The Twist and Shear of Helically Buckled Pipe
- Robert F. Mitchell (Landmark Graphics)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling & Completion
- Publication Date
- March 2004
- Document Type
- Journal Paper
- 20 - 28
- 2004. Society of Petroleum Engineers
- 1.14.1 Casing Design, 1.14 Casing and Cementing, 1.10 Drilling Equipment, 1.6.1 Drilling Operation Management, 1.6 Drilling Operations
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The effect of torque on buckling was first recognized while designing hollow propeller shafts for ships. If buckling could be caused by torque, then perhaps torque could be induced by helical buckling. This effect has not been extensively studied because the assumption has always been that the effect was too small to be of concern. This problem required an exact large-displacement solution to the helically buckled pipe because conventional small-displacement analysis has the built-in assumption that these effects are not important. How significant are these effects? Lubinski et al. ,1 in their paper on helical buckling, worked a sample problem of a squeeze cementing operation in 2 7/8-in., 6.5-lb/ft tubing. The induced torque caused by helical buckling may exceed the makeup torque (in the range of 600 to 2,300 ft-lbf, depending on the grade) for large radial clearances, so induced torque in this problem may be high enough to break the connection!
This paper gives complete details for the torque and shear calculation in the Appendices. The body of the paper examines several typical tubing-load situations to evaluate the effect of induced torque and shear on a conventional design.
The effect of torque on pipe stability was first encountered in the late nineteenth century, during the development of hollow propeller shafts for steamships.2,3
Goodier4 showed that torque buckling was associated with the geometrical effects of finite displacements, particularly finite rotations. The problem was further studied in the context of large-displacement elasticity during 1950-70.4-6 All these papers were concerned with elastic stability and critical buckling loads.
Following Lubinski,1 the problem focus shifted from elastic stability to post-buckling equilibrium. The first attempt to address the equilibrium problem with torque was made by Mitchell,7 combining concepts from Timoshenko, Goodier, and Williams.3,4,6 This analysis primarily investigates stability characteristics but does not adequately address post-buckling equilibrium. Mitchell later revisited the buckling-equilibrium problem in a more comprehensive way but did not consider the effects of torque.8 Miska and Cunha9 addressed the effect of applied torque for post-buckling equilibrium using the techniques developed by Lubinski.
In this paper, a large-displacement formulation for a helically buckled slender beam is developed. What is meant by "large displacement"? In this analysis, there have been no assumptions made about the magnitude of the pipe displacements. In typical "strength of materials" beam calculations, the lateral deflections are assumed to be small relative to the length of the beam.10 Instead, we have used Nordgren's formulation11 to model the pipe deflection, and no displacement terms have been neglected in the analysis. The details of this analysis are presented in the Appendices. For the case of zero externally applied torque, there is an induced torque in the pipe that is surprisingly large in some cases.
Induced Torque and Shear in Buckled Pipe
In this formulation of the pipe equilibrium equations, the lateral displacements are given by:
in which ?=ßs is the helix angle, s=the measured depth, and r=the tubing/casing radial clearance. This configuration is illustrated in Fig. 1. ß>0 implies that the helix rotates to the right as s increases.
As shown in Appendix B, a weightless pipe buckled into a pure helix induces the following twisting moment and shear force in the pipe.
Here, Mt=the induced torque, Vb=the induced shear force, P=the axial buckling force, and EI=the bending stiffness. For typical buckling forces, a is approximately equal to 1. The minus sign in Eq. 2 indicates that the induced torque twists to the left for ß>0. A helix turning to the right induces torque to unscrew the connection.
Note that the radial clearance, r, determines the magnitude of the lateral displacement and that the shear is linear in r while the moment is quadratic. Small-displacement analysis would consider Eq. 2 to be second order and, therefore, negligible. As shown here, this assumption is not strictly true.
Following Lubinski,1 we consider these formulas reasonably correct when evaluated at a point in a real buckled tube.
The variation of induced shear with buckling force P is shown in Fig. 2. In this figure, the horizontal axis is a dimensionless buckling force, and the vertical axis shows the ratio of shear/buckling force. For oilfield tubulars, the dimensionless buckling force is generally less than 0.10, so this figure indicates that the induced shear will generally be less than 10% of the buckling force. Note that the buckling force for cylinders is given by:
in which Fa=the actual pipe force (tension positive), p i=the internal pressure, ri=the pipe inside radius, po=the external pressure, and ro =the pipe outside radius. There is no buckling for P<0. Because the buckling force is pressure dependent, the induced shear may be greater in magnitude than the actual pipe force. The induced shear increases quickly for small values of the dimensionless buckling force r2P/EI and exceeds 10% for a dimensionless buckling force of approximately 0.075.
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