Introduction

Background

Timber is a material with an excellent strength–weight ratio, the possibility of being sustainably sourced, and usually perceived as aesthetically very pleasant. However, human influence on its physical and mechanical properties is generally very limited due to the natural growth, leading to a high variability in timber properties. Without taking measures to reduce the variability, large safety factors would be needed when designing timber structures, limiting the efficiency in utilising the material. For instance, during the production of glued laminated timber (GLT) or cross-laminated timber (CLT), the raw material needs to be strength graded to ensure minimum performance requirements and optimise its use. Strength grading of timber into different strength classes based on parameters obtained from non-destructive tests allows for a reduction in the material’s variability and, thus, for a more efficient utilisation of the raw material. Therefore, strength grading depends mainly on the correlations between the measured (indicating) parameters and the target mechanical parameters.

State of the art

Anatomical characteristics, such as large single knots, knot clusters, bark inclusions and fibre deviations, cause the most significant reduction in the strength of timber. Regarding glued laminated timber, the tensile strength of the laminations is the key property for allocating timber boards to a certain strength class (e.g. Ehlbeck et al. 1985; Brandner and Schickhofer 2008; Fink 2014). Close to knots and in knot clusters, the tensile strength of a board is reduced due to local deviations to the mainly longitudinal fibre direction and due to the reduction in cross-sectional area available for these fibres in the presence of knots. This results in non-uniform stress distributions in these cross sections, especially in case of eccentric knots (Foley 2001). Limits regarding the knot size are provided for different grades in current strength grading standards, such as DIN 4074-1 (2012), for strength grading of coniferous sawn timber, and DIN 4074-5 (2008), for strength grading of deciduous sawn timber. For boards with fewer and smaller knots, the influence and importance of other characteristics increase. In particular, the local fibre direction becomes more important. DIN 4074-1 (2012) and DIN 4074-5 (2008) specify threshold values of this grading parameter that correspond to the various grades too. The local fibre direction also has a high influence on the strength of finger joints in beech wood laminations (Aicher et al. 2001). The detection and quantification of the fibre direction are much more complex than the determination of the size of knots and are particularly relevant to machine grading (Ridley-Ellis et al. 2016). On the one hand, the fibre direction is often difficult to detect with the naked eye; on the other hand, the fibre direction changes continuously along the length and width of a board. To overcome these problems, procedures and instruments have been developed to automatically evaluate the fibre direction (Schlotzhauer et al. 2018).

Multiple physical principles are employed to evaluate the fibre direction. Among them are the dielectric properties of wood (Baradit et al. 2006; Denzler and Weidenhiller 2015; Norimoto and Yamada 1972), the electrical field strength (Cramer and McDonald 1989; Norton et al. 1974; Steele et al. 1991), the thermal conductivity (Belkacemi et al. 2016; Daval et al. 2015; Krapez et al. 1996), and the so-called tracheid effect (Matthews et al. 1976, 1986; Metcalfe et al. 2002; Nyström 2003; Sarén et al. 2006; Simonaho et al. 2004; Soest 1997). The method based on the tracheid effect has been frequently used in descriptions and models of timber and its anatomical structure (Briggert et al. 2016; Foley, 2001; Lukacevic and Füssl 2014). The tracheid effect has been used to improve the strength grading of coniferous timber (Brännström et al. 2008; Olsson et al. 2013; Olsson and Oscarsson 2017; Viguier et al. 2015, 2017). More recently, it has also been applied successfully to fibre orientation measurements on hardwoods (Besseau et al. 2020), on beech veneers to estimate the modulus of elasticity of LVL (Viguier et al. 2018), and to strength grading of oak (Olsson et al. 2018).

The vast majority of the mentioned investigations focussed on coniferous timber species, mainly Norway spruce (Picea abies (L.) Karst.). For European beech (Fagus sylvatica L.) timber, significantly fewer studies are available. In Switzerland, Germany, and Austria, European beech amounts to 18, 15, and 10% of the total forest stock, respectively, corresponding to shares between 50 and 80% of the hardwood stock in these countries (FOEN 2018; Sauter and Breinig 2016). European beech timber is also among the European native species with the highest strength and stiffness properties. Compared to timber from coniferous trees, European beech timber exhibits significantly fewer knots and knot-free boards predominate in high-strength classes. Consequently, the strength grading parameter fibre direction is of utmost importance.

Ehrhart et al. (2018a) presented a non-contact method for the identification, quantification, and documentation of the fibre direction in European beech timber based on the analysis of the medullary rays. Making use of image-analysis techniques, the spindle pattern formed by the medullary rays was used to predict the fibre direction in European beech boards. Curti et al. (2018) adapted this method to investigate the effect of fibre direction on cutting forces and chip geometry during machining of beech timber in the fibre-saturated state.

To further increase the reliability and utilisation efficiency of timber in general, improved machine grading procedures are required to measure various grading parameters at high speed, with greater accuracy and precision. In general, the correlation between the measured grading parameters and the structural properties is performed using simple linear or multilinear correlations (Ehlbeck et al. 1985; Fink 2014), neglecting that the structure of timber is complex and that these correlations are often nonlinear. Machine-learning techniques are particularly suited to overcome challenges related to finding patterns in complex systems. These data-driven approaches are able to identify hidden patterns in data of different type and create classification and predictive models (Fathi et al. 2020; Schubert and Kläusler 2020). Combining machine-learning techniques with non-destructive optical and digital image analysis methods is therefore a promising strategy for grading high-strength timber boards with few obvious visual defects. These advanced grading techniques can be further complemented with mechanical constitutive models, which establish the relation between material parameters and external loads (e.g. Sarnaghi and van de Kuilen 2019), leading to hybrid strength grading approaches.

