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Shou-Jun Chene-mail: [email protected]

Qi An1

e-mail: [email protected]

School of Mechanical and Power Engineering,East China University of Science and Technology,

130 Meilong Road,Xuhui District,

Shanghai 200237, P. R. China

Yi ZhangWuxi Seamless Oil Pipe Co., Ltd.,

Wuxi 214028, P. R. China

Lian-Xin Gao

Qiang Lie-mail: [email protected]

School of Mechanical and Power Engineering,East China University of Science and Technology,

130 Meilong Road, Xuhui District,Shanghai 200237, P. R. China

Loading Analysis on the ThreadTeeth in Cylindrical Pipe ThreadConnectionThis paper presents a new analytical method that can calculate the load distribution onthe thread teeth in cylindrical pipe threaded connection. The new method was developedby analyzing each male and female thread tooth from the connection on the basis ofelastic mechanics. By using this method, the load distribution on each thread tooth canbe calculated with the tightening torque and thread numbers. By applying the newmethod on the sample of API 88.9 mm round threaded connection, the obtained resultsshow that the load on thread tooth mainly concentrates on the last four or five threadsengaged. By using the finite element analysis method to the same sample validates thenew method. The new method proposed in this paper is practical and convenient becauseit can be applied to calculate the load and deformation on each thread tooth just withtightening torque and thread numbers, which is easier to implement in practice.�DOI: 10.1115/1.4000729�

Keywords: cylindrical pipe threaded connection, analytical method, elastic mechanics,load distribution, FEA method

IntroductionFailures occurring in threaded connections are the main conse-

uences leading to oil tubing and casing accidents. Surveys �1–3�how that 80% of thread failure occur at the joint of threadedonnections and highlight the joint as the weakest point in oilubing and casing. Improving the quality of threaded connectionsill consequently have a positive effect on deep oil drillingrojects.

Previous references indicate that each thread tooth is under dif-erent load situation, which leads to the damage or destruction ofhreaded connection. The research of load distribution on threadooth was analyzed in the last century in a lot of fine analytical,xperimental, and finite element analysis �FEA� papers. The firstpproach is developed on the basis of analytical method. The firstnalytical models were developed by Maduschka �4� and Sopwith5� and enabled the designers to know the load distribution inylindrical screw threads. The results clearly show how more thanalf of the total load induced by the make-up torque is carried byhe first three or four threads engaged. Yazawa and Hongo �6�erived an expression for the distribution of load in the screw-hread of a bolt-nut connection that was subjected not only toxial forces but also simultaneously to tangential forces and bend-ng moments. By using an extensometer on the outside of a nut,oodier �7� investigated the distribution of thread loads and de-

ormations. He showed that the load distribution was governed bynumber of mechanisms: stretching of the bolt, compression in

he nut, bending of the threads, and circumferential stretching.ang and Marshek �8� developed a spring model to predict the

oad distribution along the threaded portions of a bolt and nutonnection and a pipe threaded connection. Heywood �9� devel-

1Corresponding authorContributed by the Pressure Vessel and Piping Division of ASME for publication

n the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received April 6, 2009;nal manuscript received November 18, 2009; published online May 4, 2010. Assoc.
ditor: Donald Mackenzie.
ournal of Pressure Vessel Technology Copyright © 20

aded 25 Nov 2010 to 193.48.36.131. Redistribution subject to ASME

oped methods for calculating thread fillet stress, joint efficiency,and load carrying capacity. He showed that determination andmanipulation of load distribution played an important role in com-puting cyclic and static load capacity. The references mentionedabove have developed some calculation models but these are com-paratively complicated and not convenient for application.

The second approach is developed on the basis of experimentaltest. Yuan et al. �10� made a nonlinear finite element analysis inAPI �American Petroleum Institute� round threaded connectionunder make-up torque and axial tensile load. In their experimentaltest, foil gauges were placed on the inner surface of the malethread and the outer surface of the female thread along the axialand circumferential direction. The results show that the operationsof make-up and break-out in oilfields have a strong impact on theservice life of oil tubing threaded connection. Kim et al. �11�made a twist compression test to investigate the galling in forminggalvanized advanced high strength steels. Placido �12� conductedsome experiments on full and reduced scale sample of aluminumdrill pipes under cyclic bending and constant tensile loads to in-vestigate their fatigue mechanism. Stoeckly and Macke �13� de-veloped a testing apparatus that actually measured the axial dis-placement of the threads. They modified the Sopwith equations toaccommodate tapered screw-thread connections in order to com-pare their experimental data with Sopwith’s. They concluded thatthe coefficient of friction between the lubricated bolt and nut haslittle effect on the thread load distribution. However, these experi-mental tests focus on limited aspects of threaded connection prob-lems and these experimental methods are difficult for applicationin practice.

