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Research Article
Stiffness Calculation Model of Thread Connection
Considering Friction Factors
Shi-kun Lu ,1,2 Deng-xin Hua ,1Yan Li ,1Fang-yuan Cui ,1and Peng-yang Li 1
1Xi’an University of Technology, Xi’an 710048, China
2Laiwu Vocational and Technical College, Laiwu 271100, China
Correspondence should be addressed to Deng-xin Hua; dengxinhua@xaut.edu.cn and Yan Li; jyxy-ly@xaut.edu.cn
Received 9 September 2018; Revised 12 December 2018; Accepted 1 January 2019; Published 23 January 2019
Guest Editor: Yingyot Aue-u-lan
Copyright © Shi-kun Lu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In order to design a reasonable thread connection structure, it is necessary to understand the axial force distribution of threaded
connections. For the application of bolted connection in mechanical design, it is necessary to estimate the stiness of threaded
connections. A calculation model for the distribution of axial force and stiness considering the friction factor of the threaded
connection is established in this paper. e method regards the thread as a tapered cantilever beam. Under the action of the
thread axial force, in the consideration of friction, the two cantilever beams interact and the beam will be deformed, these
deformations include bending deformation, shear deformation, inclination deformation of cantilever beam root, shear deformation
of cantilever beam root, radial expansion deformation and radial shrinkage deformation, etc.; calculate each deformation of the
thread, respectively, and sum them, that is, the total deformation of the thread. In this paper, on the one hand, the threaded
connection stiness was measured by experiments; on the other hand, the nite element models were established to calculate
the thread stiness; the calculation results of the method of this paper, the test results, and the nite element analysis (FEA) results
were compared, respectively; the results were found to be in a reasonable range; therefore, the validity of the calculation of the
method of this paper is veried.
1. Introduction
e bolts connect the equipment parts into a whole, which
is used to transmit force, moment, torque, or movement. e
bolt connection is widely used in various engineering elds,
such as aviation machine tools, precision instruments, etc.
e precision of the threaded connection aects the quality
of the equipment. Especially for high-end CNC machine
tools, the precision of the thread connection is very high.
erefore, it is important to study the stiness of the threaded
connection to improve the precision of the device. Many
researchers have conducted research in this area.
Dongmei Zhang et al. [], propose a method which can
compute the engaged screw stiness, and the validity of the
method was veried by FEA and experiments. Maruyama et
al. [] used the point matching method and the FEM, based
on the experimental results of Boenick and the assumptions
made by Fernlund [] in calculating the pressure distribution
between joint plates. Motash [] assumed that the pressure
distribution on any plane perpendicular to the bolt axis had
zero gradient at r=ℎ/2and r=𝑜,whereitalsovanishes.
ey mainly use numerical methods to calculate the inuence
of dierent parameters on the stiness of bolted connections.
Kenny B, Patterson E A. [] introduced a method for
measuring thread strains and stresses. Kenny B [] et al.
reviewed the distribution of loads and stresses in fastening
threads. Miller D L et al. [] established the spring model of
thread force analysis and, combined with the mathematical
theory, analyzed the stress of the thread and compared
with the FEA results and experimental results to verify the
correctness of the spring mathematical model. Wang W
and Marshek K M. et al. [] proposed an improved spring
model to analyze the thread load distribution, compared
the load distributions of elastic threads and yielding thread
joints, and discussed the eect of the yield line on the load
distribution. Wileman etal. [] p erformed a two-dimensional
(D) FEA for members stiness of joint connection. De
Agostinis M et al. [] studied the eect of lubrication on
Hindawi
Mathematical Problems in Engineering
Volume 2019, Article ID 8424283, 19 pages
https://doi.org/10.1155/2019/8424283
Mathematical Problems in Engineering
thread friction characteristics or torque. Dario Croccolo
et al. [–] studied the eect of Engagement Ratio (ER,
namely, the thread length over the thread diameter) on the
tightening and untightening torque and friction coecient
of threaded joints using medium strength threaded locking
devices. Zou Q. et al. [, ] studied the use of contact
mechanics to determine the eective radius of the bolted
joint and also studied the eect of lubrication on friction
and torque-tension relationship in threaded fasteners. Nassar
S.A.etal.[,]studiedthethreadfrictionandthread
friction torque in thread connection. Nassar S. A. et al.
[, ] also investigated the eects of tightening speed and
coating on the torque-tension relationship and wear pattern
in threaded fastener applications in order to improve the
reliability of the clamping load estimation in bolted joints.
Kopfer et al. [] believe that suitable formulations should
consider contact pressure and sliding speed; based on this, the
contribution shows experimental examples for main uncer-
tainties of frictional behavior during tightening with dierent
material combinations (results from assembly test stand).
