Evans4724Using The Weibull Distribution.indd

Using the Weibull Distribution to Estimate
Lumber Properties
Results of mechanical tests on lumber, wood
composites, and wood structures are often summarized
by a distribution function ﬁt to the data. This ﬁtted
distribution can then be used in calculating properties
based on percentiles of the data, in reliability-
based design calculations, or in simulations of
the performance of wood structures. The Weibull
distribution (named after Waloddi Weibull, a Swedish
physicist who used it in 1939 to describe the breaking
strength of material) is playing an increasingly
important role in this type of research and has become
a part of several American Society of Testing and
Materials (ASTM) standards. Its popularity with
researchers is due in part to one of the parameters—the
shape parameter—which allows it to look like a variety
Weibull density functions for different shape parameters.
of other distributions, such as the normal, lognormal,
and exponential distributions. The ﬁgure illustrates
1. Methods must be available to ﬁt the distribution to
how the shape of the distribution changes as the
a data set and provide statistically sound estimates
shape parameter changes. This ﬂexibility to model
of the parameters of the distribution that uniquely
experimental results makes the Weibull distribution a
deﬁne the distribution, such as the mean and
powerful tool for Forest Products Laboratory (FPL)
variance of a normal distribution. A great deal
researchers as they work on projects designed to
of research has been published in the statistical
meet major Forest Service goals, such as improved
literature on this question. However, the effect
utilization of wood from small-diameter trees.
that different ways of estimating a parameter
has on estimating lower tail percentiles from
Background
censored data has not been widely researched. This
To summarize research results and get property
information is needed for many types of wood
estimates or simulation results needed in wood
research.
utilization research, the distribution ﬁtted to the data
2. The variability of parameter estimates needs to
must offer several capabilities. These capabilities can
be known for any type of estimation procedure
be grouped into six categories:
used. This is especially important if the ﬁtted
U. S. Department of Agriculture • Forest Service

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distribution is going to be used in simulations of
powerful and readily available tool for researchers
the structural performance of wood assemblies and
working on improved wood utilization.
if the estimates come from censored data. Again,
the statistical literature has focused primarily on
Approach
complete samples.
The approach to developing new capabilities for
3. We need to be able to estimate percentiles of the
the Weibull distribution is a mixture of theoretical
population from the ﬁtted distribution and provide
statistical development and computer simulations.
tolerance limits and conﬁdence intervals for
Methods of estimating tolerance limits, goodness-
these estimates. FPL research has developed such
of-ﬁt, and other aspects of the Weibull distribution
procedures for complete data sets. Work is needed
usually begin with theoretic development of possible
to extend this to censored data sets and to compare
solutions. However, these solutions often need to be
FPL methods to some other methods that have been
veriﬁed or critical values for tests need to be found
suggested.
through extensive computer simulations. For example,
4. After ﬁtting the distribution to a data set, we need
tests of the goodness-of-ﬁt for a Weibull distribution ﬁt
to know how to evaluate if it really ﬁts the data
to a complete data set can be developed theoretically.
or is an inappropriate distributional form for the
However, the critical values for such a test depend upon
data. FPL research has developed these goodness-
the shape parameter and the sample size. This requires
of-ﬁt tests for complete data sets. This needs to be
extensive computer simulations to develop the critical
extended to censored data sets.
value and then statistical modeling of the critical values
5. A computer program with user manual needs
to provide a method of predicting the critical values for
to be readily available for researchers to get the
any combination of sample size and shape parameter.
information discussed in categories 1 to 4 for both
complete and censored data sets. FPL has written
Expected Outcomes
such a program for complete data sets, but it
Results of this research will provide a powerful tool
needs to be documented. An extension to censored
that can be used to improve our ability to summarize
data sets will require completion of the research
wood utilization research results, provide improved
discussed in each category.
estimates of material properties used to design wood
6. Because mechanical properties of lumber are often
structures, and allow more realistic simulations of
related in some way to each other (for example,
wood performance in structural applications. The
specimens with above-average bending strength
results should also affect ASTM standards on wood
often have above-average tensile strength), the
properties and design.
distributions should be expandable to at least two
variables with some type of relationship between
Timeline
them. This bivariate distribution is especially
needed for simulations of the performance of
Research results are expected periodically over the
specimens under combined loading situations.
period from June 2003 through the end of 2007.
Initial work on a bivariate Weibull distribution has
been done at FPL, but the theoretical form of such
Contact Information
a distribution must be evaluated using real data.
Dr. James W. Evans
This will require looking at aspects of categories 1
USDA Forest Service
to 5 above for the bivariate case for both complete
Forest Products Laboratory
and censored samples.
Madison, WI 53726-2398
(608) 231-9332
Objective
jwevans@fs.fed.us
The objective of this study is to develop capabilities
for the Weibull distribution that allow it to be a more