Introduction

Elastic constants, including the Young’s modulus, are the basic properties that define the deformation behaviours of a material; therefore, it is desirable to determine the elastic constants accurately and conveniently. Among the various methods to determine the elastic constants, conducting vibration tests is advantageous because the force required to generate the vibration is usually sufficiently low to enable non-destructive testing. In addition, the measurement time for vibration tests is usually smaller than for static tests, including tension, compression, and bending tests. Among the vibration tests, flexural vibration (FV) tests are frequently conducted to determine the Young’s modulus. It is desirable to conduct FV tests because the fundamental frequency of the FV mode can be easily identified owing to its large amplitude. Consequently, the FV test methods have been standardised for refractory materials (ASTM C1548-02 2020), metallic materials at elevated temperatures (JIS Z 2280-1993), fine ceramics (JIS R 1602-1995), and concrete (JIS A 1127-2010). Moreover, the FV test method for solid wood has been standardised as ASTM D6874-12 by ASTM International. In addition to this standard, there exist several examples for the conduction of FV tests (Hearmon 1958, 1966; Matsumoto 1962; Ono and Kataoka 1979a, b; Ono 1983; Sobue 1986; Chui and Smith 1990; Haines et al. 1996; Divós et al. 1998; Brancheriau and Bailleres 2002; Brancheriau and Baillères 2003; Divos et al. 2005; Brancheriau 2006; Murata and Kanazawa 2007; Roohnia et al. 2010, 2011; Sohi et al. 2011; Yoshihara 2011, 2012a, b, c; Shamaiirani and Roohnia 2016; Roohnia 2019) to obtain the Young’s modulus of solid wood and wood-based materials.

In FV tests, the Young’s modulus is often calculated considering only the fundamental frequency, based on the Euler–Bernoulli’s theory (JIS Z 2280-1993; JIS R 1602-1995; JIS A 1127-2010; ASTM D6874-12; Hearmon 1958, 1966; Matsumoto 1962). In the Euler–Bernoulli’s theory, only the flexure caused by the bending moment is taken into account, and the equation to derive the Young’s modulus is represented in a simple form. However, in an actual FV test, the flexural components caused by the shearing force and rotatory inertia are inevitably induced, along with those induced by the bending moment. These components become more pronounced when the length/depth ratio of the specimen decreases; therefore, to obtain the accurate Young’s modulus by the Euler–Bernoulli equation while reducing these components, the length/depth ratio of the specimen should be sufficiently large. As an alternative method to obtain the Young’s modulus value, multiple resonance frequencies for the FV modes can be measured, and the Young’s modulus can be calculated based on Timoshenko’s vibration theory (Timoshenko 1921). This method is advantageous in that the length/depth ratio of the specimen does not necessarily need to be large. Furthermore, the shear modulus in the length-depth plane of the specimen can be obtained along with the Young’s modulus in the length direction. However, measuring multiple resonance frequencies is time-consuming when the shear modulus does not need to be determined. It is more convenient and practical to determine the accurate value of the Young’s modulus by conducting the FV test and identifying only the fundamental frequency alone.

In this study, FV tests were conducted using Sitka spruce specimens with various length/depth ratios, and Young’s moduli were obtained by measuring the fundamental and multiple resonance frequencies of the FV modes. An equation to obtain the accurate value of the Young’s modulus value considering only the fundamental frequency was proposed, and the validity of the equation was examined by conducting the analyses based on a statistical method and the experimentally obtained results. There are several attempts to provide such an equation for isotropic materials (Pickett 1945; Spinner et al. 1960; Spinner and Tefft 1961), the details of which are described below. However, it is difficult to find any examples in the literature listed above that propose the equation for solid wood, which possesses a strong orthotropy. The novelty of the work conducted here is the proposal of the equation for determining the Young’s modulus from the fundamental frequency in a certain range of the length/depth ratio of the specimen.

