Torsion Spring Calculator

How to use the torsion spring calculator

What this calculator does

This calculator estimates the behavior of a round-wire helical torsion spring under an applied moment. It combines coil geometry, active turns, material stiffness, and load to calculate the spring index, inner and outer stress-correction factors, corrected bending stress, angular displacement, and torsional rate. It is useful for preliminary sizing, comparing candidate geometries, checking a hand calculation, and translating a force applied at a lever arm into torque. It does not determine allowable stress, fatigue life, permanent set, manufacturability, leg-end concentration, or a complete safety factor; those require material certification and application-specific engineering review.

When to use it

Use the calculator when selecting a spring for a hinge or latch, estimating the rotation caused by a known torque, comparing wire diameters or coil diameters, or checking whether a trial geometry produces an obviously excessive elastic stress. The torque unit is the newton metre, consistent with the NIST description of SI torque. Stress is reported in megapascals, a multiple of the pascal within the International System of Units.

How to calculate

1. The calculator opens with a complete steel-spring demonstration: D = 12 mm, d = 1 mm, Na = 5, E = 200 GPa, F = 10 N, and r = 5 mm. Results and a validated example XLSX are immediately available.
2. Choose a Geometry input method. Enter either Mean coil diameter (D) with Wire diameter (d), or Spring index (C) with wire diameter. Switching methods converts the current valid geometry rather than merely changing the label.
3. Enter Number of active turns (Na) and Young modulus (E). Then choose a Torque input method: Applied force (F) multiplied by Arm length (r), or Applied torque (M) directly.
4. Read Angular displacement (θ) as the primary result, then review Bending stress (σ), Spring rate (k), Inner correction (Ki), and Outer correction (Ko). The Calculation detail table shows every intermediate relationship.
5. Select Download Excel to export the current validated inputs and canonical results to a real XLSX workbook. Reset clears the demonstration and all result state; Download Excel is then disabled until a complete valid state is entered again.

Input guide

Geometry input method is required and accepts one of two listed options. In diameter mode, Mean coil diameter (D) is a required positive decimal in millimetres, for example 12, and must produce a D/d ratio from 1.01 through 100. Increasing D while other values stay fixed lowers spring rate, increases angular displacement, and changes the stress correction through C. Do not substitute outer diameter: mean diameter follows the wire centreline. In index mode, Spring index (C) is a required dimensionless decimal from 1.01 through 100, for example 12. Values near 4 – 12 are common in many practical spring contexts, although the acceptable range depends on material and manufacturing; a Virginia Tech engineering study discusses the practical significance of spring-index selection.

Wire diameter (d) is required, positive, and entered in millimetres, for example 1. Because d appears to the fourth power in rate and deflection and to the third power in stress, small changes have a large effect; entering radius instead of diameter is a serious error. Number of active turns (Na) is a required positive decimal, for example 5. More active turns reduce spring rate and increase angular displacement. Count only turns that participate elastically; end contributions depend on the actual leg geometry. Young modulus (E) is a required positive decimal in gigapascals, for example 200 for a typical steel assumption. A higher modulus makes the spring stiffer and reduces deflection. Use a value for the actual alloy and condition rather than assuming all metals share the same modulus.

Torque input method is required. In force mode, Applied force (F) is a required nonnegative decimal in newtons, for example 10, and Arm length (r) is a required nonnegative decimal in millimetres, for example 5. The calculator assumes the force acts perpendicular to the arm, so M = F × r after converting millimetres to metres. Using the full leg length when the force line is not perpendicular overstates torque. In direct mode, Applied torque (M) is a required nonnegative decimal in N·m, for example 0.05. Increasing torque raises stress and angular displacement proportionally but does not change the geometry-controlled spring rate.

Output guide

Angular displacement (θ) is the estimated elastic rotation at the applied torque, shown in degrees and radians. Zero torque produces zero displacement; a high value may signal that the assumed linear range or available travel needs review. Bending stress (σ) is the inner-fibre corrected nominal bending stress in MPa. It rises with torque and falls sharply as wire diameter increases. It is an estimate, not an allowable-stress verdict. Spring rate (k) is the exact ratio within this idealized linear model, shown in N·m/rad and N·m/turn; a higher rate means more torque is required for the same rotation.

Spring index (C) is D/d. Inner correction (Ki) and Outer correction (Ko) are dimensionless curvature factors. The inner factor is used for the reported corrected bending stress because the inner surface is more highly stressed in this model. The top summary pills repeat Spring index, Torque, Deflection, and Rate from the same canonical calculation. The Calculation detail table lists Quantity, Current value, and Relationship so the inputs, intermediate factors, and outputs can be audited without relying on a graphic.

Worked example

For the startup example, force mode gives M = 10 N × 0.005 m = 0.0500 N·m. The spring index is C = 12 mm ÷ 1 mm = 12. With Ki = 1.066288, corrected bending stress is about 543.06 MPa. Angular displacement is θ = 64 × 0.05 × 0.012 × 5 ÷ [200 × 10⁹ × (0.001)⁴] = 0.9600 rad, or 55.00°. The spring rate is k = M/θ = 0.05208 N·m/rad, equivalent to 0.32725 N·m/turn. These values match the first-open cards, table, summary pills, and workbook checkpoints.