Research ArticleStiffness Calculation Model of Thread ConnectionConsidering Friction Factors
Shi-kun Lu ,1,2 Deng-xin Hua ,1 Yan Li ,1 Fang-yuan Cui ,1 and Peng-yang Li 1
1Xiโan University of Technology, Xiโan 710048, China2Laiwu Vocational and Technical College, Laiwu 271100, China
Correspondence should be addressed to Deng-xin Hua; [email protected] and Yan Li; [email protected]
Received 9 September 2018; Revised 12 December 2018; Accepted 1 January 2019; Published 23 January 2019
Guest Editor: Yingyot Aue-u-lan
Copyright ยฉ 2019 Shi-kun Lu et al. This 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 threadedconnections. For the application of bolted connection in mechanical design, it is necessary to estimate the stiffness of threadedconnections. A calculation model for the distribution of axial force and stiffness considering the friction factor of the threadedconnection is established in this paper. The method regards the thread as a tapered cantilever beam. Under the action of thethread axial force, in the consideration of friction, the two cantilever beams interact and the beam will be deformed, thesedeformations include bending deformation, shear deformation, inclinationdeformation of cantilever beam root, shear deformationof cantilever beam root, radial expansion deformation and radial shrinkage deformation, etc.; calculate each deformation of thethread, respectively, and sum them, that is, the total deformation of the thread. In this paper, on the one hand, the threadedconnection stiffness was measured by experiments; on the other hand, the finite element models were established to calculatethe thread stiffness; the calculation results of the method of this paper, the test results, and the finite element analysis (FEA) resultswere compared, respectively; the results were found to be in a reasonable range; therefore, the validity of the calculation of themethod of this paper is verified.
1. Introduction
The bolts connect the equipment parts into a whole, whichis used to transmit force, moment, torque, or movement. Thebolt connection is widely used in various engineering fields,such as aviation machine tools, precision instruments, etc.The precision of the threaded connection affects the qualityof the equipment. Especially for high-end CNC machinetools, the precision of the thread connection is very high.Therefore, it is important to study the stiffness of the threadedconnection to improve the precision of the device. Manyresearchers have conducted research in this area.
Dongmei Zhang et al. [1], propose a method which cancompute the engaged screw stiffness, and the validity of themethod was verified by FEA and experiments. Maruyama etal. [2] used the point matching method and the FEM, basedon the experimental results of Boenick and the assumptionsmade by Fernlund [3] in calculating the pressure distributionbetween joint plates. Motash [4] assumed that the pressure
distribution on any plane perpendicular to the bolt axis hadzero gradient at r = ๐โ/2 and r = ๐๐, where it also vanishes.Theymainly use numericalmethods to calculate the influenceof different parameters on the stiffness of bolted connections.Kenny B, Patterson E A. [5] introduced a method formeasuring thread strains and stresses. Kenny B [6] et al.reviewed the distribution of loads and stresses in fasteningthreads. Miller D L et al. [7] established the spring model ofthread force analysis and, combined with the mathematicaltheory, analyzed the stress of the thread and comparedwith the FEA results and experimental results to verify thecorrectness of the spring mathematical model. Wang Wand Marshek K M. et al. [8] proposed an improved springmodel to analyze the thread load distribution, comparedthe load distributions of elastic threads and yielding threadjoints, and discussed the effect of the yield line on the loaddistribution.Wileman et al. [9] performed a two-dimensional(2D) FEA for members stiffness of joint connection. DeAgostinis M et al. [10] studied the effect of lubrication on
HindawiMathematical Problems in EngineeringVolume 2019, Article ID 8424283, 19 pageshttps://doi.org/10.1155/2019/8424283
2 Mathematical Problems in Engineering
thread friction characteristics or torque. Dario Croccoloet al. [11โ13] studied the effect of Engagement Ratio (ER,namely, the thread length over the thread diameter) on thetightening and untightening torque and friction coefficientof threaded joints using medium strength threaded lockingdevices. Zou Q. et al. [14, 15] studied the use of contactmechanics to determine the effective radius of the boltedjoint and also studied the effect of lubrication on frictionand torque-tension relationship in threaded fasteners. NassarS. A. et al. [16, 17] studied the thread friction and threadfriction torque in thread connection. Nassar S. A. et al.[18, 19] also investigated the effects of tightening speed andcoating on the torque-tension relationship and wear patternin threaded fastener applications in order to improve thereliability of the clamping load estimation in bolted joints.Kopfer et al. [20] believe that suitable formulations shouldconsider contact pressure and sliding speed; based on this, thecontribution shows experimental examples for main uncer-tainties of frictional behavior during tightening with differentmaterial combinations (results from assembly test stand).Kenny B et al. [21] reviewed the distribution of load andstress in the threads of fasteners. Shigley et al. [22] presentedan analytical solution for member stiffness, based on thework of Lehnhoff and Wistehuff [23]. Nasser [24], Mustoand Konkle [25], Nawras [26], and Nassar and Abbound[27] also proposed mathematical model for the bolted-jointstiffness. Qin et al. [28] established an analytical model ofbolted disk-drum joints and introduced its application todynamic analysis of joint rotor. Liu et al. [29] conductedexperimental and numerical studies on axially excited boltconnections.
There are also several authors that, starting from thenature of thread stiffness, from the perspective of threaddeformation, established a mathematical model of the calcu-lation of the distribution of thread axial force. The Sopwithmethod [30] and the Yamamoto method [31] received exten-sive recognition. The 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 stiffness of threadedconnections. The assumption for Yamamoto method is thatthe load per unit width along the helix direction is uniformlydistributed. In fact, for the three-dimensional (3D) helixthread, the load distribution is not uniform.Therefore, basedon the Yamamoto method, Dongmei Zhang et al. [1] proposea method which can compute the engaged screw stiffnessby considering the load distribution, and the validity of themethod was verified by FEA and experiments. The methodof Zhang Dongmei does not consider the influence of thefriction coefficient of the thread contact surface. In fact,the friction coefficient of the contact surface of the threadconnection has an influence on the distribution of the axialforce of the thread and the stiffness of the thread. Therefore,we propose a new method which can compute the engagedscrew stiffness more accurately by considering the effects offriction and the load distribution.The accuracy of themethodwas verified by the FEA and bolt tensile test. The flow chartof the article is shown in Figure 2.
