Experimental study on glued laminated timber beams with well-known knot morphology

Five different types of GLT beam setups, with ten specimens each, were produced. The five GLT beam types A, B, C, D, and E differ in number of laminations, grading class, and length. In Table 1, the parameters of the different types are summarised.

Since the study is devoted to the impact of knots on the mechanical behaviour, all beams were produced of full length lamellas. In this way, finger joints, which would have represented additional weak-sections in the GLT assembly, could be avoided.

Norway spruce boards of grades LS15 and LS22—delivered by a sawmill located in Töreboda, Sweden—were used. The original dimensions were 35 mm height, 95 mm width and 5400 mm length. Each board was marked with a unique ID, so the measured data could always be linked to the corresponding board.

For each board, the dynamic modulus of elasticity was obtained from Olsson et al. (2012)where \(\rho\) is the density, \(f_1\) is the lowest axial eigenfrequency and L is the length of the board. The eigenfrequency \(f_1\) was obtained using a hammer, a microphone (GRAS, type 26 CA) and a DataPhysics DP700-60 FFT-analyser.

The used boards were inspected using a tracheid effect-based laser scanning device, yielding the fibre angle on all surfaces (Nyström 2003; Simonaho et al. 2004; Petersson 2010). For a more detailed description of the procedure it is referred to Olsson et al. (2013) and Kandler et al. (2015), where the fibre angle measurements have been used to obtain a stiffness profile for each board.

After inspection, all boards were sent to a GLT manufacturer for gluing using a MUF-adhesive. Prior to gluing, all boards were planed to a height of 33 mm. Using the unique board number, the position of each board within the GLT assembly could be traced. The orientation of the boards was chosen to meet the Swedish standards for GLT production. That is, the boards were oriented such that the local pith of the top-most and bottom-most boards would be located above and below of the beam, respectively. After gluing, the beams were planed to the final width of 90 mm.

The lamella strength classes T14 (LS15) and T22 (LS22) correspond to GLT strength classes GL 24h and GL 30h, respectively (EN 14080 2013).

In a first step, a geometrical description of the knot morphology is determined by employing the automated approach described in Kandler et al. (2016). Therein, fibre angle measurements were experimentally obtained from laser scans based on the so-called tracheid effect, which describes the light propagation patterns in wood. From the fibre angle measurements, the knot areas are estimated. In combination with the pith location, all knots were reconstructed using rotationally symmetric cones. It has been observed that this algorithm also works well for small knots. In Fig. 1a–c, the results for an exemplary board are shown. In Fig. 1b, all recognised knots are displayed, whereas in Fig. 1c only the significant knot groups according to a modified version of the criterion presented in Kandler et al. (2016) are shown.

The modified criterion is based on omitting knots with an opening angle \(\alpha <1.5^\mathrm {o}\). Then, for all knots, the corresponding knot areas visible on the board surface are evaluated. Subsequently, all knot area values are sorted according to their value, resulting in a histogram. Then the value situated at the 70% quantile is chosen to be the critical value. All knot areas larger than this value are from now on defined as large knots, forming the 30% largest-knots quantile. These knots are assumed to be mechanically significant and are retained. In case the mutual distance in longitudinal direction x of two such knots is below 200 mm, these knots are grouped together. It is assumed that smaller knots are not important for the mechanical model and are therefore omitted. However, in case a smaller knot is in close vicinity to one of the larger knots, it is retained in the model, since it is assumed to lead to interactions in terms of fibre angle course and failure modes. As measure for the vicinity, a distance value in x-direction of 100 mm has been observed to yield good results.

The automatic nature of the reconstruction algorithm lends itself particularly useful for big sample sizes. In this study, a total number of 350 boards was investigated using this procedure.

In this study, two different types of stiffness profiles were assessed. A stiffness profile describes the fluctuation of the local modulus of elasticity E in the longitudinal direction x, i.e. \(E \equiv E(x)\). The procedure for the first type of stiffness profile is presented in Kandler et al. (2015), where fibre angle measurements obtained through laser scanning are directly used to transform the clearwood stiffness tensor. This approach is similar to the approach presented in Olsson et al. (2013), where the clearwood stiffness tensor was obtained by combining dynamic eigenfrequency measurements with a dynamic finite element analysis. Therein, the components of the stiffness tensor were calibrated so that the lowest eigenfrequency of the finite element model would match the experimentally obtained eigenfrequency. Subsequently, in each point the longitudinal stiffness component is extracted from the transformed stiffness tensor. Employing a nearest-neighbour interpolation, the stiffness values are homogenised over each cross section, resulting in a stiffness profile for each board. In Olsson and Oscarsson (2017), it has been shown that the weakest value in such a stiffness profile correlates well to the strength of a single board.

