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Journal of Natural Gas Science and Engineering 18 (2014) 237e246

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Journal of Natural Gas Science and Engineering

journal homepage: www.elsevier .com/locate/ jngse

Sinusoidal buckling of a thin rod with connectors constrained in acylinder

Wenjun Huang, Deli Gao*

College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

a r t i c l e i n f o

Article history:Received 17 January 2014Received in revised form28 February 2014Accepted 3 March 2014Available online

Keywords:Sinusoidal bucklingAxial compressionConnectorBeam-column model

* Corresponding author.E-mail address: gaodeli@cup.edu.cn (D. Gao).

http://dx.doi.org/10.1016/j.jngse.2014.03.0031875-5100/� 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

Kinds of tubular strings with connectors are widely used in petroleum exploration and developmentfield. In previous research, tubular strings are usually treated as homogeneous thin elastic rods con-strained in a cylinder and few studies have been made on the effect of connectors.

This paper builds a model to describe the sinusoidal buckling behaviors of a rod with connectors. Thethree-dimensional buckling problem is divided into two lateral buckling problems in two mutual-perpendicular planes and the buckling solutions are obtained by solving the two lateral bucklingequations.

There are two buckling states for the rod, including initial plane configuration and sinusoidalconfiguration. And there are three contact cases between the rod and the cylinder, including no contact,point contact and wrap contact. The critical condition from initial plane configuration to sinusoidalconfiguration is approximately obtained from the results of no connector rod, and the critical conditionsbetween different contact states are obtained by solving the model numerically. The effects of geometricparameters of the connectors and weight of the rod on critical conditions are analyzed. For the no contactand point contact cases, the deflection curves of a rod in the initial plane configuration and sinusoidalconfiguration are calculated in this paper. The effects of connectors on bending stress and contact force ofthe rod are analyzed.

The results indicate that buckling states and contact states depend on the combination of thedimensionless axial compression Ku, the ratio of rod/cylinder radial clearance to connector/cylinderradial clearance Kr and the dimensionless weight per unit length of the rod m. Compared with the noconnector case, connectors have a great influence on the deflection curve, bending moment and contactforce of the rod. All the results from the model indicate that the effects of connectors cannot be ignoredin sinusoidal buckling analyses.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Tubular strings such as drill string, casing, and tubing play animportant role in oil exploration & development. Take the drillstring, for example. The drill string transmits mechanical power(torque and rotation), hydraulic power (pressure and flow rate) andweight of the drill string to the bit to break underground rocks(Jorge and Sampaio, 2007) as shown in Fig.1. The outer diameters ofthese tubular columns are usually smaller than 0.5 m, but theyextend for thousands of meters in the extended reach wells. Thesetubular strings have quite small radial displacements under theconstraint of thewellbore of which the diameter is usually less than

1 m. Therefore, these tubular columns are usually taken as thinelastic rods and the wellbore as a cylinder which constrains thedeflection of the rod.

Euler solved the lateral buckling problem of a thin rod undercompression for the first time. But the solution was not applicablefor the case where a rod is constrained in a cylinder. The followingresearch was focused more on the buckling behaviors of a rodconstrained in a cylinder.

Lubinski and Althouse (1962) studied the helical buckling of aweightless rod by virtual work principle and derived the relation-ship between the axial compression and the deflections of rods.Paslay and Bogy (1964) analyzed the stability of a rod constrained inan inclined cylinder by virtual work principle and gave the criticalload for sinusoidal buckling. Mitchell (1986, 1988, 1996), Robert(1999), Gao and Miska (2010, 2009), Gao (2006) and Liu (1999)solved the buckling problems while considering the effects of

Fig. 1. Drill string in the wellbore.

Fig. 2. Schematic of deflection of a rod with connectors in the vertical plane.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246238

weight, friction and so on based on the general differential equa-tions of buckling of a rod. All these results have beenwidely used inengineering practice.