2. Mathematical Model
2.1. Axial Load Distribution. According to Yamamoto [31],the thread is regarded as a cantilever beam, and the thread isdeformed under axial force and preload. These deformationsinclude the following (shown in Figure 3): thread bendingdeformation, thread shear deformation, thread root incli-nation deformation, thread root shear deformation, radialdirection extended deformation, or radial shrinkage defor-mation.
For the ISO thread, the axial deformation of the threadat ๐ฅ at the axial unit width force ๐คz is thread bendingdeformation ๐ฟ1, thread shear deformation ๐ฟ2, thread rootinclination deformation ๐ฟ3, thread root shear deformation๐ฟ4, and radial direction extended deformation ๐ฟ5 (nut) orradial shrinkage deformation ๐ฟ5 (screw), and calculate thesedeformations 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 frictioncoefficient is๐, when the sliding force along the inclined planeis greater than the friction force along the inclined plane, therelative sliding occurs between the two inclined planes, andthe axial unit width force (shown in Figure 3) is ๐ค๐ง; if theinfluence of the lead angle is ignored, the force per unit widthperpendicular to the thread surface can be expressed as
๐ค = ๐ค๐ง๐ sin ๐ผ + cos ๐ผ (1)
The force per unit width perpendicular to the threadsurface can be decomposed into the x-direction componentforce and the y-direction component force, respectively
๐ค cos ๐ผ = ๐ค๐ง cos ๐ผ๐ sin ๐ผ + cos ๐ผ (2)
and
๐ค sin ๐ผ = ๐ค๐ง sin ๐ผ๐ sin ๐ผ + cos๐ผ (3)
The friction generated along the slope is w๐; i.e.,๐ค๐ = ๐ค๐ง๐๐ sin ๐ผ + cos ๐ผ (4)
The force w๐ is also decomposed into x-direction forceand y-direction force, which are ๐ sin ๐ผ and ๐ cos ๐ผ, respec-tively.
๐ค๐ sin ๐ผ = ๐ค๐ง๐ sin ๐ผ๐ sin ๐ผ + cos ๐ผ (5)
๐ค๐ cos๐ผ = ๐ค๐ง๐ cos ๐ผ๐ sin ๐ผ + cos ๐ผ (6)
In the unit width, the thread is regarded as a rectangularvariable-section cantilever beam. Under the action of the
Mathematical Problems in Engineering 3
above-mentioned force, the thread undergoes bending defor-mation, and the virtual work done by the bending moment ๐ธon the beam ๐๐ฆ section is
๐ธ๐๐ = ๐ธ ๐๐ค๐ธ๐๐ผ (๐ฆ)๐๐ฆ (7)
According to the principle of virtual work, the deflection๐ฟ1 (see Figure 3(a)) of the beam subjected to the load is
๐ฟ1 = โซ๐0
๐ธ๐๐ค๐ธ๐๐ผ (๐ฆ)๐๐ฆ (8)
where ๐ธ is the bending moment of the unit load beam.๐๐ค is the bending moment of the beam under the actualload. I(y) is the area moment of inertia of the beam at ๐ฆ. ๐ธ๐ isYoungโs Modulus of the material. c is the length of the beam.Here, the forces are assumed as acting on the mean diameterof the thread.
As shown in Figure 5, the height โ(๐ฆ) of the beam sectionper unit width and the area moment of inertia ๐ผ(๐ฆ) of thesection can be expressed by using the function interpolation.
โ (๐ฆ) = [1 + (๐ฝ1 โ 1) (๐ โ ๐ฆ)๐ ] โ, (0 โค ๐ฆ โค ๐)
๐ผ (๐ฆ) = 112๐โ3 [1 + (๐ฝ1 โ 1) (๐ โ ๐ฆ)๐ ]3 , (0 โค ๐ฆ โค ๐)
๐ฝ1 = ๐โ
(9)
where h is the beam end section height; b is the beamsection width; ๐ฝ1 is the beam root section height and thebeam end section height ratio; see Figure 5.
FromFigure 5, the bendingmoment of the beam is relatedto the y-axis component of ๐ค and w๐, and these componentscause the beam to bend; therefore, the analytical solutionshows that the bending moment of the unit width beamsubjected to the friction force and the vertical load of thethread surface is
๐๐ค = ๐ค๐ง๐ sin ๐ผ + cos ๐ผ โ [cos๐ผ โ c + ๐ sin ๐ผ โ ๐โ sin ๐ผ(๐2 โ ๐ tan ๐ผ) + ๐ cos ๐ผ(๐2 โ ๐ tan ๐ผ)]
(10)
Substituting (10) and (9) to (8) and integrating to obtainthe analytical expression of the deflection ๐ฟ1 (shown inFigure 3(a)) of the cantilever beamwith variable cross-sectionunder load one has
