The one equation that matters
Here's the thing nobody prints on the tooling rack. Your endmill is a cantilever beam with a cutting edge on the end, and it bends exactly like the beam bending in a college mechanics class. The deflection — how far the tip wanders sideways off true — is:
where F is the tangential cutting force (lbf), L is stickout (in), E is Young's modulus of the tool material (psi), and I is the area moment of inertia. For a round shank, I = π·d⁴/64.
Deflection calculator
Plug in your real numbers — the ones from the job on the table right now. This gives you the static tip deflection under a roughing cut. Finish passes cut lighter; if your finish tool passes 0.004" deflection you're already scraping the tolerance envelope.
| Cantilever stiffness k = 3EI/L³ | — |
| I = πd⁴/64 | — |
| Deflection / spindle rev period ratio | — |
Worked example — the phantom wall
Roughing a 0.75" slot in 1018, 3/8" solid carbide, 2" stickout (v2 — tolerance envelope tightened per challenge)
Where the model breaks
I trust this equation about as far as I trust a mid-range driver to hold tolerance — which is to say, it's the starting point, not the last word. The failure modes I've actually hit:
Chatter outweighs statics
Once the flute hits the cut, regenerative chatter is a dynamic problem. Deflection here is static/DC; chatter is an oscillation at the tool's natural frequency. A tool that looks fine at 0.004" static can still scream if your RPM hits a harmonic. This calc won't save you from that — it just tells you which lever to pull first.
The tip isn't a point
Cutting force isn't one vector at the end; it's spread along the engaged flute length and moves with the helix. For long cuts the real deflection profile is a curve, not a triangle. My rule: run this calc, then add 30% for roughing because force distribution is never as tidy as the model.
E isn't constant
Above ~500°F carbide's modulus doesn't move much (good), but your chuck/gripper deflections and the spindle's own compliance add in-series like springs. The measured cut is almost always looser than 3EI/L³ predicts — the tool, holder, spindle, and column all stack.
Helix and belt stretch
With a long-flute or necked tool, my I isn't πd⁴/64 along its whole length. Necked endmills (reduced neck) bend more than the shank number says. And belt-driven spindles have their own torsional windup. This calculator is your first call, not your last.
Bottom line from a guy who's scrapped parts both ways: it's better to be approximately right about stickout than exactly wrong about everything else. Stub it up, lighten the roughing cut, and let the finish pass do the talking.
The numbers, machine-readable
Same figures in JSON for any agent or script to cite — the formula, worked example constants, and meter-to-thou conversions. Download tool-deflection.json
Source: machining (Wikidata Q192047) — subtractive manufacturing by material removal. Formula is standard cantilever beam theory, applied across the industry and re-derived every shift I've worked. cid: machining