Effects of root radius and pitch difference on fatigue strength and anti-loosening performance for high strength bolt

Abstract

The fatigue strength improvement and anti-loosening performance are studied experimentally and analytically for JIS M16 bolt–nut connections. Three different root radii are considered coupled with three different pitch differences. The enlarging the bolt root radius improves the fatigue limit of the bolt by more than 30% because both stress amplitude and mean stress can be reduced. Furthermore, suitable pitch difference improves the fatigue limit by more than 25%. This is because under no pitch difference the crack initiation always occurs at No.1 or No.2 threads close to the bolt head causing the final failure; however, under a suitable pitch difference the crack initiation occurs at No.6 or No.7 threads far away from the bolt head. Good anti-loosening performance can be expected for the bolt–nut connections having enlarged root radius because the prevailing torque $T_{P}$ = 19 Nm and the residual prevailing torque $T_{P}^{u} = 10 Nm$ are not smaller compared to other special bolt–nut connections.

Keywords

anti-loosening bolt–nut connections fatigue strength pitch difference root radius

Introduction

Bolt nut connections are one of the essential mechanical elements used in various industrial fields; for instance, about 2000 bolt nut connections are needed in one vehicle (Ujita, 2010). When subject to dynamic and impact loading with vibrations, self-loosening, and fatigue failure of bolt nut connections may happen and even causes serious accidents. Several kinds of bolt failure under complex working environments are discussed in recent studies (Elliott et al., 2019; Roy et al., 2018, 2019; Sivapathasundaram and Mahendran, 2018; Teh et al., 2017). As shown in these references, excellent anti-loosening performance is always required for bolt–nut connections. Some previous studies suggested that self-loosening is mainly caused by the dynamic transverse loading (Junker, 1972; Ramey et al., 1995). Several useful studies are available for preventing self-loosening (Noda et al., 2008; Panja and Das, 2017; Sase et al., 1996; Xiao et al., 2010; Zhou et al., 2019). Typically, Wakabayashi designed a nut pair that consists of an upper nut and a lower nut that eccentrically engage with each other to avoid loosening, named Hard Lock. This kind of bolt nut connection has witnessed the success of Japanese Shinkansen for several decades (Wakabayashi, 2002). Previous studies for bolt–nut connections mainly focused on anti-loosening performance (Chen et al., 2012; Izumi et al., 2005a; Noda et al., 2008; Ranjan et al., 2013; Wakabayashi, 2002) and relatively few studies contributed to improvements in the fatigue strength (Chakherlou et al., 2013; Hirai et al., 2005; Lee et al., 2014; Li et al., 2014; Nishida, 1980; Noda et al., 2011; Pedersen, 2013; Zhou et al., 2015). This is because high stress concentration appearing at the No. 1 thread in Figure 1(a) is difficult to be reduced. Usually, the anti-loosening ability affects the fatigue strength and the cost significantly. Furthermore, most previous studies concern either anti-loosening or high fatigue life and few studies aimed at improving both anti-loosening and fatigue strength. Previously the authors found that the fatigue life and the anti-loosening performance of bolt nut connections can be improved by introducing a suitable pitch difference between the bolt and the nut (Chen et al., 2015, 2016; Noda et al., 2016).

Figure 1.

Schematic illustration of: (a) bolted joint, (b) nut chamfer at nut ends, (c) threads contact when α = 0, and (d) thread contact when α > 0.

Since no research is available, this paper will focus on the coupled effect of the root radius $ρ$ and the pitch difference $α$ on the fatigue strength in bolt nut connections. Based on the authors’ previous researches, three kinds of root radii α for M16 bolt nut connections are chosen. Furthermore, at the same time, enlarging the thread root radius $ρ$ to reduce the concentration aiming at improving the fatigue life and fatigue limit. The fatigue strength improvement will be discussed from the S-N curve and fatigue limit diagram based on the experiments and FEM analysis. The anti-loosening effect will be confirmed since the coupled effect of the thread root radius $ρ$ and the pitch difference α may affect the anti-loosening as well as the fatigue strength.

Fatigue strength improvement

Previous studies for bolt fatigue strength

The bolt fatigue failure sometimes happened without the nut loosening. This is because the large stress concentration always appears at the bolt thread root. Since it is difficult to reduce the stress concentration, few researches are available for improving the fatigue strength compared to anti-loosening.

Walker et al. studied the thread parameters effects on the fatigue life, and they found that the reduced stress concentration may improve the fatigue strength (Walker et al., 1970). Also, Yoshimoto et al. found increasing the flank angle by 5° in the pressure flank and reducing the pitch by 0.15% for M24 may improve the fatigue strength of bolt nut connections (Yoshimoto et al., 1978). These studies discussed the root radius effect on the fatigue strength without providing pitch difference. Instead, the authors previously discussed the pitch difference effect without considering thread shape modification. (Chen et al., 2015, 2016; Noda et al., 2016). Nishida studied the effect of root radius on fatigue strength; in his research, three different kinds of root radii 0.3, 0.5, and 0.7 mm of JIS M25 are used, and found that the root radius has little effect on fatigue strength and fatigue limit.

Nishida et al. (1997) also designed a new bolt named CD bolt (Critical Design for Fracture) and indicated that the fatigue life can be improved almost twice. Honarmandi et al. (2005) studied the effects of four geometrical parameters, namely, flank type, root radius, thread run-out, and head fillet radius on fatigue life. Honarmandi et al. (2005) also studied the CD bolt. Noda et al. (2008) clarified the CD bolt geometries to improve fatigue strength. Majzoobi et al. (2005) investigated the effect of nut shape with/without washer on the fatigue life of bolt nut connections and reported that the fatigue life of slotted tapered nut is much longer than that of hexagonal nut and tapered nut. Wang et al. (2020) indicated that the shear capacity of Hollo bolts is much higher than the normal bolt but the fatigue life is slightly higher than that of normal bolts. To improve both anti-loosening and fatigue strength, Shinbutsu et al. (2017) designed a new bolt fastener with the double nut structure, named DTB-II, having practically sufficient tensile strength and excellent anti-loosening performance. The authors’ previous researches indicated that a slight pitch difference in bolt–nut improves the fatigue life and the anti-loosening performance (Chen et al., 2015, 2016; Noda et al., 2016). For JIS M16 bolt nut connections, the pitch difference α = 15 μm improves the fatigue life increases by about 1.5 times compared to α = 0 although the fatigue limit does not increase (Chen et al., 2016).