๐ฟ1 = 12๐๐ค๐2๐ธ๐๐โ3 โ [12 ( 1๐ฝ21 + 1) โ 1๐ฝ1] โ 1(๐ฝ1 โ 1)2 (11)
2.1.2. Shear Deformation. Assume that the distribution ofshear stress on any section is distributed according to the
parabola [31] and the deformation ๐ฟ2 (see Figure 3(b)) causedby the shear force within the width of unit 1 is
๐ฟ2 = log๐ (๐๐) โ 6 (1 + V) (cos๐ผ + ๐ sin ๐ผ) cot ๐ผ5๐ธ๐
โ ๐ค๐ง๐ sin ๐ผ + cos ๐ผ(12)
2.1.3. Inclination Deformation of the Thread Root. Underthe action of the load, the thread surface is subjected to abending moment, and the root of the thread is tilted, asshown in Figure 3(c). Due to the inclination of the thread,axial displacement occurs at the point of action of the threadsurface force, and the axial displacement can be expressed as[31]
๐ฟ3 = ๐ค๐ง๐ sin ๐ผ + cos๐ผ โ 12๐ (1 โ V2)๐๐ธ๐๐2 โ [cos ๐ผ โ ๐
+ ๐ sin ๐ผ โ ๐ โ sin ๐ผ(๐2 โ ๐ tan ๐ผ)+ ๐ cos ๐ผ(๐2 โ ๐ tan ๐ผ)]
(13)
2.1.4. Deformation due to Radial Expansion and RadialShrinkage. According to the static analysis, the thread is sub-jected to radial force ๐ค sin ๐ผ โ ๐ค๐ cos ๐ผ (shown in Figure 4),and it is known from the literature [31] that the internal andexternal thread radial deformation (shown in Figure 3(d)) are
๐ฟ4b = (1 โ ]๐) ๐๐2๐ธ๐๐ tan ๐ผ โ ๐ค๐ง (sin ๐ผ โ ๐ cos ๐ผ)๐ sin ๐ผ + cos๐ผ (14)
and
๐ฟ4n = (๐ท20 + ๐2๐๐ท20 โ ๐2๐ + ]๐) ๐๐2๐ธ๐๐ โ tan ๐ผ
โ ๐ค๐ง (sin ๐ผ โ ๐ cos ๐ผ)๐ sin ๐ผ + cos ๐ผ
(15)
2.1.5. Shear Deformation of the Root. Assuming that theshear stress of the root section is evenly distributed, thedisplacement of the ๐ point in the ๐ฅ direction caused bythe shear deformation (shown in Figure 3(e)) is the sameas the displacement of the thread in the ๐ฅ direction; thisdisplacement can be expressed as [31]
๐ฟ5 = ๐ค๐ง๐ sin ๐ผ + cos ๐ผ โ 2 (1 โ V2) โ (cos ๐ผ + ๐ sin ๐ผ)๐๐ธ๐
โ {๐๐ log๐ (๐ + ๐/2๐ โ ๐/2) + 12 log๐ (4๐2๐2 โ 1)}(16)
4 Mathematical Problems in Engineering
For ISO internal threads, the relationship between a, b, c,and pitch ๐ is
๐ = 0.833๐๐ = 0.5๐๐ = 0.289๐
(17)
Substituting (17) into (9), (10), (11), (12), (13), (14), and (16)one gets the relation
๐ฟ1๐ = 3.468 ๐๐ค๐๐ธ๐๐2๐ฝ3๐ โ [12 ( 1๐ฝ2๐
+ 1) โ 1๐ฝ๐ ]โ 1(๐ฝ๐ โ 1)2 , ๐ฝ๐ = 1.6684
(18)
๐๐ค๐ = ๐ค๐ง๐ sin ๐ผ + cos ๐ผ [0.289๐ โ (cos ๐ผ + ๐ โ sin ๐ผ)โ (0.4165๐ โ 0.289๐ โ tan ๐ผ) (sin ๐ผ โ ๐ โ cos ๐ผ)]
(19)
๐ฟ2๐ = 0.51 โ 6 (1 + V) (cos ๐ผ + ๐ sin ๐ผ) cot ๐ผ5๐ธ๐
โ ๐ค๐ง๐ sin ๐ผ + cos๐ผ(20)
๐ฟ3๐ = ๐ค๐ง๐ sin ๐ผ + cos๐ผ โ 12๐ (1 โ V2)๐๐ธ๐๐2 โ [0.289๐
โ (cos ๐ผ + ๐ โ sin ๐ผ) โ (0.4165๐ โ 0.289๐ โ tan ๐ผ)โ (sin ๐ผ โ ๐ โ cos๐ผ)]
(21)
๐ฟ4b = (1 โ ]๐) ๐๐2๐ธ๐๐ (sin ๐ผ โ ๐ cos ๐ผ) tan ๐ผโ ๐ค๐ง๐ sin ๐ผ + cos๐ผ
(22)
๐ฟ5๐ = 1.8449๐ค๐ง๐ sin ๐ผ + cos๐ผ โ 2 (1 โ V2) โ (cos ๐ผ + ๐ sin ๐ผ)๐๐ธ๐ (23)
For ISO internal threads, the relationship between a, b, c,and pitch ๐ is
๐ = 0.875๐๐ = 0.5๐๐ = 0.325๐
(24)
Substituting (24) into (9), (10), (11), (12), (13), (15), and (16)type one gets the relation
๐ฟ1๐ = 3.784 ๐๐ค๐๐ธ๐๐2๐ฝ2๐ โ [12 ( 1๐ฝ2๐ + 1) โ 1๐ฝ๐ ]
โ 1(๐ฝ๐ โ 1)2 , ๐ฝ๐ = 1.751
(25)
๐๐ค๐ = ๐ค๐ง๐ sin ๐ผ + cos ๐ผ [0.325๐ โ (cos ๐ผ + ๐ โ sin ๐ผ)โ (0.4375๐ โ 0.325๐ โ tan ๐ผ) (sin ๐ผ โ ๐ โ cos๐ผ)]
(26)
๐ฟ2๐ = 0.55962 โ 6 (1 + V) (cos ๐ผ + ๐ sin ๐ผ) cot ๐ผ5๐ธ๐
โ ๐ค๐ง๐ sin ๐ผ + cos ๐ผ(27)
๐ฟ3๐ = ๐ค๐ง๐ sin ๐ผ + cos ๐ผ โ 12๐ (1 โ V2)๐๐ธ๐๐2 โ [0.325๐
โ (cos ๐ผ + ๐ โ sin ๐ผ)โ (0.4375๐ โ 0.325๐ โ tan ๐ผ) (sin ๐ผ โ ๐ โ cos๐ผ)]
(28)
๐ฟ4n = (๐ท20 + ๐2๐๐ท20 โ ๐2๐ + ]๐) ๐๐2๐ธ๐๐ โ (sin ๐ผ โ ๐ cos ๐ผ) tan ๐ผโ ๐ค๐ง๐ sin ๐ผ + cos ๐ผ
(29)
๐ฟ5๐ = 1.7928๐ค๐ง๐ sin ๐ผ + cos ๐ผ โ 2 (1 โ V2) โ (cos ๐ผ + ๐ sin ๐ผ)๐๐ธ๐ (30)
By adding these deformations separately, the total defor-mation (shown in Figure 4) of screw thread and nut threadcan be obtained under the action of force ๐ค๐ง.
