Torsion Spring Calculator
Estimate stress, angular rotation, tip deflection, stiffness, and stored energy for round-wire helical torsion springs or straight torsion bars.
Engineering screening tool—verify inputs and results independently. Use at your own risk.
Spring model and inputs
Choose the geometry first. All dimensions are positive magnitudes in millimetres.
Calculated response
Stress is a nominal circular-wire bending measure. Compare it with an independently established allowable value including fatigue, stress concentration, material condition, and safety factor.
How to use
- Select Helical coil or Straight torsion bar. The selected mode determines the additional geometry fields and equations.
- Enter round-wire diameter d, load-arm length r, model elastic modulus E, and applied force magnitude F in the displayed units.
- For a helical coil, enter mean coil diameter D and active coil count n. For a straight bar, enter effective active length Lₑ.
- Select Calculate, or edit a field for an immediate update. Select Reset to restore the helical example.
- Read nominal stress, rotation, tip deflection, and stiffness. The helical mode also reports active coil length and stored energy; all displayed outputs are rounded to two decimals.
- Correct any red warning before using a result. Force is an unsigned magnitude; assign rotation direction separately from the actual load orientation.
- Do not use this simplified model for rectangular wire, large rotation, coil clash, leg/contact flexibility, dynamic impact, buckling, yielding, fatigue-life qualification, or safety-critical final design.
Worked example
1. Select Helical coil and enter d = 2 mm, D = 10 mm, r = 30 mm, n = 1.5, E = 210 GPa, and F = 10 N. 2. The result is σ = 381.97 MPa, Δφ = 4.91° (0.09 rad), f = 2.57 mm, kφ = 2.04 N/deg, krad = 116.67 N/rad, kf = 3.89 N/mm, Lc = 3.00 mm, and Ep = 0.01 J. 3. The high nominal stress indicates that material strength and fatigue should be checked before using this geometry.
Equations used
Both modes: σ = 32Fr/(πd³)
Helical coil: Lc = nd
Helical coil rotation: Δφcoil(deg) = 2nD·180·σ/(Ed); equivalently, Δφrad = 2πnDσ/(Ed) = Δφcoilπ/180
Helical coil tip movement: fcoil = rΔφrad
Straight-bar tip movement: fbar = 2rLₑσ/(Ed)
Straight-bar rotation: Δφrad = fbar/r; Δφdeg = Δφrad(180/π)
Stiffness: kdeg = F/Δφdeg; krad = F/Δφrad; kf = F/f
Helical stored energy: Ep = FrΔφrad/2
d is round-wire diameter; D is mean coil diameter; r is perpendicular load arm; n is active coil count; Lₑ is effective straight-bar length; E is the model elastic modulus; F is positive force magnitude; σ is nominal stress; Δφ is rotation; f is tip movement; k is secant stiffness; and Ep is stored energy. Lengths must be consistent, E and σ use compatible stress units, and radians are dimensionless. The calculator converts mm, GPa, and N to SI internally.
Theory and method
The model applies the moment M = Fr to a solid circular section and uses the nominal surface stress σ = 32M/(πd³). The two modes then use different compliance relations: helical-coil flexibility depends on active turns and mean coil diameter, while the straight-bar relation depends on active length. Stiffness values are secant ratios at the entered load. Positive inputs report response magnitudes; the physical rotation sign follows the actual force direction.
Assumptions and limits: homogeneous isotropic linear elasticity, circular constant-diameter wire, small deformation, static loading, ideal restraint, no initial torque, and no contact or leg flexibility. The modulus E is used exactly as defined in these implemented relations; do not substitute a shear-modulus torsion formula and expect the same result. Curvature and stress-concentration corrections, residual forming stress, relaxation, temperature, corrosion, surface condition, fatigue, surging, and manufacturing tolerances are omitted.
References: A. van Beek, Advanced Engineering Design: Lifetime Performance and Reliability, torsion-spring calculation collection; equations are shown above. For production helical torsion springs, consult EN 13906-3 and verified material/fatigue data, and apply an appropriate curvature correction and safety factor.
Continue in TriboSolver to refine loading, material behavior, stress concentration, or nonlinear response with a higher-fidelity workflow.
Related calculators