Reliability Solutions for the Aerospace Industry
Powerful solutions for high-performing components

Reliability Solutions for the Airline Industry
Reliability solutions that make your profits soar

Reliability Solutions for the Automotive Industry
Reliability from R&D to the road

Reliability Solutions for the Chemical and Process Industry
Build processes with reliable results

Reliability Solutions for the Defense Industry
A strong reliability program breeds strong defense

Reliability Solutions for the Electronics and Appliances Industry
Reliable components ensure reliable business

Reliability Solutions for the Energy Industry
Let reliability light the way

Reliability Solutions for the IT Hardware
Reliability keeps the world connected

Reliability Solutions for the Medical and Healthcare Industry
Reliability for life

Reliability Solutions for Military Organizations
Dependability begins with reliability

Reliability Solutions for the Oil and Gas Industry
Streamline processes along the pipeline

Reliability Solutions for the Semiconductor Industry
Big results for small materials

Reliability Solutions for the Telecommunications Industry
Reliable solutions to better help you connect the world

Reliability Solutions for the Transportation Industry
Results that get you where you need to be

Reliability Solutions for the Trucks and Heavy Equipment
Reliability solutions that do the heavy lifting

Ordering Software from ReliaSoft
Choose the software license and configuration that's best fit for your organization

The Synthesis Platform
Integration to Empower the Reliability Organization

ALTA — Accelerated Life Testing Data Analysis Software
Accelerated life tests... quantifiable resultsTM

RENO — Simulation Software for Probabilistic Event and Risk Analysis
If you can flowchart it, you can simulate itTM

RBI — Risk Based Inspection (RBI) Software
Full-featured RCM with risk based inspection

ReliaSoft Offers Unparalleled Technical Support and Documentation
Connect with ReliaSoft's acclaimed technical support team via phone, email and live chat at locations throughout the world

The Synthesis Enterprise Portal
Open up the Synthesis Platform, and your work in the platform, to your whole organization with this web-based Portal

DOE++ — Software Tool for Experiment Design and Analysis (DOE)
DOE software designed with reliability in mindTM

Lambda Predict — Standards Based Reliability Prediction Software
MIL-217 * Bellcore/Telcordia * FIDES * NSWC Mechanical

MPC — MSG-3 Aircraft Maintenance Program Creation Software
An MSG-3 compliant maintenance program creator

Register and Activate Software from ReliaSoft
Start using ReliaSoft's helpful applications by registering and activating your software today!

The Synthesis API
Application Programming Interfaces (APIs) for the Synthesis Platform

RGA — Reliability Growth Analysis and Repairable System Analysis
Reliability growth and repairable system analysisTM

Xfmea — FMEA and FMECA Software Tool
Expert support for all types of FMEA and FMECATM

XFRACAS — Web-Based Failure Reporting Software (FRACAS System)
Highly configurable, web-based FRACAS

Download Free Software Demos and Free Service Releases
Get a free trial of the reliability engineering software trusted by thousands of companies, or install the latest update

Weibull++ — Life Data Analysis (Weibull Analysis) Software
The standard for reliability life data analysisTM

BlockSim — System Reliability and Maintainability Analysis Software
The ultimate system visualization and analysis toolTM

RCM++ — Reliability Centered Maintenance (RCM) Software
Putting the reliability back into reliability centered maintenanceTM

Orion eAPI — Enterprise System Focused on Asset Performance Management
Reliability Business IntelligenceTM

Reliability Education Course List
Peruse the variety of introductory and advanced courses offered for reliability practitioners of all experience levels

The New Master the Subject, Master the Tools!SM Course Lineup
Choose the reliability engineering education track that best fits your needs: Product Development or Asset Management

Worldwide Public Seminar Calendar
Find out which courses are being taught near you, and register today!

Why Attend ReliaSoft's Reliability Training Seminars?
Learn about ReliaSoft's commitment to reliability engineering education, as well as the benefits of attending

Seminar Registration
Register for a course today, and learn about seminar options and other details

Seminar Instructors
Meet the esteemed group of professionals that bring ReliaSoft's training seminars to life

Reliability Training Center (RTC)
Take a glimpse inside ReliaSoft's newly built, multimillion dollar state-of-the-art training facility

Frequently Asked Questions about ReliaSoft's Seminars
Get answers to the most common questions about the finer details of attending ReliaSoft's courses

Available Reliability Consulting Services
ReliaSoft's expert consulting team can accommodate requests of any size or complexity in a wide variety of areas

Customers that Benefited from ReliaSoft's Consulting Services
View some of the organizations that took advantage of ReliaSoft's on-site consulting and education services

ReliaSoft's Expert Reliability Consultants
Meet the esteemed group of professionals that organizations worldwide have trusted for their consulting needs

Frequently Asked Questions about ReliaSoft's Consulting Services
Get answers to the most common questions about the finer details of ReliaSoft's consulting services

About ReliaSoft Corporation
Learn more about ReliaSoft and its commitment to empowering the reliability professional

ReliaSoft Corporation Online Store
Purchase new software licenses, upgrades or resource books, and register for training seminars or ARS events

Companies that Rely on ReliaSoft's Reliability Software and Services
See some of the hundreds of companies that look to ReliaSoft for the tools necessary to power their reliability programs

Press Releases
Read about the latest exciting ReliaSoft product releases and announcements

Print-Ready Publications and Media Links
Browse for publications and videos relating to ReliaSoft's products and services

Career Opportunities at ReliaSoft Corporation
Want to work for the leading provider of reliability engineering software tools and services? Check for openings today!