๐ฟ๐ = ๐ฟ1๐ + ๐ฟ2๐ + ๐ฟ3๐ + ๐ฟ4๐ + ๐ฟ5๐ (31)
๐ฟ๐ = ๐ฟ1๐ + ๐ฟ2๐ + ๐ฟ3๐ + ๐ฟ4๐ + ๐ฟ5๐ (32)
The unit force per unit width of the axial direction can beexpressed as
๐ฮ = ๐ค๐ง๐ค๐ง = 1 (33)
Under the action of unit force of axial unit width, the totaldeformation of external thread and internal thread is
๐ฟ๐1 = ๐ฟ1๐ + ๐ฟ2๐ + ๐ฟ3๐ + ๐ฟ4๐ + ๐ฟ5๐๐ค๐ง (34)
and
๐ฟ๐1 = ๐ฟ1๐ + ๐ฟ2๐ + ๐ฟ3๐ + ๐ฟ4๐ + ๐ฟ5๐๐ค๐ง (35)
For threaded connections, at the x-axis of the load F, theaxial deformation of screws and nuts can be expressed as
๐ง๐ (๐) = ๐ฟ๐1 โ ๐๐น๐๐ (36)
๐ง๐ (๐) = ๐ฟ๐1 โ ๐๐น๐๐ (37)
Mathematical Problems in Engineering 5
x
L
NutL
w
Screw
๏ฟฝread axial force distribution
Fx
Fb
(a)
0.5a
0.5a
w
P
w
y
x
o
Nut
Screw
Partial view
(b)
Figure 1: Thread force.
Here ๐ is the length along the helical direction, and therelation between the axial height ๐ฅ and the length along thehelix direction can be represented by the following formulaaccording to the geometric relation shown in Figure 6.
๐ = ๐ฅsin ๐ฝ (38)
Here, ๐ฝ is the lead angle of the thread shown in Figure 6,and then
๐ง๐ (๐ฅ) = ๐ฟ๐1 โ ๐๐น๐๐ = ๐ฟ๐1 โ ๐๐น๐๐ฅ โ ๐๐ฅ๐๐ = ๐ฟ๐1 โ sin ๐ฝ โ ๐๐น๐๐ฅ (39)
๐ง๐ (๐ฅ) = ๐ฟ๐1 โ ๐๐น๐๐ = ๐ฟ๐1 โ ๐๐น๐๐ฅ โ ๐๐ฅ๐๐ = ๐ฟ๐1 โ sin ๐ฝ โ ๐๐น๐๐ฅ (40)
Assume๐๐น๐๐ฅ โ ๐ง๐ (๐ฅ) = 1๐ฟ๐1 โ sin ๐ฝ = ๐๐ข๐๐ฅ (๐ฅ) (41)
๐๐น๐๐ฅ โ ๐ง๐ (๐ฅ) = 1๐ฟ๐1 โ sin ๐ฝ = ๐๐ข๐๐ฅ (๐ฅ) (42)
Here, ๐๐๐ฅ(x) and ๐๐๐ฅ(x) represent the stiffness of the unitaxial length of the nut and the screw, respectively, for the unitforce.
The axial total deformation of the threaded connection at๐ฅ is denoted as
๐ง๐ฅ (๐ฅ) = ๐ง๐ (๐ฅ) + ๐ง๐ (๐ฅ) (43)
The stiffness of the unit axial length of the threadedconnection is expressed as
๐๐ข๐ฅ (๐ฅ) = ๐๐น๐๐ฅ โ ๐ง๐ฅ (๐ฅ)= ๐๐น๐๐ฅ รท (๐ฟ๐1 โ sin ๐ฝ โ ๐๐น๐๐ฅ + ๐ฟ๐1 โ sin ๐ฝ โ ๐๐น๐๐ฅ)= 1(๐ฟ๐1 + ๐ฟ๐1) โ sin ๐ฝ
(44)
As shown in Figure 1(a), the threaded connection struc-ture includes a nut body and a screw body. The nut is fixed,
the screw is subjected to pulling force, the total axial forceis ๐น๐, and the axial force at the threaded connection screw ๐ฅis F(x). If the position of the bottom end face of the nut isthe origin 0 and, at the ๐ฅ position, the axial force is F(x), thescrew elongation amount ๐๐ and the nut compression ๐๐ canbe obtained from the following:
๐๐ (๐ฅ) = ๐น (๐ฅ)๐ด๐ (๐ฅ) ๐ธ๐ (45)
๐๐ (๐ฅ) = ๐น (๐ฅ)๐ด๐ (๐ฅ) ๐ธ๐ (46)
where ๐ด๐(x) and ๐ด๐(x) are the vertical cross-sectionalareas of screws and nuts at the ๐ฅ position. ๐ธ๐ and ๐ธ๐ are,respectively, Youngโs modulus of the screw body and Youngโsmodulus of the nut body. Find the displacement gradient forthe expression, which is, respectively, expressed as
๐๐ง๐ (๐ฅ)๐๐ฅ = 1๐๐ข๐๐ฅ (๐ฅ) โ ๐2๐น๐๐ฅ2 (47)
๐๐ง๐ (๐ฅ)๐๐ฅ = 1๐๐ข๐๐ฅ (๐ฅ) โ ๐2๐น๐๐ฅ2 (48)
Here, ๐๐ข๐๐ฅ(๐ฅ) = 1/(๐ฟ๐1 โ sin ๐ฝ), and ๐๐ข๐๐ฅ(๐ฅ) = 1/(๐ฟ๐1 โ sin ๐ฝ).
As shown in Figure 1(a), the screw is subjected to thetensile force ๐นb, with the bottom of the nut as the coordinateorigin, and the force at the x position is ๐น๐ฅ, and thenthe elongation of the screw at x is โซ๐ฟ
๐ฅ๐๐(๐ฅ)๐๐ฅ = ๐ค๐,
and the compressed shortening amount of the nut at x isโซ๐ฟ๐ฅ๐๐(๐ฅ)๐๐ฅ = ๐ค๐. The relationship between ๐คb, ๐คn, ๐งb, and๐งn is โซ๐ฟ๐ฅ ๐๐(๐ฅ)๐๐ฅ + โซ๐ฟ
๐ฅ๐๐(๐ฅ)๐๐ฅ = [๐ง๐(๐ฅ) + ๐ง๐(๐ฅ)]๐ฅ=๐ฟ โ [๐ง๐(๐ฅ) +๐ง๐(๐ฅ)]๐ฅ=๐ฅ (see Figures 7 and 1(a),), and the partial derivative
of this relation can be obtained by the following formula:
๐๐ (๐ฅ) + ๐๐ (๐ฅ) = ๐๐ง๐ (๐ฅ)๐๐ฅ + ๐๐ง๐ (๐ฅ)๐๐ฅ (49)
Substituting (45), (46), (47), and (48) into (49) andsimplifying it
(๐ด๐๐ธ๐ + ๐ด๐๐ธ๐)๐ด๐๐ธ๐๐ด๐๐ธ๐ โ ๐๐ข๐๐ฅ๐๐ข๐๐ฅ(๐๐ข๐๐ฅ + ๐๐ข๐๐ฅ)๐น (๐ฅ) = ๐2๐น (๐ฅ)๐๐ฅ2 (50)
6 Mathematical Problems in Engineering
ww Force 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 ๏ฟฝread 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
โซ0
kx(x) =L
โซ0
f(x)kux(x)dx
zb(x) = b1 ยทF
r= b1 ยท
F
xยทx
r= b1 ยท ๏ผญ๏ผฃ๏ผจ ยท
F
x
zn(x) = n1 ยทF
r= n1 ยท
F
xยทx
r= n1 ยท ๏ผญ๏ผฃ๏ผจ ยท
F
x
Figure 2: Schematic diagram of the article.
