IntroductionThe study of the variation of mechanical properties along boards has been a recurrent subject since the beginning of the 1980s. The fluctuation in the modulus of elasticity (MOE), as well as in the tensile and compressive strengths along boards, has mostly been investigated in the context of numerical strength models for glued laminated timber (GLT) beams, for which a descriptive model for these variables is needed as input. Foschi and Barrett (1980) initiated this field of material science with a simulation model for the variation of MOE and tensile strength of Douglas fir boards based on Weibull’s weakest link (WL) theory (Weibull 1951). Investigations on the variability of mechanical properties of spruce in cells 150 mm in length by Ehlbeck et al. (1984) set the foundations for what is known as the Karlsruher Rechenmodel—a numerical finite-element–based GLT strength model. This model was reimplemented by Frese (2006), when it was calibrated for GLT beams of the hardwood species beech.Investigations by Kline et al. (1986) and Taylor and Bender (1988) added the spatial component to the variation analysis by considering the lag-correlations of both MOE and tensile strength. For this, boards were subdivided into segments of constant lengths, where the studied variables were measured. The obtained results were then used to calibrate an autoregressive (AR) model for the variables, later implemented in a GLT strength model (Taylor and Bender 1991). A related model was presented by Lam and Varoglu (1991b) for tensile strength profiles of spruce–pine–fir specimens for segments 610 mm in length. Based on experimental results (Lam and Varoglu 1991a), a moving average process of order 3 [MA(3)] was then developed.Isaksson (1999) focused on the variation of edgewise bending strength in spruce boards and developed a model based on the concept of weak zones, defined as those areas in a board that contained knots of significant size (defined by a relative threshold). In essence, the model considered the alternate arrangement of weak and strong sections along boards. Strong sections were attributed throughout to the same strength value in each individual board. The strength of weak sections and the distance between each consecutive weak section followed a stochastic model. This approach was then adopted by Fink (2014) in his study of spruce boards, in which a censored regression analysis was also used to estimate the parameters governing the variation of strength along the board.A rather neglected aspect of the variation of tensile strength in timber boards in general relates to the theoretical framework considered when analyzing the experimental strength distributions. Although it is common to assume WL theory for the study of timber members, thus a Weibull distribution, there is enough scientific evidence pointing at the inapplicability of this theory to quasibrittle materials, such as wood [see, e.g., Pang et al. (2008), Bažant and Le (2017)]. Although the weakest link concept, that is, structural failure triggered by the failure of the weakest element, applies generically, the Weibull distribution requires a very large, in fact infinite, number of links. This condition is met by perfectly brittle materials because their fracture process zone (FPZ) is negligible compared to the size of the structure. Thus, the number of representative volume elements (RVEs) is large enough for the strength to be approximated by the Weibull distribution. However, for quasibrittle materials, having a FPZ of nonnegligible size, the relatively low number of RVEs prevents the full development of the Weibull distribution, which is constrained to the lower tail of the distribution. On the opposite side of the failure spectrum, ductile failures occur, which are governed by the simultaneous mobilization of the strengths of all RVEs on the failure surfaces. This implies that the failure load is a weighted sum of each of these strengths, which, by the central limit theorem of probability, must converge to the Gaussian distribution (Pang et al. 2008). It has been experimentally shown (Bažant and Le 2017) that quasibrittle materials move from a predominantly Gaussian behavior, for relatively small sizes, to a predominantly Weibull behavior as the relative size increases, thus bridging plastic and brittle failure.Bažant and Pang (2007) considered this duality and developed the so-called finite weakest-link (FWL) theory—emphasizing the difference with the typical infinite weakest link theory—which introduces a smooth transition between plastic and brittle failure and is best described by a grafted Weibull–Gaussian distribution (discussed subsequently). The shortcomings of the Weibull theory for quasibrittle materials were previously addressed by Duxbury et al. (1987), too, who derived a statistical distribution for the strength of quasibrittle materials based on a fuse network model. This model has been proven to accurately describe experimental data (van den Born et al. 1991) and simulation results (Bertalan et al. 2014).In the first step of an experimental campaign on European oak boards (Quercus petraea, Q. robur), the MOE variation along the boards was investigated (Tapia and Aicher 2018, 2019). The free length of oak boards, tested in tension, was virtually subdivided into 15 segments (cells) 100 mm in length each, and the local MOE of each cell was measured. The MOE variation was studied based on serial correlation analysis and is explained in detail in Tapia and Aicher (2021b). In a second step, the boards were tested in tension until failure, and the remnants were tested, if possible, in secondary tests until failure.This paper presents an analysis of the multiple fracture tests, which is a prerequisite for a stochastic GLT strength model. The data of the measured local tensile strengths were analyzed using survival analysis, which permits accounting for the missing information in the data set, that is, for the tensile strength of the unbroken cells. The estimated parameters for the different models were then used to simulate tensile strength profiles along oak boards, which in turn allows accounting for the correlation between MOE and tensile strength profiles during the simulation process. Finally, the stochastic cell model was employed to simulate the effect of the board’s length on its global tensile strength.Materials and MethodsOak Boards, Grading, Global and Local MOEInvestigations into MOE variations, tensile strength, and density were performed using European oak boards (Q. robur or Q. petraea), originating