Simulation of the Localized Modulus of Elasticity of Hardwood Boards by Means of an Autoregressive Model
Publication: Journal of Materials in Civil Engineering
Volume 33, Issue 6
Abstract
An autoregressive (AR) model for the simulation of modulus of elasticity (MOE) fluctuations along hardwood boards without pronounced knot periodicity is presented. The model is calibrated based on contiguous, localized (100 mm gauge length) empirical MOEs from European white oak boards. The calibration involves the computation of the sample autocorrelation function (SACF). For this computation, the stationary data resulting from a two-step transformation method are used: (1) first, board-internal MOE normalization suppresses the interboard bias; and (2) the rather left-skewed normalized data are described by a log-gamma distribution, which is then mapped to to obtain the needed stationary data. A first-order AR process suffices to accurately describe the data and is thus used as a basis to simulate MOE profiles, followed by the sketched transformation procedure in reverse order. The quality of the simulated MOE profiles and of the methodological approach is shown by comparison with the experimental MOE profiles. It was found that the simulation model correctly reproduces the statistical information of the MOE variations while also producing very plausible profiles, with similar characteristics as the empirical MOE profiles. This model can be used to better study the size effect of glued-laminated timber by means of numerical simulations.
Introduction
The variation in the modulus of elasticity (MOE) along lumber boards has been studied since the early 1980s, when the first comprehensive model to simulate the variation in the MOE and tensile strength of Douglas fir boards was introduced by Foschi and Barrett (1980). In that model, the intraboard variation in mechanical properties was described by means of a Weibull approach (Weibull’s weakest link theory), together with an explicit consideration of the knot diameters simulated from frequency data. To study and later simulate the variation in mechanical properties, the boards were virtually subdivided lengthwise in segments 152 mm (6 in.) in length, then termed cells. The localized MOE was accounted for only in a theoretical manner, as the experimental data used to calibrate the model corresponded to segments with lengths of 762 mm (30 in.), with a single knot in the middle.
The basic concepts defined by Foschi and Barret (1980) became the basis for future studies on variations in mechanical properties along boards. One of these concepts, presented a few years later, was introduced by Ehlbeck et al. (1984), who developed a model based on cells 150 mm in length as part of what later would be known as the Karlsruher Rechenmodel—a numerical model used to simulate glued-laminated timber (GLT) beams. In their model, the variation in the MOE along boards was described in terms of a correlation with knot indicators and density values. A random component was added to consider a certain stochastic variation for defect-free regions (cells without knots), as the density was assumed to be constant along each board. Throughout the years, this model has been adapted to various datasets and species (i.e., Colling and Scherberger 1987; Blaß et al. 2005; Frese 2006).
A different approach was introduced by Kline et al. (1986), who described MOE variations in terms of a second-order autoregressive [AR(2)] model. In that study, southern pine boards were analyzed with cells 762 mm (30 in.) in length. The rather long cells used in this model certainly reduced the localized nature of the model when compared to the model by Ehlbeck et al. (1984), in which the cell length is 150 mm. Nevertheless, the novel approach was attractive, as the generation of knot indicators was no longer necessary, and it was purely concentrated on the observed statistical characteristics of the MOE variations. Similar approaches were followed (i.e., Showalter et al. 1987) for the simulation of tensile strength along the boards in which an autoregressive model with a moving average component was applied. Lam and Varoglu (1991) presented another model to simulate tensile strength along boards by using a third-order moving average model [MA(3)] and considering cells 610 mm in length. Based on the experimental data from previous studies, Bulleit and Chapman (2004) developed an first-order autoregressive [AR(1)] model for both MOE and tensile strength, in which the size of the cells may be different from the one used to obtain the data. These four last models can be considered to belong to the same kind of linear models that work on stationary processes, and are related to the model presented in this paper.
