Loads vary, strengths vary, and reliability usually declines for mechanical systems, electronic systems, and electrical systems.  The cause of failures is a load-strength interference problem frequently describing problems which go bump in the middle of the night.

Bumps in the night occur when loads are higher than strengths, or strengths are lower than the loads.  So it might be a good idea to work out these details with facts rather than praying for help from the Lord to cover our ignorance.

Things don’t fail on the basis of averages (assuming average loads are widely separated from average strengths), parts fail on high loads/weak strengths.  To get your thinking adjusted to this concept, people don’t usually drown in average depths in a river, they drown in deep water!

Seldom is the load fixed.  When loads vary to the low side everything is failure free.  When loads vary to the high side, failures occur and reliability is lost.

Seldom is the strength fixed.  When strength varies to the low side, often because of lack of homogenous substances, failures occur and thus reliability is lost. When strengths vary to the high side everything is failures free.

The condition of unreliability is described in Figure 1.  Loads and strengths interfere in the overlap area and this is the area of concern for failures.

 Figure 1—Load-Strength Interference The overlap of load-strength in Figure 1 is not literally the calculated area.  It’s a joint probability of occurrence problem or roughly ½ of the literal area.  The joint probabilities are described by integral calculus for reliability.  If large safety margins are used, the probability of incurring failure from load-strength interference is usually very low when both loads and strengths are well known.  If low safety margins are used, ignorance abounds for of load distributions, or ignorance of strength distributions, it’s another sad story.  Failures occur where loads/strengths overlap.

Patrick D. T. O’Connor’s book (see Practical Reliability Engineering, by, John Wiley Chapter 4 Load-strength Interference points out where loading roughness is low (i.e. small standard deviation of loads) and strengths are well behaved (i.e., small standard deviations of strengths) in addition to strengths displaced widely to the right of the loads you can achieve intrinsic reliability.  Where the probability of failure is low large safety margins of 3-5 are used.  Whereas if the load curve is skewed to the right and the strength curve is skewed to the left, the two distributions begin to interfere and reliability declines.

In short, you need to know the load curves and the strength curves and keep them widely separated to achieve high reliability by avoiding load-strength interference.  Safety margins of 3-5 are suitable for pressure vessels but not airplanes (if airplanes have too much weight they can’t fly)!  Keep in mind the class of equipment you’re designing and maintaining.  Use appropriate strategies to achieve reliability by limiting loads for a given strength.

Component reliability can be determined as the probability of load being less than the strength for all possible values of strength (see  Reliability-Based Design, by S. S. Rao, McGraw-Hill, Inc., 1992, ISBN 0-07-051192-6, Chapter 8: Strength Based Reliability and Interference Theory, page 235-273).  Rao’s equations of importance are: This is equivalent to Rao’s Eq. 8.9

Where fS(s) is the probability density function of the strength, and fL(l) is the probability density function of the load.  FL(s) is the cumulative distribution function of the load in units of the strength.  The reliability statement is a statement of success.

The statement of unreliability UR = 1- R = pof is a statement of probability of failure.

You have three obvious ways to solve this complicated and convoluted calculus problem:
1) Use Mathcad to solve the integral.  See figure 7 for an example.
2) Use SuperSMITH Weibull to solve the problem using Monte Carlo simulation--
(download the demonstration program to solve the problem, click on the calculator
icon, click on load-strength interference and input the statistical data).  Monte
Carlo simulation methodology is illustrated below for the Excel simulation.
3) Use Excel to solve the problem (download a Monte Carlo simulation).  The
simulation draws a random load and a random strength from the described
distributions for 5 different statistical distributions for loads and 5 different
statistical distributions for strength.  If the strength is greater than the load you
have success.  If the random load exceeds the random strength, you have a failure.
The Monte Carlo reliability calculation is R = (successes)/(successes + failures).
With the Excel simulation file, you can see the equations used for the probability density function and cumulative distribution function.