Let
๐ = โ (๐ด๐๐ธ๐ + ๐ด๐๐ธ๐)๐ด๐๐ธ๐๐ด๐๐ธ๐ โ ๐๐ข๐๐ฅ๐๐ข๐๐ฅ(๐๐ข๐๐ฅ + ๐๐ข๐๐ฅ) (51)
Then
๐2๐น (๐ฅ) = ๐2๐น (๐ฅ)๐๐ฅ2 (52)
From mathematical knowledge, the equation is a differ-ential equation. The general solution of the equation can beexpressed as
๐น (๐ฅ) = ๐ถ1 sinh ๐๐ฅ + ๐ถ2 cosh ๐๐ฅ (53)
As can be seen from Figure 1, the axial force at the firstthread at the connection surface of the nut and the screwis ๐น๐, and the axial force at the last thread at the lower endof the thread joint surface of the nut and screw is 0; that is,the boundary condition is F(x=0)=๐น๐ and F(x=L)=0. Takingthese boundary conditions into the equation will give ๐ถ1=-๐น๐(cosh(๐๐ฟ))/sinh(๐๐ฟ) and ๐ถ2=๐น๐, so we get the expressionof the threaded connection axial load as
๐น (๐ฅ) = ๐น๐ (cosh ๐๐ฅ โ cosh ๐๐ฟsinh ๐๐ฟ sinh ๐๐ฅ) (54)
Therefore, the axial force distribution density of thethread connection along the ๐ฅ direction can be expressed as
๐ (๐ฅ) = ๐น (๐ฅ)๐น๐ = (cosh ๐๐ฅ โ cosh ๐๐ฟsinh ๐๐ฟ sinh ๐๐ฅ) (55)
2.2. Thread Connection Stiffness. The stiffness in the axialdirection ๐ฅ of the bolted connection is equal to the axial forcedistribution of the threaded connectionmultiplied by the unitstiffness; i.e.,
๐๐ฅ (๐ฅ) = ๐ (๐ฅ) ๐๐ข๐ฅ (๐ฅ) (56)
The overall stiffness of the bolt connection can beexpressed as
๐พ๐ = โซ๐ฟ0๐๐ฅ (๐ฅ) = โซ๐ฟ
0๐ (๐ฅ) ๐๐ข๐ฅ (๐ฅ) ๐๐ฅ (57)
Mathematical Problems in Engineering 7
ww
Screw
x
x
Bending deformation
1
(a)
Screw
x
x
ww
Shear deformation
2
(b)
Screw
x
x
ww
3
Inclination deformation of thread root(c)
Screw
ww
x
x
4
Radial direction extended deformation and radial shrinkage deformation
(d)
Screw
x
x
ww
Shear deformation of the root of the thread
5
(e)
Figure 3: Thread deformation caused by various reasons.
Substituting (44) and (55) into (57), the stiffness of thebolt connection is expressed as
๐พ๐ = โซ๐ฟ0๐๐ฅ (๐ฅ) = โซ๐ฟ
0๐ (๐ฅ) ๐๐ข๐ฅ (๐ฅ) ๐๐ฅ
= 1๐ (๐ฟ๐1 + ๐ฟ๐1) โ sin ๐ฝ โ cosh ๐๐ฟ โ 1sinh ๐๐ฟ
(58)
3. FEA Model
A 3D finite element model (shown in Figure 10) was estab-lished, and FEA was performed to analyze the influence ofvarious parameters of the thread on the thread stiffness.Theseparameters include material, thread length, pitch, etc.
The FEA software ANSYS 14.0 was used for analysis.During the analysis, the end face of the nut was fixed (shownin Figure 9), the initial state of themodel is shown in Figure 8,and an axial displacement ฮ๐ฅ was forced to the end face ofthe screw. Then, the axial force ๐น๐ฅ of the screw end face wasextracted. The axial stiffness of the threaded connection wascalculated by the FEM.The friction coefficient ๐ of the thread
contact surface is set first. In FEA, the contact algorithmused is Augmented Lagrange. Figures 11(a)โ11(c) are the forceconvergence curves for FEA of threaded connections. Figures12(a)โ12(c) are the effect of the reciprocal of the mesh sizeon the axial force obtained by FEA. We can see from Figures12(a)โ12(c) that as the mesh size decreases, the resulting axialforce gradually decreases, but when the mesh size is small toa certain extent, the resulting axial force will hardly decrease.The axial force at this time is the axial force required bythe author. With known displacements and axial force, thestiffness of the threaded connection can be calculated usingthe formula
๐พ๐ = ๐น๐ฅฮ๐ฅ (59)
4. Tensile Test of Threaded Connections [1]
In order to verify the effectiveness of this paper method,the experimental data of the experimental device in [1]are used. In [1], the electronic universal testing machine isused to measure the load-defection data of samples, and the
8 Mathematical Problems in Engineering
y
x
o
c
Nut
w
y
x
o
c
Nut
wcos
wsinw
w
wsina
w
wcosa
wsina-wcosa
Comprehensive deformation of nut
n=1n+2n+3n+4n+5n wz=wcos+wsin
(a)
w
w
y o
c
wcos
wsin
w
w
y o
wsina
wcosa
wsina-wcosa
cx x
Screw Screw
Comprehensive deformation of screw
n=1n+2n+3n+4n+5nwz=wcos+wsin
(b)
Figure 4: The comprehensive deformation of the thread.
0.5a
0.5a
y
x
o
h
c-yc
w
w
Figure 5: The force on the thread.
test sample is made of brass. The tension value ๐น๐ฅ๐ก can beread from the test machine. The axial deflection of threadconnection can be represented by the displacement variation๐ฟ๐ฟ between two lines as shown in Figure 14, which can bemeasured by a video gauge [1]. In [1], in order to obtain the
P
d2
d2
Figure 6: The lead angle of the thread.
most accurate data possible, each size of the thread is in asmall range of deformation during the tensile test, and eachsize of the thread tensile test is performed 10 times, and theaverage value is calculated as the final calculated data. Somesamples in the experiment are shown in Figure 13.
