Research ArticleLoading and Contact Stress Analysis on the Thread Teeth inTubing and Casing Premium Threaded Connection

Honglin Xu1 Taihe Shi1 Zhi Zhang1 and Bin Shi2

1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation School of Petroleum and Natural Gas EngineeringSouthwest Petroleum University Chengdu 610500 China

2 RampD Center of Tianjin Pipe (Group) Corporation Ltd Tianjin 300301 China

Correspondence should be addressed to Honglin Xu xuhlaca1986 jy163com

Received 20 April 2014 Revised 15 August 2014 Accepted 21 August 2014 Published 13 October 2014

Academic Editor Chunlin Chen

Copyright copy 2014 Honglin Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Loading and contact stress distribution on the thread teeth in tubing and casing premium threaded connections are of greatimportance for design optimization pretightening force control and thread failure prevention This paper proposes an analyticalmethod based on the elastic mechanics This is quite different from other papers which mainly rely on finite element analysisThe differential equation of load distribution on the thread teeth was established according to equal pitch of the engaged threadafter deformation and solved by finite difference method Furthermore the relation between load acting on each engaged threadand mean contact stress on its load flank is set up based on the geometric description of thread surface By comparison thisnew analytical method with the finite element analysis for a modified API 1778mm premium threaded connection is approvedComparison of the contact stress on the last engaged thread between analyticalmodel and FEMshows that the accuracy of analyticalmodel will decline with the increase of pretightening force after thematerial enters into plastic deformation However the analyticalmethod can meet the needs of engineering to some extent because its relative error is about 62sim181 for the in-service level ofpretightening force

1 Introduction

Premium threaded connections have been widely used formaintaining structural and sealing integrity of tubing andcasing strings in HPHT wells The overwhelming majorityof premium threaded connections are characterized by atapered metal-to-metal seal structure for leakage resistanceand an interior rotary torque shoulder to control finalmake-up torques providing additional sealing performance(shown in Figure 1) These characteristics are different fromAPI buttress thread connections and extreme-line threadconnections [1ndash3] In general the thread teeth of premiumthreaded connection are used to bear axial tensile load dueto the load flank angle 120572 design (shown in Figure 3) of usuallyminus4

∘

sim3

∘ We also noticed that the taper angle of metal-to-metal seal is usually 179

∘

sim286

∘ so the axial componentof sealing contact stress could be neglected and the actionforce of sealing and shoulder replaced approximatively by anaxial pretightening force 119865

119905acting on the A-B cross section

plane (shown in Figure 1) Consequently it is significantto investigate the load and contact stress on thread teethcaused by this pretightening force 119865

119905for design optimization

pretightening torque control and thread failure preventionThe load distribution and contact stress on thread teeth

were investigated in many previous references in the lastcentury whose methods can mainly be divided into threecategories The first category is developed on the basis ofanalytical method Maduschka [4] and Sopwith [5] firstproposed the models of the load distribution in cylindricalscrew threads whose results clearly indicate that the firstthree or four engaged threads carry more than half of thepreload induced by the make-up torque Taking axial andtangential forces aswell as bendingmoments into considera-tion Yazawa and Hongo [6] developed a model to investigatethe load distribution of a bolt-nut connection Wang andMarshek [7] also established a spring model to predictthe load distribution of bolt-nut connection FurthermoreBruschelli and Latorrata [8] developed an analytical model to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 287076 11 pageshttpdxdoiorg1011552014287076

2 Mathematical Problems in Engineering

Metal-to-metal seal Torque shoulder

CouplingPinPin

Thread tooth

B

Axis of connectionA

Figure 1 The schematic of a premium threaded connection

study the load distribution for conical threads connectionsChen et al [9] presented a practical and convenient modelfor cylindrical pipe threaded connection which can beapplied to calculate the load and deformation on each threadtooth just with tightening torque and thread numbers Thevast majority of the analytical models mentioned above areestablished from screw threaded connection or cylindricalpipe threaded connection and therefore are not applicableto premium connection its tapered thread connection typecontains special metal-to metal seal interior rotary torqueshoulder and the specialized tooth form of buttress

The second category method is developed based on finiteelement method (FEM) Capable of considering materialnonlinear behavior and complex boundary conditions easilyFEMmakes studying the load distribution contact stress andstress concentration on the thread tooth possible OrsquoHara [10]established a 2D axisymmetric model of thread connectionbased on thread characteristic lengths and the numericalresults were generally validated by Heywoodrsquos equationsBretl and Cook [11] proposed a unique FEM technique forthreaded connections and their simulated results agreed withthose from analytical and experimental methods both forconventional and for tapered threads Zhao [12 13] developedan elastic-plastic FEM and investigated the load distributionand SCFs (stress concentration factors) of the bolt-nut con-nection Macdonald and Deans [14] acquired the SCFs andinduced stresses on the drilling string thread and they foundout that the high local stresses are located at the root of thefirst engaged tooth in the pin and the last engaged tooth inthe box Bahai et al [15] also calculated the SCFs of APIdrill string threaded connector subjected to preloading axialloading and bending loads Chen and Shih [16] developed a3DFEM for bolted joints and analyzed the helical and frictioneffect on the load distribution of each thread Baragetti [17]proposed a numerical finite element procedure for rotaryshouldered connections (RSCs) and quantified the effectsof taper on the loads carried by each engaged thread andcontact stress on the thread flanks Fukuoka and Nomura[18] established the FE model with accurate thread geometryby deriving a mathematical expression of the helical threadgeometry of a single-thread screwThenumerical results indi-cate that axial load along engaged threads shows a differentdistribution pattern from those obtained by axisymmetricFE analysis and elastic theory Although FE models have

incomparable advantages for threaded connections analysismost of them are based on 2D axisymmetric assumptions andthe helix angle is usually ignored

The third category method is developed based on exper-imental testing Stoeckly and Macke [19] developed a testingapparatus to measure the axial displacement of the threadsand compare their experimental data by modifying Sopwithequations They found out that the frictional coefficientbetween the lubricated bolt and nut has little effect on thethread load distribution Yuan et al [20] tested the make-up and break-out operations in oilfields by placing the foilgauges on the inner surface of the male thread and the outersurface of the female thread along the axial and tangentialdirection and concluded that those operations have a strongimpact on the service life of oil tubing threaded connectionPlacido et al [21] carried out some full and reduced scaleexperimental tests on aluminum drill pipes to investigatetheir fatigue mechanism under cyclic bending and constanttensile loads Slack et al [22] proposed the ultrasonic reflec-tion techniques to directlymeasure the contact stress betweenthread and sealing surface It turned out to be an accurate andeffective method to evaluate the sealing performance and todetect sealing damage in premium connections [23] Thoughexperimental testing can be an effectivemethod for validatinganalytical models and FEM those tests focused on limitedaspects of threaded connection problems and are difficult forapplication in practice

Based on elastic mechanics theory and some simplifica-tions this paper proposed an analytical method to calculatethe load distribution and contact stress on each engagedthread tooth for premium threaded connections The 2DFEM validated the analytical model and the comparison ofthe contact stress on the last engaged thread between twomethods investigated the accuracy of the analytical modeland its practicality in engineering

2 Model Development and Solution

21 Loading Analysis on the Premium Threaded Connec-tion During tightening the engaged threads of a premiumconnection will transfer the pretightening force 119865

119905caused

mainly by shoulder contact thus leading to the compressedpin and the stretched coupling To facilitate the analyticalmodel we assumed all thread teeth were intact and let

Mathematical Problems in Engineering 3

Pin

Coupling

XO

1 2 3 i

Contact stress between load flanks

NtFt

FtR0

r0

Axis of connection

Figure 2 The schematic of load distribution for premium connection

the axis of premium threaded connection as 119909 directionto establish a one-dimensional coordinate system OX withitsorigin being at the big end of the pin thread Meanwhilethe engaged thread teeth were numbered 1sim119873

119905consecutively

from origin to the small end of the pin thread (shown inFigure 2)

The contact stresses between load flanks of engagedthread interact with each other as action and reaction so theaxial load distribution 119865(119909) along 119909 direction is equal withinpin thread and coupling thread Let 119891(119909) denote the loaddistribution intensity then

119891 (119909) =

d119865 (119909)d119909

(1)

The load distribution intensity 119891(119909) equals the axial loadacting on the engaged thread within unit axial length whichreflects the distribution characteristics of pretightening forcealong the engaged thread It is obvious that axial loaddistribution 119865(119909) can be calculated by the integral formulaas follows

119865 (119909) = int

119909

0

119891 (119909) d119909 (2)

The axial load acting on each engaged thread can befurther obtained as

119865

119894119875= 119865 [119894119875] minus 119865 [(119894 minus 1) 119875]

= int

119894119875

(119894minus1)119875

119891 (119909) d119909 (119894 = 1 2 119873

119905)

(3)

22 Deformation of Thread Tooth in Premium ThreadedConnection The thread tooth profile of most premium con-nections typically adopts modified API buttress type threadwhose deformation can be calculated by simplifying itself intoa cantilever beammodel of an isosceles trapezoid [24] (shownin Figure 3)

Thenecessary parameters for calculating the elastic defor-mation of simplified thread tooth can be obtained from thisbasic geometry parameter of buttress type thread

119886 =

119875 minus 119865rscos 120574

119887 = 119886 minus 119888 [tan (120572 minus 120574) + tan (120573 + 120574)]

119888 =

ℎ

119861cos 1205742

(4)

According to [24] the deformations of simplified threadtooth are mainly caused by five reasons (shown in Figure 4)Now let angle 120572 minus 120574 be equal to 120599 in Figure 4 and thethread tooth deformation 120575

119886caused by tooth bending can be

expressed as

120575

119886(119909) =

119873 (119909) cos (120572 minus 120574)119864

3 (1 minus ]2)4

sdot [1 minus (2 minus

119887

119886

)

2

+ 2 ln(119886119887

)] 119888 tan3 (120572 minus 120574)

minus 4(

119888

119886

)

2

tan (120572 minus 120574)

(5)

The thread tooth deformation 120575

119887caused by shear force

can be expressed as

120575

119887(119909)

=

119873 (119909) cos (120572 minus 120574)119864

6 (1 + ])5

119888 tan (120572 minus 120574) ln(119886119887

)

(6)

The thread tooth deformation 120575

119888caused by tooth root

incline can be expressed as

120575

119888(119909)

=

119873 (119909) cos (120572 minus 120574)119864

12119888 (1 minus ]2)120587119886

2

[119888 minus

119887

2

tan (120572 minus 120574)] (7)

