Estimation of the perpendicular-to-the-grain tensile strength of Scots pine glued laminated timber via three-point bending tests

Three-point bending tests were performed on specimens of glued laminated timber with different specimen heights to failure to determine the relationship between specimen height and bending strength under tension perpendicular to the grain. For the three-point bending tests, two types of glued laminated timber composed of homogeneous grade timber, as specified in the Japanese Agricultural Standard, were used. The laminae used for the glued laminated timber were L80 grade Scots pine and L110 grade Scots pine. The specimens used in the three-point bending tests had dimensions of 105 mm (width) and 10–300 mm (height). The experimental results showed that the bending strength decreased as the specimen height increased, but the rate of decrease in the bending strength decreased with increasing specimen height when the specimen height exceeded 100 mm. From the relationship between the bending strength and specimen height, parameters that fit Bažant’s size-effect law were derived, and for a specimen height of approximately 100 mm, the bending strength was equal to the perpendicular-to-the-grain tensile strength.


Introduction
The tensile strength of wood is extremely high parallel to the grain and extremely low perpendicular to the grain. The structural design standards for timber structures in Japan specify the parallel-to-the-grain tensile strength, but do not specify the perpendicular-to-the-grain tensile strength. One reason for this is the experimental difficulty of evaluating the perpendicular-to-the-grain tensile strength.
Test methods used for evaluating the perpendicular-tothe-grain tensile strength of timber via tensile and bending experiments have been described in JIS Z 2101 [1], ASTM D143-14 [2] and ISO 13910 [3]. The tension test method is shown in Fig. 1. In the method shown in Fig. 1a [1], the stress is uniformly applied at the center of the specimen; however, the stress becomes nonuniform and the stress concentration increases with increasing distance from the center [4]. Experimental studies have also reported that the tension test specimens shown in Fig. 1a [1] break mainly in the arc region [5]. In the method shown in Fig. 1b [2], the stress does not act uniformly, and the stress concentration is very high [4]. The method shown in Fig. 1c [3] yields different tensile strength values depending on the dimensions [6,7]. This is due to the influence of the dimensional effects of Weibull's statistical theory. ISO 8375 [8] also specifies the method shown in Fig. 1c, where the specimen has the following dimensions: h = 400 mm, b ≥ 100 mm, and b × l = 25,000 mm 2 .
The bending test method described in ISO 13910 [3] is shown in Fig. 2 (1)

Open Access
Journal of Wood Science *Correspondence: okamoto@osaka-cu.ac.jp 1 Graduate School of Human Life Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan Full list of author information is available at the end of the article where f t90 is the perpendicular-to-the-grain tensile strength, F ult is the value of the applied load at failure. Equation (1) is equal to 1/3 of the maximum bending stress according to the Bernoulli-Euler theory multiplied by the size-effect factor. Multiplying by the dimensional effect factor normalizes the tensile strength to the equivalent value for a cube of timber with a side length equal to 800 mm. This test method yields a tensile strength of the lowest boundary limit.
Kuwamura [5] performed the bending test shown in Fig. 2 (for a specimen height 20 mm, specimen width 30 mm, and distance between the fulcrums of 100 mm) and the tension test shown in Fig. 1a on cuts from the same piece of wood. They reported a flexural strength result that was slightly higher but not significantly different from the tensile strength. Yokobori [9] explained that the dimensional effect in the case of stress gradients such as those due to bending is due to the difference in the average value of stress over a constant length t, as shown in Fig. 3. These results indicate that it is difficult to determine the perpendicular-to-the-grain tensile strength by using a tension test and that by using a bending test, in addition to the effect of specimen size, it is necessary to consider the effect of the difference in the average value of the stresses over a constant length.
In a perfectly brittle material containing a crack, the entire fracture process under a tensile loading takes place at the tip of the crack. However, in many materials, such as concrete and wood, the fracture process is not confined to a point and occurs within a certain length, called the fracture process zone, which extends beyond the tip of the crack. This fracture process zone is said to be one of the causes of the size effect on the bending strength.
Hillerborg et al. [10] proposed a fictitious crack model, which is an analytical method that can evaluate the influence of this fracture process zone. Figure 4 shows the analytical results of the fictitious crack model for the unnotched three-point bending tests of concrete [11], where f t is the tensile strength, f f is the bending strength, l ch is a characteristic length, and σ i is the initial stress. In the case of σ i = 0, the bending strength decreases as the beam height d increases and asymptotically approaches the tensile strength. The fictitious crack model has also been applied to wood, and Boström [12] applied the fictitious crack model to the compact tension specimen of wood and reported the influence of different material properties on the experimental results.
Bažant proposed a variety of general equations for describing the size effect of the fracture process zone by asymptotic analysis of the energy release. Bažant first proposed a size-effect law for a structure containing a notch or a stably growing large crack [13]. Subsequently, Bažant proposed Eq. (2) [14] and Eq. (3) [15] by extending the size-effect law [13] for a structure with a crack to a structure without a crack: where B is the specimen width, D is the specimen height, f r is the nominal bending strength (f r = 6M max /BD 2 ), M max is the maximum bending moment, f r∞ is the nominal (3)  where r is an arbitrary positive constant. Equation (2) is used when r = 1, and Eq. (3) is used when r = 2.
Equations (2)(3)(4) show that as the specimen height D/D b increases, f r / f r∞ decreases and approaches 1.0, similar to the results for σ i = 0 in Fig. 4.
The probability of bending failure of a structure without a crack increases as the cross-sectional area of the beam increases because of the presence of a large number of defects at locations of high stress. Therefore, the effect of specimen size must be considered in addition to the effect of the fracture process zone. Therefore, Bažant proposed Eq. (5), which adds the effect of size to Eq. (4) [17]: where m is the Weibull modulus and n is the number of spatial dimensions (n = 1, 2, or 3; in the present calculations, 2). According to existing test data for concrete, f r∞ = 3.68 N/mm 2 , D b = 15.53 mm, r = 1.14 and m/n = 12 are optimal [17].
Aicher [18] evaluated the effect of size on the perpendicular-to-the grain tensile strength in European spruce (Picea abies) notched beam specimens with the Bažant's size-effect law [13] and determined the fracture process zone length. However, no previous studies have applied Bažant's size-effect law [17] to timber without a crack. (4) , Therefore, in this study, the relationship between specimen height and bending strength was investigated by performing three-point bending tests on Scots pine (Pinus sylvestris) glued laminated timber specimens with different heights. In addition, parameters were derived from the relationship between the specimen height and bending strength to fit Eq. (5). Since f r∞ in Eq. (5) is considered to be very close to the tensile strength, we also examined the specimen height at which the bending strength is approximately equal to the tensile strength.