Table 1 shows a comparison of some special bolt–nut connections. Most special bolt–nuts have either more components or very special geometry, leading to a complex manufacturing process and a high cost, which is usually more than three times of the normal bolt–nut. The suggested nut in this study can be manufactured as the same way as the normal nut, and the cost is predicted to be about 1.5 times of the normal nut considering the modification of thread tap as well as the checking procedure on the pitch difference. To improve the fatigue limit as well as the fatigue life, in this study, bolt nut connections are designed by enlarging root radius and providing pitch difference. The details of fatigue test specimens and experimental conditions are described in the next section.

Table 1.

Comparison of some special bolt–nut connections.

Method	Anti-loosening performance improvement	Fatigue limit improvement	Fatigue life improvement	Machinability	Low cost
This study
Pitch difference nut (Chen et al., 2015; Noda et al., 2016)
SPR (Noda et al., 2008)
Hard lock nut (Wakabayashi, 2002)
CD bolt (Nishida et al., 1997)
Low strength nut (SNCM630→S20C) (Nishida, 2014)
Double structure bolt (DTB-II) (Shinbutsu et al., 2017)
Standard bolt–nut

: bad; : fair; : pretty; : remarkable.

Fatigue test specimen and experimental conditions

This study focuses on high strength Japanese Industrial Standards (JIS) M16 bolt nut connections. Three kinds of new bolt shapes whose root radii are larger than that of JIS M16 bolt are prepared, aiming at improving both the fatigue life and fatigue limit. Japanese Industrial Standards (JIS) specifies the standards used for industrial activities in Japan and ISO, JIS, and DIN standards are based upon the metric system and are closely related. The screw thread specifications based on JIS also apply to ISO and DIN threads. Although the nut height dimension H = 16 mm in Figure 1 varies depending on the standard, when there is no pitch difference between the bolt and the nut, the effect of nut height on fatigue strength is negligible. This study focuses on the pitch difference nut when H = 16 mm is based on JIS M16. The effect of the nut height on the fatigue life and anti-loosening performance for the nut height under a small pitch difference will be discussed in further studies.

Figure 1(a) illustrates a bolted joint whose thread is numbered as −3, −2, ... 7, 8 from the bolt head side to the other side. As shown in Figure 1(b), since nut chamfers are commonly used, the threads are neglected at two ends, as shown in Figure 1(a) in the simulation. Instead of the standard JIS M16 pitch p = 2000 $μ m$ , the nut pitch is α $μ m$ larger than that of the bolt pitch p = 2000 $μ m$ in this study. When α = 0, the thread contact is shown in Figure 1(c) where the largest stress appears at the root of No.2 thread. Instead, when α > 0, the thread contact is shown in Figure 1(d) where the largest stress appears at the root of No.7 thread (Chen et al., 2015, 2016; Noda et al., 2016). In previous studies, Patters showed that the longer nut has the longer the fatigue (Patterson, 1990). Griza described that bolts with longer length tend to have high fatigue strength for bolt joints under the same tightening torque (Griza et al., 2013). According to JIS B 1181 - 1993 Type 2, Grade A, Hexagon Nuts, the nut height is between 15.7 and 16.4 mm. To eliminate the effect of nut length, all the height of the nuts used in this study are 16 mm. Figure 2 illustrates the shape and dimension of the thread designed in the study. Instead of the standard root radius JIS M16 $ρ = ρ_{0} =$ 0.29 $mm$ , the newly designed bolt specimens have a root radius $ρ = 2 ρ_{0}$ and $ρ = 3 ρ_{0}$ . To evaluate the stress reduction conveniently, we consider the stress concentration factor $K_{t}$ of a round bar having a circumferential 60° notch shown in Figure 2(d) whose $K_{t}$ formula is available (Noda et al., 1999, 2006). The stress concentration factor $K_{t}$ decreases from 4.53 to 2.90 by increasing the thread root radius from $ρ = ρ_{0}$ to $ρ = 2 ρ_{0}$ , and the stress concentration factor $K_{t}$ is reduced to 2.4 when the bolt root radius becomes $ρ = 3 ρ_{0}$ .

Figure 2.

Three types of bolt specimens with different thread shapes: (a) ρ = ρ₀ (K_t = 4.53 in Figure 2(d)), (b) ρ = 2ρ₀ (K_t = 2.90 in Figure 2(d)), (c) ρ = 3ρ₀ (K_t = 2.40 in Figure 2(d)), and (d) notched bar.

Figure 3 illustrates the assembled state of a bolt nut connection in the fatigue experiment. As shown in Figure 3, the bolt head side and the nut side are in the two different frames on the device. The lower side frame is fixed on the machine, and the upper side subjects to a cyclic loading F. Table 2 shows the bolt–nut materials JIS SCM435 steel and JIS S45C steel. Similar to the previous papers (Chen et al., 2016; Noda et al., 2016), a 392 kN (400 ton) servo fatigue tester is used, the frequency is set to 5 or 10 Hz, and the mean load $F_{m}$ is 30 kN. Esmaeili et al. found that the tightening torque T can affect the fatigue strength of bolt joints (Esmaeili et al., 2014). To eliminate the effect of tightening torque T, in the fatigue experiments in this study, the tightening torques T = 0. Table 3 shows the loading conditions set in the experiments. The S-N curve was obtained from the experimental results under five levels of stress amplitude with a fatigue limit of 2 × 10⁶ cycles.

Table 2.

Material properties of bolt and nut.

