Proceedings of the 5th International Conference on Integrity-Reliability-Failure, Porto/Portugal 24-28 July 2016

Editors J.F. Silva Gomes and S.A. Meguid

Publ. INEGI/FEUP (2016)

-1203-

PAPER REF: 6299

FATIGUE MECHANISM OF BOLTED JOINTS UNDER MULTI-AXIAL

VIBRATION

Shinji Hashimura1(*)

, Tomotaka Tanaka2, Takefumi Otsu

3

1Department of Engineering Science and Mechanics, Shibaura Institute of Technology, Tokyo, Japan 2Dep. Mechanical Engineering, Graduate School of Nagaoka University of Technology, Nagaoka, Japan 3Department of Mechanical Engineering, Kurume National College of Technology, Fukuoka, Japan (*)

Email: hasimura@shibaura-it.ac.jp

ABSTRACT

Many bolted joints and bolt/nut assemblies are used in various fields. However it is also

undeniable that there are many accidents caused by bolts failure. Many fatigue tests of bolted

joints which are subjected to axial vibration have been conducted. Fatigue tests of bolted

joints which are subjected to transverse vibration have been also conducted in recent years.

However the actual bolted joints are subjected to complex vibration. In this study, fatigue

tests of bolted joints which was simultaneously subjected to axial vibration and transverse

vibration were performed to reveal fatigue phenomenon of a bolted joint subjected to complex

vibrations. In the experiments, the amplitude of axial vibration force was a constant and the

amplitude of transverse vibration force was only changed. The experiments for the bolted

joint subjected to only transverse vibration were also conducted to compare. The results

showed that the apparent transverse fatigue limits, the highest amplitude of transverse

vibration which the bolted joint did not fail due to fatigue, were different according to the

phase difference of superposition of vibrations. If the two vibrations were superposed with the

same phase, the apparent transverse fatigue limit decreased because principal stress at the first

thread root became the highest.

Keywords: Bolted joint, fatigue mechanism, axial vibration, transverse vibration.

INTRODUCTION

Many Bolted joints and threaded fasteners are used for assembling machines and structures.

But it is also undeniable that there are many accidents caused by bolts failure (Ministry of

Land, 2013). A lot of researches about fatigue of bolted joints have been investigated so far.

Especially, fatigue characteristics and mechanisms for the bolted joints subjected to axial

vibration had been revealed (Stephens, 2007) (Yoshimoto, 1984) (Ohashi, 1985) (Ohashi,

1994) (Alexander, 2000). In recent years, investigation of fatigue and self-loosening of bolted

joints which are subjected to transverse vibration have been also conducted (Jiang, 2003)

(Jiang, 2002) (Hashimura, 2006) (Hashimura, 2007). In the most cases, fatigue cracks occurs

at a root of the first thread, which external threads begin to engage internal threads. A nominal

stress at the root of the first thread can be easily calculated from the axial vibration force and

the effective section area if the bolted joint was subjected to axial vibration. A method to

calculate a nominal stress at the root of the first thread occurred by transverse force had also

been proposed if the bolted joint was subjected to only transverse vibration (Hashimura,

2010). However a calculation method of the stress at the root of the first thread of bolted

joints, which was simultaneously subjected to axial vibration and transverse vibration, have

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not been proposed yet. Jiang et al. investigated an influence of loading direction on self-

loosening of bolted joint (Zhang, 2006). In their experiments, angles of cyclic loads applied to

the bolted joints were 30 degree, 15 degree and 0 degree from pure shearing direction. The

results said that the angle of the cyclic load from pure shearing direction resulted in an

increase in self-loosening resistance. In their experiments, relative displacement of the two

clamped parts was a controlling parameter, and axial vibration and transverse vibration had

not been separately controlled. They have focused not on fatigue but on self-loosening.