The thread tooth deformation 120575119889caused by shear defor-

mation of tooth root can be expressed as

120575

119889(119909) =

119873 (119909) cos (120572 minus 120574)119864

2 (1 minus ]2)120587

sdot [

119875

119886

ln(119875 + 05119886

119875 minus 05119886

) +

1

2

ln(41198752

119886

2

minus 1)]

(8)

4 Mathematical Problems in Engineering

Simplified tooth Root and crest parallel to pitch line

Pitch lineb

c

P

a

Z

Parallel to pitch line

Axis of connection

120574120572 minus 120574

120572

hB

120573

120573

+ 120574

Frs

Figure 3 The schematic of simplified cantilever beam model of an isosceles trapezoid for buttress type thread tooth

b

c

z

ya

Ncos120599

Nsin120599

120599

120575a120575b

(a)

a b

c

z

y

Ncos120599

Nsin120599

120575c

(b)

a b

c

z

y

120575d

Ncos120599

(c)

a b

c

z

y

Nsin120599120575r

(d)

Figure 4The elastic deformations of simplified thread tooth (a) deformation caused by tooth bending or shear force (b) deformation causedby tooth root incline (c) deformation caused by shear deformation of tooth root (d) deformation caused by radial effect

The deformations 120575rp 120575rc caused by radial effect for pinthread tooth and coupling thread tooth can be respectivelyexpressed as

120575rp (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot [

119889

2

2

(119909) + 4119903

2

0

119889

2

2

(119909) minus 4119903

2

0

minus ]]1198892(119909)

(9)

120575rc (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot (

4119877

2

0

+ 119889

2

2

(119909)

4119877

2

0

minus 119889

2

2

(119909)

minus ])119889

2(119909)

(10)

Consequently the total elastic deformations 120575tp 120575tc forpin thread tooth and coupling thread tooth along its conedirection (119885 direction) are respectively as follows

120575tp (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rp

=

[119896 + 119896rp (119909)]119873 (119909) cos (120572 minus 120574)119864

(11)

120575tc (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rc

=

[119896 + 119896rc (119909)]119873 (119909) cos (120572 minus 120574)119864

(12)

In (11) and (12) 119896 = 119896

119886+ 119896

119887+ 119896

119888+ 119896

119889 The deformation

coefficients 119896119886 119896119887 119896119888 119896119889 119896rp and 119896rc can be determined from

(5) to (10)In (5)sim(12) 119873(119909) denotes the normal load acting on

the load flank of thread tooth within unit axial length d119909Because the load flank angle and thread taper are both verysmall in premium threaded connection we ignored the axialcomponents of frictional force on load flank and that of thecontact stress between root and crest The relation between119873(119909) and 119891(119909) can be briefly expressed as

119873(119909)

cos120572tan [120595 (119909)]

d119909 = 119891 (119909) d119909 (13)

In (13) the helix angle 120595(119909) is

120595 (119909) = arc tan [ 119875

120587119889

2(119909)

]

119889

2(119909) = 119889

2119904minus 2119909 tan 120574

(14)

Mathematical Problems in Engineering 5

By submitting (1) and (14) into (13) we can get

119873(119909) =

119875

120587119889

2(119909) cos120572

d119865 (119909)d119909

(15)

Then by submitting (15) into (11) and (12) the total elasticdeformations for pin thread tooth and coupling thread toothare respectively as follows

120575tp (119909) = 120582p (119909)d119865 (119909)d119909

=

[119896 + 119896rp (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(16)

120575tc (119909) = 120582c (119909)d119865 (119909)d119909

=

[119896 + 119896rc (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(17)

In (16) and (17) the coefficients 120582p and 120582c depend on thegeometry parameters of connection and thread tooth as wellas the used material properties whose derivatives for 119909 arerespectively as follows

120582

1015840

p

=

d120582p

d119909=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rp) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rp)d119909

119875

120587119889

2

+ (119896 + 119896rp)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

119889

2

2

+ 4119903

2

0

119889

2

2

minus 4119903

2

0

minus

16119889

2

2

119903

2

0

(119889

2

2

minus 4119903

2

0

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rp)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

119889

4

2

(119909) minus 16119903

4

0

minus 16119903

2

0

119889

2

2

(119909)

(119889

2

2

(119909) minus 4119903

2

0

)

2

minus ]]

minus (119896 + 119896rp)2119875

119889

2(119909)

(18)

120582

1015840

c

=

d120582cd119909

=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rc) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rc)d119909

119875

120587119889

2

+ (119896 + 119896rc)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

4119877

2

0

+ 119889

2

2

4119877

2

0

minus 119889

2

2

+

16119877

2

0

119889

2

2

(4119877

2

0

minus 119889

2

2

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rc)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

16119877

4

0

minus 119889

4

2

(119909) + 16119877

2

0

119889

2

2

(119909)

(4119877

2

0

minus 119889

2

2

(119909))

2

minus ]]

minus (119896 + 119896rc)2119875

119889

2(119909)

(19)

23 Differential Equation of the Axial Load 119865(119909) Under theaction of pretightening force the pin thread and couplingthread are always engaged to each other indicating thattheir thread pitches are equal after deformation Howeverthe compressed pin will make its own thread pitch decreasewhile the stretched coupling will make its own thread pitchincreaseThedeformation of pin thread tooth should increaseits own pitch while the deformation of coupling thread toothshould reduce its ownpitch thus guaranteeing the equal pitchand the perfectly engaged state for pin thread and couplingthread

According to Hookersquos law the compressed and stretchedelastic deformations of pin and coupling body along thethread cone direction are respectively as follows

120594p =4

120587119864 cos 120574int

119909

0

119865 (119909)

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

d119909 (20)

120594c =4

120587119864 cos 120574int

119909

0

119865 (119909)

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

d119909 (21)

Moreover the deformation compatibility equation canbe obtained from the equal pitch after deformation for pinthread and coupling thread

120594p (119909) + 120594c (119909) = [120575tp (119909) + 120575tc (119909)]

minus [120575tp (0) + 120575tc (0)] (22)

6 Mathematical Problems in Engineering

Then by submitting (16) (17) (20) and (21) into (22) weget

4

120587119864 cos 120574int

119909

0

[

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

]119865 (119909) d119909

= (120582p + 120582c)d119865 (119909)d119909

(23)

Taking the derivative with respect to 119909 for (23) andconsidering that 120582p + 120582c = 0 the differential equation of axialload distribution can be obtained as

d2119865 (119909)d1199092

+ 119898 (119909)

d119865 (119909)d119909

+ 119899 (119909) 119865 (119909) = 0

119898 (119909) = (

120582

1015840

p + 1205821015840

c

120582p + 120582c)

119899 (119909) = minus

4

120587119864 (120582p + 120582c) cos 120574

sdot

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

(24)

The boundary conditions for (24) are

119865 (0) = 0

119865 (119871) = 119865

119905

(25)

24 Solution of Differential Equation The axial load distri-bution equation (24) is a nonlinear differential equation andits analytical solution is very difficult to get so we solved itsnumerical solution with finite difference method First wedivided the engaged thread according to unit length of threadpitch 119875 and numbered the node 0sim119873

119905successively from the

big end of the pin thread so the coordinate for node 119894 was119909

119894= 119894119875 Then we took 119865

119894to denote the node force 119865(119909

119894) and

considered the boundary conditions equation (25) Finallythe difference equation of (24) can be obtained as

119860[119865

1119865

2sdot sdot sdot 119865

119873minus1]

⊤

= 119861 (26)

In (26) the coefficient matrix is

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus

2

119875

2

+ 119899 (119909

1)

1

119875

2

+

119898 (119909

1)

2119875

0

1

119875

2

minus

119898 (119909

2)

2119875

minus

2

119875

2

+ 119899 (119909

2)

1

119875

2

+

119898 (119909

2)

2119875

sdot sdot sdot

1

119875

2

minus

119898 (119909

119894)

2119875

minus

2

119875

2

+ 119899 (119909

119894)

1

119875

2

+

119898 (119909

119894)

2119875

sdot sdot sdot

sdot sdot sdot

0

1

119875

2

minus

119898 (119909

119873minus1)

2119875

minus

2

119875

2

+ 119899 (119909

119873minus1)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

And the constant term column matrix is

119861 = [0 0 sdot sdot sdot minus (

1

119875

2

+

119898 (119909

119873minus1)

2119875

)119865

119905]

⊤

(28)

The difference equation (26) was solved based on Gaus-sian main elimination method

25 Geometric Description for Load Flank of Thread ToothFor calculating the mean contact stress on the load flank ofthread tooth the geometric description of thread surface isnecessary Setting the premium thread connection as the 119909-axis and its intersection with the plane of thread load flank asthe origin we established global coordinate system 119874-119909119910119911

whose unit vectors are i j and k respectively shown inFigure 5

For determining the unit normal vector n and unittangent vector 120591 of thread surface the cylindrical coordinatesystem 119874-119903120599119909 was also established and its vectors are

u = minusj

k = minusk

w = i

(29)

The plane 1198741198612119862

2119863

2 which stands for thread load flank

can be obtained by rotating the quadrangle 119874119861119862119863 in 119910119900119911

plane 120572 around the 119910-axis and then 120595 around the 119911-axissuccessively If quadrangle 119874119861119862119863 is a 1 times 1 square its

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Metal-to-metal seal Torque shoulder

CouplingPinPin

Thread tooth

B

Axis of connectionA

Figure 1 The schematic of a premium threaded connection

study the load distribution for conical threads connectionsChen et al [9] presented a practical and convenient modelfor cylindrical pipe threaded connection which can beapplied to calculate the load and deformation on each threadtooth just with tightening torque and thread numbers Thevast majority of the analytical models mentioned above areestablished from screw threaded connection or cylindricalpipe threaded connection and therefore are not applicableto premium connection its tapered thread connection typecontains special metal-to metal seal interior rotary torqueshoulder and the specialized tooth form of buttress

