Shaft under Combined Bending + Torsion

stressmass-cost

The torsion shaft lived in a polite world where shafts only twist. Real shafts carry pulleys and gears, and everything that twists them also pushes them sideways: the belt whose tensions $T_1 + T_2$ bend the shaft toward the pulley while their difference torques it, the gear whose separating force does the same. Bending and torsion arrive on the same surface element, and the question “which stress matters?” needs an honest answer:

Three things to notice:

• The worst plane is one you didn’t draw. Neither the bending stress nor the torsion stress alone is the danger — rotate the element (Mohr’s circle is the bookkeeping) and the extreme shear lives on a tilted plane, with magnitude $\sqrt{(\sigma_b/2)^2 + \tau_t^2}$. The widget’s circle is drawn live from your knobs; the radius IS $\tau_{max}$.
• Two criteria, one element, different verdicts. Tresca asks “worst shear plane?”, von Mises asks “how much shape-changing energy?” — and they disagree by up to 15 %, worst in pure torsion, agreeing exactly in pure bending. The widget shows both safety factors side by side; watch their gap breathe as you trade $M$ against $T$. Which is right? Test data sides with von Mises for ductile metals; Tresca errs safe. They bracket the truth, and the bracket is the honest answer.
• The old shortcut falls out of the algebra. $\tau_{max} \propto \sqrt{M^2+T^2}$ means the shaft fails in shear as if carrying a single “equivalent torque” $T_e = \sqrt{M^2+T^2}$ — pre-computer designers sized shafts with that one number, and the size-diameter configuration runs the same line backwards today.

One loud caveat, carried by the first relation’s assumptions: this page is a static snapshot. A rotating shaft under a stationary bending moment sees $\sigma_b$ fully reversed every revolution — that problem is fatigue, it is the actual sizing problem for most shafts, and it belongs to a future THING. Treat these margins as the first-cycle check they are.

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 compares two static yield criteria on a perfect surface element. Rotating machinery has richer ways to die:

• Rotation turns bending into fatigue — the headline. Hold the bending moment still and spin the shaft: every surface point rides $+\sigma_b \to -\sigma_b$ once per revolution while the torsional stress sits steady. That combination — fully-reversed bending plus steady torsion — is THE canonical fatigue problem (Shigley’s ch. 7 exists almost entirely for it), with allowables far below yield once cycles reach the millions. This page’s static margins are the first-cycle sanity check, never the life prediction.
• Stress concentrations own the failure site. Real shafts carry keyways, shoulder fillets, retaining-ring grooves, and press-fitted hubs; each multiplies the local stress by 1.5–3×. Combined with fatigue, the failure address is almost always a feature, not the smooth mid-span this model analyzes.
• The criteria gap is small next to the load uncertainty. Tresca vs von Mises differ ≈15 % at most; belt-tension estimates, shock factors, and misalignment loads routinely differ by more. The bracket the two criteria draw is tight compared to how poorly $M$ and $T$ are usually known — which is why service factors exist.
• Deflection and whirl. A shaft sized for stress can still deflect enough to misalign gear meshes and chew bearings — and at the speed where rotation meets the shaft’s lateral natural frequency (critical speed), deflection grows without new load. Stiffness ($E$, absent from this page on purpose) and bearing spacing decide that, not strength.
• Combined is worse than either load alone suggests. $\tau_{max}$ under $M$ and $T$ together exceeds the worst stress of either load by itself — but the interaction is geometric ($\sqrt{M^2+T^2}$), so the cross-load enters quadratically: the first bit of torque on a bending-critical shaft is nearly free (20 % of $M$ as torque raises $\tau_{max}$ only 2 %), and each further increment costs more than the last. Checking bending and torsion separately under-predicts the pair; adding their stresses over-predicts it. The Pythagorean middle is the truth.

Sources

• Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 3 (plane-stress transformation and Mohr's circle: the principal stresses σ_A,B = (σ_x+σ_y)/2 ± √(((σ_x−σ_y)/2)² + τ_xy²) and the max-shear radius), §5-4 (maximum-shear-stress theory), §5-5 (distortion-energy theory, the plane-stress von Mises form).
• Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — flexure formula (§5.5) and circular-shaft torsion (§3.3–3.5), the two single-load formulas this THING combines.