Materials and methods
For the three-point bending test, two types of glued laminated timber composed of homogeneous grade timber, as specified in the Japanese Agricultural Standard (JAS) [19], were used. The laminae used for the glued laminated timber were L80 grade (8.0-10.0 GPa) Scots pine (L80) and L110 grade (11.0-14.0 GPa) Scots pine (L110). The laminae were 30.3 to 31.6 mm thick and bonded with phenol-resorcinol-formaldehyde resin. The strength grade of the glued laminated timber made from the L80 laminae is E75-F270 [19]. The strength grade of the glued laminated timber made from the L110 laminae is E105-F345 [19]. wood manufactured with cross-sections with 105 mm widths and 150, 300 and 750 mm heights. We did not use a constant temperature and humidity testing chamber, and we did not control the temperature and humidity when the specimens were stored. The mean ± standard deviation of the density was 480 ± 14 kg/m 3 for L80 and 518 ± 12 kg/m 3 for L110. The mean ± standard deviation of the moisture content was 9.6 ± 3.1% for L80 and 11.6 ± 2.7% for L110. The moisture content was measured using a moisture tester (HM-520, Kett Electric Laboratory). The moisture content considered was the value measured for specimens with D ≥ 60 mm. The surface on which the tension of the specimen acted (the underside of the specimen) is shown in Fig. 6. For the experiment, the load was applied at a constant crosshead speed. The crosshead speed was adjusted from 1 to 2 min, from the start of loading to bending failure.
The bending strength f r was determined by Eq. (6) using the maximum load obtained in the experiment: where P max is the value of the maximum load.
The parameter fitting in Eq. (5) is derived by an iterative method using curve fitting with changing r or m/n.
The optimal values of the parameters are obtained by the least squares method using Eq. where Curve fitting is determined by the minimum value of the residual sum of squares of f r /f r∞ obtained by Eq. (9): where is RSS is the residual sum of squares of f r /f r∞ , f r, cal. is the calculated value obtained by Eqs. (4) and (5). f r, exp. is the experimental value obtained by Eq. (6). N is the number of all specimens (N = 198).
The method of this iterative calculation is described later in this paper.

Experimental results
The final failure state of the side under tension is shown in Fig. 7. Most of the specimens in the three-point bending tests failed at the pith located near the crosshead (Fig. 7a) or at the adhesive layer located on the pith side near the crosshead (Fig. 7b). Figure 8 shows the relationship between the force point load P and the force point displacement δ of a typical specimen. Table 1 shows the f r (average and standard deviation) of the experimental results. Figure 9 shows the relationship between f r and D in the experimental results. When D was between 10 and 100 mm, the value of f r decreased significantly as D increased. However, when D was between 100 and 300 mm, the values of f r were almost the same, although there was a slight tendency of f r to decrease. Regarding the difference in L, for L80, f r was slightly smaller when L was larger, even though D was the same. However, for L110, there were cases where there was no difference in f r due to the difference in L or where a larger L value resulted in a larger f r . Therefore, the variation in strength had a greater effect than the difference in L did on the experimental results. Fig. 6 Lower surfaces of representative specimens

The parameters of the size-effect law
Bažant et al. [17] assumed that the greatest change from one concrete or one testing approach to another can be described by the values of f r∞ and D b and that the values of r and n/m are approximately the same for different concretes and test series. In this study, this assumption was also applied to wood, and it was assumed that the values of f r∞ and D b are different, between L80 and L110, but the values of r and n/m are the same. We also assumed that f r∞ and D b would be the same for different L because we thought that the effect of different L would be very small in the scope of the present study. The parameters of Eq. (5) were then derived in steps 1-4. Step 1 Using the experimental data of L80 and L110 as different data sets, obtain f r∞ and D b for r = 0.01-3.50 by the linear regression approximation of Eq. (7). Next, obtain f r using Eq. (4), and obtain the residual sum of squares of f r /f r∞ using Eq. (9) with L80 and L110 as one data set. The optimal value is the parameter when the residual sum of squares of f r /f r∞ obtained by changing r is minimized. Figure 10 shows the relationship between r and RSS/N.