	Strength grade	Elastic modulus E (GPa)	Poisson’s ratio ν	Yield strength $σ_{sl}$ (MPa)	Tensile strength $σ_{B}$ (MPa)
SCM435 (Bolt)	10.9	206	0.3	800	1200
S45C (Nut)	–	206	0.3	530	980

Table 3.

Experimental conditions.

Load (kN)		Stress (MPa)		$R = \frac{σ_{m} - σ_{a}}{σ_{m} + σ_{a}}$
Mean load	Load amplitude	Mean stress $σ_{m}$	Stress amplitude $σ_{a}$	$R = \frac{σ_{m} - σ_{a}}{σ_{m} + σ_{a}}$
30	22.6	213	160	0.14
30	18.3	213	130	0.24
30	14.1	213	100	0.36
30	11.3	213	80	0.45
30	8.5	213	60	0.56

Figure 3.

Schematic illustration of fatigue test.

Fatigue strength improvement due to enlarged root radius

Previously, Walker et al. (1970), Yoshimoto et al. (1978), and Nishida et al. (1997) studied the root radius effect on the fatigue strength without providing pitch difference. Those results did not show significant fatigue strength improvement although the stress concentration can be reduced. To clarify the root radius effect and to verify the fatigue limit improvement, a series of fatigue tests are conducted on the specimen in Figure 2. Figure 4 shows the S-N curve obtained by varying the thread root radius as $ρ = ρ_{0}$ , $ρ = 2 ρ_{0}$ , and $ρ = 3 ρ_{0}$ under the pitch difference (a) $α$ = 0 $μ m$ and (b) $α$ = 15 $μ m .$

Figure 4.

S-N curves for bolt nut connections by varying the thread root radius: (a) $α$ = 0 µm and (b) $α$ = 15 µm.

From Figure 4(a), when $α$ = 0 $μ m,$ it can be seen that by enlarging the root radius from standard $ρ = 1 ρ_{0}$ to $ρ = 2 ρ_{0}$ and $ρ = 3 ρ_{0}$ , the fatigue limit of the bolt nut connections increases from 60 to 80 MPa. Besides, the fatigue life of $ρ = 3 ρ_{0}$ is 3.00 times larger than that of $ρ = ρ_{0}$ . From Figure 4(b), when $α$ = 15 $μ m$ , it can be seen that the fatigue life of $ρ = 2 ρ_{0}$ was improved more than two times than that of $ρ = ρ_{0}$ when the stress amplitude is 130 MPa. When increasing the root radius from $ρ = ρ_{0}$ to $ρ = 2 ρ_{0}$ , the fatigue limit was improved from 60 MPa to 100 MPa. The fatigue limit increases to 110 MPa when the root radius increases to $ρ = 3 ρ_{0}$ . That is to say by enlarging the root radius to $ρ = 2 ρ_{0}$ and $ρ = 3 ρ_{0}$ the fatigue limit of the bolt nut connections can be improved by 67% and 83%, respectively. It may be concluded that enlarging bolt root radius and choosing suitable pitch differences may improve the fatigue life and fatigue limit of the bolt nut connections efficiently.

Fatigue strength improvement due to pitch difference

In the authors’ previous papers, the pitch difference effect on the fatigue strength was discussed experimentally and theoretically (Chen et al., 2016; Noda et al., 2016). In this paper, the pitch difference effect is discussed coupled with the root radius effect to improve both the fatigue life and the fatigue limit. Figure 5 shows the S-N curves of bolt nut connections when $α$ = 0 and 15 μm coupled with $ρ = 1 ρ_{0}$ and $ρ = 2 ρ_{0}$ . In Figure 6, when $ρ = ρ_{0}$ with increasing the pitch difference from α = 0 to 15 μm, the fatigue life can be significantly improved although the fatigue limit remains the same. For example, the fatigue life for a bolt with a root radius of $ρ = 2 ρ_{0}$ is twice that of the bolt with a root radius of $ρ = ρ_{0}$ when α = 0 μm, and besides, the fatigue limit is improved from 60 to 80 MPa by 33%. Moreover, by introducing a pitch difference of α = 15 μm coupled with enlarging the root radius to $ρ = 2 ρ_{0}$ , the fatigue limit is improved by 67%, and besides, the fatigue life is improved by 162% when the stress amplitude is 160 MPa. In a word, both the fatigue life and fatigue limit of bolt nut connections can be significantly enhanced by enlarging the root radius and introducing an appropriate pitch difference between the bolt and the nut at the same time.

Figure 5.

S-N curves for bolt nut connections when $α$ = 0 and 15 µm.

Figure 6.

Crack configuration observed from the fractured specimen surface: (a) ρ = 2ρ₀, α = 0 μm (σ_a = 130 MPa), (b) ρ = 3ρ₀, α = 0 μm (σ_a = 160 MPa), (c) ρ = 2ρ₀, α = 15 μm (σ_a = 130 MPa), (d) ρ = 3ρ₀, α = 15 μm (σ_a = 160 MPa), (e) ρ = ρ₀, α = 15 μm (σ_a = 130 MPa), and (f) ρ = 2ρ₀, α = 15 μm (σ_a = 130 MPa).

Stress and crack appearing at bolt threads

Crack observation

Figure 6(a) to (d) illustrates the crack configuration observed from the bolt outer surface after the fatigue experiments. When α = 0, as shown in Figure 6(a) and (b), a crack is observed only at No.1–No.3 threads. This is because No.1–No.3 threads carry most of the load as shown in Figure 1(c). Then, the crack initiated at No.1 or No.2 thread propagates and causes the final bolt fracture at the same thread without extending to other threads.

On the other hand, when α = 15 µm, as shown in Figure 6(c) and (d), cracks can be observed between No. 2 and No. 7. When α = 15 μm, cracks initiate at No. 6 and No. 7 threads because those threads carry most of the load as shown in Figure 1(d). After the cracks propagate at No.6 and No.7 threads, another crack initiates and propagates at No. 5, No. 4… toward No. 1 thread consecutively until causing the final bolt failure. The consecutive crack extension is caused by the crack initiation and propagation at each thread, which changes the thread contact state between the bolt and nut due to the pitch difference (Noda et al., 2016).

