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 1.15x. 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 Wood’s (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 (60X ) 0.04635 

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 Wood’s 

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 Wood’s 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 

cumulative damage model. All of these methods 

2 
Figure 1 - Relation of working stress to duration of load

(Fig. 4 of Wood 1951). (M 89 380)

Figure 2 - Relation of strength to time of loading in rapid-loading bending tests of small, clear

Douglas-fir specimens (Fig. 2 of Wood 1951). (M 89 381)

3
Figure 3 - Relation of duration of constant stress to level of stress in long-time loading bending 

tests of Douglas-fir specimens (Fig. 1 of Wood 1951). (ML87 5570) 

require specifying a standard rate of loading. The 

first two methods require data from a group of 

specimens that have failed under the standard 

rate of loading. This group of specimens is 

usually matched with specimens tested under 

constant load, using non- destructive measures 

such as strength ratio and modulus of elasticity. 

All three methods require an equal rank 

assumption. That is, the order of failure in any 

group of specimens is the same under ROL and 

DOL tests. 

These methods can be illustrated by applying 

them to a set of data from the “edge knot” study 

of Gerhards and Link (1987). The data consisted 

of 294 pieces of Douglas-fir 2 by 4 lumber chosen 

such that each piece contained an edge knot of a 

specified size as the strength-controlling defect. 

Using modulus of elasticity, the specimens were 

split into six matched groups. Three of these 

groups were tested under ramp loading, and 

the other three groups were tested under 

constant load. 

4 
Figure 4 - Estimated stress levels for edge knot study using matched order statistics and 

estimated failure distribution (+, high constant load; 0, medium constant load; x, low constant 

load). y axis: stress level (proportion), x axis: natural logarithm of time-to-failure in minutes. 

(ML88 5412) 

The first method for estimating stress levelmatching specimens by order of failure (Johns 

and Madsen 1982)-can be most easily illustrated 

if the sample size of the group that failed under 

ramp loading is the same as the sample size of 

the group tested under constant load. The pieces 

in the ramp failure group are ordered by actual 

static strength (which is also the order of failure 

time if each piece has the same ramp rate), and 

the pieces of the constant load group (including 

initial ramp failures) are ordered by time of 

failure. The stress level of the ith failure of the 

constant load test is then the load at failure 

divided by the load of the ith failure under ramp 

(short-term static strength test) load. Using the 

edge knot study as an example, let the middle 

ramp rate (approximately 6 pounds per minute) be 

chosen as the standard rate. For failures under 

constant load, the resulting stress level versus 

log time-to-failure plot can be seen in figure 4 

(preliminary ramp failures not shown). If the 

sample sizes of the constant load group and the 

standard ramp group are different, one may use 

this method by interpolating estimated static 

strength between adjacent failure times. 

5 
In the second method for estimating stress level 

(Gerhards 1986, 1988), a failure distribution is 

specified for the standard rate of loading, and the 

parameters of this distribution are then 

estimated. The stress level of the ith failure (out 

of n specimens) is then the i × 100/n percentile 

of the fitted distribution. Using the edge knot 

data, if the standard ROL data are assumed to 

have a lognormal distribution, the estimated 

parameters are the usual mean and standard 

deviation of the logarithms of the static strength. 

Gerhards (1988) fit a regression line to a plot of 

normal scores and failure loads (similar to fig. 5). 

For one ROL test, this method should give 

comparable results to the usual mean and 

standard deviation. However, Gerhards added 

initial ramp failures from the constant load tests 

to presumably improve the estimation procedure. 

Since added failures in one region of the data 

may bias the regression, initial failures should 

not be added to the data. Using an estimated 

underlying failure distribution, the resulting 

stress level plot for the edge knot data is very 

similar to the plot obtained through matched 

order statistics (fig. 4). Unless the ramp data 

deviate greatly from the fitted failure distribution, 

methods 1 and 2 should give similar stress 

level plots. 

