failure Forecast Plot: Return to the WSW Abernethy Risk menu and select confidence, 90%, upper. Select the table and the bottom icon again. In WSV click on the data set title and copy the set. Return to the open WSV first plot of the risk and paste (pastebottle) this data into set 2. Note the scale the risk plot goes to 60 months while the plot in the overview is cut off at 25 months. Use the magnifying glass icon, Zoom, to change the x scale to 0-25 months.

To obtain cost forecasts return to the WSW Abernethy risk menu and select cost and enter $1000 as the cost of a warranty claim.

The optimal replacement icon is selected to determine the optimal mileage to replace the steering link. Enter the planned replacement cost as $95 and the unplanned cost as $1850. Section 4.8. To obtain the plot of the table, click on the bottom icon. Labels may be added to all plots in WSW and WSV using the label icon.

To obtain the risk plot return to the WSW risk menu and enter 22,000 miles as the Planned Replacement Interval. Overplot the original risk by copying and pasting the data set on the other WSV plot.

There are many new options for "Abernethy Risk Forecasting" described in Chapter 4:

• Usage may be a constant rate per month, seasonal by month or randomly sampled from a distribution.
• Production of new units added to the fleet may be a constant rate per month, seasonal by month or randomly sampled from a distribution. The Automotive Industry Action Group standard truck usage histograms are included in the software.
• Predictive intervals (similar to confidence intervals) may be added to the forecast and plot.
• Predictions may be for events like failures, warranty claims, or for costs. Similarly cost rates may be forecast from failure rates.
• Failed parts may be replaced (renewal) or not replaced.
• Replacement parts may have a different Weibull life.
• To reduce the number of forecast failures, parts replacement intervals may be evaluated.

Please send any questions to me via email.