 Original Article
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Study on the strength of glued laminated timber beams with round holes: difference in structural performance between homogeneousgrade and heterogeneousgrade timber
Journal of Wood Science volume 67, Article number: 8 (2021)
Abstract
Various design codes and design proposals have been proposed for glued laminated timber beams with round holes, assuming that the entire beam is composed of homogeneousgrade timber. However, in Japan, glued laminated timber composed of homogeneousgrade timber is rarely used for beams. In this study, the difference in the loadbearing capacity of glued laminated beams composed of homogeneousgrade timber and heterogeneousgrade timber with round holes when fractured by cracking was investigated experimentally and analytically. The materials used in the tests were glued laminated beams composed of homogeneousgrade Scots pine timber with a strength grade of E105F345 and heterogeneousgrade Scots pine timber with a strength grade of E105F300. Experiments confirmed that although the glued laminated beams composed of heterogeneousgrade timber have a lower material strength in the lamina with holes, its resistance to fracturing due to cracks associated with the holes is almost the same as that of the glued laminated beams composed of homogeneousgrade timber. The stresses acting on the holes in the laminated timber with holes of less than half the beam height were lower in the glued laminated beams composed of heterogeneousgrade timber than in the glued laminated beams composed of homogeneousgrade timber. The ratio of the stresses was found to be approximately equal to the ratio of the maximum bending stress or the maximum shear stress acting on the inner layer lamina, as determined by Bernoulli–Euler theory.
Introduction
When the shear force and bending moment act on a glued laminated timber beam with a round hole, the beam will undergo flexural fracture if the ratio of the bending moment is high. However, if the rate of shear forces acting on the beam is high, it will fracture through the development of cracks associated with the holes [1]. In real buildings, holes are often located near the support points of beams. Therefore, holes are located where high shear forces and small bending moments occur.
As shown in Fig. 1, a large tensile stress perpendicular to the grain acts on the crack plane of the hole in the beam affected by the shear force or bending moment, and the tensile stress decreases exponentially with distance from the hole [2]. Various design codes and design proposals for glued laminated timber beams with round holes have been presented [1,2,3,4,5,6]. The design code in the Swedish Glulam handbook [4] and DIN EN 199511/NA [5] is a method for estimating the fracture strength by calculating the tensile stress perpendicular to the grain acting on the 45° portion shown in Fig. 1. The design proposal of Aicher [2], 2] incorporates the size effect of Weibull's statistical theory into the methods of the design code [4, 5]. The design proposal of Noguchi [3] is a method for estimating the fracture strength using the fracture energy of fracture mode 1. The design proposal of Hijikata [1] is a method for estimating the fracture strength by calculating the maximum principal stress acting around holes. These design codes and design proposals calculate the stresses acting on the holes, assuming that the beams are entirely composed of homogeneous wood.
In Japan, glued laminated beams composed of heterogeneousgrade timber, as specified in Japanese Agricultural Standard (JAS) [7], are very often used as beams, and it is not known whether the presented design codes and design proposals [1,2,3,4,5,6] can be used. Therefore, in [8], it was confirmed that the stresses acting on a hole in a glued laminated beam composed of heterogeneousgrade timber can be accurately estimated by finite element analysis (FEA) to estimate the loadbearing capacity associated with fracturing due to cracking. In [8], a design equation was proposed to compensate for the average length of the mean stress method [9] (Fig. 2), a generalization of linear elastic fracture mechanics, to take into account the size effect of the portions of the design with large tensile stresses (Eqs. 1, 2). Equations (1, 2) can be used to estimate the bearing capacity related to small and large holes during crackrelated fracturing. The analysis in [8] shows that the stress acting on the hole is smaller in the case of a glued laminated beam composed of heterogeneousgrade timber than in the case of a glued laminated beam composed of homogeneousgrade timber:
where D is the diameter of the hole, F_{t90} is the tensile strength perpendicular to the grain, F_{s} is the shear strength, \(\overline{\sigma }\) and \(\overline{\tau }\) are the mean values of the tensile stress σ and the shear stress τ perpendicular to the grain in the potential fracture area of length a_{ms}′ (Fig. 2), k_{vol} is the factor of size effect, G_{Ic} is the mode I fracture energy, E_{x} is Young’s modulus parallel to the grain, E_{y} is Young’s modulus perpendicular to the grain, G_{xy} is the shear modulus, ν_{xy} is Poisson’s ratio, α is a constant (1 for D/H ≤ 1/3), and H is the beam height. Since \(\overline{\sigma }\) is the dominant stress at failure, a_{ms} is set to the value of pure mode I.