Kenny B et al. [] reviewed the distribution of load and
stress in the threads of fasteners. Shigley et al. [] presented
an analytical solution for member stiness, based on the
work of Lehnho and Wistehu []. Nasser [], Musto
and Konkle [], Nawras [], and Nassar and Abbound
[] also proposed mathematical model for the bolted-joint
stiness. Qin et al. [] established an analytical model of
bolted disk-drum joints and introduced its application to
dynamic analysis of joint rotor. Liu et al. [] conducted
experimental and numerical studies on axially excited bolt
connections.
ere are also several authors that, starting from the
nature of thread stiness, from the perspective of thread
deformation, established a mathematical model of the calcu-
lation of the distribution of thread axial force. e Sopwith
method [] and the Yamamoto method [] received exten-
sive recognition. e Sopwith method gave a method for cal-
culating the axial force distribution of threaded connections.
Yamamoto method can not only calculate the axial force dis-
tribution of threads but also calculate the stiness of threaded
connections. e assumption for Yamamoto method is that
the load per unit width along the helix direction is uniformly
distributed. In fact, for the three-dimensional (D) helix
thread, the load distribution is not uniform. erefore, based
on the Yamamoto method, Dongmei Zhang et al. [] propose
a method which can compute the engaged screw stiness
by considering the load distribution, and the validity of the
method was veried by FEA and experiments. e method
of Zhang Dongmei does not consider the inuence of the
friction coecient of the thread contact surface. In fact,
the friction coecient of the contact surface of the thread
connection has an inuence on the distribution of the axial
force of the thread and the stiness of the thread. erefore,
we propose a new method which can compute the engaged
screw stiness more accurately by considering the eects of
friction and the load distribution. e accuracy of the method
was veried by the FEA and bolt tensile test. e ow chart
of the article is shown in Figure .
2. Mathematical Model
2.1. Axial Load Distribution. According to Yamamoto [],
the thread is regarded as a cantilever beam, and the thread is
deformed under axial force and preload. ese deformations
include the following (shown in Figure ): thread bending
deformation, thread shear deformation, thread root incli-
nation deformation, thread root shear deformation, radial
direction extended deformation, or radial shrinkage defor-
mation.
For the ISO thread, the axial deformation of the thread
at at the axial unit width force zis thread bending
deformation 1, thread shear deformation 2,threadroot
inclination deformation 3, thread root shear deformation
4, and radial direction extended deformation 5(nut) or
radial shrinkage deformation 5(screw), and calculate these
deformations of the thread, respectively, and then sum them,
that is, the total deformation.
2.1.1. Bending Deformation. In the threaded connection,
under the action of the load, the contact surface friction
coecient is , when the sliding force along the inclined plane
is greater than the friction force along the inclined plane, the
relative sliding occurs between the two inclined planes, and
the axial unit width force (shown in Figure ) is 𝑧;ifthe
inuence of the lead angle is ignored, the force per unit width
perpendicular to the thread surface can be expressed as
= 𝑧
sin +cos ()
e force per unit width perpendicular to the thread
surface can be decomposed into the x-direction component
force and the y-direction component force, respectively
cos = 𝑧cos
sin +cos ()
and
sin = 𝑧sin
sin +cos ()
e friction generated along the slope is w; i.e.,
= 𝑧
sin +cos ()
e force wis also decomposed into x-direction force
and y-direction force, which are sin and cos ,respec-
tively.
sin = 𝑧sin
sin +cos ()
cos = 𝑧cos
sin +cos ()
In the unit width, the thread is regarded as a rectangular
variable-section cantilever beam. Under the action of the
Mathematical Problems in Engineering
above-mentioned force, the thread undergoes bending defor-
mation, and the virtual work done by the bending moment
on the beam section is
=𝑤
𝑏 ()
According to the principle of virtual work, the deection
1(see Figure (a)) of the beam subjected to the load is
1=𝑐
0𝑤
𝑏 ()
where is the bending moment of the unit load beam.
𝑤is the bending moment of the beam under the actual
load. I(y) is the area moment of inertia of the beam at .𝑏is
Young’s Modulus of the material. cis the length of the beam.
Here, the forces are assumed as acting on the mean diameter
of the thread.
AsshowninFigure,theheight()of the beam section
per unit width and the area moment of inertia () of the
section can be expressed by using the function interpolation.
=1+1−1−
, 0≤≤
= 1
1231+ 1−1−
3,0≤≤
1=
()
where his the beam end section height; bis the beam
section width; 1is the beam root section height and the
beam end section height ratio; see Figure .
