Proceedings of 4th International Conference on Fracture Fatigue and Wear, FFW 2015, Ghent University, Belgium, 27-28 August 2015The high strength bolts and nuts are widely used in various fields. In this study the effect of slight
pitch difference is considered when the nut pitch is αμm larger than the bolt pitch. In the first place, the fatigue experiment is conducted with varying pitch difference. The results show that the fatigue life is
extended to about 1.5 times by introducing the suitable pitch difference under the high stress amplitude.
Next, the detail observation is performed on the fractured specimens including the fractured positions and
the crack configurations. It is found that the fractured positions and the crack distributions vary depending on the pitch difference. Finally, to clarify the improvement mechanism of the fatigue strength, the finite element method is applied to calculate the stress amplitude and mean stress at each bolt threads, and the
incomplete threads at the nut ends are also considered to obtain the accurate analytical results

Abdel Wahab, Magd

Chen, Xin

Fujisawa, Ryota

Maruyama, Hikaru

Noda, Nao-Aki

Sano, Yoshikazu

Takase, Yasushi

English

Wahab, Magd Abdel

Kyushu Institute of Technology of Academic Repository

Kyushu Institute of Technology Academic Repository
九州工業大学学術機関リポジトリ
Title Fatigue strength evaluation for bolt-nut connections havingslight pitch difference considering incomplete threads of nut
Author(s)Takase, Yasushi; Noda, Nao-Aki; Chen, Xin; Sano, Yoshikazu;Fujisawa, Ryota; Maruyama, Hikaru; Wahab, Magd Abdel
Issue Date2015
URL http://hdl.handle.net/10228/5420
Rights
 International Journal of Fracture Fatigue and Wear, Volume 3 
43 
 
Proceedings of the 4th International Conference on  
Fracture Fatigue and Wear, pp. 43-49, 2015 
FATIGUE STRENGTH EVALUATION FOR BOLT-NUT CONNECTIONS 
HAVING SLIGHT PITCH DIFFERENCE CONSIDERING INCOMPLETE 
THREADS OF NUT 
Yasushi Takasea, Nao-Aki Nodaa, Xin Chena,b, Yoshikazu Sanoa,  
Ryota Fujisawaa, Hikaru Maruyamaa and Magd Abdel Wahabb 
aDepartment of Mechanical Engineering, Kyushu Institute of Technology,Kitakyushu 804-8550, Japan 
bDepartment of Mechanical Construction and Production, Faculty of Engineering and Architecture, 
Ghent University, 9000 Ghent, Belgium 
 
Abstract: The high strength bolts and nuts are widely used in various fields. In this study the effect of slight 
pitch difference is considered when the nut pitch is α µm larger than the bolt pitch. In the first place, the 
fatigue experiment is conducted with varying pitch difference. The results show that the fatigue life is 
extended to about 1.5 times by introducing the suitable pitch difference under the high stress amplitude. 
Next, the detail observation is performed on the fractured specimens including the fractured positions and 
the crack configurations. It is found that the fractured positions and the crack distributions vary depending on 
the pitch difference. Finally, to clarify the improvement mechanism of the fatigue strength, the finite element 
method is applied to calculate the stress amplitude and mean stress at each bolt threads, and the 
incomplete threads at the nut ends are also considered to obtain the accurate analytical results.  
Keywords: Bolt-Nut connections, Pitch Difference, Fatigue Life, Finite Element Analysis 
1 INTRODUCTION  
Bolt-nut connections are commonly used in engineering structures due to the advantages they offer, such as 
the ability of clamping, and the ease of disassembly for maintenance or repair. However, fatigue failure 
problems of bolt are always of concern. High stress concentration factors always occur at the root of bolt 
thread and it is not easy to improve the fatigue strength of screws. Most previous studies are focusing on the 
anti-loosening performance for newly developed bolt and nut [1-3]. 
The effect of the thread shape on the fatigue life of bolt has been investigated [4-6]. A previous study 
indicated that the fatigue strength may be improved depending on the pitch error [7]. Our previous 
experiment clarified that the fatigue life is improved by introducing suitable pitch difference under a certain 
level of stress amplitude [8]. 
As further work, in the present study, two types of specimens will be investigated systematically, including 
the standard bolt-nut connections and the connections having a slight pitch difference α. Figure 1 shows the 
schematic diagram of bolt and nut connection. First, the fatigue experiment will be carried out for the two 
 International Journal of Fracture Fatigue and Wear, Volume 3 
44 
 