The stiffness calculation formula is
๐พ๐ = ๐น๐ฅ๐ก๐ฟ๐ฟ . (60)
Mathematical Problems in Engineering 9
L
ScrewL
Nut fixing surface
Nut
Fb
(wn) x
=x
(wb)
x=x
[z๏ผ(x
)+z ๏ผจ
(x)]
x=x
[z๏ผ(x
)+z ๏ผจ
(x)]
x=L
Figure 7: Illustration of the elastic deformation of the screwedportion of the threaded connection.
Figure 8: Initial state.
The materials used to make nuts and screws are brass.Youngโsmodulus of brass is 107GPa, and Poissonโs ratio is 0.32[1].
5. Results and Discussion
5.1. Stiffness of Threaded Connections. Croccolo, D. [12],Nassar SA [19], and Zou Q [32] studied the coefficient offriction of the thread. According to the study by Zou Q andNassar SA, in the case of lubricating oil on the thread surface,the friction coefficient of the steel-steel thread connectionthread is 0.08, and the friction coefficient of aluminum-aluminum thread connection thread is 0.1.
In order to verify the correctness of the calculation resultsof the theory presented in this paper, a variety of threadedconnections were used to calculate an experimental test.
In the finite element analysis and theoretical calculationsof this paper, Youngโs modulus of steel is ๐ธb= ๐ธn =200Gpa,and Poissonโs ratio of steel is 0.3, and the friction coefficient[12, 19, 32] is set to 0.08.
Fixed support surfaceFx
Figure 9: Threaded connection deformed by axial forces.
The experimental data, FEA data, and Yamamotomethoddata in Tables 1 and 5 are from literature [1]. As can beseen from Tables 1 and 5, the calculated values obtainedin this paper are all higher than the experimental results.Perhaps the error is caused by the presence of a smallamount of impurities on the surface of the thread andpartial deformation of the thread inevitably and there is aslip between the threaded contact surfaces. The theoreticalcalculation results and FEA results in this paper have a smallerror.
In Table 2, the effect of thread length on stiffness ispresented. It can be seen that when the same nominaldiameter M10, the same pitch P=1.5, and the same materialsteel are taken, when the thread engaged length is taken as 14mm, 9mm, and 6mm, respectively, it is found that the longerthe thread engaged length, the greater the stiffness and thesmaller the length of the bond, the smaller the stiffness.
In the FEA, the method of this paper and the Yamamotomethod, Youngโs Modulus of aluminum alloy is E=68.9GPa;Poissonโs ratio of the aluminum alloy is 0.34. The frictioncoefficient of the steel- steel threaded connection [12, 21, 22,32] is set to 0.08, and the friction coefficient of aluminum-aluminum threaded connection [12, 19, 32] is set to 0.1. Table 3shows the effect of different materials on the stiffness ofthreaded connections. The two types of threaded connectionsare made of two different materials, the steel and aluminumalloys. It can be seen from the table that, under the conditionof the same pitch, the same nominal diameter, and the sameengaged length, the stiffness of the steel thread connection islarger than that when the material is aluminum.
In Table 4, it also shows the influence of different pitcheson the stiffness of the thread connection. It can be seenthat with the same engaged length, the same material, andthe same nominal diameter, the pitch ๐ is 1.5, 1.25, and 1,respectively, and we find that the smaller the pitch, the greaterthe stiffness.
When using FEM to analyze the influence of frictionfactors on the stiffness of threaded connections, the thread
10 Mathematical Problems in Engineering
Table1:Stiffnessof
threaded
conn
ectio
nswith
different
engagedleng
ths[1](kN/m
m).
No.
Size
code
ofthreads
Material
Exp.
Theory
FEA
Thisstu
dyYamam
otometho
d1
M36
ร4ร32
Brass
3627.6
4201.9
4282.1
3630.9
2M36
ร4ร20
2664
.33152.9
3946
.12761.6
3M36
ร4ร12
1801.3
2074.2
3129.3
1816.8
Mathematical Problems in Engineering 11
(a) Nut (b) Screw
Figure 10: Finite element meshing of thread-bonded 3D finite element models.
Table 2: Stiffness of threaded connections with different engaged lengths (kN/mm).
No. Size code of threads Material Theory FEAThis study Yamamoto method
1 M10ร1.5ร14Steel
2095.26 2076.6 1965.212 M10ร1.5ร9 1759.23 2006.2 1770.023 M10ร1.5ร6 1354.70 1804.6 1551.27
Table 3: Stiffness of threaded connections with different material (kN/mm).
No. Size code of thread Material Theory FEAThis study Yamamoto method
1 M10ร1.5ร9 Steel 1759.23 2006.2 1770.022 M10ร1.5ร9 Aluminum alloy 607.51 682.11 597.99
Table 4: Stiffness of threaded connections with different pitch (kN/mm).
No. Size code of threads Material TheoryThis study Yamamoto method
1 M10ร1.5ร9Steel
1759.23 2006.22 M10ร1.25ร9 2083.17 2177.13 M10ร1ร9 2324.30 2375.2
Table 5: Stiffness of threaded connections with different engaged lengths [1] (kN/mm).