Materials and methods

Materials

The FV tests were conducted on Sitka spruce (Picea sitchensis Carr.) lumber with the initial dimensions of 900 mm (L) × 110 mm (T) × 110 mm (R). Prior to the test, the lumber was stored for approximately three years at a constant temperature of 20 °C and relative humidity of 65%, under air-dried condition. The equivalent moisture content (EMC) was 12%. These conditions were maintained throughout the duration of the tests. The density of the specimen was 404 ± 17 kg/m3, and the moisture content was determined to be 10% by using a moisture tester (HM-530, Kett Electric Laboratory, Tokyo, Japan). All the specimens were cut from the aforementioned lumber such that they were side-matched. During the preparation of the specimens, the defects contained in the lumber, including knots, checks, and fibre distortions, were reduced; therefore, the specimens could be regarded as ‘small and clear’. The specimen contained approximately 10–12 annual rings per 10 mm. The specimen was a straight beam with a square cross section, and the length, width, and depth directions, which coincided with the longitudinal (L), radial (R), and tangential (T) directions, respectively, were defined as L, B, and H, respectively. According to the method defined in JIS Z 2101-(2009), twelve specimens with an L value of 400 mm; and B and H values of 8 mm × 8 mm, 12 mm × 12 mm, 16 mm × 16 mm, and 20 mm × 20 mm were cut from the aforementioned lumber.

Free-free flexural vibration tests

In the FV test standardised in ASTM D6874-12 (2012), it is determined that the specimen should be simply supported with the span which exceeds 0.98 times the length. However, in this condition, it is difficult to excite the vibration with high resonance modes. To overcome this obstacle, the specimen was set upon a pair of extruded polystyrene (XPS) foams fixed on a medium-density fibreboard (MDF) plate. As shown in Fig. 1, the nodal positions of the free-free resonance vibration mode fn, were supported using a pair of XPS foams, and the specimen was excited along the depth direction (T direction) by using a hammer. The locations of the nodal positions are listed in Table 1 (Roohnia 2019). The aerial vibration generated from the specimen was detected by a microphone (UC-53A, Rion Co. Tokyo), preamplified by NH-22 (Rion Co. Tokyo), and the resonance frequencies were analysed using a fast Fourier transform (FFT) analyser (SA-78, Rion Co. Tokyo). The frequencies from the 1st to 4th resonance modes of flexural vibrations were measured in the FV tests. The Young’s moduli were determined from the following Euler–Bernoulli’s and Timoshenko’s vibration theories, and they were defined as ELEB and ELTGH, respectively.

Fig. 1
figure 1

Free-free flexural vibration test setup

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Table 1 Location of the nodal position adopted for supporting the specimen in the FV test (Roohnia 2019)
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When considering the Euler–Bernoulli’s theory, the ELEB value was calculated using only the fundamental frequency for the FV mode, as follows (JIS Z 2280-1993; JIS R 1602-1995; JIS A 1127-2010; ASTM D6874-12; Hearmon 1958, 1966; Matsumoto 1962):

$$E_{L}^{{{\text{EB}}}} = \frac{{48\pi^{2} \rho L^{2} f_{1}^{2} }}{{m_{1}^{4} H^{2} }}$$
(1)

where ρ is the density of the specimen, f1 is the fundamental frequency, and m1 = 4.730. ELEB was derived using Eq. (1) considering only the bending moment induced in the specimen. However, in an actual FV test, flexural components are also induced by the shearing force and rotatory inertia, and these effects are enhanced as the length/depth ratio of the specimen L/H decreases. Consequently, the value of ELEB usually decreases as L/H value decreases. Timoshenko proposed a differential equation including the effects of the shearing force and rotatory inertia (Timoshenko 1921). Later, Goens (1931) derived an approximate solution for the Timoshenko differential equation, denoted as ELTGH, as follows:

$$E_{L}^{{{\text{TGH}}}} = \frac{{48\pi^{2} \rho L^{2} f_{n}^{2} }}{{m_{n}^{4} H^{2} }}T_{n}$$
(2)

where n is the mode number for the FV, fn is the resonance frequency at the nth mode, and Tn is derived as follows:

$$T_{n} = 1 + 2m_{n} F\left( {m_{n} } \right)\frac{1}{12}\left( \frac{H}{L} \right)^{2} \left( {3 - s\frac{{E_{L}^{{{\text{TGH}}}} }}{{G_{{{\text{LT}}}} }}} \right) + m_{n}^{2} F^{2} \left( {m_{n} } \right)\frac{1}{12}\left( \frac{H}{L} \right)^{2} \left( {1 + s\frac{{E_{L}^{{{\text{TGH}}}} }}{{G_{{{\text{LT}}}} }}} \right) - s\frac{{4\pi^{2} \rho L^{2} f_{n}^{2} }}{{G_{{{\text{LT}}}} }}$$
(3)

