### Residual axial bolt force

Figure 4 shows typical results for the axial bolt force ratio over time for a test piece with a thickness *T*_{h} of 30 mm and a bolt spacing *S* of 48 mm. Here, the axial force ratio is calculated using

$$alpha =frac{{F}_{T}}{{F}_{mathrm{f}}}$$

(1)

where *F*_{f} is the initial axial force (2 kN) and *F*_{T} is the axial force *T* [s] thereafter.

First, looking at Se.1, it can be seen that when the bolts are tightened in this order, the axial force of the previously tightened bolts is reduced by approximately 16% upon completion of tightening. This trend is also observed in Se.2. After all the bolts have been tightened in Se. 1 and Se.2, the bolt tightened last (i.e., bolt D and C in Se. 1 and Se. 2, respectively) has different axial force ratio of other bolts. However, looking at C in Se.3 and B in Se.4 reveals that tightening the two bolts adjacent to bolts results in the axial force ratio decreasing twice, and unlike in Se.1 and Se.2, where the axial force ratio was not uniform after all the bolts had been tightened, the variation in axial force ratio was large. Furthermore, a difference can be observed between the first and the second decreases in the axial force ratio of bolt B in Se.4.

Tables 2, 3, and 4 show the axial force ratios of all bolts after tightening. The tables also include the ratio between the maximum and minimum axial force ratio as *Max*./*Min*. First, with regard to *T*_{h} = 30 mm in Table 2, there is a tendency for *Max*./*Min*. to decrease as the bolt spacing *S* is increased. This indicates that the axial forces of all the bolts are approximately uniform. Next, as shown in Table 3, *Max*./*Min*. for *T*_{h} = 45 mm becomes higher than that for *T*_{h} = 30 mm, and for *S* = 48 mm, the *Max*./*Min*. value in Se.3 and Se.4 is four times higher than for *T*_{h} = 30 mm. In particular, both of bolt C of Se. 3 and bolt B of Se. 4 are affected by two decreases in axial force, resulting in respective axial forces of 0.17 and 0.16, which can be understood to cause the large *Max*./*Min*. However, the *Max*./*Min*. for *S* = 84 mm ranged from 1 to 1.1, regardless of the tightening sequence, which was the same as for *T*_{h} = 30 mm. However, it is difficult to conclude that there is an explicit relationship between bolt spacing and *Max*./*Min*. Similarly, as shown in Table 4, *Max*./*Min*. for *T*_{h} = 60 mm becomes higher than that for *T*_{h} = 30 mm in a similar manner as for *T*_{h} = 45 mm. For *S* = 84 mm, the *Max*./*Min*. range is 1.26 to 1.68, which is a large variation compared to the ranges for *S* = 84 mm for *T*_{h} = 30 mm or *T*_{h} = 45 mm.

Figure 5 shows the relationship between *Max*./*Min*. and *S* for each tightening sequences. According to Fig. 5, *Max*./*Min*. tended to decrease as *S* increased regardless of *T*_{h} and tightening sequences. On the other hand, no clear trend was observed between *Max*./*Min*. and *T*_{h}. These results suggest that, within the scope of these experiments, when *S* is increased, a suppression in variation of axial forces due to elastic interactions can be expected, regardless of which tightening sequence is employed.

### Calculation of axial bolt force ratio under elastic interaction

As mentioned above, bolt spacing greatly affect variations in axial force due to elastic interactions and it is assumed that the embedment stiffness of a metal washer is closely related to it. Washer embedment stiffness greatly depends on the side length ratio of the washer (washer’s side length to thickness) [27, 28]. The side length ratios of the washers used in these experiments were 7.8 (35 mm/4.5 mm ≈ 7.8) and for such side length ratios, it is considered that there is almost no bending deformation of the washers due to bolt tightening [11, 27, 28]. It is also assumed that the washer became embedded into the wood side while maintaining an almost rectangular shape (a state in which the washer can be regarded as a rigid body). Figure 6 shows a diagram of the deformation when bolts are tightened with washers, which are assumed to be rigid bodies. As mentioned above experimental results, when evaluating the bolt axial force ratio, it is sufficient to consider the influence of two adjacent bolts, and three bolts are joined. First, as shown in Fig. 6a, when bolt 1 is tightened by an initial axial force *F*_{1}, a deformation *δ*_{1} occurs directly beneath the washer. Next, as shown in (b), when bolt 2 is tightened by a force *F*_{1}, the deformation directly beneath the washer of bolt 1 is affected by the deformation directly beneath the washer of bolt 2, thus changing the deformation *δ*_{1} to *δ*_{1,2} and changing the axial force to *F*_{1, 2}. Furthermore, as shown in (c), when bolt 3 is tightened by a force *F*_{1}, the deformation directly beneath the washer of bolt 1 is affected by the deformation directly beneath the washer of bolt 3, thus changing the deformation *δ*_{1,2} to *δ*_{2,3} and changing the axial force to *F*_{2,3}. An evaluation formula for the axial force ratio *α* is derived from this deformation diagram. Note that changes in axial force due to stress relaxation and creep of wood were not taken into account when deriving the evaluation formula in this study. First, the deformed shape of the additional length part in the direction of the grain from the washer end is expressed as an exponential curve, and if it is further assumed that the length from the end of the washer to a position, where becomes almost zero is 1.5 times the wood thickness *T*_{h}, then the function can be expressed as [29]

$$fleft(xright)={delta }_{i} {e}^{-frac{3}{2{T}_{text{h}}}x}$$

(2)

where *x* is the distance from the washer end and *δ*_{i} is the amount of deformation directly beneath the washer. This can be calculated using the embedment stiffness *K*_{ewi} of the washer and initial axial force *F*_{i}:

$${delta }_{i}=frac{{F}_{i}}{{K}_{mathrm{ew}i}}$$

(3)

Here, *δ*_{1,2} of Fig. 6 can be calculated from Eq. (4) using the amount of change in axial force Δ*F*_{1,2} [22] of bolt 1 due to tightening of bolt 2 and Eqs. (2) and (3):

$${delta }_{mathrm{1,2}}=frac{{Delta F}_{mathrm{1,2}}}{{K}_{mathrm{ew}1}}+frac{{F}_{1}}{{K}_{mathrm{ew}2}}{e}^{-frac{3}{2{T}_{text{h}}}left(S-varphi right)}$$

(4)

Similarly, *δ*_{2,3} can be calculated using the amount of change in axial force Δ*F*_{2,3} [22] of bolt 1 due to tightening of bolt 3 after tightening bolt 2:

$${delta }_{mathrm{2,3}}=frac{{Delta F}_{mathrm{2,3}}}{{K}_{mathrm{ew}1}}+frac{{F}_{1}}{{K}_{mathrm{ew}2}}{e}^{-frac{3}{2{T}_{text{h}}}left(S-varphi right)}+frac{{F}_{1}}{{K}_{mathrm{ew}3}}{e}^{-frac{3}{2{T}_{text{h}}}left(S-varphi right)}$$

(5)

From the compatibility conditions of the elastic deformation, Eqs. (6) and (7) hold using the bolt elongation *δ*_{b1}:

$${delta }_{mathrm{1,2}}+{delta }_{{text{b}}1}=frac{{Delta F}_{mathrm{1,2}}}{{K}_{mathrm{ew}1}}+frac{{F}_{1}}{{K}_{mathrm{ew}2}}{e}^{-frac{3}{2{T}_{text{h}}}left(S-varphi right)}+frac{{Delta F}_{mathrm{1,2}}}{{K}_{{text{b}}1}}=0$$

(6)

$${delta }_{mathrm{2,3}}+{delta }_{{text{b}}1}=frac{{Delta F}_{mathrm{2,3}}}{{K}_{mathrm{ew}1}}+frac{{F}_{1}}{{K}_{mathrm{ew}2}}{e}^{-frac{3}{2{T}_{text{h}}}left(S-varphi right)}+frac{{F}_{1}}{{K}_{mathrm{ew}3}}{e}^{-frac{3}{2{T}_{text{h}}}left(S-varphi right)}+frac{{Delta F}_{mathrm{2,3}}}{{K}_{{text{b}}1}}=0$$

(7)

Here, *K*_{b1} is the stiffness of the bolt and can be calculated by the equation in Ref. [30]. The sums *δ*_{1,2} + *δ*_{b1} and *δ*_{2,3} + *δ*_{b1} can be calculated from Eqs. (6) and (7), respectively, and Δ*F*_{1,2} and Δ*F*_{2,3} can be calculated from Eqs. (8) and (9), respectively:

$$Delta {F}_{mathrm{1,2}}=-{F}_{1}{e}^{-frac{3}{2{T}_{mathrm{h}}}left(S-varphi right)}frac{1}{left(frac{1}{{K}_{mathrm{ew}1}}+frac{1}{{K}_{mathrm{b}1}}right){K}_{mathrm{ew}2}}$$

(8)

$$Delta {F}_{mathrm{2,3}}=-{F}_{1}{e}^{-frac{3}{2{T}_{mathrm{h}}}left(S-varphi right)}frac{left(frac{1}{{K}_{mathrm{ew}2}}+frac{1}{{K}_{mathrm{ew}3}}right)}{left(frac{1}{{K}_{mathrm{ew}1}}+frac{1}{{K}_{mathrm{b}1}}right)}$$

(9)

Therefore, the axial force ratio *α* of bolt 1 in Fig. 6b, c can be calculated from Eqs. (10) and (11), respectively.

The axial force ratio *α* of bolt 1 in Fig. 6b is

$$alpha =frac{left({F}_{1}+{Delta F}_{mathrm{1,2}}right)}{{F}_{1}}$$

(10)

and the axial force ratio α of bolt 1 in Fig. 6c is

$$alpha =frac{left({F}_{1}+{Delta F}_{mathrm{2,3}}right)}{{F}_{1}}$$

(11)

The axial force ratio α was calculated from the above series of equations and compared with the experimental results. Table 5 shows a list of values for the washer embedment stiffness *K*_{ew} obtained from the washer embedment experiment for use in the calculations. Tables 6,7 and 8 show a comparison between calculation values and experimental values. Note that Se.3 (bolt C) and Se.4 (bolt B) for *T*_{h} = 30 and 45 mm and *S* = 48 mm, and Se.3 (bolt C) and Se. 4 (bolt B) for *T*_{h} = 60 mm and *S* = 48 and 60 mm produced a negative value, and Cal./Exp. is recorded as “0.00”. Furthermore, after tightening, a decrease in axial force of several percent due to stress relaxation and creep over time is confirmed, as shown in Fig. 4. However, it is considered that this decrease in axial force due to stress relaxation and creep is also included in the fluctuations in axial force due to elastic interactions. As such, when comparing the experimental values with the calculated values from the proposed evaluation formula, which does not consider stress relaxation and creep into account, it is appropriate for the experimental value to also exclude this stress relaxation and creep. However, it is assumed that the stress relaxation and creep characteristics differ depending on the variation of the wood material directly beneath each washer or on the tightening speed [31] and conducting an evaluation excluding stress relaxation and creep is very difficult. As such, in this study, the experimental values were set to include stress relaxation and creep and were set to the values after tightening was completed (the same values as in Tables 2, 3, and 4).

According to Tables 6, 7 and 8, in the case of *T*_{h} = 30 mm, *S* = 72, 84 mm, it was found that the calculated value tended to capture the experimental value with approximately ± 10%. However, in other cases, the accuracy of the calculation decreased and was underestimated, but as *S* increased, the calculated value tended to be slightly closer to the experimental value. This is considered to be, because, when *S* is 72 mm or greater, the value of *S* − *φ* is sufficiently large, and after the adjacent bolt is tightened, the vertical displacement added to the compressed wood part directly beneath the target bolt is almost 0. On the other hand, when *S* = 72 mm or less, the effect received by tightening adjacent bolts is significant, and the vertical displacement added to the compressed wood part directly beneath the target bolt becomes significant. Since Eq. (2) is a function for when the compressed part of the wood has a flat shape, it is considered that it cannot be applied to the geometric change of the compressed part of wood due to the tightening of adjacent bolts. From the above, when *S* is large, the proposed evaluation formula can ignore the influence of geometric changes. However, the evaluation formula for when *S* is small will be left as a future subject. In addition, this calculation method does not consider the effects of stress relaxation on wood. It is very difficult to quantitatively evaluate how much stress relaxation affects the fluctuation of axial force due to elastic interactions, but it is speculated that the effect may be greater depending on the material and this could also be investigated in future studies.

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