LoadStrength
Interference 
Loads vary, strengths vary, and reliability usually
declines for mechanical systems, electronic systems, and electrical
systems. The cause of failures is a
loadstrength 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—LoadStrength Interference 

The overlap of loadstrength 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 loadstrength 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 Loadstrength 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 35 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 loadstrength interference. Safety margins of 35 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 ReliabilityBased Design, by S. S. Rao, McGrawHill, Inc., 1992, ISBN 0070511926, Chapter 8: Strength Based Reliability and Interference Theory, page 235273). Rao’s equations of importance are:
This is equivalent to Rao’s Eq. 8.9
Where f_{S}(s) is the probability density function of the strength, and f_{L}(l) is the probability density function of the load. F_{L}(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 loadstrength
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
failurefree zone. The failurefree
strength phenomena occur with offset of the origin of the distribution. This is described in The New Weibull Handbook as a t_{0}
shift for a 3parameter distribution.
Dr. Abernethy in The New Weibull Handbook sets four requirements for use
of a 3parameter distribution:
1. Must have at least
21 data points for a valid analysis
2. Raw data plotted on a 2parameter Weibull
probability plot will show a
concave downward appearance.
3. The goodness of fit criteria (i.e., R^{2}
or PVE%) must show substantial
improvement with use of a 3parameter
distribution compared to the
2parameter 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 2parameter probability plot
display a concave downward
appearance. A straight line occurs on a 3parameter plot
when
110,000 psi is subtracted from the raw
data to correct concavity. Be
careful with t_{0} 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 Yaxis versus the unit of measure on the Xaxis.
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 Yaxis versus the unit of measure on the Xaxis.
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 Xaxis is give in gloads. The data recorded was the maximum positive gload from each of 23 flights. The plot as made in SuperSMITH Weibull software. The Gumbel upper distribution is used because high gloads are very important for airplanes. (Notice the results from handmade 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 Xaxis is shown in gloads and the failure free interval is 2.939 gloads.
Figure 5: 3Parameter 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 3Parameter PDF 
The calculations and difficulties are shown in Figure 7.
Figure 7: Mathcad LoadStrength
Interference Calculations 

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