All the above studies are all under the assumption that the rod ishomogeneous and in continuous contact with the cylinder. How-ever, in practical applications, connectors distribute discretelyalong the rod and the diameters of connectors are larger than thatof the rod. As a result, some portions of the rod lose contact withthe cylinder for the existence of connectors. That is to say, theassumption of the continuous contact between the rod and thecylinder does not hold in this case. The effects of connectors shouldnot be neglected in buckling analyses. As to our knowledge, onlyseveral studies consider the effects of connectors. Mitchell (2000,2003) and Robert (2003) solved the buckling of a rod by assumingthat the connectors were in contact with the cylinder but the rodwere in no contact with the cylinder. In fact, with the increase of theaxial compression, the rod between two connectors bends andtends to touch the inner surface of the cylinder. Gao et al. (2012)studied the buckling of a rod with connectors in the initial verti-cal plane. Actually, the rod switches to the sinusoidal configurationfrom the initial vertical configuration when the axial compressionexceeds a certain value. However, little research has considered thesituation under which the rod is in contact with the cylinder for thesinusoidal configuration.

In this paper, a new buckling model is built to depict the contactstate between the rod with connectors and the cylinder. Thebuckling solutions in initial plane configuration and sinusoidalconfiguration are obtained based on the beam-column model. Thecritical conditions between different cases and the deflectioncurves of the rod are given. A detailed analysis is made about theeffects of connectors.

2. Model description of a thin rod with connectors

In the oil-well drilling process, the underground rock isdestroyed by the combination of the axial force F and torque Tapplied to the drill bit as shown in Fig. 1. The drill string and thewellbore can be respectively taken as a thin rod and a cylinder, andbuckling behaviors of the drill string in the horizontal wellbore canbe obtained by solving the bucklingmodel of a compressed thin rodin a horizontal cylinder.

For a compressed rod without connectors in a horizontal cyl-inder, the rod deviates from its initial straight configuration andbuckles into a sinusoidal configuration when the compression Fexceeds a certain value (Paslay and Bogy, 1964). Sinusoidalconfiguration means that the rod snakes along the lower surface ofthe cylinder and approximately forms a curve, a ¼ aM$sinðu$sÞ,where a is the angular displacement, aM is the amplitude, s is theaxial distance along the axis of the cylinder and u denotes theangular frequency.

For a rod with connectors under no axial compression, the rodbends in the vertical plane due to its gravity. For a rod with con-nectors under a small axial compression, the rods still keeps itsinitial plane configuration. The contact state between the rod andthe cylinder is divided into three cases: no contact, point contactand wrap contact as shown in Fig. 2. No contact means that the rodis in no contact with the cylinder, the point contact means that therod is in contact with the cylinder at a single point, and the wrapcontact means a section of the rod is in continuous contact with thecylinder.

A rod with connectors deviates from its initial plane configu-ration and buckles into an approximate sinusoidal configurationwhen the axial compression exceeds a certain value, which we callthe critical condition for sinusoidal buckling. The sinusoidal buck-ling of a rod with connectors is shown in Fig. 3. The connectors A, Band C are in contact with the inner surface of the cylinder, while therod tends to lose contact with the cylinder due to the existence ofconnectors. The definitions of contact states including no contact,point contact and wrap contact in sinusoidal configuration are thesame as that in its initial plane configuration.

For a low axial compression, the rod tends to suspend betweenconnectors and lose contact with the cylinder. However, as the axialcompression increases, the rod becomes in point contact or wrapcontact with the cylinder.

Fig. 4 shows the relationship between different buckling states.“I” denotes the initial vertical plane and “S” denotes the sinusoidalconfiguration. “N”, “P” and “W” denote the no contact, point contactand wrap contact respectively. “C” denotes the critical conditions.For example, “I & N” means the initial vertical plane for no contactcase, “CIS_N” means the critical condition from the initial verticalconfiguration to sinusoidal configuration in no contact case, and“CNP_I”means the critical condition fromno contact to point contactin the initial vertical plane. The arrows denote the direction of thebuckling state change when the axial compression is increasing.

A rod can change from no contact to point contact, or pointcontact to wrap contact for the initial vertical configuration andsinusoidal configuration while loading. A rod also changes frominitial vertical configuration to sinusoidal configuration for nocontact, point contact and wrap contact while loading. For a rod ininitial vertical plane of no contact or point contact, it has two

Fig. 3. Schematic of sinusoidal buckling of a rod with connectors constrained in a horizontal cylinder: (a) no contact, (b) point contact and (c) wrap contact.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246 239

possible states to change to while loading. For example, “I & N”maychange to “S & N” or “I & P”, and the final buckling state is deter-mined by values of “CIS_N” and “CNP_I”. That is to say, if “CIS_N” issmall than “CNP_I”, the rod will change to “S & N”.