From Figure , the bending moment of the beam is related
to the y-axis component of and w, and these components
cause the beam to bend; therefore, the analytical solution
shows that the bending moment of the unit width beam
subjected to the friction force and the vertical load of the
thread surface is
𝑤=𝑧
sin +cos ⋅cos ⋅c+sin ⋅
−sin
2−tan +cos
2−tan ()
Substituting () and () to () and integrating to obtain
the analytical expression of the deection 1(shown in
Figure (a)) of the cantilever beam with variable cross- section
under load one has
1=12𝑤2
𝑏3⋅1
21
2
1+1− 1
1⋅ 1
1−12()
2.1.2. Shear Deformation. Assume that the distribution of
shear stress on any section is distributed according to the
parabola [] and the deformation 2(see Figure (b)) caused
by the shear force within the width of unit is
2=log𝑒
⋅6(1+V)cos +sin cot
5𝑏
⋅𝑧
sin +cos
()
2.1.3. Inclination Deformation of the read Root. Under
the action of the load, the thread surface is subjected to a
bending moment, and the root of the thread is tilted, as
shown in Figure (c). Due to the inclination of the thread,
axial displacement occurs at the point of action of the thread
surface force, and the axial displacement can be expressed as
[]
3=𝑧
sin +cos ⋅121−V2
𝑏2⋅cos ⋅
+sin ⋅−sin
2−tan
+cos
2−tan
()
2.1.4. Deformation due to Radial Expansion and Radial
Shrinkage. According to the static analysis, the thread is sub-
jected to radial force sin −cos (shown in Figure ),
anditisknownfromtheliterature[]thattheinternaland
external thread radial deformation (shown in Figure (d)) are
4b=1−]𝑏𝑝
2𝑏tan ⋅𝑧sin −cos
sin +cos ()
and
4n=2
0+2
𝑝
2
0−2
𝑝+]𝑛𝑝
2𝑛⋅tan
⋅𝑧sin −cos
sin +cos
()
2.1.5. Shear Deformation of the Root. Assuming that the
shear stress of the root section is evenly distributed, the
displacement of the point in the direction caused by
the shear deformation (shown in Figure (e)) is the same
as the displacement of the thread in the direction; this
displacement can be expressed as []
5=𝑧
sin +cos ⋅21−V2⋅cos +sin
𝑏
⋅
log𝑒+/2
−/2+1
2log𝑒42
2−1 ()
Mathematical Problems in Engineering
For ISO internal threads, the relationship between a, b, c,
and pitch is
=0.833
=0.5
=0.289 ()
Substituting () into (), (), (), (), (), (), and ()
one gets the relation
1𝑏 =3.468 𝑤𝑏
𝑏23
𝑏⋅1
21
2
𝑏+1−1
𝑏
⋅1
𝑏−12,
𝑏=1.6684 ()
𝑤𝑏 =𝑧
sin +cos 0.289⋅cos +⋅sin
−(0.4165−0.289⋅tan )sin −⋅cos ()
2𝑏 =0.51⋅6(1+V)cos +sin cot
5𝑏
⋅𝑧
sin +cos
()
3𝑏 =𝑧
sin +cos ⋅121−V2
𝑏2⋅0.289
⋅cos +⋅sin −(0.4165−0.289⋅tan )
⋅sin −⋅cos
()
4b=1−]𝑏𝑝
2𝑏sin −cos tan
⋅𝑧
sin +cos
()
5𝑏 =1.8449𝑧
sin +cos ⋅21−V2⋅cos +sin
𝑏
()
For ISO internal threads, the relationship between a,b,c,
and pitch is
=0.875
=0.5
=0.325 ()
Substituting () into (), (), (), (), (), (), and ()
type one gets the relation
1𝑛 =3.784 𝑤𝑛
𝑛22
𝑛⋅1
21
2
𝑛+1− 1
𝑛
⋅1
𝑛−12,
𝑛=1.751 ()
𝑤𝑛 =𝑧
sin +cos 0.325⋅cos +⋅sin
−(0.4375−0.325⋅tan )sin −⋅cos ()
2𝑛 =0.55962⋅ 6(1+V)cos +sin cot
5𝑏
⋅𝑧
sin +cos
()
3𝑛 =𝑧
sin +cos ⋅121−V2
𝑏2⋅0.325
⋅cos +⋅sin
−(0.4375−0.325⋅tan )sin −⋅cos
()
4n=2
0+2
𝑝
2
0−2
𝑝+]𝑛𝑝
2𝑛⋅sin −cos tan
⋅𝑧
sin +cos
()
5𝑛 =1.7928𝑧
sin +cos ⋅21−V2⋅cos +sin
𝑏
()
By adding these deformations separately, the total defor-
mation (shown in Figure ) of screw thread and nut thread
can be obtained under the action of force 𝑧.
𝑏=1𝑏 +2𝑏 +3𝑏 +4𝑏 +5𝑏 ()
𝑛=1𝑛 +2𝑛 +3𝑛 +4𝑛 +5𝑛 ()
e unit force per unit width of the axial direction can be
expressed as
Δ=𝑧
𝑧=1 ()
Under the action of unit force of axial unit width, the total
deformation of external thread and internal thread is
𝑏1 =1𝑏 +2𝑏 +3𝑏 +4𝑏 +5𝑏
𝑧()
and
𝑛1 =1𝑛 +2𝑛 +3𝑛 +4𝑛 +5𝑛
𝑧()
For threaded connections, at the x-axis of the load F,the
axial deformation of screws and nuts can be expressed as
𝑏()=𝑏1 ⋅
()
𝑛()=𝑛1 ⋅
()
Mathematical Problems in Engineering
x
L
Nut
L
w
Screw
read axial force distribution
Fx
Fb
(a)
0.5a0.5a
w
P
w
y
x
o
Nut
Screw
Partial view
(b)
F : read force.