In a related study, Lam et al. (1994) presented a method to simulate continuous MOE profiles based on the analysis of bending profiles of spruce-pine-fir boards obtained with a continuous bending grading machine (Cook-Bolinder type). The approach was based on the previous work performed by Bechtel (1985), Foschi (1987), and Lam et al. (1993), in which the real localized bending MOE was approximated by means of a Fourier series. The rather long boards of 4.9 m studied in Lam et al. (1994) presented a clear trend in the MOE along the length of the board, making it a nonstationary process. Once the trend was removed, the process was assumed to be stationary and normally distributed. The simulation of MOE profiles was composed of two parts: (1) the generation of a stationary process by means of a sum of cosines with empiric spectral amplitudes; and (2) the addition of a trend component to reproduce the observed nonstationarity. The idea of first simulating a stationary process and then transforming it into a nonstationary process is also used here, but with a different approach.
More recently, Fink (2014) presented a model based mostly on the approach taken by Ehlbeck et al. (1984). One of the main differences consisted of considering the concept of clear wood segments (CWSs) and weak segments (WSs), previously studied by Isaksson (1999), for which a different set of relationships were established based on dynamic MOE and knot indicators.
The primary objective of most of the previously mentioned models was to provide simulated board data for numerical models of glued-laminated timber beams capable of capturing the experimentally observed and theorized lamination effect. The latter effect states that the increase in tensile and bending strength of GLT beams is related to the strength of the board material, which results from the load sharing and compensation of weak zones of the parallel-bonded laminations within the GLT with stochastic MOE fluctuations. Therefore, profiles of mechanical properties of boards [the MOE and modulus of resistance (MOR)] are simulated to assemble virtual GLT beams and use them in Monte Carlo simulations.
GLT made from hardwoods, including American and European white oak, has gained a considerable market share in Europe in recent years. Its main applications are posts and beams in structural glass façade assemblies but also outstanding large structures, e.g., the roof of the debating chamber of the Scottish Parliament or Warner Stand at Lord’s Cricket Ground, London. To extend the application range of hardwoods for structural purposes—specifically for oak GLT—beyond the present state of technical approvals and previous knowledge [e.g., ETA-13/0642 (ETA 2018); Aicher and Stapf (2014); Aicher et al. (2014)], a European research project (EU Hardwoods) was brought into existence. In this context, extensive ongoing research is conducted at the Materials Testing Institute (MPA), University of Stuttgart, focusing on stochastic numerical GLT models (Tapia and Aicher 2018). A principal aim consists of an improved prediction of the size effect for oak GLT, which is strongly influenced by the interboard fluctuations and especially the intraboard fluctuations in the MOE.
This paper presents a model to simulate profiles of localized MOE measurements along boards by means of an AR model. The empirical data used for the development and calibration of the model stem from tensile tests, including localized MOE measurements from boards made from the hardwood species European white oak (Quercus robur/Q. petraea) (Tapia and Aicher 2019). A two-step method is introduced, whereby the MOE data are first normalized and then transformed into a stationary process, which can be described by a simple autoregressive model. It is shown that if the stationary process is correctly simulated, then the two-step transformation can be reversed, resulting in realistically simulated localized MOE profiles. These profiles present fluctuation characteristics that are very similar to the features observed in the measured MOE profiles.
Materials and Methods
Boards and Measurement of the Localized MOE
Investigations into the variations in the MOE and density were performed using oak boards (Q. robur/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 deviations in the sample. The nominal dimensions (length ) of the planed boards were , respectively. The moisture content (MC) was, on average, 10.2% [coefficient of variation ]. The boards were visually graded at the MPA, University of Stuttgart, in hardwood strength grades LS7, LS10, and LS13 as specified in the German structural hardwood grading standard DIN 4074-5 (DIN 2008). The number of boards allocated to each of the mentioned grades was 4, 17, and 22, respectively; four boards were rejected. Grading was performed according to the provisions and criteria specified for boards. The process used to perform strength grading, together with the exact quantification and allocation of the growth defects of each board, is described in detail in Tapia and Aicher (2019).
The global and local moduli of elasticity were measured over a length of . The length differed from specified in EN 408 (CEN 2012), which is due to the specific objective of the tests: measuring the variation in the MOE along the longest possible free length. The global density of the total sample at 12% MC was , with very small differences between the individual strength grades. The global MOE (mean standard deviation) of all the boards, adjusted to 12% MC according to EN 384 (CEN 2019), was determined to be . For the structurally most relevant grade combination subset of LS10 and LS13, a value of was obtained, with a rather negligible difference between the individual grades.
