Torsion Spring Calculator

Angular spring rate, KB bending stress, coil diameter change, Goodman fatigue, leg geometry — SMI / IS 7906-3.

🌀 Torsion Spring Calculator

📖 What is a Torsion Spring Calculator?

A torsion spring calculator applies the standard helical torsion spring equations — as codified by the Spring Manufacturers Institute (SMI), the Indian Standard IS 7906 Part 3, and EN 13906-3 — to determine the complete mechanical performance of a helical torsion spring from its geometry and material. Engineers use it during spring design to verify angular spring rate, bending stress, coil diameter change under load, leg geometry contributions, fatigue life, and mandrel clearance before ordering or manufacturing.

Torsion springs differ fundamentally from compression and extension springs in three critical respects. First, they store energy in angular deflection — twist, not axial movement. The output is a torque (N·mm or N·m), not a force. Second, the primary stress is bending stress, not shear stress. Young's modulus E governs spring rate (not shear modulus G), and the curvature correction uses the KB factor for curved beams in bending (not the Wahl shear factor Kw). Third, deflection causes the coil diameter to change — for a spring wound tighter under load (the standard orientation), mean coil diameter decreases and body length increases as the spring deflects. All three phenomena are handled by this calculator.

The KB correction factor is defined as KB = (4C² − C − 1) / (4C(C − 1)) where C = D/d is the spring index. For C = 8, KB = 1.106. This factor corrects for the stress concentration at the inner surface of the curved wire, where bending stress peaks. Without KB, springs are systematically under-designed and inner-surface fatigue cracks develop at stress levels far below the predicted value.

The active coil count Na includes a contribution from both leg lengths. Per SMI, each straight leg deflects like a cantilever beam and contributes Na_leg = L / (3πD) coils of angular compliance. For legs up to one coil diameter long this is small (~0.1 coil); for long legs (L = 3D) the contribution reaches ~0.32 coil per leg. This calculator adds leg contributions automatically for accurate spring rate and deflection predictions.

Ten spring materials are available — from hard-drawn steel wire and music wire through chrome-vanadium, chrome-silicon, two grades of stainless steel, 17-7 PH precipitation-hardened stainless, phosphor bronze, beryllium copper, and Inconel 718 for high-temperature service. For each material, Young's modulus E, shear modulus G, density, and allowable bending stress fraction of UTS (per SMI) are applied automatically from validated material data tables.

Four interactive uPlot charts visualise the complete design: torque vs angle (showing preload and working operating points), bending stress vs angle (with KB-corrected stress and allowable limit), the modified Goodman fatigue diagram, and coil mean diameter vs angle (showing the loaded ID and OD at any deflection). All charts are expandable for detailed inspection.

This calculator is designed for preliminary engineering design and educational use. For safety-critical or high-cycle applications — automotive interior mechanisms, medical devices, aerospace latches — always validate results against the applicable design code and engage a qualified mechanical engineer.

📝 Torsion Spring Formulas

Spring Index:
C = D / d    Valid design range: 4 ≤ C ≤ 12

KB Curvature-Bending Correction Factor (SMI / IS 7906-3):
KB = (4C² − C − 1) / (4C × (C − 1))
[Corrects inner-surface bending stress for wire curvature; always KB > 1]

Active Coils (body + leg contribution):
Na = N_body + (L₁ + L₂) / (3 × π × D)
[L₁, L₂ = straight leg lengths; SMI 3.1.4 leg angular compliance correction]

Angular Spring Rate:
k_θ = (E × d⁴) / (10.8 × D × Na)    [N·mm / radian — SMI standard form]
[E = Young's modulus (MPa); factor 10.8 is the SMI-standard denominator in N·mm/rad]
Display rate: k_θ_deg = k_θ_rad × (π / 180)    [N·mm / degree]

Torque at Angular Deflection θ:
T = k_θ_deg × θ_deg = k_θ_rad × θ_rad    [N·mm]

Bending Stress (KB corrected):
σ = KB × (32 × T) / (π × d³)    [MPa; T in N·mm, d in mm]
Allowable: σ_allow = stressFraction × UTS (0.68–0.78 for steel, per SMI)

Coil Mean Diameter Under Load (winding-tighter direction):
D_loaded(θ) = D × N_body / (N_body + θ / 360)
[D decreases as θ increases; total body turns increase]

Loaded Inner Diameter and Outer Diameter:
ID_loaded = D_loaded − d    OD_loaded = D_loaded + d
[ID_loaded must remain > mandrel diameter; OD_loaded must remain < housing bore]

