Statistical models for the distribution of modulus of elasticity and modulus of rupture in lumber with implications for reliability calculations

Abstract

It 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 "pseudotruncation." The theoretical 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 Contents

A fit of a mixture of bivariate normals to lumber stiffness–strength data

“…In section 4.2 of Verrill et al (2017) we noted that graphical evidence from the 200-piece sample suggested that bivariate stiffness-strength data might have approximately the distribution of a mixture of two bivariate normal distributions. We further noted that this would explain the good separate fits of mixtures of univariate normals to the stiffness data and to the strength data.…”

“…However, these assumptions are based on the work of researchers who were fitting distributions to grades of lumber (as in the In-Grade work) rather than to mill run populations. This fact, and the need for an accurate estimate of the MOR probability density function in reliability calculations, led us to perform the experiment described in Verrill et al (2017) and Owens et al (2018) -an experiment designed to yield estimates of mill run bivariate stiffness-strength distributions (which, in turn, yield estimates of pseudo-truncated MOR distributions).…”

“…However, we felt that it would be useful to first address the fundamental question of whether a "simple" population could be modeled by a "simple" distribution such as a normal or two-parameter Weibull. In Verrill et al (2017) and Owens et al (2018), we reported stiffness-strength data and distribution fits for a sample of 200 pieces of mill run 2x4 lumber obtained from a single mill on a single day.…”

“…The models discussed in section 4.1 of Verrill et al (2017) were "univariate" models. That is, they were models for the distributions of single variables (sb MOE 1 , ecomp 1 , dire 1 , or MOR 1 ).…”

“…Dire was obtained by measuring acoustic velocity via a Fibre-gen Director HM200 (Fibre-gen, Christchurch, New Zealand). Details can be found in section 3 of Verrill et al (2017).…”