where GLT is the shear modulus in the LT plane, s is Timoshenko’s shear factor, which is considered as 1.2 for a specimen with a rectangular cross section, and mn and F(mn), which are the coefficients corresponding to each resonance mode, are defined as follows:

$$\left\{ {\begin{array}{*{20}l} {m_{1} = 4.730} \hfill \\ {m_{2} = 7.853} \hfill \\ {m_{n} = \frac{{\left( {2n + 1} \right)\pi }}{2} \left( {n \ge 3} \right)} \hfill \\ \end{array} } \right.$$
(4)

and

$$\left\{ {\begin{array}{*{20}c} {F\left( {m_{1} } \right) = 0.9825} \\ {F\left( {m_{2} } \right) = 1.0008} \\ {F\left( {m_{n} } \right) = 1 \left( {n \ge 3} \right)} \\ \end{array} } \right.$$
(5)

The ELTGH and GLT values were determined using the iterative method proposed by Hearmon using Eqs. (2) and (3). In the iterative method (Hearmon 1958, 1966), which is known as the Timoshenko-Goens-Hearmon (TGH) method, the X and Y values corresponding to each mode are derived using Eqs. (2) and (3) as follows:

$$\left\{ {\begin{array}{*{20}c} {X = \frac{{48\pi^{2} \rho L^{2} f_{n}^{2} }}{{m_{n}^{4} H^{2} }}\left[ { - 2m_{n} F\left( {m_{n} } \right) + m_{n}^{2} F^{2} \left( {m_{n} } \right)} \right] } \\ {Y = \frac{{48\pi^{2} \rho L^{2} f_{n}^{2} }}{{m_{n}^{4} H^{2} }}\left[ {\frac{{H^{2} }}{{12L^{2} }} + 6m_{n} F\left( {m_{n} } \right) + m_{n}^{2} F^{2} \left( {m_{n} } \right) - \frac{{4\pi^{2} \rho L^{2} f_{n}^{2} s}}{{G_{xy} }}} \right]} \\ \end{array} } \right.$$
(6)

The X–Y relationship corresponding to each mode is regressed into the linear function Y = q − pX, and the Young’s modulus ELTGH and shear modulus GLT can be determined as

$$\left\{ {\begin{array}{*{20}c} {E_{L}^{{{\text{TGH}}}} = q} \\ {G_{{{\text{LT}}}} = s\frac{q}{p}} \\ \end{array} } \right.$$
(7)

Subsequently, the refined GLT value is substituted into Eq. (6), and the iteration procedure is continued until the ELTGH and GLT values converge. As described above, the resonance frequencies from the 1st to 4th flexural vibration modes were measured and analysed using a fast Fourier transform (FFT) analyser (SA-78, Rion Co., Tokyo). Recently, a method for determining the Young’s modulus and shear modulus values from the multiple FV testing data was proposed as Roohnia et al. (2010), Shamaiirani and Roohnia (2016), and Roohnia (2019). However, in this study, the above-mentioned TGH method was adopted.

After the FV tests were conducted, the length of the specimen was reduced, and the succeeding FV tests were conducted using the specimen with the reduced length. In particular, the value of L was decreased from 400 to 200 mm in intervals of 50 mm. Therefore, the L/H ranged from 10 to 50.

Results and discussion

Comparison of the Young’s moduli obtained using the Euler–Bernoulli’s method and TGH method

Figure 2 shows the fn values corresponding to the L and L/H values for the 1st–4th flexural vibration modes. All the ELEB, ELTGH, and GLT values demonstrated below were calculated from the fn values shown in Fig. 2.

Fig. 2
figure 2

fn value corresponding to the L and L/H values. n = mode number. The SD values cannot be demonstrated because of the low variation in the fn value

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Before discussing the results of the Young’s modulus, the dependence of the GLT value determined using the TGH method on the L/H value is considered, as shown in Fig. 3. It can be noted that the GLT value obtained from the specimens with a similar H tends to be constant despite the variation in the L value; therefore, the GLT value obtained using the TGH method in the L/H range examined in this study can be considered to be accurate.