Here is the length along the helical direction, and the
relation between the axial height and the length along the
helix direction can be represented by the following formula
according to the geometric relation shown in Figure .
=
sin ()
Here, is the lead angle of the thread shown in Figure ,
and then
𝑛()=𝑛1 ⋅
=𝑛1 ⋅
⋅
=𝑛1 ⋅sin ⋅
()
𝑏()=𝑏1 ⋅
=𝑏1 ⋅
⋅
=𝑏1 ⋅sin ⋅
()
Assume
⋅𝑏()=1
𝑏1 ⋅sin =𝑢𝑏𝑥 ()()
⋅𝑛()=1
𝑛1 ⋅sin =𝑢𝑛𝑥 ()()
Here, 𝑏𝑥(x)and𝑛𝑥(x) represent the stiness of the unit
axial length of the nut and the screw, respectively, for the unit
force.
e axial total deformation of the threaded connection at
is denoted as
𝑥()=𝑏()+𝑛()()
e stiness of the unit axial length of the threaded
connection is expressed as
𝑢𝑥 ()=
⋅𝑥()
=
÷𝑏1 ⋅sin ⋅
+𝑛1 ⋅sin ⋅
=1
𝑛1 +𝑏1⋅sin
()
AsshowninFigure(a),thethreadedconnectionstruc-
ture includes a nut body and a screw body. e nut is xed,
the screw is subjected to pulling force, the total axial force
is 𝑏, and the axial force at the threaded connection screw
is F(x).Ifthepositionofthebottomendfaceofthenutis
the origin and, at the position, the axial force is F(x), the
screw elongation amount 𝑏and the nut compression 𝑛can
be obtained from the following:
𝑏()=()
𝑏()𝑏()
𝑛()=()
𝑛()𝑛()
where 𝑏(x)and𝑛(x) are the vertical cross-sectional
areas of screws and nuts at the position. 𝑏and 𝑛are,
respectively, Young’s modulus of the screw body and Young’s
modulus of the nut body. Find the displacement gradient for
the expression, which is, respectively, expressed as
𝑏()
=1
𝑢𝑏𝑥 ()⋅2
2()
𝑛()
=1
𝑢𝑛𝑥 ()⋅2
2()
Here, 𝑢𝑏𝑥() = 1/(𝑏1 ⋅sin ),and𝑢𝑛𝑥() = 1/(𝑛1 ⋅
sin ).
As shown in Figure (a), the screw is subjected to the
tensile force b, with the bottom of the nut as the coordinate
origin, and the force at the x position is 𝑥,andthen
the elongation of the screw at x is ∫𝐿
𝑥𝑏() = 𝑏,
and the compressed shortening amount of the nut at x is
∫𝐿
𝑥𝑛() = 𝑛. e relationship between b,n,b,and
nis ∫𝐿
𝑥𝑏()+∫𝐿
𝑥𝑛()=[𝑏()+𝑛()]𝑥=𝐿 −[𝑏()+
𝑛()]𝑥=𝑥 (see Figures and (a),), and the partial derivative
of this relation can be obtained by the following formula:
𝑏()+𝑛()=𝑏()
+𝑛()
()
Substituting (), (), (), and () into () and
simplifying it
𝑛𝑛+𝑏𝑏
𝑏𝑏𝑛𝑛⋅𝑢𝑏𝑥𝑢𝑛𝑥
𝑢𝑏𝑥 +𝑢𝑛𝑥()=2()
2()
Mathematical Problems in Engineering
w
wForce analysis of thread
Deformation of thread
Total deformation of the thread when
the helix angle is not considered
Total deformation of the thread
when considering the helix angle
Calculation of thread stiffness read stiffness test FEA of thread stiffness
Comparison of results
Influence of friction coefficient on
stiffness and load distribution
b=
1b +
2b +
3b +
4b +
5b
1b ,
2b,
3b,
4b,
5b
1n,
2n,
3n,
4n,
5n
n=
1n +
2n +
3n +
4n +
5n
Kc=
L
∫
0
kx(x) =
L
∫
0
f(x)kux(x)dx
zb(x)=
b1 ·F
r =
b1 ·F
x ·x
r =
b1 · ＭＣＨ · F
x
zn(x)=
n1 ·F
r =
n1 ·F
x ·x
r =
n1 · ＭＣＨ · F
x
F : Schematic diagram of the article.