Mitchell (2003) has studied “I & N”, “S & N”, “CNP_I”, “CIS_N” and“CNP_S”. Gao et al. (2012) have studied “I & N”, “I & P”, “I & W”,“CNP_I”, “CPW_I”, “CIS_N”, “CIS_P” and “CIS_W”. In this paper, all thebuckling states and critical conditions except “I & W” and “S & W”

are included.The geometric parameters involved in this paper include the

diameter of the cylinder Dy, the diameter of the connectors Dc andthe diameter of the rod Dd as seen in Fig. 5. Then the radial

Fig. 4. Phase diagram of buckling states and critical conditions.

clearance between the connector and the cylinder (connector/cyl-inder radial clearance) is

rc ¼ Dy � Dc

2(1)

and the radial clearance between the rod and the cylinder (rod/cylinder radial clearance) is

Fig. 5. Geometric parameters of the rod, connector and cylinder.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246240

rd ¼ Dy � Dd2

(2)

3. Buckling solutions based on the beam-column model

3.1. Assumption

The investigation in this paper is based on the followingassumptions:

1. The axis of the cylinder is straight and horizontal.2. The connectors are taken as points of support and contact with

the cylinder.3. Elastic theory is satisfied.4. Friction and torque are neglected.5. The direction of gravity is consistent with y-axis as shown in

Fig. 3.

3.2. Beam-column model

A Beam subjected to axial compressions and lateral loads arecalled a beam-column. Here we consider the case of a beam-column applied by an axial compression F and bending moments(Max, May), (Mbx, Mby) at the ends and the distributed force q andconcentrated force (Qx, Qy) on the rod as shown in Fig. 6.

For the case of a beam-column with two supports not on thesame level in xez and yez planes, the rotation angles at the twoends are

qay ¼ MayL3EI

jþMbyL6EI

fþ qL3

24EIcþ Qy

Fsa þ

hby � hayL

qby ¼ MbyL3EI

jþMayL6EI

fþ qL3

24EIcþ Qy

Fsb �

hby � hayL

(3)

qax ¼ MaxL3EI

jþMbxL6EI

fþ Qx

Fsa � hbx � hax

L

qbx ¼ MbxL3EI

jþMaxL6EI

fþ Qx

Fsb þ

hbx � haxL

(4)

Fig. 6. A beam-column with two supports.

where F is the axial force, (Max,May) and (Mbx,Mby) are the bendingmoments, ha and hb are the lateral displacements of the two ends, Lis the length of the beam-column, EI is the bending stiffness of thebeam-column, q is the distributed force, u is a dimensionless termand is calculated by

u ¼ k$L (5)

where k is equal toffiffiffiffiffiffiffiffiffiffiF=EI

p.

J, 4, c, sa and sb are introduced to simplify Eqs. (3) and (4) andtheir expressions are shown as follows:

f ¼ 6u

�1

sin u� 1u

�j ¼ 3

u

�1u� 1tan u

�

c ¼24�tan

u2� u2�

u3

sa ¼ sin uysin u

� y

sb ¼ sin uð1� yÞsin u

� 1þ y

(6)

in which y represents the dimensionless distance between theapplied concentrated force (Qx, Qy) and the right end of the beam-column and is calculated by

y ¼ cL

(7)

3.3. Buckling solutions

In Fig. 3 connector A is at the bottom of the cylinder andconnector B reaches themaximum angular displacement of the rod.Mitchell (2003) pointed out that the case in Fig. 3 represents theconfiguration with the highest possible axial force, the worst casefor bending stress and the maximum deflection of the rod. There-fore, we only consider this case in the following studies.

Let the supports A and B in Fig. 6 correspond to the connectors Aand B in Fig. 3 respectively, q denote the action of gravity of the rodand (Qx, Qy) denote the contact forces between the rod and thecylinder. For the no contact case, the contact forces Qx and Qy areboth equal to 0.