from the southwestern part of France. The sample consisted of 52 boards and contained a mixture of appearance grades QF2 and QF3, according to EN 975-1 (CEN 2009). The two QF appearance grades intentionally enabled a wide range of knottiness and grain deviation in the sample and resembled the material used by Faye et al. (2017) on an investigation of GLT beams of oak. The nominal dimensions (length l×width w×thickness  t) of the planed boards were 2,500×175×24  mm, respectively. The moisture content (MC) was, on average, 10.2% (coefficient of variation [COV]=4.6%). The density adjusted to an MC of 12% was 699±47  kg/m3. Thereafter, the sample was visually graded at the Materials Testing Institute (MPA), University of Stuttgart, into hardwood strength grades LS7, LS10, and LS13, as specified in the German structural hardwood grading standard DIN 4074-5 (DIN 2008). The details of the strength grading, together with exact quantification and allocation of the growth defects of each board, are given in Tapia and Aicher (2019).The global modulus of elasticity was measured by axial tensile loading on a length lE=8.6·w=1,500  mm, resulting in a mean and standard deviation for the entire set of 11.6±2.1  GPa when corrected to 12% MC. For the subset of LS10 and LS13 boards, the MOE was 11.8±2.0  GPa. The length employed to measure the MOE lE differs from le=5·w stipulated in EN 408 (CEN 2012), owing to the specific objective of the tests: measuring the variation of the MOE along the longest possible free length. For determining the local tensile MOE variation along the boards’ length lE, each board was virtually subdivided into cells of 100 mm in length (Fig. 1), for which the MOE was measured in a discrete manner (Tapia and Aicher 2021b). The cell length of 100 mm was chosen for two reasons: (1) it is very close to the evaluation length of 90 mm, for which Olsson and Oscarsson (2017) found the best correlation between localized MOE based on fiber orientation measurements and bending strength, and somewhat shorter than the cell size used, among others, by Ehlbeck et al. (1984) and Frese (2006) for their strength models, and (2) it was the minimum length achievable by modifying an existing extensometer used to measure the localized MOE in the same boards (Tapia and Aicher 2021b). Because the objective was to study the localized MOE and tensile strength variation jointly, the same cell size of 100 mm was used to investigate the localized tensile strength, too.Multiple Strength MeasurementsThe boards were tested in tension parallel to the grain direction until failure at a constant piston displacement rate of 0.04  mm/min with a computer-controlled servo-hydraulic testing machine. Both ends of each board were fixed by hydraulically actuated grips, which gradually introduced the clamping pressure along the clamping length (≈350  mm), minimizing stress concentrations at the transition of free length to the clamps. Each board’s first testing to failure delivered the global tensile strength ft,0,glob corresponding to a free length ls of 1620  mm=9.25·w, which conforms closely to the provisions of EN 408 (CEN 2012) and EN 384 (CEN 2018), where ls=9·w is stipulated. Because the employed free length was slightly (3%) longer than specified in the European standards, the obtained global tensile strength values can be regarded as being conservative to a very small degree, which means that the obtained strength values tended to be slightly lower (about 1%) as compared to the standard approach. The mentioned strength decrease resulted from the fact that there is a higher probability of more and larger growth-bound defects (within the limits of the respective grading or strength class) in an increased free test-length, as compared to shorter ones. The stated quantification of the tensile strength reduction resulting from the increased free test length is based on the following length effect simulation results.The remaining parts of each broken board left after the first global ultimate failure tests were tested, if possible, in a second and occasionally third and fourth tensile test, similar to those done by Lam and Varoglu (1991a), then for spruce–pine–fir boards of 6.1 m in length. The location, that is, the number of the cell responsible for the failure, was recorded. The procedure of sampling multiple tensile strength values from one board is illustrated in Fig. 1. The outlined procedure was fostered by the predominant occurrence of blunt failures of the tested oak boards, producing two remaining parts separated by the first (global) failure by fracture planes rather perpendicular to the board’s length axis [Figs. 2(a and b)]. The free lengths (ls,2,i) between the grips of these secondary tensile tests were significantly lower than the primary global test. The values for ls,2,i were on average (±std.) (3.2±1.7)·w with a minimum of 0.6·w. However, the reduced lengths of the remnants do not constitute a methodological problem because the secondary tests are intended to detect the strengths of the stronger parts of the boards in relation to the weakest section in the board to obtain a larger database for local MOE, density, and strength correlations.Finite Weakest Link TheoryThe finite weakest-link theory deals with the statistical characterization of the failure behavior of quasibrittle materials and was developed in a series of works by Bažant and Pang (2006, 2007), and Le et al. (2011), among others. The theory describes the transition zone between ductile and brittle behavior by considering a power law in the lower tail of the strength distribution (brittle failure) combined with a normally distributed core (ductile failure). Such behavior is accurately represented by the so-called grafted Weibull–Gaussian (WG) distribution (Le and Bažant 2011; Bažant and Le 2017) as given in Table 1. There, xgr is the grafting point, m and δ are the shape and scale parameters of the two-parameter Weibull distribution, and μ and σ are the mean and standard deviation of the Gaussian core. The grafting probability F(xgr)=Fgr is defined as (1) Table 1. Statistical distributions and fitted parameters for the different investigated models to describe the tensile strength variation along board lengthTable 1. Statistical distributions and fitted parameters for the different investigated models to describe the tensile strength variation along board