Objective, scope, and overview

The objective of this study was to develop a model to improve the prediction of the tensile strength of European beech (Fagus sylvatica L.) boards without obvious visual defects. The developed model considers the measured dynamic modulus of elasticity and new information on the fibre direction, obtained through a non-destructive and non-contact method developed by Ehrhart et al. (2018a). Different machine-learning techniques are applied, and their potential for maximising the prediction quality of the models is evaluated.

Materials and methods

Overview

This study was based on 47 flat sawn planed beech boards supplied by four Swiss sawmills. The dimension of the boards was lb × wb × tb = 3000 × 160 × 25 mm3. The thickness of the beech laminations tb = 25 mm is significantly smaller than usual thicknesses of softwood laminations (up to 40 mm), but this is due to delamination problems arising when beech laminations thicker than 30 mm are used (Ohnesorge et al. 2010) for glulam production. The average moisture content of the boards at the time of testing was ω = 8 ± 2%. This moisture content corresponds to an environment with 24 °C and 35% relative humidity, which is not uncommon in offices and residential buildings and also corresponded to the expected environmental conditions in the laboratories where the tests were performed. Since the focus of this study was on improving the grading of high strength boards, only boards free of knots, bark inclusions and other obvious structural characteristics affecting the tensile strength parallel to the grain were considered. The wide faces of the boards were mostly parallel to the longitudinal-tangential plane of the wood. In such knot-free boards, the influence of fibre orientation on the tensile strength becomes more important. Ehrhart et al. (2018a) have shown that the medullary ray spindle pattern, which is clearly visible on the face sides, can be used to assess the local fibre orientation. About 75% of the 47 boards were comprehensively photographed, documented, and tested to failure in tension parallel to the grain by Ehrhart et al. (2016b). The other 25% of the boards were analysed, and the density, the dynamic MOE, the surface photographs and the experimental tensile strength were documented by Jungo (2016). The data sets collected by Ehrhart et al. (2016a) and Jungo (2016) were combined and subsequently randomly split into two data sets: a training or calibration data set (used to fit models) and a test data set (used to evaluate the models). The training data set consisted of 36 boards (77%) and the test data set of 11 boards (23%). The total number of boards used in this study is limited, but was shown to be adequate to assess the ability of the proposed method of using image-based local fibre direction data to very significantly improve the prediction of the tensile strength of European beech boards.

2.2 Strength grading parameters measured using non-destructive methods

Density and dynamic modulus of elasticity

The devices Brookhuis MTG Timber Grader and Microtec ViScan were used to determine the density and the dynamic modulus of elasticity of the beech boards (Ehrhart (2019) showed that the two systems provide almost identical results). The dynamic modulus of elasticity Edyn is calculated using Eq. (1), based on the first longitudinal eigenfrequency fe, the density ρ and the length lb of the board.

$$E_{{{\text{dyn}}}} = 4 \cdot f_{{\text{e}}}^{2} \cdot l_{{\text{b}}}^{2} \cdot \rho$$
(1)

The statistical parameters of the dynamic modulus of elasticity and density are summarised in Table 1 for both the training (n = 36) and the test (n = 11) data sets. The range of density is wider for the training data set (647–817 kg/m3) than for the test data set (671–759 kg/m3). However, the mean values of density at a moisture content of ω = 8 ± 2% (ρ8,mean) and the coefficients of variation (cov) of both data sets agree well (726 and 710 kg/m3 and 0.05 and 0.04, respectively). Similar values of densities of European beech are reported by Frühwald and Schickhofer (2005), Frese (2006), Hübner (2013), and Westermayr et al. (2018).

Table 1 Statistical parameters for the density ρ, dynamic modulus of elasticity Edyn, static modulus of elasticity Et,0 and tensile strength parallel to the grain ft,0 of the training and test data sets (ω = 8 ± 2%)
Full size table

The range of the dynamic modulus of elasticity is also wider for the training (12,805–23,371 MPa) than for the test data set (13,613–20,791 MPa), and the mean value of the training data set is slightly higher than that of the test data set (16,520 and 16,291 MPa, respectively). Frese (2006) reported a range of 9656–20,613 MPa and a mean value of 14,506 MPa.

Local fibre direction

The local fibre direction was identified, quantified, and documented using the non-destructive and non-contact method developed by Ehrhart et al. (2018a). This method assumes that the medullary rays and the annual rings form corridors, along which the wood fibres run and, thus, are a proxy for the fibre direction. Due to the low shear strength of European beech wood in the longitudinal-radial plane (Ashby et al. 1985), the spindles formed by the medullary rays were found to be an excellent indicator for the critical fibre direction, i.e. the fracture path developing in destructive tension tests.