The final method, used by Gerhards and Link 

(1987), obtains stress levels through the use of a 

cumulative damage model. The numerator is still 

the constant load at failure, but the denominator 

is found by using the estimated parameters from 

the cumulative damage model. The resulting plot 

is almost identical to that shown in figure 4; a 

slightly different scale on the y axis is due to 

another choice of the standard ROL. 

DOL factors may be calculated from stress level 

plots as a ratio of the stress level at the desired 

load duration and the stress level at the baseline. 

A fitted equation can be used, but the fitted line 

changes with the addition of different failure 

times for the censored data. If sufficient data 

exist, a representative point can be chosen at the 

baseline as well as the time for the desired load 

duration, and a ratio can be derived from these 

times. This is usually impractical for load 

durations of greater than a few months because 

long-term (years) data do not exist. Also, due to 

the problems of censored data and estimated 

stress levels (versus true stress levels), 

confidence intervals on the ratio estimates are 

difficult to obtain. 

Gerhards (1988) fit a regression line to the stress 

level versus log time to failure, using the 

parameter and variance estimates from the fitted 

regression to give confidence intervals for the 

estimated DOL factors. While the ratio estimates 

may be reasonable, the reported confidence 

intervals were derived from incorrect methodology 

and are also much too small. 

Figure 5 - Lognormal probability plot for edge 

knot rate of loading (ROL) tests (+, high ROL; 0, 

medium ROL; x, low ROL). y axis: normal score 

from a standard normal distribution with mean 0 

and standard deviation 1 ( - 1.645 is 5th 

percentile, 0 is 50th percentile, etc.), x axis: 

natural logarithm of load in pounds. Lines 

indicate Gerhards’ fitted exponential cumulative 

damage model. (ML88 5413) 

6 
Cumulative Damage Models 

Several cumulative damage models have been 

developed for the DOL phenomenon. Nielson 

(1986a, 1986b) proposed a model based on 

visco-elastic fracture mechanics. Schaffer’s 

model (1973, 1982) included compensation for 

temperature of exposure. Barrett and Foschi 

(1978) and Foschi and Barrett (1982) proposed a 

cumulative damage model with stress- and 

damage-dependent factors. Gerhards (1979) 

proposed an exponential cumulative damage 

model with a stress-dependent factor based on 

Miner’s rule (1945). Caulfield (1985) obtained 

the same model as Gerhards by a chemical 

kinetics approach. 

Use of these cumulative damage models to 

obtain reliable ratio estimates and confidence 

intervals requires methodology for estimating the 

parameters and associated variability. I am 

unaware of any variability estimates for the 

Nielson or Barrett and Foschi models. For the 

Gerhards model, details of parameter estimation 

and associated variability are given in Link, 

Gerhards, and Murphy (1988). 

A brief review of the Gerhards model follows. This 

cumulative damage model describes the damage, 

a, as a function of time, t, and load history, s(t). 

Damage ranges from 0 (undamaged specimen) to 

1 (failed specimen) using the following equation:

(3)

where b, c, and w are model parameters and X is 

a random variable used to model the underlying 

variability of the strength of lumber. The 

underlying distribution of the strength of lumber 

is assumed to be either a lognormal or a Weibull 

distribution. In this discussion, the lognormal 

distribution is used. This damage model gives the 

following mean trends (X = 1) for ROL and DOL 

tests (assuming instantaneous application 

of load): 

ROL load = (1/c) + (1/b) ln(bk) (4) 

DOL ln(T) = (b/c) - b (load) or 

load = (1/c) - (1/b) ln(T) (5) 

where k is the ROL and T is the time to failure. 

Data Needed to Estimate Parameters 

ROL and DOL tests are conducted using groups 

of matched specimens. A group usually contains 

50 to 200 specimens. Matching is based on 

estimated failure strength using nondestructive 

measurements of modulus of elasticity and 

strength ratio. 

For Gerhards’ exponential cumulative damage 

model, the variability parameter, w, can be 

estimated as long as there are multiple 

specimens within a test group because it is a 

measure of variability within a test group. 

Estimation of the other two parameters, b and c, 

requires at least two groups of data from 

specimens that failed under two different ROL or 

at two different load levels. It is preferable to 

have more than two groups of data to judge the 

lack of fit of the data to the hypothesized model. 

