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Statistical models for the distribution of modulus of elasticity and modulus of rupture in lumber with implications for reliability calculationsAuthor(s): Steve P. Verrill; Frank C. Owens; David E. Kretschmann; Rubin Shmulsky
Source: Res. Pap. FPL-RP-692. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 51 p.
Publication Series: Research Paper (RP)
Station: Forest Products Laboratory
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DescriptionIt is common practice to assume that a two-parameter Weibull probability distribution is suitable for modeling lumber properties. Verrill and co-workers demonstrated theoretically and empirically that the modulus of rupture (MOR) distribution of visually graded or machine stress rated (MSR) lumber is not distributed as a Weibull. Instead, the tails of the MOR distribution are thinned via “pseudo-truncation.” The theoretically portion of Verrill’s argument was based on the assumption of a bivariate normal—Weibull (Gaussian—Weibull) MOE—MOR distribution for the full population of lumber (as opposed to the bivariate distribution of visual or MSR grades of lumber). We felt that it was important to investigate this assumption. In the absence of data sets in the literature that were drawn from the full population at a mill, we determined to obtain such a sample for analysis. In this paper, we report the results from this analysis. From the current experiment on mill run lumber, we conclude that if reliability engineers are entertaining the idea of obtaining new efficiencies via careful probability modeling of strength properties, then additional experimental research must be done on the fundamental question of valid models for stiffness and strength distributions for full populations of lumber from a single mill on a single day. Further, we suspect that even if research determines that a simple model can characterize such a distribution, further research will determine that this simple model varies from day to day, mill to mill, and region to region so that an ever-changing mixture model is the correct model. In this case, to ensure that reliability goals are efficiently met, reliability engineers might need to develop detailed computer models that yield real-time, in-line estimates of lumber strength based on measurements of stiffness, specific gravity, knot size and location, slope of grain, and other strength predictors.
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CitationVerrill, Steve P.; Owens, Frank C.; Kretschmann, David E.; Shmulsky, Rubin. 2017. Statistical models for the distribution of modulus of elasticity and modulus of rupture in lumber with implications for reliability calculations. Res. Pap. FPL-RP-692. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 51 p.
Keywordsnormal distribution, mixed normal distribution, two-parameter Weibull distribution, three-parameter beta distribution, skew normal distribution, mixed bivariate normal distribution, bivariate Gaussian-Weibull distribution, pseudo-truncated Weibull distribution, machine stress rated data, MSR data, probability density functions, goodness-of-fit, mill run, thin tail, lumber property distribution
- Maximum likelihood estimation of the parameters of a bivariate Gaussian-Weibull distribution from machine stress-rated data
- A fit of a mixture of bivariate normals to lumber stiffness—strength data
- Distributions of modulus of elasticity and modulus of rupture in four mill run lumber populations
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