The localized MOEs were measured with a gauge length of 100 mm. For this, the global length was virtually subdivided into 15 consecutive cells (Fig. 1). The measurement of the localized MOE with a specially adapted extensometer was performed by means of a series of loading-unloading cycles, after which the extensometer was manually moved to the next cell (Fig. 1). The gauge 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 (2) it was the minimum length achievable by modifying an existing extensometer. A maximum load of 30 kN was used for each cycle, equivalent to . This stress level represents about 25% of the mean strength level; hence, no noticeable effect in the mechanical properties due to the cycling loading should be expected. This is substantiated by the very good agreement between the experimental values for and the cell-based computed global MOE, (e.g., Tapia and Aicher 2019). Finally, the extensometer was adapted to measure the deformations along the global length. Hereinafter, the MOEs determined from the 100 mm cells are termed the localized MOE () values, while the MOE measured over the whole length of is the global MOE ().
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For a more detailed description of the previously mentioned experiments, as well as for an extensive statistical analysis of the obtained results, see Tapia and Aicher (2019).
Autoregressive Models: General Remarks
Autoregressive models are widely used across different science fields to model series of data in which a correlation between each element and the previous data points can be established. In its most basic form, an autoregressive model of order , AR(), is defined aswhere = model parameters; and = white noise term, generally assumed to be a zero-centered normally distributed random variable, . The white noise term needs to be a stationary process. Each autoregressive (AR) model possesses an autocorrelation function (ACF), which shows the correlation for the th lag, e.g., the th time or location interval. In general, the ACF, , is a function of the autocovariance function [], and can be expressed as
(1)
(2)
For the particular case of a first-order AR model, AR(1), the ACF is defined aswhere = model parameter; and = th lag. This function describes a geometrical progression, which is characteristic for AR(1) models.
(3)
The ACF serves as a theoretical reference point when fitting a dataset to an AR model, because it can be compared to the serial correlations computed directly from the studied data. This function is known as the sample autocorrelation function (SACF), , and is defined as (Brockwell and Davis 2002)with the sample autocovariance functionwhere = mean of the sample; and = total number of observations. Because the objective is to compute the SACF that is later used to fit an AR model, it is necessary to ensure that the data describe a stationary process. If this is not the case, a common solution is to transform the data, which produces the needed stationarity. A typical case is, for example, the consideration of a trend [as was done in Lam et al. (1994)] or seasonality (Brockwell and Davis 2002).
(4)
(5)
Transformation of the Localized MOE Data to a Stationary Process
In a previous analysis of the regarded localized MOEs (Tapia and Aicher 2019), normalization was applied to the measured MOE data of each board, which allows bringing the different boards to a comparable level. This step is necessary, as the differences in the absolute values (different values) would distort the analysis of the MOE variation within boards. Specifically, two alternative normalization approaches were then investigated: (1) mapping the data of each board into a distribution; and (2) dividing each value by the mean of the three highest values in each board ().
Both procedures were then used to compute the SACF for the normalized data, resulting in different values for the computed lags. For example, when only clear wood sections were considered, the first and second lags for the method were and , respectively, while the method produced and , respectively. Note that for the case in which no normalization was applied, the first two lags resulted in much higher values ( and ), comparable to the results obtained by Taylor and Bender (1991).
However, in the previous approaches, the stationarity of the normalized processes was not considered, which is required by an AR model and implies a stationary white noise term, . If the type of process obtained after the transformation is considered, then it is evident that from the two normalization methods, only the approach presents the needed stationarity. In contrast, both the normalization based on , as well as normalization, exhibit a rather skewed behavior, owing to the detrimental influence of knots and fiber deviations on the localized MOE. For this reason, they cannot be represented by a stationary white noise process. However, the normalization based on preserves the relative variation within board more accurately.
Bearing this finding in mind, a new method was devised in this study, whereby both normalization methods are sequentially applied to the data to achieve both a stationary process and an adequate representation of the relative MOE variation within a board. The procedure can be described as follows:
1.
