Torsion Spring Calculator — Torque, Angular Deflection & Wire Bending Stress
Calculate torsion spring torque at deflection, angular spring rate, and wire bending stress. Supports round wire helical torsion springs with leg configurations.
Quick Answer
For a torsion spring with 3mm wire, 20mm mean diameter, 6 active coils, deflected 90°: Spring Rate = 1.26 N·mm/°, Torque at 90° = 113 N·mm, Wire Bending Stress ≈ 720 MPa. Music wire at this stress level is acceptable for static service (≤80% UTS).
How Torsion Spring Calculations Work
Torsion springs store energy by twisting the wire in bending — not the coil axis. This is fundamentally different from compression springs (torsional shear in the wire).
1. Angular Spring Rate
k_θ = Ed⁴ / (10.8 D N_a) N·mm/radian, or divide by 57.3 for N·mm/°. The wire again dominates with d⁴ — a small gauge change has a large effect. More active coils make the spring softer (more wire length to twist).
2. Torque at Deflection
M = k_θ × θ, where θ is angular deflection in degrees or radians. Unlike compression springs, torsion springs operate over a defined angular range — typically 90-270° for most applications.
3. Bending Stress
σ_b = (32M) / (πd³) × K_i, where K_i is the stress correction factor for wire curvature. The stress is bending stress (not torsional shear), so it’s compared to the material’s tensile strength, not shear strength. For static service: σ_allow ≤ 0.78 UTS. For fatigue: ≤ 0.50 UTS.
Common Applications
- Garage door counterbalance mechanisms
- Clothespins, clipboards, and mouse traps (everyday torsion springs)
- Clock and watch mainsprings (spiral torsion springs)
- Vehicle suspension torsion bars
- Electrical switch and relay return mechanisms
Common Mistakes
- Applying compression spring formulas to torsion springs — Compression springs are in torsion (wire twists), torsion springs are in bending (wire bends). The formulas are completely different. Mixing them gives nonsense results.
- Not accounting for coil diameter reduction — As a torsion spring winds up, the coil diameter decreases — the spring tightens on its arbor. Allow 10-15% clearance between spring ID and arbor OD, or the spring locks up.
- Forgetting leg stress concentration — The bend radius where the leg transitions from the coil body is a stress concentration. A sharp bend can fail at half the load of a generous radius. Minimum leg bend radius ≥ 1.5× wire diameter.
- Not specifying direction correctly — Torsion springs are wound left-hand or right-hand. A spring wound to load clockwise when viewed from the left end won’t work if installed backward. Specify winding direction explicitly on the drawing.
- Ignoring friction on the arbor — The spring sliding on the arbor under load adds friction that reduces effective output torque by 5-15%. For precision applications (instruments, timers), use bearing-mounted arbors to minimize friction variation.
Frequently Asked Questions
What is the difference between torsion spring and compression spring formulas?
Compression: wire twists in torsion (τ = 8FD/πd³), spring stores energy axially. Torsion: wire bends (σ = 32M/πd³), spring stores energy angularly. Compression springs use shear modulus G, torsion springs use elastic modulus E. G ≈ 0.38E for steel — compression springs are inherently more efficient for energy storage per weight.
How do I calculate the number of active coils for a torsion spring?
Active coils for torsion springs = body coils between the legs. Unlike compression springs, torsion spring end coils don’t affect rate significantly because they’re not in the coiled body. Count all full 360° wraps between leg tangency points.
What is the maximum deflection angle for a torsion spring?
Depends on stress limit and geometry. Typical working range: 90-270°. Below 90°, you’re underutilizing the spring. Above 270°, the coil tightening effect becomes severe — the spring may lock on the arbor. Maximum theoretical is limited by solid condition (coils touching).
Why does coil diameter decrease as the spring winds up?
The wire helix angle changes as you wind — the coil wraps tighter around the arbor. A spring with 20mm mean diameter at free position may tighten to 18.5mm at 180° deflection. This is why arbor clearance matters — see our Spring Rate Calculator for compression spring comparisons.
Can I make a double torsion spring?
Yes — two coils wound in opposite directions from a single wire, with a connecting bar between them. This doubles torque for the same deflection angle and eliminates side thrust. Common in automotive hood hinges and industrial clamps. The calculation is 2× single spring torque.
How do I specify leg length and configuration?
Legs come in many styles: straight, straight offset, hook, loop, hinge. Leg length affects installation but not the spring rate — the rate depends on coil geometry only. Specify leg length from coil center: L = (desired linear travel) / (sin of angular deflection).
What material for high-cycle torsion springs?
Music wire (ASTM A228) for general purpose to ~250°C. Chrome-silicon (ASTM A401) for high stress or >250°C — better fatigue and relaxation resistance. Stainless 17-7PH for corrosion + high strength. For >10⁷ cycles, shot peening adds 20-50% life. Avoid plating — hydrogen embrittlement kills torsion springs faster than compression springs.
How do I measure torsion spring torque for quality control?
Torque test at a specified deflection angle — not at free position. Use a torque wrench or torque tester. Check torque at min and max deflection — the slope between them is the rate. If rate is within ±10% of design and free position is within ±5°, the spring is acceptable.