3-d fem stress analysis of screw threads in bolted joints

1 Copyright © 2014 by ASME

3-D FEM STRESS ANALYSIS OF SCREW THREADS IN BOLTED JOINTS UNDER STATIC TENSILE LOADINGS

Shunichiro SAWA HARDLOCK Industry Co., Ltd. 5-9, 2-chome, Higashi Ueno,

Taito-ku, Tokyo, Japan

Mitsutoshi ISHIMURA Shonan Institute of Technology Tsujido-nishikaigan, Fujisawa,

Kanagawa, Japan

Yuya OMIYA Okayama University

Tsushimanaka, Kita-ku, Okayama, Japan

Toshiyuki SAWA Hiroshima University

Kagamiyama, Higashi-Hiroshima Hiroshima, Japan

KEYWORDS Bolted joint, Stress Analysis, Stress concentration, Finite

element method (FEM) calculation, Fracture, Static load,

Fatigue.

ABSTRACT The stress concentration factor (SCF) for the roots of

screw threads in bolted joints under static loadings is analyzed

using 3-D elastic FEM taking account the spiral of screw

threads. At first, the stress states at the roots of screw threads in

initial clamping state in a bolted joint where two hollow

cylinders were clamped with a bolt and a nut were analyzed in

initial clamping. The elastic FEM result of SCF for the first

root was obtained as SCF=3.2. When the bolt was clamped in

initial clamping (preload) at the 60 % of bolt yield stress, the

plastic deformations were found at the first and the second

roots, and non-engaged screw threads. It was found that as the

external tensile loads increased, the development in plastic

deformation region increased from the first root to the other

roots as well as the non-engaged screw threads. It was found

that the rupture occurred from the non-engaged screw threaded

part while the plastic deformation increased at each root of

screw threads. The numerical result was coincided with the

experimental result. In the experiments, it was observed that the

rupture occurred from the non-engaged screw thread and not

from the first root of screw thread. Also, the bolt fatigue was

predicted from FEM and it was shown that a fatigue fracture

occurred from the first root.

1. INTRODUCTION Bolted joints have been widely used in a lot of industries

such as mechanical structures, rail way, aerospace, automobile

and so on. However, some accidents have occurred in the world

due to rupture and fatigue of bolts, bolt loosening and so on. It

was well known that the bolt fatigue initiated from the first root

of engaged screw threads in bolted joints under repeated

loadings. The screw threads are continuous notches and then

the stress concentrates at the roots of screw threads. It is

necessary to know the stress concentration factor (SCF) at the

first root of screw thread for better designing the bolted joints

from reliable design standpoint and for preventing accidents.

Some studies have been carried out on SCF from

experimentally and numerically. Hetenyi(1)

conducted the

photo-elastic experiments to measure SCF at the first root in

screw threads and he obtained SCF as α＝3.85 for the mean

stress using the cross sectional area of the diameter at the bolt

shank. He also showed SCF as α=2.73 for the mean stress using

the root area. Maruyama(2)

carried out the Copper-

Electroplating stress measurement method to measure SCF at

the first root of screw thread and showed α＝4.5 for M24, pitch

P=3.0mm and the radius of the root r=0.4mm. However, some

problems were in the experiments and the accuracy was a big

problem in the experiments. In addition, the reference cross

sectional area is also a problem. 3-D FEM(3)

was done on the

stress distributions in screw threads.

Big issues are 1) SCF was obtained in two dimensional

stress state and three dimensional stress analysis is not carried

out sufficiently, 2) Axi-symmetrical analyses for screw threads

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-38089


were conducted, however, spiral of screw threads were not

taken into account, 3) Some researches on SCF were done in

the initial clamping, however, the bolts were ruptured under

external static loadings. The stress distributions at the roots of

screw threads should be examined, 4) the bolts were known to

be fractured from the first root of screw thread under repeated

external loadings. It is necessary to know why the fracture

initiates from the first root of screw thread under repeated

external tensile loadings.

