287076
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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Complex AnalysisJournal of
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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
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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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
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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
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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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
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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
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