In Kandler et al. (2015), the approach is based on the same principle of transforming the clearwood stiffness tensor according to the fibre angle. Therein, in addition to the in-plane fibre angle, an empirical model for the dive-angle was employed and the clearwood stiffness tensor was obtained from a micromechanical model based on a multiscale homogenisation procedure (Hofstetter et al. 2007). Based on knowledge of the stiffness profiles in combination with the position of each board, an accurate mechanical model for assessing the stiffness of GLT beams could be developed. In Kandler et al. (2015), a linear 2D finite element code was employed to predict the effective stiffness of GLT beams. In this study, on the other hand, it is investigated how well these data can be correlated to failure mechanisms in GLT.

Complementing the approach described in the previous paragraph, the reconstructed 3D knot geometries as described in the previous section are also considered. The 3D knot geometries of each knot group were used to compute the 3D fibre angle course within the volume of the board using the procedure outlined in Hackspiel et al. (2014) and Lukacevic and Füssl (2014). This procedure follows the so-called grain-flow analogy (Foley 2003). Therein, the fibre angles in the longitudinal-tangential plane are assumed to follow the trajectories of a fluid in laminar flow where knots constitute elliptical obstacles. For the dive-angle, polynomials fitted to photographs of knot sections are used. Subsequently, the knot geometries and numerically obtained 3D fibre angles are used within a linear elastic 3D finite element analysis. Within this analysis, each knot group (in total more than 3000 groups) was loaded in tension to estimate its effective stiffness. At one end, the knot group is fixed in longitudinal direction and at the other end all nodes are exposed to a predefined displacement in longitudinal direction. The remaining boundary conditions were applied such that the specimen could freely expand in the other directions. From the resultant forces, the effective elastic modulus in longitudinal direction of each knot group could be computed. Based thereon, an alternative stiffness profile was developed, as can be seen in Fig. 1d. From Fig. 1d, it can be seen that for the knot sections a notable stiffness difference exists. The main reason is that the approach by Kandler et al. (2015) considers the local fibre deviation in each measured point but it does not consider interactions between knots. The 3D approach, on the other hand, inherently takes account of interactions between knots. Therefore, the 3D approach generally leads to lower stiffness values.

For the purpose of assessing the strength properties of individual boards, a set of indicating properties is computed from the reconstructed knot geometry. Based on the study presented in Lukacevic et al. (2015), different combinations of the following parameters were considered:In total, the four parameter setswere employed to predict the effective tensile strength \(f_t\) of each knot group.

For example, in Fig. 2, the total knot volume in the maximum bending moment region (between the two points of load application) of all tension lamellas is displayed.

Similar to the stiffness profiles, for each knot group the strength was estimated employing indicating properties 1–4. For the clearwood sections, the tensile strength is computed following the approach proposed in Hackspiel et al. (2014), which is based on scaling empirically based material properties according to results obtained by the micromechanical model for elastic behaviour.

For each GLT beam, the dynamic stiffness was obtained using the same procedure as for single boards described in Sect. 2.1. Thereafter, each beam specimen was tested under quasi-static four-point-bending until failure. The four-point-bending setup was chosen to closely resemble the procedure described in EN 408 (2010). The bending span was adjusted to the length of each type, see Table 1 and Fig. 3. At load application locations, a load cell recorded the corresponding forces \(F_\mathrm {left}\) and \(F_\mathrm {right}\), respectively. The displacement w was recorded at the center of the beam at the compression (bottom) side. The rate of loading was approximately \({5}{\hbox {kN} \hbox {min}^{-1}}\).

For all GLT specimens, continuous deflection measurements at the midspan point of the tension side of the beam were conducted during the loading process.

After testing, the geometry of all cracks occurred was manually recorded and documented for all sides of the 50 GLT beams. For cases where the point of crack initiation was not straighforward to identify, all possible crack initiation locations weres documented. In Fig. 4, the documented crack pattern is displayed exemplarily for one beam. Figures of all documented results for types A to E are available as electronic supplementary material, where within each type, beams are numbered from 1 to 10.

The recorded crack measurements were used to estimate the crack area \(A_\mathrm {crack}\). Assuming that all cracks range from one side of the beam to the other side of the beam, the crack area was computed according to \(A_\mathrm {crack} = L_\mathrm {crack} b/2\), where \(L_\mathrm {crack}\) is the accumulated crack length on all sides of the beam. For all beams, the width was b = 90 mm.