Therefore, in this study, the difference in loadbearing capacity when fracturing occurs by cracks in beams with round holes was investigated using glued laminated beams composed of heterogeneousgrade timber and homogeneousgrade timber with the same Young's modulus of bending as specified in JAS [7]. First, bending tests were performed on the beams to experimentally investigate the difference in loadbearing capacity when fractured due to cracks. We then estimated the experimental values using Eqs. (1, 2) to analytically investigate the differences in structural performance. Finally, the stresses acting on the holes in glued laminated beams composed of heterogeneousgrade timber and glued laminated beams composed of homogeneousgrade timber were determined by FEA and compared. We then examined the reduction rate of stress acting on the holes in the case of glued laminated beams composed of heterogeneousgrade timber, which is specified in JAS [7].
The results of the bending tests of beams, material experiments and experimental specification FEA for glued laminated beams composed of heterogeneousgrade timber are the same as those reported in [8]. In this paper, a new study conducted on glued laminated timber beams composed of homogeneousgrade to determine the difference in structural performance between homogeneousgrade and heterogeneousgrade glued laminated timber beams with round holes in comparison with the results reported previously is presented.
Materials and methods
Bending tests of beams
The materials used are glued laminated beams composed of heterogeneousgrade timber and homogeneousgrade timber as specified by JAS [7] for Scots pine. The strength grade of the glued laminated beams composed of heterogeneousgrade timber is E105F300, and the strength grade of the glued laminated beams composed of homogeneousgrade timber is E105F345. A schematic diagram and the specifications of the test specimens are shown in Fig. 3. The lamina arrangement of the glued laminated timber used for the test specimens is shown in Fig. 4. The beam width B of the test specimens is 105 mm, and the beam heights H are 150 mm, 300 mm and 450 mm. The hole diameters D are H/3, H/5, H/10 (H = 300 mm only) and H/15 (H = 450 mm only). In [8], we conducted bending tests on glued laminated timber composed of heterogeneousgrade with D/H of 2/3 and 4/5, but since it is considered difficult to incorporate glued laminated timber with large hole diameters into the design code, these D/H values were excluded from this study. The number of test specimens is two for each type of glued laminated beam composed of homogeneousgrade timber and three for each type of glued laminated beam composed of heterogeneousgrade timber. The mean and standard deviation values of density were 531 ± 6 kg/m^{3} for the glued laminated beams composed of homogeneousgrade timber and 522 ± 11 kg/m^{3} for the glued laminated beams composed of heterogeneousgrade timber. Density is the weight of the test object divided by the volume. The mean and standard deviation values of moisture content were 12.7 ± 4.2% for the glued laminated beams composed of homogeneousgrade timber and 13.1 ± 3.2% for the glued laminated beams composed of heterogeneousgrade timber. The moisture content was measured by a moisture meter (HM530, Kett Electric Laboratory). The experiment was carried out at a crosshead speed of 6 mm/min to ensure that the load was applied for more than 1 min from the start of loading to fracturing. Displacement gauges were placed at the load and support points to obtain the deflection of the load points.
Material tests
Tests were conducted to determine the shear strength, tensile strength perpendicular to the grain, and mode I fracture energy at the crack locations associated with the holes in the glued laminated timber beams. The materials used were L80grade and L110grade Scots pine in the inner layer position shown in Fig. 4.