As mentioned above, the fatigue investigations of bolted joints subjected to either axial

vibration or transverse vibration have been performed. However the actual bolted joints are

subjected to axial vibration and transverse vibration simultaneously. Fatigue tests of bolted

joint subjected to axial vibration and transverse vibration have been conducted in this study to

reveal the bolt fatigue mechanisms in detail. In the fatigue tests, the amplitude of axial

vibration force was a constant and the amplitude of transverse vibration force was only

changed in each experiment. In the fatigue tests, axial vibration and transverse vibration were

applied to the bolted joint with the same phase and the different phase. The fatigue tests for

the bolted joint subjected to only transverse vibration were also conducted to compare. The

fatigue characteristics were evaluated with an apparent transverse fatigue limit, the highest

amplitude of transverse vibration that the bolt does not break due to fatigue.

EXPERIMENTAL APPARATUS AND PROCEDURES

Figure 1(a) shows a test bolt and Fig. 1(b) shows a tightening situation of a tightened test bolt

situation subjected to transverse vibration and axial vibration. The test bolt was a commercial

hexagon head bolt M10, thread pitch 1.5mm, nominal length l=45mm, property class 8.8. The

corners of the bearing surface under the bolt head were machined to ignore an influence of the

scratch due to corners of the bearing surface during a fatigue test. A new test bolt was used in

each experiment.

Fig. 1 - A test bolt and a tightening situation

In the experiments, grip length lg of the bolted joint was 35 mm and engaging thread length le

was 10 mm. After the test bolt was tightened with the initial clamping force Fi into the

internal thread as shown in Fig. 1(b), axial vibration was applied to the upper clamped part

and transverse vibration was applied to the lower clamped part. Consequently the test bolt

received axial vibration and transverse vibration simultaneously.

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Figure 2 shows a schematic illustration of an experimental apparatus for the multi-axial

fatigue tests. The apparatus was designed to simulate a two-plate structure without a nut. The

test bolt was tightened into the internal thread adapter through the upper clamped part and a

load cell to measure clamping force F in the center of the apparatus. The upper clamped part

was supported by a ball retainer to move in only axial direction. The contact surfaces around

the upper clamped part were hardened to reduce the friction. Frictional losses were measured

to be less than 1% of the axial vibration force were neglected in this study. The upper

clamped part was also supported by a disk spring in axial direction to increase the load factor

of the bolt.

Fig. 2 - An experimental apparatus

The lower clamped part was supported by four linear rollers to move in only transverse

direction. The internal thread adaptor was attached to the lower clamped part. Since the

contact surfaces around the lower clamped part were hardened to reduce the friction, frictional

losses around the lower clamped part were also were neglected. The displacement of vibrated

clamped part was measured by a non-contact laser displacement transducer.

Axial vibration was applied using an eccentric cam and several disk springs. The amplitude of

axial vibration force was (∆Pa/2)=2.3 kN constant and the stress ratio Ra was Ra=0. The axial

vibration force Pa was measured by a load cell attached on the upper clamped part. Transverse

vibration was also applied using an eccentric cam and leaf springs. The amplitude of

transverse vibration force was (∆Pt/2) controlled changing number of leaf springs. The stress

ratio Rt was Rt=-1. The transverse vibration force Pt was measured by a load cell attached on

the right side of the lower clamped part. The rotations of the eccentric cams for axial vibration

and for transverse vibration were synchronized by timing belts. If phases between the axial

vibration and the transverse vibration were different, it is considered that the fatigue

characteristics might vary depending on the phases. The combination of the two phases come

down to two cases approximately as shown in Fig.2(b). The first case was a same phase, and

we defined the case as phase A. The second case was a phase gap 90 degree, and we defined

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the case as phase B. Incidentally, if the phase gap is 180 degree, the phase gap equals to phase

A. In this case, the point, at which the maximum stress generated, change to the opposite root

of threads.

The internal thread adaptor plate part was made of a chromium-molybdenum steel JIS

SCM435 and manufactured by tapping. The tap was inserted into the internal threads to check

the deformation of the internal threads before each experiment. If an abnormality of the

internal thread were detected, the adaptor would be replaced with a new one. Otherwise, the

adaptor was repeatedly used in the experiments. The upper clamped part and the lower

clamped part were also made of a chromium-molybdenum steel JIS SCM435 and a hole

diameter Dh in the bearing surface part was 11mm.