Fig. 3
figure 3

GLT value corresponding to the L and L/H values determined using the TGH method. The results represent the average ± SD

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Figure 4 shows the comparison of the ELEB and ELTGH values corresponding to the L and L/H values. Similar to the results obtained in previous studies (Hearmon 1958, 1966; Matsumoto 1962; Ono and Kataoka 1979a, b; Ono 1983; Sobue 1986; Chui and Smith 1990; Haines et al. 1996; Divós et al. 1998; Brancheriau and Bailleres 2002; Brancheriau and Baillères 2003; Divos et al. 2005; Brancheriau 2006; Murata and Kanazawa 2007; Roohnia et al. 2011; Sohi et al. 2011; Yoshihara 2011, 2012a, b, c), the ELTGH value tends to be constant in the range of L/H examined in this work. Therefore, the Young’s modulus determined using the TGH method while reducing the effects of the shearing force and rotatory inertia can be considered to be accurate and recommended to be used as the reference value in discussing the accuracy of the ELEB value. In this case, ELTGH/GLT is 11.6 ± 1.0. In contrast, the difference between the ELEB values decreases as the L/H value decreases. The results of the unpaired t tests, which are discussed in detail in the following subsection, indicate that the difference between ELEB and ELTGH is often significant when the L/H value is less than 20. The experimental results also suggest that the ELEB value converges to the ELTGH value when the L/H value is greater than 20. However, when the ELTGH/GLT value is higher than that obtained in this study, there is a concern that the ELEB value does not agree with the ELTGH value even if L/H greater than 20.

Fig. 4
figure 4

Comparisons of the ELEB-L (L/H) and ELTGH-L (L/H) relationships obtained from the FV tests. The results represent the average ± SD

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Modification of Euler–Bernoulli’s equation

As shown in Fig. 4, when Eq. (1) is used for the calculation, the dependence of ELEB on the value of L/H is significant for small L/H values. However, it is more convenient to determine the Young’s modulus by measuring only the fundamental frequency alone while reducing this dependence. Based on this concept, Eq. (1) can be modified to a form that can be used to calculate the Young’s modulus of the specimen in a wide range of L/H.

In the FV test methods for refractory materials and fine ceramics, standardised in ASTM C1548-02 and JIS R 1602-(1995), respectively, the Young’s modulus modified using the ELEB value, defined as ELMOD, is derived as follows:

$$E_{L}^{{{\text{MOD}}}} = E_{L}^{{{\text{EB}}}} T_{c}$$
(8)

where Tc is the correction factor. According to the analyses by Pickett (1945), Spinner et al. (1960), and Spinner and Tefft (1961), Tc can be expressed as Tc = 1 + 6.59(H/L)2 on the condition that L/H ≥ 20. However, this relationship cannot be directly used to determine the correction factor for solid wood because the orthotropy of solid wood is usually stronger than those of refractory materials and fine ceramics. In addition, the degree of orthotropy varies among the wood species. Considering the unique characteristics of solid wood, Tc was modified as:

$$T_{c} = 1 + \alpha \left( \frac{H}{L} \right)^{\beta }$$
(9)

The α and β values appropriate for solid wood were determined by comparing the values with those pertaining to the correction factor used in the TGH method. From Eqs. (1) and (2), ELTGH was represented using ELEB as follows:

$$E_{L}^{{{\text{TGH}}}} = E_{L}^{{{\text{EB}}}} T_{1}$$
(10)

Equations (3)–(5) can be used to determine T1, which is the value of Tn at the fundamental frequency, as follows:

$$T_{1} = 1 + \frac{1}{12}\left( \frac{H}{L} \right)^{2} \left[ {49.48 + 12.3s\frac{{E_{L}^{{{\text{TGH}}}} }}{{G_{{{\text{LT}}}} }} - \frac{40.49}{{T_{1} }}\left( \frac{H}{L} \right)^{2} } \right]$$
(11)

The modification using Eq. (9) can be effectively used when ELMOD is sufficiently close to ELTGH; therefore, according to Eqs. (8) and (11), Tc should be similar to T1. However, as described above, the value of ELTGH/GLT in Eq. (11) varies among various wood species; therefore, the α and β values in Eq. (9) are inevitably influenced by the variation in ELTGH/GLT. In this study, a probable value of ELTGH/GLT was determined using the results demonstrated in several existing studies (Hearmon 1948; Kollmann and Côté 1968; Handbook of Wood Industry 1982; Sawada 1983; Guitard 1987), and the α and β values were subsequently determined by the method described below.