The two opposite face sides of the middle 2000 mm (= tested free-length of the boards in the tension tests, see Subsection Tensile strength parallel to the grain measured in tension tests to failure) of the 3000 mm long boards were documented by taking four consecutive photographs of 500-mm-long segments. Hence, eight photographs were taken of each board. Figure 1a shows a part of one of the resulting digital photographs. Subsequently, these digital images were batch-processed to isolate and identify the medullary ray spindles using the programme Adobe Photoshop 2020. The steps of this pre-processing were: (i) smart sharpening (amount: 500%, radius: 7.5 pixels); (ii) convert mode (to grayscale); (iii) apply a cut-out filter (levels = 4, edge simplicity = 1; edge fidelity = 3); set a threshold (level: 75). Each file covers a board length of 500 mm, and the resulting PNG-files have an approximate size of 100 kB (the original size of the photograph was approximately 5 MB). The black-and-white image showing the isolated medullary ray spindles of the photograph in Fig. 1a is shown in Fig. 1b.

Fig. 1
figure 1

Non-contact determination of the fibre direction: based on a photograph (a), the medullary ray spindles are isolated (b), and their coordinates and geometrical properties are documented. Post-processing of the resulting data allows generating discretised fields of fibre direction, e.g. average local angle between the fibre directions and the longitudinal direction of the boards (c)

Full size image

The black-and-white PNG images were analysed using the programme ImageJ (Image Processing and Analysis in Java) (Schindelin et al. 2012). By applying a threshold, the medullary ray spindles were identified as objects. (An example based on an image of segment no. 4 of board no. 1004 is shown in Fig. 2a.) Subsequently, a list of the properties of all objects was generated, including: coordinates xM and yM of the centre of mass (in pixels); area A (in pixels); circularity c (calculated using Eq. (2), in which P is the perimeter of the object in pixels); and angle α between the major axis of an ellipse fitted to the object and the longitudinal direction of the board (in degrees). The angle α was measured in counter-clockwise direction; objects with angles above 90° were considered as having negative orientations, i.e. α = αmeasured − 180 (Fig. 2b).

$$c = 4 \cdot \pi \cdot \frac{A}{{P^{2} }}$$
(2)
Fig. 2
figure 2

Detail image of board no. 1004, segment no. 4, after isolation of the medullary ray spindles (a). As an example, information on the coordinates, the area, the circularity, and the angle of object no. 6478 is shown (b)

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Only objects that represent medullary ray spindles should be considered in the analysis of the field of fibre direction. Thus, objects resulting from dirt, markings or other sources were removed by filtering the resulting object list. The filtering criteria to define valid objects were: object area A between 50 and 500 pixels (approximately 0.45 to 4.50 mm2); circularity c between 0.02 and 0.60 and; absolute orientation α below 45°. Fields of local fibre direction were then obtained by discretising the boards into surface elements with a size of (e × e =) 20 × 20 mm2 and calculating the average fibre direction in each element taking into account the results for both face sides. This element size was chosen as a compromise between enough resolution to reliably detect fibre orientation and avoiding "empty" grid elements, for which no fibre orientation could be determined. If less than five elements were present in an element, no fibre direction was calculated. Figure 1c presents the resulting field of fibre direction for the segment shown in Fig. 1a and b.

Tensile strength parallel to the grain measured in tension tests to failure

Test set-up

The tensile strength parallel to the grain of the European beech boards was determined in accordance with EN 408 (2012), using a Gehzu 850 horizontal testing machine (Fig. 3). The boards had a total length lb = 3000 mm, but the free testing-length between the clamps (Fig. 3b) was ltest = 2060 mm, which was very approximately the part of the boards for which the local fibre direction was analysed (see Subsection Local fibre direction). In addition to the tensile strength parallel to the grain ft,0, also the static modulus of elasticity parallel to the grain Et,0 was determined in accordance with EN 408 (2012), based on measurements performed using linear variable differential transformers (LVDT) attached to both edge faces of the boards (Fig. 3c).

Fig. 3
figure 3

Tensile testing machine Gehzu 850, used to determine the tensile strength parallel to the grain ft,0 (a). Vertical hydraulic jacks that apply the clamping pressure and horizontal load cells to measure the applied tensile force (b). Displacement transducers on the edge faces, used to determine the static modulus of elasticity parallel to the grain Et,0 (c)

Full size image

Test results

The statistical parameters of the tensile strength and static MOE parallel to the grain are summarised in Table 1 for the training (n = 36) and the test (n = 11) data sets. The range of tensile strength is slightly wider for the training data set (42.2–132 MPa; cov = 0.25) compared to the test data set (49.5–119 MPa; cov = 0.26). The mean value of tensile strength is about 6% lower in the training data set (ft,0,mean = 85.4 MPa) compared to the test data set (ft,0,mean = 90.4 MPa). Similarly, a wider range but a slightly lower mean value of the static MOE was found for the training data set (Et,0,mean = 15,428 MPa) compared to the test data set (Et,0,mean = 15,646 MPa).

The results of previous tensile testing campaigns on European beech boards are summarised in Table 2. The tensile strengths were between 4.3 and 142 MPa (Blaß et al. 2005; Ehrhart 2019; Frühwald and Schickhofer 2005; Glos and Lederer 2000; Plos et al. 2018; and Weidenhiller et al. 2019). Since only flat sawn boards free of knots were investigated in this study, the variation of results is significantly lower. No strength values below 40 MPa were observed. Nevertheless, the maximum values in the data sets are close to those reported in other studies (Table 2).