The second category method is developed based on finiteelement method (FEM) Capable of considering materialnonlinear behavior and complex boundary conditions easilyFEMmakes studying the load distribution contact stress andstress concentration on the thread tooth possible OrsquoHara [10]established a 2D axisymmetric model of thread connectionbased on thread characteristic lengths and the numericalresults were generally validated by Heywoodrsquos equationsBretl and Cook [11] proposed a unique FEM technique forthreaded connections and their simulated results agreed withthose from analytical and experimental methods both forconventional and for tapered threads Zhao [12 13] developedan elastic-plastic FEM and investigated the load distributionand SCFs (stress concentration factors) of the bolt-nut con-nection Macdonald and Deans [14] acquired the SCFs andinduced stresses on the drilling string thread and they foundout that the high local stresses are located at the root of thefirst engaged tooth in the pin and the last engaged tooth inthe box Bahai et al [15] also calculated the SCFs of APIdrill string threaded connector subjected to preloading axialloading and bending loads Chen and Shih [16] developed a3DFEM for bolted joints and analyzed the helical and frictioneffect on the load distribution of each thread Baragetti [17]proposed a numerical finite element procedure for rotaryshouldered connections (RSCs) and quantified the effectsof taper on the loads carried by each engaged thread andcontact stress on the thread flanks Fukuoka and Nomura[18] established the FE model with accurate thread geometryby deriving a mathematical expression of the helical threadgeometry of a single-thread screwThenumerical results indi-cate that axial load along engaged threads shows a differentdistribution pattern from those obtained by axisymmetricFE analysis and elastic theory Although FE models have

incomparable advantages for threaded connections analysismost of them are based on 2D axisymmetric assumptions andthe helix angle is usually ignored

The third category method is developed based on exper-imental testing Stoeckly and Macke [19] developed a testingapparatus to measure the axial displacement of the threadsand compare their experimental data by modifying Sopwithequations They found out that the frictional coefficientbetween the lubricated bolt and nut has little effect on thethread load distribution Yuan et al [20] tested the make-up and break-out operations in oilfields by placing the foilgauges on the inner surface of the male thread and the outersurface of the female thread along the axial and tangentialdirection and concluded that those operations have a strongimpact on the service life of oil tubing threaded connectionPlacido et al [21] carried out some full and reduced scaleexperimental tests on aluminum drill pipes to investigatetheir fatigue mechanism under cyclic bending and constanttensile loads Slack et al [22] proposed the ultrasonic reflec-tion techniques to directlymeasure the contact stress betweenthread and sealing surface It turned out to be an accurate andeffective method to evaluate the sealing performance and todetect sealing damage in premium connections [23] Thoughexperimental testing can be an effectivemethod for validatinganalytical models and FEM those tests focused on limitedaspects of threaded connection problems and are difficult forapplication in practice

Based on elastic mechanics theory and some simplifica-tions this paper proposed an analytical method to calculatethe load distribution and contact stress on each engagedthread tooth for premium threaded connections The 2DFEM validated the analytical model and the comparison ofthe contact stress on the last engaged thread between twomethods investigated the accuracy of the analytical modeland its practicality in engineering

2 Model Development and Solution

21 Loading Analysis on the Premium Threaded Connec-tion During tightening the engaged threads of a premiumconnection will transfer the pretightening force 119865

119905caused

mainly by shoulder contact thus leading to the compressedpin and the stretched coupling To facilitate the analyticalmodel we assumed all thread teeth were intact and let

Mathematical Problems in Engineering 3

Pin

Coupling

XO

1 2 3 i

Contact stress between load flanks

NtFt

FtR0

r0

Axis of connection

Figure 2 The schematic of load distribution for premium connection

the axis of premium threaded connection as 119909 directionto establish a one-dimensional coordinate system OX withitsorigin being at the big end of the pin thread Meanwhilethe engaged thread teeth were numbered 1sim119873

119905consecutively

from origin to the small end of the pin thread (shown inFigure 2)

The contact stresses between load flanks of engagedthread interact with each other as action and reaction so theaxial load distribution 119865(119909) along 119909 direction is equal withinpin thread and coupling thread Let 119891(119909) denote the loaddistribution intensity then

119891 (119909) =

d119865 (119909)d119909

(1)

The load distribution intensity 119891(119909) equals the axial loadacting on the engaged thread within unit axial length whichreflects the distribution characteristics of pretightening forcealong the engaged thread It is obvious that axial loaddistribution 119865(119909) can be calculated by the integral formulaas follows

119865 (119909) = int

119909

0

119891 (119909) d119909 (2)

The axial load acting on each engaged thread can befurther obtained as

119865

119894119875= 119865 [119894119875] minus 119865 [(119894 minus 1) 119875]

= int

119894119875

(119894minus1)119875

119891 (119909) d119909 (119894 = 1 2 119873

119905)

(3)

22 Deformation of Thread Tooth in Premium ThreadedConnection The thread tooth profile of most premium con-nections typically adopts modified API buttress type threadwhose deformation can be calculated by simplifying itself intoa cantilever beammodel of an isosceles trapezoid [24] (shownin Figure 3)

Thenecessary parameters for calculating the elastic defor-mation of simplified thread tooth can be obtained from thisbasic geometry parameter of buttress type thread

119886 =

119875 minus 119865rscos 120574

119887 = 119886 minus 119888 [tan (120572 minus 120574) + tan (120573 + 120574)]

119888 =

ℎ

119861cos 1205742

(4)

According to [24] the deformations of simplified threadtooth are mainly caused by five reasons (shown in Figure 4)Now let angle 120572 minus 120574 be equal to 120599 in Figure 4 and thethread tooth deformation 120575

119886caused by tooth bending can be

expressed as

120575

119886(119909) =

119873 (119909) cos (120572 minus 120574)119864

3 (1 minus ]2)4

sdot [1 minus (2 minus

119887

119886

)

2

+ 2 ln(119886119887

)] 119888 tan3 (120572 minus 120574)

minus 4(

119888

119886

)

2

tan (120572 minus 120574)

(5)

The thread tooth deformation 120575

119887caused by shear force

can be expressed as

120575

119887(119909)

=

119873 (119909) cos (120572 minus 120574)119864

6 (1 + ])5

119888 tan (120572 minus 120574) ln(119886119887

)

(6)

The thread tooth deformation 120575

119888caused by tooth root

incline can be expressed as

120575

119888(119909)

=

119873 (119909) cos (120572 minus 120574)119864

12119888 (1 minus ]2)120587119886

2

[119888 minus

119887

2

tan (120572 minus 120574)] (7)

The thread tooth deformation 120575119889caused by shear defor-

mation of tooth root can be expressed as

120575

119889(119909) =

119873 (119909) cos (120572 minus 120574)119864

2 (1 minus ]2)120587

sdot [

119875

119886

ln(119875 + 05119886

119875 minus 05119886

) +

1

2

ln(41198752

119886

2

minus 1)]

(8)

4 Mathematical Problems in Engineering

Simplified tooth Root and crest parallel to pitch line

Pitch lineb

c

P

a

Z

Parallel to pitch line

Axis of connection

120574120572 minus 120574

120572

hB

120573

120573

+ 120574

Frs

Figure 3 The schematic of simplified cantilever beam model of an isosceles trapezoid for buttress type thread tooth

b

c

z

ya

Ncos120599

Nsin120599

120599

120575a120575b

(a)

a b

c

z

y

Ncos120599

Nsin120599

120575c

(b)

a b

c

z

y

120575d

Ncos120599

(c)

a b

c

z

y

Nsin120599120575r

(d)

Figure 4The elastic deformations of simplified thread tooth (a) deformation caused by tooth bending or shear force (b) deformation causedby tooth root incline (c) deformation caused by shear deformation of tooth root (d) deformation caused by radial effect

The deformations 120575rp 120575rc caused by radial effect for pinthread tooth and coupling thread tooth can be respectivelyexpressed as

120575rp (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot [

119889

2

2

(119909) + 4119903

2

0

119889

2

2

(119909) minus 4119903

2

0

minus ]]1198892(119909)

(9)

120575rc (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot (

4119877

2

0

+ 119889

2

2

(119909)

4119877

2

0

minus 119889

2

2

(119909)

minus ])119889

2(119909)

(10)

Consequently the total elastic deformations 120575tp 120575tc forpin thread tooth and coupling thread tooth along its conedirection (119885 direction) are respectively as follows

120575tp (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rp

=

[119896 + 119896rp (119909)]119873 (119909) cos (120572 minus 120574)119864

(11)

120575tc (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rc

=

[119896 + 119896rc (119909)]119873 (119909) cos (120572 minus 120574)119864

(12)

In (11) and (12) 119896 = 119896

119886+ 119896

119887+ 119896

119888+ 119896

119889 The deformation

coefficients 119896119886 119896119887 119896119888 119896119889 119896rp and 119896rc can be determined from

(5) to (10)In (5)sim(12) 119873(119909) denotes the normal load acting on

the load flank of thread tooth within unit axial length d119909Because the load flank angle and thread taper are both verysmall in premium threaded connection we ignored the axialcomponents of frictional force on load flank and that of thecontact stress between root and crest The relation between119873(119909) and 119891(119909) can be briefly expressed as

119873(119909)

cos120572tan [120595 (119909)]

d119909 = 119891 (119909) d119909 (13)

In (13) the helix angle 120595(119909) is

120595 (119909) = arc tan [ 119875

120587119889

2(119909)

]

119889

2(119909) = 119889

2119904minus 2119909 tan 120574

(14)

Mathematical Problems in Engineering 5

By submitting (1) and (14) into (13) we can get

119873(119909) =

119875

120587119889

2(119909) cos120572

d119865 (119909)d119909

(15)

Then by submitting (15) into (11) and (12) the total elasticdeformations for pin thread tooth and coupling thread toothare respectively as follows

120575tp (119909) = 120582p (119909)d119865 (119909)d119909

=

[119896 + 119896rp (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(16)

120575tc (119909) = 120582c (119909)d119865 (119909)d119909

=

[119896 + 119896rc (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(17)

In (16) and (17) the coefficients 120582p and 120582c depend on thegeometry parameters of connection and thread tooth as wellas the used material properties whose derivatives for 119909 arerespectively as follows

120582

1015840

p

=

d120582p

d119909=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rp) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rp)d119909

119875

120587119889

2

+ (119896 + 119896rp)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

119889

2

2

+ 4119903

2

0

119889

2

2

minus 4119903

2

0

minus

16119889

2

2

119903

2

0

(119889

2

2

minus 4119903

2

0

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rp)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

119889

4

2

(119909) minus 16119903

4

0

minus 16119903

2

0

119889

2

2

(119909)

(119889

2

2

(119909) minus 4119903

2

0

)

2

minus ]]

minus (119896 + 119896rp)2119875

119889

2(119909)

(18)

120582

1015840

c

=

d120582cd119909

=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rc) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rc)d119909