In this research, the objectives of the FEM analyses are as

follows; 1) the stress concentration factor (SCF) at the root of

the screw threads in an elastic deformation range is analyzed

and compared to the previous results. 2) The stress distributions

are analyzed and find out a position where the maximum stress

occurs close to the root in the screw threds, 3)As the bolt

preload increases such as 20, 60, and 80% of the bolt yield

stress, the stress at the roots increases and yields. An elasto-

plastic FEM analysis is carried out and the stress distributions

at each root in the engaged screw threads are examined. 4)

After clamped the clamped parts with the bolt preload

mentioned above, the stress state is examined at each root of

engaged screw threads in elasto-plastic deformation range when

external tensile loadings are applied. As the external loadings

increase, the elasto-plastic stress state at each root in the

engaged screw threads and that at the non-engaged screw

threads are examined. Finally, a position where a rupture occurs

is predicted. 5) In the initial clamping state, the smaller external

repeated load is applied to the joint, the amplitude of the stress

at the bolt roots is examined and a position where a fracture of

bolt occurs is predicted using FEM analysis. In the present

three dimensional (3-D) FEM analyses, spiral of screw threads

is taken into account.

NOMENCLATURE Ff : bolt preload[N]

Ft : increment in axial bolt force[N]

Fc : decrement in clamp force[N]

W : external tensile load[N]

2a : the inside diameter of the hollow cylinder[mm]

2b : the outside diameter of the hollow cylinder[mm]

h : the height of the hollow cylinder[mm]

r : the radius of the root of screw thread[mm]

P : pitch[mm]

As : the effective cross sectional area[mm2]

ds : the diameter of the cross section[mm]

α : the stress concentration factor (SCF)

σ1 : the maximum principal stress[MPa]

σn : the mean stress {=(tensile bolt load)/(π/4×ds 2)}[MPa]

σM : Mises’ stress[MPa]

σy : the bolt yield stress[MPa]

σZ :the normal stress[MPa]

2. MODEL OF BOLTED JOINTS FOR FEM ANALYSIS Figure 1 (a) shows a bolted joint where two hollow

cylinders are clamped with a bolt preload Ff and Fig.1(b) shows

the case where an external tensile loading W is applied to the

joint and an increment in axial bolt force increases with Ft. The

outside diameter of the hollow cylinder is denoted by 2b, the

inside diameter by 2a and the height by h, respectively. Figure

2 shows a FEM model of the bolted joint for analysis. Two

hollow cylinders are clamped with a bolt preload Ff. Cylindrical coordinates (r,θ,z) is used. The nominal diameter of bolt is M12

×60×1.75 Figure.2(a),M12×70×1.75 Figure.2(b). Figure2

(b) shows a case where an external tensile loading W is applied

to the bolted joint. The inside diameter of the hollow cylinder is

chosen as 2a=14 mm, the outside diameter as 2b=27.6mm and

the height of the cylinder as h=23.5mm. The material of the

hollow cylinders is mild steel. Their Young’s modulus and

Poisson’s ratio are 206GPa and 0.3, respectively. The boundary

conditions are as follow: All elements at the interfaces of the

clamped parts (hollow cylinder) are fixed in the axial direction

as shown in Fig.2 (a) and tensile stresses are applied at all

elements at the lower surface of the bolt axis shown in Fig.2 (a)

such as 20,40,60 and 80% of the yield stress of the bolt. In the

case where an external tensile loading is applied as shown in

Fig.1(b), the load is applied to the upper surface of the clamped

(a)Initial clamping state

(b) The case where an external tensile loading is applied

Fig.1 Bolted joints under external tensile loading


part which is close to bearing surfaces of nut as shown in Fig.2

b). In FEM analyses, 8-node hexahedron elements are used.

FEM code employed is ANSYS ver.14. The elements and node

employed for Fig.2 a) are 172176 and 169702, respectively and

they are 242180 and 239306 for Fig.2 b). Figure 3 shows a

stress - strain curves of the bolt material. The yield stress is

obtained as 840MPa. The friction coefficient is assumed 0.1 at

the bearing surfaces and at the contacted surfaces in the

engaged screw threads. Bolt pretension element is employed to

apply bolt preloads.

3. FEM RESULTS IN ELASTIC DEFORMATION RANGE In the previous researches, the value of stress

concentration factor (SCF) was around 3.5 from Hetenyi(1)

which was obtained from two dimensional analyses.