Considering that the beam-column is constrained in a cylinderas shown in Fig. 3, we obtain

hay ¼ rc

hby ¼ rc cos aM

hax ¼ 0

hbx ¼ rc sin aM

(8)

where aM is the angular displacement of the right end.The following dimensionless terms are introduced:

may ¼ MayL2

EIrcmby ¼ Mby

L2

EIrc

max ¼ MaxL2

EIrcmbx ¼ Mbx

L2

EIrc

lx ¼ QxL3

EIrcly ¼ Qy

L3

EIrc

m ¼ qEIrc

�2Lp

�2

3¼ rcL

(9)

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246 241

Substituting Eqs. (8) and (9) to Eqs. (3) and (4), we obtain thedimensionless form:

qay ¼ 13may 3jþ1

6mby 3fþ 1

384EIm 3p4cþ 1

u2ly 3saþ 3ðcosaM�1Þ

qby ¼ 13mby 3jþ1

6may 3fþ 1

384EIm 3p4cþ 1

u2ly 3sb� 3ðcosaM�1Þ

qax ¼ 13max 3jþ1

6mbx 3fþ 1

u2lx 3saþ 3sinaM

qbx ¼ 13mbx 3jþ1

6max 3fþ 1

u2lx 3sb� 3sinaM

(10)

According to the boundary conditions at the ends A and B, weobtain

max ¼ 0qay ¼ 0qbx ¼ 0qby ¼ 0

(11)

Substituting Eq. (11) into (10), we obtain the bending moments atthe two ends:

max ¼ 0

mbx ¼ �3�lxsb þ u2 sin aM

�u2j

may ¼ �384u2ðfþ 2jÞð1� cos aMÞ þ ðf� 2jÞmp4cu2 þ 384lyð � 2jsa þ sbfÞ64u2

��4j2 þ f2

�mby ¼ ��384u2ðfþ 2jÞð1� cos aMÞ þ ðf� 2jÞmp4cu2 þ 384lyð � 2jsb þ safÞ

64u2��4j2 þ f2

�(12)

For the no connector case (Qx ¼ 0 and Qy ¼ 0), the deflectioncurve of the beam-column is

bx0 ¼ �max

�sin uð1� hÞsin u

� 1þ h

�þmbx

�sin uhsin u

� h

�u2

þ h sin aM

by0 ¼may

�sin uð1� hÞsin u

� 1þ h

�þmby

�sin uhsin u

� h

�u2

þ m 3p4

16u4

cos

u2ð2h� 1Þ

cosu2

� 1

!� m 3p4hð1� hÞ

32u2þ 1þ ðcos aM � 1Þh

(13)

where h is the dimensionless distance and equal to s/L, in which sdenotes the distance from any point to the left end of the rod alongz-axis, and bx0 and by0 are the dimensionless lateral displacementsand calculated by

bx0 ¼ x0rcby0 ¼ y0rc

(14)

in which x0 and y0 are the lateral displacements of the beam-column in xez and yez planes respectively.

For the point contact case (Qxs0 or Qys0), the deflection curveof the beam-column is

bx ¼ bx0 � lx sin uy sin uhu3 sin u

þ lxyh

u2

by ¼ by0 þ ly sin uy sin uhu3 sin u

� lyyh

u2

h< 1� y

bx ¼ bx0 � lx sin uð1� yÞsin uð1� hÞu3 sin u

þ lxð1� yÞð1� hÞu2

by ¼ by0 þ ly sin uð1� yÞsin uð1�hÞu3 sin u

� lyð1� yÞð1� hÞu2

h� 1� y

(15)

4. Critical conditions for a rod with connectors in ahorizontal cylinder

From the preceding analyses we know that the axis of a rod withconnectors is a 2D curve in the vertical plane under a low axial

compression. As the axial compression increases, the rod maybuckle and transform from its initial plane configuration (2D) to an

approximate sinusoidal (3D) configuration. Whether in the initialplane configuration or in the sinusoidal configuration, the rod hasthree contact cases-no contact, point contact and wrap contact.Therefore, there are two types of critical conditions for a rod withconnectors: the critical condition from the initial plane configura-tion to sinusoidal configuration (or critical condition for sinusoidalbuckling) and the critical condition between two different contactcases (no contact to point contact and point contact to wrapcontact).

Fig. 7. Phase diagram of contact cases.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246242

4.1. Critical conditions for sinusoidal buckling

Here the critical condition of a rod with connectors is equivalentto that of a rod without connectors, where the rod/cylinder radialclearance rd in the no connect case is replaced by the connector/cylinder radial clearance rc. The deflection curve of the no connectrod is

r ¼ rc

a ¼ aM sin�ps2L� (16)

in which r is the radial displacement:

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

q(17)

and a is the angular displacement:

a ¼ arctanxy

(18)

inwhich x and y represent the lateral displacements in xez and yezplanes respectively.