The Barrett and Foschi model (1978) contains six 

parameters; two are variability parameters. The 

model, as estimated by Foschi and Barrett (1982), 

is used with only two groups of data (using a 

larger sample size). It is not clear whether the two 

variability parameters or the other four 

parameters can be uniquely estimated with only 

two groups of data. Thus, with a fixed total 

sample size, it would be preferable to test more 

load levels with fewer specimens than to test only 

two load levels. 

Data Plots From ROL and DOL Tests 

Data plots are useful for checking the 

distributional assumptions and the fit of the 

damage models. One of the most useful plots for 

checking distributional assumptions is a 

probability plot, which plots the failure data by 

the cumulative probability distributional function. 

A cumulative probability distribution function 

ranges from 0 to 1 (0 to 100 pct). If the failure 

data come from the hypothesized distribution, 

then the probability plot should look like a 

straight line. Therefore, the scale on the 

cumulative probability distribution axis depends 

on the distributional assumption. For example, if 

one assumes that the data come from a normal 

distribution, the failure data are plotted by a 

normal score. A normal score is a function of the 

cumulative probability of a standard normal 

7 
distribution: - 1.645 is the 5th percentile, 0 is the 

median, and 1.645 is the 95th percentile, etc. If 

one assumes that the data come from a 

lognormal distribution, then the logarithm of the 

failure data is plotted versus the normal score. A 

good discussion of probability-plots can be found 

in Nelson (1982). 

ROL tests provide information about failure load, 

failure time, ROL, and order of failure. For ROL 

tests, an appropriate plot of the data would be a 

function of failure load versus some cumulative 

probability distribution. If one assumes that the 

underlying strength distribution is lognormal, 

then the appropriate plot would be of the log load 

versus a normal score. The ramp test data from 

the edge knot study can be used to illustrate this 

kind of plot (fig. 5). The plot used by many 

researchers of log time to failure versus logarithm 

of loading rate (fig. 6) is inappropriate for ROL 

data; there is no way to gauge the fit of the 

model because the difference in failure times 

overwhelms anything else. Foschi and Barrett 

(1982) used log time to failure versus logarithm of 

loading rate to demonstrate that the data are 

linear. But a plot of the median failure load 

versus logarithm of ROL (which should also be 

linear by their model) shows that the data are 

not linear. 

A DOL test is made up of three segments: 

(1) initial ramp loading of the specimen to the 

desired constant load; (2) period during which the 

specimen is under constant load, given that it has 

survived the initial ramp loading; (3) second ramp 

loading (from 0 to failure) which determines the 

residual strength of a specimen that survives the 

constant load segment. In general, some 

specimens fail in each segment. DOL tests 

provide information about failure load, failure 

time, order of failure, and the segment in which 

the specimen failed. 

For DOL tests, an appropriate plot of the data is a 

function of the logarithm of the time on test 

versus a cumulative probability distribution. The 

calculation of time on test may vary from one 

segment to another. I distinguish between total 

time on test (from initial ramp loading to failure) 

and time on test in a particular segment of 

the test. 

Figure 6 - Log rate of loading (ROL) versus log 

failure time for the edge knot ROL test (+, high 

ROL; 0, medium ROL; x, low ROL). y axis: 

natural logarithm of ROL in pounds per minute, 

x axis: natural logarithm of time to failure in 

minutes. (ML88 5414) 

For specimens that fail during the first segment, 

that time is the total time on test. For specimens 

that fail during the second load segment, total 

time on test is either initial ramp time plus 

constant load time or the time on constant load 

only. If the latter is designated as total time on 

test, the time clock is reset to zero, creating a 

discontinuity in the plots between the initial ramp 

and constant load failures. The choice of time to 

failure under constant load affects the parameter 

estimation. If time on constant load is chosen, 

the first few failures under constant load (within 

seconds of uploading) may force the fitted model 

to be closer to these early failures than later 

failures; a very early failure may have an arbitrary 

short failure time that is subject only to the 

accuracy of the test equipment. The logarithm of 

this time on test is then an arbitrary large 

negative number. This influences the convergence 

criterion of the estimation procedure, which is to 

minimize the sum of squares of the difference 

8 
between actual and predicted logarithms of time. 