types of specimens with varying stress amplitude systematically. Afterwards, the fractured specimens will be 
investigated. Finally, to clarify the improvement mechanism of the fatigue strength, the finite element 
analysis will be applied to calculate the stress amplitude and mean stress at each bolt threads. In order to 
obtain the accurate analytical results, the longitudinal section of nut specimen was investigated as shown in 
Fig. 2. It’s noticeable the incomplete end threads may have an effect on the contact status between bolt and 
nut, therefore, we will pay attention to the incomplete threads of nut in the FE analysis. 
        
Fig. 1 Schematic diagram of bolt-nut connection          Fig. 2 Incomplete threads at nut ends 
2 EXPERIMENTAL SET-UP 
2.1 Materials 
In this study, the Japanese Industrial Standard (JIS) M16 bolts and nuts are employed. The strength grade of 
bolt and nut is 8.8 and 8, respectively. Table 1 presents the material property of bolt and nut. The standard 
M16 bolt and nut have the same pitch dimension as 2000 µm. Herein, the nut pitch is assumed to be 
(2000+α) µm, and α is named as pitch difference. Motivated by the results of previous study, two types of 
pitch differences, i.e. α=0 (standard bolt nut connection) and α=αm are considered in this study. The 
clearance between bolt and nut has a standard dimension as 125 µm as shown in Fig. 1. 
Table 1 Material Property of Bolt and Nut 
 
Young’s modulus 
(GPa) 
Poison’s 
ratio 
Yield strength 
(MPa) 
Tensile strength 
(MPa) 
SCM435 ( Bolt) 206 0.3 800 1200 
S45C ( Nut) 206 0.3 530 980 
2.2 Fatigue tests 
The fatigue experiment is conducted by using 392 kN Servo Fatigue Testing Machine. A series cyclic loading 
were applied to the specimens. Table 2 summarizes the experimental loading conditions and the 
corresponding stress with considering the bottom cross sectional area of bolt as AR=141 mm2. The cycling 
frequency of the loadings is 8 Hz.  
2.3 Experimental Results 
Figure 3 shows the fractured specimens which underwent a stress amplitude of σa=100 MPa. For the 
 International Journal of Fracture Fatigue and Wear, Volume 3 
45 
 
standard bolt-nut connection, it is confirmed that the fracture always occurs at the bottom of thread No.1. For 
the specimens of α=αm, the fracture takes place nearby No.1 thread, and the fracture surface shows 
noticeably different characteristics.   
  Table 2 Experimental loading conditions 
Load (kN) Stress (MPa) 
Mean load Load amplitude Mean stress Stress amplitude 
30 22.6 213 160 
30 18.3 213 130 
30 14.1 213 100 
30 11.3 213 80 
30 8.5 213 60 
30 7.1 213 50 
 
                         
                 (a) α=0, σa=100MPa                          (b) α=αm, σa=100MPa 
                           Fig. 3 Fractured specimens 
The S-N curves with fatigue limit at N=2×106 stress cycles are obtained as depicted in Fig.4. It is found that 
the pitch differences effects the fatigue life significantly. When the stress amplitude is above 80 MPa, the 
fatigue life for α=αm is about 1.5 times of the standard bolt-nut connections (α=0). However, near the fatigue 
limit, the fatigue lives of the two types of specimens are similar, and the fatigue limits remain at the same 
value of 60 MPa.  
            