Figure 6(e) and (f) shows an example of the bolt thread surface with the nut cross section after the fatigue tests when $σ_{m} = 213 MPa$ and $σ_{a} = 130 MPa$ . As shown in Figure 2, increasing the root radius of the bolt thread necessitates truncating the internal thread, which may reduce the shear strength of the bolt thread. However, the thread surface status of $α = 15 μ m$ is nearly the same in Figure 6(e) when $ρ = 1 ρ_{0}$ and in Figure 6(f) when $ρ = 2 ρ_{0}$ . No thread stripping can be seen within this fatigue strength study by enlarging thread root radius. Although slight wear can be seen at the nut thread in Figure 6(f), the wear site is far away from the crack initiation site in Figure 6(c) at No.5 thread root. It should be noted that in this study, high strength nut with a height of 16 mm is used. For common M16, of which the nut height is 13 mm and the yield strength is lower than that used in this study, more works are needed to check if thread stripping occurs or not.

FEM modeling and boundary conditions

The stress at the thread root is calculated by applying the FEM software MSC.Marc/Mentat, 2012. Figure 7 shows an example of FEM mesh used in the axisymmetric analysis where 4-node QUAD elements are used, and the minimum element size near the bolt root is about 0.01 mm × 0.01 mm. The material properties are shown in Table 2. The elasto-plastic analysis of the FEM model is performed under the same loading conditions as the experiments. The bolt head side of the clamped body is fixed, and axial force $F$ = 30 ± 14.1 $kN$ is applied to the bolt head. Thus, the corresponding stress amplitude and the nominal stress amplitude at the bolt roots are $σ$ _a = 213 $MPa$ and $σ$ _m = 100 $MPa$ , respectively.

Figure 7.

Axisymmetric FEM model when ρ = 1ρ₀, $α$ = 0.

Stress amplitude versus mean stress under no pitch difference

Figure 8 shows the endurance limit diagrams with the Soderberg line of the plane specimen. Here, $σ$ _w represents the fatigue strength under reversal stress $σ$ _m = 0 and $σ$ _sl is the yield strength. The maximum tangential stress at each thread obtained by the FEM is plotted in terms of the stress amplitude $σ$ _a = ( $σ$ _max − $σ$ _min)/2 and the mean stress $σ$ _m = ( $σ$ _max + $σ$ _min)/2. Due to no stress gradient in plain specimens, the fracture stress in notched specimens is always larger than that in the plain specimens. Therefore, it should be noted that the stress data plotted beyond the line (( $σ$ _m $/ σ$ _sl)+ $(σ$ _a $/ σ$ _w) > 1) does not represent the real fracture at the bolt thread. Several previous studies investigated the endurance limit after a certain amount of mean stress level (Burguete et al., 1995; Gunn, 1955; Hashimura et al., 2019; Munn et al., 2010). In this paper, the Soderberg line of plain specimen is used to discuss the relative hazard at the bolt roots.

Figure 8.

Endurance limit diagram when F_a = 14.1 kN: (a) ρ = ρ₀, 2ρ₀ when $α$ = 0 µm and (b) ρ = ρ₀, 2ρ₀ when $α$ = 15 µm.

When $α$ = 0, from Figure 8(a), the most dangerous part locates at No.2 thread. By enlarging the root radius from $ρ = ρ_{0}$ to $ρ = 2 ρ_{0}$ , at No.2 thread, the mean stress decreases by about 6% and the stress amplitude decreases by about 38%. When α = 15 μm, from Figure 8(b), the most dangerous part locates around No.6 and No.7 threads of the bolt. By enlarging the root radius from $ρ = ρ_{0}$ to $ρ = 2 ρ_{0},$ at No. 6 thread, the mean stress decreases by about 8% and the stress amplitude decreases by about 26%. The same trend can be found when the pitch difference $α$ = 33 μm. Those analytical results in Figure 8 are in good agreement with the fracture surface observation in Figure 6. For bolt nut connections without pitch difference $α$ = 0, both initial cracks and final breaks of the bolt occur at No. 2 thread close to the bolt head side. For bolt nut connections with pitch differences $α \neq$ 0, the initial crack occurs No. 6 and No. 7 threads far from the bolt head and then extend to No. 2 thread near the bolt head side, where the final break occurs.

Stress amplitude versus mean stress under pitch differences

Figure 9 is an endurance limit diagram obtained using the analytical results. Here, the mean stresses and stress amplitude under two different kinds of pitch difference when root radius $ρ = ρ_{0}$ and $ρ = 2 ρ_{0}$ are compared. From Figure 9, it can be seen that when increasing the pitch difference from α = 15–33 μm, the stress distributions for the bolt with a root radius of $ρ = ρ_{0}$ and $ρ = 2 ρ_{0}$ are the same. The maximum stress amplitude and the mean stress occurred at the side far from the bolt head are consistent with the experimental results.

Figure 9.

Endurance limit diagram when F_a = 14.1 kN: (a) ρ = ρ₀ when α = 15 and 33 µm and (b) ρ = 2ρ₀ when α = 15 and 33 µm.

Anti-loosening performance under enlarged root radius

As described in the above sections, the fatigue limit improvement was experimentally verified, and the stress reduction was analytically clarified by enlarging the thread root radius $ρ$ and providing suitable pitch difference $α$ . However, those results can be expected only when the nut loosening does not happen. In this section, therefore, anti-loosening is confirmed by applying three-dimensional FEM simulation to the nut screwing and tightening processes.