lengthParametersa(a)Weibull(δ,m)(b)Beta(λ,δ,a,b)(c)Weibull–Gaussian(μ,σ,δ,m)(d)DLBb(δ,κ,ρ)(e)Weibull(α0,α1,β0,β1)(f)Beta(α0,α1,β0,β1,γ0,γ1)with:λ=0δ=exp(α0+α1·Et,0,glob)m,a=exp(β0+β1·Et,0,glob)b=exp(γ0+γ1·Et,0,glob)Weibull:F(x)=1−exp[−(xδ)m]Beta:c,df(x)=Γ(a+b)Γ(a)Γ(b)·(x−λδ)a−1(1−x−λδ)b−1Weibull–Gaussian:eF(x)={1−exp[−(xδ)m],x≤xgrFgr+rfσ2π∬xgrxexp[−(x′−μ2σ)2]dx′,x>xgrDLB:F(x)=1−exp[−κ·exp(−δ·x−1/ρ)]To ensure that F(x) does not exceed 1, the factor rf needs to be scaled (Bažant and Pang 2007) as (2) rf=1−Fgr1−Φgr(μ,σ)where Φgr(μ,σ) = normal distribution N(μ,σ) evaluated at the grafting point. Furthermore, to ensure continuity of the PDF at xgr, the condition (3) must hold.A cornerstone in this theory is the representative volume element. A RVE can be defined as the smallest volume whose failure triggers the collapse of the whole structure (Pang et al. 2008), that is, the weakest link in a (finite) chain. In the present work, no effort was made to try to find the real RVE of the studied material. Instead, the 100-mm cells were assumed to work as RVEs, and their suitability was assessed by means of extreme value theory, as described subsequently. It should be clear, however, that each cell should be thought of as a hierarchical model of links connected in series and parallel. This means that local failures may occur within this RVE prior to achieving ultimate load because load sharing occurs internally.Application of Survival Analysis to the Tensile Strength ResultsSurvival analysis deals with assessing what is commonly known as censored data and is extensively used in different fields of science and technology. A common case of censoring in civil and mechanical engineering arises, for example, in the duration of load tests, when, at some given time ti, some specimens have failed, whereas others are still intact. For the survivor specimens, it is only known that their respective failure times lie above the time ti (Nelson and Hahn 1972). For the present case of studied oak boards, the censored data correspond to the tensile strength ft,0,cell associated with the unbroken cells, that is, those cells characterized only by an intrinsic tensile strength that is larger than a certain value. Survival analysis has been used before in a similar study by Fink (2014); however, the methodological approach was very different. Specifically, the likelihood function (see subsequently) was not considered, and, instead, an iterative method was applied.The problem consists of estimating the set of parameters θ of a candidate distribution function F(x) that describe the ft,0,cell variation. For this, maximum likelihood estimation (MLE) is commonly used, where the usual likelihood function (L(θ)) is modified to consider the censored data (Odell et al. 1992) according to (4) L(θ|X)=∏i=1nf(xi)δi·[1−F(xi)](1−δi)where f(x) = probability density function (PDF); and δi = indicator parameter signalizing whether the data were observed (censored), that is (5) δi={1,if  xi is an observed data point0,if  xi is a censored data pointThe main characteristic of the likelihood function [Eq. (4)] is the use of the survival function S(x)=[1−F(x)] for the censored data instead of the PDF. For practical reasons, it is typical to work with the logarithm of the likelihood function, then reading (6) logL(θ|X)=∑i=1nδi·log[f(xi)]+(1−δi)·log[1−F(xi)]Maximizing this function yields the maximum likelihood estimates θ^ for the tested distributions.In the presented analysis, six different strength distribution models were fitted to the ft,0,cell data. First, four censored parametric models were chosen: (a) a two-parameter (2p) Weibull distribution, (b) a beta distribution, (c) a grafted Weibull–Gaussian distribution, and (d) a Duxbury–Leath–Beale (DLB) distribution (Duxbury et al. 1987). Second, two censored regression models were considered: (e) Weibull and (f) beta, where the scale (δ) and shape parameters (m, and a, b) are functions of the global MOE of each board Et,0,glob. The inclusion of the 2p-Weibull distribution in the analysis is bound to the general assumption in timber engineering of adopting the weakest-link theory, although it is questionable due to the quasibrittle nature of the specific material. Nevertheless, the 2p-Weibull distribution shall serve as a reference point with the models derived specifically for quasibrittle materials, that is, models (c) and (d), evidencing possible problems of applying the Weibull distribution to analyze the experimental results. The inclusion of the beta distribution is not related to any underlying theory for the distribution of strength. It was chosen for its characteristic of being bounded at both sides, thus defining a lower and upper limit, which might be a desirable feature for stochastic finite-element (FE) simulations. These six models are presented in Table 1.The data fed to the models were constructed in the following way: 1.A binary vector δ of size 15·N (N = total number of tested boards; multiplier 15 represents the number of 100-mm-long cells per board) was assembled with either “1” or “0,” corresponding to whether a failure has been observed in the cell or not, respectively.2.A second vector of tensile strengths ft,0 of size 15·N is created in the following manner: •if δi=1, then the observed ft,0,i value is inserted, and•if δi=0, then the value assigned corresponds to the highest tensile strength registered during a test in which the ith cell was part of the free length of the board. This is interpreted as “this cell has at least a tensile strength ft,0,i.”3.For the case of models (e) and (f), representing the censored regressions, a third vector was created with the corresponding Et,0,glob value for each cell.The censored analysis was implemented with the Julia programming language (Bezanson et al. 2017), in which the Optim.jl package (Mogensen and Riseth 2018) was used to minimize the log-likelihood of each model. For the case of the grafted Weibull–Gaussian model (c), the likelihood function [Eq. (6)] was modified to consider the compatibility condition at the grafting point, that is, f(xgr−)=f(xgr+), as (7) logL(θ|X)=∑i=1nδi·log[f(xi)]+(1−δi)·log[1−F(xi)]−λ0·Δgrwhere Δgr=[f(xgr−)−f(xgr+)]2; and λ0 = Lagrange multiplier. Therefore, the grafting point is considered an additional parameter to be fitted.Within the presented analysis, the data were available for two lengths: the full free length of the boards of 1,500 mm and the cells of 100 mm in length because, under the framework of the FWL theory, the Weibull exponent m of the grafted Weibull–Gaussian distribution should be the same for the different sizes, Eq. (7) is maximized simultaneously for both lengths, where the Weibull exponent m is shared by both lengths, whereas the rest of the parameters are independent