The third approach to calculate the load distribution on threadtooth is based on finite element method �FEM�. In the last fewdecades, the finite element method has represented a powerful toolused to evaluate both the load distribution and stress concentrationon a thread tooth simultaneously by taking the nonlinear behaviorof a material into consideration. O’Hara �14�, for example, used
two-dimensional axisymmetric models of hypothetical thread pro-
JUNE 2010, Vol. 132 / 031202-110 by ASME

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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les to compare his analysis with Heywood’s. The thread projec-ions were based on characteristic lengths and not on the dimen-ions of the real components. There were indications of generalgreement with Heywood’s equations, except for the root load.retl and Cook �15� developed a unique finite element modeling

echnique for threaded connections. Their numerical results foronventional and tapered thread agreed well with theoretical andxperimental results in literature. Tafreshi and Dover �16� ana-yzed the threaded connections using FEM to obtain the locationf maximum stress concentration at the drilling string tool jointnder axial loading, pure torsion, and bending. Finally, they foundhe most concentration of stress is under the action of axial load-ng. They used node to node contact elements to model the contactegion between pin and box but contact stresses were not reported.

acdonald and Deans �17� obtained SCFs �stress concentrationactors� and induced stresses at the threaded connection of drillingtring using an axisymmetric 2D model via the finite elementethod. In this modeling, connection threaded area was modeled

s a set of parallel teeth. They found out that the location of highocal stresses is at the root of the first engaged tooth in the pin andhe last engaged tooth in the box. Bahai �18� used a 2D finitelement modeling to calculate the SCFs for the case of APIhreaded connectors used in drill string applications. The threadedoint was subjected to preloading, axial loading, and bendingoads. Due to the nonlinear governing equation of materials andhe complex geometry structure being easily incorporated into theE model, FEM is usually adopted for stress analysis of threadedonnections.

However, most of the existing methods were developed on theasis of Lame stress analysis and the conclusions could just cal-ulate the joint strength of threaded connections. These methodsould just analyze the deformation of male and female threads byssuming them as a column and the load distribution on eachhread tooth is also treated as a concentrated load. These result inhe inaccuracy of calculation on the teeth loading in threaded con-ection.

This paper applied the theory of elastic mechanics and estab-ished a new analytical method, which was based on the actualhread contact surface and can calculate the load distribution onhread surface easily. This method makes a loading analysis on aingle thread tooth in round threaded connection. By applying thisroposed method on the sample of API 88.9 mm round threadedonnection and using the FEA method to verify the calculatedesults, validates the analytical method proposed in this paper.

Model Development

2.1 Loading Analysis on Threaded Connection. The loadituation on threaded connection is complicated and it is difficulto establish the mechanical model that can obtain the calculationf deformation and load distribution on each thread tooth. In thisaper, the model was established by some assumptions. The ana-ytical assumptions in this paper are as follows:

�1� the thread deformation is in the range of elastic mechanics�2� do not consider the influence of pitch angle�3� do not consider the bending deformation on each thread

tooth�4� the load distribution on each thread tooth is a uniform load

Figure 1 shows the schematic of round threaded connection.hen the male and female thread contact smoothly with each

ther, there are no loads occurring on the thread tooth. If we stillcrew the male thread after the male thread reach the last femalehread under the tightening torque T, the elastic deformation �nill create on the last male and female thread tooth. When the

ightening torque is vanished, the counter force Fn caused by the
eformation will have an effect on the last male thread tooth and
31202-2 / Vol. 132, JUNE 2010


that result in the deformation of thread tooth �shown in Fig. 2�. Inthis condition, the male thread is compressed and the femalethread is stretched.

2.2 Loading Analysis on Male Thread Teeth. The first stepis to analyze the male thread as well as the load distribution oneach male thread tooth.

