Combined Torsion & Bending
With simultaneous bending M and torque T, find diameter by equivalent bending-moment and equivalent torque methods; take the larger.
Inputs
Formula
Me=√(M²+(αT)²); Te=√((αM)²+T²); d=max(∛(32Me/(π[σ])), ∛(16Te/(π[τ])))
Fundamentals
Shafts usually see bending and torsion together (combined loading), checked by a strength theory.
Third theory: equivalent stress $\sigma_\text{eq}=\sqrt{\sigma^2+3\tau^2}$, $\sigma$ bending normal, $\tau$ torsional shear; safe when $\sigma_\text{eq}\le[\sigma]$.
History
The 3rd and 4th strength theories for combined bending-torsion were given by Tresca (1864) and von Mises (1913).
Engineering applications
Used for shafts under both bending and torque, e.g. reducer shafts and wheel axles, to check safety factors.
Glossary
| Equivalent $\sigma_\text{eq}$ | Combined stress reduced to an equivalent uniaxial one. |
| 3rd theory | Max-shear-stress theory, for ductile materials. |
| Combined load | Bending and torsion acting together. |
How to use
- Fill in Bending M, Torque T, Allow. bend. [σ], Allow. shear [τ], Equiv. factor α in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read By equiv. moment, By equiv. torque, Recommended in the results area.
Formula notesCombined bending+torsion (max shear): σ_eq = √(σ_b² + 4τ²) ≤ [σ]; σ_b = M/W, τ = T/W_p.
Formula · Worked Example · Knowledge
Formula
With bending M and torque T, use equivalent moment $M_e=\sqrt{M^2+(\alpha T)^2}$ and equivalent torque $T_e=\sqrt{(\alpha M)^2+T^2}$; take the larger diameter.
Worked Example
M=400 N·m, T=300 N·m, α=0.6, [σ]=80, [τ]=45 MPa → recommended ≈36.6 mm.
Key Points
- α is the equivalent-factor (0.6 pulsating, 1 reversing torque).
- Combined loading is the standard shaft-design rule.
- Long shafts also need deflection and critical-speed checks.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Bending M (N·m) | M | N·m | 400 |
| Torque T (N·m) | T | N·m | 300 |
| Allow. bend. [σ] (MPa) | sigma | MPa | 80 |
| Allow. shear [τ] (MPa) | tau | MPa | 45 |
| Equiv. factor α | alpha | 0.6 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| By equiv. moment | d_bend | mm |
| By equiv. torque | d_tor | mm |
| Recommended | d_rec | mm |
Applications
- Common engineering use cases
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: Me=√(M²+(αT)²); Te=√((αM)²+T²); d=max(∛(32Me/(π[σ])), ∛(16Te/(π[τ])))
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Common engineering use cases