In the first step, the localized MOE data of each board are normalized with respect to as
(6)
This normalization method preserves the original shape of the underlying distribution as long as the support of the distribution is positive (which is true in this case, because MOE values must be positive).
2.
Then, in the second step, all of the values are fitted to a statistical distribution, , by means of maximum likelihood estimation (MLE).
3.
The fitted distribution, , is used to map the values to a distribution aswhere = inverse of the cumulative distribution function (CDF) of the distribution.
(7)
The first transformation brings the data of each board to the same level, while the needed stationarity of the process is achieved in the third step. An illustration of this procedure is presented in Fig. 2. In the following section, the values are referred to as the normalized data, while the values are termed the stationary data.
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Computation of Autocorrelations for the Stationary Data
The computation of the sample autocorrelation function is performed in the same way as described in Tapia and Aicher (2019), where the data of all boards are considered simultaneously. Specifically, the tuples used for the computation of each lag- correlation always belong to a specific board; that is, the normalized values are not concatenated into a single vector to perform the correlation analysis (Fig. 2). The SACF is then computed according to Eqs. (4) and (5) for the first five lags, in which two different cases are considered: (1) computing the autocorrelations considering all cells, and (2) analyzing only the cells belonging to clear wood (CW) segments. These CW segments are defined as cells with knot area ratio (KAR) values and a minimum length of 700 mm (seven contiguous cells). The CW segments were defined to simulate the rather short, mainly knot-free oak boards used by some European GLT producers, where the length of the boards is maximally about 700 mm [e.g., ETA-13/0642 (ETA 2018)]. For details on the computation of the used KAR values and the definition of CW segments (see Tapia and Aicher 2019).
Based on the computed serial correlations, the order of the AR model can be determined. The corresponding model parameters, , are then obtained by solving the linear system of equations defined by evaluating Eq. (1) with the stationary data
(8)
The equation system is solved by means of the ordinary least-squares method.
Model for the Simulation of MOE Profiles
The simulation method is based on an AR(1) model and consists of three main steps, which are analogous to the steps described to obtain the stationary data of the localized empiric MOEs, but reversed. The procedure is described by the following steps:
1.
Generate an autoregressive process with the model parameter estimated from the solution of Eq. (8). For this, an AR(1) model is used to simulate values aswhere = white noise component , with . To avoid a strong influence from the initial conditions, the first 30 generated values per board are discarded.
(9)
2.
The generated values are mapped to the distribution of the normalized cell MOE data, aswhere = inverse CDF of the distribution describing the normalized MOE data; and = standard normal distribution, , which is the exact inverse of Eq. (7).
(10)
3.
Finally, given a value for the global MOE of the board (), a factor is computed to scale aswhere = simulated MOE profile of the virtual board. The factor is computed aswhere = result after applying the equation for serially arranged springs to the generated values. The choice of is not important at this time; it can be either manually defined or randomly sampled from a suitable distribution.
(11)
(12)
Because the only random term in the method is the white noise in Eq. (9), it is possible to use the same sequence of to observe the effect of different sets of parameters in the simulated profiles. This effect is demonstrated subsequently by comparing simulations realized with parameters fitted considering all cells and only clear wood segments.
Results and Discussion
Statistical Distribution for the Normalized MOE Variation
The normalized values, , of all boards were fitted by four different statistical distributions: (1) the reversed Gumbel, (2) the Weibull, (3) the log-gamma, and (4) the beta distributions. These distributions were selected because all four of them should be able to reproduce the observed left skewness of the data (Fig 3). Note that both log-gamma and reversed Gumbel distributions were truncated at zero to avoid negative values, as the support of both distributions begins at . Accordingly, the location parameter of the Weibull and beta distributions was fixed at 0, which enables zero or close to zero MOE values. An improved truncation limit, which should be , could be .