This problem does not occur if the time on test 

includes the ramp loading time, which is in 

minutes not seconds. 

For specimens that fail in the third load segment, 

the time used is the failure load divided by the 

ROL during the third segment. This would be 

equivalent to the failure time obtained by setting 

the clock to zero at the beginning of the second 

ramp loading. This will appear as a continuation 

of the curve defined by the initial ramp loading 

failures, assuming there is no constant load 

segment and that rates of loading are the same 

for the first and third segments. There is no way 

to avoid the discontinuity of the constant load 

failures and the residual strength times, as total 

time on test would essentially appear as a 

straight vertical line. The plot for the edge knot 

study along with Gerhards’ fitted exponential 

cumulative damage model (fig. 7) uses total time 

on test for the constant load failures. 

Total Time on Test Versus Time on 

Constant Load 

The edge knot study had quite a few early failures 

in the constant load phase of the test (some 

failures occurred within seconds of uploading). 

The difference between the log total time on test 

and the log time on constant load was trivial after 

some time had elapsed (minutes or hours), but 

this difference was substantial for early constant 

load failures. This is seen in figure 8, which 

plots only the failures under constant load. The 

solid lines designate the same model shown in 

figure 7, and the dashed lines are the estimated 

model when the dependent variable is time on 

constant load. Note that the dashed lines fit the 

early failures quite well but are less suited to 

later failures, whereas the reverse is true for the 

solid lines. This discrepancy occurs only when 

there are a large number of early failures. The use 

of a different model will not change this 

situation, as the analysis must use log time to 

failure rather than the actual time. Of the data 

sets currently available at FPL, the total time on 

test versus the time on constant load matters in 

only the data set from the edge knot study 

because of the early failures. 

Figure 7 - Results from edge knot duration of 

load (DOL) study using total time on test 

(+, high constant load; 0, medium constant 

load; x, low constant load). y axis: normal score 

from a standard normal distribution with mean 0 

and standard deviation 1 ( - 1.645 is 5th 

percentile, 0 is 50th percentile, etc.), x axis: 

natural logarithm of time to failure in minutes. 

Lines indicate Gerhards’ fitted exponential 

cumulative damage model. (ML88 5415) 

The early failures in the edge knot study occurred 

because the specimens sustained more damage 

during ramp uploading than predicted in 

Gerhards’ model or any other model currently 

under consideration. DOL factors are needed for 

periods ranging from minutes to years. Using 

total time on constant load forces the fitted 

models to early failures (seconds, minutes, and 

hours), which flattens the estimated model (fig. 8). 

DOL factors computed with a flatter line (using 

total time on constant load) will-be larger (e.g., 

DOL has a smaller effect) than those calculated 

with the steeper line (using total time on test). 

This is unlikely to be a conservative approach 

toward estimating DOL factors. Results of DOL 

research at FPL have often been reported using 

time on constant load. Because I prefer to fit 

9 
Figure 8 - Results from edge knot duration of 

load (DOL) study: x and solid line designate total 

time on test; 0 and dashed line designate time 

on constant load. y axis: normal score from a 

standard normal distribution with mean 0 and 

standard deviation 1 ( - 1.645 is 5th percentile, 0 

is 50th percentile, etc.), x axis: natural logarithm 

of time to failure in minutes. (ML88 5416) 

later failures rather than early failures, this paper 

presents only plots and fitted models that use 

total time on test. 

Calculation of DOL Factors 

Estimated parameters and their variance 

estimates can be used to draw a line to the 

stress level versus log time-to-failure plots (fig. 9). 

Note that this curve is not fit to the data shown 

Figure 9 - Estimated stress levels for edge knot 

study using a cumulative damage model, with 

estimated parameters from Gerhards’ model 

(+, high constant load; 0, medium constant 

load; x, low constant load). y axis: stress level 

(proportion), x axis: natural logarithm of time to 

failure in minutes. (ML88 5417) 

but is the result of the parameter estimation. The 

appendix to this paper contains technical details 

of how to estimate DOL factors and their 

confidence intervals. As explained in the 

appendix, the stress level versus log 

time-to-failure plots are a function of a standard 

ROL, but the DOL factors are not. 

