Calculator Inputs
Example Data Table
These examples help you sanity-check units and scale. They are illustrative, not specifications.
| Case | Section | T (N·m) | L (m) | G (GPa) | Key size | Angle θ (deg) |
|---|---|---|---|---|---|---|
| 1 | Solid round | 120 | 1.2 | 80 | d = 35 mm | ≈ 3.03 |
| 2 | Hollow round | 250 | 0.9 | 26 | Do = 40 mm, Di = 25 mm | ≈ 3.47 |
| 3 | Rectangular (approx.) | 60 | 0.5 | 80 | b = 60 mm, t = 10 mm | ≈ 3.58 |
| 4 | Manual J | 75 | 1.0 | 80 | J = 4.8e-7 m⁴ | ≈ 1.12 |
Formula Used
For a prismatic shaft in elastic torsion, the angle of twist is:
θ = (T · L) / (J · G)
- θ is the angle of twist (radians).
- T is the applied torque (N·m).
- L is shaft length (m).
- G is shear modulus (Pa).
- J is the torsion constant (m⁴). For circular shafts, J is the polar second moment.
Section formulas
- Solid round: J = π d⁴ / 32
- Hollow round: J = π (Do⁴ − Di⁴) / 32
- Rectangular (approx.): J ≈ b t³(1/3 − 0.21β(1 − β⁴/12)), where β = t/b and b ≥ t
How to Use This Calculator
- Select what you want to solve for (θ, T, L, G, or J).
- Enter the known values and choose units for each field.
- Choose how you will provide J: dimensions or direct entry.
- Pick the section type and fill its required dimensions.
- Click Calculate to show results above the form.
- Use Download CSV or Download PDF to save a report.
Angle of Twist Guide
1) What the calculator estimates
This tool applies elastic torsion for straight, prismatic shafts: θ = T·L/(J·G). It is best for preliminary sizing, comparison studies, and quick unit-checked calculations before detailed FEA or testing.
2) Typical shear modulus values
Shear modulus drives stiffness. Common engineering values are about 80 GPa for steel, 26 GPa for aluminum, 44 GPa for titanium, and 3 to 4 GPa for many rigid plastics. Using the correct G can change twist by 2× or more.
3) How torque and length scale twist
Twist is directly proportional to torque and length. If torque doubles, θ doubles. If length increases from 0.5 m to 1.0 m with all else fixed, θ also doubles. This linearity makes sensitivity checks easy.
4) Why the torsion constant J matters
J captures how the cross section resists torsion. For a solid circular shaft, J = πd⁴/32, so diameter has a fourth-power effect. Increasing d by 10% raises J by roughly 46%, cutting twist substantially.
5) Hollow shafts and weight efficiency
For circular tubes, J = π(Do⁴ − Di⁴)/32. Removing material near the center reduces weight with a smaller stiffness penalty than reducing the outer diameter. Many driveline and aerospace parts use this stiffness-to-mass advantage.
6) Rectangular sections: expect higher twist
Rectangles warp more than circles, so their torsion constant is lower than a same-area circle. The included Saint‑Venant approximation (with β = t/b) is useful for quick estimates, but thin rectangles can be especially flexible.
7) Practical design checks
Engineers often limit twist per length to protect alignment and performance. A quick check is degrees per meter (or rad/m). The calculator also reports torsional stiffness k = JG/L, a convenient way to compare candidate shafts.
8) Stress note for circular shafts
For circular sections, the maximum shear stress is τmax = T·r/J. This page shows τmax when geometry is circular and J is computed from dimensions. Combine a stress check with an angle limit for a balanced design decision.
FAQs
1) What units does the calculator use internally?
Inputs are converted to SI (N·m, m, Pa, m⁴) for calculation, then displayed back in your selected units. This prevents mixed-unit errors when you switch between metric and imperial fields.
2) Should I enter θ in degrees or radians?
Either works. Choose the θ unit selector to match your value. Results always show both degrees and radians so you can cross-check with hand calculations or standards that specify limits in radians.
3) When should I enter J manually?
Use manual J when the cross section is nonstandard, has keyways/splines, or you already have a handbook value. Manual mode avoids geometry assumptions while keeping the same torsion equation.
4) Is the rectangular J formula exact?
No. It is a common Saint‑Venant approximation and is most reliable when b ≥ t and the rectangle is not extremely thin. For thin-walled or complex sections, consider specialized formulas or FEA.
5) Why does diameter affect twist so strongly?
For solid round shafts, J scales with d⁴. A small diameter increase dramatically increases stiffness and reduces twist. This is why modest size changes can meet an angle limit without changing material.
6) Does this cover plasticity or large twists?
No. The equation assumes linear elastic behavior and small strains. If you expect yielding, nonlinearity, or large rotation effects, use material curves and a nonlinear analysis method.
7) Why is τmax shown only for circular shafts?
The simple τmax = T·r/J expression is exact for circular shafts in Saint‑Venant torsion. Noncircular sections have nonuniform stress distributions, so a single closed-form peak value is not generally accurate.