How do you find the correct statistical distributions to use for the load analysis?
1) Measure and record the loads over time.  Treat the data as samples.  Construct a probability distribution in SuperSMITH Weibull.  Use good common sense and good engineering judgment to select the appropriate distribution.  The distribution will allow you to predict loads above/below the actual data recorded when you treat the data as a sample.
2) Measure the strengths for many samples as described above for the loads.  If you have many data, you may find strength data displays a failure-free zone.  The failure-free strength phenomena occur with offset of the origin of the distribution.  This is described in The New Weibull Handbook as a t0 shift for a 3-parameter distribution.  Dr. Abernethy in The New Weibull Handbook sets four requirements for use of a 3-parameter distribution:
1.  Must have at least 21 data points for a valid analysis
2.  Raw data plotted on a 2-parameter Weibull probability plot will show a
concave downward appearance.
3.  The goodness of fit criteria (i.e., R2 or PVE%) must show substantial
improvement with use of a 3-parameter distribution compared to the
2-parameter curve fit.
4.  A physical explanation of the reason for shifting the origin of the
distribution must be obvious (getting a better curve fit is not a valid
physical explanation).
For example, if the yield strength of the
steel grade is 110,000 psi, then no steel is released by the steel mill
if strength is less than 110,000 psi.  Thus yield strengths when
plotted on a 2-parameter probability plot display a concave downward
appearance.  A straight line occurs on a 3-parameter plot when
110,000 psi is subtracted from the raw data to correct concavity.  Be
careful with t0 data as the numbers can be misleading unless all details
are kept in the datum as recorded.
By the way, many data collected often are recorded in convenient units which lack sufficient decimals.  Convenient units often results in stacks of data on a probability plot.  This requires the use of special regression techniques (such as the inspection method) to achieve the correct statistics and the method can be selected in SuperSMITH Weibull under the methods icon.

Loads and strengths are often described by the following common distributions:
1.  Normal (Gaussian statistics of the bell shaped curve generally for error data).
2.  Weibull (Weak links in the chain of failures).
3.  Lognormal (Events accelerate with many small data and some large data).
4.  Gumbel smaller distribution (Where small data are of concern and recorded
such as thin remaining wall thickness for tubing that is corroded).
5.  Gumbel upper distribution (Where large data are of concern and such as
flood data recorded for greatest flood height each year).
Other distributions can be envisioned and used when appropriate but these mentioned distributions will cover most situations.  Use common sense and good engineering judgment in selecting the statistical distributions.  Make sure the distributions display reasonable graphics for comprehension by the engineering community.  The comprehension criterion for engineers is simple:  No graphics—No comprehension!

Equations for each distribution are shown below in Figure 2 along with equations for use with Excel.

The PDF (probability density function) has an area under the curve of unity.  The PDF shows the shape of the curve you would get if you made a tally sheet of occurrences on the Y-axis versus the unit of measure on the X-axis.

The CDF (cumulative distribution frequency) integrates the area under the PDF (which is then subtracted from unity to predict the % of the population that will occur on the Y-axis versus the unit of measure on the X-axis.

 Figure 2: PDF and CDF Equations Sometimes the problems are difficult to solve in closed form solutions (where a failure free interval exists).  Others are difficult to solve where the probability of failure is very low and a very large number of iterations must be run to get an answer that is not zero.

Consider the gust loads in Figure 3 where the X-axis is give in g-loads.  The data recorded was the maximum positive g-load from each of 23 flights.  The plot as made in SuperSMITH Weibull software.  The Gumbel upper distribution is used because high g-loads are very important for airplanes. (Notice the results from hand-made plots from 1954 are slightly different results than when computers are used.

 Figure 3: Gust Loads Described With A Gumbel Upper Distribution The PDF curve of the load is described in Figure 4.  The data was generated from SuperSMITH Weibull and plotted in SuperSMITH Visual.  Notice the long tail to the right with this actual data from a 1954 NACA document.

 Figure 4: Gust Loads As A PDF Strengths were obtained and plotted in a Weibull plot in Figure 5.  Again, note the X-axis is shown in g-loads and the failure free interval is 2.939 g-loads.

 Figure 5: 3-Parameter Weibull Strength Plot This strength curves is selected on purpose because of the calculation difficulties.

The PDF for Figure 5 is shown in Figure 6.

 Figure 6: Weibull 3-Parameter PDF The calculations and difficulties are shown in Figure 7.

 Figure 7: Mathcad Load-Strength Interference Calculations Use good engineering judgment and practical experience in interpreting the answer.  By the way, if the structure only has a 2-Parameter Weibull plot for strength, you will get a significantly different probability for failure so strength distributions are very important!

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