Table 2 lists the minimum, mean, and maximum values of ELTGH/GLT for various wood species, as reported in Hearmon (1948), Kollmann and Côté, (1968), Handbook of Wood Industry (1982), Sawada (1983), and Guitard (1987). The probable value of ELTGH/GLT was provisionally derived as 18.5 by averaging the mean values listed in Table 2. Subsequently, T1 was calculated by substituting ELTGH/GLT = 18.5 into Eq. (11) until its convergence. The converged T1-L/H relationship was regressed into Eq. (9) with L/H ranging from 10 to 50, which is similar to that adopted in the actual FV test in this study. The α and β values were obtained as 8.72 and 1.65, respectively, and the T1-L/H and Tc-L/H relationships are shown in Fig. 5. From Eqs. (1), (8), and (10), ELMOD was derived as follows:

$$E_{L}^{{{\text{MOD}}}} = \frac{{48\pi^{2} \rho L^{2} f_{1}^{2} }}{{m_{1}^{4} H^{2} }}\left[ {1 + 8.72\left( \frac{H}{L} \right)^{1.65} } \right] = \frac{{48\pi^{2} \rho L^{2} f_{1}^{2} }}{{m_{1}^{4} H^{2} }}T_{c}$$
(12)
Table 2 Minimum, mean, and maximum ELTGH/GLT for various wood species, reported in the literature
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Fig. 5
figure 5

T1-L/H and Tc-L/H relationships. ELTGH/GLT = 18.5 to determine the T1-L/H relationship

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Figure 6 shows the comparisons of the ELTGH-L (ELTGH-L/H) and ELMOD-L (ELMOD-L/H) relationships derived using Eqs. (7) and (12), respectively. As previously mentioned, the ELTGH/GLT value of the material used in this study was 11.6 ± 1.0, which significantly deviates from the value of 18.5 provisionally used to derive Eq. (12). However, Fig. 6 suggests the satisfactory agreement of the ELMOD and ELTGH values despite the discrepancy in the ELTGH/GLT values.

Fig. 6
figure 6

Comparisons of the ELMOD-L (L/H) and ELTGH-L (L/H) relationships obtained from the FV tests. The results represent the average ± SD

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Figure 7 shows the coefficient of variations (COV) for the ELEB, ELMOD, and ELTGH values. The COV value was used for the statistical analysis, which is discussed in detail in the following subsection. It should be noted that the COVs corresponding to the ELEB, ELMOD, and ELTGH values are close to each other under similar test conditions.

Fig. 7
figure 7

Coefficient of variations (COV) for the ELEB, ELMOD, and ELTGH values

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Figure 8 shows the p-values obtained from the unpaired t tests of the difference between the averages of the values of (a) ELEB and ELTGH and (b) ELMOD and ELTGH corresponding to L/H. The difference between ELEB and ELTGH is often significant when the L/H is less than 20 because the p-value is often less than 0.05 in this L/H range. In contrast, the difference between ELMOD and ELTGH is not significant over the entire range of L/H examined in this study. Therefore, the Young’s modulus can be derived effectively as ELMOD with the L/H ranging from 10 to 50.

Fig. 8
figure 8

Relationships between the p-value and L/H value obtained from the unpaired t tests of the average values of a ELEB and ELTGH values and b ELMOD and ELTGH values

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Examining the applicability of the modified equation by conducting a statistical analysis

As discussed, Eq. (12) can be used to determine the Young’s modulus of the material examined in this study, even though its ELTGH/GLT value is significantly different from 18.5. However, Eq. (12) should be applicable to various materials with various ELTGH/GLT values. This applicability was examined by conducting an unpaired t test on the Young’s moduli determined by measuring only the fundamental frequency alone (ELEB and ELMOD) and that obtained using the TGH method, ELTGH.