Maruyama(2)

carried out experiments to measure SCF at the

roots of screw threads using Copper Electroplating stress

measurement method and he showed SCF as σ1/σmean＝4.5,

where the mean stress is obtained using the diameter of the root

and σ1 is the maximum principal stress which is expressed

as the equation 𝜎1 = √𝜎2 + 4𝜏2,where σ is the normal stress

and τ is the shear stress. In the present paper, the elastic FEM

analysis was carried out for M12 bolt and nut (Hexagon head

bolts and hexagon head screws JIS B 1180(4)

, Hexagon nuts and

hexagon thin nuts JIS B 1181(5)

). Figure 4 shows the

normalized maximum principal stress distributions along the

distance z, where σn is the mean stress for the effective cross

sectional area As (the diameter of the cross section is ds), that is,

σn is defined as σn=(tensile bolt load)/(π/4×ds 2). The abscissa

is the distance z along the screw threads and the ordinate is the

normalized maximum principal stress σ1/σn. The length of

screw threads are 21mm, the height of nut is 10.5mm, the

engagement length is from 7.0 to 17.5mm. In Fig.4, the preload

Ff is changed such as 20, 40, 60,and 80% of σy. The maximum

value of SCF is obtained as 3.2 at the first root in engaged

screw threads using the nominal bolt diameter d. In addition, it

is shown that the maximum value occurs at the root of the

distance z=7.875mm (1/2 pitch, θ＝180 degree). The obtained

value of SCF is smaller than the experimental results from

Maruyama(2)

, however, it is close to Hetenyi’s result(1)

. It is a

problem how choose the mean stress, that is, how to choose the

cross sectional area in the bolts. In addition, In the FEM

analysis, it is assumed that the mesh size in FEM analysis is a little bit coarse in the present study. The value of the

normalized maximum principal stress decreases as the distance

z increases. The obtained results are coincided with the

conventional results. The stress distributions are independent of

the bolt preloads because the present FEM analysis is done in

the elastic deformation range.

4. RESULTS OF ELASTO-PLASTIC FEM ANALYSIS From the above results, it is found that the maximum principal

stress reaches a yield stress at the first root of screw thread

when the bolt preload is over 35% of bolt yield stress because

(a) In the initial clamping state

(b) The case where an external tensile loading is applied

(c) Mesh divisions for the screw threads

Fig.2 Model of bolted joint for FEM analysis

and Examples of mesh divisions


(b)Normalized Mises stress distributions

the value of SCF is over 3.0. Thus, it is necessary to carry out

the elasto-plastic FEM analysis for the stress state at each root

in screw threads. In this chapter, the elasto-plastic FEM

analysis is conducted taking account the stress-strain curve of

bolt shown in Fig.3 and the development of plastic deformation

region is examined. In the elasto-plastic FEM analyses, the bolt

preloads Ff are chosen as 20, 60 and 80 % of the bolt yield

stress σy, that is, Ff=0.2σy As, 0.6σy As and 0.8σy ×As, where As is

the effective sectional area of bolts.

Figure 5(b) shows the elasto-plastic stress distributions at

the roots along the threads in the z direction. The ordinate is the

distance z and the abscissa is normalized Mises’ stress σM/σy,

where σM is Mises’ stress which is expressed as the equation

𝜎𝑀 = √𝜎2 + 3𝜏2 ,and σy is the bolt yield stress. The

engagement length shows from 7.0 to 17.5mm. The region

where the value of σM/σy is over 1.0 means the plastic

deformation occurs. In Fig.5 (b), it is found that no plastic

deformation occurs as well as in the elastic FEM when the bolt

preload is 20% of bolt yield stress, however, the maximum

value of σM/σy, is 0.83 and a position is z=7.875mm (1/2 pitch

in the first root). In the case of 60% of bolt yield stress, the

plastic deformation is observed at the most region in the non-

engagement and the development of the plastic deformation is

observed from the first to the fourth roots in screw threads,

where the position of third root is z=10.5 and the fourth is

z=12.25mm. From the results, it is noticed that the maximum

value of the normalized σM/σy occurs at the position of 1/3 pitch

(z=7.875mm) in the z direction.