How to Contact ReliaSoft Corporation
ReliaSoft offers unparalleled phone, e-mail and live chat support at locations throughout the world

ReliaSoft Customer Testimonials
"Amazing," "User Friendly," "Cutting Edge..." See what people are saying about ReliaSoft's software!

Weibull.com
The most complete website devoted entirely to the topic of reliability engineering, theory, data analysis and modeling

ReliaSoft.tv
A multimedia resource portal for both users of our software tools as well as all reliability engineering professionals

Synthesis Applications eDocuments & ePublications Library
Access useful support documents for ReliaSoft's Synthesis Applications, including user's guides and reference books

ReliaWiki
A resource portal and a wiki for professionals in reliability engineering and related fields

Certified Reliability Professional (CRP)
Distinguishing reliability engineers who have gained and successfully demonstrated unquestionable expertise in the field

International Applied Reliability Symposium (ARS)
Sharing applications, success stories and lessons learned in reliability and maintainability engineering

Reliability Discussion Forum
Connect, discuss and share with the reliability engineering community

Reliability Edge Newsletter
In-depth articles on reliability engineering theories and principles, as well as ReliaSoft updates

Reliability HotWire
Your source of information and tips on how to improve your reliability practices

Volume 2, Issue 2

Reliability Edge Home

Classic Case Studies in Reliability Analysis: The Weibull Distribution

It has been exactly 50 years since Professor Waloddi Weibull published his groundbreaking paper, "A Statistical Distribution Function of Wide Applicability," in the Journal of Applied Mechanics. Dr. Bob Abernethy has called this paper "the hallmark paper of Weibull analysis." Although Weibull had published his initial work in 1939 in the Proceedings of the Royal Swedish Institute for Engineering Research, his 1951 paper was the one that would expose his work to a wide audience.

The 1951 paper contained seven case studies. Three of the case studies were modeled using a three parameter Weibull distribution and four were modeled using a two population mixed Weibull. It is interesting that only one of the seven examples involved life data; the others involved the distribution of physical characteristics, such as strength and size. We would like to revisit the first of these examples using modern computer tools.

Weibull's Strength of Bofors Steel Example
Professor Weibull was Scientific Advisor to AB Bofors, the major Swedish armaments company. The first example in his paper concerned the yield strength of Bofors steel, measured in kg/mm2. Professor Weibull had the advantage of working with very large sample sizes; in this case he had 388 observations. Professor Weibull made an unusual choice of scale for the strength values. He modeled these strengths using the distribution we now call the three-parameter Weibull distribution.

As modern day analysts, we enter this data set into Weibull++ 6 and our first point of interest is the goodness of fit of the three-parameter Weibull distribution. We can use the Distribution Wizard to determine that the three-parameter Weibull distribution is an excellent model for this data, dominating the other possible choices handily. Professor Weibull used a cumbersome chi-squared test, computed by hand, to convince his readers of the goodness of fit of the distribution.

After confirming the applicability of the three-parameter Weibull distribution for the data, we fit the recommended model and obtain results very similar to those found by Professor Weibull. The small differences can be attributed to our modern use of median ranks in the calculations.

It is interesting to examine the graphs used by Professor Weibull to present his results. In Figure 1, we see an original graph from Weibull's 1951 paper. The vertical axis uses the ln(ln(1/(1-p)) transformation and the horizontal axis uses the ln(x) transformation. This agrees with modern practice, except that we now re-label those scales in the original units. This makes it much easier to read the graph.

Figure 1: Prof. Weibull's probability plot for Bofors steel

Note that Professor Weibull has plotted the translated values (strength minus the threshold parameter) and that the data is remarkably linear, indicating an excellent fit. Also note how difficult it would be to extract useful information, such as the median yield strength, from this graph.

The Weibull++ 6 plot of the Bofors steel data is presented in Figure 2. You can see that it is much easier to read the Weibull++ 6 graph than the original graph from Professor Weibull's paper. The essentials (choice of scale, estimation of parameters) are the same, but the user interface is greatly simplified. Both the translated data (on the left, represented by red circles) and the original data (on the right, blue triangles) are presented in the Weibull++ 6 plot. The median yield strength can be obtained from the plot and is 45.45 kg/mm2.

Professor Weibull did not present an analysis of the fitted model in his paper. Using Weibull++ 6, we are able to quickly learn that the median yield strength is 45.45, the mean yield strength is 45.52 and that only 0.68% of the units in the population are expected to have a yield strength of 40 or lower. The percent failing below a yield strength of 40 can be easily calculated using the Weibull++ QCP, also shown in Figure 2.

Figure 2: Weibull++ probability plot and QCP results

It is unpleasant to imagine how long it took Professor Weibull, working by hand, to generate Figure 1. It was but the work of a few minutes using the Weibull++ software, and most of that was data entry.

 

 

ReliaSoft.com Footer

Copyright © 1992 - ReliaSoft Corporation. All Rights Reserved.
Privacy Statement | Terms of Use | Site Map | Contact | About Us    ReliaSoft on Facebook  ReliaSoft on Twitter  ReliaSoft on LinkedIn  ReliaSoft on Google+