10 
Results of Tests on Douglas-Fir 

2 by 4 Structural Lumber by Grades 

The FPL performed additional ROL (Gerhards and 

Link 1986) and DOL (Gerhards 1988) tests on 

Douglas-fir 2 by 4’s. The lumber was graded, 

using strength ratio, into Select Structural, No. 1, 

No. 2, and No. 3 samples. No. 1 lumber was not 

tested. There were up to three ROL tests (ROL1, 

ROL2, and ROL3) and up to three DOL tests 

(DOL1, DOL2, and DOL3) for each grade. The 

sample sizes and test conditions are given in 

table 2. The rates of loading were 300, 3, and 

0.03 pounds per minute for the ramp tests. The 

constant load tests contained several load levels 

in each test. These tests are known as step 

constant load tests and are used to get additional 

constant load failures. The specimens were 

loaded to the first constant load level (300 pounds 

per minute), held on that constant load level for a 

predetermined time, loaded to the second load 

level, etc. The steps used in the constant load 

tests were estimated percentiles of the underlying 

failure distribution. The constant load tests used 

step constant loads with a combination of 40th 

and 70th (DOL1), 5th, 15th, and 40th (DOL2), and 

5th and 15th (DOL3) percentile loads. 

Table 2 - Sample sizes and test conditions of Douglas-fir 

2 by 4’s 

Sample size 

Grade 

Rate of loading 

testsa 

Duration of load 

testsb 

1 2 3 1 2 3 

Select Structural 100 50 100 50 50 50 

No. 2 100 50 100 50 50 50 

No. 3 100 — — 50 50 — 

a

 Test conditions for rate of loading (ROL) tests 

were: ROL1, 300 Ib/min; ROL2, 3 Ib/min; and ROL3, 

0.03 Ib/min. 

b

 Step constant loads for duration of load (DOL) tests 

were: DOL1, steps at 40th and 70th percentiles; DOL2, 

steps at 5th, 15th, and 40th percentiles; and DOL3, 

steps at 5th and 15th percentiles. 

The fitted model for ROL data uses only failures 

from ROL1, ROL2, and ROL3 tests. Likewise, the 

fitted model for DOL data uses only constant 

load failures from DOL1, DOL2, and DOL3 

constant load tests. For the FPL DOL tests, the 

parameter estimates do not change substantially 

if one uses the constant load failures from all the 

constant load phases or just the first set of 

failures, nor do they change substantially if one 

uses total time on test or time on constant load. 

Therefore, the plotted fit is that of total time on 

test using only the constant load failures in the 

first “step.” Plots of the fitted damage model for 

the ROL data (fig. 10) and the DOL data (fig. 11) 

give a visual impression of the model fit to the 

data. No model was fit to the No. 3 ROL test as 

data exist for only one ROL which is insufficient 

to estimate the model parameters. Confidence 

intervals can be plotted for these fitted curves 

(Link, Gerhards, and Murphy 1988). The overall 

impression is that the model generally fits the 

data, but there appear to be systematic 

departures from the model which indicate 

additional modeling might be useful. 

The stress level versus log time-to-failure plots 

(fig. 12) show the fitted DOL model and the first 

step constant load failures. The corresponding 

DOL factors, along with 95 percent confidence 

intervals, for these data as well as edge knot data 

are given in table 3. The DOL and ROL curves are 

plotted along with Wood’s equation (fig. 13). 

Wood’s equation does not match up well with the 

experimental data because of the definition of 

the standard ROL. The relative vertical position 

Table 3 - Duration of load factors for Forest Products 

Laboratory Douglas-fir 2 by 4’s point estimatesa,b 

Duration of load 

Source Type 1 Day 1 Week 2 Months 10 Years 50 Years 

a

 Baseline = 5 minutes. 

b

 Ninety-five percent confidence intervals 

in parentheses. 

c

 ROL = rate of loading. 

d

 DOL = duration of load. 

