Stepped Shaft — Shoulder-Fillet Stress Concentration

stress

Almost every real shaft changes diameter along its length: a large section steps down to a smaller one to seat a bearing, a gear, or a coupling against the shoulder. That step cannot be a sharp corner — a sharp re-entrant corner is a stress singularity — so it is blended with a fillet of radius $r$. The fillet is where the shaft is most likely to crack, and this page is about how much the geometry there amplifies the stress. It is the shoulder that the spur gear pair is pressed against, and the same section whose nominal stresses the cantilever beam and torsion shaft pages compute.

Away from the shoulder, the smaller shaft of diameter $d$ carries an ordinary nominal stress — the elementary result for that section, depending on how it is loaded:

$\sigma_{nom} = \frac{4F}{\pi d^2}\ \text{(axial)}, \qquad \sigma_{nom} = \frac{32M}{\pi d^3}\ \text{(bending)}, \qquad \tau_{nom} = \frac{16T}{\pi d^3}\ \text{(torsion)}$

Enter that nominal stress in the widget (the combined shaft page shows where a bending-plus-torsion state comes from). At the fillet the real stress is higher, by a stress-concentration factor $K_t$ that depends only on the shape:

$\sigma_{max} = K_t\,\sigma_{nom}, \qquad K_t = A\left(\frac{r}{d}\right)^{b}$

$K_t$ is geometry, not strength

The two ratios that set $K_t$ are the diameter ratio $D/d$ (how big the step is) and the fillet ratio $r/d$ (how gently it is blended). Plotted against $r/d$ on log-log axes at fixed $D/d$, the concentration factor is very nearly a straight line, so Norton fits each curve as the power law $K_t = A\,(r/d)^{b}$ and tabulates the pair $(A, b)$ by $D/d$. A shoulder fillet under axial, bending, and torsional load concentrates stress differently, so there are three tables — pick the loading in the widget and watch $K_t$ change while the geometry stays put.

The single most important thing to see here: $K_t$ contains no material property. It is pure geometry. Switch from aluminium to steel and the peak stress $\sigma_{max}$ at a given nominal load is identical — only the safety factor $\mathrm{SF} = \sigma_y/\sigma_{max}$ moves, because only the allowable stress changed. The amplification is baked into the shape; the way to fix a fillet that is working too hard is a larger $r/d$, not a stronger alloy. Drag the $r/d$ knob up and watch $K_t$ fall toward 1; drag it down toward a sharp corner and watch it climb.

Cited data, and where it stops

The $(A, b)$ coefficients are cited data with provenance, read digit-for-digit from Norton’s Appendix C (Figs C-1 to C-3, credited to Peterson’s charts) — not derived on this page. The widget looks them up by $D/d$, interpolating linearly between the published rows, and that interpolation is machine-checked against the browser every build. Two honest boundaries are enforced, each refusing $K_t$ and everything downstream while the geometry and nominal stress keep standing:

• $D/d$ outside a table’s published rows — there are no coefficients there, so the lookup refuses. The three tables cover different spans (axial $1.01$–$2.00$, bending $1.01$–$6.00$, torsion $1.09$–$2.00$), so a $D/d$ that is fine for bending may be off the table for torsion.
• $r/d$ above $0.30$ — Norton plots the curves only to $r/d = 0.30$; beyond that the power-law fit has no published basis. Below about $r/d = 0.05$ a warning fires instead: the fit is least certain where the fillet is sharpest and $K_t$ is largest.

The /verification/ page states exactly what is machine-proven here (the lookup, the interpolation, the two refusals, the algebra) and what rests on citation (the coefficients themselves, independently cross-checked against Roark’s closed-form fit).

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

The widget answers one narrow question — by how much does this shoulder geometry amplify the nominal stress? — and reports a static safety factor on that peak. That number is the start of a strength check, not the end of one. The ways a stepped shaft actually fails mostly live in what this page deliberately does not model.

• $K_t$ is where fatigue cracks start — and fatigue is the real killer (the deliberate non-model). Shafts almost never fail by static yielding at a fillet; they fail by fatigue, and they start at the fillet precisely because that is where the stress peaks. Under a fluctuating load the number that matters is the fatigue stress-concentration factor $K_f = 1 + q\,(K_t - 1)$, where the notch sensitivity $q$ (between 0 and 1) softens the geometric $K_t$ by how much the material actually feels a sharp notch. This page computes $K_t$ and stops: it has no endurance limit, no notch sensitivity $q$, and none of the Marin surface/size/reliability factors. A comfortable static $\mathrm{SF}$ here says nothing about how many million cycles the shaft survives — that is the fatigue story a later phase will build on top of exactly this $K_t$.

• Ductile static loading hides the concentration; brittle and cyclic loading do not. Under a static load a ductile shaft yields locally at the fillet, and that little zone of yielding redistributes the stress so the part carries far more than $\sigma_{max} = K_t\sigma_{nom}$ would suggest — which is why full $K_t$ is usually not applied to static ductile design. Where $K_t$ bites in full is (a) brittle materials, which cannot redistribute, and (b) fatigue, where the peak drives crack initiation cycle after cycle. Read this page as geometry pedagogy — which knob moves the peak stress, and by how much — rather than a static design margin.

• The torsion safety factor is illustrative. For the torsion case the “nominal stress” is a shear stress, so comparing it to the tensile yield strength $\sigma_y$ mixes a shear quantity with a normal-stress allowable. A proper check uses a shear yield ($\tau_y \approx 0.577\,\sigma_y$ by von Mises), the same distinction the combined shaft page draws with its Tresca and von Mises circles. The factor here shows the amplification, not a defensible torsional margin.

• The nominal stress is only as good as the model behind it. You type $\sigma_{nom}$ in; it comes from an elementary result for the smaller section — $32M/\pi d^3$ in bending, $16T/\pi d^3$ in torsion, $4F/\pi d^2$ axial — and inherits every assumption those make (linear-elastic, slender, load applied where you think it is). A real shoulder often carries bending and torsion at once; the concentration factors do not simply add, and the interaction is its own topic.

• A fillet you cannot actually cut is a fiction. The $r/d$ knob assumes the fillet radius you dial in is really there. Grinding relief, tool-nose radius, and a bearing race that must seat flat against the shoulder all fight a generous fillet, so real shoulders are frequently sharper than the drawing — pushing $K_t$ up the steep part of the curve where the warn banner already flags the fit as least certain. This is the same shoulder the spur gear pair seats against; a fillet chosen for the gear’s convenience can quietly become the shaft’s weakest point.

Sources

• Norton, R. L., Machine Design: An Integrated Approach, Appendix C "Stress-Concentration Factors", Figs C-1 (axial tension), C-2 (bending), C-3 (torsion), pp. 1028-1029 — fitted equations K_t = A·(r/d)^b for a shoulder fillet in a stepped circular shaft, credited to Peterson's charts.