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Table 1 presents the estimated parameters for each distribution for both considered cases: all cells, and only the CW segments. When only the CW segments were considered a total of 328 cells were left (out of the total of 705). Additionally, Table 1 shows the Akaike information criterion (AIC), which serves to discriminate between the different fitted distributions based on the obtained likelihood and number of parameters, defined as , where is the likelihood, and is number of fitted parameters (Akaike 1974). The estimated parameters that produce the smallest AIC should be chosen. In this case, the log-gamma distribution achieves by far the lowest AIC when considering all the cells. For the CW segments, the reversed Gumbel distribution achieves the smallest AIC (), closely followed by the log-gamma distribution ().
| Distribution | Cells | loc | Scale | log | AIC | ||
|---|---|---|---|---|---|---|---|
| Gumbel rev. | All | 0.928 | 0.0842 | — | — | 506 | |
| 0.944 | 0.0717 | — | — | 504 | |||
| Weibull | All | 0. | 0.923 | 10.1 | — | 476 | |
| 0. | 0.941 | 12.7 | — | 500 | |||
| Log-gamma | All | 0.988 | 0.0481 | 0.437 | — | 515 | |
| 0.963 | 0.0618 | 0.790 | — | 505 | |||
| Beta | All | 0. | 1.07 | 10.0 | 2.29 | 490 | |
| 0. | 1.08 | 15.5 | 3.07 | 503 |
Note: loc = location parameter; and rev. = reversed Gumbel distribution.
A comparison between the normalized values can be extracted from Figs. 3(a and b), which give the PDFs of the four investigated distributions for both analyzed cases (all cells and only the CW segments). It is evident that for the CW segments, the variation is smaller, while the spread is larger if cells with larger knots are considered. This is a reasonable result that confirms intuitive expectations. Namely, the presence of knots increases the variation in the localized MOE within a board.
Obtained Autocorrelation
The sample autocorrelation function from the stationary data, , was computed according to Eqs. (4) and (5). The results for the first five lags are presented in Table 2 for all the cells and for the CW segments. Figs. 4(a and b) show examples of the data used for the computation of the first and second lags, respectively, considering all the cells. The correlation is evident for the first lag, while for the second lag, it is almost nondiscernible (quantitatively expressed by a low sample autocorrelation value of ). Note that the sample autocorrelation, , specified in Figs. 4(a and b), is similar to the coefficient of correlation, , for the first lag, but not exactly the same. The difference between these values is the denominator in Eq. (5), which, for the case of , should be (Brockwell and Davis 2002).
| Cells | Method | lag-1 | lag-2 | lag-3 | lag-4 | lag-5 |
|---|---|---|---|---|---|---|
| Clear wood | SACF | 0.58 | 0.35 | 0.16 | 0.06 | |
| ACF: AR(1) | 0.58 | 0.33 | 0.19 | 0.11 | 0.06 | |
| All cells | SACF | 0.42 | 0.17 | 0.08 | 0.03 | 0.00 |
| ACF: AR(1) | 0.42 | 0.18 | 0.07 | 0.03 | 0.01 |
Note: The lag-1 of the SACF is used as the model parameter.
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Fig. 4(c) presents a graphical representation of the first five lags considering all the cells, where a rough geometric decay, characteristic of an AR process, is evident. A comparison between the computed SACF and the theoretical ACF for an AR(1) [see Eq. (3)] proves very good agreement in the results for at least the first three lags (compare the values in Table 2). This finding means that a first-order AR model is most likely an adequate model to describe the stationary data , which is analyzed in more detail subsequently.
It is also of interest to compare these results with the serial correlations computed in Tapia and Aicher (2019) for both normalization approaches studied there. For the case in which all cells are considered, the obtained lag-1 for the and normalizations were 0.32 and 0.34, respectively. This finding means that the two-step process introduced in this paper increases the lag-1 by 24%–31%, depending on the method. For the case of the CW segments, the previous analysis produced lag-1 values of 0.41 and 0.61 for the and methods, respectively. Considering that the present analysis produced a lag-1 of 0.58, only a minor decrease of 5% is observed compared to the method, whereas an increase of 41% is obtained compared to the pure method.
Parameters of the Autoregressive Model
The model parameter was determined by solving Eq. (8) for . To ensure that AR(1) is the appropriate AR model to describe the specific data, a second-order AR model, i.e., AR(2), was fitted as well. The information obtained from calibrating both models enables the choice of the right model.