Fig.3 Measured Stress-strain curve of bolt

engaged threads

Fig.4 Distributions of normalized maximum principal stress

σ1/σn along the distance z (Elastic FEM results)

Nut

(a) Contour figure of Mises stress around the roots

in screw threads in the case of Ff=0.6σy As

Fig.5 Normalized Mises stress σM/σy distributions along

the distance z in elasto-plastic FEM


Figure 6 shows the values of Mises’ stress at the roots in the

threads in z direction in the case of 80% yield stress. The

ordinate is the distance z, and the abscissa is normalized Mises’

stress σM/σy. In the case of 80% of bolt yield stress, the plastic

zone is extended to the fifth root in screw threads and the

normalized Mises’ stress σM/σy increases from the first to the third root in screw threads. It is also found that it increases at

the non-engagement region. The maximum value of σM/σy

occurs sometimes close to the roots of screw threads, where the

bolt yield stress is 840MPa. It is shown that the plastic zone is

extended from the first root to the third root as well as the

screw threads in the non-engagement. As the results, it is found

that the plastic deformation occurs at the roots of screw threads

in the initial clamping with lower bolt preloads (the bolt initial

clamping stress is over 35% of bolt yield stress). It is also found

that as the bolt preload increases the plastic zone increases at

the roots in screw threads as well as at the screw threads in non-

engagement. Thus, it can be assumed that the bearing forces at

each screw thread change when the plastic deformations occur

at the roots of screw threads.

4.1 The stress distributions at the roots of screw threads when an external tensile loading is applied

The elasto-plastic stress distributions at the roots of screw

threads are analyzed when external tensile loadings are applied

to the joint, where the bolt preload Ff is 0.6σy As. The external

tensile loads W are the same as the bolt preload with 40 and

60% of yield stress, that is, W=0.4σy As and 0.6σy As. The

external tensile load is applied shown as in Fig.2. Figure 7

shows the contour figure of stress in the initial clamping state

with Ff=0.6σy As and Fig.8 shows the contour when the external

load W=0.6σy As is applied. Figure 9 shows the stress

distributions σZ in the z direction obtained from the elastro-

plastic FEM analysis. The abscissa is the distance z along the

spiral distance and the ordinate is the stress component σZ. The

distance z from 7.0mm to 17.5mm is in the engagement. From

the comparison between the contours Fig.7 and Fig.8, the stress

increases at the screw threads in the non-engagement. It is

found in Fig.9 that the maximum value of σz occurs at the position of 1/2 pitch of the first root in screw thread when the

external tensile loading is applied. This result is the same as the

results obtained from the elastic and the elastro-plastic FEM. In

addition, it is observed that the difference in the stress between the initial clamping state and the state when the external load is

applied is maximal around the first root, where the stress

amplitude in this case is 100 MPa as shown in Fig.9. Thus, it

can be assumed that the fatigue crack initiates from the first

root in screw threads, that is, the fracture surface is the first root.

Figure 10 shows a photograph of a fractured bolt where

the first root is ruptured due to fatigue. The FEM result on

fatigue crack is coincided with the experimental fractured

surface.

Figure 11 shows the contour figures of deformation in the

screw threads when external loads W is applied to the joint such

as W=0.2σy As, 0.4σy As, 0.6σy As and 0.8σy As, where the bolt

preload Ff is Ff=0.6σy As. The deformation is enlarged 5 times

Nut

(a) Contour figure of Mises stress around the root

in the screw threads in the case of Ff=0.8σy As

(b) Normalized Mises stress distributions

Fig.6 Mises stress distribution at

all nodal points around the roots


for the actual deformation. It is found that the deformation of

screw thread in the non-engagement increases as the external

loading W increases. Thus, it can be assumed that a bolt rupture

in bolted joints under static tensile loadings occurs from the

screw threads in the non-engagement. Figure 12 shows an

example of bolt rupture under static tensile loading. It is

observed that a rupture occurs from the screw thread in the non-

engagement. The FEM results of the bolt rupture under static

tensile loadings is coincided with the experimental result.