Analysis method

Figure 10 illustrates the nut screwing process from A to D, and the nut tightening process from E to G. Figure 10 also illustrates the nut untightening process from Gu to Eu and the nut unscrewing process from Du to Au. When the pitch difference $α$ = 0, the prevailing torque $T_{P}$ required in the nut screwing is zero as $T_{P}$ = 0, but when $α \neq 0,$ the prevailing torque $T_{P}$ is no longer 0 as $T_{P} \neq 0$ . This prevailing torque $T_{P}$ is commonly used to characterize the anti-loosening performance of bolt nut connections. As an example, Eccles et al. explained that all prevailing torque type nuts detach from the bolt when the external load is larger than the bolt tightening force (Eccles et al., 2010). Under a larger tightening force, therefore, the prevailing torque type nuts can be used safer.

Figure 10.

Illustration of the screwing, tightening, untightening, and unscrewing of pitch difference nut.

This paper discusses the effect of the root radius on the anti-loosening performance focusing on the prevailing torque obtained from the three-dimensional FEM simulation. To clarify the impact of root radius on loosening performance, the dimensions of M16 bolt–nut are chosen to be the same. The length of the bolts used for anti-loosening experiments is 76 mm, and the thread length and the grip length are 42 and 18 mm, respectively. The recent study showed the 3D FEM results are in good agreement with the experimental results for JIS M12 bolt–nuts (Noda et al., 2019).

Three-dimensional FEM models were simulated by ANSYS WORKBENCH 16.2. As shown in Figure 11, the bolt head and the nut shapes are simplified by cylinders to save the calculation time. Figure 11(a) shows the FEM mesh for M16 bolt–nut when the pitch difference $α$ = 0 and the root radius $ρ$ = $ρ$ ₀ with the total element number 17,5907 and the total node number 34,6712. The results are confirmed to be accurate enough since the maximum relative error is less than a few percent by applying the refined smaller mesh to the contact surface with a much larger calculation time. Figure 11(b) shows the boundary condition assuming the initial nut location at 0.1 mm away from the clamped body. The bolt head and the left side of the clamped body are fixed, and the nut is screwed onto the bolt in the clockwise tightening direction. The prevailing torque $T_{P}$ can be obtained before the nut contacts to the clamped body. Since the pitch of a standard JIS M16 bolt is 2 mm, when rotating the nut by the angle $θ$ = 18°, the nut moves to the left by 0.1 mm and touches the clamped body. The material property in Table 2 is used in FEM analysis, assuming the clamped body is also SCM435. Bilinear elasto-plastic stress–strain relations are applied instead of multi-linear relations to save the calculation time. Then, the Newton-Raphson approach is used to solve nonlinear problems. The upper thread of the bolt is supposed to contact to the lower thread of the nut, and the lower thread of the bolt is supposed to contact to the upper thread of the nut. The friction coefficient between the threads is denoted as $μ_{s}$ , and the friction coefficient between the nut and the clamped body is denoted as $μ_{w}$ . Previous studies have proven that the thread friction coefficient $μ_{s}$ is in the range 0.11–0.15, and the underhead friction coefficient $μ_{w}$ is in the range 0.16–0.18 (Udagawa, 1982). In this study, $μ_{s} = 0.12$ and $μ_{w} = 0.17$ are used in the FEM analysis by considering the previous study where the molybdenum disulfide paste spray was used in the nut screwing process (Noda et al., 2019). Elliott et al. studied the behavior and strength of bolted connections failing in shear, and found that the shear resistance of a bolted connection is not significantly affected by the hole clearances (Elliott et al., 2019). In this section of this study, all the diameters of bolt holes are 16.8 mm.

Figure 11.

FEM model and boundary conditions for tightening and untightening process: (a) FEM mesh model and (b) boundary conditions.

After the nut contacts the clamped body, the tightening force $F_{t}$ v.s. tightening torque $T relation, that is the F_{t} - T relation can be obtained$ from point E to point G in Figure 10. After the tightening process is finished, the loosening process can be analyzed subsequently from point G to point E as shown in Figure 10. The nut rotation angle θ to produce specified fastening force $F_{t 25 %}$ , which is 25% bolt axial yielding force, cannot be known beforehand. Therefore, the analysis is performed as shown in the following step (i) and step (ii).

(i) Applying a sufficiently large rotation angle $θ$ to the nut. When the rotation angle $θ$ is sufficiently large, the tensile stress of the bolt may exceed the yield stress. Thus, a rotation angle $θ_{25 %}$ corresponding to the fastening force $F_{t 25 %}$ can be obtained.

(ii) Tightening the nut by using the rotation angle $θ_{25 %}$ obtained in the process (i). After the tightening force reaches $F_{t 25 %}$ , untighten the nut by an anti-tightening angle $- θ_{25 %}$ .

Results and discussion for T– $θ$ relation

Figure 12 shows the T– $θ$ relation, that is, the tightening torque T versus the nut rotation angle $θ$ relation during the screwing and tightening processes when the bolt root radius $ρ$ = $1 ρ$ ₀. During the nut screwing, $T_{P} = 0$ when $α$ =0; however, $T_{P} \neq 0$ when $α \neq$ 0. During the nut tightening, it should be noted that the gradient $dT / d θ$ varies depending on the pitch difference $α . T$ he gradient $dT / d θ$ when $α$ = 0 is larger than the gradient $dT / d θ$ when $α$ = 33 µm. During the entire nut tightening process, $dT / d θ = const$ when $α$ = 0; however, when $α \neq$ 0, the gradient $dT / d θ = const$ but the constant value is slightly changed at point F₂ as in Figure 12. In Figure 12, the slope ratio of the lines E₁G₁, E₂F₂, and F₂G₂ are denoted by $\tan θ_{1}$ , $\tan θ_{2}$ , and $\tan θ_{3}$ , respectively. It is found that the angles have the relation $θ_{1} > θ_{3} > θ_{2}$ . This can be explained in the following way.

Figure 12.

Relationship between the and tightening torque T and the nut rotation angle θ during the screwing and tightening process when $ρ$ = $1 ρ$ ₀.