for each size. In this case, the typical definition of likelihood is used for the ft,0,glob data, modified into a Lagrangian function with the extra term λ0·Δgr. The source code of the models is available online in Tapia and Aicher (2021a).Model for the Simulation of Tensile Strength ProfilesThe simulation of localized tensile strength values ft,0,cell requires the previous generation of MOE profiles along the board Et,0,cell, as described by Tapia and Aicher (2021b). In a very brief summary, the presented simulation of Et,0,cell profiles consists of three steps: (1) generation of a Gaussian N(0,1) first-order autoregressive [AR(1)] process, creating normalized autocorrelated MOEs, termed Zt,0,cell, then (2) mapping the Zt,0,cell process into the so-called normalized MOE distribution, resulting in a new stochastic process termed E˜t,0,cell, and finally (3) the values E˜t,0,cell are multiplied by a scaling factor m0 to obtain the simulated Et,0,cell values.The currently proposed method to simulate localized ft,0,cell strengths was based on the concept of a slightly modified vector autoregressive (VAR) model, in which a set of stochastic processes are cross-correlated. In this case, a Gaussian process N(0,1) for normalized tensile strengths St,0,cell was generated based on a cross-correlation model with the previously simulated Zt,0,cell values. The values St,0,cell were then mapped into the correct distribution for ft,0,cell, estimated according to the previously described survival analysis methodology. The relevant steps for the simulation of ft,0,cell are as follows: •First, an AR(1) model for the stationary process Zt,0 was simulated as (8) Zt,0,i=φ1·Zt,0,i−1+εiwhere φ1 = Gaussian parameter of the model; and εi = white noise component with a standard deviation σ1=1−φ12.•Second, the normalized tensile strength profile St,0,cell, which depends on Zt,0,cell, was simulated as (9) where θ0 = cross-correlation coefficient between the normalized values of MOE and ft,0,cell. The white-noise term εi had, for this case, a standard deviation σ2=1−θ02. The parameters φ1 and θ0, which have been previously determined (Tapia and Aicher 2018, 2021b), are given in Table 2.•Third, the vector St,0,cell, which corresponds to all the St,0,i values in a board and was assumed to belong to a N(0,1) distribution, was mapped into the ft,0,cell distribution, which may be any of the fitted distributions F(x) presented in Table 1 by means of the transformation (10) ft,0,cell=F−1(Φ(St,0,cell))where F−1 = inverse of the cumulative distribution function (CDF); and Φ = CDF of the standard normal distribution. The transformation of Eq. (10) is commonly used to translate data from one distribution into another by preserving the original CDF [see, among others, Grigoriu (1984), Taylor and Bender (1988), Chen and Gopinath (2000), and Deodatis and Micaletti (2001)].Table 2. Parameters derived previously for the simulation model for the local tensile strengthTable 2. Parameters derived previously for the simulation model for the local tensile strengthParameter descriptionParameterAll cellsKAR < 0.05Coefficients for the AR(1) process for Zt,0,cellφ10.440.58Localized cross-correlation coefficient MOE versus ft,0θ0The vector ft,0,cell obtained via the outlined procedure corresponded to the simulated tensile strength profile, which has the following properties: (1) cross-correlation with Et,0,cell through the θ0 term, (2) implicit autocorrelation owed to the cross-correlation with Et,0,cell, and (3) statistical behavior equal to that obtained from the survival analysis performed with the experimental data. These characteristics, especially the last one, are analyzed subsequently in more detail.Order Statistics and Extreme Value TheoryOrder statistics addresses the statistical analysis of the rth smallest value in a sample of size n. More formally, if X1,X2,…,Xn constitute a sample drawn from a given distribution F(x) and are sorted in ascending order, such that X1: n≤⋯≤Xn: n, then the rth element of this sequence is defined as the r th-order statistic of the sample (Castillo et al. 2005). Of special interest are the first and last elements, that is, the minimum and maximum of X1,…,Xn, respectively, generally referred to as the extreme values.In the current study, the tensile strength is analyzed, which means that the minimum value, that is, the first-order statistic, has the highest relevance. In the considered empiric database, each board, subdivided into 15 cells of 100 mm in length, represents a sample of size 15, from which at least the minimum value (ft,0,glob) is known. If the parent distribution F(x), that is, the distribution considering n=15 cells, is known, then the distribution of the minimum value Fmin(x) can be simply derived (Castillo et al. 2005) from F(x) as (11) Bound to a correct fitting of the parametric models (a) to (d), as shown in Table 1, the application of Eq. (11) to the ft,0,cell distribution must yield the distribution for ft,0,glob, as shown subsequently.Results and DiscussionObserved Variation of Tensile Strength for All BoardsThe results obtained from the tensile tests for the global tensile strength ft,0,glob are presented in Table 3. The strength results of the secondary loading ft,0,i are omitted from the table because they cannot be presented in a meaningful, aggregated manner due to the different free lengths of each specimen (discussed subsequently). The mean and standard deviation of ft,0,glob for the LS10 and LS13 grades were 27.4±10.6 and 30.5±11.3  MPa, respectively. The characteristic values, computed according to EN 14358 (CEN 2016) for a lognormal distribution, were 12.8 and 13.9 MPa for LS10 and SL13 boards, respectively.Table 3. Statistical evaluation of the global tensile strength values of the oak boards, separately for the different hardwood strength grades according to DIN 4074-5 (DIN 2008)Table 3. Statistical evaluation of the global tensile strength values of the oak boards, separately for the different hardwood strength grades according to DIN 4074-5 (DIN 2008)GradeNMean (MPa)Std (MPa)ft,0,k (MPa)COV (%)Minimum (MPa)Maximum (MPa)Reject4.020.84.1—20.014.925.1LS74.029.612.7—43.08.640.4LS1017.027.410.611.839.013.149.2LS1322.030.511.613.038.014.655.3LS10+LS1339.029.111.313. a first approach to the analysis, Fig. 3(a) presents all (95) tensile strength values of those boards (n=37) where multiple strength measurements were possible. These data were obtained either in the first loading to fracture ft,0,glob or in the secondary loading tests ft,0,sec. Fig. 3(b) presents the same results, now