Shaft design: why rotating shafts fail by fatigue
A shaft that turns under a steady load is bent back and forth once every revolution. How bending and torque combine, why the endurance limit is corrected for surface, size and reliability, what shoulders and keyseats do to it, and how the Goodman line decides the diameter.
A steady load, a reversing stress
A shaft carrying gears or pulleys is bent by their forces, and the bending does not rotate with it. Follow one point on the surface: at the top of each turn it is squeezed in compression, at the bottom it is stretched in tension, and in between it passes through zero. A load that never changes becomes a stress that reverses completely once per revolution.
That is a lot of cycles. A shaft turning at 1500 revolutions per minute is bent back and forth more than two million times a day, so even a modest stress soon reaches the numbers where fatigue, not a single overload, decides whether it survives.
Bending and torque, combined
For a solid shaft of diameter d, a bending moment M gives a surface stress of 32·M/(π·d³) and a torque T gives a shear stress of 16·T/(π·d³). Both depend on the cube of the diameter, so a shaft only slightly too small is far too weak: halving the diameter multiplies both stresses by eight.
The two stresses act together on the same surface, so they are combined into one equivalent stress by the distortion-energy (von Mises) criterion, σ' = √(σ² + 3τ²). In a turning shaft the bending part usually alternates and a steady torque stays constant, so the calculation keeps an alternating von Mises stress and a mean one.
The endurance limit, and why it is corrected
Many steels have an endurance limit: a stress amplitude below which they survive indefinitely. Rotating-beam tests on small, polished specimens put it at roughly half the ultimate tensile strength, up to about 700 MPa for very strong steels.
A real shaft is not a polished laboratory specimen, so that value is multiplied down by correction factors named after Joseph Marin. A rougher surface starts cracks sooner — a machined surface keeps perhaps 80% of the polished value, a forged one far less. A larger shaft has more material in which a flaw can sit. And because endurance strength scatters, asking for 99% reliability instead of 50% trims it further, by about a fifth.
Shoulders, keyseats and grooves
Shafts are stepped to locate bearings and gears, cut with keyseats to drive them, and grooved for retaining rings. Every change of section concentrates stress: a sharp shoulder can nearly triple the local bending stress and a retaining-ring groove can multiply it about five times. Fatigue cracks almost always start at these features, not in the plain length between them.
That is why the check is made section by section, at each shoulder, keyseat and groove, with its own stress concentration factor. Generous fillet radii are among the cheapest fatigue improvements in machine design.
The Goodman line
A mean stress on top of the alternating one shortens fatigue life, because it holds cracks open. The modified Goodman criterion draws a straight line on a chart of alternating against mean stress, from the endurance limit on one axis to the ultimate strength on the other. A shaft whose stress point lies inside the line is safe for infinite life, and the safety factor is how far the point could move outward along a line from the origin before it reaches the line: 1/n = σ'_a/S_e + σ'_m/S_ut.
Solving that equation for the diameter gives the smallest shaft that meets a chosen factor. A separate check makes sure the peak stress on the very first cycle stays below yield, and the shaft is sized by whichever check needs the larger diameter.
How engineers learned this
On 8 May 1842 a train returning to Paris from Versailles derailed near Meudon when a locomotive axle broke, and the wooden carriages, locked shut, caught fire. Dozens of passengers died. The axle had been strong enough for the load it carried; it failed after many turns without any sign of distress.
The next year the Scottish engineer W. J. M. Rankine argued that railway axles were breaking at their sharp shoulders and recommended generous rounded fillets. In the decades that followed, the German engineer August Wöhler tested railway axles and specimens through millions of cycles, drew the first stress–life curves, and showed that steel has a stress below which it lasts indefinitely. Modern shaft design still rests on that work.
One point on a turning shaft
The bending load stays put while the shaft turns, so a point on its surface is compressed at the top of each turn and stretched at the bottom. The stress it feels reverses completely once per revolution — millions of cycles a day — which is why shafts are designed for fatigue.
Check yourself
1. Why does a shaft that turns under a steady bending load still fail by fatigue?
2. A solid shaft's diameter is doubled. Its bending stress for the same moment becomes…
3. What does the modified Goodman line connect on a fatigue diagram?
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