Let
=𝑛𝑛+𝑏𝑏
𝑏𝑏𝑛𝑛⋅𝑢𝑏𝑥𝑢𝑛𝑥
𝑢𝑏𝑥 +𝑢𝑛𝑥()
en
2()=2()
2()
From mathematical knowledge, the equation is a dier-
ential equation. e general solution of the equation can be
expressed as ()=1sinh +2cosh ()
As can be seen from Figure , the axial force at the rst
thread at the connection surface of the nut and the screw
is 𝑏, and the axial force at the last thread at the lower end
of the thread joint surface of the nut and screw is ; that is,
the boundary condition is F(x=)=𝑏and F(x=L)=. Taking
these boundary conditions into the equation will give 1=-
𝑏(cosh())/sinh()and 2=𝑏, so we get the expression
of the threaded connection axial load as
()=
𝑏cosh − cosh
sinh sinh ()
erefore, the axial force distribution density of the
thread connection along the direction can be expressed as
()=()
𝑏=cosh − cosh
sinh sinh ()
2.2. read Connection Stiness. e stiness in the axial
direction of the bolted connection is equal to the axial force
distribution of the threaded connection multiplied by the unit
stiness; i.e.,
𝑥()=()𝑢𝑥 ()()
e overall stiness of the bolt connection can be
expressed as
𝑐=𝐿
0𝑥()=𝐿
0()𝑢𝑥 () ()
Mathematical Problems in Engineering
w
w
Screw
x
x
Bending deformation
1
(a)
Screw
x
x
w
w
Shear deformation
2
(b)
Screw
x
x
w
w
3
Inclination deformation of thread root
(c)
Screw
w
w
x
x
4
Radial direction extended deformation and radial shrinkage deformation
(d)
Screw
x
x
w
w
Shear deformation of the root of the thread
5
(e)
F : read deformation caused by various reasons.
Substituting () and () into (), the stiness of the
bolt connection is expressed as
𝑐=𝐿
0𝑥()=𝐿
0()𝑢𝑥 ()
=1
𝑏1 +𝑛1⋅sin ⋅cosh −1
sinh
()
3. FEA Model
A D nite element model (shown in Figure ) was estab-
lished, and FEA was performed to analyze the inuence of
various parameters of the thread on the thread stiness. ese
parameters include material, thread length, pitch, etc.
e FEA soware ANSYS . was used for analysis.
During the analysis, the end face of the nut was xed (shown
in Figure ), the initial state of the model is shown in Figure ,
and an axial displacement 𝑥wasforcedtotheendfaceof
the screw. en, the axial force 𝑥ofthescrewendfacewas
extracted. e axial stiness of the threaded connection was
calculated by the FEM. e friction coecient of the thread
contact surface is set rst. In FEA, the contact algorithm
used is Augmented Lagrange. Figures (a)–(c) are the force
convergence curves for FEA of threaded connections. Figures
(a)–(c) are the eect of the reciprocal of the mesh size
ontheaxialforceobtainedbyFEA.WecanseefromFigures
(a)–(c) that as the mesh size decreases, the resulting axial
force gradually decreases, but when the mesh size is small to
a certain extent, the resulting axial force will hardly decrease.
e axial force at this time is the axial force required by
the author. With known displacements and axial force, the
stiness of the threaded connection can be calculated using
the formula
𝑐=𝑥
𝑥()
4. Tensile Test of Threaded Connections [1]
In order to verify the eectiveness of this paper method,
the experimental data of the experimental device in []
are used. In [], the electronic universal testing machine is
used to measure the load-defection data of samples, and the
Mathematical Problems in Engineering
y
x
o
c
Nut
w
y
x
o
c
Nut
wcos
wsin
w
w
wsina
w
wcosa
wsina-wcosa
Comprehensive deformation of nut
n=1n+2n+3n +4n+5n wz=wcos+wsin
(a)
w
w
yo
c
wcos
wsin
w
w
yo
wsina
wcosa
wsina-wcosa
c
x x
Screw Screw
Comprehensive deformation of screw
n=1n+2n+3n +4n+5n
wz=wcos+wsin
(b)
F : e comprehensive deformation of the thread.
0.5a 0.5a
y
x
o
h
c-y
c
w
w
F : e force on the thread.
test sample is made of brass. e tension value 𝑥𝑡 can be
read from the test machine. e axial deection of thread
connection can be represented by the displacement variation
𝐿between two lines as shown in Figure , which can be
measured by a video gauge []. In [], in order to obtain the
P
d2
d2
F : e lead angle of the thread.
most accurate data possible, each size of the thread is in a
small range of deformation during the tensile test, and each
size of the thread tensile test is performed times, and the
average value is calculated as the nal calculated data. Some
samples in the experiment are shown in Figure .
e stiness calculation formula is
𝑐=𝑥𝑡
𝐿.()
Mathematical Problems in Engineering
L
Screw
L
Nut fixing surface
Nut
Fb
(wn)x=x
(wb)x=x
[z＜(x)+zＨ(x)]x=x
[z＜(x)+zＨ(x)]x=L
F : Illustration of the elastic deformation of the screwed
portion of the threaded connection.
F : Initial state.
e materials used to make nuts and screws are brass.
Young’smodulusofbrassisGPa,andPoisson’sratiois.
[].
5. Results and Discussion
5.1. Stiness of readed Connections. Croccolo, D. [],
Nassar SA [], and Zou Q [] studied the coecient of
friction of the thread. According to the study by Zou Q and
Nassar SA, in the case of lubricating oil on the thread surface,
the friction coecient of the steel-steel thread connection
thread is ., and the friction coecient of aluminum-
aluminum thread connection thread is ..