The average values of the Young’s modulus obtained from the fundamental frequency, ELEB and ELMOD, are denoted as \(\overline{{E_{L}^{{{\text{EB}}}} }}\) and \(\overline{{E_{L}^{{{\text{MOD}}}} }}\), respectively, whereas the Young’s modulus obtained using the TGH method is defined as \(\overline{{E_{L}^{{{\text{TGH}}}} }}\). Hereafter, ELEB and ELMOD are represented as ELFF. Similarly, \(\overline{{E_{L}^{{{\text{EB}}}} }}\) and \(\overline{{E_{L}^{{{\text{MOD}}}} }}\) are represented as \(\overline{{E_{L}^{{{\text{FF}}}} }}\). The unpaired t test of the difference between \(\overline{{E_{L}^{{{\text{FF}}}} }}\) and \(\overline{{E_{L}^{{{\text{TGH}}}} }}\) is conducted as follows. When the number of specimens is n, the test statistic indicating the difference in the \(\overline{{E_{L}^{{{\text{FF}}}} }}\) and \(\overline{{E_{L}^{{{\text{TGH}}}} }}\) values, defined as t0, is obtained as follows (Fisher and Yates 1963):

$$t_{0} = \frac{{\left| {\overline{{E_{L}^{{{\text{TGH}}}} }} - \overline{{E_{L}^{{{\text{FF}}}} }} } \right|}}{{s_{p} \sqrt {\frac{2}{n}} }}$$
(13)

where sp is the pooled standard deviations. As shown in Fig. 7, the COV for the ELFF values is close enough to that for the ELTGH values; therefore, these COVs are considered to be similar and are collectively denoted as V. Using V, sp is derived as follows:

$$s_{p} = \sqrt {\frac{{\left( {\overline{{E_{L}^{{{\text{TGH}}}} V}} } \right)^{2} + \left( {\overline{{E_{L}^{{{\text{FF}}}} V}} } \right)^{2} }}{2}} = V\sqrt {\frac{{\left( {\overline{{E_{L}^{{{\text{TGH}}}} }} } \right)^{2} + \left( {\overline{{E_{L}^{{{\text{FF}}}} }} } \right)^{2} }}{2}}$$
(14)

ELFF/ELTGH value is defined as c obtained using Eqs. (8) and (10), as follows:

$$c = \frac{{E_{L}^{{{\text{FF}}}} }}{{E_{L}^{{{\text{TGH}}}} }} = \left\{ {\begin{array}{*{20}c} {\frac{{E_{L}^{{{\text{EB}}}} }}{{E_{L}^{{{\text{TGH}}}} }} = \frac{1}{{T_{1} }} \left( {E_{L}^{{{\text{FF}}}} = E_{L}^{{{\text{EB}}}} } \right)} \\ {\frac{{E_{L}^{{{\text{MOD}}}} }}{{E_{L}^{{{\text{TGH}}}} }} = \frac{{T_{c} }}{{T_{1} }} \left( {E_{L}^{{{\text{FF}}}} = E_{L}^{{{\text{MOD}}}} } \right)} \\ \end{array} } \right.$$
(15)

Consequently, Eqs. (13) and (14) can be transformed using the c value as:

$$t_{0} = \overline{{E_{L}^{{{\text{TGH}}}} }} \frac{{\left| {1 - c} \right|}}{{s_{p} \sqrt {\frac{2}{n}} }}$$
(16)

and

$$s_{p} = \overline{{E_{L}^{{{\text{TGH}}}} }} V\sqrt {\frac{{1 + c^{2} }}{2}}$$
(17)

From Eqs. (16) and (17), t0 can be obtained by eliminating \(\overline{{E_{L}^{{{\text{TGH}}}} }}\) as follows:

$$t_{0} = \frac{{\left| {1 - c} \right|}}{{V\sqrt {\frac{{1 + c^{2} }}{n}} }}$$
(18)

The value of c is calculated by determining the n, V, and t0 values in Eq. (18). Subsequently, the p-value corresponding to c is obtained from the t-distribution. The \(\overline{{E_{L}^{{{\text{FF}}}} }}\) and \(\overline{{E_{L}^{{{\text{TGH}}}} }}\) values at the corresponding c value exhibit no significant difference when the p-value obtained from this procedure is greater than 0.05.