119875

120587119889

2

+ (119896 + 119896rc)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

4119877

2

0

+ 119889

2

2

4119877

2

0

minus 119889

2

2

+

16119877

2

0

119889

2

2

(4119877

2

0

minus 119889

2

2

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rc)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

16119877

4

0

minus 119889

4

2

(119909) + 16119877

2

0

119889

2

2

(119909)

(4119877

2

0

minus 119889

2

2

(119909))

2

minus ]]

minus (119896 + 119896rc)2119875

119889

2(119909)

(19)

23 Differential Equation of the Axial Load 119865(119909) Under theaction of pretightening force the pin thread and couplingthread are always engaged to each other indicating thattheir thread pitches are equal after deformation Howeverthe compressed pin will make its own thread pitch decreasewhile the stretched coupling will make its own thread pitchincreaseThedeformation of pin thread tooth should increaseits own pitch while the deformation of coupling thread toothshould reduce its ownpitch thus guaranteeing the equal pitchand the perfectly engaged state for pin thread and couplingthread

According to Hookersquos law the compressed and stretchedelastic deformations of pin and coupling body along thethread cone direction are respectively as follows

120594p =4

120587119864 cos 120574int

119909

0

119865 (119909)

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

d119909 (20)

120594c =4

120587119864 cos 120574int

119909

0

119865 (119909)

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

d119909 (21)

Moreover the deformation compatibility equation canbe obtained from the equal pitch after deformation for pinthread and coupling thread

120594p (119909) + 120594c (119909) = [120575tp (119909) + 120575tc (119909)]

minus [120575tp (0) + 120575tc (0)] (22)

6 Mathematical Problems in Engineering

Then by submitting (16) (17) (20) and (21) into (22) weget

4

120587119864 cos 120574int

119909

0

[

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

]119865 (119909) d119909

= (120582p + 120582c)d119865 (119909)d119909

(23)

Taking the derivative with respect to 119909 for (23) andconsidering that 120582p + 120582c = 0 the differential equation of axialload distribution can be obtained as

d2119865 (119909)d1199092

+ 119898 (119909)

d119865 (119909)d119909

+ 119899 (119909) 119865 (119909) = 0

119898 (119909) = (

120582

1015840

p + 1205821015840

c

120582p + 120582c)

119899 (119909) = minus

4

120587119864 (120582p + 120582c) cos 120574

sdot

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

(24)

The boundary conditions for (24) are

119865 (0) = 0

119865 (119871) = 119865

119905

(25)

24 Solution of Differential Equation The axial load distri-bution equation (24) is a nonlinear differential equation andits analytical solution is very difficult to get so we solved itsnumerical solution with finite difference method First wedivided the engaged thread according to unit length of threadpitch 119875 and numbered the node 0sim119873

119905successively from the

big end of the pin thread so the coordinate for node 119894 was119909

119894= 119894119875 Then we took 119865

119894to denote the node force 119865(119909

119894) and

considered the boundary conditions equation (25) Finallythe difference equation of (24) can be obtained as

119860[119865

1119865

2sdot sdot sdot 119865

119873minus1]

⊤

= 119861 (26)

In (26) the coefficient matrix is

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus

2

119875

2

+ 119899 (119909

1)

1

119875

2

+

119898 (119909

1)

2119875

0

1

119875

2

minus

119898 (119909

2)

2119875

minus

2

119875

2

+ 119899 (119909

2)

1

119875

2

+

119898 (119909

2)

2119875

sdot sdot sdot

1

119875

2

minus

119898 (119909

119894)

2119875

minus

2

119875

2

+ 119899 (119909

119894)

1

119875

2

+

119898 (119909

119894)

2119875

sdot sdot sdot

sdot sdot sdot

0

1

119875

2

minus

119898 (119909

119873minus1)

2119875

minus

2

119875

2

+ 119899 (119909

119873minus1)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

And the constant term column matrix is

119861 = [0 0 sdot sdot sdot minus (

1

119875

2

+

119898 (119909

119873minus1)

2119875

)119865

119905]

⊤

(28)

The difference equation (26) was solved based on Gaus-sian main elimination method

25 Geometric Description for Load Flank of Thread ToothFor calculating the mean contact stress on the load flank ofthread tooth the geometric description of thread surface isnecessary Setting the premium thread connection as the 119909-axis and its intersection with the plane of thread load flank asthe origin we established global coordinate system 119874-119909119910119911

whose unit vectors are i j and k respectively shown inFigure 5

For determining the unit normal vector n and unittangent vector 120591 of thread surface the cylindrical coordinatesystem 119874-119903120599119909 was also established and its vectors are

u = minusj

k = minusk

w = i

(29)

The plane 1198741198612119862

2119863

2 which stands for thread load flank

can be obtained by rotating the quadrangle 119874119861119862119863 in 119910119900119911

plane 120572 around the 119910-axis and then 120595 around the 119911-axissuccessively If quadrangle 119874119861119862119863 is a 1 times 1 square its

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Pin

Coupling

XO

1 2 3 i

Contact stress between load flanks

NtFt

FtR0

r0

Axis of connection

Figure 2 The schematic of load distribution for premium connection

the axis of premium threaded connection as 119909 directionto establish a one-dimensional coordinate system OX withitsorigin being at the big end of the pin thread Meanwhilethe engaged thread teeth were numbered 1sim119873

119905consecutively

from origin to the small end of the pin thread (shown inFigure 2)

The contact stresses between load flanks of engagedthread interact with each other as action and reaction so theaxial load distribution 119865(119909) along 119909 direction is equal withinpin thread and coupling thread Let 119891(119909) denote the loaddistribution intensity then

119891 (119909) =

d119865 (119909)d119909

(1)

The load distribution intensity 119891(119909) equals the axial loadacting on the engaged thread within unit axial length whichreflects the distribution characteristics of pretightening forcealong the engaged thread It is obvious that axial loaddistribution 119865(119909) can be calculated by the integral formulaas follows

119865 (119909) = int

119909

0

119891 (119909) d119909 (2)

The axial load acting on each engaged thread can befurther obtained as

119865

119894119875= 119865 [119894119875] minus 119865 [(119894 minus 1) 119875]

= int

119894119875

(119894minus1)119875

119891 (119909) d119909 (119894 = 1 2 119873

119905)

(3)

22 Deformation of Thread Tooth in Premium ThreadedConnection The thread tooth profile of most premium con-nections typically adopts modified API buttress type threadwhose deformation can be calculated by simplifying itself intoa cantilever beammodel of an isosceles trapezoid [24] (shownin Figure 3)

Thenecessary parameters for calculating the elastic defor-mation of simplified thread tooth can be obtained from thisbasic geometry parameter of buttress type thread

119886 =

119875 minus 119865rscos 120574

119887 = 119886 minus 119888 [tan (120572 minus 120574) + tan (120573 + 120574)]

119888 =

ℎ

119861cos 1205742

(4)

According to [24] the deformations of simplified threadtooth are mainly caused by five reasons (shown in Figure 4)Now let angle 120572 minus 120574 be equal to 120599 in Figure 4 and thethread tooth deformation 120575

119886caused by tooth bending can be

expressed as

120575

119886(119909) =

119873 (119909) cos (120572 minus 120574)119864

3 (1 minus ]2)4

sdot [1 minus (2 minus

119887

119886

)

2

+ 2 ln(119886119887

)] 119888 tan3 (120572 minus 120574)

minus 4(

119888

119886

)

2

tan (120572 minus 120574)

(5)

The thread tooth deformation 120575

119887caused by shear force

can be expressed as

120575

119887(119909)

=

119873 (119909) cos (120572 minus 120574)119864

6 (1 + ])5

119888 tan (120572 minus 120574) ln(119886119887

)

(6)

The thread tooth deformation 120575

119888caused by tooth root

incline can be expressed as

120575

119888(119909)

=

119873 (119909) cos (120572 minus 120574)119864

12119888 (1 minus ]2)120587119886

2

[119888 minus

119887

2

tan (120572 minus 120574)] (7)

The thread tooth deformation 120575119889caused by shear defor-

mation of tooth root can be expressed as

120575

119889(119909) =

119873 (119909) cos (120572 minus 120574)119864

2 (1 minus ]2)120587

sdot [

119875

119886

ln(119875 + 05119886

119875 minus 05119886

) +

1

2

ln(41198752

119886

2

minus 1)]

(8)

4 Mathematical Problems in Engineering

Simplified tooth Root and crest parallel to pitch line

Pitch lineb

c

P

a

Z

Parallel to pitch line

Axis of connection

120574120572 minus 120574

120572

hB

120573

120573

+ 120574

Frs

Figure 3 The schematic of simplified cantilever beam model of an isosceles trapezoid for buttress type thread tooth

b

c

z

ya

Ncos120599

Nsin120599

120599

120575a120575b

(a)

a b

c

z

y

Ncos120599

Nsin120599

120575c

(b)

a b

c

z

y

120575d

Ncos120599

(c)

a b

c

z

y

Nsin120599120575r

(d)

Figure 4The elastic deformations of simplified thread tooth (a) deformation caused by tooth bending or shear force (b) deformation causedby tooth root incline (c) deformation caused by shear deformation of tooth root (d) deformation caused by radial effect

The deformations 120575rp 120575rc caused by radial effect for pinthread tooth and coupling thread tooth can be respectivelyexpressed as

120575rp (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot [

119889

2

2

(119909) + 4119903

2

0

119889

2

2

(119909) minus 4119903

2

0

minus ]]1198892(119909)

(9)

120575rc (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot (

4119877

2

0

+ 119889

2

2

(119909)

4119877

2

0

minus 119889

2

2

(119909)

minus ])119889

2(119909)

(10)

Consequently the total elastic deformations 120575tp 120575tc forpin thread tooth and coupling thread tooth along its conedirection (119885 direction) are respectively as follows

120575tp (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rp

=

[119896 + 119896rp (119909)]119873 (119909) cos (120572 minus 120574)119864

(11)

120575tc (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rc

=

[119896 + 119896rc (119909)]119873 (119909) cos (120572 minus 120574)119864

(12)

In (11) and (12) 119896 = 119896

119886+ 119896

119887+ 119896

119888+ 119896

119889 The deformation

coefficients 119896119886 119896119887 119896119888 119896119889 119896rp and 119896rc can be determined from

(5) to (10)In (5)sim(12) 119873(119909) denotes the normal load acting on

the load flank of thread tooth within unit axial length d119909Because the load flank angle and thread taper are both verysmall in premium threaded connection we ignored the axialcomponents of frictional force on load flank and that of thecontact stress between root and crest The relation between119873(119909) and 119891(119909) can be briefly expressed as