The shear strength parallel to the grain was determined by the test method with a standard shear block specimen (Fig. 5a) according to JIS Z 2101 [10]. Twenty L80grade specimens for and ten L110grade specimens were cut from the beams used in the bending tests (Fig. 5b). The size of the shear surface was 30 mm × 30 mm. The experiment was conducted at a constant crosshead speed, and the load was applied so that the time between the start of the load loading and fracturing was approximately 1 min. The mean and standard deviation values of density were 463 ± 12 kg/m^{3} for L80grade and 530 ± 18 kg/m^{3} for L110grade. The moisture content of the specimen was not measured because the moisture content of the beams was measured.
It has been reported that the flexural strength of concrete decreases with increasing specimen height and approaches the tensile strength [11]. In [8], timber specimen heights of 20 mm, 40 mm, 150 mm, and 300 mm were tested in flexural tests, and it was shown that the flexural strengths of timber specimens with heights of 150 mm and 300 mm were nearly the same. Based on these facts, in [8], the bending strength obtained in bending tests with a specimen height of 150 mm was taken as the tensile strength perpendicular to the grain of the timber. Although not a standard test method, in this study, the bending strength obtained by the bending test method shown in Fig. 6 was also used as the tensile strength perpendicular to the grain of the timber. Bending test methods are also specified in ISO 13910 [12], although they yield a fairly low tensile strength at the lower boundary limit. The tensile strength perpendicular to the grain was determined using the same bending test method as in [8], as shown in Fig. 6. The test specimens were glued laminated beams composed of L80grade and L110grade timber. The number of specimens was six each. The experiment was conducted at a constant crosshead speed and applied load, and the speed was adjusted so that 1 to 2 min elapsed from the start of load loading to fracturing. The mean and standard deviation values of density were 485 ± 6 kg/m^{3} for the L80grade timber and 510 ± 2 kg/m^{3} for the L110grade timber. The mean and standard deviation values of moisture content were 10.7 ± 2.0% for the L80grade timber and 8.0 ± 0.7% for the L110grade timber.
The test method for obtaining the fracture energy G_{Ic} of fracture mode I fractures is the singleedgenotched bending test [13] shown in Fig. 7. The height of the specimens was in accordance with the specifications in [13], and the width of the specimens was 105 mm, the same as the specimens in the bending tests. The test specimens were glued laminated beams composed of L80grade timber and glued laminated beams composed of L110grade timber. The number of specimens was set to 10 each. The experiment was conducted at a constant crosshead speed, and the load was applied so that the time between the start of the load loading and fracture was approximately 3 min. The mean and standard deviation values of density were 475 ± 6 kg/m^{3} for the L80grade timber and 532 ± 8 kg/m^{3} for the L80grade timber. The mean and standard deviation values of moisture content were 8.6 ± 1.8% for the L80grade timber and 8.7 ± 1.1% for the L80grade timber.
Finite element analysis
The stresses acting on the holes in the beam were determined by twodimensional finite element analysis (2DFEA). ANSYS 18.2 was used for 2DFEA. The specifications applied in the 2DFEA were those of the bending tests and cases in which only the shear force or bending moment acts on the center of the hole. The details of the 2DFEA are described below:

1.
Specifications of the bending tests
The numerical finite element (FE) model was set up such that the stress acting on the hole was the same as that in the bending tests, as shown in Fig. 8. The numerical FE model is representative of the same glued laminated beam composed of heterogeneousgrade timber as in [8] and the glued laminated beam composed of homogeneousgrade timber with modified laminate material properties. The specimen specifications in the bending tests in which the holes were fractured by cracks were analyzed. The material parameters of the elements were determined in the same way as in [8]. The Young's modulus parallel to the grain was set to the Young's modulus of the bending of lamina as shown in JAS [7] (E_{x} = 11,000 N/mm^{2} for L110). The Young's modulus perpendicular to the grain was set to E_{y} = E_{x}/25, the shear modulus was set to G = E_{x}/15, and the Poisson's ratio was set to ν_{xy} = 0.4. The lamina arrangement of the glued laminated timber is shown in Fig. 4. The wood was treated as a fournode planar element, and the elements were less than 10 mm per side (H = 450 mm, H = 300 mm) or less than 5 mm per side (H = 150 mm). The range of 10 mm from the hole was divided into small sections so that the nodes are placed at one degree. To determine the stress distribution at the location of fracturing by the crack, the edges and nodes of the elements were placed in the X direction (direction parallel to the grain) from the θ = 45° position. The mesh size of the FE model is shown in Fig. 9. We wanted to divide the elements very finely around only the hole, so some 3node planar elements are placed. Compared with the numerical FE results when all the elements were reduced in size and the number of elements was doubled, the error in the stress around the holes was less than 1%, so we concluded that the optimal solution could be obtained by the element division shown in Fig. 9.