As the results, it can be concluded that when the external

load W is applied in the range where W is less than Ff and the

stress amplitude is over the fatigue stress for the bolt, the

fatigue crack initiates from the first root in screw thread in the

engagement. In addition, under static tensile loadings, the

rupture occurs at the screw threads in the non-engagement. The

types of bolt rupture are different between the static loadings

and the repeated loadings.

Fig.8 Contour figure of σZ

when an external tensile load is applied

Fig.10 Photograph of fractured bolt under repeated loadings

engaged threads

Fig.7 Contour figure of stress σZ

in initial clamping state Fig.9 Stress σz distribution along the distance z


CONCLUSIONS In the present paper, three dimensional FEM stress

analysis was newly carried out for bolted joints under tensile

loadings taking into the spiral of screw threads. At first, the

stress distributions at the roots of screw threads in initial

clamping state were examined and the stress concentration

factor (SCF) was obtained. The stress distributions in bolts

were also examined in elastic and elastro-plastic deformation

range. In addition, the stress distributions were also examined

when external tensile loadings were applied to the bolted joints.

The obtained results are summarized as follows.

1) The stress distributions at the roots of screw threads (M12,

JIS) in initial clamping were analyzed using elastic FEM.

The effect of the bolt preload was examined on the stress

distributions. It was found that the maximum principal stress

occurred at the distance of 1/2 pitch (z=7.875mm, θ=180

degree) in the first root independent of the bolt preload. In

addition, the stress concentration factor (SCF) for M12

obtained from the present study was obtained as 0.32. This

result was found to be smaller than that obtained from

Maruyama, however it was close to Hetenyi’s result.

2) Also, the elasto-plastic FEM analysis was carried out for the

bolted joint in initial clamping state. The effect of bolt

preload was examined on the stress distributions at the roots

of screw thread in the engagement. It was found that the

stress distribution was in elastic deformation when the bolt

preload was 0.2σy As, however, when the bolt preloads Ff

were chosen as Ff =0.6σy As and 0.8σy As, the plastic

deformations occurred. It is noticed that the plastic

deformation should be taken into account when bolts are

clamped with higher bolt preloads.

3) When external loadings were applied to the bolted joint, the

stress distributions at the roots of screw threads were

examined. As the external loadings increase the deformation

of bolt at the non-engagement increased. In addition, the

deformation (strain) increased from the non-engagement

while the deformation increased at the roots of screw threads

in the engagement. However, the deformation at the screw

thread in the non-engagement was larger than that in the

engagement. Thus, it was assumed that a rupture occurred

from the screw thread in the non-engagement under static

tensile loadings. This result was coincided with the

experimental result.

4) When external tensile loading was applied, the maximum

stress occurred at the first root of screw thread in the

engagement. When a repeated load was applied, it was

shown that a fatigue fracture occurred from the first root of

screw thread in the engagement. This result was confirmed

with the experimental result.

REFERENCES 1. M. Hetenyi: A Photoelastic Study of Bolt and Nut

Fastening,J.Appl.Mech.,Trans.ASME,10,2(1943)A-93

2. Maruyama,K., Stress Analysis of a Bolt-Nut Joint by the

c)the case of W (= 0.4σy・As)

b)the case of W (= 0.2σy・As)

e)the case of W (= 0.8σy・As)

d)the case of W (= 0.6σy・As)

Fig.11 Contour figures of stress distributions of bolts in bolted joints under tensile loadings

a)the initial clamping state

Fig.12 Photograph of ruptured bolt under static tensile

loading


Finite Element Method and the Copper-Electroplating

Method : 2nd Report, Stress at the Root of a Bolt Thread

under a Tensile Load, Transactions of the Japan Society of

Mechanical Engineers, Vol.39,No.324(1973),pp.2340-2349.

3. Fukuoka, T., Nomura, M. and Morimoto, Y., Proposition of

Helical thread Modeling with Accurate Geometry and Finite

Element Analysis, Transactions of the Japan Society of

Mechanical Engineers, Series A, Vol.72, No723 (2006),

pp.1639-1645.

4. JIS B 1180, “Hexagon head bolts and hexagon head screws”,

Japanese Industrial Standards, (2009)

5. JIS B 1181, “Hexagon nuts and hexagon thin nuts”, Japanese

Industrial Standards, (2004)

3-d fem stress analysis of screw threads in bolted joints

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