The fatigue tests were started after the test bolt was tightened by a wrench with Fi=15kN and

a device to apply the axial vibration was attached on the upper clamped part. To reveal

influences of an interaction between two vibrations on bolted joint fatigue, the fatigue tests

were conducted in three conditions. In the first condition, the bolted joints were applied the

two vibrations with phase A. In the second condition, the bolted joints were applied the two

vibrations with phase B. The third condition was the fatigue tests under only transverse

vibration. The amplitude of axial vibration force was fixed with constant (∆Pa/2)=2.3 kN, and

the amplitude of transverse vibration force was changed from (∆Pt/2)=0.7 kN to 0.27 kN in

each experiment to reveal fatigue limits. The fatigue tests were stopped when the clamping

force F reached zero or the loading cycle exceeded 7101× cycles. In all fatigue tests, the

thread surface and bearing surface were lubricated by MoS2 grease.

In this study, we defined the highest amplitude of transverse vibration that the bolt does not

break due to fatigue as an apparent transverse fatigue limit (∆Pt/2)w when the bolt received

only transverse force. When the test bolt received the axial vibration in addition to the

transverse vibration in phase A, the apparent transverse fatigue limit was denoted by

(∆Pt/2)w-A. When the test bolt received the axial vibration in addition to the transverse

vibration in phase B, the apparent transverse fatigue limit was also denoted by (∆Pt/2)w-B. The

fatigue characteristics were evaluated by these apparent transverse fatigue limits although the

axial vibration was applied to the test bolt.

RESULTS OF MULTI-AXIAL FATIGUE TESTS

Figure 3 shows the results of fatigue tests. The ordinate is the amplitude of transverse force

(∆Pt/2), and abscissa is the number of cycles to failure Nf. In Fig. 3, black circular symbols

indicate the fatigue lives Nf of the test bolts subjected to transverse vibration only. And white

circular symbols show the results that the test bolts did not break due to fatigue. The apparent

fatigue limit due to transverse vibration only was (∆Pt/2)w=0.43 kN. Black rhombus symbols

indicate the fatigue lives Nf of the bolts subjected to axial vibration with phase A in addition

to transverse vibration. And a white rhombus symbol shows the results that the test bolts did

not break. The apparent fatigue limit of phase A was (∆Pt/2)w-A=0.27 kN. Black square

symbols indicate the fatigue lives Nf of phase B. A white square symbol shows the results that

the test bolts did not break. The apparent fatigue limit of phase B was (∆Pt/2)w-B=0.34 kN.

It can be seen in Fig. 3 that the fatigue limits for each loading condition were different. As a

matter of course, the apparent fatigue limit (∆Pt/2)w by only transverse vibration were the

highest in all conditions because the bolted joint in phase A and phase B was subjected to the

axial vibration in addition to the transverse vibration. The number of cycles to failure Nf,

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which the bolted joint was subjected to the transverse vibration only, were also longer than

the other conditions. Fracture cracks occurred at the root of the first thread in most cases.

Fig. 3 - Relationships between amplitude of transverse vibration force ∆Pt/2 and

number of cycles to failure Nf.

It can be also seen that the apparent fatigue limit (∆Pt/2)w-B of phase B was about 25 % higher

than the apparent fatigue limit (∆Pt/2)w-A of phase A although the same axial vibration was

applied to the bolted joint. The difference of these conditions was the phase gap of vibrations

only. The results indicate that the stresses at the root of the first thread were different in each

loading condition even if the amplitude of axial vibration force and the amplitude of

transverse vibration were the same. Therefore we have to understand relationships between

the stresses at the root of the first thread and the amplitudes of each vibration force.

Incidentally, the difference between (∆Pt/2)w and (∆Pt/2)w-A was 0.16 kN although the axial

vibration was applied to the bolted joint in addition to transverse vibration in the experiments

of phase A. The difference between (∆Pt/2)w and (∆Pt/2)w-B was also 0.09 kN. These two

differences were very small. It is seen from the results that transverse vibration has a great

influence on the bolt fatigue in comparison with axial vibration.

In our experimental apparatus, there were linear rollers between clamped parts. Hence it is

considered that the actual apparent fatigue limits are larger than the results in this study

because the actual apparent fatigue limits must be included friction force between bearing

surfaces. However the actual apparent transverse fatigue limits are not so strong even if the

friction force between clamped parts was included. Therefore we have to pay enough

attention to transverse vibration of bolted joints.