In order to verify the correctness of the calculation results
of the theory presented in this paper, a variety of threaded
connections were used to calculate an experimental test.
In the nite element analysis and theoretical calculations
of this paper, Young’s modulus of steel is b=n=Gpa,
and Poisson’s ratio of steel is ., and the friction coecient
[, , ] is set to ..
Fixed support surface
Fx
F : readed connection deformed by axial forces.
e experimental data, FEA data, and Yamamoto method
data in Tables and are from literature []. As can be
seen from Tables and , the calculated values obtained
in this paper are all higher than the experimental results.
Perhaps the error is caused by the presence of a small
amount of impurities on the surface of the thread and
partial deformation of the thread inevitably and there is a
slip between the threaded contact surfaces. e theoretical
calculation results and FEA results in this paper have a small
error.
In Table , the eect of thread length on stiness is
presented. It can be seen that when the same nominal
diameter M, the same pitch P=., and the same material
steel are taken, when the thread engaged length is taken as
mm, mm, and mm, respectively, it is found that the longer
the thread engaged length, the greater the stiness and the
smaller the length of the bond, the smaller the stiness.
In the FEA, the method of this paper and the Yamamoto
method, Young’s Modulus of aluminum alloy is E=.GPa;
Poisson’s ratio of the aluminum alloy is .. e friction
coecient of the steel- steel threaded connection [, , ,
] is set to ., and the friction coecient of aluminum-
aluminum threaded connection [, , ] is set to .. Table
shows the eect of dierent materials on the stiness of
threaded connections. e two typ es of threaded connections
are made of two dierent materials, the steel and aluminum
alloys. It can be seen from the table that, under the condition
of the same pitch, the same nominal diameter, and the same
engaged length, the stiness of the steel thread connection is
larger than that when the material is aluminum.
In Table , it also shows the inuence of dierent pitches
on the stiness of the thread connection. It can be seen
that with the same engaged length, the same material, and
the same nominal diameter, the pitch is ., ., and ,
respectively, and we nd that the smaller the pitch, the greater
the stiness.
When using FEM to analyze the inuence of friction
factors on the stiness of threaded connections, the thread
Mathematical Problems in Engineering
T : Stiness of threaded connections with dierent engaged lengths [] (kN/mm).
No. Sizecodeofthreads Material Exp. eory FEA
is study Yamamoto method
M××
Brass
. . . .
M×× . . . .
M×× . . . .
Mathematical Problems in Engineering
(a) Nut (b) Screw
F : Finite element meshing of thread-bonded D nite element models.
T : Stiness of threaded connections with dierent engaged lengths (kN/mm).
No. Sizecodeofthreads Material eory FEA
is study Yamamoto method
M×.×
Steel
. . .
M×.× . . .
M×.× . . .
T : Stiness of threaded connections with dierent material (kN/mm).
No. Size code of thread Material eory FEA
is study Yamamoto method
M×.× Steel . . .
M×.× Aluminum alloy . . .
T : Stiness of threaded connections with dierent pitch (kN/mm).
No. Sizecodeofthreads Material eory
is study Yamamoto method
M×.×
Steel
. .
M×.×..
M××..
T : Stiness of threaded connections with dierent engaged lengths [] (kN/mm).
No. Sizecodeofthreads Material Exp. eory
is study Yamamoto method
M×× Brass . . .
M×× Brass . . .
specication is M××. and the friction coecients are .,
., ., ., ., and .. (as shown in Figures –)
Figures and show the results of stiness calcu-
lations. e thread size is M×.×andM××., the
material is steel, Poisson’s ratio of the material is ., Young’s
Modulus of the material is GPa, and the thread surface
friction coecient is taken as ., ., ., ., ., and
.. Calculated using the theories of this paper, FEA and
Yamamoto, respectively, and from Figures and , we can
see that the results of FEA are very similar to the results of the
theoretical calculations of this paper, the variation trend of
stiness with friction coecient is the same, and it increases
with the increase of friction coecient, and the results of
the FEA are in good agreement with those of the FEA;
Mathematical Problems in Engineering
Force convergence Force Criterion
97.1
21.4
4.47
2.21
0.448
0.229
0.05
Force (N)Time (s)
1
0
1234
1234
Cumulative Iteration
(a) M×.×, 𝜇=., Δ𝑥=.mm
108
20.2
0.711
0.448
0.133
0.025
Force (N)
Time (s)
1
0
1234
1234
Cumulative Iteration
Force convergence Force Criterion
(b) M×.×., 𝜇=., Δ𝑥=.mm
18.7
6.7
2.39
0.855
0.305
0.109
Force (N)
Time (s)
1
0
1234
1234
Cumulative Iteration
Force convergence Force Criterion
(c) M×.×., 𝜇=., Δ𝑥=.mm
F : FEA convergence curve.
however, Yamamoto theory does not consider the inuence
of the friction coecient on stiness, and this is obviously
unreasonable.