Free Body Length (closely-wound):
Lf = N_body × d    [mm; coils packed solid, no pitch gap in free state]
Loaded body length: L_body(θ) = (N_body + θ / 360) × d

Wire Mass:
m = ρ × (π / 4) × d² × (π × D × Na) × 10⁻⁶    [kg; ρ in kg/m³, d and D in mm]

Energy Stored:
W = 0.5 × k_θ_rad × (θ₂_rad² − θ₁_rad²)    [N·mm = mJ]

Modified Goodman Fatigue Safety Factor:
σ_mean = (σ₂ + σ₁) / 2    σ_alt = (σ₂ − σ₁) / 2
SF = 1 / (σ_alt / S_e + σ_mean / S_ut)
S_e ≈ 0.56 × UTS (bending endurance limit for steel wire)    S_ut = UTS
[Goodman for bending; note: S_e fraction higher than torsion springs due to bending mode]

✍️ How to Use This Calculator

1
Enter wire diameter d and mean coil diameter D. Mean diameter D = OD − d = ID + d. The spring index C = D/d should be 4–12; aim for 6–9 for best balance of stress, fatigue life, and manufacturing.
2
Enter body coils N — the number of helical turns in the coil body (not counting legs). Select winding direction — right-hand winds tighter when the torque is clockwise (as viewed from the leg-1 end); left-hand winds tighter with counter-clockwise torque.
3
Enter leg lengths L₁ and L₂ — the straight length from the last body coil tangent point to the load application point on each leg. These are added as angular compliance per SMI equation Na_eff = N_body + (L₁ + L₂) / (3πD).
4
Enter preload angle θ₁ (installed angular position from free angle; use 0 if spring is unloaded at installation) and working angle θ₂ (maximum service deflection). Both are measured from the free (unloaded) spring position.
5
Enter the mandrel diameter if the spring operates over a mandrel or arbor. The calculator will warn if the loaded inner diameter ID_loaded(θ₂) is less than the mandrel diameter (coil-on-mandrel contact). Enter 0 to skip this check.
6
Select the material. Young's modulus E, density, UTS, and allowable bending stress fraction are applied automatically from SMI and IS 7906 material tables.
7
Click Calculate Torsion Spring. Check bending stress PASS/FAIL, fatigue safety factor (aim > 1.3), and any mandrel clearance or spring index warnings.
8
Review the four charts: Torque vs Angle, Bending Stress vs Angle, Goodman Fatigue Diagram, and Coil Diameter vs Angle (to verify OD/ID clearance throughout the deflection range).

📄 Example Calculations

Example 1 — Music wire precision instrument torsion spring

1
Inputs: d = 1.2 mm, D = 8 mm, N_body = 6 turns, RH winding, L₁ = L₂ = 8 mm, θ₁ = 3°, θ₂ = 10°, no mandrel, Music Wire (E = 210,000 MPa, UTS = 1960 MPa)
2
Spring index and KB: C = 8 / 1.2 = 6.67 ✓  |  KB = (4 × 44.49 − 6.67 − 1) / (4 × 6.67 × 5.67) = 170.28 / 151.26 = 1.126
3
Active coils: Na_leg = (8 + 8) / (3 × π × 8) = 16 / 75.40 = 0.212  |  Na = 6 + 0.212 = 6.212
4
Angular spring rate: k_θ_rad = (210,000 × 1.2⁴) / (10.8 × 8 × 6.212) = 435,456 / 536.7 = 811.3 N·mm/rad  |  k_θ_deg = 811.3 × π/180 = 14.16 N·mm/°
5
Torques: T₁ = 14.16 × 3 = 42.5 N·mm  |  T₂ = 14.16 × 10 = 141.6 N·mm
6
Bending stresses: σ₁ = 1.126 × (32 × 42.5) / (π × 1.728) = 281 MPa  |  σ₂ = 1.126 × (32 × 141.6) / (π × 1.728) = 937 MPa  |  Allowable = 0.78 × 1960 = 1529 MPa → PASS (61.3% utilisation)
7
Goodman fatigue: σ_mean = (937 + 281) / 2 = 609 MPa  |  σ_alt = (937 − 281) / 2 = 328 MPa  |  S_e = 0.56 × 1960 = 1098 MPa  |  SF = 1 / (328/1098 + 609/1960) = 1 / (0.299 + 0.311) = SF 1.64 ✓ Good fatigue life
8
Coil diameter at θ₂: D_loaded = 8 × 6 / (6 + 10/360) = 48 / 6.028 = 7.963 mm  |  ID_loaded = 6.763 mm  |  OD_loaded = 9.163 mm
Free body length = 6 × 1.2 = 7.2 mm  |  Loaded body length = 6.028 × 1.2 = 7.23 mm  |  Energy = 0.5 × 811.3 × ((10 × π/180)² − (3 × π/180)²) = 11.3 mJ