                                Fig. 4 S-N curves 
 International Journal of Fracture Fatigue and Wear, Volume 3 
46 
 
3 CRACK OBSERVATION 
Figure 5 and Figure 6 show the observed trajectory of cracks along the longitudinal cross section of the 
specimens at the fatigue stress amplitudes σa=60 MPa and 100 MPa. For the standard bolt-nut connections, 
it can be seen that the initial crack may occur at thread No.2, and final fracture happens at No.1 thread. For 
α=αm, the initial crack may start at No.5 thread or No.6 thread, extending toward thread No.1 and finally 
fracture happen nearby thread No.1. From the S-N curves and the observations of crack trajectories, we can 
conclude that the fatigue life of the bolt-nut connections may be extended by introducing a pitch difference 
because the changes in crack propagation trajectory may take place.  
         
               (a) α=0                                               (b) α=αm 
Fig. 5 Crack configuration observed from the surface and 
at the longitudinal section of the bolt when σa=60MPa 
              
  (a) α=0                                             (b) α=αm 
Fig. 6 Crack configuration observed from the surface and 
at the longitudinal section of the bolt when σa=100MPa 
4 FINITE ELEMENT ANALYSIS 
To analyse the stress at the bolt threads, the axisymmetric model of the bolt-nut connection is created by 
using FEM code MSC.Marc/Mentat 2012. The elastic-plastic FE analysis is performed. The multifrontal 
sparse solver is used. Friction coefficient of 0.3 is entered in contact table and Coulomb friction is used. 
 International Journal of Fracture Fatigue and Wear, Volume 3 
47 
 
Considering the incomplete threads of nut shown in Fig. 2, here, the complete thread model and the 
incomplete thread model are created as shown in Fig. 7. In accordance with the experimental specimens, 
two types of pitch differences, i.e. α=0 and α=αm are analysed by using both the complete thread model and 
incomplete thread model. At the roots of bolt threads a very fine mesh with a minimum size of 0.01mm is 
used. The clamped plate is fixed in the horizontal direction and the bolt head is subjected to cyclic load F. 
For each bolt thread from No.-2 thread to No.8 thread, the maximum stress amplitude and the mean stress 
are investigated at the point where the maximum stress amplitude appears. According to these stress 
components, the endurance limit diagrams are obtained to evaluate the relative danger level of the bolt 
threads. Figure 8 shows the endurance limit diagrams analysed by using the complete thread model when 
the load F=30±14.1kN. In the endurance limit diagram, the Soderberg line is plotted. It should be noted that 
because of the stress gradient, the maximum stress amplitude for fracture of notched specimens is always 
larger than that of the plain specimens. Therefore, the stress data plotted beyond the Soderberg line does 
not represent the real fracture at the bolt thread.  
For α=0, Fig. 8 (a) indicates that thread No.1 has the highest stress amplitude. On the other hand, for α=αm, 
Fig. 8 (b) shows that No.7 and No.8 thread become more dangerous. However, these analytical results do 
not correspond to the crack observation results very well, because the crack may initiate at No.2 thread for 
α=0, and large crack occurs at No.5 thread for α=αm when σa=100 MPa as illustrated in section 3.  
Figure 9 shows the endurance limit diagrams analysed by using the incomplete thread model. For α=0, 
thread No.2 becomes the most dangerous thread in accordance with the observed crack. For α=αm, the 
stress amplitude at No.6 thread increases while that at No.7 thread decreases, and these results become 
more closer to the crack configuration in Fig.6 (b) where the large crack occurs at No.5 thread. 
 
