Buckling, fatigue and triaxial stress in the drillstring

A pipe in compression inside an inclined hole does not buckle all at once: it first passes through a sinusoidal shape, predicted by Dawson and Paslay, touching the low side of the hole at alternating points without wrapping around it. Only past a second, higher limit does the shape become a full helix — the Wu and Chen criterion — with the pipe in continuous contact against the wall along the whole helix pitch.

Rotary torque feeds into that limit two ways. On its own, with no axial compression, there is a critical torque that would trigger buckling by itself — but for conventional drill pipe that value sits far above real operating torque, so in practice torque alone rarely buckles the string.

The effect that matters in practice is the second one: operating torque, even below critical, knocks down the axial load that triggers helical buckling — typically a few percentage points, but that is exactly the margin the design was counting on.

A string rotating inside a helical buckle is not just in compression — every point on the pipe wall cycles through tension and compression in bending once per revolution, because the pipe follows a helical path rather than a straight one. That bending curvature imposed by the helix itself adds, by root-sum-of-squares, to the well's natural dogleg severity when computing the fatigue ratio.

Fatigue itself is judged against the official API RP 7G curves — Lubinski's modified Goodman diagram — which relate allowable alternating bending stress to the axial stress present, by pipe grade. A pipe under more tension tolerates less alternating bending before accumulating damage; a more relaxed pipe tolerates more.

Damage accumulated across multiple cycles and operating conditions follows Miner's rule: each cycle consumes a fraction of the material's fatigue life, and those fractions add linearly until the pipe reaches the limit.

A pipe's yield limit is not set by a single stress component but by the combination of three principal stresses acting at the wall: axial, bending, and circumferential — the hoop stress produced by the difference between internal and external pressure. The von Mises criterion combines the three into a single number comparable to the material's yield strength.

Under low differential pressure, hoop stress is small and the triaxial criterion collapses to the simple uniaxial limit — tension plus bending. The difference shows up in high-pressure, high-temperature wells: hoop stress starts eating a meaningful share of the yield margin, and a point that looked safe on axial tension alone can sit close to the limit once differential pressure is accounted for.

Checking von Mises at both the inner and the outer pipe wall — not just one — matters because internal and external pressure push hoop stress in opposite directions; the governing wall changes depending on whether the well is being drilled or is being controlled through a kick.