119873(119909)

cos120572tan [120595 (119909)]

d119909 = 119891 (119909) d119909 (13)

In (13) the helix angle 120595(119909) is

120595 (119909) = arc tan [ 119875

120587119889

2(119909)

]

119889

2(119909) = 119889

2119904minus 2119909 tan 120574

(14)

Mathematical Problems in Engineering 5

By submitting (1) and (14) into (13) we can get

119873(119909) =

119875

120587119889

2(119909) cos120572

d119865 (119909)d119909

(15)

Then by submitting (15) into (11) and (12) the total elasticdeformations for pin thread tooth and coupling thread toothare respectively as follows

120575tp (119909) = 120582p (119909)d119865 (119909)d119909

=

[119896 + 119896rp (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(16)

120575tc (119909) = 120582c (119909)d119865 (119909)d119909

=

[119896 + 119896rc (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(17)

In (16) and (17) the coefficients 120582p and 120582c depend on thegeometry parameters of connection and thread tooth as wellas the used material properties whose derivatives for 119909 arerespectively as follows

120582

1015840

p

=

d120582p

d119909=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rp) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rp)d119909

119875

120587119889

2

+ (119896 + 119896rp)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

119889

2

2

+ 4119903

2

0

119889

2

2

minus 4119903

2

0

minus

16119889

2

2

119903

2

0

(119889

2

2

minus 4119903

2

0

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rp)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

119889

4

2

(119909) minus 16119903

4

0

minus 16119903

2

0

119889

2

2

(119909)

(119889

2

2

(119909) minus 4119903

2

0

)

2

minus ]]

minus (119896 + 119896rp)2119875

119889

2(119909)

(18)

120582

1015840

c

=

d120582cd119909

=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rc) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rc)d119909

119875

120587119889

2

+ (119896 + 119896rc)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

4119877

2

0

+ 119889

2

2

4119877

2

0

minus 119889

2

2

+

16119877

2

0

119889

2

2

(4119877

2

0

minus 119889

2

2

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rc)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

16119877

4

0

minus 119889

4

2

(119909) + 16119877

2

0

119889

2

2

(119909)

(4119877

2

0

minus 119889

2

2

(119909))

2

minus ]]

minus (119896 + 119896rc)2119875

119889

2(119909)

(19)

23 Differential Equation of the Axial Load 119865(119909) Under theaction of pretightening force the pin thread and couplingthread are always engaged to each other indicating thattheir thread pitches are equal after deformation Howeverthe compressed pin will make its own thread pitch decreasewhile the stretched coupling will make its own thread pitchincreaseThedeformation of pin thread tooth should increaseits own pitch while the deformation of coupling thread toothshould reduce its ownpitch thus guaranteeing the equal pitchand the perfectly engaged state for pin thread and couplingthread

According to Hookersquos law the compressed and stretchedelastic deformations of pin and coupling body along thethread cone direction are respectively as follows

120594p =4

120587119864 cos 120574int

119909

0

119865 (119909)

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

d119909 (20)

120594c =4

120587119864 cos 120574int

119909

0

119865 (119909)

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

d119909 (21)

Moreover the deformation compatibility equation canbe obtained from the equal pitch after deformation for pinthread and coupling thread

120594p (119909) + 120594c (119909) = [120575tp (119909) + 120575tc (119909)]

minus [120575tp (0) + 120575tc (0)] (22)

6 Mathematical Problems in Engineering

Then by submitting (16) (17) (20) and (21) into (22) weget

4

120587119864 cos 120574int

119909

0

[

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

]119865 (119909) d119909

= (120582p + 120582c)d119865 (119909)d119909

(23)

Taking the derivative with respect to 119909 for (23) andconsidering that 120582p + 120582c = 0 the differential equation of axialload distribution can be obtained as

d2119865 (119909)d1199092

+ 119898 (119909)

d119865 (119909)d119909

+ 119899 (119909) 119865 (119909) = 0

119898 (119909) = (

120582

1015840

p + 1205821015840

c

120582p + 120582c)

119899 (119909) = minus

4

120587119864 (120582p + 120582c) cos 120574

sdot

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

(24)

The boundary conditions for (24) are

119865 (0) = 0

119865 (119871) = 119865

119905

(25)

24 Solution of Differential Equation The axial load distri-bution equation (24) is a nonlinear differential equation andits analytical solution is very difficult to get so we solved itsnumerical solution with finite difference method First wedivided the engaged thread according to unit length of threadpitch 119875 and numbered the node 0sim119873

119905successively from the

big end of the pin thread so the coordinate for node 119894 was119909

119894= 119894119875 Then we took 119865

119894to denote the node force 119865(119909

119894) and

considered the boundary conditions equation (25) Finallythe difference equation of (24) can be obtained as

119860[119865

1119865

2sdot sdot sdot 119865

119873minus1]

⊤

= 119861 (26)

In (26) the coefficient matrix is

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus

2

119875

2

+ 119899 (119909

1)

1

119875

2

+

119898 (119909

1)

2119875

0

1

119875

2

minus

119898 (119909

2)

2119875

minus

2

119875

2

+ 119899 (119909

2)

1

119875

2

+

119898 (119909

2)

2119875

sdot sdot sdot

1

119875

2

minus

119898 (119909

119894)

2119875

minus

2

119875

2

+ 119899 (119909

119894)

1

119875

2

+

119898 (119909

119894)

2119875

sdot sdot sdot

sdot sdot sdot

0

1

119875

2

minus

119898 (119909

119873minus1)

2119875

minus

2

119875

2

+ 119899 (119909

119873minus1)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

And the constant term column matrix is

119861 = [0 0 sdot sdot sdot minus (

1

119875

2

+

119898 (119909

119873minus1)

2119875

)119865

119905]

⊤

(28)

The difference equation (26) was solved based on Gaus-sian main elimination method

25 Geometric Description for Load Flank of Thread ToothFor calculating the mean contact stress on the load flank ofthread tooth the geometric description of thread surface isnecessary Setting the premium thread connection as the 119909-axis and its intersection with the plane of thread load flank asthe origin we established global coordinate system 119874-119909119910119911

whose unit vectors are i j and k respectively shown inFigure 5

For determining the unit normal vector n and unittangent vector 120591 of thread surface the cylindrical coordinatesystem 119874-119903120599119909 was also established and its vectors are

u = minusj

k = minusk

w = i

(29)

The plane 1198741198612119862

2119863

2 which stands for thread load flank

can be obtained by rotating the quadrangle 119874119861119862119863 in 119910119900119911

plane 120572 around the 119910-axis and then 120595 around the 119911-axissuccessively If quadrangle 119874119861119862119863 is a 1 times 1 square its

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Simplified tooth Root and crest parallel to pitch line

Pitch lineb

c

P

a

Z

Parallel to pitch line

Axis of connection

120574120572 minus 120574

120572

hB

120573

120573

+ 120574

Frs

Figure 3 The schematic of simplified cantilever beam model of an isosceles trapezoid for buttress type thread tooth

b

c

z

ya

Ncos120599

Nsin120599

120599

120575a120575b

(a)

a b

c

z

y

Ncos120599

Nsin120599

120575c

(b)

a b

c

z

y

120575d

Ncos120599

(c)

a b

c

z

y

Nsin120599120575r

(d)

Figure 4The elastic deformations of simplified thread tooth (a) deformation caused by tooth bending or shear force (b) deformation causedby tooth root incline (c) deformation caused by shear deformation of tooth root (d) deformation caused by radial effect

The deformations 120575rp 120575rc caused by radial effect for pinthread tooth and coupling thread tooth can be respectivelyexpressed as

120575rp (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot [

119889

2

2

(119909) + 4119903

2

0

119889

2

2

(119909) minus 4119903

2

0

minus ]]1198892(119909)

(9)

120575rc (119909) =119873 (119909) cos (120572 minus 120574)

119864

tan2 (120572 minus 120574)2119875

sdot (

4119877

2

0

+ 119889

2

2

(119909)

4119877

2

0

minus 119889

2

2

(119909)

minus ])119889

2(119909)

(10)

Consequently the total elastic deformations 120575tp 120575tc forpin thread tooth and coupling thread tooth along its conedirection (119885 direction) are respectively as follows

120575tp (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rp

=

[119896 + 119896rp (119909)]119873 (119909) cos (120572 minus 120574)119864

(11)

120575tc (119909) = 120575

119886+ 120575

119887+ 120575

119888+ 120575

119889+ 120575rc

=

[119896 + 119896rc (119909)]119873 (119909) cos (120572 minus 120574)119864

(12)

In (11) and (12) 119896 = 119896

119886+ 119896

119887+ 119896

119888+ 119896

119889 The deformation

coefficients 119896119886 119896119887 119896119888 119896119889 119896rp and 119896rc can be determined from

(5) to (10)In (5)sim(12) 119873(119909) denotes the normal load acting on

the load flank of thread tooth within unit axial length d119909Because the load flank angle and thread taper are both verysmall in premium threaded connection we ignored the axialcomponents of frictional force on load flank and that of thecontact stress between root and crest The relation between119873(119909) and 119891(119909) can be briefly expressed as

119873(119909)

cos120572tan [120595 (119909)]

d119909 = 119891 (119909) d119909 (13)

In (13) the helix angle 120595(119909) is

120595 (119909) = arc tan [ 119875

120587119889

2(119909)

]

119889

2(119909) = 119889

2119904minus 2119909 tan 120574

(14)

Mathematical Problems in Engineering 5

By submitting (1) and (14) into (13) we can get

119873(119909) =

119875

120587119889

2(119909) cos120572

d119865 (119909)d119909

(15)

Then by submitting (15) into (11) and (12) the total elasticdeformations for pin thread tooth and coupling thread toothare respectively as follows

120575tp (119909) = 120582p (119909)d119865 (119909)d119909

=

[119896 + 119896rp (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(16)

120575tc (119909) = 120582c (119909)d119865 (119909)d119909

=

[119896 + 119896rc (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(17)

In (16) and (17) the coefficients 120582p and 120582c depend on thegeometry parameters of connection and thread tooth as wellas the used material properties whose derivatives for 119909 arerespectively as follows

120582

1015840

p

=

d120582p

d119909=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rp) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rp)d119909

119875

120587119889

2

+ (119896 + 119896rp)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

119889

2

2

+ 4119903

2

0

119889

2

2

minus 4119903

2

0

minus

16119889

2

2

119903

2

0

(119889

2

2

minus 4119903

2

0

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rp)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