Figure 3 shows the loading analysis on each male thread tooth.Figure 3�a� describes the loading on thread 1 and Fig. 3�b� showsthe loading on thread i−1. In Fig. 3�c�, the load on the last malethread tooth consists of �1� the uniform load f�zn� from the femalethread to the contact surface, �2� the force Fn−1 from the threadn−1, and �3� the force Fn exerted by the right side. Except for thethread 1, all the other threads have the similar force condition asthe last one �shown in Fig. 3�b��. The thread 1 just has the uniformload f�z1� and the force F1 from the second male thread �shown inFig. 3�a��.

According to this relation, we can obtain equations as follows:

Fn−1 = Fn − f�zn�D − d

2

Fn−2 = Fn−1 − f�zn−1�D − d

2

Fi = Fi+1 − f�zi+1�D − d

2�1�

F1 = F2 − f�z2�D − d

2

F1 = f�z1�D − d

2

Now, take the single thread tooth into consideration and select thelast thread as the research object.

In Fig. 4, to the left side of male thread tooth, the force condi-tion on thread tooth consists of �1� the uniform load f�zn�, �2� the

Fig. 1 The schematic of round threaded connection

Fig. 2 Load and deformation on last male thread tooth

Transactions of the ASME


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orce Fn−1 from the former thread, and �3� the force Fn from theight side.

The compound force of the uniform load f�zn� in the direction xs

f�zn�x

cos �sin � = f�zn�x tan �

he deformation �n in the location x should meet the equation asollows:

�n

�x=

Fn − f�zn�x tan �

�b��D

2− x tan ��2

− �d0

2�2�

hen, the axial deformation in left side could be expressed as

�n =0

D−d2 tan � Fn − f�zn�x tan �

�b��D

2− x tan ��2

− �d0

2�2�dx �2�

The loading analysis on the right side of male thread is shownn Fig. 5 and the force Fn caused by the deformation just exerts onhe right thread tooth. Like the left side of male thread tooth, inhe direction x, the deformation �n� on the right side of male threadooth is

�n� =0

D−d2 tan � Fn

�b��D

2− x tan ��2

− �d0

2�2�dx �3�

ccording to Eqs. �2� and �3�, the total deformation of the lastale thread tooth could be expressed as

Fig. 3 Loading analysis on each male thread t

ig. 4 Loading analysis on the left side of last male thread
ooth
ournal of Pressure Vessel Technology


�n + �n� =0

D−d2 tan � 2Fn − f�zn�x tan �

�b��D

2− x tan ��2

− �d0

2�2�dx

= −1

tan �·

1

Eb��− 4Fn + �D + d0�f�zn�

2d0In

d + d0

D + d0

+4Fn − �D − d0�f�zn�

2d0In

d − d0

D − d0� �4�

By applying the same method and make the same analysis toother threads, it could be found that the load distribution on otherthreads is similar to the last one. So the load situation on eachmale thread tooth can be obtained as follows:

�n + �n� =0


�b��D

2− x tan ��2

− �d0

2�2�dx

�n−1 + �n−1� =0

D−d2 tan � 2Fn−1 − f�zn−1�x tan �

�b��D

2− x tan ��2

− �d0

2�2�dx

�i + �i� =0

D−d2 tan � 2Fi − f�zi�x tan �

�b��D

2− x tan ��2

− �d0

2�2�dx

h: „a… thread 1, „b… thread i-1, and „c… thread n

Fig. 5 Loading analysis on the right side of last male thread

oot

tooth

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�1 + �1� =0

D−d2 tan � 2F1 − f�z1�x tan �

�b��D

2− x tan ��2

− �d0

2�2�dx

ccording to the derivation above, the total axial deformation of male thread teeth could be expressed as

� = �n + �n� + �n−1 + �n−1� + ¯ + �i + �i� + ¯ + �1 + �1�

=0

D−d2 tan � 2�Fn + Fn−1 + ¯ + Fi + ¯ + F1� − �f�zn� + f�zn−1� + ¯ + f�zi� + ¯ + f�z1��x tan �

Eb��D

2− x tan ��2

− �d0

2�2� dx �5�

Fig. 6 Loading analysis on last female thread tooth: „a… thread n, „b… the left side, and „c… the right side