According to the mechanical sampling method defined in JIS Z 2101-2009 and ISO 3129:2012, the minimum number of specimens, nmin, can be obtained as follows:

$$n_{\min } \ge \left( {\frac{{Vt_{\inf } }}{{p_{i} }}} \right)^{2}$$
(19)

where pi is the index of the test precision, which is set as 0.05, and tinf is the index of the result authenticity, which is 1.960 when pi = 0.05 (Fisher and Yates 1963). In JIS Z 2101-2009, the number of specimens is considered to be 12 unless otherwise specified. Based on this concept, n and nmin are set as 12. By substituting nmin = 12 in Eq. (19), the value of V can be obtained as 8.837 × 10–2. Additionally, when n = 12, t0 in Eq. (18) is 2.074 (Fisher and Yates 1963). The range of c values to ensure that the p-value is larger than 0.05 is 0.927–1.078. The critical length/depth ratio (L/H)c is defined as the L/H value at c = 0.927 or 1.078.

The c value corresponding to the L/H value was calculated by varying the ELTGH/GLT values. Table 3 lists the ELTGH/GLT values used to calculate the c value. These values were determined with reference to the ELTGH/GLT values listed in Table 3. This method was similar to those adopted for determining the specimen configuration to obtain the Young’s modulus by the longitudinal vibration (LV) test (Yoshihara and Maruta 2021) and shear modulus by the torsional vibration (TV) test (Yoshihara and Maruta 2020). Figure 9 represents the c-L/H relationships corresponding to the ELTGH/GLT values listed in Table 3. The difference between ELFF (= ELEB or ELMOD) and ELTGH is not significant when the c value is located between the semi-solid lines, and the (L/H)c value can be determined considering the intersectional point of the c-L/H curve and a semi-solid line. Figure 9 indicates that the range of L/H to determine the accurate Young’s modulus is markedly extended by the modification.

Table 3 ELTGH/GLT values used to calculate c
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Fig. 9
figure 9

The c-L/H relationships obtained using Eq. (18) under various ELTGH/GLT values. a c = ELEB/ELTGH and b c = ELMOD/ELTGH

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Table 4 lists the (L/H)c values when ELFF = ELEB and ELFF = ELMOD. Even when the L/H value is greater than 20, as indicated in ASTM D6874-12, the ELEB value often deviates from the ELTGH value when the ELTGH/GLT value is high. Therefore, the Euler–Bernoulli equation (Eq. (1)) is not always suitable to obtain the accurate value of the Young’s modulus. In contrast, this table indicates the reasonable agreement between ELMOD and ELTGH when L/H is determined according to ASTM D6874-12 (2012). Additionally, the L/H value standardised for the static bending test methods, as reported in ISO 3133:(1975a); ISO 3349:(1975b); JIS Z 2101-(2009); and ASTM D143-14 (2016) ranges from 16 to 18 (L = span length + overhang lengths). The (L/H)c values for the ELMOD, listed in Table 4, indicate that the specimens having the configurations reported in the standards for the static bending tests can be used to conduct the FV test. This aspect renders it convenient to adopt the modified equation. The aforementioned results demonstrate that Eq. (12) can be used to determine the accurate values of the Young’s modulus by measuring only the fundamental frequency in a wide range of L/H values.

Table 4 Critical length/depth ratio (L/H)c
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To enhance the accuracy of the modified equation, further research should be conducted considering various wood species with various configurations. If the accuracy can be obtained in all the cases, the FV test using the modified equation can be established as a standardised method to measure the Young’s modulus of solid wood, in addition to the existing standards.

Conclusion

FV tests were conducted on Sitka spruce specimens to determine the Young’s modulus in the longitudinal direction. The Young’s moduli were calculated using the resonance frequency at only the 1st FV mode (fundamental frequency) and the frequencies pertaining to the 1st–4th FV modes. By comparing these values, the method for determining the Young’s modulus using only the fundamental frequency was evaluated.

The results of the FV tests indicated that the Young’s modulus can be determined using only the fundamental frequency by modifying the Euler–Bernoulli equation. The results of the statistical analyses based on the sampling methods recommended in JIS Z 2101-(2009) indicated that the modified equation could be used to determine the Young’s modulus of various kinds of solid wood. Therefore, to standardise the FV test, the modified equation is a promising candidate to enable the determination of the Young’s modulus of solid wood using only the fundamental frequency.