All collected data was assembled in a database and statistically evaluated in the commercial software tool optiSLang\(^\mathrm {TM}\).

In addition, an approach of directly employing the stiffness profiles E(x) was pursued. This approach is inspired by the promising procedure presented in Olsson et al. (2013), Oscarsson et al. (2014), and Olsson and Oscarsson (2017). Therein, the minimum value of the stiffness profile E(x) of a board is assumed to present the weakest defect. Consequently, this value is used as indicating property for the effective strength of the whole timber board, resulting in remarkable prediction results with values of \(R^2 \approx 0.7\).

Since for most GLT beams failure is initiated in a weak section on the tension side of the beam, here the minimum value of E(x) of the outer-most lamination on the side exposed to tension is considered. It should be noted that here the stiffness profiles computed in Kandler et al. (2015) are used, which, although based on a different procedure, should yield similar results. Furthermore, the minimum stiffness value not of the whole board length but in the maximum moment range is considered. For the present load case of four-point bending this means that the minimum stiffness value between the two load application locations is considered. To account for the size of the beam, also the beam height h and the beam bending span L are considered, resulting in the linear modelThe model was trained using stepwise regression, where at each iteration step, predictor terms are added or removed according to a given criterion. Herein, for each term an F-test of the change in the sum of squared errors (SSE) through addition or removal of the corresponding term is used. In case the p-value of the F-statistic is smaller than 5%, the term is added to the model. Conversely, if the p-value exceeds 10%, the term is removed from the model. The bending span L and quadratic terms have been considered but did not result in significant improvements and, therefore, were omitted, resulting in the estimated bending strengthwhere the regression coefficients, standard errors (SE) and p-values areThe stepwise regression approach helps in systematically reducing the number of parameters to the statistically significant ones. In this example, the stepwise regression procedure revealed that for the investigated sample, the bending span L does not have a statistically significant impact on the resulting bending strength.

From the plots shown in Fig. 5a, e, it can be seen that the two grading classes reach different maximum load peaks.

In accordance with EN 408 (2010), the system stiffness \(k = \varDelta F/ \varDelta w\) is computed from a linear regression of the load displacement curve in the range of \(0.1 F_{\max }\) and \(0.4 F_{\max }\), where a coefficient of determination \(R^2> 0.99\) is guaranteed. In Fig. 5b, d, f, the relation between system stiffness and maximum load are displayed. While a trend can be clearly observed, the values within each type are not very strongly correlated as the \(R^2\) values given in Table 1 suggest.

For four-point-bending tests, the bending strength is computed according to (EN 408 2010)where \(F_{\max }\) is the maximum total load, a is the horizontal distance between support and load, b is the width of the beam, and h is its height. In Fig. 6, it can be seen that the bending strength \(f_b\) decreases with increasing lamination number. Not surprisingly, for beams of same lamination number, the bending strength of grading class LS22 is higher than the bending strength of grading class LS15. As can be seen from the confidence intervals indicated by the dotted lines, the difference between the median values of the two grading classes is significant at the 5%-level for the beams of 10 laminations. For the beams of 4 laminations, the 95% confidence intervals show a slight overlap. Therefore, strictly speaking, the null-hypothesis of different medians cannot be rejected at the 5%-level. However, looking at the individual sample values, still a notable difference is observed for the 4-lamella beams. The boxplots indicate three outliers: C4, C10 and E9.

The system-related stiffness k can be translated to the material-related effective modulus of elasticity (EN 408 2010)where L denotes the distance between the supports and G is the elastic shear modulus. In Kandler et al. (2015), the value for G was obtained from a micromechanical model. However, the study of Kandler et al. (2015) and more recently also Balduzzi et al. (2018) revealed that the shear modulus only has minor effect on the outcome of Eq. (5) for the investigated beams. For this reason and also to avoid introducing unnecessary error, here a constant value of \(G={650}{\hbox {MPa}}\) in accordance with EN 408 (2010) is used.

For both stiffness and strength, the transition of the system-related quantities \(F_\mathrm {max}\) and k to the material-related quantities \(f_m\) and \(E_\mathrm {GLT}\), respectively, ‘compresses’ the data. This ‘compression’ leads to a reduced coefficient of determination \(R^2\) for the whole sample, as can be observed in Fig. 7. However, within each individual type, \(R^2\) remains the same as a linear relationship is maintained.