No. Size code of threads Material Exp. TheoryThis study Yamamoto method
1 M36ร3ร12 Brass 2085.3 2593.2 3525.12 M36ร2ร12 Brass 2396.2 3418.3 4008.9
specification isM6ร1ร6.1 and the friction coefficients are 0.01,0.05, 0.1, 0.2, 0.25, and 0.3. (as shown in Figures 15โ20)
Figures 21 and 22 show the results of stiffness calcu-lations. The thread size is M10ร1.5ร9 and M6ร1ร6.1, thematerial is steel, Poissonโs ratio of the material is 0.3, YoungโsModulus of the material is 200 GPa, and the thread surfacefriction coefficient is taken as 0.01, 0.05, 0.1, 0.2, 0.25, and
0.3. Calculated using the theories of this paper, FEA andYamamoto, respectively, and from Figures 21 and 22, we cansee that the results of FEA are very similar to the results of thetheoretical calculations of this paper, the variation trend ofstiffness with friction coefficient is the same, and it increaseswith the increase of friction coefficient, and the results ofthe FEA are in good agreement with those of the FEA;
12 Mathematical Problems in Engineering
Force convergence Force Criterion
97.121.44.472.21
0.4480.229
0.05Fo
rce (
N)
Tim
e (s) 1
0
1 2 3 4
1 2 3 4Cumulative Iteration
(a) M10ร1.5ร9, ๐=0.08, ฮ๐ฅ=0.001mm
10820.2
0.7110.4480.1330.025
Forc
e (N
)Ti
me (
s) 10
1 2 3 4
1 2 3 4Cumulative Iteration
Force convergence Force Criterion
(b) M8ร1.25ร6.5, ๐=0.08, ฮ๐ฅ=0.001mm
18.76.7
2.390.8550.3050.109
Forc
e (N
)Ti
me (
s) 10
1 2 3 4
1 2 3 4Cumulative Iteration
Force convergence Force Criterion
(c) M6ร1.0ร6.1, ๐=0.08, ฮ๐ฅ=0.001mm
Figure 11: FEA convergence curve.
however, Yamamoto theory does not consider the influenceof the friction coefficient on stiffness, and this is obviouslyunreasonable.
5.2. Effect of Friction Coefficient on Axial Force Distribution.Take the thread size asM6ร0.75ร6.1, the axial load ๐นb is takenas 100N, 350N, and 550N, respectively, and take the frictioncoefficients 0, 0.3, 0.6, and 1, respectively, to calculate theaxial force distribution of the thread. As can be seen fromFigure 23, when the friction coefficient is 1, the curve bendingdegree is the greatest, when the friction coefficient is 0, thecurve bending degree is the lightest, the curve bending degree
is greater, indicating that the more uneven the distribution ofaxial force, the smaller the curve bending degree, indicatingthat the more uniform the distribution of axial force. We cansee that the friction coefficient of thread surface has an effecton the distribution of axial force.
6. Conclusion
This study provides a new method of calculating the threadstiffness considering the friction coefficient and analyzes theinfluence of the thread geometry and material parameterson the thread stiffness and also analyzes the influence of
Mathematical Problems in Engineering 13
1750
1800
1850
1900
1950
2000 A
xial
load
F (N
)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50Reciprocal of mesh size
(a) M10ร1.5ร9, ๐=0.08, ฮ๐ฅ=0.001mm
1300
1350
1400
1450
1500
1550
1600
Axi
al lo
ad F
(N)
1 2 3 4 5 6 7 8 9 100Reciprocal of mesh size
(b) M8ร1.25ร6.5, ๐=0.08, ฮ๐ฅ=0.001mm
2 4 6 8 10 12 14 16 180Reciprocal of mesh size
1090
1100
1110
1120
1130
1140
1150
1160
1170
1180
Axi
al lo
ad F
(N)
(c) M6ร1.0ร6.1, u=0.08, ฮ๐ฅ=0.001mm
Figure 12: Influence of the reciprocal of finite element mesh size on axial force.
the friction coefficient on the thread stiffness and axial forcedistribution.
(1) The results of the calculation of the thread stiffnesscalculated by the theoretical calculation method ofthis study are basically consistent with the results ofthe FEA. The results obtained by the test are smallerthan the calculated results.This is due to the influenceof the thread manufacturing on the experimentalresults.
(2) Thread-stiffness is closely related to material proper-ties, pitch, and thread length. We can obtain higherstiffness by increasing Youngโs modulus of the mate-rial, increasing the length of the thread, and reducingthe pitch.
(3) We can also increase the friction coefficient of thethread joint surface to increase the stiffness of thethread connection, but we have found that using thismethod to increase the thread stiffness is limited.
(4) In order to make the axial load distribution of thethread uniform, we can reduce the friction coefficientof the thread surface, but we found that the use of thismethod to improve the distribution of the axial forceof the thread has limited effectiveness.
Nomenclature
๐: Contact surface friction coefficient๐ค๐ง: Axial unit width force, N๐ฟ1: Thread bending deformation, mm๐ฟ2: Thread shear deformation, mm๐ฟ3: Thread root inclination deformation, mm๐ฟ4: Radial direction extended deformation orradial shrinkage deformation, mm๐ฟ5: Thread root shear deformation, mm๐ธ: Bending moment of the unit load beam,Nโmm
14 Mathematical Problems in Engineering
๏ฟฝread Screwing Length Nut
Screw
Internal thread
External threads
ca
0.125P
0.125H
0.125H
0.167H0.167P
P
P
a
H
c
b
0.25H
b
Figure 13: Part of experimental threaded connection samples and ISO internal thread and ISO external thread.
Collet
Specimen
Initial State End State
L
Figure 14: Tensile test [1].
Mathematical Problems in Engineering 15
0.0010007 Max0.000941740.000882730.000823710.00076470.000705690.000646680.000587670.000528660.000469650.000410640.000351630.000292620.000233610.0001746 Min
Unit:mm
Figure 15: Axial displacement for screws with a friction coefficient of ๐=0.01. Total force reaction=1086.00 N.
0.0010008Max0.000941570.000882390.00082320.000764020.000704840.000645660.000586480.000527290.000468110.000408930.000349750.000290560.000231380.0001722 Min
Unit:mm
Figure 16: Axial displacement for screws with a friction coefficient of ๐=0.05. Total force reaction=1092.00 N.
