Torsional Spring Calculator - Stress, Rate & Deflection
This torsional spring calculator computes bending stress, spring rate, angular deflection, and correction factors Ki and Ko from torque and coil dimensions.
Torsional Spring Calculator
Results
What Is a Torsional Spring Calculator?
A torsional spring calculator computes the bending stress, spring rate, angular deflection, and stress correction factors of a helical torsion spring from its geometry and the applied load. Engineers, students, and machinists use this torsional spring calculator to size springs for clothespins, garage-door counterbalances, return-to-centre mechanisms, and any device that must deliver a controlled rotational force.
- • Mechanical design: Size a torsion spring for a hinge, latch, or clutch by entering the wire diameter, coil diameter, number of active turns, and expected force to check that the bending stress stays below the material's allowable limit.
- • Physics coursework: Work through a torsion spring problem step by step, computing the spring index, stress correction factors Ki and Ko, torque, angular deflection in radians and degrees, and the spring rate in N-m/rad.
- • Product prototyping: Predict how many degrees a spring will rotate under a known arm force before building a physical prototype, and adjust the wire or coil diameter to hit a target deflection.
- • Quality and inspection: Compare the measured angular deflection of a production spring against the theoretical value to confirm the wire diameter and material match the drawing.
Helical torsion springs are common in both metric and imperial designs. Enter dimensions in millimetres and newtons, and the tool converts internally to SI base units before computing stress in megapascals and deflection in both radians and degrees.
For axial springs that resist linear force rather than torque, the Spring Constant & Deflection Calculator covers the companion Hooke's-law calculation.
How the Torsional Spring Formulas Work
The torsional spring calculator chains four core formulas together: the spring index, the stress correction factors, the bending stress, and the angular deflection. Each formula feeds the next, and the spring rate falls out as the ratio of torque to angular deflection.
- D: Mean coil diameter of the spring in millimetres, equal to the inner diameter plus one wire diameter.
- d: Wire diameter in millimetres. Appears to the fourth power in deflection and to the third power in stress.
- Na: Number of active turns that deflect under load. More turns produce proportionally more angular deflection.
- E: Young's modulus of the spring material in gigapascals. Steel is roughly 200 GPa and stainless steel about 190 GPa.
- F: Force applied at the end of the spring arm in newtons.
- r: Arm length, the distance from the spring centre to the point where force is applied, in millimetres.
- M: Torque, calculated as F times r, in newton-metres.
- C: Spring index, the ratio D/d. Typical values range from 4 to 16.
- Ki: Inner stress correction factor, always larger than Ko. Equals (4C² − C − 1) / (4C(C − 1)).
- Ko: Outer stress correction factor. Equals (4C² + C − 1) / (4C(C + 1)).
- θ: Angular deflection in radians. Multiply by 180/π to convert to degrees.
- k: Spring rate in N-m/rad, a constant within the elastic range of the spring.
The spring index C controls the stress correction factors. At low C values the inner surface of the coil sees more stress than the outer surface because the wire is bent more tightly. As C increases above about 12, both Ki and Ko converge toward 1.0.
Steel torsion spring, 12 mm coil, 1 mm wire, 5 active turns
D = 12 mm, d = 1 mm, Na = 5, E = 200 GPa, F = 5 N, r = 10 mm
C = 12/1 = 12. Ki = (4×144 − 12 − 1) / (4×12×11) = 563/528 = 1.0663. M = 5 × 0.010 = 0.05 N-m. σb = 1.0663 × 32 × 0.05 / (π × 0.001³) = 543.06 MPa. θ = 64 × 0.05 × 0.012 × 5 / (200e9 × 1e-12) = 0.96 rad. k = 0.05 / 0.96 = 0.05208 N-m/rad.
Angular deflection = 0.96 rad = 55.00°. Spring rate = 0.052083 N-m/rad. Bending stress = 543.06 MPa.
A 5 N force at 10 mm produces about 55 degrees of rotation in this small steel spring, with bending stress of 543 MPa well below the typical 1800 MPa allowable for music wire.
According to the Spring Manufacturers Institute (SMI), the angular deflection of a helical torsion spring equals 64 times the torque times the mean coil diameter times the number of active turns divided by Young's modulus times the wire diameter to the fourth power, and the stress correction factor Ki on the inner surface is always larger than Ko on the outer surface.
To convert the torque computed here into rotational power at a given RPM, use the Torque, Power & Speed Calculator.
Key Concepts Explained
Four ideas underpin every torsion spring calculation: the spring index, the stress correction factors, the angular deflection formula, and the spring rate. Understanding how they interact helps you adjust a design when the first pass does not hit the target deflection or stress.
Spring index C = D/d
The spring index is the ratio of the mean coil diameter to the wire diameter. Values between 4 and 16 keep the spring manufacturable. Below 4 the wire is so tightly coiled that coiling introduces residual stress; above 16 the spring loses its shape.
Stress correction factors Ki and Ko
Bending stress is not uniform across the wire cross-section because the inner surface of the coil is more tightly curved. Ki corrects for the higher inner stress and Ko for the lower outer stress. Design uses Ki because it governs the allowable load.
Angular deflection θ = 64MDNa/(Ed⁴)
The angular deflection in radians grows linearly with torque, mean coil diameter, and active turns, and shrinks with Young's modulus and the fourth power of the wire diameter.
Spring rate k = M/θ
The spring rate is the torque per radian of angular deflection and stays constant within the elastic range. Express it in N-m/rad or multiply by 2π for N-m per revolution.
When you increase the wire diameter to raise the spring rate, the bending stress drops at the same time. Iterate on both outputs together when adjusting geometry.
For solid and hollow shafts under pure torsion, the Angle Of Twist Calculator solves the companion theta = TL/(GJ) equation.
