Estimate the required smooth solid-shaft diameter from simultaneous static bending and torque, yield strength, and required safety.
Set the inputs and run the calculation.
Estimate the minimum smooth solid-shaft diameter from bending, torque and yield strength.
Bending moment Mᵦ, torque T, material yield strength and required safety define the static case.
Bending and torsion use different section resistances and cannot simply be added as moments. Diameter changes both stresses nonlinearly, so a few millimeters can provide substantial reserve.
Bending and torsional nominal stress are combined into a von Mises stress for a circular section; diameter is iterated until allowable stress is met.
A safety factor of 2 halves allowable stress and noticeably increases required diameter compared with factor 1.
Sources and limits: Classical elasticity and DIN 743 preliminary design; notches, keyways and fatigue need the complete verification.
Estimate the minimum smooth solid-shaft diameter from bending, torque and yield strength.
The calculator finds the smallest diameter of a smooth solid shaft for which the equivalent stress from bending and torsion does not exceed Re/S. Bending moment produces normal stress sigma_b = 32·Mb/(pi·d^3), while torque produces shear stress tau_t = 16·Mt/(pi·d^3). For ductile shaft steel, the distortion-energy criterion combines them as sigma_v = sqrt(sigma_b^2 + 3 tau_t^2).
σv = √(σb² + 3·τt²) ≤ Re/S
σb = 32 · Mb / (π · d³)τt = 16 · Mt / (π · d³)dmin = ³√[(16 · S/(π · Re)) · √(4·Mb² + 3·Mt²)]| Symbol / input | Meaning |
|---|---|
| Mb | Bending moment at the critical shaft section; on a rotating shaft, a spatially fixed transverse load can produce alternating bending stress. |
| Mt | Torque transmitted as shear stress through the circular section. |
| Re, S | Yield strength or Rp0.2 for the actual material condition and the required safety factor. |
| σb, τt | Nominal bending and torsional stresses at the outer surface of the idealized smooth solid shaft. |
| σv | Equivalent stress from the distortion-energy criterion for combined static loading. |
| dmin | Calculated minimum diameter; round it up and then perform the complete DIN 743 verification for the real shaft. |
Bending moment Mᵦ, torque T, material yield strength and required safety define the static case. Bending and torsion use different section resistances and cannot simply be added as moments. Diameter changes both stresses nonlinearly, so a few millimeters can provide substantial reserve.
Enter bending and torsional moment at the critical section, the shaft material's yield strength and the required safety factor. The calculator iterates to find the minimum diameter of a smooth solid shaft.
A safety factor of 2 halves allowable stress and noticeably increases required diameter compared with factor 1.
Minimum diameter is a calculated boundary for the entered load case. Round up to an available diameter for design, then verify the real geometry. Separate bending and torsional stresses show which load component dominates; equivalent stress is their common assessment quantity. At 100% utilization Re/S is fully used, but notch, fatigue and stiffness checks are still absent.
Moments in N·m, yield strength in MPa, diameter in millimetres.
A quick first estimate of shaft diameter in an early design phase, before geometry details such as shoulders and keyways are fixed.
Classical elasticity and DIN 743 preliminary design; notches, keyways and fatigue need the complete verification.
Common mistake: Do not take this value as a final manufacturing diameter – it applies to a smooth, notch-free shaft. Shoulders, keyways, holes and fillets noticeably raise local stress and require the complete DIN 743 strength verification with notch effect, size and surface influence and separate fatigue and static safety.
It addresses only the static limit case of a smooth shaft. Notch effects, fatigue strength, load variation and real shaft shoulders are only captured by the complete DIN 743 verification.
Since stress falls with the cube of diameter, a higher required safety factor produces a disproportionately larger required diameter.
They produce different stress types (normal and shear); only an equivalent-stress hypothesis such as von Mises combines them correctly into a single equivalent stress.
Use the internal moments acting simultaneously at the likely critical section in the governing load case. Do not combine maxima from separate load cases that can never occur together without justification.
dmin is the exact mathematical boundary of the idealized model. Rounding down would exceed allowable equivalent stress; round up to an available stock or finished size with machining and verification margin.
The calculator combines nominal stresses in a solid circular section into an equivalent stress. It supports an initial packaging estimate, for example for a drive shaft between a gear and coupling.
Notches, keyways, shoulders, surface condition, size effects, and fluctuating loads can materially increase the required diameter. Follow with a complete DIN 743 verification.