119889

4

2

(119909) minus 16119903

4

0

minus 16119903

2

0

119889

2

2

(119909)

(119889

2

2

(119909) minus 4119903

2

0

)

2

minus ]]

minus (119896 + 119896rp)2119875

119889

2(119909)

(18)

120582

1015840

c

=

d120582cd119909

=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rc) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rc)d119909

119875

120587119889

2

+ (119896 + 119896rc)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

4119877

2

0

+ 119889

2

2

4119877

2

0

minus 119889

2

2

+

16119877

2

0

119889

2

2

(4119877

2

0

minus 119889

2

2

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rc)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

16119877

4

0

minus 119889

4

2

(119909) + 16119877

2

0

119889

2

2

(119909)

(4119877

2

0

minus 119889

2

2

(119909))

2

minus ]]

minus (119896 + 119896rc)2119875

119889

2(119909)

(19)

23 Differential Equation of the Axial Load 119865(119909) Under theaction of pretightening force the pin thread and couplingthread are always engaged to each other indicating thattheir thread pitches are equal after deformation Howeverthe compressed pin will make its own thread pitch decreasewhile the stretched coupling will make its own thread pitchincreaseThedeformation of pin thread tooth should increaseits own pitch while the deformation of coupling thread toothshould reduce its ownpitch thus guaranteeing the equal pitchand the perfectly engaged state for pin thread and couplingthread

According to Hookersquos law the compressed and stretchedelastic deformations of pin and coupling body along thethread cone direction are respectively as follows

120594p =4

120587119864 cos 120574int

119909

0

119865 (119909)

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

d119909 (20)

120594c =4

120587119864 cos 120574int

119909

0

119865 (119909)

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

d119909 (21)

Moreover the deformation compatibility equation canbe obtained from the equal pitch after deformation for pinthread and coupling thread

120594p (119909) + 120594c (119909) = [120575tp (119909) + 120575tc (119909)]

minus [120575tp (0) + 120575tc (0)] (22)

6 Mathematical Problems in Engineering

Then by submitting (16) (17) (20) and (21) into (22) weget

4

120587119864 cos 120574int

119909

0

[

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

]119865 (119909) d119909

= (120582p + 120582c)d119865 (119909)d119909

(23)

Taking the derivative with respect to 119909 for (23) andconsidering that 120582p + 120582c = 0 the differential equation of axialload distribution can be obtained as

d2119865 (119909)d1199092

+ 119898 (119909)

d119865 (119909)d119909

+ 119899 (119909) 119865 (119909) = 0

119898 (119909) = (

120582

1015840

p + 1205821015840

c

120582p + 120582c)

119899 (119909) = minus

4

120587119864 (120582p + 120582c) cos 120574

sdot

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

(24)

The boundary conditions for (24) are

119865 (0) = 0

119865 (119871) = 119865

119905

(25)

24 Solution of Differential Equation The axial load distri-bution equation (24) is a nonlinear differential equation andits analytical solution is very difficult to get so we solved itsnumerical solution with finite difference method First wedivided the engaged thread according to unit length of threadpitch 119875 and numbered the node 0sim119873

119905successively from the

big end of the pin thread so the coordinate for node 119894 was119909

119894= 119894119875 Then we took 119865

119894to denote the node force 119865(119909

119894) and

considered the boundary conditions equation (25) Finallythe difference equation of (24) can be obtained as

119860[119865

1119865

2sdot sdot sdot 119865

119873minus1]

⊤

= 119861 (26)

In (26) the coefficient matrix is

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus

2

119875

2

+ 119899 (119909

1)

1

119875

2

+

119898 (119909

1)

2119875

0

1

119875

2

minus

119898 (119909

2)

2119875

minus

2

119875

2

+ 119899 (119909

2)

1

119875

2

+

119898 (119909

2)

2119875

sdot sdot sdot

1

119875

2

minus

119898 (119909

119894)

2119875

minus

2

119875

2

+ 119899 (119909

119894)

1

119875

2

+

119898 (119909

119894)

2119875

sdot sdot sdot

sdot sdot sdot

0

1

119875

2

minus

119898 (119909

119873minus1)

2119875

minus

2

119875

2

+ 119899 (119909

119873minus1)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

And the constant term column matrix is

119861 = [0 0 sdot sdot sdot minus (

1

119875

2

+

119898 (119909

119873minus1)

2119875

)119865

119905]

⊤

(28)

The difference equation (26) was solved based on Gaus-sian main elimination method

25 Geometric Description for Load Flank of Thread ToothFor calculating the mean contact stress on the load flank ofthread tooth the geometric description of thread surface isnecessary Setting the premium thread connection as the 119909-axis and its intersection with the plane of thread load flank asthe origin we established global coordinate system 119874-119909119910119911

whose unit vectors are i j and k respectively shown inFigure 5

For determining the unit normal vector n and unittangent vector 120591 of thread surface the cylindrical coordinatesystem 119874-119903120599119909 was also established and its vectors are

u = minusj

k = minusk

w = i

(29)

The plane 1198741198612119862

2119863

2 which stands for thread load flank

can be obtained by rotating the quadrangle 119874119861119862119863 in 119910119900119911

plane 120572 around the 119910-axis and then 120595 around the 119911-axissuccessively If quadrangle 119874119861119862119863 is a 1 times 1 square its

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

By submitting (1) and (14) into (13) we can get

119873(119909) =

119875

120587119889

2(119909) cos120572

d119865 (119909)d119909

(15)

Then by submitting (15) into (11) and (12) the total elasticdeformations for pin thread tooth and coupling thread toothare respectively as follows

120575tp (119909) = 120582p (119909)d119865 (119909)d119909

=

[119896 + 119896rp (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(16)

120575tc (119909) = 120582c (119909)d119865 (119909)d119909

=

[119896 + 119896rc (119909)]

119864

cos (120572 minus 120574)cos120572

119875

120587119889

2(119909)

d119865 (119909)d119909

(17)

In (16) and (17) the coefficients 120582p and 120582c depend on thegeometry parameters of connection and thread tooth as wellas the used material properties whose derivatives for 119909 arerespectively as follows

120582

1015840

p

=

d120582p

d119909=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rp) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rp)d119909

119875

120587119889

2

+ (119896 + 119896rp)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

119889

2

2

+ 4119903

2

0

119889

2

2

minus 4119903

2

0

minus

16119889

2

2

119903

2

0

(119889

2

2

minus 4119903

2

0

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rp)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

119889

4

2

(119909) minus 16119903

4

0

minus 16119903

2

0

119889

2

2

(119909)

(119889

2

2

(119909) minus 4119903

2

0

)

2

minus ]]

minus (119896 + 119896rp)2119875

119889

2(119909)

(18)

120582

1015840

c

=

d120582cd119909

=

cos (120572 minus 120574)119864 cos120572

d [(119896 + 119896rc) (1198751205871198892)]d119909

=

cos (120572 minus 120574)119864 cos120572

[

d (119896 + 119896rc)d119909

119875

120587119889

2

+ (119896 + 119896rc)d (119875120587119889

2)

d119909]

=

cos (120572 minus 120574)119864 cos120572

tan2 (120572 minus 120574) tan 120574119875

sdot [

4119877

2

0

+ 119889

2

2

4119877

2

0

minus 119889

2

2

+

16119877

2

0

119889

2

2

(4119877

2

0

minus 119889

2

2

)

2

minus ]]

sdot

119875

120587119889

2

minus (119896 + 119896rc)2119875 tan 120574120587119889

2

2

=

tan 120574 cos (120572 minus 120574)120587119864119889

2(119909) cos120572

tan2 (120572 minus 120574)

sdot [

16119877

4

0

minus 119889

4

2

(119909) + 16119877

2

0

119889

2

2

(119909)

(4119877

2

0

minus 119889

2

2

(119909))

2

minus ]]

minus (119896 + 119896rc)2119875

119889

2(119909)

(19)

23 Differential Equation of the Axial Load 119865(119909) Under theaction of pretightening force the pin thread and couplingthread are always engaged to each other indicating thattheir thread pitches are equal after deformation Howeverthe compressed pin will make its own thread pitch decreasewhile the stretched coupling will make its own thread pitchincreaseThedeformation of pin thread tooth should increaseits own pitch while the deformation of coupling thread toothshould reduce its ownpitch thus guaranteeing the equal pitchand the perfectly engaged state for pin thread and couplingthread

According to Hookersquos law the compressed and stretchedelastic deformations of pin and coupling body along thethread cone direction are respectively as follows

120594p =4

120587119864 cos 120574int

119909

0

119865 (119909)

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

d119909 (20)

120594c =4

120587119864 cos 120574int

119909

0

119865 (119909)

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

d119909 (21)

Moreover the deformation compatibility equation canbe obtained from the equal pitch after deformation for pinthread and coupling thread

120594p (119909) + 120594c (119909) = [120575tp (119909) + 120575tc (119909)]

minus [120575tp (0) + 120575tc (0)] (22)

6 Mathematical Problems in Engineering

Then by submitting (16) (17) (20) and (21) into (22) weget

4

120587119864 cos 120574int

119909

0

[

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

]119865 (119909) d119909

= (120582p + 120582c)d119865 (119909)d119909

(23)

Taking the derivative with respect to 119909 for (23) andconsidering that 120582p + 120582c = 0 the differential equation of axialload distribution can be obtained as

d2119865 (119909)d1199092

+ 119898 (119909)

d119865 (119909)d119909

+ 119899 (119909) 119865 (119909) = 0

119898 (119909) = (

120582

1015840

p + 1205821015840

c

120582p + 120582c)

119899 (119909) = minus

4

120587119864 (120582p + 120582c) cos 120574

sdot

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

(24)

The boundary conditions for (24) are

119865 (0) = 0

119865 (119871) = 119865

119905

(25)

24 Solution of Differential Equation The axial load distri-bution equation (24) is a nonlinear differential equation andits analytical solution is very difficult to get so we solved itsnumerical solution with finite difference method First wedivided the engaged thread according to unit length of threadpitch 119875 and numbered the node 0sim119873

119905successively from the

big end of the pin thread so the coordinate for node 119894 was119909

119894= 119894119875 Then we took 119865

119894to denote the node force 119865(119909

119894) and

considered the boundary conditions equation (25) Finallythe difference equation of (24) can be obtained as

119860[119865

1119865

2sdot sdot sdot 119865

119873minus1]

⊤

= 119861 (26)