11 
Figure 10 - Lognormal probability plots for rate of loading (ROL) tests of Douglas-fir Select 

Structural and No. 2 lumber (+, high ROL; 0, medium ROL; x, low ROL). y axis: normal score 

from a standard normal distribution with mean 0 and standard deviation 1 ( - 1.645 is 5th 

percentile, 0 is 50th percentile, etc.), x axis: natural logarithm of load in pounds. (ML88 5418) 

is not important; it is the ratio of two values 

which matters. 

Several results using Gerhards’ exponential 

cumulative damage model are immediately 

apparent from table 3 and figure 13: 

(1) The parameter estimates and DOL factors are 

not the same for DOL and ROL tests. Although, 

theoretically (by Gerhards’ (1979) model or any of 

the other models currently under consideration 

for modeling the effect of DOL) they should give 

the same result; in practice they do not. Therefore 

one cannot substitute ROL tests, which can be 

done much more quickly, for DOL tests. ROL 

tests predict a much smaller DOL effect. If both 

the ROL and DOL data sets are used to estimate 

the cumulative damage parameters, the ROL tests 

will determine the parameter estimates because 

the variability of log time to failure in a ROL test 

is at least 20 times smaller than the variability of 

log time to failure under DOL tests. 

(2) The DOL factors for lumber DOL tests appear 

to be smaller than those derived by Wood (1951). 

(3) Stress level curves using DOL tests were 

steeper as the quality of the material 

decreases, and those using ROL tests had the 

reverse trend. However, these trends are not 

statistically significant. 

(4) Confidence intervals increase as the load 

duration increases. This is to be expected as 

there is less information available at longer 

load times. 

(5) Confidence intervals for ROL tests were 

smaller than the corresponding intervals for DOL 

tests because (a) sample size was larger in the 

ROL tests and (b) variability of the parameter 

estimates was larger for DOL tests compared to 

ROL tests. 

12 
Figure 11 - Results from the Douglas-fir constant load study of Select Structural, No. 2, and No. 3 

lumber using total time on test (+, high constant load; 0, medium constant load; x, low constant 

load). y axis: normal score from a standard normal distribution with mean 0 and standard 

deviation 1 ( - 1.645 is 5th percentile, 0 is 50th percentile, etc.), x axis: natural logarithm of time 

to failure in minutes. (ML88 5419) 

13 
Figure 12 - Estimated stress levels using cumulative damage model, with estimated parameters 

from Gerhards’ model, for Douglas-fir constant load study of Select Structural, No. 2, and No. 3 

lumber; failures from first constant load level only (+, high constant load; 0, medium constant 

load; and x, low constant load). y axis: stress level (proportion), x axis: natural logarithm of time 

to failure in minutes. (ML88 5420) 

14 
Other Aspects of DOL Research 

Figure 13 - Estimated stress level equations for 

Forest Products Laboratory Douglas-fir 2 by 4 

studies and Wood’s duration of load curve (solid 

lines: constant load studies, dashed lines: rate 

of loading studies). y axis: stress level 

(proportion), x axis: natural logarithm of time to 

failure in minutes. (ML88 5421) 

Three other aspects of damage modeling and 

design of DOL tests deserve comment: (1) the 

need for additional modeling, (2) usefulness of 

step loads and required percentage of failures, 

and (3) matching of lumber samples. 

Additional Modeling 

Systematic departures of the data from Gerhards’ 

exponential cumulative damage model suggest 

the need for continued effort in modeling the 

effect of DOL. All of the current models use an 

exponential function for damage accumulation 

which implies that the damage accumulates only 

near the failure time or load. It has been 

recognized that more damage accumulates at 

lower stress levels than the models predict. Two 

suggested modifications to Gerhards’ model, 

which model damage as a power function rather 

than an exponential function, have not solved the 

problem. The first modification is 

It is impossible to estimate n using ROL tests. 

Although an estimate of n was obtained for 

DOL tests, it did not change the plot of the 

fitted model. 