5.2. Eect of Friction Coecient on Axial Force Distribution.
TakethethreadsizeasM×.×., the axial load bis taken
as N, N, and N, respectively, and take the friction
coecients , ., ., and , respectively, to calculate the
axial force distribution of the thread. As can be seen from
Figure , when the friction coecient is , the curve bending
degree is the greatest, when the friction coecient is , the
curve bending degree is the lightest, the curve bending degree
is greater, indicating that the more uneven the distribution of
axial force, the smaller the curve bending degree, indicating
that the more uniform the distribution of axial force. We can
see that the friction coecient of thread surface has an eect
on the distribution of axial force.
6. Conclusion
is study provides a new method of calculating the thread
stiness considering the friction coecient and analyzes the
inuence of the thread geometry and material parameters
on the thread stiness and also analyzes the inuence of
Mathematical Problems in Engineering
1750
1800
1850
1900
1950
2000
Axial load F (N)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
Reciprocal of mesh size
(a) M×.×, 𝜇=., Δ𝑥=.mm
1300
1350
1400
1450
1500
1550
1600
Axial load F (N)
123456789100
Reciprocal of mesh size
(b) M×.×., 𝜇=., Δ𝑥=.mm
2 4 6 8 10121416180
Reciprocal of mesh size
1090
1100
1110
1120
1130
1140
1150
1160
1170
1180
Axial load F (N)
(c) M×.×., u=., Δ𝑥=.mm
F : Inuence of the reciprocal of nite element mesh size on axial force.
the friction coecient on the thread stiness and axial force
distribution.
() e results of the calculation of the thread stiness
calculated by the theoretical calculation method of
this study are basically consistent with the results of
the FEA. e results obtained by the test are smaller
than the calculated results. is is due to the inuence
of the thread manufacturing on the experimental
results.
() read-stiness is closely related to material proper-
ties, pitch, and thread length. We can obtain higher
stiness by increasing Young’s modulus of the mate-
rial, increasing the length of the thread, and reducing
the pitch.
() We can also increase the friction coecient of the
thread joint surface to increase the stiness of the
thread connection, but we have found that using this
method to increase the thread stiness is limited.
() In order to make the axial load distribution of the
thread uniform, we can reduce the friction coecient
of the thread surface, but we found that the use of this
method to improve the distribution of the axial force
of the thread has limited eectiveness.
Nomenclature
: Contact surface friction coecient
𝑧:Axialunitwidthforce,N
1: read bending deformation, mm
2: read shear deformation, mm
3: read root inclination deformation, mm
4: Radial direction extended deformation or
radial shrinkage deformation, mm
5: read root shear deformation, mm
: Bending moment of the unit load beam,
Nmm
Mathematical Problems in Engineering
read Screwing Length Nut
Screw
Internal thread
External threads
c
a
0.125P
0.125H
0.125H
0.167H
0.167P
P
P
a
H
c
b
0.25H
b
F : Part of experimental threaded connection samples and ISO internal thread and ISO external thread.
Collet
Specimen
Initial State End State
L
F : Tensile test [].
Mathematical Problems in Engineering
0.0010007 Max
0.00094174
0.00088273
0.00082371
0.0007647
0.00070569
0.00064668
0.00058767
0.00052866
0.00046965
0.00041064
0.00035163
0.00029262
0.00023361
0.0001746 Min
Unit:mm
F : Axial displacement for screws with a friction coecient of =.. Total force reaction=. N.
0.0010008Max
0.00094157
0.00088239
0.0008232
0.00076402
0.00070484
0.00064566
0.00058648
0.00052729
0.00046811
0.00040893
0.00034975
0.00029056
0.00023138
0.0001722 Min
Unit:mm
F : Axial displacement for screws with a friction coecient of =.. Total force reaction=. N.
𝑤: Bending moment of the beam under the
actual load, Nmm
:Areamomentofinertia,mm
4
𝑏:Young’smodulusofthescrewmaterial,
N/mm2
: Length of the beam, thread pitch line
height, mm
: Beam end section height, mm
: Beam section width, mm
1: Beam root section height and the beam end
section height ratio
𝑏: Ratio of the height of the screw thread
root section to the section height at the
middiameter
𝑛: Ratiooftheheightofthenutthread
root section to the section height at the
middiameter
4𝑏: Radial shrinkage deformation of the screw
thread, mm
4𝑛: Radial direction extended deformation of
the nut thread, mm
: Width of the thread root, mm
D0: Cylinder (nut) outer diameter, mm
dp: Eective diameter of the thread, mm
V𝑛: Poisson’s ratio of nut material
P: Pitch, mm
𝑤𝑏: Bending moment of the screw thread,
Nmm
1𝑏: read bending deformation of the screw
thread, mm
2𝑏: read shear deformation of the screw
thread, mm
3𝑏: read root inclination deformation of the
screw thread, mm
4𝑏: Radial direction extended deformation of
the screw thread, mm
5𝑏: read root shear deformation of the screw
thread, mm
𝑤𝑛: Bending moment of the nut thread, Nmm
1𝑛: read bending deformation of the nut
thread, mm
2𝑛: read shear deformation of the nut
thread, mm
3𝑛: read root inclination deformation of the
nut thread, mm
4𝑛: Radial direction extended deformation of
the nut thread, mm
5𝑛: read root shear deformation of the nut
thread, mm
Δ: Unit force per unit width of the axial
direction
Mathematical Problems in Engineering
0.0010008Max
0.00094147
0.00088219
0.0008229
0.00076362
0.00070433
0.00064505
0.00058576
0.00052648
0.00046719
0.00040791
0.00034862
0.00028934
0.00023005
0.00017077 Min
Unit:mm
F : Axial displacement for screws with a friction coecient of =.. Total force reaction=. N.
