Combined Stress Shaft⚠ unverified

Understanding the Parameters

• Bending moment $M$ — produces a normal stress $\sigma = 32M/(\pi d^3)$ that is maximum at the surface and fully reverses each revolution in a rotating shaft.

• Torque $T$ — produces a shear stress $\tau = 16T/(\pi d^3)$, also maximum at the surface.

• Diameter $d$ — both stresses scale as $1/d^3$, so the equivalent stress does too; the equation is most often used inverted, to solve for the diameter that keeps $\sigma_{\text{eq}}$ below the allowable (see Shaft Diameter Combined).

• The factor 3 on $\tau^2$ — the signature of the distortion-energy (von Mises) theory: shear stress is "three times as damaging" (in the squared sense) as normal stress toward causing yield, because pure shear reaches yield at $\tau_y = \sigma_y/\sqrt{3}$.

Derivation (Approaching a Proof)

At the surface of a shaft under bending and torsion, the stress element has a normal stress $\sigma$ (from bending) on one face and a shear stress $\tau$ (from torsion) — a plane-stress state with $\sigma_x = \sigma$, $\sigma_y = 0$, $\tau_{xy} = \tau$.

The von Mises (distortion-energy) criterion states that yielding begins when the distortion strain energy reaches the value it has at uniaxial yield. For a general plane-stress state the equivalent stress is

$$\sigma_{\text{eq}} = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2}.$$

Substituting $\sigma_y = 0$, $\sigma_x = \sigma$, $\tau_{xy} = \tau$ gives the shaft form:

$$\sigma_{\text{eq}} = \sqrt{\sigma^2 + 3\tau^2}.$$

Inserting the shaft stresses $\sigma = 32M/(\pi d^3)$ and $\tau = 16T/(\pi d^3)$ and factoring out $16/(\pi d^3)$:

$$\sigma_{\text{eq}} = \frac{16}{\pi d^3}\sqrt{(2M)^2 + 3T^2} = \frac{16}{\pi d^3}\sqrt{4M^2 + 3T^2},$$

which is exactly the group inverted in the Shaft Diameter Combined sizing formula. The factor 3 on $\tau^2$ comes directly from the distortion-energy derivation, in which hydrostatic stress does not cause yielding — only the distortional part does.

Dimensional check. Each term under the root is a stress squared, so $[\sigma_{\text{eq}}] = \text{Pa}$. ✓

History and Development

The distortion-energy yield criterion is due to von Mises (1913), building on Huber (1904) and later Hencky — a refinement of the earlier maximum-shear (Tresca) theory that better matches ductile-metal test data. Its application to combined bending-and-torsion shafts is a cornerstone of the ASME shaft-design code and Shigley. An alternative shaft form uses the maximum-shear theory, giving $\sqrt{M^2 + T^2}$ inside the equivalent-torque expression.

Related Concepts: Von Mises Stress, Torsional Shear Stress, Beam Bending Stress, Shaft Diameter Combined, Shaft Fatigue Factor

Notes: Von Mises (distortion-energy) equivalent stress; internally uses $\sigma = 32M/\pi d^3$ and $\tau = 16T/\pi d^3$. Compare against yield (static) or use with fatigue factors for cyclic loading. The maximum-shear theory gives a slightly more conservative alternative.