2.3 Loading Analysis on Female Thread Teeth. In Fig. 6�a�,e consider the load on the last female thread tooth and the forcen exerts on the single female thread tooth. By apply the samenalytical method, the load situation on the left side is shown inig. 6�b� and the right side is shown in Fig. 6�c�.As the left side on female thread tooth, in the direction x, the

eformation �n on the left side of female thread tooth is

�n =0

D−d2 tan � Fn − f�zn�x tan �

�n��D0

2�2

− �D

2− x tan ��2�dx �6�

As the right side of female thread tooth in the direction x, theeformation �n� on the right side of female thread tooth is

�n� =0

D−d2 tan � Fn

�n��D0

2�2

− �D

2− x tan ��2�dx �7�

Then, according to Eqs. �6� and �7�, the total deformation ofnal female thread tooth could be expressed as

�n + �n� =0


�n��D0

2�2

− �D

2− x tan ��2�dx

=1

tan �·

1

En��− 4Fn + �D0 + D�f�zn�

2D0In

D0 + d

D0 + D

+4Fn + �D0 − D�f�zn�

2D0In

D0 − d

D0 − D� �8�

31202-4 / Vol. 132, JUNE 2010


The load situation on each female thread tooth could be ex-pressed as follows:

�n + �n� =0


�n��D0

2�2

− �D


�n−1 + �n−1� =0

D−d2 tan � 2Fn−1 − f�zn−1�x tan �

�n��D0

2�2

− �D


�i + �i� =0

D−d2 tan � 2Fi − f�zi�x tan �

�n��D0

2�2

− �D


�1 + �1� =0

D−d2 tan � 2F1 − f�z1�x tan �

�n��D0

2�2

− �D


where the force Fn is the function in Eq. �1�.So according to the analysis above, the total axial deformation

of female thread teeth could be expressed as



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� = �n + �n� + �n−1 + �n−1� + ¯ + �i + �i� + ¯ + �1 + �1�

=0

D−d2 tan � 2�Fn + Fn−1 + ¯ + Fi + ¯ + F1� − �f�zn� + f�zn−1� + ¯ + f�zi� + ¯ + f�z1��x tan �

En��D0

2�2

− �D

2− x tan ��2� dx �9�

2.4 Derivation of Deformation and Covariant Equation. Consider that the total deformation of male thread tooth is equal to thene of female thread, we can find the deformation and covariant equation about male and female threads

�n + �n� = �n + �n� −1

Eb·

4FnIn�d − d0��D + d0��D − d0��d + d0�

2d0−

1

Eb

f�zn��D + d0�Ind + d0

D + d0− �D − d0�In

d − d0

D − d0�

2d0

=1

En·

4FnIn�D0 − d��D0 + D��D0 − D��D0 + d�

2D0+

1

En·

f�zn��D0 + D�InD0 + d

D0 + D+ �D0 − D�In

D0 − d

D0 − D�

2D0

y simplification, it becomes

f�zn� = kFn �10�here

k =

−En

Eb·

D0

d0In

�d − d0��D + d0��D − d0��d + d0�

− 4In�D0 − d��D0 + D��D0 − D��D0 + d�

En

Eb·

D0

d0��D + d0�In

d + d0

D + d0− �D − d0�In

d − d0

D − d0� + �D0 + D�In

D0 + d

D0 + D− �D0 − D�In

D0 − d

D0 − D

Equation �10� shows the linear relation of uniform load f�zn�nd Fn.

2.5 Relation Between Load on Thread Tooth and Tighten-ng Torque. According to the calculating method on frictionalorque, the torque on each thread tooth could be expressed as

