Loading and Contact Stress Analysis on the Thread Teeth in Tubing and Casing Premium Threaded Connection

Loading and contact stress distribution on the thread teeth in tubing and casing premium threaded connections are of great importance for design optimization, pretightening force control, and thread failure prevention. This paper proposes an analytical method based on the elastic mechanics. This is quite different from other papers, which mainly rely on finite element analysis. The differential equation of load distribution on the thread teeth was established according to equal pitch of the engaged thread after deformation and solved by finite difference method. Furthermore, the relation between load acting on each engaged thread and mean contact stress on its load flank is set up based on the geometric description of thread surface. By comparison, this new analytical method with the finite element analysis for a modified API 177.8mm premium threaded connection is approved. Comparison of the contact stress on the last engaged thread between analyticalmodel and FEMshows that the accuracy of analytical model will decline with the increase of pretightening force after thematerial enters into plastic deformation. However, the analytical method can meet the needs of engineering to some extent because its relative error is about 6.2%∼18.1% for the in-service level of pretightening force.


Introduction
Premium threaded connections have been widely used for maintaining structural and sealing integrity of tubing and casing strings in HPHT wells.The overwhelming majority of premium threaded connections are characterized by a tapered metal-to-metal seal structure for leakage resistance and an interior rotary torque shoulder to control final make-up torques, providing additional sealing performance (shown in Figure 1).These characteristics are different from API buttress thread connections and extreme-line thread connections [1][2][3].In general, the thread teeth of premium threaded connection are used to bear axial tensile load, due to the load flank angle  design (shown in Figure 3) of usually −4 ∘ ∼3 ∘ .We also noticed that the taper angle of metal-tometal seal is usually 1.79 ∘ ∼2.86 ∘ , so the axial component of sealing contact stress could be neglected and the action force of sealing and shoulder replaced approximatively by an axial pretightening force   acting on the A-B cross section plane (shown in Figure 1).Consequently, it is significant to investigate the load and contact stress on thread teeth caused by this pretightening force   for design optimization, pretightening torque control, and thread failure prevention.
The load distribution and contact stress on thread teeth were investigated in many previous references in the last century, whose methods can mainly be divided into three categories.The first category is developed on the basis of analytical method.Maduschka [4] and Sopwith [5] first proposed the models of the load distribution in cylindrical screw threads, whose results clearly indicate that the first three or four engaged threads carry more than half of the preload induced by the make-up torque.Taking axial and tangential forces, as well as bending moments, into consideration, Yazawa and Hongo [6] developed a model to investigate the load distribution of a bolt-nut connection.Wang and Marshek [7] also established a spring model to predict the load distribution of bolt-nut connection.Furthermore, Bruschelli and Latorrata [8] developed an analytical model to study the load distribution for conical threads connections.Chen et al. [9] presented a practical and convenient model for cylindrical pipe threaded connection, which can be applied to calculate the load and deformation on each thread tooth just with tightening torque and thread numbers.The vast majority of the analytical models mentioned above are established from screw threaded connection or cylindrical pipe threaded connection and therefore are not applicable to premium connection: its tapered thread connection type contains special metal-to metal seal, interior rotary torque shoulder, and the specialized tooth form of buttress.
The second category method is developed based on finite element method (FEM).Capable of considering material nonlinear behavior and complex boundary conditions easily, FEM makes studying the load distribution, contact stress, and stress concentration on the thread tooth possible.O'Hara [10] established a 2D axisymmetric model of thread connection based on thread characteristic lengths and the numerical results were generally validated by Heywood's equations.Bretl and Cook [11] proposed a unique FEM technique for threaded connections and their simulated results agreed with those from analytical and experimental methods both for conventional and for tapered threads.Zhao [12,13] developed an elastic-plastic FEM and investigated the load distribution and SCFs (stress concentration factors) of the bolt-nut connection.Macdonald and Deans [14] acquired the SCFs and induced stresses on the drilling string thread and they found out that the high local stresses are located at the root of the first engaged tooth in the pin and the last engaged tooth in the box.Bahai et al. [15] also calculated the SCFs of API drill string threaded connector subjected to preloading, axial loading, and bending loads.Chen and Shih [16] developed a 3D FEM for bolted joints and analyzed the helical and friction effect on the load distribution of each thread.Baragetti [17] proposed a numerical finite element procedure for rotary shouldered connections (RSCs) and quantified the effects of taper on the loads carried by each engaged thread and contact stress on the thread flanks.Fukuoka and Nomura [18] established the FE model with accurate thread geometry by deriving a mathematical expression of the helical thread geometry of a single-thread screw.The numerical results indicate that axial load along engaged threads shows a different distribution pattern from those obtained by axisymmetric FE analysis and elastic theory.Although FE models have incomparable advantages for threaded connections analysis, most of them are based on 2D axisymmetric assumptions and the helix angle is usually ignored.
The third category method is developed based on experimental testing.Stoeckly and Macke [19] developed a testing apparatus to measure the axial displacement of the threads and compare their experimental data by modifying Sopwith equations.They found out that the frictional coefficient between the lubricated bolt and nut has little effect on the thread load distribution.Yuan et al. [20] tested the makeup and break-out operations in oilfields by placing the foil gauges on the inner surface of the male thread and the outer surface of the female thread along the axial and tangential direction and concluded that those operations have a strong impact on the service life of oil tubing threaded connection.Plácido et al. [21] carried out some full and reduced scale experimental tests on aluminum drill pipes to investigate their fatigue mechanism under cyclic bending and constant tensile loads.Slack et al. [22] proposed the ultrasonic reflection techniques to directly measure the contact stress between thread and sealing surface.It turned out to be an accurate and effective method to evaluate the sealing performance and to detect sealing damage in premium connections [23].Though experimental testing can be an effective method for validating analytical models and FEM, those tests focused on limited aspects of threaded connection problems and are difficult for application in practice.
Based on elastic mechanics theory and some simplifications, this paper proposed an analytical method to calculate the load distribution and contact stress on each engaged thread tooth for premium threaded connections.The 2D FEM validated the analytical model, and the comparison of the contact stress on the last engaged thread between two methods investigated the accuracy of the analytical model and its practicality in engineering.