The fitted parameters are presented in Table 3 for the two models [AR(1) and AR(2)], as well as for the two studied cases, considering all cells and only the CW segments. It is immediately evident that the additional parameter has almost no relevance in the prediction quality of the stationary data (see the parameters and the values in Table 3). This irrelevance becomes even more clear when the AIC is considered, where in both cases, the AR(1) presents the lowest value. Furthermore, the fact that the difference of the AIC values for AR(1) and AR(2) is exactly 2 means that the extra parameter has no notable effect in the prediction. Note that the same number of data points were used to compute the AIC for AR(1), as were available for the AR(2) fitting process, because the AIC depends on the number of data points used (owing to the likelihood).
| Model | Residual | AIC | |||
|---|---|---|---|---|---|
| AR(1)Aii | 0.440 | — | 0.43 | 0.96 | 1722 |
| AR(2)Aii | 0.446 | 0.43 | 0.96 | 1724 | |
| AR(1)CW | 0.584 | — | 0.60 | 0.77 | 637 |
| AR(2)CW | 0.586 | 0.60 | 0.77 | 639 |
Simulation of MOE Profiles with the Proposed Model
For the simulation of MOE profiles of oak boards, the procedure described previously in conjunction with Eqs. (9)–(12) is followed. For the simulation of the values, the parameters , corresponding to the AR(1) models, are taken from Table 3. For the distribution of the normalized MOE, , the log-gamma distribution is chosen with the corresponding parameters from Table 1. Simulations for both studied cases are performed: (1) with the parameters determined on the basis of all cells; and (2) considering only clear wood segments. To make both simulated cases comparable at the board level, the same white noise sequences, , were used for each case.
Examples of four generated MOE profiles are presented in Figs. 5(a–d). Focusing first on the solid lines (Parameter set 1, all cells), it can be seen that a wide range of typical empirical MOE profiles can be reproduced. These generated profiles can be compared to a selection of the empirically determined values, presented in Figs. 6(a–d), which were chosen based on their apparent (visual) similarity with the simulated profiles of Figs. 5(a–d). In this sense, Fig. 6(a) somehow resembles Fig. 5(a), Fig. 6(b) is similar to Fig. 5(b), as well as Figs. 6(c and d) and Figs. 5(c and d), respectively. It can be seen that in general terms, the simulated profiles present very similar characteristics as the empirical profiles. It should be emphasized that the presented generated profiles are not intended to match the empirical measurements exactly, but rather that the model is capable to reproduce profiles belonging to the same category of variation characteristics. This is achieved by maintaining the autocorrelation function (Fig. 7) and reproducing the distributions of normalized MOE, , and global MOE, .
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In detail, Fig. 5(a) shows rather low variation in the MOE, while the profile depicted in Fig. 5(b) presents higher variation, where a steady increase in MOE is observed for approximately 800 mm. One could imagine that such a behavior arises in a real board due to a steady decrease in the fiber inclination. Fig. 5(c) presents a board with a few downward steps in the MOE in an otherwise rather moderately varying sequence of values. This finding would simulate the presence of local defects (e.g., knots). Finally, Fig. 5(d) illustrates the case in which a very large local defect is present, considerably lowering the local MOE in a narrow region.
Prior to the discussion of the simulated MOE profiles for the CW segments (Parameter set 2), Figs. 6(c and d) should be considered in relation to the empirical data. There, the measured clear wood area ratio (CWAR) values () for each cell show clearly that regions of high MOE variation not only occur in the vicinity of knots, but also in knot-free regions. It can be seen that the simulated MOEs of the CW cells [dashed lines in Figs. 5(a–d)] move very similarly as compared to the lines based on all cells. This results from the fact that the same white noise sequence was used. However, due to the different parameters, some differences in the variation can be observed [e.g., Fig. 5(d)]. Overall, the variation in the simulated CW segments is slightly smaller than that in all the cells, which can be explained primarily by the difference in the distributions for the normalized MOEs, . However, it should be mentioned that the experimental results for the clear wood segments exhibit large variations, which is captured by the fitted distribution. This phenomenon is why clear wood segments do not represent an especially low variation in the studied case—as the concept of clear wood might otherwise suggest. In this context, it should be recalled that clear wood, as defined in this paper, relates exclusively to an almost complete exclusion of knots (), but includes any local or global fiber deviation, as it can be assumed that this is the dominant reason for CW MOE fluctuations.