Step 2
Using the experimental data of L80 and L110 as different data sets, obtain f r∞ for m/n = 4.1-90.0 by the linear regression approximation of Eq. (8) using the optimal values of r and D b . Next, obtain f r using Eq. (5), and obtain the residual sum of squares of f r /f r∞ using Eq. (9) with L80 and L110 as one data set. The optimal value is the parameter when the residual sum of squares of f r /f r∞ obtained by changing m/n is minimized. Figure 11 shows the relationship between m/n and RSS/N.
Step 3 Using the experimental data of L80 and L110 as different data sets, obtain f r∞ and D b for r = 0.01-3.50 by the linear regression approximation in Eq. (8) using the optimal value of m/n. Next, obtain f r using Eq. (5), and obtain the residual sum of squares of f r /f r∞ using Eq. (9) with L80 and L110 as one data set. The optimal value is the parameter when the residual sum of squares of f r /f r∞ obtained by changing r is minimized. Figure 12 shows the relationship between r and RSS/N.

Step 4
Using the optimal values of r and D b obtained in step 3, obtain the optimal values of f r∞ and m/n in step 2. Then, using the optimal value of m/n obtained in step 2, obtain the optimal values of r, f r∞ and D b in step 3. This is repeated until r, f r∞ , D b and m/n do not change.
Steps 2 and 3 were calculated repeatedly, and 28 calculations converged with no change in value. Figure 13 shows the relationship between m/n and RSS/N obtained in step 2 after convergence, and Fig. 14 shows the relationship between r and RSS/N obtained in step 3 after convergence.
According to existing concrete test data, m/n = 12 is optimal [17]. Barrett [20] reported a size parameter of 7.68 for the perpendicular-to-the-grain tensile strength of Douglas fir by volume change and a theoretical size parameter of 9 for a beam loaded along a constant width. Based on the above, the m/n = 10.3 obtained in Fig. 13 is considered to be a reasonable value.
The relationship between f r /f r∞ and D/D b for the experimental and calculated values (Eq. 5) using the optimal parameters is shown in Fig. 15. The calculated values generally captured the experimental values, confirming the applicability of Eq. (5) to Scots pine glued laminated timber. Figure 16 plots the horizontal axis of the calculated values in Fig. 15 as a function of D. When the specimen height is D = 80-120 mm, the error of f r /f r∞ is within 5%. Therefore, the bending strength is almost equal to the tensile strength for D = 80-120 mm.  Figure 17 compares f r∞ and f r (D = 100 mm) obtained in the three-point bending test with existing data on perpendicular-to-the-grain tensile strength. The density of f r∞ is the average value of all the specimens. The existing data are the perpendicular-to-the-grain tensile strengths of softwood species grown in the United States [21] and that of softwood species grown in North Korea [22]. The perpendicular-to-the-grain tensile strengths of the softwood species grown in the United States [21] are the averages of the radial and tangential strengths evaluated according to ASTM D143 [2], and the wood species were bald cypress (Taxodium distichum), cedar (Chamaecyparis thyoides, Calocedrus, Thuja occidentalis, Chamaecyparis lawsoniana, Thuja plicata, and Cupressus nootkatensis), Douglas fir (Pseudotsuga menziesii), fir (Abies balsamea, Abies magnifica, Abies grandis, Abies procera, and Abies concolor), hemlock (Tsuga heterophylla),   Min. Fig. 12 The residual sum of squares and the optimal values of the parameters for step 3 (first time) not directly comparable due to differences in testing methods, the existing data and f r∞ were approximately the same.

Conclusion
In this study, a three-point bending test was conducted on specimens of Scots pine glued laminated timber with different specimen heights to determine the relationship between specimen height and bending strength to failure in tension perpendicular to grain. The experimental results showed that the bending strength decreased as the specimen height increased, but it did not decrease considerably as the specimen height exceeded 100 mm. From the relationship between the bending strength and specimen height, parameters were then derived that fit Bažant's size-effect law. It was found that for the specimen specifications in this study, the bending strength is equal to the perpendicular-to-the-grain tensile strength at a specimen height of approximately 100 mm. However, it is assumed that the bending strength does not decrease considerably above a certain specimen height, although the specimen height at which the bending strength is approximately equal to the tensile strength perpendicular to the grain, is likely to be different for other species. Thus, it may be possible to establish a bending test method that can measure a slightly conservative perpendicular-to-thegrain tensile strength by investigating specimen heights