as ratios ft,0,sec/ft,0,glob. The chosen presentation of the data enables a first assessment of the variability of ft,0,cell in the studied oak boards.In both Figs. 3(a and b), the boards are ordered from left to right by the assigned hardwood strength grade (LS) and within each LS grade group by ascending strength ratio ft,0,sec/ft,0,glob within each board. Substantial variations denoted by strength ratios up to almost 6 can be observed. However, the mean and standard deviation of the regarded strength ratio of 2.2±1.2 fosters the impression that this variation usually is not that extreme. Nevertheless, such an assessment is biased because the data are highly incomplete, owing to the physical impossibility of testing the tensile strength of each cell. As such, the real variation should probably be higher. This is analyzed subsequently in more depth.Estimated Parameters of the Local Tensile Strength ModelsTable 4 presents the estimated parameters of the six studied strength distribution models, which were fitted to the tensile strength data ft,0,cell of the whole set (N=47) and to the subset of grades LS10 and LS13 boards (N=39). Further, the logL and the Akaike information criterion (AIC) (Akaike 1974) are given for each model. The AIC serves to discriminate between different models based on the likelihood and number of parameters (AIC=2k−2logL, with k parameters). For the computation of the AIC of the Weibull–Gaussian (c) model, k=4 because the model is fully defined by the parameters δ, m, μ, and σ, whereas xgr is only needed for the fitting process. Due to numerical problems, the beta regression model could not be fitted to the data subset of LS10 and LS13 boards.Table 4. Parameters and statistical indicators for ft,0,cell estimated for the different fitted distributionsTable 4. Parameters and statistical indicators for ft,0,cell estimated for the different fitted distributionsLocScalelogLAICλ, μδm, a, κb, σ, ρ (a)Weibull—7.72×1012.90—−601.51,207.1 (b)Beta6.939.34×1012.151.23−600.31,208.5 (c)Wei.-Gauss.67.93.58×1015.802.80×101−599.81,207.6 (d)DLB—2.96×1011.19×1054.73−601.11,208.1 (e)Weibull (reg.)0α0: 4.04β0: 2.22×10−1—−591.81,191.5α1: 2.36×10−5β1: 8.01×10−5 (f)Beta (reg.)0α0: 5.90β0: 2.02×10−1γ0: 2.01−590.51,193.1α1: 5.50×10−5β1: 1.19×10−4γ1: 1.48×10−4 (a)Weibull—7.39×1013.15—−492.4988.7 (b)Beta10.67.93×1011.971.00−490.1988.1 (c)Wei.-Gauss.65.82.72×1019.322.52×101−489.6987.2 (d)DLB—3.17×1012.73×1054.68−492.3990.6 (e)Weibull (reg.)0α0: 3.91β0: 4.84×10−1—−484.9977.9α1: 3.11×10−5β1: 6.34×10−5According to the AIC, the Weibull regression model (e) is the best of the six models to simulate the censored ft,0,cell data (smallest AIC). This means that the incorporation of the modulus of elasticity data Et,0,glob has a positive and meaningful impact on the description of ft,0,cell. This observation applies to the subgroup of LS10+LS13 grades, too. Considering only the parametric models, the Weibull–Gaussian (c) presents the maximum logL value, meaning that the quality of the fitting is better for this model, with the beta (b) model also being relatively close.Figs. 4 and 5 show the CDF and PDF distributions of localized tensile strength according to all four fitted parametric models, for all boards and the subsample of LS10+LS13 boards, respectively. The CDF distributions in Figs. 4(a) and 5(a) are given in a Weibull plot representation and compared to the nonparametric Kaplan–Meier (KM) estimator (Kaplan and Meier 1958).Good agreement can be observed between all model curves and the KM estimator on the upper part of the strength distribution. However, as apparent, the models behaved fundamentally differently at the lower tail of the distributions. As predicted by the FWL theory, a rather abrupt change in the slope of the data was observed—most pronounced in the LS10+LS13 subgroup [Fig. 5(a)]. It is clear that the 2p-Weibull distribution could not follow this curve because the equation represents a straight line in the Weibull plot. The beta distribution seems to fit the data much better. However, because it imposes a lower limit λ=6.9  MPa and 10.6 MPa for the case of all boards and the LS10+LS13 subgroup, respectively, this model is expected to present similar problems as the 3p-Weibull distribution [see Bažant and Le (2017), and Pang et al. (2008)], which is explained further subsequently. The WG model (c) presents a fit that is visually similar to the beta model, with matching likelihood values, too. The grafting point—marked with a circle in Figs. 4(a) and 5(a)—was determined as xgr=15.0  MPa and 14.6 MPa for all boards and the LS10+LS13 subgroup, respectively. Finally, the DLB distribution, although in principle able to follow the specific nonlinear type of curve, could not be fitted to the same degree as, for example, the WG distribution (c), which is graphically and numerically evident from Figs. 4(a) and 5(a) and Table 4.The differences on the midupper region of the distributions are evidenced in Figs. 4(b) and 5(b), where models (a), (c), and (d) show similar behaviors, whereas model (b), beta, is characterized by a sudden drop in its PDF, marking the end of its support. This drop was estimated for the case of all boards [Fig. 4(a)] as loc+scale≈100  MPa. For the case of the LS10+LS13 subgroup, the beta model (b) exhibited a sharp end at about 90 MPa, determined by the shape parameter b=1 [Fig. 5(a)]. The beta model was the only model presenting a clear skewness “to the left,” which is expressed by an abrupt descent of the right tail, whereas all other models showed more symmetric behaviors, even slightly right-skewed, reaching higher ft,0,cell values.Figs. 6(a and b) illustrate the results of the fitted Weibull and beta regression models (e) and (f) for all grades, respectively, by means of several PDF curves corresponding to different Et,0,glob values in the range of 8–16 GPa. The expected mean strength values of the regression models fitted to the LS10+LS13 subset are shown in Fig. 6(a), too. Here, a clear increase in the expected value of ft,0,cell was observed from both models with growing Et,0,glob. Thus, these two models capture both the variation of ft,0,cell and its correlation with the global MOE of boards. The behavior of both models was, however, different. The Weibull model (e) started with a rather right-skewed (s=0.42) and widespread distribution for lower Et,0,glob and evolved toward a more symmetric distribution (s=−0.18) with less spread for higher Et,0,glob values. The beta regression model (f) started with a widespread, right-skewed distribution for lower Et,0,glob