2.
Specifications where only shear force or bending moment acts on the center of the hole
The numerical FE model was set up for the case where the stress at the center of the hole was shear force only (Q = τ_{1}BH (N), M = 0, τ_{1} = 1 N/mm^{2}) and bending moment only (Q = 0, M = τ_{1}BH^{2} (Nmm), τ_{1} = 1 N/mm^{2}), as shown in Fig. 10. The values of the shear force and bending moment were set so that the stresses acting on them would be the same at different beam heights H. The D/H values were set to 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5, and the beam height H was set to 300 mm. The material parameters of the elements were the same as in (1). The lamina arrangement of the glued laminated timber is shown in Fig. 11 according to JAS [7]. The strength grade specified in JAS [7] for the glued laminated beam composed of homogeneousgrade timber was E105F345. However, the stresses acting on the beams constructed of homogeneousgrade timber are the same for different strength grades. The strength grades specified in JAS [7] for the glued laminated beams composed of heterogeneousgrade timber were E105F300 and E55F200. The wood was treated as a fournode planar element, and the elements were less than 5 mm per side. The range of 10 mm from the hole was divided into small sections so that the nodes are placed at one degree. To obtain the stress distribution at the location of the high stress, the edges and nodes of the elements were placed in the X direction (parallel to the grain direction) from the θ = 45° and 60° positions. The mesh size of the FE model is shown in Fig. 12. Compared with the numerical FE results when all the elements were reduced in size and the number of elements was doubled, the error in the stress around the holes was less than 1%, so we concluded that the optimal solution could be obtained by the element division shown in Fig. 12.
Results and discussion
Material tests
Data on the shear strength parallel to the grain obtained from the material experiments, existing data on the shear strength parallel to the grain for coniferous trees [14], and the relationship between density ρ and shear strength parallel to the grain with a shear surface of 30 mm × 30 mm for 20 tree species (Eq. 3) [15] are shown in Fig. 13. Data on the tensile strength perpendicular to the grain obtained from material experiments, existing data on the tensile strength perpendicular to the grain for coniferous trees [14], and the relationship between density ρ and tensile strength perpendicular to the grain with a tensile cross section of 25 mm × 50 mm for three tree species (Eq. 4) [15] are shown in Fig. 14. The mode I fracture energy obtained from material experiments and the relationship between density ρ and mode I fracture energy for Scots pine (Eq. 5) [16] are shown in Fig. 15. From these comparisons, it was confirmed that the values obtained in the material experiments corresponded approximately to the lower limit of the existing strength data and were in general agreement with the relationships described by Eqs. (3–5). The maximum, minimum, mean and standard deviation of the values obtained in the material experiments are shown in Table 1.
where ρ (g/cm^{3}) is the density.
Bending tests of beams
All specimens with D/H = 1/15 and one of the glued laminated beams composed of heterogeneousgrade timber with D/H = 1/10 fractured during bending without holerelated cracking. Other specimens fractured due to cracks in the holes, as shown in Fig. 16. The specimens with cracks in the hole first exhibited a short crack in the upper part of the force side, and then after the crack had extended to some length, a crack developed in the lower part of the fulcrum side, and the crack extended to the end surface, reducing the bearing capacity. Some specimens also fractured with the formation of all cracks occurring at the same time.
