When drill string in a gradual dog leg is subjected to either tensile or compressive axial loads, the maximum curvature of the drill pipe body exceeds that of the hole axis curvature. Lubinski recognized this fact and published an analysis in 1961 for the tensile case. His results are given in the API RP7G manual.

Contemporary technology uses compressive axial loads in some extended reach and horizontal drilling applications. In addition, when casing is set, it is common to have the casing in a dog leg under compression. An analysis for compressive loads, analogous to Lubinski's analysis, is required to evaluate the bending stresses in these situations accurately.

This work presents the analytical solution for maximum drill pipe curvature when an axial compressive load, in the practical range, is applied to the drill string. Similar to the tensile case, contact (point and wrap) between the drill pipe and the hole wall as well as tool joint contact load must be taken into account.

Dimensionless curves are presented for bending stress calculation for both tensile and compressive axial loads. The curves allow easy evaluation of the bending stress magnification (BSM). The results show it is important to take the bending stress and the BSM into account in practical design particularly because of coupling with axial tension and compression loads. The results given in this paper complement calculations based on torque and drag models of drill strings. The results also complement and build on the earlier work of Lubinski.

The results are extended to burst and collapse analysis in casing and to fatigue analysis in drill pipe. The important influences of BSM are made clear in design curves for these applications.

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