STRESS AT A ROOT OF THE FIRST THREAD

To reveal an influence of the phase gap on the fatigue strengths, we measured the stresses at

the root of the first thread applied by axial vibration and transverse vibration using strain

gages. And then we have also proposed a method to calculate the stresses at the root of the

first thread by axial vibration and transverse vibration analytically.

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At first we calculated the relationship between axial force Pa and the stress σa at the root of

the first thread. When the bolted joint received axial force Pa, the additional axial force Qa of

the bolt becomes a following equation.

aa PQ ⋅=Φ (1)

where Φ is load factor of the bolted joint and expressed by Eq.(2).

cb

b

CC

C

+=Φ

(2)

where Cb is spring constant of the bolt and Cc is spring constant of the clamped part. Let Sa be

the load to the clamped part due to axial force Pa, Φ is expressed as

a

aa

a

a

P

SP

P

Q −==Φ

(3)

The nominal stress σa caused by axial force Pa is expressed by a following equation.

s

aa

s

aa

A

SP

A

P −=

⋅=

Φσ

(4)

where As is an effective section area of the bolt thread portion. In the experimental apparatus,

axial force Pa was measured by the load cell for axial force and the load Sa to the clamped part

was measured by the load cell for clamping force. As a result of preliminary experiments,

load factor Φ of the apparatus was Φ=0.68.

Next, the relationship between the transverse force and the stress at the root of the first thread

is calculated. Fig. 4 shows a schematic illustration of a deformed bolt which is subjected to

transverse vibration only. In Fig.4, point O is a point under the bolt head, point A is a border

point between the shank and threaded portion. Point B is the first bolt thread which begins to

engaging the internal thread.

When the bolt receives transverse force Pt, the bolt deforms S-shape as shown in Fig.4

because the engaged bolt threads are constrained by the internal threads. At this time, the bolt

head has an inclination ϕo at point O. The thread surface slips at point B in the transverse

direction, δs-slip. The threaded portion simultaneously inclines at point B, ϕB, and the bending

moment MB at point B is generated. The right side of Fig.4 shows a bending moment diagram

of the bolt subjected to transverse force Pt. The bending moment M on the bolt is expressed

by a following equation by beam theory.

( )BMxlPM gt +−⋅−= (5)

Where where lg is a grip length of the bolted joint. The bending moment MB is expressed as

gt lPCM ⋅⋅=B (6)

where C is a coefficient of the bending moment. If ϕB=0 and ϕo=0, the coefficient C becomes

0.5. However the coefficient C does not become zero because ϕB and ϕo are existing actually.

Hence it is not easily to determine the coefficient C because C depends on the inclination ϕB

at point B and the slippage δs-slip between thread surfaces.

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Fig. 4 - A schematic illustration of deformed bolt due to transverse force

If the coefficient C is known, the relationship between the transverse force Pt and the nominal

stress σt at the root of the first thread can be expressed as a follow.

22

33B d

I

lPCd

I

M

b

gt

b

t ⋅⋅⋅

=⋅=σ (7)

where d3 is the root diameter of bolt thread and Ib is the moment of inertia of effective section

area of the bolt thread.

If we could know either the bending moment MB or coefficient C, the nominal stress σt at the

root of the first thread can be calculated from transverse force Pt. Then we measured the

bending moment MB at point B using a test bolt attached strain gages.

Fig.5 shows a test bolt to measure the bending moment MB and a bending moment diagram.

The thread portion, which did not engage with the internal threads, was removed by a lathe.