values, too. However, in contrast to the Weibull model, the shape of the PDFs stayed rather constant for higher Et,0,glob, only shifted upward toward higher ft,0,cell values.Models also considering the location parameter λ as a free parameter (i.e., λ≠0, both dependent and independent of Et,0,glob) did not render satisfactory results but, in fact, resulted in mechanically unlikely distributions. This could be related to insufficient data points or numerical instabilities, preventing finding the optimum result. For highly nonlinear problems with many parameters to fit, choosing the right initial values has an important influence on the outcome. For the analyzed models with λ=0, the estimated parameters represented the optimal ones, which was proven by checking multiple different combinations of the initial conditions.Verification of the Parametric Models by Extreme Value TheoryThe goodness of the fitted parametric beta (b) and Weibull–Gaussian (c) models can be checked by extreme value theory. As previously mentioned, if the fitted models describe the tensile strength variation along the board, then the application of Eq. (11) to the CDF of the fitted models should match the experimental values of ft,0,glob (minimum value of each board). This is demonstrated graphically in Figs. 7 and 8 for all boards and for the LS10+LS13 subsample, respectively. The experimental results were plotted by applying the midpoint position method P(x≤xi)=(i−0.5)/n (Rinne 2009).It is apparent that the theoretical minimum CDF curve Fmin(x) associated with model (c) matched rather exactly both the experimental results and the WG distribution fitted to the global results (fitted simultaneously with local values, as described previously) using an exponent n=15 in Eq. (11). This is the desired behavior in a stochastic model for ft,0,cell because this means that the model should be able to capture the size effect (discussed subsequently) in a relatively good manner. It further ensures that a power law describes the size effect of the lower tail. The behavior of model (b), illustrated in Figs. 7(b) and 8(b), showed in general good agreement with the global data. Nevertheless, the observed behavior at the lower tail indicates that the size effect will not be governed by a power law, which is the same problem as observed with the 3p-Weibull distribution; that is, the model will tend to an asymptotic limit for larger lengths (Bažant and Le 2017). However, for realistic board lengths of the regarded hardwood species, limited to a few meters (max.≈3  m), the problem might not be completely evident from simulation results (see subsequently).Regarding the censored regression models (e) and (f), a similar analysis could not be performed because their parameters depend on Et,0,glob. This would need a comparison against experimental data for different specific Et,0,glob values, which is not possible for this data set due to the rather small sample size.Simulation of Tensile Strength Profiles along Board LengthBased on the estimated parameters of the studied models for ft,0,cell, tensile strength profiles of virtual boards can be generated using Eqs. (8)–(10) of the described vector autoregressive model. Figs. 9(a and b) show profiles of ft,0,cell along two boards (I, II) generated with the beta model (b). Furthermore, the figures include the corresponding Et,0,cell values, simulated according to Tapia and Aicher (2021b).For both simulated profiles, a clear correlation between Et,0,cell and ft,0,cell can be observed. Both curves moved similarly, but differences arose due to the introduced white noise component εi in Eq. (9). In general, regions of lower Et,0,cell values also present relatively low ft,0,cell values, and vice versa. The Et,0,glob value needed for MOE profile simulation was taken randomly from the statistical distribution for global MOE given in Tapia and Aicher (2021b).The relatively high variation in both Et,0,cell and ft,0,cell was due to the characteristics of the studied oak board material, which made up in essence two strength grades, LS10 confined at the lower and upper bounds by grades LS7 and LS13, respectively, and LS13 and better. Nevertheless, the presented method is independent of the specific board material and considered applicable to different data sets showing more or less internal MOE or tensile strength variation.Simulation of Length Effect for Tensile StrengthThe decrease of strength with the growing size of wooden (compound) materials was first addressed by Newlin and Trayer (1924) and Freas and Selbo (1954) in a phenomenological manner and related to bending strength as dependent on cross-sectional depth. In addition, Weibull (1951) derived a universal mechanical–mathematical model for the effect of size on strength of brittle materials, based on a higher probability of occurrence of a more detrimental defect, also termed weakest link (WL) in a larger structure. Subsequently, Barrett (1974) introduced the (WL) approach for characterizing quasibrittle material wood, and then, specifically, the size effect in tension perpendicular to the fiber direction. First investigations on the effect of length of boards on tensile strength parallel to fiber—all related (implicitly) to the WL concept—were reported by Showalter et al. (1987), Bechtel (1988), Lam and Varoglu (1990), Barrett and Fewell (1990), Madsen (1990), Rouger and Barrett (1995), Burger and Glos (1996), and Zhou et al. (2010) and, more recently, for example, by Fink (2014) and Blank et al. (2017). Findings of some of the mentioned studies are discussed subsequently in comparison with the results derived here, including the finite weakest-link theory (Bažant and Pang 2007), which so far has not been applied for the description and simulation of length effects in timber.The fact that the previously presented modified vector autoregressive model can simulate the effect of size, that is, of board length, on the tensile strength results directly from the “cell”-centered stochastic nature of the model and is tightly related to its compliance with the extreme value theory (shown previously). In order to demonstrate the size effect produced by the simulation method, virtual boards of seven different lengths, 500–3,500 mm in steps of 500 mm, were generated applying the fitted models (b), (c), (e), and (f) for ft,0,cell considering all grades and the subgroup LS10+LS13. For each length, a total of 50,000 simulations per ft,0,cell model were performed. The simulation of different lengths does not require any special consideration beyond generating the needed