In Fig. 5, the load deflection curves \(F = F_\mathrm {left} + F_\mathrm {right}\) of all types are displayed. It can be seen that after an initially linear curve, 12 beams show nonlinear behaviour before the system load bearing capacity \(F_{\max }\) is reached. These nonlinearities are on the one hand cracks on the tension side, resulting in a spike in the load-displacement curve and, on the other hand, plastifications on the compression side of the specimen, leading to a reduction of the load-displacement gradient. Thereafter, brittle system failure due to initiation of cracks is observed. The transition from a linear to a nonlinear curve can be explained by local plastification-like effects in the compression zone of the beams, as can be seen in Fig. 4a. Computing \(f_b\) according to Eq. (4) does not reflect these local plastifications which lead to a non-linear normal stress distribution over the cross-section height, therefore rendering Eq. (4) inaccurate. Rather, \(f_b\) has system character and represents a stress quantity corresponding to conventional brittle strength theory (Blank et al. 2017).

After a first crack has formed, some beams reach a higher load bearing capacity. This behaviour is observed for 2 beams of type A, 4 beams of type B and 3 beams of type C. For type E, one specimen showed abnormal behaviour. While the sample size may be too small to give a reliable assessment, the bigger beam types seem to show a more brittle failure behaviour than the smaller beam types, as was also found in Blank et al. (2017) (Table 2).

To identify patterns in the crack directions, for each segment of the recorded crack geometries the height difference \(\varDelta z\) between end and start point were computed. Subsequently, for each beam the sum of those values was computed to obtain a measure for the steepness of the cracks: \(L_{\mathrm {crack},z} = \sum \varDelta z\). Similarly, the component related to the x-direction, \(L_{\mathrm {crack},x}\), was computed from the sum of differences \(\varDelta x\). In Fig. 8, the ratio of those two results \(L_{\mathrm {crack},z}/L_{\mathrm {crack},x}\) is displayed for each beam. Therein, it can be seen that this ratio lies in the same range for beams of grading class LS22 and seems to be independent of the number of laminations. For example, a crack that spans 1000 mm in x-direction is, on average, accompanied by a z-increment of 40 mm. Thus, such a crack crosses approximately one lamination (recall that all laminations have a thickness of 33 mm). Conversely, for grading class LS15, the ratio \(L_{\mathrm {crack},z}/L_{\mathrm {crack},x}\) is significantly larger and a crack with \(\varDelta X = {1000}{\hbox {mm}}\), on average, crosses at least 2 lamellas.

This behaviour can also be observed by comparing the visualizations of the crack patterns of the two grading classes for the same numbers of laminations, i.e. Figures E.1 with E.2 and Figures E.4 with Figures E.5, respectively. For the lower grading class of LS15 crack patterns seem to remain more localized with respect to their extension in longitudinal direction, which can be explained by the higher propability of adjacent weak sections compared to the higher grading class of LS22, which is emphasized by the higher density of colored patches within the plots of the former, showing the locations of knots, and also by the higher amount of blueish/darker colors, denoting higher single knot volumes and, thus, bigger knots.

The comparison of GLT beams for the lower grading class of LS15 (cf. Figures E.1 and E.4) shows that the difference in dimensions and bending spans, about twice the length and bending span for the bigger GLT beams, leads to almost twice the steepness of the cracks. This behaviour might be explained by the fact that for the smaller dimensions, the extension of the crack in vertical direction is limited by their height, as after failure of just two laminations already half of the beam’s cross section is cracked. For the bigger GLT beams of LS15, the crack, which as explained for this grading class tends to be more localized and, thus, to propagate more likely in vertical direction, leads to comparatively more failed laminations.

Interestingly, as mentioned above, the steepness measure for the beams of LS22 (cf. Figures E.2, E.3 and E.5) seems to be the same for all dimensions and numbers of laminations. This means in the present case that as the bending span gets bigger and also the extension of the crack in longitudinal direction \(L_{\mathrm {crack},x}\) is getting bigger, more laminations fail. But still even for the biggest beam type only the first few tensile-loaded laminations fail, such that the steepness measure stays the same and the size of the crack only depends on the dimensions and/or bending span.

For the linear regression model based on minimum values of the stiffness profiles shown in Eqs. (2) and (3), an adjusted coefficient of determination of \(R_{\mathrm {adj}}^2=0.6\) and root mean squared error (RMSE) of \(\sqrt{\mathrm {MSE}}={5.62}{\hbox {MPa}}\) were obtained. In Fig. 10a, the values predicted from the regression model are plotted against the actual values. It can be seen that lower strength values tend to be overestimated while higher strength values tend to be underestimated by the criterion.

