DISTRIBUTIONS OF MODULUS OF ELASTICITY AND MODULUS OF RUPTURE IN FOUR MILL RUN LUMBER POPULATIONS
Keywords:full lumber population, structural lumber, mill run, modulus of elasticity, modulus of rupture, normal distribution, Weibull, lognormal, skew normal, mixed normal, pseudo-truncation
The modulus of elasticity (MOE) and modulus of rupture (MOR) of graded lumber populations are commonly modeled by normal, lognormal, or Weibull distributions, but recent research has cast doubt on the appropriateness of these models. Such modeling has implications for ultimate performance and efficiency of resource use. It has been shown mathematically that the distribution of MOR in a graded subpopulation does not have the same theoretical form as the full, ungraded (or “mill-run”) population from which it was drawn; rather, its form is pseudo-truncated, exhibiting thinned tails. Although the phenomenon of pseudo-truncation in graded populations has been well substantiated, the form of the underlying full distribution—an essential factor in characterizing the distribution of the graded population—remains unsettled. The objective of this study was to characterize the distributions of both MOE and MOR in four diverse mill-run lumber populations to determine if and to what extent the distributions of strength and stiffness in mill-run lumber are similar from mill to mill. The authors collected a mill-run sample of 200 southern pine 24 specimens from each of four sawmills, for a total of 800 test pieces. After measuring MOE and MOR, they fit candidate distributions to those data by mill and evaluated each distribution for goodness of fit. Results suggest that perhaps none of the traditional distributions of normal, lognormal, or Weibull is adequate to model MOE or MOR across all four mills; rather, MOE and MOR in full lumber populations might be better modeled by skew normal or mixed normal distributions.
ASTM (2015) Standard test methods of static tests of lumber in structural sizes. D198-15. Annual book of ASTM standards. ASTM, West Conshohocken, PA.
ASTM (2016) Standard practice for establishing allowable properties for visually-graded dimension lumber from ingrade tests of full-size specimens. D1990-16. Annual book of ASTM standards. ASTM, West Conshohocken, PA.
ASTM (2017a) Standard practice for sampling and data analysis for structural wood and wood-based products. D2915-17. Annual book of ASTM standards. ASTM, West Conshohocken, PA.
ASTM (2017b) Standard specification for computing reference resistance of wood-based materials and structural connections for load and resistance factor design. D5457-17. Annual book of ASTM standards. ASTM, West Conshohocken, PA.
D’Agostino RB, Stephens MA (1986) Goodness-of-fit techniques. Marcel Dekker, New York, NY.
Evans JW, Johnson RA, Green DW(1997) Goodness-of-fit tests for two-parameter and three-parameter Weibull distributions. Pages 159-178 in NL Johnson and N Balakrishnan, eds. Advances in the theory and practice of statistics: A volume in honor of Samuel Kotz. Wiley, New York, NY.
Galligan WL, Hoyle RJ, Pellerin RF, Haskell JH, Taylor JR (1986) Characterizing the properties of 2-inch softwood dimension lumber with regressions and probability distributions: Project completion report. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory, Madison, WI.
Green DW, Evans JW (1987) Mechanical properties of visually graded lumber: Volumes 1-8. U.S. Department of Commerce, National Technical Information Service, Springfield, VA.
Gross J, Ligges U (2015) nortest: Tests for normality. R package version 1.0-4. https://CRAN.R-project.org/package.nortest. (31 May 2018).
Krit M (2017) EWGoF: Goodness-of-fit tests for the exponential and two-parameter Weibull distributions. R package version 2.2.1. https://CRAN.R-project.org/package.EWGoF. (31 May 2018).
Owens FC, Verrill SP, Kretschmann DE, Shmulsky R (2018) Distributions of MOE and MOR in a full lumber population. Wood Fiber Sci 50(3):265-279.
R Core Team (2013) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org (31 May 2018).
Ross RJ, (ed.) (2015) Nondestructive evaluation of wood: Second edition. General Technical Report FPL-GTR-238. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory, Madison, WI. 169 pp.
Verrill SP, Evans JW, Kretschmann DE, Hatfield CA (2012) Asymptotically efficient estimation of a bivariate Gaussian-Weibull distribution and an introduction to the associated pseudo-truncated Weibull. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory Research Paper FPL-RP-666, Madison, WI. 76 pp.
Verrill SP, Evans JW, Kretschmann DE, Hatfield CA (2013) An evaluation of a proposed revision of the ASTM D 1990 grouping procedure. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory Research Paper FPL-RP-671, Madison, WI. 34 pp.
Verrill SP, Evans JW, Kretschmann DE, Hatfield CA (2014) Reliability implications in wood systems of a bivariate Gaussian-Weibull distribution and the associated univariate pseudo-truncated Weibull. ASTM J Test Eval 42(2):412-419.
Verrill SP, Evans JW, Kretschmann DE, Hatfield CA (2015) Asymptotically efficient estimation of a bivariate Gaussian-Weibull distribution and an introduction to the associated pseudo-truncated Weibull. Commun Stat Theor Methods44:2957-2975.
Verrill SP, Owens FC, Kretschmann DE, Shmulsky R (2017) Statistical models for the distribution of modulus of elasticity and modulus of rupture in lumber with implications for reliability calculations. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory Research Paper FPL-RP-692, Madison, WI. 51 pp.
Verrill SP, Owens FC, Kretschmann DE, Shmulsky R (2018) A fit of a mixture of bivariate normals to lumber stiffness strength data. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory Research Paper FPLRP-696, Madison, WI. 44 pp.
Verrill SP, Owens FC, Kretschmann DE, Shmulsky R, Brown L (2019) Visual and MSR grades of lumber are not two-parameter Weibulls and why it matters (with a discussion of censored data fitting). Under review. http:// www1.fpl.fs.fed.us/weib2.new.pdf (12 November 2018).
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