Six strain gages were attached to the both opposite sides on the bolt shank as shown in an

illustration in Fig.5. The outputs of the strain gages were calibrated applying bending moment

to the test bolt in advance. A graph in Fig.5 shows a bending moment diagram when

transverse vibration (∆Pt/2) was applied to the bolted joint. The bending moment diagrams

were drawn for three amplitudes of transverse vibration, (∆Pt/2)=0.29 kN, (∆Pt/2)=0.46 kN

and (∆Pt/2)=0.55 kN. As can be seen in Fig.5, all bending moment diagrams were linear

against the bolt position in axial direction. It can be also seen in Fig.5 that the bolt deformed

like S-shape. The bending moment MB at point B was calculated by extrapolation. Coefficient

C became C=0.37 substituting MB for Eq.(6). And this coefficient C was almost the same for

the all amplitudes of transverse vibration. The stress (σa+σt) at the root of the first thread due

to axial vibration Pa and transverse vibration Pt were calculated substituting (∆Pa/2) and

(∆Pt/2) for Eq.(4) and Eq.(7).

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Fig. 5 - A test bolt attached strain gages and bending moment diagrams of the test bolt

Figure 6 shows the measured stress σn-exp and the calculated stress (σa+σt)n-cal by Eq.(4)

and Eq.(7) when the test bolt was subjected to (∆Pt/2)=0.46 kN and (∆Pa/2)=2.3 kN in phase

A. The measured stress σn-exp was measured from the output of M1, B1, M2 and B2 by

extrapolation. In Fig.6, the ordinate is the nominal stress σn at the root of the first thread and

the abscissa is time. The black wide solid line and the gray wide solid line show the stress at

the left side root of the first thread, T1. The black thin solid line and the gray thin solid line

show the stress at the right side root of the first thread, T2. The black lines show the results of

an experiment using the test bolt attached the strain gages. The gray lines show the calculated

stress (σa+σt)n-cal. In the bolt situation of Fig.6, when the bolt pulled with the maximum

axial force such as a wide arrow, the lower clamped part received the maximum transverse

force in left direction. Hence the amplitude of the stress at T1 was higher than that at T2.

Fig. 6 - Comparisons of the stresses which were measured by the strain gages and the stresses which were

calculated using the axial vibration and the transverse vibration

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It can be seen in Fig.6 that the experimental stress σn-exp and the calculated stress (σa+σt)n-

cal were almost the same although the experimental stress had included electrical noise a

little. Consequently, the validity of a superposition of the stresses calculated using Eq.(4) and

Eq.(7) were confirmed. The results indicate that there is almost no interaction between axial

vibration and transverse vibration. Namely it is considered that axial vibration has almost no

influence on the slippage between thread surfaces occurred by transverse vibration. In Fig.6,

the amplitude of stress σn at S1 determines the fatigue life of the bolt because the amplitude

of σn at S1 was larger than that of σn at S2. Based on these results, the maximum amplitudes

of stress at the root of the first thread were calculated for each experimental condition, and the

experimental results in Fig.3 were re-drawn by the maximum amplitudes of stress at the root

of the first thread.

Figure 7 shows the experimental results which were drawn by the maximum amplitudes of

stress (σa+σt)n-cal at the root of the first thread. In Fig.7, circular symbols indicate the fatigue

characteristic of the bolts subjected to transverse vibration only. Its fatigue limit was σtw=97

MPa. Rhombus symbols indicate the fatigue characteristic of phase A. Its fatigue limit was

(σt+σa)w-A=88 MPa. Square symbols indicate the fatigue characteristic of phase B. The fatigue

limit was (σt+ σa)w-B=81 MPa. It can be seen that the true fatigue limits in Fig.7 are almost the

same although the apparent fatigue limits are different in Fig.3. The finite fatigue lives Nf

became also almost the same. The results show that the true fatigue limits never change by the

loading conditions although the apparent fatigue limits change depending on the loading

conditions.

Fig. 7 - Relationships between nominal stresses at the root of the first thread and

number of cycles to failure

CONCLUSIONS

In this study, we conducted the fatigue tests of bolted joints simultaneously subjected to axial

vibration and transverse vibration of which phases were controlled. The main conclusions

obtained in this study are summarized as follows.

1. The apparent transverse fatigue limit, the highest amplitude of transverse vibration that

the bolt does not break due to fatigue, changes depending on phase difference between

axial vibration and transverse vibration.

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2. The true fatigue limits do not change depending on phase difference between axial

vibration and transverse vibration.

3. The nominal stress at the root of the first thread, which is a fracture point, can calculate by

a superposition of the stresses due to axial vibration and transverse vibration.

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