   
(a) Complete thread model                          (b) Incomplete thread model 
Fig. 7 Axi-symmetric finite element model of bolt-nut connection  
 
 International Journal of Fracture Fatigue and Wear, Volume 3 
48 
 
             
              (a) α=0                                      (b)  α=αm 
Fig. 8 Endurance limit diagram, σa=100MPa (Complete thread model) 
             
(a) α=0                                          (b) α=αm 
Fig. 9 Endurance limit diagram, σa=100MPa (Complete thread model vs Incomplete thread model) 
5 CONCLUSIONS 
A slight pitch difference α is considered between the M16 bolt-nut connections. The fatigue experiment has 
been conducted for two types of pitch difference, i.e. α=0 and α=αm. The stress amplitude and mean stress 
at each bolt threads have been numerically analysed using the finite element method with considering the 
incomplete threads of nut. The conclusions can be summarized as follows:  
(1) For α=0, the incomplete thread model explains that the crack initiates at No.2 thread as shown in Fig. 9 
(a). 
(2) For α=αm, the incomplete thread model explains the crack initiates at No.5 or No.6 threads as shown in 
Fig. 9 (b). 
 
 
 
 
 International Journal of Fracture Fatigue and Wear, Volume 3 
49 
 
6 REFERENCES 
[1] Hard Lock Industry KK, Looseness Preventing Nut, Patent Abstracts of Japan, Publication No.: 
2002–195236, 2002-7-10.  
[2] Izumi, S., Yokoyama, T., Teraoka, T., Iwasaki, A., Sakai, S., Saito, K., Nagawa, M., Noda, 
H.,Verification of anti-loosening performance of super slit nut by finite element method, Transaction 
of the Japan Society of Mechanical Engineers, Series A, Vol.71, No.703(2005), pp.380-386 (in 
Japanese). 
[3] Xiao, Y., Kuhara, M., Noda, N.-A., Saito, K., Nagawa, M. and Yumoto, A., Optimum dimensions of 
thin walled tube on the mechanical performance of super stud bolt, Transaction of the Japan Society 
of Mechanical Engineers, Series A, Vol.74, No.743(2008), pp.954-960 (in Japanese). 
[4] Kawahara, S., et al., A study on the thread shape of the high strength bolt, Summaries of technical 
papers of annual meeting Architectural Institute of Japan, (1990), pp.1545-1546 (in Japanese). 
[5] Nishida, S-I., Urashima, C., and Tamasaki, H., A new method for fatigue life improvement of screws, 
European Structural Integrity Society, No.22(1997), pp.215-225. 
[6] Xiao, Y., Wan, Q., Noda, N.-A., Akaishi, Y.-I., Takase, Y. and Nishida, S.-I., Stress reduction effect of 
tapering thread bolts and nuts which have slightly different pitches, Transactions of Society of 
Automotive Engineers of Japan, Vol.42, No.4(2011), pp.927-933 (in Japanese).  
[7] Maruyama, K., Stress analysis of a bolt-nut Joint by the finite element method and the 
copper-electroplating method (3rd report), Transaction of the Japan Society of Mechanical 
Engineers, Vol.19, No.130(1976), pp.360-368. 
[8] Akaishi, Y.-I., Chen, X., Yu, Y., Tamasaki, H., Noda, N.-A., Sano, Y. and Takase, Y., Fatigue strength 
analysis for bolts and nuts which have slightly different pitches considering clearance, Transactions 
of Society of Automotive Engineers of Japan, Vol.4, No.44(2013), pp.1111-1117 (in Japanese). 


Fatigue strength evaluation for bolt-nut connections having slight pitch difference considering incomplete threads of nut