๐๐ค: Bending moment of the beam under theactual load, Nโmm๐ผ: Area moment of inertia, mm4๐ธ๐: Youngโs modulus of the screw material,N/mm2๐: Length of the beam, thread pitch lineheight, mmโ: Beam end section height, mm๐ต: Beam section width, mm๐ฝ1: Beam root section height and the beam endsection height ratio๐ต๐: Ratio of the height of the screw threadroot section to the section height at themiddiameter๐ต๐: Ratio of the height of the nut threadroot section to the section height at themiddiameter๐ฟ4๐: Radial shrinkage deformation of the screwthread, mm๐ฟ4๐: Radial direction extended deformation ofthe nut thread, mm๐: Width of the thread root, mm
D0: Cylinder (nut) outer diameter, mmdp: Effective diameter of the thread, mmV๐: Poissonโs ratio of nut material
P: Pitch, mm๐๐ค๐: Bending moment of the screw thread,Nโmm๐ฟ1๐: Thread bending deformation of the screwthread, mm๐ฟ2๐: Thread shear deformation of the screwthread, mm๐ฟ3๐: Thread root inclination deformation of thescrew thread, mm๐ฟ4๐: Radial direction extended deformation ofthe screw thread, mm๐ฟ5๐: Thread root shear deformation of the screwthread, mm๐๐ค๐: Bending moment of the nut thread, Nโmm๐ฟ1๐: Thread bending deformation of the nutthread, mm๐ฟ2๐: Thread shear deformation of the nutthread, mm๐ฟ3๐: Thread root inclination deformation of thenut thread, mm๐ฟ4๐: Radial direction extended deformation ofthe nut thread, mm๐ฟ5๐: Thread root shear deformation of the nutthread, mm๐ฮ: Unit force per unit width of the axialdirection
16 Mathematical Problems in Engineering
0.0010008Max0.000941470.000882190.00082290.000763620.000704330.000645050.000585760.000526480.000467190.000407910.000348620.000289340.000230050.00017077 Min
Unit:mm
Figure 17: Axial displacement for screws with a friction coefficient of ๐=0.1. Total force reaction=1098.00 N.
0.0010008Max0.00094140.000882020.000822640.000763250.000703870.000644490.000585110.000525730.000466350.000406970.000347590.000288210.000228820.00016944 Min
Unit:mm
Figure 18: Axial displacement for screws with a friction coefficient of ๐=0.2. Total force reaction=1108.40 N.
๐ฟ๐1: Total deformation of external (screw)thread, mm๐ฟ๐1: Total deformation of internal (nut) thread,mm๐ง๐: The load on somewhere on the x-axis is F,where the screw thread axial deformation,mm๐ง๐: The load on somewhere on the x-axis isF, where the nut thread axial deformation,mm๐: Length along the helical direction, mm๐ฝ: Lead angle of the thread, degree๐๐๐ฅ(๐ฅ): Stiffness of the unit axial length of thescrew, N/mm๐๐๐ฅ(๐ฅ): Stiffness of the unit axial length of the nut,N/mm2๐ง๐ฅ: Axial total deformation of the threadedconnection, mm๐๐ข๐ฅ(๐ฅ): Stiffness of the unit axial length of thethreaded connection, N/mm2๐น๐: Total axial force (load), N๐๐: At the ๐ฅ position, the axial force is๐น(๐ฅ), thescrew elongation amount
๐๐: At the ๐ฅ position, the axialforce is ๐น(๐ฅ), the nutcompression amount๐ด๐(๐ฅ): Vertical cross-sectionalareas of screw at the ๐ฅposition, mm2๐ด๐(๐ฅ): Vertical cross-sectionalareas of nut at the ๐ฅposition, mm2๐ธ๐: Youngโs modulus of thescrew body, N/mm2๐ธ๐: Youngโs modulus of the nutbody, N/mm2๐๐ฅ(๐ฅ): Stiffness in the axialdirection ๐ฅ, N/mm๐พ๐: Overall stiffness of thethreaded connection,N/mmฮ๐ฅ: Axial displacement, mm๐น๐ฅ: Total axial force, N๐น๐ฅ๐ก: Axial tension load, N๐พ๐ก: Overall stiffness of thethreaded connection,N/mm
Mathematical Problems in Engineering 17
0.0010008Max0.00094370.000881950.000822530.00076310.000703680.000644260.000584840.000525420.0004660.000406570.000347150.000287730.000228310.00016889 Min
Unit:mm
Figure 19: Axial displacement for screws with a friction coefficient of ๐=0.25. Total force reaction=1110.6 N.
0.0010008Max0.000941350.00088190.000822440.000762990.000703540.000644090.000584640.000525190.000465740.000406280.000346830.000287380.000227930.00016848 Min
Unit:mm
Figure 20: Axial displacement for screws with a friction coefficient of ๐=0.3. Total force reaction=1114.2 N.
FEM๏ฟฝeory of this articleSopwith method
ร 106
1.75
1.8
1.85
1.9
1.95
2
2.05
Stiff
ness
๏ผ๏ผ
(N/m
m)
0.05 0.1 0.15 0.2 0.25 0.30Coefficient of friction
Figure 21: Effect of friction coefficient on stiffness. M10ร1.5ร9.
FEM๏ฟฝeory of this articleSopwith method
ร 106
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
Stiff
ness
๏ผ๏ผ
(N/m
m)
0.05 0.1 0.15 0.2 0.25 0.30Coefficient of friction
Figure 22: Effect of friction coefficient on stiffness. M6ร1ร6.1.
18 Mathematical Problems in Engineering
u=0u=0.3u=0.6u=1
0
10
20
30
40
50
60
70
80
90
100Fo
rce F
(N)
1 2 3 4 5 60Length L (mm)
2.46 2.47 2.482.45Length L (mm)
32
33
34
35
36
Forc
e F (N
)
(a) Fb=100N
0
50
100
150
200
250
300
350
Forc
e F (N
)
1 2 3 4 5 60Length L (mm)
130
135
140
Forc
e F (N
)
2.19 2.2 2.212.18Length L (mm)
u=0u=0.3u=0.6u=1
(b) Fb=350N
u=0u=0.3u=0.6u=1
0
100
200
300
400
500
600
Forc
e F (N
)
1 2 3 4 5 60Length L (mm)
2.71 2.72 2.73 2.742.7Length L (mm)
160
165
170
175
Forc
e F (N
)
(c) Fb=550N
Figure 23: Effect of Friction Coefficient on Axial Force Distribution.
๐ฟ๐ฟ: Axial deformation of thethread, mm๐ป: Thread original trianglehigh, mm.
Data Availability
The data used to support the findings of this study areincluded within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to acknowledge support fromthe National Natural Science Foundation of China [Grantnos. 51675422, 51475366, and 51475146] and Science &
Mathematical Problems in Engineering 19
Technology Planning Project of Shaanxi Province [Grant no.2016JM5074].
References
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[3] I. Fernlund, A Method to Calculate the Pressure Between Boltedor TU, 1962. Riveted Plates, Chalmers University of Technology,1961.
[4] N. Motosh, โDetermination of joint stiffness in bolted connec-tions,โ Journal of Engineering for Industry, vol. 98, no. 3, pp. 858โ861, 1976.
[5] B. Kenny and E. A. Patterson, โLoad and stress distribution inscrew threads,โ Experimental Mechanics, vol. 25, no. 3, pp. 208โ213, 1985.
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