Loading Analysis on the Premium Threaded Connection.
During tightening, the engaged threads of a premium connection will transfer the pretightening force   caused mainly by shoulder contact, thus leading to the compressed pin and the stretched coupling.To facilitate the analytical model, we assumed all thread teeth were intact and let  the axis of premium threaded connection as  direction to establish a one-dimensional coordinate system OX with itsorigin being at the big end of the pin thread.Meanwhile, the engaged thread teeth were numbered 1∼  consecutively from origin to the small end of the pin thread (shown in Figure 2).The contact stresses between load flanks of engaged thread interact with each other as action and reaction, so the axial load distribution () along  direction is equal within pin thread and coupling thread.Let () denote the load distribution intensity; then

Axis of connection
The load distribution intensity () equals the axial load acting on the engaged thread within unit axial length, which reflects the distribution characteristics of pretightening force along the engaged thread.It is obvious that axial load distribution () can be calculated by the integral formula as follows: ( The axial load acting on each engaged thread can be further obtained as (3)

Deformation of Thread Tooth in Premium Threaded
Connection.The thread tooth profile of most premium connections typically adopts modified API buttress type thread, whose deformation can be calculated by simplifying itself into a cantilever beam model of an isosceles trapezoid [24] (shown in Figure 3).
The necessary parameters for calculating the elastic deformation of simplified thread tooth can be obtained from this basic geometry parameter of buttress type thread: According to [24], the deformations of simplified thread tooth are mainly caused by five reasons (shown in Figure 4).Now, let angle  −  be equal to  in Figure 4, and the thread tooth deformation   caused by tooth bending can be expressed as The thread tooth deformation   caused by shear force can be expressed as The thread tooth deformation   caused by tooth root incline can be expressed as The thread tooth deformation   caused by shear deformation of tooth root can be expressed as The deformations  rp ,  rc caused by radial effect for pin thread tooth and coupling thread tooth can be, respectively, expressed as Consequently, the total elastic deformations  tp ,  tc for pin thread tooth and coupling thread tooth along its cone direction ( direction) are, respectively, as follows: In ( 11) and ( 12),  =   +   +   +   .The deformation coefficients   ,   ,   ,   ,  rp , and  rc can be determined from ( 5) to (10).
In ( 5)∼( 12), () denotes the normal load acting on the load flank of thread tooth within unit axial length d.Because the load flank angle and thread taper are both very small in premium threaded connection, we ignored the axial components of frictional force on load flank and that of the contact stress between root and crest.The relation between () and () can be briefly expressed as In (13), the helix angle () is By submitting ( 1) and ( 14) into (13), we can get Then, by submitting ( 15) into ( 11) and ( 12), the total elastic deformations for pin thread tooth and coupling thread tooth are, respectively, as follows: In ( 16) and ( 17), the coefficients  p and  c depend on the geometry parameters of connection and thread tooth as well as the used material properties, whose derivatives for  are, respectively, as follows:

