### A Cumulative Damage Model to Predict Load Duration Characteristics of Lumber

#### Abstract

The exponential damage model dα/dt = exp[-a + bσ_{s}(t)/σ_{s}] is used in this paper to describe duration-of-load data on lumber tested in bending where dα/dt is rate of damage, σ_{s} is static strength, σ(t) represents applied load history, and a and b are parameters. A specially selected set of Douglas-fir 2 by 4s was divided into six 49-specimen groups having similar distributions of edge knot size and modulus of elasticity. Each group was randomly assigned to one of three rates of ramp loading or one of three levels of constant loading.

The lognormal distribution σ_{s} = σ_{o}exp(wR) provided a reasonable description of static strength of the 2 by 4s where σ_{o} is the median static strength, w is a measure of variability, and R is a normal random variable. With b' = b/σ_{o}, the model used to fit the ramp and constant load experimental data by nonlinear least squares was dα/dt = exp[-a + b'σ_{o}(t)/exp(wR)]; thus a, b', and w were parameters that were estimated. The model fit some but not all of the ramp and constant load data reasonably well. The estimates of variability (w) were slightly greater under ramp loading than under constant loading. Residual strength of specimens surviving constant load was less than expected. A greater duration-of-load effect was observed for the edge knot 2 by 4 lumber than that previously indicated for small clear-wood specimens; however, the difference does not appear to be statistically significant.

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