The load–displacement relationship is shown in Fig. 17. The load is the value measured at the force point, and the displacement is the amount of deflection at the force point. In 80% of the specimens where initial cracks could be visually identified, the initial cracks occurred when the deflection at the force point was offset by 0.2 to 0.5 mm. In particular, 0.2 mm was the most common at 36%. The offset values did not change much for different specimen sizes. In [8], the initial crack load was defined as the load when the deflection at the force point was offset by 0.2 mm for this reason. In this study, the load at a deflection at the force point of 0.2 mm was defined as the initial crack load in the same way. The maximum loads for all specimens and the initial cracking loads for specimens that failed due to cracking are shown in Table 2. Figure 18 shows the relationship between the ratio of the maximum load or initial crack load to D/H for the glued laminated beams composed of homogeneousgrade and heterogeneousgrade timber. The ratio of the material strength of the inner lamina is also shown as a straight line. Except for the initial cracking load of D/H = 1/5 at H = 150 mm, the ratio of the bending tests was smaller than the material properties, and the maximum load and the initial cracking load were not much different between the glued laminated beams made up of homogeneousgrade and heterogeneousgrade timber.
Estimation of experimental values
To analytically investigate the differences in structural performance between the glued laminated beams composed of heterogeneousgrade timber and homogeneousgrade timber, the experimental values were estimated by Eqs. (1, 2). The tensile stress perpendicular to the grain acting around the hole is shown in Fig. 19, which was obtained from the analysis in Fig. 8. The results of the analysis show that when the shear force acting on the hole is Q = BHτ_{1} and the hole diameter ratio D/H is constant, the distribution of σ is almost the same. The maximum values of σ are all at θ = 45°, and it is assumed that cracks form and extend parallel to the grain direction (X direction) from this position. This is generally consistent with the location of the first crack in the bending tests of beams. The maximum values of σ in Fig. 19 were approximately 14% lower for the glued laminated beams composed of heterogeneousgrade timber than for the glued laminated beams composed of homogeneousgrade timber at D/H = 1/3, 13% lower at D/H = 1/5, and 10% lower at D/H = 1/10.
The distribution of tensile stresses perpendicular to the grain and shear stresses at the assumed fracture location are shown in Fig. 20. The position around the hole is set to x = 0, and the horizontal axis is x/D. The results of the analysis show that the stress distribution is almost the same at different beam heights. The ratio of the stresses in the glued laminated beams composed of homogeneousgrade timber and heterogeneousgrade timber did not change much with the position of x/D, and the ratio was somewhat constant. The average stress for use in Eq. (1) can be found in Eq. (6) using the coefficients in each term in Fig. 20:
Figure 21 shows a comparison of the calculated and experimental cracking loads in the bending tests calculated by Eqs. (1, 2) using the average values in Table 1. The calculated values for the glued laminated beams composed of homogeneousgrade timber (cal. Eq. (1), Homo. Eq. (2), L110) and for the glued laminated beams composed of heterogeneousgrade timber (cal. Eq. (1), Hetero. Eq. (2), L80) are given. In addition, the calculated values for the stresses in the glued laminated beams composed of homogeneousgrade timber using the material strength of the L80 grade are also given as reference values (cal. Eq. (1), Homo. Eq. (2), L80). The calculated values of D/H = 1/3 and D/H = 1/5 are good estimates of the bearing capacity at the initial cracking, and the calculated value of D/H = 1/10 is approximately halfway between the initial and maximum bearing capacities. The ratio of (calculated value of Homo. L110)/(calculated value of Homo. L80) is 1.21, which is the same as (F_{t90} for L110)/(F_{t90} for L80). However, the ratio (calculated value of Homo. L110)/(calculated value of Hetero. L80) is smaller than the ratio of material properties, as shown in Fig. 18, which is 1.05 for D/H = 1/3, 1.08 for D/H = 1/5 and 1.09 for D/H = 1/10.