Table 2 Tensile strength parallel to the grain of European beech boards reported in other studies
Full size table

Failure patterns

As described in Ehrhart et al. (2018a), the dominating failure mode was shear failure along the spindles formed by the medullary rays. As an example of a board with the spindles formed by the medullary rays running mostly parallel to the x-axis, i.e. the longitudinal axis of the board, a segment of board no. 3002 is shown in Fig. 4a. The objects identified in this board segment are shown in Fig. 4b, where the almost horizontal orientation of the spindles is even clearer. The stain at the bottom part of the image is not considered in the calculation of the fibre direction matrix due to its circularity (> 0.6) and the orientation of the main axis of the fitted ellipse (> 45°) (see filtering criteria in Subsection Local fibre direction). Failure cracks created in the tension tests are shown in Fig. 4c. It can be seen that these cracks run mostly parallel to the x-axis, meaning that the fibres are quite straight (α ≈ 0). Due to the small fibre inclination, a relatively high tensile strength of ft,experimental = 112 MPa was observed with board no. 3002.

Fig. 4
figure 4

Detail image of a segment of board no. 3002, the spindles formed by the medullary rays running mostly parallel to the longitudinal axis of the board (a), identified objects (b) and fracture path in the same segment (c). This board exhibited a tensile strength of 112 MPa in the tensile test to failure

Full size image

As an example of a board with a higher fibre inclination, a segment of board no. 2058 is shown in Fig. 5. Both in the original image (Fig. 5a) and the image showing the objects (Fig. 5b), a negative inclination of the spindles (α ≈ − 15°) is clearly seen. The actual fibre direction, revealed by the failure pattern shown in Fig. 5c, agrees well with this observation. A significantly lower tensile strength of ft,0,experimental = 42 MPa was observed in the tensile test.

Fig. 5
figure 5

Detail image of a segment of board no. 2058, with the spindles formed by the medullary rays running mostly at an angle of 15° to the longitudinal axis of the board (a), identified objects (b) and fracture path in the same segment (c). This board exhibited a tensile strength of 42 MPa in the tensile test to failure

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Numerical methods

Data-fitting methods

The regression parameters were calibrated by minimising the residual sum of squares (RSS) between the estimated and the experimentally determined tensile strengths. This was done using the generalised reduced gradient (GRG) nonlinear solver implemented in MS Excel 2016. Only the training dataset (36 boards) was used in the model calibration.

Machine-learning techniques

The main objective of using machine-learning techniques is to properly adjust important hyperparameters of the algorithms and to avoid overfitting, in order to achieve a good generalisation with respect to the unseen test data (Mansfield et al. 2007). Thus, the choice of a proper experimental methodology is crucial. Therefore, as mentioned before, the data set (N = 47) was divided in: a training data set (n = 36) and a test data set (n = 11). The training data set was used to identify and adjust important model hyperparameters by applying a (k =) sixfold cross-validation approach (Kohavi 1995). Cross-validation is a resampling procedure used to effectively evaluate machine-learning models on a limited data set (Kohavi 1995). In the process of cross-validation, one part of the training data set (n = 36) was set as the validation data set, while the remaining k−1 = 5 parts were used as the training data set.

After training the machine-learning algorithms using the cross-validation method, the generalisation ability was assessed with the test data set (hold-out evaluation) based on the accuracy measures: coefficient of determination (R2); root mean square error (RMSE); and mean absolute percentage error (MAPE). The results from this evaluation serve as a quality measure (generalisation ability) for these models. Typical MAPE values for performance evaluation are categorised as follows (Lewis 1982): MAPE ≤ 10% corresponds to a ‘high accuracy prediction’; 10% < MAPE ≤ 20% corresponds to a ‘good prediction’; and 20 < MAPE ≤ 50% corresponds to a ‘reasonable prediction’. A MAPE above 50% indicates an ‘inaccurate prediction’. The input parameters for the machine-learning techniques were the same that were used in the developed physically based model.

Deep learning approaches, for example convolutional neural networks, would be particularly suited to predict the tensile strength based solely on images of the boards, but require extremely large data sets and are computationally intensive. Therefore, shallow machine-learning algorithms were used instead. Shallow algorithms require the data to be described by pre-defined features, which in this case are the grading parameters and the local fibre direction.

The machine-learning techniques applied in this study were Support-Vector Machines (e.g. Vapnik 1995), Decision Trees (e.g. Kotsiantis 2013), Random Forest (e.g. Breiman 2001), and Artificial Neural Network (e.g. Hinton and Salakhutdinov 2006).

