Engineering CalculatorsInvolute Splines: Geometry, Measurement Over Pins & Torque Capacity
A plain-language guide to involute splines to ISO 4156 / ANSI B92.2M: how module and pressure angle set the geometry, how measurement over pins inspects tooth thickness, and how the SAE method rates torque capacity.
Why splines, and why involute ones
A spline is a set of teeth cut straight along a shaft that mesh with matching internal teeth in a hub — a way to transmit torque while letting the two parts slide axially and self-centre, which a single key can't do. You could cut those teeth with straight (parallel) sides, but almost all modern splines use an involute profile: the same tooth form as a gear. It's stronger at the root, it centres itself under load, and — the practical clincher — it's made with ordinary gear-cutting tools. ISO 4156 (identical in substance to ANSI B92.2M, and sharing its 30° geometry with ANSI B92.1 and DIN 5480) is the standard that defines these splines by metric module.
Module and pressure angle set everything
An involute spline is defined by just a few numbers. The module m is the tooth size (bigger m = bigger, fewer teeth); the number of teeth z; and the pressure angle α — the slope of the tooth flank, standardised at 30°, 37.5° or 45°. From those, the geometry falls out directly:
- Pitch (reference) diameter D = m·z — the notional circle where tooth and space are equal.
- Base diameter Db = m·z·cos α — the circle the involute is generated from.
- Major (tip) diameter Dee = m·(z + 1) for a 30° external spline, with the tip stubbier at higher pressure angles.
- Circular tooth thickness s = ½·π·m at the pitch circle — exactly half the pitch, by definition.
The 30° profile is the workhorse; 37.5° and 45° use shorter, stubbier teeth (fillet root only) that suit thin-walled hubs and high tooth counts. A flat root (available at 30°) is a little stronger in bending; a fillet root is more common and easier to hob. Because it's a side-fitspline, the flanks carry the torque and do the centring — the major and minor diameters clear each other and don't locate anything.
Measurement over pins: how you actually inspect one
You can't put a caliper on a single spline tooth and get a meaningful thickness. Instead, splines are inspected by measurement over pins: drop two precision balls or pins into opposite tooth spaces and measure across them. That dimension is a proxy for the tooth thickness at the pitch circle — the thing that actually controls the fit — and it's what goes on the drawing for the inspector to check.
The geometry is pure involute. A pin of diameter DR seated in a space contacts both flanks at a pressure angle φ found from inv φ = s/D + inv α + D_R/D_b − π/z, where inv α = tan α − α is the involute function. The measurement over two pins is then M = D_b/cos φ + D_R for an even tooth count (odd counts multiply the first term by cos(90°/z), because opposite a space is a tooth). The standard tabulates a pin size for each spline; the key point is that the measurement follows exactly from the tooth thickness, so it's a genuine functional check, not an approximation.
Rating a spline for torque
Geometry aside, the design question is: will it carry the torque? A spline has three ways to give way, and the classic SAE / ANSI B92.1 (Dudley) method checks each:
- Tooth shear — the teeth shear off at the pitch line:
τ = 2·T·Ks / (L·z·t·D). - Flank bearing (compressive) stress — the flanks crush or wear:
σc = 2·T·Ks / (L·z·h·D), with engagement height h ≈ one module. - Shaft core — the shaft itself twists off at the minor diameter:
τ = 16·T·Ks / (π·D_ie³).
Here T is torque, L the engagement length, z the tooth count, t the tooth thickness, D the pitch diameter and Die the minor diameter. The capacity is the lowest of the three — the governing failure mode — and more teeth, more length, a bigger module or a harder material all raise it. Notice length and diameter matter as much as the teeth: a longer, larger spline is a stronger one.
The service factor: fixed vs flexible splines
The Ks in those formulas is a service factor that bundles the duty into one number, and it splits on how the spline is used. A fixed (non-sliding) spline is limited by fatigue: Ks = Ka / Kf, where Ka is an application/shock factor and Kf a fatigue-life factor that shrinks with cycle count. A flexible (sliding) spline is limited by wear instead: Ks = Ka·Km·Kd / Kw, adding a misalignment load-distribution factor Km and a wear-life factor Kw. Same spline, same torque — a sliding one is derated harder because it frets and wears where a fixed one just fatigues.