Differential Equation of the Axial Load 𝐹(𝑥).
Under the action of pretightening force, the pin thread and coupling thread are always engaged to each other, indicating that their thread pitches are equal after deformation.However, the compressed pin will make its own thread pitch decrease while the stretched coupling will make its own thread pitch increase.The deformation of pin thread tooth should increase its own pitch while the deformation of coupling thread tooth should reduce its own pitch, thus guaranteeing the equal pitch and the perfectly engaged state for pin thread and coupling thread.
According to Hooke's law, the compressed and stretched elastic deformations of pin and coupling body along the thread cone direction are, respectively, as follows: Moreover, the deformation compatibility equation can be obtained from the equal pitch after deformation for pin thread and coupling thread: Mathematical Problems in Engineering Then, by submitting ( 16), ( 17), (20), and ( 21) into (22), we get Taking the derivative with respect to  for ( 23) and considering that  p +  c ̸ = 0, the differential equation of axial load distribution can be obtained as The boundary conditions for (24) are 2.4.Solution of Differential Equation .The axial load distribution equation ( 24) is a nonlinear differential equation and its analytical solution is very difficult to get, so we solved its numerical solution with finite difference method.First, we divided the engaged thread according to unit length of thread pitch  and numbered the node 0∼  successively from the big end of the pin thread, so the coordinate for node  was   = .Then we took   to denote the node force (  ) and considered the boundary conditions equation (25).Finally, the difference equation of ( 24) can be obtained as In ( 26), the coefficient matrix is And the constant term column matrix is The difference equation ( 26) was solved based on Gaussian main elimination method.

Geometric Description for Load Flank of Thread Tooth.
For calculating the mean contact stress on the load flank of thread tooth, the geometric description of thread surface is necessary.Setting the premium thread connection as the axis and its intersection with the plane of thread load flank as the origin, we established global coordinate system -, whose unit vectors are i, j, and k, respectively, shown in Figure 5.
For determining the unit normal vector n and unit tangent vector  of thread surface, the cylindrical coordinate system - was also established and its vectors are The plane  2  2  2 , which stands for thread load flank, can be obtained by rotating the quadrangle  in  plane  around the -axis and then  around the -axis successively.If quadrangle  is a 1 × 1 square, its vertices after 2 rotations are (0, 0, 0),  2 (sin , 0, cos ),  2 (sin  + sin , cos , cos ), and  2 (sin , cos , 0).Thus, the equation of load flank plane can be obtained as  −  tan  −  tan  = 0. (30) The unit normal vector n and unit tangent vector  of thread load flank are And the infinitesimal area of thread surface dΩ can be also expressed with the infinitesimal area d in the axial cross section as dΩ = √ 1 + tan 2  + tan 2  d. (32)

Mean Contact Stress on Load Flank of Thread Tooth.
Supposing that the contact stress on load flank was uniform and that the helix angle was constant in each engaged thread, the axial component of contact stress integral on the load flank within each engaged thread should be equal to the axial load acting on itself; that is, By submitting (31) and ( 32) into (33), the mean contact stress on load flank of th tooth is In (34), At last, the mean contact stress on the load flank of th tooth is

Calculation Example
3.1.Analytical Method.For simplicity, the geometrical parameters of a premium connection are mainly taken from API 177.8 mm, P110 grade, buttress thread casing in API Spec.5B.Its geometry parameters and material properties are listed in Table 1.
Suppose the pretightening force   from sealing surface and shoulder contact is 1000 KN; the axial load acting on each engaged thread   and the mean contact stress on load flank   can be calculated according to the analytical model proposed in this paper; the results are shown in Figures 6 and  7, respectively.
Figure 6 shows that the load acting on engaged thread tooth increases with the increase of thread number and that the last 4∼6 engaged threads bear more than 90% of the pretightening force.Particularly for the last engaged thread, it carries 44.54% of the pretightening force.The load distribution is similar to the practical situation in threaded connection, indicating that the proposed analytical method is reasonable.Meanwhile, contact stress distribution on load flank of thread teeth shows a similar pattern to the load distribution, whose maximum value is 522.74MPa on the last engaged thread, shown in Figure 7.It is obvious that thread surface would sustain a galling risk if the pretightening force is increased to some extent.