amount of cells according to Eqs. (8)–(10). After this, the smallest ft,0,cell value of each virtual board was taken as the ft,0,glob value.Fig. 10 shows the results of the simulations for the seven different board lengths, based on the parameters specified in Table 4 for all grades. A pronounced size effect can be observed for the four studied models, with diminishing strength values as the length increases. Table 5 gives the means and standard deviations of the tensile strength of the different board lengths for the fitted ft,0,cell models. It is evident from Table 5 that the parametric beta (b) and Weibull–Gaussian (c) models show a very similar evolution in both mean strength values and spread. More importantly, regarding the agreement with the empiric values (Table 3), both models simulate the mean tensile strength of the reference length (n=15=1500  mm) almost exactly. Similarly, the standard deviation is just slightly underestimated by 4% for models (b) and (c).Table 5. Simulated global tensile strength values ft,0,glob (mean±std.), all grades, for models (b), (c), (e), and (f) and different board lengthsTable 5. Simulated global tensile strength values ft,0,glob (mean±std.), all grades, for models (b), (c), (e), and (f) and different board lengthslboard (mm)5001,0001,5002,0002,5003,0003,500 (b)41.8±16.732.7±12.928.3±10.825.6±9.423.7±8.522.3±7.821.2±7.3 (c)41.6±16.832.5±12.828.1±10.725.4±9.323.5±8.422.2±7.621.1±7.1 (e)42.3±16.433.9±13.629.8±12.227.1±11.325.2±10.723.8±10.222.7±9.8 (f)42.6±16.934.1±13.130.1±11.527.7±10.526.0±9.924.7±9.523.6±9.11′′Regarding the results of the regression-based ft,0,cell models, the Weibull model (e) showed a similar size effect trend as those obtained for models (b) and (c). However, the mean and standard deviation for the reference length deviated from the empiric results by 5% and 9%, respectively. Similar behavior was observed for model (d), in which the mean and standard deviation differed from the experiments by 6% and 3%, respectively. The reason for the slightly higher deviations of the regression models is probably related to the specific Et,0,glob distribution used to generate the Et,0,glob input data because the models directly depend on it. A larger data set would probably help improve the model prediction quality because both the parameters estimated for the distribution of Et,0,glob and models (e) and (f) would be determined with higher confidence.To quantify the simulated length effect comprehensively, a power law of the form (12) was fitted to the results of each of the four models, where ft,i = tensile strength for a length li; and the reference length lref=1,500  mm. The quantification of the size effect by means of a power law as a generalized description of the size impact, irrespective of underlying failure mechanisms (WL-concept or fracture mechanics), has become an internationally adopted approach in test evaluation and timber design codes, for example, EN 384 (CEN 2018), EN 1995-1-1 (CEN 2014), ASTM D245 (ASTM 2019), and ASTM D6570-18a (ASTM 2018). The obtained results for the exponent ξ are presented in Table 6 and Fig. 11. Figs. 11(a and b) illustrate the mean and 5%-quantile results for the simulated data. Disregarding first the differences between the slopes at the mean and 5%-quantile levels, a clear linear trend was observed in the log-log plot of the ft,0 values versus length for all cases. This implies that the assumed power law of Eq. (12) is adequate for the investigated lengths at both distribution levels. Considering all boards, the size effect exponents ξ obtained on the mean level ranged from 0.31 to 0.35, whereas for the 5% quantile, ξ moved between 0.18 and 0.34. For the subset of grades LS10 and LS13 only, the exponents ξ were slightly lower, ranging from 0.30 to 0.33 on the mean level and between 0.14 and 0.31 for the 5% quantile.Table 6. Size exponents ξ describing the length effect of the tensile strength for the sets of boards simulated by models (b), (c), (e), and (f)Table 6. Size exponents ξ describing the length effect of the tensile strength for the sets of boards simulated by models (b), (c), (e), and (f)ξ-level(b)(c)(e)(f) ξ–mean0.350.350.320.31 ξ–5%-quant. ξ–mean0.320.320.30— ξ–5%-quant.—The differences observed for the ξ values between the mean and 5%-quantile levels are best explained in terms of FWL theory. FWL predicts that a power law such as that described in Eq. (12) is only valid in the asymptotic limit, that is, for large enough structures in which the Weibull part of the Weibull–Gaussian distribution clearly dominates. In this limit, the relation ξ=1/m must hold, where m is the Weibull module of the fitted WG distribution. It is immediately evident that the 5%-quantile level has already reached or is very close to the asymptotic limit, given that for all boards, ξ=0.17=1/5.9≈1/5.5=1/m, and for the LS10+LS13 subgroup, ξ=0.14=1/7.1≈1/9.0=1/m.The general expression for the size effect of quasibrittle materials as proposed by Bažant et al. (2009), Le et al. (2011), and Carloni et al. (2019), is defined as (13) σNc=fr∞[(DaD)rn/m+rDbD]1/rwhere σNc = global strength; fr∞ = strength parameter related to the fracture process zone and material energy; Da, Db, and r = scaling parameters; m = Weibull modulus; and n = number of spatial dimensions in which the structure is scaled. Eq. (13) describes a monotonically decreasing function with an asymptotically decreasing slope that converges to 1/m (Fig. 12). This relation has been experimentally verified for different quasibrittle materials [e.g., Le et al. (2013)]. It can be deduced, then, that the mean values are still in the transition zone of the curve defined by Eq. (13) because the sizes are not large enough to make the Weibull component of the WG relevant. However, it is reasonable to think that this is not necessarily true for the 5%-quantile level, given that there is a higher probability for these values to be bound to the lower tail of the distribution, that is, to the Weibull region of the WG distribution. For example, the parameter Db in Eq. (13) may be different at the mean and 5%-quantile level, which is illustrated in Fig. 12 for arbitrary parameters. In this manner, the observed difference between the size effects computed for the mean and 5%-quantile levels is coherently explained, and the observed difference in ξ is only a consequence of the quasibrittle nature of the studied material and the simplified scaling law of Eq. (12).Regarding an assessment of the derived exponents for oak boards and any other hardwood species, no reference values