Ghent University Academic Bibliography

 International Journal of Fracture Fatigue and Wear, Volume 3 43  Proceedings of the 4th International Conference on  Fracture Fatigue and Wear, pp. 43-49, 2015 FATIGUE STRENGTH EVALUATION FOR BOLT-NUT CONNECTIONS HAVING SLIGHT PITCH DIFFERENCE CONSIDERING INCOMPLETE THREADS OF NUT Yasushi Takasea, Nao-Aki Nodaa, Xin Chena,b, Yoshikazu Sanoa,  Ryota Fujisawaa, Hikaru Maruyamaa and Magd Abdel Wahabb aDepartment of Mechanical Engineering, Kyushu Institute of Technology,Kitakyushu 804-8550, Japan bDepartment of Mechanical Construction and Production, Faculty of Engineering and Architecture, Ghent University, 9000 Ghent, Belgium  Abstract: The high strength bolts and nuts are widely used in various fields. In this study the effect of slight pitch difference is considered when the nut pitch is α µm larger than the bolt pitch. In the first place, the fatigue experiment is conducted with varying pitch difference. The results show that the fatigue life is extended to about 1.5 times by introducing the suitable pitch difference under the high stress amplitude. Next, the detail observation is performed on the fractured specimens including the fractured positions and the crack configurations. It is found that the fractured positions and the crack distributions vary depending on the pitch difference. Finally, to clarify the improvement mechanism of the fatigue strength, the finite element method is applied to calculate the stress amplitude and mean stress at each bolt threads, and the incomplete threads at the nut ends are also considered to obtain the accurate analytical results.  Keywords: Bolt-Nut connections, Pitch Difference, Fatigue Life, Finite Element Analysis 1 INTRODUCTION  Bolt-nut connections are commonly used in engineering structures due to the advantages they offer, such as the ability of clamping, and the ease of disassembly for maintenance or repair. However, fatigue failure problems of bolt are always of concern. High stress concentration factors always occur at the root of bolt thread and it is not easy to improve the fatigue strength of screws. Most previous studies are focusing on the anti-loosening performance for newly developed bolt and nut [1-3]. The effect of the thread shape on the fatigue life of bolt has been investigated [4-6]. A previous study indicated that the fatigue strength may be improved depending on the pitch error [7]. Our previous experiment clarified that the fatigue life is improved by introducing suitable pitch difference under a certain level of stress amplitude [8]. As further work, in the present study, two types of specimens will be investigated systematically, including the standard bolt-nut connections and the connections having a slight pitch difference α. Figure 1 shows the schematic diagram of bolt and nut connection. First, the fatigue experiment will be carried out for the two  International Journal of Fracture Fatigue and Wear, Volume 3 44  types of specimens with varying stress amplitude systematically. Afterwards, the fractured specimens will be investigated. Finally, to clarify the improvement mechanism of the fatigue strength, the finite element analysis will be applied to calculate the stress amplitude and mean stress at each bolt threads. In order to obtain the accurate analytical results, the longitudinal section of nut specimen was investigated as shown in Fig. 2. It’s noticeable the incomplete end threads may have an effect on the contact status between bolt and nut, therefore, we will pay attention to the incomplete threads of nut in the FE analysis.         Fig. 1 Schematic diagram of bolt-nut connection          Fig. 2 Incomplete threads at nut ends 2 EXPERIMENTAL SET-UP 2.1 Materials In this study, the Japanese Industrial Standard (JIS) M16 bolts and nuts are employed. The strength grade of bolt and nut is 8.8 and 8, respectively. Table 1 presents the material property of bolt and nut. The standard M16 bolt and nut have the same pitch dimension as 2000 µm. Herein, the nut pitch is assumed to be (2000+α) µm, and α is named as pitch difference. Motivated by the results of previous study, two types of pitch differences, i.e. α=0 (standard bolt nut connection) and α=αm are considered in this study. The clearance between bolt and nut has a standard dimension as 125 µm as shown in Fig. 1. Table 1 Material Property of Bolt and Nut  Young’s modulus (GPa) Poison’s ratio