Introduced by Vapnik (1995), Support-Vector Machines are supervised learning models for classification and regression. Since support-vector machine regression relies on kernel functions, it is considered a nonparametric technique. In this study, the scaling factor, i.e. the kernel scale, and the cost function were optimised using sequential minimal optimisation (SMO). Decision Trees build regression or classification models in the form of a tree structure (Kotsiantis 2013). Thereby, the data are split into at least two homogeneous sets (also called sub-populations) based on the most significant features of the input variables. An associated decision tree, featuring decision nodes and leaf nodes, is incrementally developed. Decision trees are precursors to Random Forests (Kotsiantis 2013). Operated by constructing a multitude of decision trees, a Random Forest is an ensemble learning method for (e.g.) classification and regression. Thereby, each tree is generated using a random variable subset from the candidate’s predictor variables and a random subset of data, generated by means of a bootstrap (Breiman 2001). Artificial Neural Networks comprise three functional parts (Hinton and Salakhutdinov 2006): (i) input layer; (ii) hidden layers; and (iii) an output layer. The hidden part can consist of many layers that enable multi-level and nonlinear operations (Hinton and Salakhutdinov 2006). The interconnections between nodes, the parameters that represent the strengths of the interconnections (weights), and the activation function, which is a nonlinear function that converts the weighted sum of the input signals to output at each node, are important features for the performance of the network and are adjusted through learning (LeCun et al. 2015). The learning process for updating weights is called “training”, which occurs in an iterative way. During each optimisation step, the weights are adjusted in order to reduce the difference between the prediction and the measured activity (Hinton and Salakhutdinov 2006).

All algorithms were programmed in MATLAB R2018b (using the statistics and machine-learning toolbox). The hyperparameters of the Support-Vector Machines, Decision Trees, and Random Forests were optimised using Bayesian optimisation (Snoek et al. 2012). The optimal configuration, i.e. the decision about the type of training algorithm, learning rate, activation function (e.g. radial basis, tanh, ReLU) and number of layers and neurons per layer for the artificial neural network, was obtained by a trial and error approach until no further improvement could be achieved. A detailed mathematical description of the algorithms has been presented by Bishop (2006).

Development and evaluation of strength models

Overview

This study focussed on grading high-strength European beech boards free of knots. Thus, knot-related parameters, such as the single knot (DEB) and the knot cluster (DAB) parameter, or the knot area ratio (KAR) and the total knot area ratio (tKAR), could not be used in the models to estimate the tensile strength.

The potential of using the board density (ρ) and the dynamic modulus of elasticity (Edyn) as grading parameters for the tensile strength parallel to the grain (ft,0) of beech boards was investigated in several studies (Glos and Lederer 2000; Frühwald et al. 2003; Frühwald and Schickhofer 2005; Plos et al. 2018; Weidenhiller et al. 2019; Ehrhart, 2019). The reported coefficients of determination between the density and the tensile strength parallel to the grain, i.e. R2(ρ, ft,0), were consistently very low (R2 < 0.20) (Table 3). Low coefficients of determination of R2 = 0.06 were also found in the present study (Fig. 6a).

Table 3 Coefficients of determination R2 between the potential strength grading parameters density (ρ) and dynamic MOE (Edyn) and the target parameter tensile strength parallel to the grain (ft,0) reported in previous studies (listed in the first column of the table) and found for the data training and test sets used in the present study
Full size table
Fig. 6
figure 6

Correlation of potential grading parameters density (R2 = 0.06) (a) and dynamic modulus of elasticity (R2 = 0.48) (b) with the target parameter tensile strength parallel to the grain (ft,0)

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The coefficients of determination for a linear correlation between the dynamic MOE and the tensile strength parallel to the grain reported in the literature are much more diverse (Table 3). While Ehrhart (2019), Frühwald et al. (2003), and Frühwald and Schickhofer (2005) reported relatively low coefficients of determination (R2 between 0.19 and 0.27), Weidenhiller et al. (2019) observed much higher coefficients of determination (R2 ≈ 0.60). In this study, a coefficient of determination between the dynamic MOE and the tensile strength parallel to the grain, i.e. R2(Edyn, ft,0), of 0.48 (Fig. 6b) was found.

Sub-model to estimate the minimum local tensile strength of a board

As described above, the fibre orientation within a board is not constant. It changes continuously both along the longitudinal as well as along the transverse axis of a board. Generally, the tensile strength is locally reduced in areas with high fibre inclination. However, not only the tensile strength but also the static MOE is smaller for elements with high fibre inclination. Consequently, the stresses in a cross section are distributed depending on the stiffnesses of its elements and failure will not necessarily occur in the element with the highest fibre inclination. To take this into account, a sub-model to calculate the minimum local tensile strength of a board min(ft,ij) is implemented. A preliminary study based on this approach was presented by Ehrhart et al. (2018b). The minimum local tensile strength corresponds to the external stress that leads to the failure of the first element.

For calculating min(ft,ij), the local fibre orientation matrices (see Subsection Local fibre direction) are taken into account. Constant strain across each cross section is assumed, i.e. the stresses within a cross section are distributed according to the local static MOE of each element in the cross section. The local MOE used herein (Et,ij) is calculated based on the local fibre direction αij of each element (Fig. 7a) using the Hankinson model (Eq. 3) (Hankinson 1921). The resulting matrix of Et,ij has a size of m (= wb/e = 160 mm/20 mm = 8) × n (= lrelevant/e = 2000 mm/20 mm = 100).