Figure 13 illustrates why the T– $θ$ relation can be depicted as shown in Figure 12 by illustrating the thread contact status. From Figure 13, it is seen that when $α$ = 0, the slope ratio is closely related to the clamped body deformation whose height is $H_{c}$ . This is because No.1 or No.2 nut thread near the bolt head may carry most of the load. When $α \neq$ 0, however, the slope ratio is closely related to the deformation due to the clamped body and the nut whose height is $H_{c} + H_{n}$ . This is because No.6 or No.7 nut thread near the nut end may carry most of the load. At the nut position from E₂ to the nut position F₂ in Figure 13(b), the thread contact status changes at No. 1 or No. 2 nut thread. This is the reason why the slope slightly changes at Point F₂ in Figure 12. In the case of Figure 12, the angle $θ_{1} = 165 degree > θ_{3} = 157 degree > θ_{2} = 156 degree .$

Figure 13.

Illustration why the T–θ relation can be depicted as shown in Figure 12 during the tightening and untightening processes: (a) α = 0 µm and (b) α > 0 µm.

Results and discussion for F_t–T relation

Figure 14 shows the tightening force $F_{t}$ versus the torque $T$ relation when $ρ = 1 ρ_{0}$ and $ρ = 2 ρ_{0}$ with α = 0 and α = 33 µm. In the previous study, the authors conducted impact-vibration test prescribed in NAS3350 (National Aerospace Standard) for M16 pitch difference nut. Then, anti-loosening is confirmed for $α \geq$ 33 µm showing the prevailing torque $T_{P}$ = 30–39 Nm for JIS M16 (Chen et al., 2016). Such good anti-loosening was also reported for “U-nut” widely used in the world although U-nut has relatively smaller prevailing torque $T_{P}$ = 1.5 Nm for JIS M12 (Fuji Seimitsu Co., Ltd, 2018). Another special nut named “Super Slit Nut” is also accessible in the market showing $T_{P}$ = 15–19 Nm for JIS M16 with good anti-loosening (Izumi et al., 2005b). As shown in Figure 14, the FEM analysis shows that the pitch difference nut has $T_{P}$ = 19 Nm even when $ρ$ = $2 ρ$ ₀, α = 33 μm. Considering other special nuts such as U-nut and Super Slit Nut, good anti-loosening performance can be expected for the enlarged root radius of pitch difference nut considered in this paper.

Figure 14.

Tightening torque T versus clamping force $F_{t}$ : (a) ρ = 1ρ₀ and (b) ρ = 2ρ₀.

As shown in Figure 14, when α = 0, the tightening force $F_{t}$ appears once $T$ applies. However, when α = 33 µm, the tightening force $F_{t}$ appears only after the torque exceeds the prevailing torque as $T \geq T_{p}$ . After the tightening force reaches $F_{t}$ = $F_{t 25 %}$ producing 25% of the bolt yield stress $σ_{sl}$ = 800 MPa (see Table 2), untightening process starts by reversing torque $T$ . During this process, tightening force $F_{t}$ decreases with decreasing the magnitude of $T .$ When α = 33 μm the magnitude of $T$ is always larger than that of α = 0. And even when $F_{t}$ = 0 the reverse torque is not zero as $T$ = $T_{p}^{u}$ > 0, which is named as residual prevailing torque and discussed in the recent paper (Noda et al., 2020a). In Figure 14, the light green zone illustrates the difference between α = 0 and 33 µm defined in equation (1). Such torque difference in equation (1) can be regarded as the loosening resistance torque $T_{p}^{u}$ contributing to anti-loosening. (Noda et al., 2020a)

T_{P}^{u} = {| T_{P} |}_{α \neq 0} - {| T_{P} |}_{α = 0}

(1)

As shown in Figure 14(b), when $ρ = 2 ρ_{0}$ and α = 33 μm, the light green zone region $T_{p}^{u}$ in equation (1) is comparatively smaller. However both $T_{p}$ and $T_{p}^{u}$ appear in a similar way of $ρ = 1 ρ_{0}$ and α = 33 μm in Figure 14(a) and those values $T_{p}$ and $T_{p}^{u}$ are not smaller compared to U- nut and Super Slit Nut as described above. The recent study showed that the loosening resistance torque $T_{p}^{u}$ increases with increasing the pitch difference α as well as the prevailing torque $T_{p}$ (Noda et al., 2020b, 2020c). Therefore, Figure 14(b) suggests that even when the bolt root radius is enlarged, the anti-loosening performance can be expected under the suitable pitch difference.

It should be noted that common bolted joints α = 0 are loosening resistant only when sufficient tightening force $F_{t}$ is provided; therefore, for example, no loosening resistance at $F_{t}$ = 0 in Figure 14. Instead, α = 33 µm has larger loosening resistance as $T_{P}^{u} = {| T_{P} |}_{α \neq 0} - {| T_{P} |}_{α = 0} \geq 0$ and even at $F_{t}$ = 0, $T_{P}^{u} = 10 Nm$ in Figure 14.

Conclusion

Since the authors’ previous research showed that a slight pitch difference might improve the fatigue life, in this paper, the effect of root radius on the fatigue limit was investigated by varying the pitch difference in JIS M16 bolt–nut connections. Although increasing the root radius of the bolt thread necessitates truncating the internal thread, which may reduce the shear strength of the bolt thread, no thread stripping can be seen for the enlarged root radius of the bolt within this experiment. Conclusions can be summarized as follows.

Fatigue tests were performed when the root radius is twice or three times larger than the standard bolt–nut root radius. This experiment verified that the fatigue limit could be improved by more than 30% by enlarging the bolt root radius. The FEM analysis showed that when the bolt root radius is enlarged, both stress amplitude and mean stress at each thread root can be reduced significantly.

When the root radius is larger than the standard one, the fatigue limit can be improved further by more than 25% by introducing a suitable pitch difference. When there is no pitch difference, the crack initiation always occurs at No.1 or No.2 threads close to the bolt head causing the final failure at the same thread. When there is a pitch difference, the crack initiation occurs at No.6 or No.7 threads far away from the bolt head. Then the crack initiation and propagation expand toward the bolt head until the final failure occurs at No.1 or No.2 threads.