How to Use This Calculator
Enter the spring geometry, material, and applied load into the torsional spring calculator to get bending stress, angular deflection, spring rate, and stress correction factors in one step.
- 1 Enter the mean coil diameter: Type the average diameter of the spring coil in millimetres.
- 2 Enter the wire diameter: Type the diameter of the spring wire in millimetres. Deflection scales with d to the fourth power.
- 3 Set the number of active turns: Count the coils that deflect freely under load. Clamped or ground-flat end coils do not count.
- 4 Choose a material preset or enter Young's modulus: Select a common spring material from the preset list, or pick Custom and type the modulus in GPa.
- 5 Enter the applied force and arm length: Type the force in newtons and the distance from the coil centre to the force application point in millimetres.
- 6 Read the results: Check the angular deflection in degrees and radians, the spring rate, torque, bending stress, spring index, and both stress correction factors.
Try D = 12 mm, d = 1 mm, Na = 5, E = 200 GPa, F = 5 N, r = 10 mm. The calculator should report about 55 degrees of deflection, a spring rate of 0.052 N-m/rad, and a bending stress near 543 MPa.
If the spring wire also carries transverse shear, cross-check the combined stress state with the Shear Stress Calculator.
Benefits of Using This Calculator
A dedicated torsional spring tool saves time and reduces errors compared with hand-calculation spreadsheets.
- • Full result chain in one view: Spring index, stress correction factors, torque, bending stress, angular deflection, and spring rate are displayed together.
- • Material presets for common spring alloys: Steel, stainless steel, music wire, phosphor bronze, and beryllium copper presets fill in Young's modulus.
- • Deflection in both radians and degrees: Read the angular deflection in the unit your report or course requires.
- • Stress correction factors shown explicitly: Both Ki and Ko are displayed so you can confirm the correction is applied correctly.
- • Real-time recalculation: Every input change recalculates all outputs immediately.
Use this calculator as a first-pass design tool before running a detailed fatigue analysis or building a prototype. The closed-form formulas match standard mechanical engineering references.
To estimate the rotational energy stored in the spring at a given deflection, pair this result with the Elastic Potential Energy Calculator.
Factors That Affect Your Results
Five geometry and material factors control the deflection and stress of a torsion spring, and two modelling assumptions limit how closely the closed-form result matches a real part.
Wire diameter
Wire diameter appears to the fourth power in the deflection formula and to the third power in the stress formula. Doubling the wire diameter reduces deflection by a factor of 16 and increases the torque capacity by a factor of 8.
Mean coil diameter
Larger coil diameter increases angular deflection linearly. Higher D raises the spring index C, lowering Ki toward 1.0 and reducing bending stress for the same torque.
Number of active turns
Angular deflection grows linearly with active turns. Doubling Na doubles the deflection and halves the spring rate.
Young's modulus
Stiffer materials with higher E produce less deflection. Steel at 200 GPa deflects about 5 per cent less than stainless steel at 190 GPa.
Applied force and arm length
Torque equals force times arm length. Doubling either doubles the torque, deflection, and stress.
- • The formulas assume linear elastic behaviour. If the bending stress exceeds yield strength, the spring will take a permanent set.
- • Stress concentrations at the ends where the wire exits the coil, at arm bends, and at surface defects are not captured. Fatigue life in cyclic applications depends on those local effects.
For a spring that must operate over millions of cycles, treat the closed-form stress as a nominal value and apply a fatigue factor before comparing against the endurance limit.
According to ScienceDirect helical spring reference, the bending stress in a helical torsion spring depends on a stress correction factor that accounts for the different loading on the inside and outside of the coil, and the spring index C = D/d typically ranges from 4 to 16 for manufactured springs.
To compare the torsional stiffness of this spring against a shaft or other rotational element in the same assembly, use the Rotational Stiffness Calculator.
Frequently Asked Questions
Q: What is a torsional spring?
A: A torsional spring is a coil spring that stores and releases rotational energy by resisting angular deflection. When a force acts on the spring arm at a distance from the coil centre, the spring produces a restoring torque proportional to the angular displacement, with the spring rate k equal to torque divided by angular deflection.
Q: What is the formula for torsional spring bending stress?
A: The bending stress of a helical torsion spring equals sigma equals K times 32 times M divided by pi times d cubed, where K is the stress correction factor for the inner surface of the coil, M is the applied torque, and d is the wire diameter. The correction factor Ki accounts for the higher stress on the inside of the coil compared with the outside.
Q: How do you calculate the spring rate of a torsion spring?
A: The spring rate k equals the applied torque M divided by the angular deflection theta in radians. You can also compute it from geometry as k equals E times d to the fourth power divided by 64 times D times Na, where E is Young's modulus, d is the wire diameter, D is the mean coil diameter, and Na is the number of active turns.
Q: What is the spring index of a torsion spring?
A: The spring index C is the ratio of the mean coil diameter D to the wire diameter d. A spring index between 4 and 16 is typical for manufactured torsion springs. Values below 4 make coiling difficult and raise stress correction factors, while values above 16 produce a spring that is too floppy to hold its shape.
Q: What is the difference between a torsion spring and a compression spring?
A: A compression spring resists axial force and deflects linearly along its coil axis, following Hooke's law F equals k times x. A torsion spring resists torque applied at its ends and deflects angularly, following M equals k times theta. The two spring types use different formulas and serve different mechanical functions.
Q: How does wire diameter affect torsional spring performance?
A: Wire diameter appears to the fourth power in the angular deflection formula and to the third power in the stress formula, so doubling the wire diameter reduces deflection by a factor of 16 and increases the torque capacity by a factor of 8. Small changes in wire diameter therefore produce large changes in spring rate and stress.