In (26) the coefficient matrix is

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus

2

119875

2

+ 119899 (119909

1)

1

119875

2

+

119898 (119909

1)

2119875

0

1

119875

2

minus

119898 (119909

2)

2119875

minus

2

119875

2

+ 119899 (119909

2)

1

119875

2

+

119898 (119909

2)

2119875

sdot sdot sdot

1

119875

2

minus

119898 (119909

119894)

2119875

minus

2

119875

2

+ 119899 (119909

119894)

1

119875

2

+

119898 (119909

119894)

2119875

sdot sdot sdot

sdot sdot sdot

0

1

119875

2

minus

119898 (119909

119873minus1)

2119875

minus

2

119875

2

+ 119899 (119909

119873minus1)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

And the constant term column matrix is

119861 = [0 0 sdot sdot sdot minus (

1

119875

2

+

119898 (119909

119873minus1)

2119875

)119865

119905]

⊤

(28)

The difference equation (26) was solved based on Gaus-sian main elimination method

25 Geometric Description for Load Flank of Thread ToothFor calculating the mean contact stress on the load flank ofthread tooth the geometric description of thread surface isnecessary Setting the premium thread connection as the 119909-axis and its intersection with the plane of thread load flank asthe origin we established global coordinate system 119874-119909119910119911

whose unit vectors are i j and k respectively shown inFigure 5

For determining the unit normal vector n and unittangent vector 120591 of thread surface the cylindrical coordinatesystem 119874-119903120599119909 was also established and its vectors are

u = minusj

k = minusk

w = i

(29)

The plane 1198741198612119862

2119863

2 which stands for thread load flank

can be obtained by rotating the quadrangle 119874119861119862119863 in 119910119900119911

plane 120572 around the 119910-axis and then 120595 around the 119911-axissuccessively If quadrangle 119874119861119862119863 is a 1 times 1 square its

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Then by submitting (16) (17) (20) and (21) into (22) weget

4

120587119864 cos 120574int

119909

0

[

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

]119865 (119909) d119909

= (120582p + 120582c)d119865 (119909)d119909

(23)

Taking the derivative with respect to 119909 for (23) andconsidering that 120582p + 120582c = 0 the differential equation of axialload distribution can be obtained as

d2119865 (119909)d1199092

+ 119898 (119909)

d119865 (119909)d119909

+ 119899 (119909) 119865 (119909) = 0

119898 (119909) = (

120582

1015840

p + 1205821015840

c

120582p + 120582c)

119899 (119909) = minus

4

120587119864 (120582p + 120582c) cos 120574

sdot

1

[119889

2(119909) minus ℎ

119861]

2

minus 4119903

2

0

+

1

4119877

2

0

minus [119889

2(119909) + ℎ

119861]

2

(24)

The boundary conditions for (24) are

119865 (0) = 0

119865 (119871) = 119865

119905

(25)

24 Solution of Differential Equation The axial load distri-bution equation (24) is a nonlinear differential equation andits analytical solution is very difficult to get so we solved itsnumerical solution with finite difference method First wedivided the engaged thread according to unit length of threadpitch 119875 and numbered the node 0sim119873

119905successively from the

big end of the pin thread so the coordinate for node 119894 was119909

119894= 119894119875 Then we took 119865

119894to denote the node force 119865(119909

119894) and

considered the boundary conditions equation (25) Finallythe difference equation of (24) can be obtained as

119860[119865

1119865

2sdot sdot sdot 119865

119873minus1]

⊤

= 119861 (26)

In (26) the coefficient matrix is

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus

2

119875

2

+ 119899 (119909

1)

1

119875

2

+

119898 (119909

1)

2119875

0

1

119875

2

minus

119898 (119909

2)

2119875

minus

2

119875

2

+ 119899 (119909

2)

1

119875

2

+

119898 (119909

2)

2119875

sdot sdot sdot

1

119875

2

minus

119898 (119909

119894)

2119875

minus

2

119875

2

+ 119899 (119909

119894)

1

119875

2

+

119898 (119909

119894)

2119875

sdot sdot sdot

sdot sdot sdot

0

1

119875

2

minus

119898 (119909

119873minus1)

2119875

minus

2

119875

2

+ 119899 (119909

119873minus1)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

And the constant term column matrix is

119861 = [0 0 sdot sdot sdot minus (

1

119875

2

+

119898 (119909

119873minus1)

2119875

)119865

119905]

⊤

(28)

The difference equation (26) was solved based on Gaus-sian main elimination method

25 Geometric Description for Load Flank of Thread ToothFor calculating the mean contact stress on the load flank ofthread tooth the geometric description of thread surface isnecessary Setting the premium thread connection as the 119909-axis and its intersection with the plane of thread load flank asthe origin we established global coordinate system 119874-119909119910119911

whose unit vectors are i j and k respectively shown inFigure 5

For determining the unit normal vector n and unittangent vector 120591 of thread surface the cylindrical coordinatesystem 119874-119903120599119909 was also established and its vectors are

u = minusj

k = minusk

w = i

(29)

The plane 1198741198612119862

2119863

2 which stands for thread load flank

can be obtained by rotating the quadrangle 119874119861119862119863 in 119910119900119911

plane 120572 around the 119910-axis and then 120595 around the 119911-axissuccessively If quadrangle 119874119861119862119863 is a 1 times 1 square its

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

zB

r

C

Dy

x

Axis ofconnection

C2

D2

B2

O

n

120595

120572

120579

τ

Figure 5 Geometric description schematic of thread load flank

vertices after 2 rotations are 119874(0 0 0) 1198612(sin120572 0 cos120572)

119862

2(sin120572 + sin120595 cos120595 cos120572) and 119863

2(sin120595 cos120595 0) Thus

the equation of load flank plane can be obtained as

119909 minus 119910 tan120595 minus 119911 tan120572 = 0 (30)

The unit normal vector n and unit tangent vector 120591 ofthread load flank are

n =

i minus tan120595j minus tan120572k

radic1 + tan2120595 + tan2120572=

tan120595u + tan120572k + w

radic1 + tan2120595 + tan2120572

120591 = minus sin120595i minus cos120595j = cos120595u minus sin120595w

(31)

And the infinitesimal area of thread surface dΩ can bealso expressedwith the infinitesimal area d119860 in the axial crosssection as

dΩ =radic1 + tan2120595 + tan2120572 d119860 (32)

26 Mean Contact Stress on Load Flank of Thread ToothSupposing that the contact stress on load flank was uniformand that the helix angle was constant in each engaged threadthe axial component of contact stress integral on the loadflank within each engaged thread should be equal to the axialload acting on itself that is

∬

Ω119894

[(119901

119894n119894+ 120583

119905119901

119894120591119894) dΩ] sdot i = 119865

119894119875 (33)

By submitting (31) and (32) into (33) the mean contactstress on load flank of 119894th tooth is

119901

119894=

119865

119894119875

(1 minus 120583

119905

radic1 + tan2120595119894+ tan2120572 sin120595

119894)∬

119860119894

d119860 (34)

In (34)

∬

119860119894

d119860 = int

2120587

0

int

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587+ℎ1198612

1198892[(119894minus1)119875]minus120599(119901 tan 120574)120587minusℎ1198612

119903 d119903 d120599

= 120587ℎ

119861[119889

2119904minus (2119894 + 1) 119875 tan 120574]

(35)

At last the mean contact stress on the load flank of 119894thtooth is

119901

119894

= 119865

119894119875(120587ℎ

119861(1 minus 120583

119905

radic1 + tan2120595

119894+ tan2120572 sin120595

119894)

sdot [119889

2119904minus (2119894 + 1) 119875 tan 120574] )

minus1

(36)

3 Calculation Example

31 Analytical Method For simplicity the geometricalparameters of a premium connection are mainly taken fromAPI 1778mm P110 grade buttress thread casing in APISpec5B Its geometry parameters and material properties arelisted in Table 1

Suppose the pretightening force 119865119905from sealing surface

and shoulder contact is 1000KN the axial load acting oneach engaged thread 119865

119894119875and the mean contact stress on load

flank 119901119894can be calculated according to the analytical model

proposed in this paper the results are shown in Figures 6 and7 respectively

Figure 6 shows that the load acting on engaged threadtooth increases with the increase of thread number andthat the last 4sim6 engaged threads bear more than 90 ofthe pretightening force Particularly for the last engagedthread it carries 4454 of the pretightening force The loaddistribution is similar to the practical situation in threadedconnection indicating that the proposed analytical methodis reasonable Meanwhile contact stress distribution on loadflank of thread teeth shows a similar pattern to the loaddistribution whosemaximum value is 52274MPa on the lastengaged thread shown in Figure 7 It is obvious that threadsurface would sustain a galling risk if the pretightening forceis increased to some extent

32 Finite ElementMethod In order to validate the proposedanalytical model in this paper we analyzed the same samplewith the finite element software ANSYS 145 Considering thecharacteristics of axisymmetric load and boundary condi-tions a 2D finite element model of modified API 1778mmpremium thread connection has been established (shown inFigure 8)

To make the simulated results more approachable topractical situations the materials of pin and coupling wereboth defined as bilinear isotropic hardening model in theFEM The frictional contact type between pin thread andcoupling thread has been selected indicating that the contactbetween thread teeth is taken as an elastic-plastic and fric-tional contact problemThe eight-node quadrilateral elementQuad 8 node-183 which can be used as an axisymmetricelement to simulate plastic deformation was selected for thisanalysisThe element size was controlled and then local meshrefinement was applied to the thread tooth edges Figures9 and 10 show the mesh on the whole model and the teethedges Because the contact between engaged thread teeth is amaterial and boundary nonlinear problem static analysis hasbeen selected for solving it The boundary conditions set for

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 Geometry parameters and material properties of premium connection

Symbol Value Unit Symbol Value Unit119877

0

98 mm 119865rs 239 mm119903

0

7854 mm ℎ

119861

158 mm119875 508 mm 119873

119905

15 Integer119889

2119904

17663 mm 119864 206 GPa120574 179 Degree ] 03 Dimensionless120572 3 Degree 120583

119905

008 Dimensionless120573 10 Degree 119891ymn 75842 MPa

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50

100

150

200

250

300

350

400

450

500

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Load

actin

g on

each

enga

ged

thre

ad (k

N)

Thread number

Analytical method

Figure 6 Load acting on each engaged thread by the proposedanalytical model

this analysis are an axial displacement confined at the big endof coupling thread and application of 31393MPa compressedstress which is equivalent to 1000KN of pretightening forceon the small end of pin thread By using ANSYS we canacquire the stress distribution on the thread teeth (shown inFigure 11)