In addition, a ‘stiffness-profile-curvature’ was introduced to also model the spatial vicinity of neighbouring weak spots (see Fig. 9). Starting at the topmost lamella 0 (at the tension side), the lowest stiffness value in the region of the maximum bending moment is determined. For the next lamella 1, all local minima are determined, and the one located closest to the original weak spot is selected. Starting from lamella 1, the next weak spot in lamella 2 is searched and so on. Finally, a gradient is estimated by linear regression through the determined points. The idea of this approach is that the gradient represents the crack pattern responsible for failure.

Alternatively, a more elaborate model employing a 2D finite element analysis is employed. For this, an approach similar to the mechanical model in Kandler et al. (2015) was chosen. Instead of the continuous laser-scan based stiffness profiles, the 3D FE based stiffness profiles according to Fig. 1d are used for describing the longitudinal stiffness of each lamella. In addition, the strength profiles are used to describe the tensile strength of each lamella.

The material properties are extracted from the set of stiffness and strength profiles that are provided for the FE procedure. The stiffness profiles provide the value for the longitudinal stiffness \(E_L\). The remaining parameters of the transversal isotropic material are defined in accordance with the values found in EN 14081 (2009) and Neuhaus (1981):The values for \(E_L(x)\) are obtained from the stiffness profile of the corresponding lamella. Thus, for the plane stress problem, in each integration point the elasticity matrix \(\mathbb {C}\) is computed fromwhere \(E_L(x)\) is the stiffness profile value of the corresponding lamella. Similarly, each integration point is associated with strength parameters which are obtained from the corresponding strength profile. The results returned by the FE solver comprise the displacement values of all nodes as well as the stresses in all integration points. In addition, a Tsai-Wu criterion (Dorn 2012) is used to assess the utilization and load bearing capacity of the beam. Retaining the terms for the plane stress state results in the Tsai-Wu criterion as followsThereby, L corresponds to x and R corresponds to the z-direction. Since the tensile strength values, represented by strength profiles, vary spatially, the corresponding parameters are dependent on the location of the integration point. The components in L-direction are computed in each integration point according towhere \(f_{c,L}= -52.0{\hbox {N}}/{\hbox {mm}^{2}}\) and \(f_{t,L}(x)\) is obtained from the strength profile of the corresponding lamella. The coefficients which are constant within the beam are (Dorn 2012)In accordance with the findings presented in Serrano and Gustafsson (2007), a mean stress approach is pursued. Consequently, the stress components \(\sigma _{11}\), \(\sigma _{22}\), and \(\sigma _{12}\) are not compared directly within each integration point. Rather, the mean values of these components are computed for each cell of a rectangular grid (cell height of 43 mm and cell length of 79 mm). The cell dimensions were found by employing an optimization scheme, where the chosen dimensions were found to yield the highest coefficient of determination \(R^2\). The resulting mean values are subsequently used within the Tsai-Wu failure criterion. In comparison to a strictly point-related evaluation, the mean stress approach leads to higher estimations of the total system load bearing capacity.

A comparison of the corresponding numerical and the experimental results for the bending strength \(f_b\) is given in Fig. 10b. Therein, the results using the procedure with the four different IPs for the tensile strength property are shown. Results for IP 1 were omitted as the results did not show usable agreement. However, while the correct trend can be observed for IPs 3 and 4, the prediction quality is unsatisfying, the highest coefficient of determination being \(R^2=0.40\) (IP 3). Using the mean stress approach (see Fig. 10c), higher numerical bending strength values are achieved, leading to a coefficient of determination of \(R^2=0.54\), which is still not reliable enough. Therefore, it can be concluded that although the system failure behaviour can be interpreted as brittle failure, such brittle mechanical models do not concur well with the experimental observations. This observation also agrees with the findings presented in the work of Blank et al. (2017).

Subsequently, a statistical evaluation of the data is performed using the commercial software package optiSLang\(^\mathrm {TM}\). A general overview of linear correlation coefficients \(\delta\) and distributions of both parameters and results \(E_\mathrm {GLT,exp}\), \(f_b\) and \(n_\mathrm {lam,failed}\) is given in Fig. 11. The linear correlation coefficient between two quantities X and Y is computed according towhere \(\mathrm {COV}(X,Y)\) is the covariance between two variables, \(\sigma _X\) the standard deviation, \(x_i\) the i-th measurement of variable X, N is the sample size and \(\hat{\mu }_X\) the estimated mean value and \(\hat{\sigma }_X\) the estimated standard deviation of the corresponding variable. As for the medium-sized GLT beams (Type C) no experiments for the lower grading class were conducted and, in addition, not all parameters for the higher grading class were available, in Fig. 11, which correspond to the 3D FE and the strength profile parameters, the results only for Types A, B, D and E are shown. The data is clustered into general parameters and specific parameter groups as follows: