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# Evaluation method for embedment stiffness of metal washers in bolted timber joints using torque gradient and nut factor

*Journal of Wood Science*
**volumeÂ 70**, ArticleÂ number:Â 10 (2024)

## Abstract

Assessing the residual performance of timber joints affected by decay is important for seismic diagnosis and reinforcement of timber structures. This study introduces a method to evaluate the embedment stiffness of metal washers in bolted timber joints using a torque wrench. Formulas for the embedment stiffness of metal washers using the torque gradient (the gradient of the tightening torque against the tightening rotation angle of the nut) and the nut factor are presented and the calculated values are compared with experimental values obtained by conducting tightening tests with a torque wrench and embedment tests of metal washers in bolted joints made from Japanese cedar, Hiba, and Japanese cypress. The results show that the experimental values of the embedment stiffness of metal washers in Japanese cedar and Hiba are generally within the range of the values calculated from the minimum and maximum values of the nut factor. However, for Japanese cypress, the values calculated from the maximum value of the nut factor exceed the experimental values. This was presumably due to locally large frictional forces generated on the bearing surface or threaded part.

## Introduction

Wood is easily affected by moisture and, especially in high-humidity environments, wood-rot fungus can lead to biological deterioration, resulting in a significant decrease in strength, etc. [1,2,3,4,5,6]. Wooden buildings with deteriorated members are likely to exhibit reduced structural performance, so, to ensure structural safety, it is necessary to replace or reinforce the deteriorated members. Particularly in the case of wooden structures, the performance of joints often dictates the overall performance of the structure. Therefore, when planning the reinforcement, it is necessary to assess the residual strength and stiffness in the joints affected by decay. Thus far, research has focused on the survival performance of joints affected by wood decay, including nail joints [7,8,9,10,11,12], screw joints [13,14,15,16], and bolt/drift pin joints [17]. Many of these studies evaluated the relationship between decay and joint shear performance.

In contrast, the present study focuses on bolted connections, which are the most common connection type in wooden structures. There are two forms of resistance mechanisms, where either tensile force or shear force is applied to the bolt [18]. For the tensile force case, joint performance is dominated by the embedment of metal washers, whereas the shear force case involves the bending yield. For this, when the diameter length ratio (main material thickness/bolt diameter) is large and the bending deformation of the bolt increases, the shear resistance increases because of the boltâ€™s axial tensile force, resulting in the so-called rope effect becoming significant. In this situation, an embedment force is generated on metal washer that balances the bolt's tensile force, and the rope effect becomes more pronounced as embedment stiffness of the metal washer increases [19].

Therefore, embedment of metal washers is one of the governing factors for the strength performance of bolted joints and a comprehensive understanding of the embedment behavior of metal washers is crucial. For example, wooden houses with poor underfloor ventilation or which have not been treated with preservatives are at risk of wood rot and this results in reduced performance of the anchor joints. Therefore, a method to comprehend and evaluate the residual performance of the embedment of metal washers is essential. Therefore, this study aimed to establish a method to accurately evaluate the residual performance of bolted joints affected by wood decay and present a method to directly measure and evaluate the embedment stiffness of metal washers on-site. It first focused on the relationship between bolt tightening torque, axial force, and nut rotation angle. It theoretically demonstrated that the embedment stiffness of metal washers can be calculated by combining these relationships and derived a formula for theoretical calculations. Bolt joint tightening tests were conducted using a torque wrench and embedment tests of metal washers were performed using Japanese cedar, Hiba, and Japanese cypress of different densities and the agreement between the calculated and the experimental values was examined.

## Theory

This study examined the bolted timber joint shown in Fig.Â 1. It consists of a stud bolt, metal washer, nut, timber member, and steel plate (rigid body). The derivation of the following equations is based on the work of Fukuoka et al. [20].

First, the relationship between tightening torque *T*_{f} and clamping force *F*_{b} when the nut is tightened is expressed as:

where *K*_{n} is the nut factor and *d* is the bolt diameter. Here, *K*_{n} can also be expressed using Eq.Â (2) below [21]:

where *Î¼*_{th} is the coefficient of friction between threads, *d*_{2} is the effective thread diameter, *Î¼*_{b} is the coefficient of friction between bearing surfaces and *D*_{b} is the equivalent frictional diameter on the bearing surface. The terms inside parentheses in Eq.Â (2) show the breakdown of torque consumption: the first term is the torque expended on clamping force of thread pitchâ€™s inclined plane, the second term is the torque expended on thread friction, and the third term is the torque expended on bearing surface friction [22]. Therefore, *K*_{n} is highly dependent on the friction between the bearing surface and the threaded part.

Furthermore, if the steel plate is regarded as a rigid body, the rotation angle *Ï†* when the bolt is tightened is expressed as [23]:

where *K*_{ew} is the embedment stiffness of the metal washer, *K*_{b} is the stiffness of the bolt, *P* is the thread pitch. Here, *K*_{b} can be calculated as a series combination of the spring constants *K*_{th}, *K*_{s}, and *K*_{cyl} for thread engagement, thread play, and bolt cylinder, respectively, using the following equation:

where *K*_{th}, *K*_{s}, and *K*_{cyl} can be calculated as follows:

where *A*_{s} is the effective cross-sectional area of the thread; *A* is the cross-sectional area of the cylinder; *E*_{b} is Youngâ€™s modulus, which is set to 205,000Â N/mm^{2} here; *L*_{s} is the thread play length; *L*_{cyl} is the length of the cylindrical part of the bolt (See Fig.Â 2); and *L*_{th} is equivalent length, which can be calculated using the bolt's nominal diameter *d* as follows [24]:

EquationÂ (1) can be solved for *F*_{b}, substituted into Eq.Â (3), and rearranged for *K*_{ew} as follows:

Therefore, from Eq.Â (9), if the torque gradient *T*_{f}/*Ï†*, which is the gradient of the tightening torque with respect to rotation angle, and the nut factor *K*_{n} are known, then the embedment stiffness of the metal washer, *K*_{ew}, can be calculated.

## Materials and methods

### Materials

Japanese cedar (*Cryptomeria japonica*), Hiba (*Thujopsis dolabrata*), and Japanese cypress (*Chamaecyparis obtusa*) wood specimens with dimensions of 105Â mm (T: Tangential direction)â€‰Ã—â€‰140Â mm (L: Longitudinal direction)â€‰Ã—â€‰30Â mm (R: Radial direction) and with a *Ï†* 13Â mm bolt hole in the center were used. Fifteen specimens of each species were prepared and six specimens of each species were used in each experiment to investigate the embedment yield load of metal washer *P*_{y} and the embedment stiffness of metal washers *K*_{ew} and the torque gradient *T*_{f}/*Ï†*. The densities and moisture content of the specimens, presented as the meanâ€‰Â±â€‰standard deviation, were 314â€‰Â±â€‰4Â kg/m^{3}, 9.5â€‰Â±â€‰0.7% for Japanese cedar, 401â€‰Â±â€‰4Â kg/m^{3}, 12.2â€‰Â±â€‰0.9% for Hiba, and 504â€‰Â±â€‰9Â kg/m^{3}, 13.1â€‰Â±â€‰0.3% for Japanese cypress. M12 stud bolts made of S45C material, with a thread pitch *P* of 1.75Â mm, thread length of 38Â mm, cylindrical length *L*_{cyL} of 24Â mm, and a total length of 100Â mm. The nut was a hexagonal nut of S45C material, with a bilateral width of 18.6Â mm and a height of 10Â mm. The washer was 42Â mm in diameter and 3.2Â mm thickness, and made from SWCH with a *Ï†* 13Â mm bolt hole in the center.

### Embedment test for measuring *P*
_{y}

FigureÂ 3 shows the experimental apparatus. The specimen was placed on a steel component and loaded monotonically in the R direction up to a displacement of 3Â mm using a universal testing machine (AG-Xplus 50Â kN, Shimadzu Corp.). Displacement was defined as the travel of the crosshead. The test speed was 2Â mm/min. From the obtained loadâ€“displacement relationship, the linear slope in the elastic range and the secondary slope in the plastic range were calculated using the least-squares method and the intersection point of the two lines was designated as *P*_{y} [25]_{.}

### Tightening test for measuring nut factor *K*
_{n}

FigureÂ 4 shows the experimental apparatus. To investigate the nut factor *K*_{n}, a bolt tightening test was conducted using a tightening test machine (NST-500Nm, Japan Measuring System Co.). Each material was placed in the testing machine, the nut was tightened, and the tightening torque of nut and clamping force was measured with a load cell. The clamping force was limited to 10Â kN. Five tightening speeds (0.5, 1, 4, 10, and 20Â rpm) were utilized, with six specimens for each condition. *K*_{n} was calculated from the relationship between the obtained tightening torque and clamping force at 5Â kN using Eq.Â (1).

### Tightening test for measuring *T*
_{f}/*Ï†* and embedment test for measuring *K*
_{ew}

FigureÂ 5 shows the experimental apparatus. To obtain *T*_{f}/*Ï†*, the target tightening torque *T*_{A} was first determined as set out below.

The maximum tightening torque needed to obtain the target maximum clamping force *F*_{max} is *T*_{max}, the minimum tightening torque needed to obtain the target minimum clamping force *F*_{min} is *T*_{min}, and *T*_{A} is the average of these values:

*T*_{max} and *T*_{min} are expressed based on Eq.Â (1) as, respectively:

where *K*_{n-min} and *K*_{n-max} are the minimum and maximum nut factors, respectively, using the minimum and maximum values obtained from the bolt tightening test using the tightening test machine as described above.

On the other hand, *T*_{max} and *T*_{min} have the following relationship when a torque wrench is used for the torque method. The error rate of torque for a torque wrench is *m* and the target tightening torque *T*_{A} must be less than *T*_{max} and more than *T*_{min}, taking into account the error of the torque wrench. *T*_{max} and *T*_{min} are expressed, respectively, as:

where, since the equals sign holds when the upper and lower limits of *T*_{A} are reached, taking into account the error of the torque wrench. The *F*_{min} corresponding to *T*_{max} and *T*_{min} can be derived from Eqs.Â (11, 12, 13, 14), as:

In this study, *F*_{max} was set to 70% of the mean value of *P*_{y} for each species of tree. The error rate *m* was set at 1% [26].

Using a digital torque wrench (DPW-50-P, Adrec Corp.), which can measure the tightening torque and rotation angle in real time using dedicated software, the nut was slowly tightened and the tightening was stopped when *T*_{A} was reached. The tightening operation was performed while checking the tightening torque values monitored on the dedicated software. *T*_{f}/*Ï†* was then calculated from the relationship between the obtained tightening torque and the rotation angle.

Next, after investigating *T*_{f}/*Ï†* for each specimen, an embedment test of the metal washers was conducted to examine the embedment stiffness of the metal washer, *K*_{ew}. The experimental method was the same as for *P*_{y} above. Force was applied until *F*_{max} was reached and the linear slope was obtained from the loadâ€“displacement relationship by the least-squares method, and this was used as *K*_{ew}.

## Results and discussion

### Results of *P*
_{y} and *K*
_{n}

*P*

*K*

FigureÂ 6 shows the average loadâ€“displacement relationship obtained from the embedment tests of metal washers (to experimentally determine *P*_{y}). It shows that, although the initial slip is slightly larger for Hiba, the load increases linearly, reaches a relatively clear yield point, and then tends to increase in a nonlinear manner. Based on this relationship, the linear slope in the elastic range (*K*_{ew}) was defined at load interval of 2â€“3Â kN for Japanese cedar, 3â€“4Â kN for Hiba, and 5â€“6Â kN for Japanese cypress. The linear slope in the plastic range (*K*_{pw}) was defined as 1â€“3Â mm displacement for all species. *P*_{y} was calculated as the intersection of both slopes by the least-squares method. The calculated characteristic values are listed in TableÂ 1. The table also shows the determined *F*_{max} (=â€‰0.7â€‰Ã—â€‰*P*_{y}). According to the results, *K*_{ew} and *P*_{y} were largest for Japanese cypress, followed by Hiba and then Japanese cedar.

FigureÂ 7 shows the relationship between nut factor *K*_{n} and tightening speed obtained from the bolt tightening tests. According to this, at a tightening speed of 20Â rpm, the variation in *K*_{n} was smaller than at other tightening speeds, but no clear relationship between tightening speed and *K*_{n} was observed. *K*_{n-min} was 0.25 and *K*_{n-max} was 0.37. This large variation in *K*_{n} under no-lubrication conditions has been reported in several previous studies [27, 28].

### Comparison of calculated and experimental values

Table 2 shows *T*_{A} values calculated using the determined *F*_{max}, *K*_{n-min}, and *K*_{n-max} values. FigureÂ 8 shows a typical relationship between *T*_{f} and *Ï†*. As shown in the figure, for Hiba, a large *Ï†* value was required for *T*_{f} to increase linearly similarly to what was observed in Fig.Â 6. Based on the obtained results, the *T*_{A}â€‰Ã—â€‰0.5Â âˆ’Â *T*_{A}â€‰Ã—â€‰0.8 Nm intervals for Japanese cedar, Hiba, and Japanese cypress were calculated using the least-squares method for *T*_{f}/*Ï†*.

Table 3 shows the *K*_{ew-cal} values of each specimen calculated from Eq.Â (8) using these results, including *K*_{ew-cal.1} and *K*_{ew-cal.2} calculated from *K*_{n-min} (=â€‰0.25) and *K*_{n-max} (=â€‰0.37), in addition to *K*_{ew-exp} calculated from the washer penetration experiment. Here, the same method used to calculate *P*_{y} was used to calculate *K*_{ew-exp}. As shown in TableÂ 3, for all Japanese cedar and Hiba specimens, except one, *K*_{ew-exp} was within the range of *K*_{ew-cal.1} and *K*_{ew-cal.2} calculated from *K*_{n-min} and *K*_{n-max}. However, five out of the six Japanese cypress specimens had lower *K*_{ew-exp} values than *K*_{ew-cal.1} values calculated from *K*_{n-max}. A reason for *K*_{ew-exp} being lower than *K*_{ew-cal.1} calculated from *K*_{n-max} may be that *K*_{n} was actually larger than *K*_{n-max}. In addition, TableÂ 3 shows the results of *K*_{n-cal} also calculated using Eq.Â (9). It can be seen that the specimens whose *K*_{ew-exp} values are lower than the *K*_{ew-cal.1} values calculated from *K*_{n-max} are higher than the *K*_{n-max} value of 0.37. Since the tightening work was performed using a manual torque wrench, it is assumed that, when tightening with a large tightening force, a large frictional force is generated locally on the bearing surface or at the thread, resulting in an increase in *K*_{n}.

From the above, *K*_{ew-cal} was evaluated as being overestimate depending on the *K*_{n} setting and the species of wood. In future, the use of automatic tools may be explored, such as nut runners that allow tightening to occur at a constant speed, to stabilize the friction on the bearing surface or thread and prevent variations in *K*_{n}. Furthermore, in this study, lubricating oil was not applied to metal objects such as bolts. But by applying lubricating oil to the bearing surface and threads, friction against torque could be reduced and a more accurate *K*_{ew} value could be expected regardless of the type of tool used for tightening.

## Conclusions

Formulas for the embedment stiffness of metal washers using torque gradient and nut factor were presented for bolted timber joints. Tightening and embedment tests of metal washers were conducted using a torque wrench on bolted joints made of three species of wood: Japanese cedar, Hiba, and Japanese cypress. The experimental values for Japanese cedar and Hiba were found to fall within the range of embedment stiffness values for metal washers calculated from the minimum and maximum nut factors in all but one of the six specimens. However, for five of the six Japanese cypress specimens, the calculated maximum nut factors exceeded the experimental values. This was presumably due to locally large frictional forces generated on the bearing surface or threaded part.

## Availability of data and materials

Not applicable.

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## Funding

This work was supported in part by the research Grant of the Maeda Engineering Foundation.

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DM designed and performed the experiments and analyzed the data. MT and TM analyzed the data. All authors read and approved the final manuscript.

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### Cite this article

Matsubara, D., Teranishi, M. & Mori, T. Evaluation method for embedment stiffness of metal washers in bolted timber joints using torque gradient and nut factor.
*J Wood Sci* **70**, 10 (2024). https://doi.org/10.1186/s10086-024-02123-7

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DOI: https://doi.org/10.1186/s10086-024-02123-7