could be found. Hence, in a first approach, the quantitative discussion of the simulation-based ξ values is related to the size factors given in the literature for softwoods.Based on a metastudy of an extensive database (27 subsets, 3,070 specimens) by Showalter et al. (1987), Lam and Varoglu (1990), and (Madsen 1990) on Canadian spruce–pine–fir (SPF) and Southern pine (US) of different structural grades, Barrett and Fewell (1990) proposed an exponent of ξ=0.17 for the 5%-quantile level [5%-quantiles when computed according to CEN TC 124.202 (CEN 1989)]. Further, it was concluded that the length effect in tension and bending is alike. Rouger and Barrett (1995) supported the stated length effect exponent ξ=0.17 and discussed the variables that can influence the sample size effect. These are mainly: (1) the sawing pattern; and (2) the used grading system. Specific sawing patterns may lead to an apparent size factor up to 0.37. The effect of the grading system was illustrated with lower and higher quality samples graded both visually and with regard to MOE by machine grading. Whereas for the MOE-graded material, ξ=0.23 was found, the visually graded boards showed ξ values ranging from –0.5 to 0.1. Burger and Glos (1996) conducted an experimental study with 750 European spruce (Picea abies) boards with knot area ratios ranging from 0.16 to 0.61. The valid (n=730) test results from three significantly different lengths (0.15, 1.0, and 2.5 m) yielded size exponents of 0.13 and 0.22 for the mean and characteristic levels, respectively (5%-quantiles where computed by using a 3p-Weibull distribution).Considering the cited literature on softwood results, it can be stated that the simulation-based length effect for the oak boards is well within the experimental range at the 5%-quantile level. Focusing exclusively on the subset LS10+LS13, being more relevant for structural purposes, the parametric models (b) and (c) yielded ξ values of 0.19 and 0.14, rather similar to the softwood results. The Weibull regression model (e) gave a much higher value, relatively close to the mean level results. Nevertheless, the computation of experimental 5%-quantiles is complicated because it is highly sensitive to the number of specimens and to the assumed strength distribution, which will impact the computed size effect. For example, the fact that the size effect exponents at the mean level are smaller than at the 5%-quantile level in Burger and Glos (1996) goes against the predictions made here based on FWL theory. Therefore, although good agreement is obtained, caution is needed when comparing simulations with experimental results at this quantile level. At the more stable mean level, the obtained results for the size exponent ξ are up to two times higher as compared to the literature results for softwoods. This might be due to intrinsic differences between oak and softwood species; however, additional experiments are needed to enable a decisive statement on this aspect.From an overall perspective, it can be concluded that the presented order statistics approach, based on localized (cell-wise) strength values, provides conceivable results for the length effect. Nevertheless, the magnitude of the length effect exponent, including the model validation, must be investigated experimentally with larger samples.DiscussionThe two approaches used to estimate the within-board variation of tensile strength present some implications beyond simply simulating a ft,0 profile for a single board. One of the major differences between the simple parametric models (a), (b), (c), and (d) and the censored regression models (e) and (f) consists in their practical application to simulate a large data set of virtual boards. As is widely known, there is a positive correlation between global MOE and tensile strength of wooden boards, which has been repeatedly shown for different species, including oak. Consequently, an aim should be to replicate such a correlation when simulating a population of boards with the presented models. In this sense, it is evident that models (e) and (f) intrinsically carry the information needed to simulate the correlation with the global MOE because Et,0,glob is used as a variable in the model. The parametric models, however, lack any correlation with global MOE because they correlate only to the localized Et,0,cell through the variable Zz,0,cell and would require an extra step to account for the observed Et,0−ft,0, correlation.However, there is also an advantage in this latter approach; that is, by decoupling the simulation of Et,0 and ft,0 at a board level, it is easier to generate data that follow exactly the experimental distributions of Et,0 and ft,0. This was shown in the previous section, where models (e) and (f) could not reproduce the experimental distribution for ft,0,cell as precisely as models (b) and (c). Of course, this is not only model-bound but probably related, as mentioned previously, to the influence of the size of the data set.It can be argued that a model of type (b) resembles the concept of weak and strong sections used by Isaksson (1999), Fink (2014), and Blank et al. (2017), among others. This results from the fact that the fitted beta distribution will, in general, assign very similar high values to a large portion of a board (≈ strong sections) while producing a smaller number of regions with relatively low tensile strength (weak sections). The fact that this distribution is not compatible with the FWL theory raises a question about its applicability for simulation purposes. However, as shown, at least for the relevant board lengths, the results of the beta distribution are comparable to the Weibull–Gaussian approach.Finally, the presented analysis demonstrates the benefit of incorporating the finite weakest link theory to the strength analysis of timber elements for brittle failure modes. This includes tensile strength parallel and perpendicular to the fiber, as well as shear strength. The fitting of the grafted Weibull–Gaussian distribution to tensile strength data of boards will yield the asymptotic size effect coefficient 1/m, which was shown to be relevant for the 5%-quantile level of boards, but should also be relevant for the mean level of larger structures composed of individual boards such as glued laminated timber beams. This dependency should be studied in depth because it could have a large impact on the amount and dimensions of the typically required specimens needed to assess the size effect of GLT beams.References ASTM. 2018. 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