Fig. 7
figure 7

Discretisation of the boards into elements ij of e × e = 20 mm × 20 mm. For each element, the fibre direction (a), the estimated local static modulus of elasticity Et,ij (b), and the estimated local tensile strength ft,ij (c) are determined

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In Eq. (3), the local fibre direction (αij), the reference static MOE parallel (Et,0,ref) and perpendicular to the grain (Et,90,ref) as well as the exponent (a) are considered. Subsequently, the average MOE of each cross section (Et,i) was calculated. The exponent (a) was calibrated through a data-fitting procedure on the training data set (see Subsection Calibration and testing of the model).

$$E_{{{\text{t}},{ij}}} = \frac{{E_{{{\text{t}},0,{\text{ref}}}} \cdot E_{{{\text{t}},90,{\text{ref}}}} }}{{E_{{{\text{t}},0,{\text{ref}}}} \cdot \sin^{{\text{a}}} \left( {\alpha_{{{ij}}} } \right) + E_{{{\text{t}},90,{\text{ref}}}} \cdot \cos^{{\text{a}}} \left( {\alpha_{{{ij}}} } \right)}}$$
(3)

The matrix of local tensile strengths ft,ij (Fig. 1c), i.e. the external stress that would cause failure in the element eij, was calculated considering the local fibre direction αij, the local stiffness Et,ij, and the mean stiffness of the cross section i using Eq. (4). The reference tensile strength parallel (ft,0,ref) and perpendicular to the grain (ft,90,ref) were chosen according to Wagenführ (2006). The calibration parameter b was determined through a data-fitting procedure on the training data set (see Subsection Calibration and testing of the model).

$$f_{{{\text{t}},{ij}}} = \frac{{f_{{{\text{t}},0,{\text{ref}}}} \cdot f_{{{\text{t}},90,{\text{ref}}}} }}{{f_{{{\text{t}},0,{\text{ref}}}} \cdot \sin^{{\text{b}}} \left( {\alpha_{{{ij}}} } \right) + f_{{{\text{t}},90,{\text{ref}}}} \cdot \cos^{{\text{b}}} \left( {\alpha_{{{ij}}} } \right)}} \cdot \frac{{E_{{{\text{t}},{ij}}} }}{{E_{{{\text{t}},{i}}} }}$$
(4)

In the modelling of the tensile strength (ft,estimated), the grading parameter Edyn and the grading parameter min(ft,ij) are taken into account. The latter one allows considering the influence of the local fibre orientation. In accordance with previous studies (Frühwald and Schickhofer 2005; Blaß et al. 2005; Plos et al. 2018), no significant influence of the density on the tensile strength could be determined. Hence, this grading parameter is not considered in the developed model.

Sub-model to estimate the tensile strength parallel to the grain of a board

The model to estimate the tensile strength parallel to the grain of a board is based on two independent variables Edyn and min(ft,ij). It has been shown that the dynamic modulus of elasticity Edyn exhibits a medium linear correlation with the tensile strength (R2 (X1, ln(ft,0)) = 0.48; Fig. 6b) (correlation categorisation based on the JCSS Probabilistic Model Code (JCSS 2006): 0.8 ↔ high correlation, 0.6 ↔ medium correlation, 0.4 ↔ low correlation, 0.2 ↔ very low correlation). The independent variable min(ft,ij) represents the minimum of the estimated local tensile strengths of each cross section and is related to the local fibre orientation as described in Subsection Sub-model to estimate the minimum local tensile strength of a board and Fig. 7. The model assumes a multilinear correlation between the variables Edyn and min(ft,ij) and the logarithm of the estimated tensile strength (ft,estimated) and includes three calibration parameters (Ci) (Eq. 5).

$$\ln \left( {f_{{{\text{t}},0,{\text{estimated}}}} } \right) = C_{0} + C_{1} \cdot E_{{{\text{dyn}}}} + C_{2} \cdot \min \left( {f_{{{\text{t}},{ij}}} } \right)$$
(5a)
$$f_{{{\text{t}},0,{\text{estimated}}}} = e^{{\left( {C_{0} + C_{1} \cdot E_{{{\text{dyn}}}} + C_{2} \cdot \min \left( {f_{{{\text{t}},{ij}}} } \right)} \right)}}$$
(5b)
$$f_{{{\text{t}},{ij}}} = \frac{{f_{{{\text{t}},0,{\text{ref}}}} \cdot f_{{{\text{t}},90,{\text{ref}}}} }}{{f_{{{\text{t}},0,{\text{ref}}}} \cdot \sin^{{\text{b}}} \left( {\alpha_{{{ij}}} } \right) + f_{{{\text{t}},90,{\text{ref}}}} \cdot \cos^{{\text{b}}} \left( {\alpha_{{{ij}}} } \right)}} \cdot \frac{{\frac{{\frac{{E_{{{\text{t}},0,{\text{ref}}}} }}{{E_{{{\text{t}},90,{\text{ref}}}} }}}}{{\frac{{E_{{{\text{t}},0,{\text{ref}}}} }}{{E_{{{\text{t}},90,{\text{ref}}}} }} \cdot \sin^{{\text{a}}} \left( {\alpha_{{{ij}}} } \right) + \cos^{{\text{a}}} \left( {\alpha_{{{ij}}} } \right)}}}}{{\frac{1}{m} \cdot \mathop \sum \nolimits_{j = 1}^{m} \left( {\frac{{\frac{{E_{{{\text{t}},0,{\text{ref}}}} }}{{E_{{{\text{t}},90,{\text{ref}}}} }}}}{{\frac{{E_{{{\text{t}},0,{\text{ref}}}} }}{{E_{{{\text{t}},90,{\text{ref}}}} }} \cdot \sin^{{\text{a}}} \left( {\alpha_{{{ij}}} } \right) + \cos^{{\text{a}}} \left( {\alpha_{{{ij}}} } \right)}}} \right)}}$$
(6)