The generalized potential energy of the rod per unit length is(Gao, 2006)

e ¼ r2

24EIr�dads

�4

þ EIr

d2ads2

!2

� Fr�dads

�2

þ 2qð1� cos aÞ35(19)

Substituting m ¼ ðq=EIrcÞð2L=pÞ2, u ¼ k$L and k ¼ ffiffiffiffiffiffiffiffiffiffiF=EI

pinto Eq.

(19) and using the approximation cos az1� ða2=2Þ þ ða4=24Þ, thetotal potential energy can be simplified as

E ¼Z4L

s¼0

e$ds ¼ EIr2a2Mp4

L3

�m

16� 14p2u

2 þ 116

þ 364

a2M � m

256a2M

�(20)

According to the minimum potential energy principleðvE=vAÞ ¼ 0, we obtain

aM ¼ 2ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimp2 � 4u2 þ p2

p2ð�12þ mÞ

s(21)

The critical condition for sinusoidal buckling is obtained byletting aM equal to 0, that is to say

u ¼ p

2

ffiffiffiffiffiffiffiffiffiffiffiffi1þ m

p(22)

A dimensionless term is defined as follows:

Ku ¼ up2

ffiffiffiffiffiffiffiffiffiffiffiffi1þ m

p (23)

According to Eq. (22), the critical condition for sinusoidal bucklingis expressed as

Ku ¼ 1 (24)

We extend the result of Eq. (24) to the buckling of the rod withconnectors in the following analysis.

4.2. Critical conditions between contact cases

For the no contact case, the deflection curve of the beam isobtained from Eq. (13). Introducing a dimensionless term Kr ¼ rd/rc,the critical condition from no contact to point contact is obtainedby letting the maximum radial clearance equal to rod/cylinderradial clearance rd, namely

maxhbr ¼ max

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibx20 þ by20q¼ Kr (25)

For the point contact case, the deflection curve is obtained fromEq. (15), where y represents the dimensionless distance betweenthe contact point and the right end as shown in Fig. 6, (lx, ly) is thedimensionless contact force between the rod and the cylinder. Thecritical condition from point contact to wrap contact is when themiddle peak of the radial displacement curve of the rod disappears,that is to say

brjh¼1�y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibx2 þ by2q

jh¼1�y ¼ Kr

d2brdh2

h¼1�y

¼ d2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibx2 þ by2qdh2

h¼1�y

¼ 0(26)

Combining Eq. (13) with Eq. (25) or combining Eq. (15) with Eq. (26),there still exists two unknown term aM and m. For the initial planeconfiguration, aM ¼ 0 is satisfied, while for the sinusoidal configu-ration, Eq. (21) is used to calculate aM. Here we consider the case ofm¼ 1.Nowthe critical conditions fromno contact to point contact andfrom point contact towrap contact are obtained by solving the aboveequations numerically. A combination of homotopy method andNewton iterationmethod is recommended to solve Eqs. (25) and (26).

Critical conditions from Eqs. (24)e(26) are shown in Fig. 7. Thephase diagram is divided into three regions: no contact, pointcontact and wrap contact based on the critical conditions betweencontact cases. The phase diagram is divided into two regions: initialplane configuration and sinusoidal configuration based on thecritical condition for sinusoidal buckling. Therefore, there are sixregions which correspond to the six buckling states in Fig. 4respectively. The critical conditions including two solid lines andone dashed line in Fig. 7 correspond to the seven critical conditionsin Fig. 4 respectively.

Fig. 9. Deflection curves of the beam under different axial compressions: (a) radialdisplacement and (b) angular displacement.Ku¼1 represents the deflection curveunderthe critical condition for sinusoidal buckling, Ku ¼ 1.177 represents the deflection curveunder the critical condition from no contact to point contact, and Ku ¼ 1.946 representsthe deflection curve under the critical condition from point contact to wrap contact.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246 243

The term Kr is divided into three cases: 1 ¼ <Kr � 1.0053,1.0053 < Kr � 1.0158 and Kr > 1.0158 when there is no axialcompression applied on the ends of the beam, namely Ku¼ 0.When1¼<Kr � 1.0053, the rod is in continuous contact with the cylinderwhether an axial compression Ku is applied or not. For the case of1.0053 < Kr � 1.0158, the rod goes through point contact and wrapcontact with the increase of the axial compression. WhenKr> 1.0158 is satisfied, there are three contact cases. If the term Kr isrelatively large (Kr > 1.04) and the axial compression is relativelysmall (Ku < 1.414), only the no contact case are taken in consider-ation in sinusoidal buckling analysis.

