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Statistical Considerations in Duration of Load Research United States
Department of
Agriculture
Forest Service
Forest
Products
Laboratory
Research
Paper
FPL-RP-487
Statistical
Considerations in
Duration of Load
Research
Carol L. Link Abstract
Contents
Duration of load factors are the most significant
factors that reduce allowable design stresses for
lumber. This paper discusses statistical
considerations for the design of duration of load
tests and for analysis of the resulting data.
Duration of load factors along with their
associated confidence intervals are estimated for
one model using Douglas-fir data from tests at
Forest Products Laboratory.
Keywords: Duration of load, duration of load
factors, cumulative damage models.
July 1988
Link, Carol L. 1988. Statistical considerations in duration of
load research. Res. Pap. FPL-RP-487. Madison, WI: U.S.
Department of Agriculture, Forest Service, Forest Products
Laboratory. 20 p.
A limited number of free copies of this publication are
available to the public from the Forest Products Laboratory,
One Gifford Pinchot Drive, Madison, WI 53705-2398. Laboratory
publications are sent to over 1,000 libraries in the United
States and elsewhere.
The Laboratory is maintained in cooperation with the
University of Wisconsin.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculation of Duration of Load Factors. . . .
Data Variability. . . . . . . . . . . . . . . . . . . . . .
Estimation of Stress Levels. . . . . . . . . . . .
Cumulative Damage Models . . . . . . . . . . . . .
Data Needed to Estimate Parameters. . . .
Data Plots From ROL and DOL Tests . . . .
Total Time on Test Versus Time on
Constant Load . . . . . . . . . . . . . . . . . . . . . .
Calculation of DOL Factors. . . . . . . . . . . .
Results of Tests on Douglas-Fir 2 by 4
Structural Lumber by Grades . . . . . . . . . . . .
Other Aspects of DOL Research. . . . . . . . . .
Additional Modeling. . . . . . . . . . . . . . . . . .
Step Loads and Required Percentage
of Failures . . . . . . . . . . . . . . . . . . . . . . . . .
Matching of Lumber Samples . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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20 Statistical
Considerations in
Duration of Load
Research
Carol L. Link, Mathematical Statistician
Forest Products Laboratory, Madison, WI
Introduction
In determining allowable stress values for lumber,
engineers must consider duration of load. This
phenomenon is demonstrated in both rate of
loading (ROL) and duration of load (DOL) tests. In
ROL tests, the strength of lumber increases with
increased loading rate. In DOL tests, a piece of
lumber carrying a given load may be unable to
carry that load indefinitely. Building codes
account for DOL by using multiplicative DOL
factors. These factors change the allowable
stress as a function of load duration time, which
equals time under maximum load for a particular
condition. The current National Design
Specifications for Wood Construction (National
Design Specifications for Wood Construction
1986) DOL factors (table 1) assume a baseline
duration of 10 years. The following example
illustrates how to interpret DOL factors. If the
baseline allowable stress is
x, then the allowable
stress under snow load (of 2-month duration)
would be 1.15
x. Duration of load researchers
often use a baseline of 5 minutes (the duration of
a standard static-strength test) and DOL factors
that reduce allowable stress for periods longer
than 5 minutes. These factors are given in table 1.
The National Design Specifications DOL factors
(table 1) are based on Woods (1951) bending
tests of small, clear wood specimens. Current
DOL research in the United States, Canada, and
Europe includes testing of structural lumber and
development of models. The ultimate goal of this
research is to derive new DOL factors based on
tests of structural lumber in bending, tension,
and compression. This paper discusses
statistical considerations involved in designing
DOL tests, modeling the resulting data, and
computing DOL factors. These statistical
considerations are important for knowing the
variability of proposed DOL factors. Without
variability estimates, it is impossible to compare
new estimates of DOL factors to those
currently in use or compare estimates from
other experiments.
While this paper necessarily focuses on the data
and models developed in the United States, the
comments are relevant to research in other
countries as well. Also, I do not mean to imply
that the data or models developed in the United
States are the best possible. The paper
recounts the development of the current DOL
factors, describes a cumulative damage model
illustrated with data from Forest Products
Laboratory (FPL), presents additional data with
the fitted cumulative damage model, and finally
comments on testing and development of models.