�Tn = f�zn��x

cos ��d

2+ x tan ��

Tn =0

D−d2 tan � f�zn�

cos ��d

2+ x tan ��dx

o the total frictional torque in thread connection is

T = i=1

n

f�zi�0

D−d2 tan � 1

cos ��d

2+ x tan ��dx

= i=1

n

f�zi��

8 sin ��D2 − d2� �11�

et

k1 =1

8 sin ��D2 − d2�

hen Eq. �11� could be transferred as

T = k1�i=1

n

f�zi� �12�

n Eq. �1�, we can find

Fn = �f�z1� + f�z2� + f�z3� + ¯ + f�zn��D − d

2=

T

�k1·

D − d

2

�13�

y combining Eqs. �1� and �12�, we can obtain



f�zn� = kFn =kT

� · k1·

D − d

2�14�

The load on any thread tooth could be expressed as

Fi = �1 − k ·D − d

2�Fi+1 = �1 − k

D − d

2�n−i

Fn

= �1 − kD − d

2�n−iD − d

2·

T

�k1�15�

Then, the uniform load on each thread tooth is

f�zn−1� =kT

� · k1·

D − d

2�1 − k

D − d

2�

f�zn−2� =kT

� · k1·

D − d

2�1 − k

D − d

2�2

f�zn−3� =kT

� · k1·

D − d

2�1 − k

D − d

2�3

�16�

f�zi� =kT

� · k1·

D − d

2�1 − k

D − d

2�n−i

f�z2� =kT

� · k1·

D − d

2�1 − k

D − d

2�n−2

f�z1� =kT

� · k1·

D − d

2�1 − k

D − d

2�n−1

In Eq. �16�, it can be found that by the application of the tight-ening torque, the load on the last thread tooth is higher than anyother teeth and the load on the other teeth will decrease along the
axial direction.
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Calculating Example

3.1 Analytical Method. We choose the sample of API 88.9m round threaded connection and the geometric parameters are

isted in Table 1.By using Eq. �10�, it can obtain that

k = 171.32, k1 = 1.1495 � 10−4

In accordance with the calculation of Eq. �16�, the results ob-ained by the new method proposed in this paper are shown in Fig..

Figure 7 shows that by using the analytical method proposed inhis paper, the load on thread tooth increases with the increase inhread number and mainly concentrates on the last four or fivehreads engaged. Especially on the last two or three threads, theoad on the thread tooth changes rapidly. This phenomenon isimilar to the practical situation in threaded connection. It indi-ates that the new model proposed in this paper is reasonable.

3.2 Finite Element Method. In order to validate the calcula-ion of new analytical method proposed in this paper, we analyzehe same sample with the finite element software ANSYS 10.0.igure 8 shows the finite element model of API 88.9 mm round

hreaded connection.The finite element model with elastic contact threaded surfaces

as created in ANSYS. We cut the mesh area into several indi-idual parts along each thread tooth. Figure 9 shows the schematicf finite element model after cutting. The element type we se-ected for this analysis is Quad 8node-82, which means an eight-ode quadrilateral element. Figures 10 and 11 show the mesh onhe whole model and the teeth edges. Because of nonlinear naturef this contact problem, static analysis has been chosen to solvehe problem. The boundary conditions used in this analysis arelamping the end of female thread and application of an equiva-ent axial load on the opposite of male thread end.

The tightening torque T is first transferred into the equivalentxial load F using Eq. �17� then the boundary conditions are setnd the load that is exerted on the male thread is defined. By using

able 1 Geometric parameters of API 88.9 mm round threadedonnection

T�N m�

D0�mm�

D�mm�

d0�mm�

d�mm� n �

10,296 108 89.5318 69.89 84.9786 18 0.2

ig. 7 Load distribution on thread tooth by the new method
roposed in this paper
31202-6 / Vol. 132, JUNE 2010


ANSYS, we can obtain the stress distribution on the thread teeth�shown in Fig. 12�.

T = FDtan� + ��

2�17�

Figure 12 shows the stress distribution on each contact threadtooth. In Fig. 12, we can find that the load on thread tooth mainlyconcentrates on the last four or five threads engaged. This phe-nomenon is identical to the results obtained by analytical methodproposed in this paper.

3.3 Comparison and Discussion. Before making a compari-son between the solutions of new analytical method and finiteelement method, we must first transfer the load distribution oneach thread tooth into mean contact stress. By using Eq. �18�, wecan obtain the mean contact stress on each thread tooth.