Yield strength (MPa) Tensile strength (MPa) SCM435 ( Bolt) 206 0.3 800 1200 S45C ( Nut) 206 0.3 530 980 2.2 Fatigue tests The fatigue experiment is conducted by using 392 kN Servo Fatigue Testing Machine. A series cyclic loading were applied to the specimens. Table 2 summarizes the experimental loading conditions and the corresponding stress with considering the bottom cross sectional area of bolt as AR=141 mm2. The cycling frequency of the loadings is 8 Hz.  2.3 Experimental Results Figure 3 shows the fractured specimens which underwent a stress amplitude of σa=100 MPa. For the  International Journal of Fracture Fatigue and Wear, Volume 3 45  standard bolt-nut connection, it is confirmed that the fracture always occurs at the bottom of thread No.1. For the specimens of α=αm, the fracture takes place nearby No.1 thread, and the fracture surface shows noticeably different characteristics.     Table 2 Experimental loading conditions Load (kN) Stress (MPa) Mean load Load amplitude Mean stress Stress amplitude 30 22.6 213 160 30 18.3 213 130 30 14.1 213 100 30 11.3 213 80 30 8.5 213 60 30 7.1 213 50                                            (a) α=0, σa=100MPa                          (b) α=αm, σa=100MPa                            Fig. 3 Fractured specimens The S-N curves with fatigue limit at N=2×106 stress cycles are obtained as depicted in Fig.4. It is found that the pitch differences effects the fatigue life significantly. When the stress amplitude is above 80 MPa, the fatigue life for α=αm is about 1.5 times of the standard bolt-nut connections (α=0). However, near the fatigue limit, the fatigue lives of the two types of specimens are similar, and the fatigue limits remain at the same value of 60 MPa.                                              Fig. 4 S-N curves  International Journal of Fracture Fatigue and Wear, Volume 3 46  3 CRACK OBSERVATION Figure 5 and Figure 6 show the observed trajectory of cracks along the longitudinal cross section of the specimens at the fatigue stress amplitudes σa=60 MPa and 100 MPa. For the standard bolt-nut connections, it can be seen that the initial crack may occur at thread No.2, and final fracture happens at No.1 thread. For α=αm, the initial crack may start at No.5 thread or No.6 thread, extending toward thread No.1 and finally fracture happen nearby thread No.1. From the S-N curves and the observations of crack trajectories, we can conclude that the fatigue life of the bolt-nut connections may be extended by introducing a pitch difference because the changes in crack propagation trajectory may take place.                          (a) α=0                                               (b) α=αm Fig. 5 Crack configuration observed from the surface and at the longitudinal section of the bolt when σa=60MPa                 (a) α=0                                             (b) α=αm Fig. 6 Crack configuration observed from the surface and at the longitudinal section of the bolt when σa=100MPa 4 FINITE ELEMENT ANALYSIS To analyse the stress at the bolt threads, the axisymmetric model of the bolt-nut connection is created by using FEM code MSC.Marc/Mentat 2012. The elastic-plastic FE analysis is performed. The multifrontal sparse solver is used. Friction coefficient of 0.3 is entered in contact table and Coulomb friction is used.  International Journal of Fracture Fatigue and Wear, Volume 3 47  Considering the incomplete threads of nut shown in Fig. 2, here, the complete thread model and the incomplete thread model are created as shown in Fig. 7. In accordance with the experimental specimens, two types of pitch differences, i.e. α=0 and α=αm are analysed by using both the complete thread model and incomplete thread model. At the roots of bolt threads a very fine mesh with a minimum size of 0.01mm is used. The clamped plate is fixed in the horizontal direction and the bolt head is subjected to cyclic load F. For each bolt thread from No.-2 thread to No.8 thread, the maximum stress amplitude and the mean stress are investigated at the point where the maximum stress amplitude appears. According to these stress components, the endurance limit diagrams are obtained to evaluate the relative danger level of the bolt threads. Figure 8 shows the endurance limit diagrams analysed by using the complete thread model when the load F=30±14.1kN. In the endurance limit diagram, the Soderberg line is plotted. It should be noted that because of the stress gradient, the maximum stress amplitude for fracture of notched specimens is always larger than that of the plain specimens. Therefore, the stress data plotted beyond the Soderberg line does not represent the real fracture at the bolt thread.  