For H = 150 mm, the D/H = 1/3 specimen of glued laminated beams composed of heterogeneousgrade timber had an initial crack in the lamina of L100grade timber. Therefore, the calculation results using the material strength of L80grade timber differed from the actual situation. Young's modulus and density have a linear relationship [17]. The shear strength and mode I fracture energy have a linear relationship with density, as shown in Eqs. (3, 5). Additionally, tensile strength has a nonlinear relationship with density (Eq. 4), but Fig. 14 shows that it can be estimated with a linear relationship. Therefore, because no material testing was performed on the L100grade timber, it was assumed that its Young's modulus and material strength exhibited a linear relationship, and the material strength of L100grade timber was estimated from the average material strength of L80grade and L110grade timber. The estimated material strength of L100grade timber was F_{s} = 7.89 N/mm^{2}, F_{t90} = 2.45 N/mm^{2}, and G_{Ic} = 407 Nm/m^{2}. Figure 21 shows the calculated values for the glued laminated material with D/H = 1/3 based on the estimated material strength of L100grade timber. The ratio (calculated value of Homo. L110)/(calculated value of Hetero. L100) ratio for D/H = 1/3 was 0.93.
Differences in stress levels around the hole
Many design codes and design proposals [1,2,3,4,5,6] also require the tensile stress perpendicular to the grain acting on the holes, assuming that the beams are entirely composed of homogeneous wood. Therefore, the stresses acting on the holes in glued laminated beams composed of heterogeneousgrade timber and glued laminated beams composed of homogeneousgrade timber were determined by the analysis in Fig. 10 and compared. We then examined the reduction rate of stress acting on the holes in the case of the glued laminated beams composed of heterogeneousgrade timber, which is specified in JAS [7].
The tensile stresses perpendicular to the grain and the shear stresses acting on the holes in the glued laminated beams composed of homogeneousgrade timber determined by the numerical FE analysis are shown in Fig. 22. The stresses acting on the hole are symmetric or inversely symmetric with respect to the center of the hole. The tensile stresses perpendicular to the grain and the shear stresses acting on the holes in the glued laminated beams composed of homogeneousgrade timber and glued laminated beams composed of heterogeneousgrade timber (E105F300 and E55F200), as determined by the numerical FE analysis, are shown in Figs. 23 and 24. The stresses are shown for only onequarter of the holes. The ratio of stresses acting on the holes of glued laminated beams composed of the homogeneousgrade and heterogeneousgrade timber was constant with little change in position.
The bending and shear stress distributions from Bernoulli–Euler theory for glued laminated beams composed of heterogeneousgrade timber and glued laminated beams composed of homogeneousgrade timber are shown in Fig. 25a. The ratios of the maximum bending stress (σ_{Inner} in Fig. 25a) to the maximum shear stress (τ_{Max.} in Fig. 25a) in the inner lamina for the same bending moment or the same shear force in the glued laminated beams composed of heterogeneousgrade timber (Fig. 25b) and glued laminated beams composed of homogeneousgrade timber as defined by JAS [7] are shown in Fig. 25c. The ratio of bending stress is k_{M} and the ratio of shear stress is k_{Q}. Figure 26 compares the ratio of the maximum stress in Figs. 23 and 24 with k_{M} and k_{Q} (Fig. 25c). For small D/H values, the ratios for k_{Q} and the pure shear force (center of the hole) case obtained by FEA are almost equal, and the ratios for the pure bending moment case obtained by k_{M} and FEA are almost equal. However, as the D/H increases, the FEA results become smaller than those of k_{Q} and k_{M}.