One further comparison was performed to confirm that the simulated data preserve the specified autoregressive parameters. For this comparison, a single, very long board of cells was simulated. Then, the two-step normalization process was applied, and the SACF was computed. In this study, the criterion was generalized to use the highest values of the profile, where , to compensate for the different number of cells considered. The results of this analysis can be seen in Figs. 7(a and b), for all cells and CW cells, respectively. In both cases, perfect agreement between the experimental results and the generated profiles can be observed, meaning that the autocorrelation properties are correctly simulated.
Discussion
The proposed model presents high flexibility regarding the needed parameters, meaning that it should be straightforward to fit to other datasets and different hardwood species. Specifically, the distribution for , which captures the amount of variation within a board, can take different forms without affecting the described procedure. Although the data presented in this study showed a rather high intraboard variation—leading to a very left-skewed distribution—there is no problem with having a much less skewed distribution with lower variation, representing high-quality boards.
In general terms, the use of left-skewed, upper-bounded distributions to represent the intraboard MOE variation has the positive effect of setting an upper limit for the relative variation while allowing for the characteristic drop-downs of the MOE due to the presence of local defects. If a normal distribution were used to describe the intraboard variation, as is normally done in other models (e.g., Ehlbeck et al. 1984; Blaß et al. 2005; Fink 2014), unrealistically high values can occur, because the random component eventually generates such values. Additionally, by not concentrating exclusively on the knots as the main source of variation, the presented model can implicitly consider different factors affecting the localized MOE.
Conceptually, the model presented in this paper is similar to the model presented by Lam et al. (1994), as both rely on an initial stationary process and an additional transformation to obtain the MOE profile. In Lam et al. (1994), the stationary process is generated by means of a summation of cosines, in which the spectral amplitudes are obtained from the estimated spectral density. The spectral density implicitly encodes the autocorrelation present in the bending profiles (Brockwell and Davis 2002). Therefore, the generated process can be regarded as equivalent to an AR model. The main difference is that the presented model consists of the transformation of the normally distributed process into a distribution that better represents the variation of the MOE within board (described in this study by a log-gamma distribution). This is an important addition, because it allows the MOE profile to show some characteristic, large drop-downs of the MOE, but not sudden increases, which would not be realistic. It could be argued that a Fourier series, calibrated with the empirical bending profiles, should be able to reproduce the mentioned drop-downs, because any function within an interval can be approximated by a Fourier series. However, because the method of Lam et al. (1994) considers random phase-shifting for each frequency contribution, one could eventually obtain the exact reversed process as the input. This seems to not be a major problem in the measurements presented by Lam et al. (1994), because no apparent large amount of skewness is present in the residuals of the detrended process (although no data are presented in this regard), but it would certainly be a problem for the rather skewed data presented in this study.
Due to the rather short board segments of only 1.5 m investigated in the presented research, no evident trend in the evaluation of the intraboard MOE was observed. A trend, as observed and simulated by Lam et al. (1994), could be an addition to the presented model if longer boards need to be simulated. However, because the boards currently used in the production of oak GLT are rather short [e.g., ETA-13/0642 (ETA 2018)], it does not seem to be an indispensable feature at the moment. Nevertheless, tensile tests on longer oak board segments ( m in length) would certainly help in improving the presented model, because the larger number of cells would translate into a more robust computed SACF.
For stochastic simulations of the load capacity of GLT beams (e.g., Ehlbeck et al. 1984; Frese 2006; Fink 2014) the presented MOE simulation model has to be coupled with an adequate MOR model representing the strength variation along boards. Such a model is presented in a follow-up study, in which survival analysis is used to derive suitable strength models for the same sample of boards as investigated in this study.