For α=0, Fig. 8 (a) indicates that thread No.1 has the highest stress amplitude. On the other hand, for α=αm, Fig. 8 (b) shows that No.7 and No.8 thread become more dangerous. However, these analytical results do not correspond to the crack observation results very well, because the crack may initiate at No.2 thread for α=0, and large crack occurs at No.5 thread for α=αm when σa=100 MPa as illustrated in section 3.  Figure 9 shows the endurance limit diagrams analysed by using the incomplete thread model. For α=0, thread No.2 becomes the most dangerous thread in accordance with the observed crack. For α=αm, the stress amplitude at No.6 thread increases while that at No.7 thread decreases, and these results become more closer to the crack configuration in Fig.6 (b) where the large crack occurs at No.5 thread.     (a) Complete thread model                          (b) Incomplete thread model Fig. 7 Axi-symmetric finite element model of bolt-nut connection    International Journal of Fracture Fatigue and Wear, Volume 3 48                             (a) α=0                                      (b)  α=αm Fig. 8 Endurance limit diagram, σa=100MPa (Complete thread model)              (a) α=0                                          (b) α=αm Fig. 9 Endurance limit diagram, σa=100MPa (Complete thread model vs Incomplete thread model) 5 CONCLUSIONS A slight pitch difference α is considered between the M16 bolt-nut connections. The fatigue experiment has been conducted for two types of pitch difference, i.e. α=0 and α=αm. The stress amplitude and mean stress at each bolt threads have been numerically analysed using the finite element method with considering the incomplete threads of nut. The conclusions can be summarized as follows:  (1) For α=0, the incomplete thread model explains that the crack initiates at No.2 thread as shown in Fig. 9 (a). (2) For α=αm, the incomplete thread model explains the crack initiates at No.5 or No.6 threads as shown in Fig. 9 (b).      International Journal of Fracture Fatigue and Wear, Volume 3 49  6 REFERENCES [1] Hard Lock Industry KK, Looseness Preventing Nut, Patent Abstracts of Japan, Publication No.: 2002–195236, 2002-7-10.  [2] Izumi, S., Yokoyama, T., Teraoka, T., Iwasaki, A., Sakai, S., Saito, K., Nagawa, M., Noda, H.,Verification of anti-loosening performance of super slit nut by finite element method, Transaction of the Japan Society of Mechanical Engineers, Series A, Vol.71, No.703(2005), pp.380-386 (in Japanese). [3] Xiao, Y., Kuhara, M., Noda, N.-A., Saito, K., Nagawa, M. and Yumoto, A., Optimum dimensions of thin walled tube on the mechanical performance of super stud bolt, Transaction of the Japan Society of Mechanical Engineers, Series A, Vol.74, No.743(2008), pp.954-960 (in Japanese). [4] Kawahara, S., et al., A study on the thread shape of the high strength bolt, Summaries of technical papers of annual meeting Architectural Institute of Japan, (1990), pp.1545-1546 (in Japanese). [5] Nishida, S-I., Urashima, C., and Tamasaki, H., A new method for fatigue life improvement of screws, European Structural Integrity Society, No.22(1997), pp.215-225. [6] Xiao, Y., Wan, Q., Noda, N.-A., Akaishi, Y.-I., Takase, Y. and Nishida, S.-I., Stress reduction effect of tapering thread bolts and nuts which have slightly different pitches, Transactions of Society of Automotive Engineers of Japan, Vol.42, No.4(2011), pp.927-933 (in Japanese).  [7] Maruyama, K., Stress analysis of a bolt-nut Joint by the finite element method and the copper-electroplating method (3rd report), Transaction of the Japan Society of Mechanical Engineers, Vol.19, No.130(1976), pp.360-368. [8] Akaishi, Y.-I., Chen, X., Yu, Y., Tamasaki, H., Noda, N.-A., Sano, Y. and Takase, Y., Fatigue strength analysis for bolts and nuts which have slightly different pitches considering clearance, Transactions of Society of Automotive Engineers of Japan, Vol.4, No.44(2013), pp.1111-1117 (in Japanese). 

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Fatigue strength evaluation for bolt-nut connections having slight pitch difference considering incomplete threads of nut

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