0.0010008Max
0.0009414
0.00088202
0.00082264
0.00076325
0.00070387
0.00064449
0.00058511
0.00052573
0.00046635
0.00040697
0.00034759
0.00028821
0.00022882
0.00016944 Min
Unit:mm
F : Axial displacement for screws with a friction coecient of =.. Total force reaction=. N.
𝑏1:Totaldeformationofexternal(screw)
thread, mm
𝑛1: Total deformation of internal (nut) thread,
mm
𝑏: e load on somewhere on the x-axis is F,
where the screw thread axial deformation,
mm
𝑛: e load on somewhere on the x-axis is
F, where the nut thread axial deformation,
mm
: Length along the helical direction, mm
: Lead angle of the thread, degree
𝑏𝑥(): Stiness of the unit axial length of the
screw, N/mm
𝑛𝑥(): Stiness of the unit axial length of the nut,
N/mm2
𝑥: Axial total deformation of the threaded
connection, mm
𝑢𝑥(): Stiness of the unit axial length of the
threaded connection, N/mm2
𝑏: Total axial force (load), N
𝑏:Attheposition, the axial force is (),the
screw elongation amount
𝑛:Attheposition, the axial
force is (),thenut
compression amount
𝑏(): Vertical cross-sectional
areas of screw at the
position, mm2
𝑛(): Vertical cross-sectional
areas of nut at the
position, mm2
𝑏:Young’smodulusofthe
screw body, N/mm2
𝑛:Young’smodulusofthenut
body, N/mm2
𝑥():Stinessintheaxial
direction ,N/mm
𝑐: Overall stiness of the
threaded connection,
N/mm
𝑥: Axial displacement, mm
𝑥: Total axial force, N
𝑥𝑡:Axialtensionload,N
𝑡: Overall stiness of the
threaded connection,
N/mm
Mathematical Problems in Engineering
0.0010008Max
0.0009437
0.00088195
0.00082253
0.0007631
0.00070368
0.00064426
0.00058484
0.00052542
0.000466
0.00040657
0.00034715
0.00028773
0.00022831
0.00016889 Min
Unit:mm
F : Axial displacement for screws with a friction coecient of =.. Total force reaction=. N.
0.0010008Max
0.00094135
0.0008819
0.00082244
0.00076299
0.00070354
0.00064409
0.00058464
0.00052519
0.00046574
0.00040628
0.00034683
0.00028738
0.00022793
0.00016848 Min
Unit:mm
F : Axial displacement for screws with a friction coecient of =.. Total force reaction=. N.
FEM
eory of this article
Sopwith method
×10
6
1.75
1.8
1.85
1.9
1.95
2
2.05
Stiffness ＋＃(N/mm)
0.05 0.1 0.15 0.2 0.25 0.30
Coefficient of friction
F : Eect of friction coecient on stiness. M×.×.
FEM
eory of this article
Sopwith method
×10
6
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
Stiffness ＋＃(N/mm)
0.05 0.1 0.15 0.2 0.25 0.30
Coefficient of friction
F : Eect of friction coecient on stiness. M××..
Mathematical Problems in Engineering
u=0
u=0.3
u=0.6
u=1
0
10
20
30
40
50
60
70
80
90
100
Force F (N)
1234560
Length L (mm)
2.46 2.47 2.482.45
Length L (mm)
32
33
34
35
36
Force F (N)
(a) Fb=N
0
50
100
150
200
250
300
350
Force F (N)
1234560
Length L (mm)
130
135
140
Force F (N)
2.19 2.2 2.212.18
Length L (mm)
u=0
u=0.3
u=0.6
u=1
(b) Fb=N
u=0
u=0.3
u=0.6
u=1
0
100
200
300
400
500
600
Force F (N)
1234560
Length L (mm)
2.71 2.72 2.73 2.742.7
Length L (mm)
160
165
170
175
Force F (N)
(c) Fb=N
F : Eect of Friction Coecient on Axial Force Distribution.
𝐿:Axialdeformationofthe
thread, mm
: read original triangle
high, mm.
Data Availability
e data used to support the ndings of this study are
included within the article.
Conflicts of Interest
e authors declare that they have no conicts of interest.
Acknowledgments
e authors would like to acknowledge support from
the National Natural Science Foundation of China [Grant
nos. , , and ] and Science &
Mathematical Problems in Engineering
Technology Planning Project of Shaanxi Province [Grant no.
JM].
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