Engineering task and calculation objective
The WTOR module dimensions shafts subjected to a torsional moment — the typical application in pressure equipment and valve engineering is the drive or operating shaft of a valve that must transmit the breakaway torque of the closure member. The module calculates the cylindrical shaft cross-section, the two-flat (spreader) drive end (rounded or rectangular), and cross-sections weakened by blind tapped holes.
For each cross-section, the program determines the polar section modulus and compares the actual shear stress resulting from the required breakaway torque — multiplied by a safety factor — with the fatigue design strength of the cross-section. Following the shaft calculation approach (cf. DIN 743, the German shaft design standard), this strength is built up from the torsional endurance limit of the material and the influence factors of the real component: fatigue notch factor, surface factor (via the peak-to-valley roughness height), technological size factor and geometric size factor. The results are the allowable torques of the two-flat and the cylindrical cross-section as well as their ratio — making it immediately apparent which cross-section limits the transmittable torque.
Material characteristics such as tensile strength and torsional endurance limit are applied temperature-dependently for the selected material, so the shaft dimensioning remains consistent for elevated operating temperatures as well.
Calculation workflow
- Define operating data and material: Temperature, nominal pressure and the material used are specified; from these follow the tensile strength and the torsional endurance limit as the base characteristics of the strength calculation.
- Determine the governing torsional moment: The starting point is the required breakaway torque of the valve; the stress comparison is referred to a multiple of this moment in order to cover starting shocks and reserves.
- Section moduli of the cross-sections: For the cylindrical cross-section, the two-flat (rounded or rectangular) and the cross-section weakened by the blind hole, the polar section moduli are calculated from the dimensions (shaft diameter, width and long side of the two-flat, thread diameter).
- Build up the fatigue design strength for each cross-section: The torsional endurance limit of the material is converted into the fatigue design strength of the respective cross-section using the fatigue notch factor, the surface factor (from the roughness height), the technological size factor and the geometric size factor.
- Allowable torques and verification: From the fatigue design strength and the polar section modulus, the allowable torque of each cross-section follows; the actual shear stress from the design moment is compared against it, and the ratio of the torques is reported — the cross-section with the smallest allowable torque governs.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Shaft diameter | DW = | – |
| Spreader width | B = | – |
| Polar section modulus | Wp 1 = | – |
| Temperature | T = | – |
| Nominal pressure | Δp = | – |
| Material | Nummer = | – |
| Actual torsional stress due to -times breakaway axial moment | τT = | – |
| Variable 8 | = ≥ ? | – |
| Required breakaway axial moment | Mlos = | – |
| Actual shear stress | τvorh = | – |
| Shape stability of spreader | τt zul = | – |
| Variable 12 | Mlos ≤ ? | – |
| Variable 13 | Mlos ≤ ? | – |
| Thread diameter of stud hole | DG = | – |
| Polar section modulus | Wp 2 = | – |
| Polar section modulus of spreader | Wp = | – |
| Tensile strength | Rm = | – |
| Torsional endurance limit | τD = | – |
| Peak to valley height of spreader | Rz = | – |
| Surface factor spreader | b1 = | – |
| Form factor of spreader | b2 = | – |
| Fatigue notch factor of spreader | βkt = | – |
| Aux. factor cross section of spreader | kt = | – |
| Shaft diameter of cyl. cross section | DW,z = | – |
Worked example
For a valve shaft with a full circular cross-section, determine the polar section modulus and the actual shear stress when the design moment is taken as 1.0 times a breakaway torque of 500 N·m — a worked example of a shaft torsion calculation. The comparison with the fatigue design strength is then carried out in the module using the material- and surface-dependent factors.
Given values
| Shaft diameter d | 40 mm |
| Design torsional moment T | 500 N·m |
Solution
Polar section modulus of the cylindrical cross-section
Wp = π · d³ / 16 = π · 40³ / 16 = 12,566 mm³
Actual shear stress
τt = T / Wp = 500,000 N·mm / 12,566 mm³ = 39.8 N/mm²
In the module, this stress is compared with the fatigue design strength of the cross-section, which is formed from the torsional endurance limit of the material with the fatigue notch factor, surface factor and size factors. For the two-flat, the calculation must be repeated with its significantly smaller polar section modulus and higher fatigue notch factor — it usually governs.
Result
| Polar section modulus Wp | 12,566 mm³ |
| Actual shear stress τt | 39.8 N/mm² |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Why is the calculation based on a multiple of the breakaway torque?
The breakaway torque of a valve scatters widely: actuator tightening torques, sticking of the sealing faces, deposits and low temperatures can raise the actually occurring moment well above the catalog value. The factor on the breakaway torque provides the reserve needed so that the shaft neither yields nor risks fatigue failure even under the most unfavorable breakaway event. It is deliberately implemented as an input quantity, since actuator sizing and operating practice are plant-specific.
Why is the two-flat usually the critical cross-section?
The two-flat reduces the polar section modulus considerably compared with the full circle, and the re-entrant edges act as notches with a correspondingly high fatigue notch factor — more pronounced for the rectangular version than for the rounded one. The module therefore reports the allowable torques of both cross-sections separately; if the ratio of the torques is well below 1, the rounded two-flat form or a larger across-flats width is worthwhile from a design point of view.
What role do the roughness height and the size factors play?
The endurance limit data of materials are determined on small polished specimens. Real shafts have rougher surfaces (surface factor from the peak-to-valley height) and larger cross-sections, for which both the through-hardening achieved in manufacture and the stress gradient are less favorable (technological and geometric size factor). These factors reduce the sustainable stress considerably in some cases — omitting them is a classic cause of overly optimistic hand calculations.
Does the module also cover bending and combined loading?
WTOR is designed for pure torsional loading, as it dominates in operating and drive shafts of valves. If significant bending moments or axial forces are added — for example with long, laterally loaded shafts — a complete shaft calculation with equivalent stress formation per DIN 743 is required.