By varying the root radius $ρ$ and the pitch differences α, $F_{t} - T$ relation, that is, the clamping force $F_{t}$ versus the tightening torque T relation, was clarified by applying 3D FEM. When $ρ = 2 ρ_{0}$ and α = 33 µm, it was confirmed that the prevailing torque $T_{P}$ = 19Nm and the residual prevailing torque $T_{P}^{u} = 10 Nm$ even at $F_{t}$ = 0. Since those values are not smaller compared to other special nuts, good anti-loosening performance can be expected for the enlarged root radius of bolt–nut connections under the suitable pitch difference.

Footnotes

Appendix

Acknowledgements

The authors wish to express our thanks to the members of our group Mr. Tomohiko Matsunari and Mr. Kosuke Tateishi, for their kind support of the experimental studies.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by the Japanese Ministry of Education research expenses [grant number 000591, Yasushi TAKASE]; and Japan Keirin Autorace foundation for the Advancement [grant number 2019M-190, Nao-Aki NODA].

ORCID iD

Biao Wang

References

Burguete

Patterson

(1995) The effect of mean stress on the fatigue limit of high tensile bolts. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 209(4): 257–262.

Chakherlou

Maleki

Aghdam

, et al. (2013) Effect of bolt clamping force on the fracture strength of mixed mode fracture in an edge crack with different sizes: Experimental and numerical investigations. Materials & Design 45: 430–439.

Chen

Shimizu

Masuda

(2012) Relation between thread deformation and anti-loosening effect for nut with circumference slits. Transactions of the Japan Society of Mechanical Engineers Series A 78(788): 390–402.

Chen

Noda

Wahab

, et al. (2015) Fatigue failure analysis for bolt–nut connections having slight pitch differences using experimental and finite element methods. Acta Polytechnica Hungarica 12(8): 61–79.

Chen

Noda

Wahab

, et al. (2016) Fatigue life improvement by slight pitch difference in bolt–nut connections. Journal of the Chinese Society of Mechanical Engineers 37(1): 11–19.

Eccles

Sherrington

Arnell

(2010) Towards an understanding of the loosening characteristics of prevailing torque nuts. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224(2): 483–495.

Elliott

Teh

Ahmed

(2019) Behaviour and strength of bolted connections failing in shear. Journal of Constructional Steel Research 153: 320–329.

Esmaeili

Zehsaz

Chakherlou

(2014) Experimental investigations on the effects of torque tightening on the fatigue strength of double-lap simple bolted and hybrid (bolted/bonded) joints. Strain 50(4): 347–354.

Fuji Seimitsu Co., Ltd (2018) U-NUT product information. Available at: http://www.fun.co.jp/products/detail.php?no=Ng== (accessed 5 August 2018).

10.

Griza

da Silva

dos Santos

, et al. (2013) The effect of bolt length in the fatigue strength of M24× 3 bolt studs. Engineering Failure Analysis 34: 397–406.

11.

Gunn

(1955) Effect of yielding on the fatigue properties of test pieces containing stress concentrations. The Aeronautical Quarterly 6(4): 277–294.

12.

Hashimura

Kamibeppu

Nutahara

, et al. (2019) Effects of clamp force on fatigue strength of aluminum alloy bolts. Procedia Structural Integrity 19: 204–213.

13.

Hirai

Uno

(2005) Fatigue strength of super high strength bolt. Transaction of the Architectural Institute of Japan 595: 117–122.

14.

Honarmandi

Behdinan

(2005) Elasto-plastic fatigue life improvement of bolted joints and introducing FBI method. Mechanics Based Design of Structures and Machines 33(3–4): 311–330.

15.

Izumi

Yokoyama

Iwasaki

, et al. (2005a) Three-dimensional finite element analysis of tightening and loosening mechanism of threaded fastener. Engineering Failure Analysis 12(4): 604–615.

16.

Izumi

Yokoyama

Teraoka

, et al. (2005b) Verification of anti-loosening performance of super slit nut by finite element method. Transactions of the Japan Society of Mechanical Engineers Part A 17(3): 387–393. (in Japanese).

17.

Junker

(1972) Criteria for self-loosening of fasteners under vibration. Aircraft Engineering and Aerospace Technology 44(10): 14–16.

18.

Lee

Kim

Han

(2014) Mechanism for reducing stress concentrations in bolt–nut connectors. International Journal of Precision Engineering and Manufacturing 15(7): 1337–1343.

19.

Zhang

, et al. (2014) Optimization study of C/SiC threaded joints. International Journal of Applied Ceramic Technology 11(2): 289–293.

20.

Majzoobi

Farrahi

Hardy

, et al. (2005) Experimental results and finite-element predictions of the effect of nut geometry, washer and Teflon tape on the fatigue life of bolts. Fatigue & Fracture of Engineering Materials & Structures 28(6): 557–564.

21.

Munn

(2010) Investigation into the effect of thread root condition on the high cycle fatigue performance of a metric threaded fastener. In: ASME 2010 pressure vessels and piping division/K-PVP conference, Bellevue, Washington, USA, 18–22 July, pp. 381–395. American Society of Mechanical Engineers Digital Collection.

22.

Nishida

(2014) Failure Analysis in Engineering Applications. Oxford: Elsevier.

23.

Nishida

Urashima

Masumoto

; Nippon Steel Corp and Nippon Steel Bolten Co Ltd (1980) Screwed connection having improved fatigue strength. U.S. Patent 4,189,975.

24.

Nishida

Urashima

Tamasaki

(1997) A new method for fatigue life improvement of screws. European Structural Integrity Society 22: 215–225.

25.

Noda

Chen

Sano

, et al. (2016) Effect of pitch difference between the bolt–nut connections upon the anti-loosening performance and fatigue life. Materials & Design 96: 476–489.

26.

Noda

Kuhara

Xiao

, et al. (2008) Stress reduction effect and anti-loosening performance of outer cap nut by finite element method. Journal of Solid Mechanics and Materials Engineering 2(6): 801–811.

27.