Finite Element Method.
In order to validate the proposed analytical model in this paper, we analyzed the same sample with the finite element software ANSYS 14.5.Considering the characteristics of axisymmetric load and boundary conditions, a 2D finite element model of modified API 177.8 mm premium thread connection has been established (shown in Figure 8).
To make the simulated results more approachable to practical situations, the materials of pin and coupling were both defined as bilinear isotropic hardening model in the FEM.The frictional contact type between pin thread and coupling thread has been selected, indicating that the contact between thread teeth is taken as an elastic-plastic and frictional contact problem.The eight-node quadrilateral element Quad 8 node-183, which can be used as an axisymmetric element to simulate plastic deformation, was selected for this analysis.The element size was controlled and then local mesh refinement was applied to the thread tooth edges.Figures 9 and 10 show the mesh on the whole model and the teeth edges.Because the contact between engaged thread teeth is a material and boundary nonlinear problem, static analysis has been selected for solving it.The boundary conditions set for  this analysis are an axial displacement confined at the big end of coupling thread and application of 313.93 MPa compressed stress, which is equivalent to 1000 KN of pretightening force, on the small end of pin thread.By using ANSYS, we can acquire the stress distribution on the thread teeth (shown in Figure 11).Figure 11 shows the Von-Mises stress distribution on each contact thread tooth.In Figure 11, we can find that the last 4∼6 engaged threads bear almost all the pretightening force, which is identical to the load distribution calculated by the proposed analytical model in this paper.

Comparison and Discussion
. The mean contact stress on the load flank of thread teeth calculated from the proposed analytical model and that obtained by FEM are compared in Figure 7.One can find that both curves are identical in trend and that the minimum and maximum values obtained by the proposed analytical method are also close to those of FEM.Meanwhile, we can also find that there exists some discrepancy in the middle region of the two curves, which may be attributed to the simplifications and assumptions of the analytical model.The helix angle in the analytical model may be a reason because the 2D FEM cannot contain this factor.However, the analytical model can still predict a  relatively meaningful stress value for the last engaged threads, which is the easiest one to fail.
Figure 12 shows the comparison of mean contact stresses on the last engaged threads under different pretightening forces.It can be seen from Figure 12 that mean contact stress on the last engaged thread obtained by analytical model keeps increasing linearly with the increase of pretightening force, while that calculated from FEM tends to be stable after a section of linear increase.This discrepancy can mainly because that elastic deformation has been supposed for the analytical model but elastic-plastic deformation considered in FEM.Because the allowable in-service pretightening force is about 1350 KN∼1750 KN, the error of the analytical model for the stress on the last engaged thread is about 6.2%∼ 18.1%.Consequently, the proposed analytical method has certain accuracy and can, to some extent, meet the needs of engineering.in tubing and casing premium threaded connection on the basis of elastic mechanics.

Conclusions
( pretightening force mainly applies to the last 4∼6 engaged threads, about 50% of that to the last engaged threads. (3) Comparing the contact stress of the last engaged thread between the analytical model and FEM shows that the accuracy of the analytical model will decline with the increase of pretightening force after the material enters into plastic deformation.In practice, the relative error is about 6.2%∼18.1%.This indicates that the analytical method can, to some extent, meet the needs of engineering.M e a n h e l i x a n g l e o n th tooth.

Figure 1 :
Figure 1: The schematic of a premium threaded connection.

Figure 2 :
Figure 2: The schematic of load distribution for premium connection.

Figure 3 :Figure 4 :
Figure 3: The schematic of simplified cantilever beam model of an isosceles trapezoid for buttress type thread tooth.

Figure 5 :
Figure 5: Geometric description schematic of thread load flank.

Figure 6 :
Figure 6: Load acting on each engaged thread by the proposed analytical model.

Figure 7 :
Figure 7: Mean contact stress on load flank of thread tooth in premium connection obtained by analytical method and finite element method.

Figure 9 :Figure 10 :
Figure 9: Mesh on the whole finite element model.

Figure 11 :
Figure 11: Von-Mises stress distribution on the thread teeth in ANSYS.

Figure 12 :
Figure 12: Comparison of mean contact stresses on the last engaged thread tooth under different pretightening forces.

Table 1 :
Geometry parameters and material properties of premium connection.
,   ,   : Elastic deformation of thread tooth caused by tooth bending, shear force, root incline, and shear deformation of tooth root, respectively  rp ,  rc : Elastic deformation of thread tooth caused by radial effect for pin thread tooth and coupling thread tooth, respectively   ,   ,   ,   ,  rp ,  rc , ,  p ,  c : Relevant deformation