Every calculator on this site so far checks one failure mode at a time: burst, or collapse, in isolation. Real strings don’t fail that way — pressure and axial load act together, and a pipe that comfortably passes a burst check and a collapse check separately can still yield when both loads are on at once. This post covers the tool that checks them together: the triaxial, or von Mises, load envelope.
Why burst and collapse alone aren’t enough
Our burst and collapse calculators each answer a narrower question than it sounds like: “at zero axial load, what pressure causes this failure mode?” But a string hanging in a well is never at zero axial load — it’s carrying its own weight (or someone else’s above it), and that axial stress interacts with whatever pressure is also acting on the pipe. Tension makes a pipe body slightly easier to burst and meaningfully easier to collapse; compression works the other way. Treat the checks as independent and you can convince yourself a design is fine when the combination of loads actually pushes it into yield.
One equation, one boundary
The von Mises criterion combines axial stress (σa) and hoop stress (σt) into a single equivalent stress, and compares that against the material’s yield strength:
σvme = √(σa² − σa·σt + σt²)Once σvme reaches the pipe’s yield strength, the pipe body yields — regardless of whether that stress state came from pressure, axial load, or (as is almost always the case downhole) both. Axial stress converts to axial load through the pipe’s cross-sectional area (σa = Fa ÷ As); hoop stress converts to pressure through the same thin-wall relation the burst and collapse calculators use.
Plotted on a chart of axial load versus pressure, the equation σvme = Yp traces a closed ellipse — the pipe body’s full yield envelope. Anywhere inside it, the pipe body is elastic. Cross the line, in any direction, and it yields. That’s what the Triaxial Load Envelope calculator draws for you, with your actual design load plotted as a point against it.
A worked example
Take 9-5/8 in., N-80 casing (Yp = 80,000 psi, wall 0.472 in.) carrying 200,000 lbf of tension with 3,000 psi internal pressure:
σa = 200,000 ÷ 13.57 in² = 14,736 psi
σt = 3,000 × (9.625 − 0.472) ÷ (2 × 0.472) = 29,088 psi
σvme = √(14,736² − 14,736×29,088 + 29,088²) = 25,192 psi — 31% of the pipe body’s yield strength
Well inside the envelope. The calculator plots this point directly on the chart alongside the 100% and 95% yield curves, so you can see at a glance how much margin is left — not just whether you passed.
What this number doesn’t tell you
This is the pipe body only, and it’s a biaxial simplification (axial and hoop stress; radial stress is dropped under the standard thin-wall assumption, which is fine for typical D/t but loses some accuracy on thick-wall pipe). It doesn’t rate the connection — on a made-up string, the connection is very often the weaker link — and it doesn’t include temperature derating, buckling, or bending stress from doglegs. Treat a point that lands inside the envelope as confirmation the pipe body itself is fine under that combined load, not a full string design.
Try the Triaxial Load Envelope calculator yourself, or get in touch for a full combined-load design case.