Figures 27 and 28 show the stresses of glued laminated beams composed of homogeneousgrade timber in Figs. 23 and 24 multiplied by k_{Q} or k_{M}. Figures 29 and 30 show the distribution of stresses in the direction parallel to the grain (tensile stresses perpendicular to the grain σ and shear stresses τ) at θ = 60° (y/D = 0.25) and θ = 45° (y/D = 0.35). The stresses in the glued laminated beams composed of homogeneousgrade timber are multiplied by k_{Q} or k_{M}. In the case of small D/H values, the results are almost equal to the stresses in the glued laminated beams composed of heterogeneousgrade timber. However, as the D/H increases, the stresses in the glued laminated beams composed of homogeneousgrade timber multiplied by k_{Q} or k_{M} tend to be higher.
It can be concluded that when the D/H is low, the stresses in the glued laminated beams composed of homogeneousgrade timber can be multiplied by k_{Q} or k_{M} in Fig. 25c to obtain the stresses in the glued laminated beams composed of heterogeneousgrade timber. In addition, if the D/H is less than 0.5, the stresses in the glued laminated beams composed of heterogeneousgrade timber will not be underestimated even when the D/H is large.
Conclusion
In this study, the difference in the loadbearing capacity of glued laminated beams composed of homogeneousgrade timber and heterogeneousgrade timber with round holes when fractured by cracking was investigated experimentally and analytically. The following results were obtained:

1.
The ratio of material strength (shear strength, tensile stress perpendicular to the grain, and mode I fracture energy) of the inner lamina of glued laminated beams composed of homogeneousgrade timber (E105F345) and heterogeneousgrade timber (E105F300) with the same Young's modulus of bending as specified by JAS [7] was 1.13–1.21 in terms of (strength of inner layer lamina (L110) of homogeneousgrade timber)/(strength of inner layer lamina (L80) of heterogeneousgrade timber). However, the (loadbearing capacity of homogeneousgrade timber)/(loadbearing capacity of heterogeneousgrade timber) ratio of resistance to fracturing due to cracking in the bending tests for glued laminated timber beams with round holes ranged from 0.87 to 1.08.

2.
The loadbearing capacity of the beams obtained in the bending tests was calculated using FEA and material strength, and it was found that the ratio of (loadbearing capacity of homogeneousgrade timber)/(loadbearing capacity of heterogeneousgrade timber) was 1.05–1.09. The difference in the ratio of material strength to loadbearing capacity of the beams was found to be due to the reduced stresses acting on the holes located in the inner lamina when the glued laminated beams were composed of heterogeneousgrade timber.

3.
The tensile stress perpendicular to the grain and shear stresses acting on the holes in the laminated timber with holes of less than half the beam height were lower in the glued laminated beams composed of heterogeneousgrade timber than in the glued laminated beams composed of homogeneousgrade timber. The ratio of the stresses was found to be approximately equal to the ratio of the maximum bending stress or the maximum shear stress acting on the inner layer lamina as determined by Bernoulli–Euler theory. The ratio obtained by the FEA tended to be lower than that obtained by the Bernoulli–Euler theory as the hole size increased. However, using the ratio by Bernoulli–Euler theory to determine the stresses in glued laminated beams composed of heterogeneousgrade timber does not underestimate the stresses and can be applied by the design code.
Availability of data and materials
The test materials, test method, analysis method, and data were recorded as shown in the manuscript. Additional data are available from the corresponding author upon reasonable request.
Abbreviations
 JAS:

Japanese Agricultural Standard
 FEA:

Finite element analysis
 2DFEA:

Twodimensional finite element analysis
 FE:

Finite element
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Acknowledgements
We would like to thank the Hiroshima Prefectural Technology Research Institute Forestry Research Center for conducting the bending tests on the beams and American Journal Experts (https://www.aje.com/) for proofreading this manuscript.
Funding
This work was supported by JSPS KAKENHI (Grant numbers 18K13867 and 20K04796).
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SO planned and implemented the research and wrote the manuscript. All authors analyzed the results. All authors read and approved the final manuscript.
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Okamoto, S., Akiyama, N., Araki, Y. et al. Study on the strength of glued laminated timber beams with round holes: difference in structural performance between homogeneousgrade and heterogeneousgrade timber. J Wood Sci 67, 8 (2021). https://doi.org/10.1186/s10086021019413
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