Noda

Liu

Sano

(2019) Three-dimensional finite element analysis for prevailing torque in the screwing process of bolt and nut connections with pitch difference. Transactions of the JSME 85(876): 19-00149. (in Japanese).

28.

Noda

Liu

Sano

, et al. (2020a) Prevailing torque and residual prevailing torque of bolt–nut connections having slight pitch difference. Mechanics Based Design of Structures and Machines. Epub ahead of print 10 June 2020. DOI: 10.1080/15397734.2020.1768114.

29.

Noda

Liu

Sano

, et al. (2020b) Three-dimensional finite element analysis for prevailing torque of bolt–nut connection having slight pitch difference. Journal of Mechanical Science and Technology 34(6): 2469–2476.

30.

Noda

Liu

Sano

, et al. (2020c) Three-dimensional finite element analysis of tightening and untightening process of bolt and nut connections with slight pitch difference. The Japan Society of Mechanical Engineers 86(886): 1–12. (in Japanese).

31.

Noda

Takase

(1999) Stress concentration formulae useful for any shape of notch in a round test specimen under tension and under bending. Fatigue & Fracture of Engineering Materials & Structures 22(12): 1071–1082.

32.

Noda

Takase

(2006) Stress concentration formula useful for all notch shape in a round bar (comparison between torsion, tension and bending). International Journal of Fatigue 28(2): 151–163.

33.

Noda

Xiao

Kuhara

(2011) The reduction of stress concentration by tapering threads. Journal of Solid Mechanics and Materials Engineering 5(8): 397–408.

34.

Panja

Das

(2017) Development of an anti-loosening fastener and comparing its performance with different other threaded fasteners. Sādhanā 42(10): 1793–1801.

35.

Patterson

(1990) A comparative study of methods for estimating bolt fatigue limits. Fatigue & Fracture of Engineering Materials & Structures 13(1): 59–81.

36.

Pedersen

(2013) Overall bolt stress optimization. The Journal of Strain Analysis for Engineering Design 48(3): 155–165.

37.

Ramey

Jenkins

(1995) Experimental analysis of thread movement in bolted connections due to vibrations. Final report, Auburn University, Auburn.

38.

Ranjan

BSC

Vikranth

Ghosal

(2013) A novel prevailing torque threaded fastener and its analysis. Journal of Mechanical Design 135(10): 101007-1∼9.

39.

Roy

Lau

Ting

TCH

, et al. (2019) Experiments and finite element modelling of screw pattern of self-drilling screw connections for high strength cold-formed steel. Thin-Walled Structures 145: 106393.

40.

Roy

Lim

Yousefi

, et al. (2018) Low fatigue response of crest-fixed cold-formed steel drape curved roof claddings. In: Proceeding of international specialty conference on cold-formed steel structures, St. louis, Missouri, USA, 7–8 November, pp. 713–727.

41.

Sase

Koga

Nishioka

, et al. (1996) Evaluation of anti-loosening nuts for screw fasteners. Journal of Materials Processing Technology 56(1–4): 321–332.

42.

Shinbutsu

Amano

Takemasu

, et al. (2017) Thread rolling and performance evaluations of a new anti-loosening double thread bolt combining a single thread and multiple threads. Procedia Engineering 207: 603–608.

43.

Sivapathasundaram

Mahendran

(2018) Pull-out capacity of multiple screw fastener connections in cold-formed steel roof battens. Journal of Constructional Steel Research 144: 40–52.

44.

Teh

(2017) Ultimate tilt-bearing capacity of bolted connections in cold-reduced steel sheets. Journal of Structural Engineering 143(4): 04016206.

45.

Udagawa

(1982) Survey report for standardization on fastening performance of high strength bolts (Report V). Journal of the Japan Research Institute for Screw Threads and Fasteners 13(5): 165–172 (in Japanese).

46.

Ujita

(2010) Strength design of machine/structure and practical examples II: Railway rolling stock body structure. Journal of the Society of Materials Science, Japan 59(7): 575–582.

47.

Wakabayashi

(2002) Hard lock nut. Japan Patent. Hard Lock Kogyo. 195236.

48.

Walker

Finkelston

(1970) Effect of basic thread parameters on fatigue life. SAE Transactions 25: 29–39.

49.

Wang

, et al. (2020) Fatigue behaviour of stainless steel bolts in tension and shear under constant-amplitude loading. International Journal of Fatigue 133: 105401.

50.

Xiao

Noda

Kuhara

, et al. (2010) Optimum design of thin walled tube on the mechanical performance of super stud bolt. Journal of Solid Mechanics and Materials Engineering 4(10): 1455–1466.

51.

Yoshimoto

Hongo

Maruyama

, et al. (1978) Investigation on the screw threads profile to improve the fatigue strength. Research Laboratory of Precision Machinery and Electronics, Bulletin 44(528): 1508–1513. (in Japanese)

52.

Zhou

Liu

Ouyang

, et al. (2019) Self-loosening behavior of bolted joints subjected to dynamic shear load. International Journal of Modern Physics B 33(01n03): 1940009.

53.

Zhou

Zhang

, et al. (2015) Load distribution in threads of porous metal–ceramic functionally graded composite joints subjected to thermomechanical loading. Composite Structures 134: 680–688.

Effects of root radius and pitch difference on fatigue strength and anti-loosening performance for high strength bolt–nut connections

Abstract

Keywords

Introduction

Fatigue strength improvement

Previous studies for bolt fatigue strength

Fatigue test specimen and experimental conditions

Fatigue strength improvement due to enlarged root radius

Fatigue strength improvement due to pitch difference

Stress and crack appearing at bolt threads

Crack observation

FEM modeling and boundary conditions

Stress amplitude versus mean stress under no pitch difference

Stress amplitude versus mean stress under pitch differences

Anti-loosening performance under enlarged root radius

Analysis method

Results and discussion for T– θ relation

Results and discussion for Ft–T relation

Conclusion

Footnotes

Appendix

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References

Results and discussion for T– $θ$ relation

Results and discussion for F_t–T relation