One of the major problems with this model is the localized MOE measurements required for calibration. The rather time-consuming method employed to sample the data used in this study (Tapia and Aicher 2019) complicates the creation of a large database of localized MOE values. However, much more efficient methods could be used to measure the MOE variation, such as digital image correlation, surface laser scanners, X-ray methods, or other optical systems, as shown recently (i.e., Kandler et al. 2015; Viguier et al. 2017; Olsson and Oscarsson 2017; Briggert et al. 2020). The use of such systems can foreseeably simplify the procedure to obtain the needed data, leading to a larger database that can be used to calibrate this model.
Another relevant point to consider is whether the obtained parameters are applicable to different cross sections. As previously stated, it can be assumed that at a local board level the main source of variation are the fiber orientation and the presence of knots. Because the fibers tend to be similarly oriented over the cross section, it follows that small cross-sectional differences will probably show minor impact. Knots are considered generally in a relative manner (e.g., KAR values), meaning that they scale with the cross-section size. Because fiber orientation and knot indicators are normally used to classify boards, it can be argued that the obtained parameters should be applicable to different cross sections as long as they represent a similar mixture of grades as investigated here. However, a definitive answer on the cross-sectional issue, especially for boards with significantly different cross-sectional dimensions, necessitates further experimental data.
Finally, note that the model in its present state is not well suited for the simulation of MOE profiles for northern hemisphere softwood boards, as their inherent rather periodic occurrence of weak zones (Isaksson 1999; Fink 2014) is not explicitly considered. Nevertheless, with some small modifications, e.g., by adding a cyclical component to the AR(1) model for , the model could be upgraded for the simulation of softwood profiles in an adequate manner.
Conclusions
The derived autoregressive model suits the simulation of MOE profiles of wooden boards characterized by a nonperiodic occurrence of weak zones. By not focusing on any specific defect indicator (e.g., a knot indicator), the model can intrinsically reproduce different weakening factors, such as pronounced fiber deviations. Such growth features are typical for many northern hemisphere hardwoods, such as American and European white oak. The model calibration requires empirical localized MOE data of rather small cell lengths of 100 mm along the main axis of the boards.
To obtain a stationary input for the determination of the MOE sample autocorrelation function, a two-step transformation process of the empirical cell MOE data was derived. In the first step, the empiric MOEs are normalized by the averaged intraboard maxima, and the resulting data are used to fit a statistical distribution. Due to the specific hardwood growth characteristics in the regarded case (European white oak), the normalized cell data present clear left skewness, which is best approximated by a log-gamma distribution. Finally, the normalized MOE values are then mapped onto a normal distribution, thus achieving stationarity.
A first-order AR process [AR(1)] proved to be well suited for a satisfactory representation of all cell data, including knots, as well as for clear wood segments. A second-order AR process does not deliver any enhanced prediction quality of the stationary data. This phenomenon results from very low model parameters and from the comparison of the AIC between the fitted AR(1) and AR(2) models. The apparent adequacy of the simulated MOE profiles compared to the empirical fluctuations was quantitatively proven by a high agreement in the lag coefficients from the SACF for the experimental data and simulated results.
The presented autoregressive model is independent from knot indicators, which are the basis of most of the current stochastic regression-based MOE variation approaches. A drawback of these models is that they are not capable of reconstructing a strong interknot MOE variability, which is typical for many hardwood species. However, softwoods are not considered in the present model. To simulate the material characteristics of northern hemisphere softwoods with an inherent pronounced periodicity of knot occurrences, the derived model can be easily upgraded by adding a cyclic component to the AR(1) process for the transformed stationary cell data.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available in a repository or online in accordance with funder data retention policies (Tapia and Aicher 2020).
Acknowledgments
The financial support of the work by the German foundation, Fachagentur Nachwachsende Rohstoffe e.V. (FNR), contract 2200414, within the European ERA-WoodWisdom project “European hardwoods for the building sector (EU Hardwoods)” is gratefully acknowledged.
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Journal of Materials in Civil Engineering
Volume 33 • Issue 6 • June 2021
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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.
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Received: Jun 15, 2020
Accepted: Sep 28, 2020
Published online: Mar 31, 2021
Published in print: Jun 1, 2021
Discussion open until: Aug 31, 2021
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