In order to obtain the effect of rod weight on sinusoidal buck-ling, the relationship between the critical condition fromno contactto point contact Ku and the term Kr (Kr > 1.04) under differentweight per unit length m is shown in Fig. 8.

From Fig. 8 we can see that the critical condition from no contactto point contactKu growsmore andmore slowlywith the increase ofthe termKr. The critical condition fromnocontact topoint contactKudecreases with the increase of weight per unit length of the rod m

under the same value of Kr. Take the case of 1.20 > Kr > 1.04 andKu < 1.414 for example. Only the no contact case should be consid-ered in sinusoidal buckling analysis when m < 1. When 4 > m > 1,point contact case should be in consideration in sinusoidal config-uration.When the rodweight is relatively large (m>8), point contactand wrap contact may exist in initial plane configuration.

5. Other results for a rod with connectors in a horizontalcylinder

5.1. Deflection curve of a thin rod with connectors

From the above analysis we know that the critical conditions areobtained from Eqs. (25) and (26), and the deflection curves of therod are obtained from Eqs. (13) and (15). For the case of m ¼ 1 andKr ¼ 1.025, the critical conditions and the deflection curves areshown in Fig. 9. In Fig. 9 h ¼ 0 corresponds to the connector A andh ¼ 1 corresponds to the connector B in Fig. 3.

From Fig. 9 we can see that radial displacements and angular dis-placements increase roughly with the increase of the axial compres-sion. As the axial compression increases, the rod goes through fourstates: no contact in initial plane configuration (Ku� 1), no contact insinusoidal configuration (1 < Ku � 1.177), point contact

Fig. 8. Critical condition from no contact to point contact Ku vs. Kr under different m.

(1.177 < Ku � 1.946) and wrap contact (Ku > 1.946) in sinusoidalconfiguration. When the axial compression is relatively small(Ku � 1.177), the radial displacement curve is approximately sym-metricwith respect to itsmid-point.However, the radial displacementcurvebecomesasymmetricas theaxial compression increases further.

The maximum radial displacements and their correspondingpositions on the rod are shown in Fig. 10. The radial displacementincreases slowly in initial plane configuration (Ku � 1), but rapidlyin the no contact case of sinusoidal configuration (1 < Ku � 1.177).

For the no contact case of initial plane configuration (Ku� 1), theposition corresponds to the maximum radial clearance is the mid-point of the beam. The maximum radial displacement positionmoves to the right in no contact case of sinusoidal configuration(1 < Ku � 1.177). For the point contact case, the position corre-sponding to the maximum radial clearance is right the contactpoint between the beam and cylinder. The contact point moves tothe right slightly at first, and then it moves to the left with theincrease of axial compression. The contact point reaches 0.3725under the critical condition from point contact to wrap contact.

5.2. Bending stress magnification

Here we only consider the bending moment and bending stressnear the connector B. The bending moment is calculated by

M ¼ EI

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2xdz2

!2

þ d2ydz2

!2vuut (27)

Fig. 10. Maximum radial displacements and their corresponding positions underdifferent axial compressions.

Fig. 11. Bending stress magnification Km under different axial compressions.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246244

and the bending stress is

s ¼ MDd2Id

(28)

where Id is the inertia moment of the cross section of the rod.For the no connector case defined by Eq. (16), the maximum

bending moment on the rod is

M0 ¼ EIrcaML2

�p2

�2(29)

Paslay and Cernocky (1991) proposed the concept of “bendingstress magnification” to depict the effect of connectors on bendingstress for a rod constrained in a two-dimensional curved wellbore.Similar to Paslay’s definition, in this paper the bending stressmagnification Km is defined by the ratio of the maximum bendingstress of the rod with connectors to the maximum bending stress(or bending moment) of the rod without connectors. Therefore, Kmis calculated by

Km ¼ MM0

¼�2p

�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2bxdh2

!2

þ d2bydh2

!2vuut (30)

The relationship between the bending stress magnification andaxial compression is shown in Fig. 11. The results show that Km isrelatively smallwhenKu<1 andKm increases linearlywith respect toKu when Ku � 1. It indicates that the connector reduces the bendingstress when the axial compression is small (Ku � 1.25), but enlargesthe bending stress when axial compression is large (Ku > 1.25).