Calibration and testing of the model

The model to estimate the tensile strength of a board, resulting from combining the sub-models described in the previous subsections, is presented in Eqs. (5) and (6). Thereby, Edyn is the measured dynamic MOE (see Subsection Sub-model to estimate the tensile strength parallel to the grain of a board), aij is the computed local fibre orientation, and m is the number of cross sections in the considered discretisation of the board (see Subsection Sub-model to estimate the minimum local tensile strength of a board). The calibration parameters of the model are the exponents a and b (model to estimate the minimum local tensile strength of a board, Subsection Sub-model to estimate the minimum local tensile strength of a board) and C0, C1, and C2 (model to estimate the tensile strength of a board, Subsection Sub-model to estimate the tensile strength parallel to the grain of a board). To ensure that the calibration parameters remain within reasonable values, i.e. that they make physical sense, the threshold values presented in Table 4 were enforced during the calibration procedure.

Table 4 Regression coefficients Ci, the exponents for the dynamic modulus of elasticitya and the tensile strengthb used in the model
Full size table

Since it is not the actual value but the ratio between the two parameters that has an influence on the stress distribution in the cross section, the moduli of elasticity parallel and perpendicular to the grain were combined in a single parameter, which was assumed to be (Et,0,ref/Et,90,ref =) 30 (Wagenführ, 2006). The reference tensile strength parallel to the grain (ft,0,ref = 180 MPa) and the reference tensile strength perpendicular to the grain (ft,90,ref  = 7 MPa) were chosen according to Wagenführ (2006).

The calibration parameters were determined by minimising the residual sum of squares (RSS) between the estimated (ft,0,estimated) and the experimentally determined tensile strengths (ft,0,experimental) and are presented in Table 4. Only the training data set (n = 36) was considered in this procedure. The other 11 boards remained “unseen”. Equation (7) results from this procedure. It can be used to estimate the tensile strength of a board based on the dynamic MOE (Edyn), the minimum local tensile strength min(ft,ij), computed based on the local fibre direction (αij), and an error term ε. The independent variable min(ft,ij) alone, which is only based on the local fibre direction αij, shows a medium–high linear correlation with the logarithmic value of tensile strength ln(ft) (R2 = 0.65).

$$\ln \left( {f_{{{\text{t}},0,{\text{estimated}}}} } \right) = 1.91 + 5.06 \cdot 10^{ - 5} \cdot E_{{{\text{dyn}}}} + 1.21 \cdot 10^{ - 2} \cdot \min \left( {f_{{{\text{t}},{ij}}} } \right) + \varepsilon$$
(7)

Figure 8 summarises all process steps: based on photographs of the board sections (Fig. 8a), the medullary ray spindles were identified (Fig. 8b), and the field of fibre direction was generated (Fig. 8c). The dynamic MOE (constant along the board) and the minimum local tensile strength of each cross sections i min(ft,j) are shown in Fig. 8d. Such information on the tensile strength for each cross section/along the board’s axis could be especially of importance for choosing the location of finger joints during fabrication. However, for the estimation of the tensile strength parallel to the grain of the board ft,0,estimated, the lowest local tensile strength in the entire board min(ft,ij) is considered in Eq. (7).

Fig. 8
figure 8

Overview of all steps of the reference model using the example of board no. 1035: photos of the board segments between the clamping jaws (a); post-processed black-and-white image of the medullary ray spindles (b); matrix of local fibre directions (c). Based on the measured Edyn and min(ft,ij) along the board’s longitudinal axis (d), the tensile strength ft,0,estimated is predicted (e). The fracture pattern (f) agrees well with the estimated fibre direction. The modelled (ft,0,estimated = 64 MPa) and experimentally determined tensile strength (ft,0, experimental = 64.4 MPa) are on a similar level and failure occurred in the area of maximum fibre inclination

Full size image

In Fig. 9a, the tensile strength predicted by the developed model (ft,0,estimated) is compared to the tensile strength determined in the destructive tension tests (ft,0,experimental). A coefficient of determination of R2 = 0.74 was found. The model error is described by a normally distributed error term (Fink 2014) with the parameters ε ~ N(με = 0; σε2 = 0.018). Taking only the calibration data into account, the root mean square error (RMSE) of the model is 11.0, and the mean absolute percentage error (MAPE) of the model is 9.99.

Fig. 9
figure 9

Correlation analysis between the modelled tensile strength ft,0,estimated and the experimentally determined tensile strength ft,0,experimental for the training data set (a) (n = 36; R2 = 0.74) and the test data set (b) (n = 11; R2 = 0.84)

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In the training data set, the maximum deviation between the modelled and the experimentally determined tensile strength was found for board no. 2008 with a relative error of 43% (model: 82.8 MPa; experiment: 58.0 MPa). The predictions (ft,0,estimated) for all other 35 boards lie within the 95% confidence interval and show a maximum deviation from the experimentally determined tensile strength (ft,0,experimental) of ± 25%. For 22 of the 36 boards (61%), the relative error between modelled and experimentally determined tensile strength is below ± 10% (Fig. 9). Applying the calibrated model to the (unseen) test data set reveals an even better coefficient of determination of R2 = 0.84 (Fig. 9b). Furthermore, the root mean square error (RMSE = 10.2) and the mean absolute percentage error (MAPE = 9.56) are lower compared to the training data set. A maximum relative difference between the experimentally determined and the modelled tensile strength for an individual board of 20% was observed. Although, the number of boards in the test data set was small (n = 11), these results indicate that even with unseen data the model is able to generate good estimates for the tensile strength parallel to the grain.