The second alternative modification is 

One can estimate the parameters of this model 

for ROL and DOL tests. However, the fitted model 

for ROL tests does not change. For DOL tests, 

the underprediction of failure times was worse 

using the power model (dashed line) than using 

the exponential model (solid line) (fig. 14). 

Step Loads and Required 

Percentage of Failures 

DOL tests using step loads (progressively higher 

stresses) are common in the testing of electronic 

parts (Yurkowsky, Schafer, and Finkelstein 1967). 

In an effort to obtain more information from 

additional failures and to accelerate the tests, 

Gerhards (1986) used a three-step constant load 

to compare the DOL effect for southern pine 

15 
Figure 14 - Gerhards’ exponential (solid line) and 

power models (dashed lines) fitted to the edge 

knot study constant load tests. y axis: normal 

score from a standard normal distribution with 

mean 0 and standard deviation 1 ( - 1.645 is 5th 

percentile, 0 is 50th percentile, etc.) x axis: 

natural logarithm of time to failure in minutes. 

(ML88 5422) 

lumber dried conventionally and at high 

temperatures. Only one ramp test and one 

three-step constant load test were used. 

Becauses there were very few failures on either of 

the two lower steps, the experiment was 

essentially reduced to one ramp and one 

constant load test. The resultant data were 

insufficient for estimating the parameters of 

Gerhards’ exponential cumulative damage model. 

Gerhards combined these data into one data set 

and plotted a stress level versus log time to 

failure using a hypothesized distribution of static 

strength. He concluded that the data exhibited a 

much smaller DOL effect than that predicted 

using Wood’s equation. However, different results 

are obtained if the conventional and 

high-temperature data sets are kept separate 

(fig. 15). The different symbols denote drying 

temperature and load level. There are no 

consistent patterns. 

Gerhards’ (1988) study of the effect of grade on 

the DOL behavior of Douglas-fir 2 by 4’s also 

featured two or three constant load levels within 

a single test. Although these tests contained a 

reasonable number of failures at each constant 

load level, the estimated parameters and their 

standard errors do not change much if all the 

constant load failures or just those at the lowest 

step are used. The stress level versus log 

time-to-failure plots for this study (fig. 12) show 

constant load failures from only the lowest step. 

Because the damage model predicts little 

damage, it is tempting to include constant load 

failures from the higher load levels as well, but 

this is likely to increase the variability of the 

data. Figure 16 shows estimated stress levels for 

the Douglas-fir Select Structural specimens using 

failures from all constant load levels. Note that 

some damage occurred in the lowest step; the 

estimated stress levels for the second step 

failures were lower than those for the first step 

failures during the first week of constant load. 

A sufficient number of failures are needed in 

each constant load test. As mentioned earlier, 

early and late failures do not follow the same 

pattern. When the high constant load is 

terminated, later failures cannot be observed. In 

the edge knot study (Gerhards and Link 1987), the 

high constant load was held for 3 to 4 days. After 

the initial 2 hours of constant load, there were 

only 10 (out of 49) failures at the high constant 

load level versus 16 at the middle constant load 

level and 29 at the low constant load level. 

Therefore, the lower constant load levels had a 

greater influence in estimating the parameters. In 

the later Douglas-fir 2 by 4 test, the high constant 

load level was maintained for a longer period. 

The estimation procedure requires that 30 to 

50 percent of the specimens fail in the constant 

load segment of the test. 

Matching of Lumber Samples 

Matching the lumber samples for each test is 

important. Lumber contains a large amount of 

variability, and samples should therefore be 

matched in as large a group as practical. Small 

groups of samples, such as the matched groups 

of 25 specimens used in the Douglas-fir study 

(Gerhards 1988), have a high degree of variability. 

16 
Figure 15 - Estimated stress levels using a 

cumulative damage model with estimated 

parameters from Gerhards’ model for the 

high-temperature drying constant load study, 

using failures from all constant load levels. 

(Conventional drying: +, step 1; 0, step 2; 

x, step 3. High-temperature drying: , step 1; 

D , step 2; �, step 3). y axis: stress level 

(proportion), x axis: natural logarithm of time to 

failure in minutes. (ML88 5423) 

A larger matching group will decrease some of 

the variability. That is, if the sample size is 100, 

the lumber should be matched in groups of 100. 