As is known that bending stress is closely related to the fatiguefailure of rods, Km should be considered in the computation modelof rod fatigue failure.

5.3. Contact forces

The shear force on the cross section of the rod is calculated by

V ¼

� EId3xdz3

; EId3ydz3

!(31)

Then the contact force between the rod and the cylinder at positionz is

Q ¼ V jzþ � V jz� (32)

where ‘zþ’ denotes the right cross section at position z and ‘z�’denotes the left cross section at position z.

In no connector case the distributed contact force between aweightless rod and a cylinder is given in Mitchell (1988)

qd ¼ rcF2

4EI(33)

Then we define a dimensionless contact force

l ¼ kQkqdL

¼ 64

ð1þ mÞ2p4K4u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d3bxdh3

hþ

� d3bxdh3

h�

!2

þ d3bydh3

hþ

� d3bydh3

h�

!2vuuut

(34)

where k$k denotes 2-norm, ‘hþ’ and ‘h�’ correspond to ‘z þ 0’ and‘z � 0’ respectively.

We denote the contact forces on connector A, connector B andthe contact point between the rod and the cylinder as shown inFig. 3 by the terms lA, lB and lP respectively. According to Eq. (34),the results of the three contact forces are shown in Fig. 12.

When the rod is in the vertical plane configuration (Ku < 1), thecontact force on connector A is equal to that on connector B. How-ever, when the rod is in the sinusoidal configuration (Ku � 1), thecontact force on connector B becomes larger than that on connectorA. For the no contact case (Ku < 1.17685), there is no contact force.However, when the rod is in point contact with the cylinder, thecontact force rapidly increases with respect to the axial compres-sion. By comparing the three contact forces, the worst contact forceoccurs on the connector B in the sinusoidal buckling configuration.

Contact forces play an important role in the wear between therod and cylinder, when the rod rotates around its axis. Considering

Fig. 12. Contact forces under different axial compressions.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246 245

the contact area between the rodand the cylinder is verysmall in thepoint contact case, theremay be seriouswear on the contact point ofthe rod. Besides, the contact force on connector B is a conservativeresult when considering the wear between connectors and thecylinder. The existence of connectors has a great influence on thecontact forces on rod and connectors, so the effects of connectorsshould be considered in the computation model of rod wear.

6. Conclusions

This paper provides amodel for describing the sinusoidal bucklingbehaviors of a rod with connectors. Buckling states of the rod includethe initial plane configuration and sinusoidal configuration, and thecontact states include no contact, point contact and wrap contact.Given specific axial compression Ku, geometric parameter Kr, andweightperunit lengthof the rodm, thebuckling state andcontact statebetween the rod and the cylinder can be determined. The deflectioncurves of the rod in no contact andpoint contact cases are obtained bysolving the newly built buckling model. The results indicate thatconnectors have a great influence on bending moments and contactforces of the rod. For the bending moments and contact forces areclosely related to the fatigue failure and thewear of rods, the effect ofconnectors should be considered in the design computation of rods.

The thin rod model in this paper gives a comprehensivedescription of sinusoidal buckling of a compressed drill string withconnectors in a horizontal cylinder. Friction and torque of the drillstring and the curvature of the wellbore also affect the bucklingbehaviors infield operation and this is the subject of the futurework.

Acknowledgment

The authors gratefully acknowledge the financial support of theNational Natural Science Foundation of China (NSFC, 51221003).This research was also supported by the other projects (Grantnumbers: 2011ZX05009, 2010CB226703).

Appendix A

The following code is used to calculate the parameters includingu, 4, j, c, q, sa, sb,max,may,mbx,mby and the results of the deflection

curve and its first, second and third derivatives. Then we can pro-gram to obtain the buckling results referred above by the aid of thegiven code.

W. Huang, D. Gao / Journal of Natural Gas Science and Engineering 18 (2014) 237e246246

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