�n =f�zn�L

A�18�

Fig. 8 Finite element model of API 88.9 mm round threadedconnection

Fig. 9 The schematic of finite element model after cutting

Fig. 10 Mesh on the finite element model



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Figure 13 shows the comparison of mean contact stresses ob-ained by analytical method and the finite element method. We cannd that the maximum and minimum values obtained by the newnalytical method are close to the ones of FEA method and thebtained two curves are identical in trend. Although there is a

Fig. 11 Refined mesh at teeth edge

Fig. 12 Stress distribution on the thread teeth in ANSYS

ig. 13 Mean contact stress obtained by analytical method
nd finite element method


discrepancy in the middle region, the shapes of the two curves aresimilar and this discrepancy may be attributed to the analyticalmodel assumptions. The analytical method proposed in this papercould calculate the load distribution just with the tightening torqueand thread numbers without considering all the influence factorssuch as pitch angle, friction coefficient and so on. But in FEAmethod, all these important factors have been included. In practi-cal situation, if we got the essential parameters and apply theanalytical method proposed in this paper, it is convenient to cal-culate the load distribution on thread tooth in threaded connection.

4 Conclusions

�1� In this paper, we proposed a new analytical method foranalyzing the connection of male and female thread andestablished a new analytical model, which presents the re-lation between load distribution, tightening torque, andthread number.

�2� The method proposed in this paper is convenient for appli-cation since it can be used to calculate the load and defor-mation on each thread tooth just with tightening torque andthread numbers.

�3� By the comparison of the solutions of a specific samplebetween the proposed analytical method and FEA method,it could be found that the load mainly concentrates on thelast four or five threads engaged. The results obtained bythe two methods are identical, which means the newmethod proposed in this paper is reasonable.

AcknowledgmentThe author wishes to thank Professor Lian-Xin Gao and Yi

Zhang for their help in this paper and the financial support ofWuxi Seamless Oil Pipe Co., Ltd.

NomenclatureT � tightening torque

Tn � frictional torque on nth tooth� � total axial deformation on male thread tooth

�n � axial deformation on the left side of malethread tooth

�n� � axial deformation on the right side of malethread tooth

� � total axial deformation on female thread tooth�n � axial deformation on the left side of female

a… middle region and „b… end region

thread tooth

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�n� � axial deformation on the right side of femalethread tooth

Fn � force caused by axial deformation on nth toothFi � force on ith toothF1 � force on first tooth

f�zn� � uniform load on nth toothf�zi� � uniform load on ith toothf�z1� � uniform load on first tooth

D � outer diameter of male threadD0 � outer diameter of female thread at hand-tight

planed � inner diameter of male thread at hand-tight

planed0 � inner diameter of cylindrical pipen � thread number� � friction coefficient � pitch angle� � tooth shape angle

� � equivalent frictional angle�n � mean contact stress on nth toothL � length of thread toothA � area of contact thread surface

eferences�1� Andrieux, S., 1993, “Multiple Scaling Method for the Calculation of Threaded

Assemblies,” Comput. Methods Appl. Mech. Eng., 102�3�, pp. 293–317.�2� Englund, R. B., and Johnson, D. H., 1997, “Finite Element Analysis of a

Threaded Connection Compared to Experimental and Theoretical Research,” J.Eng. Technol., 14�2�, pp. 42–47.

�3� Tsai, M. Y., and Morton, J., 1990, “Stress and Failure Analysis of a Pin-Loaded
Composite Joints,” J. Compos. Mater., 24�10�, pp. 1101–1120.
31202-8 / Vol. 132, JUNE 2010


�4� Maduschka, L., 1936, “Beansprukung von Schraubenverbindungen undZweckmassige Gestaltung der Gewindertraeger,” Forsch., 7, pp. 299–305.

�5� Sopwith, D. G., 1948, “The Distribution of Load in Screw Threads,” Inst.Mech. Eng. Appl. Mech. Proc., 159, pp. 373–383.

�6� Yazawa, S., and Hongo, K., 1988, “Distribution of Load in the Screw Threadof a Bolt-Nut Connection Subjected to Tangential Forces and Bending Mo-ments,” JSME Int. J., Ser. I, 31�2�, pp. 174–180.

�7� Goodier, J. N., 1940, “The Distribution of Load on the Threads of Screws,”Trans. ASME, 62, pp. A10–A16.

�8� Wang, W., and Marshek, K. M., 1995, “Determination of the Load Distributionin a Threaded Connector Having Dissimilar Materials and Varying ThreadStiffness,” ASME J. Eng. Ind., 117�1�, pp. 1–8.

�9� Heywood, R. B., 1948, “Tensile Fillet Stresses in Loaded Projections,” Proc.Inst. Mech. Eng., 159, pp. 384–398.

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