Figure 11 shows the Von-Mises stress distribution on eachcontact thread tooth In Figure 11 we can find that the last4sim6 engaged threads bear almost all the pretightening forcewhich is identical to the load distribution calculated by theproposed analytical model in this paper

33 Comparison and Discussion Themean contact stress onthe load flank of thread teeth calculated from the proposedanalytical model and that obtained by FEM are comparedin Figure 7 One can find that both curves are identical intrend and that the minimum and maximum values obtainedby the proposed analytical method are also close to thoseof FEM Meanwhile we can also find that there exists somediscrepancy in the middle region of the two curves whichmay be attributed to the simplifications and assumptionsof the analytical model The helix angle in the analyticalmodel may be a reason because the 2D FEM cannot containthis factor However the analytical model can still predict a

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50100150200250300350400450500550600

Mea

n co

ntac

t stre

ss o

n lo

ad fl

ank

(MPa

)

Thread number

Analytical methodFinite element method

Figure 7 Mean contact stress on load flank of thread tooth inpremium connection obtained by analytical method and finiteelement method

relativelymeaningful stress value for the last engaged threadswhich is the easiest one to fail

Figure 12 shows the comparison of mean contact stresseson the last engaged threads under different pretighteningforces It can be seen from Figure 12 that mean contact stresson the last engaged thread obtained by analytical modelkeeps increasing linearly with the increase of pretighteningforce while that calculated from FEM tends to be stableafter a section of linear increaseThis discrepancy can mainlybecause that elastic deformation has been supposed for theanalytical model but elastic-plastic deformation consideredin FEM Because the allowable in-service pretightening forceis about 1350KNsim1750KN the error of the analytical modelfor the stress on the last engaged thread is about 62sim181 Consequently the proposed analytical method hascertain accuracy and can to some extent meet the needs ofengineering

4 Conclusions

(1) This paper proposes an analytical model for investi-gating loading and contact stress on the thread teeth

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Big end of coupling thread

PinSmall end of pin thread

Coupling

Figure 8 Finite element model of modified API 1778mm premium connection

Figure 9 Mesh on the whole finite element model

(a) (b)

Figure 10 Refined mesh at teeth edges (a) left region and (b) right region

8415e8

6545e8

4675e8

2552e4

2805e4

9352e7

748e8

561e8

374e8

187e8

Figure 11 Von-Mises stress distribution on the thread teeth inANSYS

in tubing and casing premium threaded connectionon the basis of elastic mechanics

(2) The proposed analytical model is validated by theapplication of the FEM to the same sample The

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

100200300400500600700800900

10001100

Mea

n co

ntac

t stre

ss o

n th

e las

t

Pretightening force (kN)

Analytical methodFinite element method

enga

ged

thre

ad (M

Pa)

Figure 12 Comparison of mean contact stresses on the last engagedthread tooth under different pretightening forces

pretightening force mainly applies to the last 4sim6engaged threads about 50 of that to the last engagedthreads

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

(3) Comparing the contact stress of the last engagedthread between the analytical model and FEM showsthat the accuracy of the analytical model will declinewith the increase of pretightening force after thematerial enters into plastic deformation In practicethe relative error is about 62sim181 This indicatesthat the analytical method can to some extent meetthe needs of engineering

Nomenclature

119865

119905 Total force transferred by

engaged threads119873

119905 Total number of engaged

thread teeth119873(119909) Normal load acting on the load

flank of thread tooth withinunit axial length d119909

119891(119909) Axial load distributionintensity

119865(119909) Axial load distribution119865

119894119875 Axial load acted on each

engaged thread119875 Pitch of thread120572 Load flank angle in the axial

cross section of thread120573 Leading flank angle in the axial

cross section of thread120595(119909) Helix angle119889

2(119909) Pitch diameter

119889

2119904 Pitch diameter at the big end of

pin thread120574 Half angle of thread cone119886 Root width of isosceles

trapezoid tooth119887 The width of isosceles

trapezoid tooth at pitchdiameter

119888 The distance from pitchdiameter of isosceles trapezoidtooth to its root

119865rs Root width of buttress typetooth

ℎ

119861 Completed thread height

120575tp Total elastic compresseddeformation of pin threadtooth

120575tc Total elastic stretcheddeformation of coupling threadtooth

120575

119886 120575119887 120575119888 120575119889 Elastic deformation of threadtooth caused by tooth bendingshear force root incline andshear deformation of toothroot respectively

120575rp 120575rc Elastic deformation of threadtooth caused by radial effect forpin thread tooth and couplingthread tooth respectively

119896

119886 119896119887 119896119888 119896119889 119896rp 119896rc 119896 120582p 120582c Relevant deformation

coefficients of threadtooth

119864 Elastic modulus ofmaterial

] Poisson ratio of material119891ymn Specified minimum yield

strength of material119903

0 Internal radius of pipe

119877

0 External radius of

coupling120594p 120594c Compressed and

stretched elasticdeformation along threadcone direction for pipeand coupling bodyrespectively

119898 119899 Coefficients of differentialequation

119865

119894 Axial load on 119894th node

119901

119894 Mean contact stress on

the load flank of 119894th tooth120583

119905 Frictional coefficient on

the thread surface120595

119894 Mean helix angle on 119894th

tooth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (Grant nos 50904050and 51244007) and the financial support of Tianjin Pipe(Group) Co Ltd

References

[1] G Carcagno ldquoThe design of tubing and casing premiumconnections for HTHP wellsrdquo in Proceedings of the SPE HighPressureHigh Temperature SourWell Design Applied TechnologyWorkshop Society of Petroleum Engineers 2005

[2] A Bradley S Nagasaku and E Verger ldquoPremium connectiondesign testing and installation for HPHT sour wellsrdquo in SPEHigh PressureHigh Temperature SourWell Design Applied Tech-nology Workshop Society of Petroleum Engineers 2005

[3] M Sugino KNakamura S Yamaguchi DDaly G Briquet andE Verger ldquoDevelopment of an innovative high-performancepremium threaded connection for OCTGrdquo in Proceedings of theOffshore Technology Conference (OTC rsquo10) pp 1883ndash1891 May2010

[4] L Maduschka ldquoBeanspruchung von Schraubenverbindungenund zweckmaszligige Gestaltung der Gewindetragerrdquo Forschungauf dem Gebiete des Ingenieurwesens A vol 7 no 6 pp 299ndash305 1936

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[5] D Sopwith ldquoThedistribution of load in screw threadsrdquoProceed-ings of the Institution of Mechanical Engineers vol 159 no 1 pp373ndash383 1948

[6] S Yazawa and K Hongo ldquoDistribution of load in the screwthread of a bolt-nut connection subjected to tangential forcesand bending momentsrdquo JSME International Journal I SolidMechanics Strength ofMaterials vol 31 no 2 pp 174ndash180 1988

[7] W Wang and K M Marshek ldquoDetermination of the loaddistribution in a threaded connector having dissimilarmaterialsand varying thread stiffnessrdquo Journal of Engineering for Industryvol 117 no 1 pp 1ndash8 1995

[8] L Bruschelli andV Latorrata ldquoThe influence of ldquoshell behaviorrdquoon load distribution for thin-walled conical jointsrdquo Journal ofApplied Mechanics vol 67 no 2 pp 298ndash306 2000

[9] S-J ChenQ An Y Zhang L-X Gao andQ Li ldquoLoading anal-ysis on the thread teeth in cylindrical pipe thread connectionrdquoJournal of Pressure Vessel Technology vol 132 no 3 pp 0312021ndash0312028 2010

[10] P OrsquoHara ldquoFinite element analysis of threaded connectionsrdquo inProceedings of the Army Symposium on Solid Mechanics DTICDocument pp 99ndash119 1974

[11] J L Bretl and R D Cook ldquoModelling the load transfer inthreaded connections by the finite element methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 14 no9 pp 1359ndash1377 1979

[12] H Zhao ldquoA numerical method for load distribution in threadedconnectionsrdquo Journal of Mechanical Design vol 118 no 2 pp274ndash279 1996

[13] H Zhao ldquoStress concentration factors within bolt-nut connec-tors under elasto-plastic deformationrdquo International Journal ofFatigue vol 20 no 9 pp 651ndash659 1998

[14] K A Macdonald andW F Deans ldquoStress analysis of drillstringthreaded connections using the finite element methodrdquo Engi-neering Failure Analysis vol 2 no 1 pp 1ndash30 1995

[15] H Bahai G Glinka and I I Esat ldquoNumerical and experimentalevaluation of SIF for threaded connectorsrdquo Engineering FractureMechanics vol 54 no 6 pp 835ndash845 1996

[16] J-J Chen and Y-S Shih ldquoStudy of the helical effect on thethread connection by three dimensional finite element analysisrdquoNuclear Engineering andDesign vol 191 no 2 pp 109ndash116 1999

[17] S Baragetti ldquoEffects of taper variation on conical threadedconnections load distributionrdquo Journal of Mechanical Designvol 124 no 2 pp 320ndash329 2002

[18] T Fukuoka and M Nomura ldquoProposition of helical threadmodeling with accurate geometry and finite element analysisrdquoJournal of Pressure Vessel Technology vol 130 no 1 Article ID011204 pp 1ndash6 2008

[19] E Stoeckly and H Macke ldquoEffect of taper on screw thread loaddistributionrdquo Transactions of the ASME vol 74 no 1 pp 103ndash112 1952

[20] G Yuan Z Yao J Han and Q Wang ldquoStress distribution ofoil tubing thread connection during make and break processrdquoEngineering Failure Analysis vol 11 no 4 pp 537ndash545 2004

[21] J C R Placido P E Valada de Miranda T A Netto I PPasqualino G FMiscow and B de Carvalho Pinheiro ldquoFatigueanalysis of aluminum drill pipesrdquoMaterials Research vol 8 no4 pp 409ndash415 2005

[22] M Slack H Salkin and F Langer ldquoTechnique to assessdirectly make-up contact stress inside tubular connectionsrdquo inSPEIADCDrilling Conference Society of Petroleum Engineers1990

[23] K A Hamilton B Wagg and T Roth ldquoUsing ultrasonic tech-niques to accurately examine seal surface contact stress in pre-mium connectionsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3461ndash3471 Society of PetroleumEngineers November 2007

[24] A YamamotoThe Theory and Computation of Thread Connec-tion Youkendo Tokyo Japan 1980 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of