Comparison with models based on machine-learning techniques

The input parameters for the ML models were the two independent variables Edyn and min(ft,ij) of each board. The density was not considered as an input parameter, since a preliminary check and the literature review indicated that the influence of this parameter on the strength of European beech wood was negligible (Table 3).

The deviation of the maximum and minimum values in the results obtained by different performance measures for each of the sixfolds for training and validating was very limited, indicating that all machine-learning models showed no tendency towards under- or overfitting (Table 5). Overfitting would mean that the analysis would correspond too closely to this particular set of data, without being able to predict subsequent (unseen) observations reliably. Theoretically, the prediction model works best when R2 equals one and both RMSE and MAPE equal zero. Table 5 summarises the performance values R2, RMSE and MAPE of the five models for the test data set, which was not considered in all prior analyses and therefore called ‘unseen’ in order to determine the generalisation ability of the models.

Table 5 Descriptive statistics for the training/validation data set (6-k cross-validation) and the test data set
Full size table

The results clearly show that the predictions of all machine-learning algorithms, according to the relevant indices (R2, RMSE, MAPE), are highly correlated with the experimental observations of the mechanical tensile strength property (Table 5). When applied to the test data set, the predictive models support-vector machine (R2 = 0.87/RMSE = 8.40/MAPE = 7.72%) and random forest (R2 = 0.88/RMSE = 8.16/MAPE = 8.48%) perform best, i.e. the predicted values were closest to the experimentally determined tensile strengths. The artificial neural network model (R2 = 0.86/RMSE = 8.46/MAPE = 7.59%) showed a similar prediction quality and performed best with regard to the MAPE. Even the worst performance of the machine-learning algorithms (Decision Tree algorithm) had acceptable accuracy indices of R2 = 0.81, RMSE = 9.73 and MAPE = 9.91 for the test data set.

Due to the small dataset, the investigations were limited to the evaluation of the applicability of specific machine-learning techniques. The use of deep learning approaches, which require very large data sets and are computationally intensive, was not possible. This study shows that the reference model as well as shallow machine-learning algorithms can be used to analyse the limited dataset and to predict the tensile strength parallel to the grain of European beech boards with high accuracy. However, the use of shallow machine-learning algorithms on a small dataset required the use of cross-validation methods and hold-out tests (Varma and Simon 2006). As discussed above, efficient strength grading of timber is important when attempting at controlling the inherent heterogeneous material structure on different scales and the high variability of wood properties. Machine learning is a powerful tool in this respect and can help to overcome the recurring challenge in wood technology to transform the very heterogeneous raw material wood into products with high value and well-defined properties (Schubert et al. 2020; Schubert and Kläusler 2020).

Conclusion

The present study explores the potential of using information on the local fibre direction in addition to the dynamic modulus of elasticity in the strength grading process of European beech (Fagus sylvatica L.) boards. Based on investigations on 36 knot-free boards assigned to a training or calibration data set, which were used to train different machine-learning techniques (Support-Vector machines, Decision Tree, Random Forest, and Artificial Neural Network using a 6-k cross-validation approach), and another 11 boards assigned to the ‘unseen’ test data set, the following conclusions can be drawn:

  • Data on the local fibre direction are very important for understanding and predicting the tensile strength parallel to the grain of European beech timber. Especially for knot-free boards, this is considered the key parameter to achieve predictions of high accuracy.

  • A model that takes into account the dynamic modulus of elasticity and a second parameter representing the critical local fibre direction was developed using a training data set (77% of the data). For this training data set, a coefficient of determination of R2 = 0.74 between the predicted tensile strength and the experimentally determined tensile strength parallel to the grain and a mean absolute percentage error (MAPE) of 9.99% was found. Applying the model to the unseen test data set (23% of the data) showed an even better coefficient of determination of R2 = 0.84 and a lower MAPE of 9.56%.

  • Shallow machine-learning algorithms in combination with cross-validation methods were successfully used to further improve the prediction accuracy. The machine-learning techniques support-vector machine (R2 = 0.87/RMSE = 8.40/MAPE = 7.7%) and random forest (R2 = 0.88/RMSE = 8.16/MAPE = 8.5%) showed the best performance when applied to the unseen test data set.

The outcomes of this study indicate that the application of combined approaches, including experimental methods and machine-learning, has a great potential for increasing the precision of the strength grading of European beech timber. However, due to the limited size of the dataset available for this study, applying the presented methods to a larger dataset would be important and is envisaged in the course of follow-up projects. In the future, such combined approaches have the potential to contribute to a more resource-efficient use of this timber species widely available in Central Europe and may open new and more demanding fields for high-level timber applications.