Table 1 - Current National Design Specifications (1996) for
wood construction duration of load (DOL) factors
Type of load
Load duration

DOL factors baseline
10 years 5 minutes
Static strength 5 minutes
1.00
Wind
1
week
1.25
.77
Snow
2 months
1.15
.71
Live
10 years
1.00
.62
Permanent
50 years
.90
.56
1 day
1.33
.82
1 Calculation of Duration of
Load Factors
Current DOL factors were derived from research
on small, clear wood specimens (Wood 1951).
Both ROL and DOL tests of primarily Douglas-fir
specimens were combined in a stress level versus
log time-to-failure plot. The stress level was
estimated; it was calculated from matched
specimens. Wood fit a hyperbolic curve to these
data (fig. 1) using the equation
SL(
X ) = (108.4) + 18.3
(60
X )
0 . 0 4 6 3 5
where
X is time in minutes and SL is stress level
in percent. Calculation of DOL factors from this
equation is possible given the baseline and the
desired load duration times. For example, the
DOL factor for 2 months given a baseline of
10 years is then the ratio of SL(2 months) to SL
(10 years) which is equal to 1.15.
SL(2 months) = SL(60
×
24
×
61 minutes)
= SL(87,840 minutes) = 71.2
(2)
SL(10 years) = SL(60
×
24
×
365
×
10)
= SL(5,256,000 minutes) = 62.1
DOL factors for other time periods can be
computed in the same manner.
Data Variability
Real data are inherently variable. While Woods
1951 curve may appear to give a precise definition
of the DOL phenomenon, one should be aware of
the data variability behind this curve. The
variability of the small clear data in ROL (fig. 2)
and DOL (fig. 3) tests is not trivial. The strength
of wood specimens is extremely variable, even for
small, clear pieces. This underlying variability is
not expressed in the point estimate of a DOL
factor. The underlying variability can only be
expressed in a confidence interval for this factor.
Confidence intervals are a measure of how
precisely one knows or can estimate a given
quantity. Point estimates without confidence
intervals ignore the underlying variability and give
the misguided illusion of precision. Without
confidence intervals, one cannot determine if
DOL factors obtained from various experiments
are actually similar or different from each other
or from currently used DOL factors.
Unfortunately, confidence intervals for DOL
factors using Woods equation are not
straightforward because there are no variability
estimates for its parameters. Such estimates
would be difficult to obtain because the stress
levels are only approximate and some of the data
are censored. Censored data arise when the
exact failure time is unknown. For specimens that
have not failed under constant load at the
conclusion of the constant load tests, the precise
time to failure is unknown, although it must
exceed the duration of that constant load test.
While a curve can be fit to the stress level versus
log time-to-failure data, this curve may vary with
the exclusion or inclusion (and plotting position)
of the censored data points. Confidence intervals
for parameters of this fitted curve cannot be
determined in the usual fashion due to the
estimated stress levels and censoring. Therefore,
some type of model must be used to obtain
confidence intervals.
Estimation of Stress Levels
Results from DOL tests are commonly shown in a
plot of stress level versus log time to failure.
Theoretically, a stress level is the ratio of the load
at which a specimen failed in a constant load
test to the load at which it would fail in a
short-term static strength test. Stress levels are
always approximate because strength of any
specimen is unknown under a loading scheme
different from the one in which it failed. Wood
determined a stress level by using the failure load
from a matched specimen that had failed in a
short-term static strength test. Underlying the
failure load of a matched specimen is some
standard rate of loading (since the strength of a
specimen varies with the rate of loading). Stress
level will vary as the standard rate of loading
is varied.
Since matched specimens are only practical for
tests of small, clear wood specimens (and even
these are not perfectly matched), another
procedure must be used to estimate the stress
level for tests of structural lumber. At least three
methods are found in the literature: (1) matching
specimens by order of failure, (2) estimating an
underlying failure distribution, and (3) using a
cu