The sample size in each group will depend for the 

Figure 16 - Estimated stress levels using a 

cumulative damage model with estimated 

parameters from Gerhards’ model for the 

Douglas-fir Select Structural constant load 

study, using failures from all constant load 

levels (+, first constant load; 0, second 

constant load). y axis: stress level (proportion), 

x axis: natural logarithm of time to failure in 

minutes. (ML88 5424) 

most part on the resources at hand. The results 

of the Douglas-fir lumber tests indicate that a 

minimum of 50 samples should be used. 

17 
Conclusions Literature Cited 

The results of current FPL tests on DOL suggest 

Inclusion of confidence intervals increases the 

usefulness of point estimates of DOL factors 

by allowing comparison of results from 

different experiments and the currently 

used values. 

Fitting regressions to stress level versus log 

time-to-failure plots does not give statistically 

correct confidence intervals for DOL factors. 

Using some cumulative damage model that 

includes estimates of both standard errors and 

parameters provides confidence intervals for 

DOL factors. 

The exponential cumulative damage model 

provides reasonable starting estimates of DOL 

factors. Additional modeling might be useful, 

since the exponential cumulative damage 

model apparently does not account for damage 

at a lower stress level, nor does the model fit 

both late and early constant load failures. 

Parameter estimates from ROL tests are 

apparently not useful for predicting the DOL 

results in structural lumber. 

Step loads within a DOL test do not appear to 

add information about the model parameters. 

DOL tests should be designed to allow for at 

least 30 to 50 percent failures under constant 

load. Also, one should run at least one more 

test than the number of tests needed to 

estimate the model parameters. This allows at 

least a visual assessment of lack of fit to 

the model. 

Matching of lumber samples is important; a 

minimum of 50 specimens should be used for 

each test sample. 

18 
19
Appendix

Estimation of DOL Factors and

Confidence Intervals Using Gerhards’

Exponential Cumulative Damage Model

Stress level is the ratio of constant load to a 

standard static strength. It is also the ratio of 

load under ramp loading to the load under 

constant load at any given time: 

SL(t) = load under constant/load under ramp 

b - c In(t) = b + c In(bk) 

using equations (4) and (5) of the main text. While 

this equation assumes mean trends and 

instantaneous constant load, it can be modified 

for the ramp time. Note that the stress level is 

still a function of the ROL, k. This equation 

defines the line on the stress level versus log 

time-to-failure plot (fig. 10). The constant load 

data can also be shown on the same plot. The 

equation used is similar but uses the variability 

parameter, w, to estimate the ramp failure load. 

DOL factors may be calculated from the stress 

level equation. Again, the DOL factor for time, t2 , 

given a baseline time, t1, is the ratio of 

SL(t2)/SL(t1): 

Note that this DOL factor is not a function of the 

ROL, k. By taking the ratio of two stress levels, 

one effectively eliminates the problem of having 

to specify a standard ROL. In that the DOL factor 

is a function of the baseline and target times as 

well as the estimated parameters, confidence 

intervals can be obtained for the DOL factor 

using the estimated variability of the parameters 

b and c. 

Standard error 

(DOL factor) = sedol = [(sec x dc)

2 + (seb × db) 2 

+ 2(rbc × sec × dc × seb 

× d b)]1/2 

where: sec = standard error of c 

seb = standard error of b 

rbc = correlation of b and c 

dc = b[ln(t2) - In(t1)]/[b - c ln(t2 )] 2 

db = c[ln(t1) - In(t2)]/[ b - c ln(t2 )] 2 

Parameter estimates, standard errors, and 

correlations of the parameter estimates may be 

found using the computer program in Link, 

Gerhards, and Murphy (1988). Confidence 

intervals for the DOL factor are then: DOL factor 

±z a (sedol), where za is the appropriate standard 

normal coefficient, e.g., 1.96 for a 95 percent 

confidence interval. 

1.5-7/88 

20