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-to-the-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. The tensile strength according to the test method in ISO 13910 [3] can be found in Eq. (1):

$$f_{t90} = \left( {\frac{{3.75 \cdot F_{{{\text{ult}}}} }}{b \cdot h}} \right) \times \left( {\frac{{0.03 \cdot b \cdot L_{h}^{2} }}{{800^{3} }}} \right)^{0.2} ,$$

(1)

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:

$$f_{r} = f_{r\infty } \cdot \left( {1 + \frac{{D_{b} }}{D}} \right),$$

(2)

$$f_{r} = \sqrt {f_{r\infty }^{2} + \frac{{2 \cdot f_{r\infty }^{2} \cdot D_{b} }}{D}} ,$$

(3)

where *B* is the specimen width, *D* is the specimen height, *f*_{r} is the nominal bending strength (*f*_{r} = 6*M*_{max}/*BD*^{2}), *M*_{max} is the maximum bending moment, *f*_{r∞} is the nominal bending strength for a very large specimen height, and *D*_{b} is a constant length.

After proposing Eqs. (2)–(3), Bažant proposed Eq. (4) [16]:

$$f_{r} = f_{r\infty } \cdot \left( {1 + \frac{{r \cdot D_{b} }}{D}} \right)^{1/r} ,$$

(4)

where *r* is an arbitrary positive constant. Equation (2) is used when *r* = 1, and Eq. (3) is used when *r* = 2.

Equations (2–4) 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]:

$$f_{r} = f_{r\infty } \cdot \left[ {\left( {\frac{{D_{b} }}{D}} \right)^{r \cdot n/m} + \frac{{r \cdot D_{b} }}{D}} \right]^{1/r} ,$$

(5)

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. 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.