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    Author(s): Steve P. Verrill; Frank C. Owens; Matthew A. Arvanitis; David E. Kretschmann; Rubin Shmulsky; Robert J. RossPatricia Lebow
    Date: 2020
    Source: Res. Pap. FPL-RP-705. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 47 p.
    Publication Series: Research Paper (RP)
    Station: Forest Products Laboratory
    PDF: Download Publication  (1.0 MB)


    To evaluate the reliability of lumber structures, we need (among many other things) good models for the strength and stiffness distributions of visual and machine-stressrated (MSR) grades of lumber. Verrill et al. established theoretically and empirically that the strength properties of visual and MSR grades of lumber are not distributed as twoparameter Weibulls. Instead, strength properties of grades of lumber must have (at least to a first approximation) “pseudo-truncated” distributions. To properly implement Verrill et al.’s pseudo-truncation theory, we must know the true mill run modulus of elasticity (MOE) and modulus of rupture (MOR) distributions. Owens et al. investigated the mill run distributions of MOE and MOR at two times for each of four mills. They found that univariate mill run MOE and MOR distributions are well-modeled by skewnormal distributions or mixtures of normal distributions, but not so well-modeled by normal, lognormal, two-parameter Weibull, or three-parameter Weibull distributions. Verrill et al. investigated a mixture of two bivariate normals model for the mill run bivariate MOE–MOR population at a single time at a single mill. (Some possible sources of two-component mixture relationships include a mixture of trees from a fast-grown plantation stand and a suppressed stand, trees of two separate species, small-diameter trees and large-diameter trees, and lumber from the pith region versus lumber from the bark region.) They found that a mixture of two bivariate normals model performed well. In this paper, we apply this model to all eight of the Owens et al. lumber samples. We find that the model continues to yield good fits. However, we also find that the fits differ from mill to mill and time to time. Some variability is, of course, to be expected. However, we find that the fitted models differ to such an extent that the calculated probability that a piece of lumber randomly drawn from a fixed “grade” breaks at a fixed load can vary by a factor as large as 35 when we permit both season and mill to vary, and as large as 15 when we permit only mill to vary. Similar factors were found when we replaced fixed loads with loads randomly drawn from fixed load distributions.

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    • This article was written and prepared by U.S. Government employees on official time, and is therefore in the public domain.


    Verrill, Steve P.; Owens, Frank C.; Arvanitis, Matthew A.; Kretschmann, David E.; Shmulsky, Rubin; Ross, Robert J.; Lebow, Patricia. 2020. Estimated probability of breakage of lumber of a fixed “grade” can vary greatly from mill to mill and time to time. Res. Pap. FPL-RP-705. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 47 p.


    Machine-stress-rated lumber, MSR lumber, MOE binned lumber, visually graded lumber, lumber property distribution, lumber reliability, bivariate normal, mixture of bivariate normals

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