Example 2 — Chrome-vanadium automotive latch spring (high-cycle fatigue, with mandrel)

1
Inputs: d = 1.6 mm, D = 10 mm, N_body = 5 turns, RH winding, L₁ = L₂ = 10 mm, θ₁ = 3°, θ₂ = 8°, mandrel = 7.5 mm, Chrome-vanadium (E = 208,000 MPa, UTS = 1720 MPa)
2
Spring index and KB: C = 10 / 1.6 = 6.25 ✓  |  KB = (156.25 − 6.25 − 1) / (4 × 6.25 × 5.25) = 149.0 / 131.25 = 1.135
3
Active coils: Na_leg = (10 + 10) / (3 × π × 10) = 20 / 94.25 = 0.212  |  Na = 5 + 0.212 = 5.212
4
Angular spring rate: k_θ_rad = (208,000 × 1.6⁴) / (10.8 × 10 × 5.212) = 1,363,149 / 562.9 = 2,421 N·mm/rad  |  k_θ_deg = 2421 × π/180 = 42.26 N·mm/°
5
Torques: T₁ = 42.26 × 3 = 126.8 N·mm  |  T₂ = 42.26 × 8 = 338.1 N·mm
6
Bending stresses: σ₁ = 1.135 × (32 × 126.8) / (π × 4.096) = 357 MPa  |  σ₂ = 1.135 × (32 × 338.1) / (π × 4.096) = 952 MPa  |  Allowable = 0.75 × 1720 = 1290 MPa → PASS (73.8% utilisation)
7
Goodman fatigue: σ_mean = (952 + 357) / 2 = 654.5 MPa  |  σ_alt = (952 − 357) / 2 = 297.5 MPa  |  S_e = 0.56 × 1720 = 963.2 MPa  |  SF = 1 / (297.5/963.2 + 654.5/1720) = 1 / (0.309 + 0.381) = SF 1.45 ✓ Acceptable for high-cycle use
8
Mandrel check at θ₂: D_loaded = 10 × 5 / (5 + 8/360) = 50 / 5.022 = 9.956 mm  |  ID_loaded = 9.956 − 1.6 = 8.356 mm  |  Mandrel = 7.5 mm → Clearance 0.856 mm ✓ OK
Free body length = 5 × 1.6 = 8.0 mm  |  Loaded body length at θ₂ = 5.022 × 1.6 = 8.04 mm  |  Energy = 0.5 × 2421 × ((8π/180)² − (3π/180)²) = 18.6 mJ

❓ Frequently Asked Questions

What is the KB correction factor and why is it used instead of the Wahl factor? +
In a torsion spring the wire is loaded primarily in bending, not torsion. The KB factor (also called the stress-correction factor for curved beams in bending) accounts for the stress concentration due to wire curvature on the inner face of the coil. KB = (4C² − C − 1) / (4C(C − 1)) where C = D/d is the spring index. For a typical C = 8, KB ≈ 1.11, meaning actual bending stress is 11% higher than the simple beam formula gives. The Wahl factor Kw applies to helical springs in torsion (compression and extension); using Kw for torsion springs gives incorrect and unconservative stress values. Always use KB for torsion spring stress calculations.
Why does the coil diameter change under load? +
When a torsion spring deflects, each coil rotates and the helix geometry must accommodate the rotation. For a spring loaded to wind tighter (the normal design orientation where deflection adds turns), the mean coil diameter decreases according to D_loaded = D × N_body / (N_body + θ/360). This diameter reduction can be significant — for a spring with 6 body coils deflecting 90°, D_loaded = D × 6/6.25 = 0.960 × D, a 4% reduction. The loaded inner diameter ID_loaded = D_loaded − d must remain greater than any mandrel diameter or the coils will drag on the mandrel, causing friction, wear, and loss of spring rate accuracy. The designer must check mandrel clearance at maximum working angle.
How is angular spring rate calculated and what affects it? +
Angular spring rate k_θ = E × d⁴ / (10.8 × D × Na) in N·mm per degree. Young's modulus E is used (not shear modulus G) because torsion spring wire is in bending. Spring rate increases with d to the fourth power — doubling wire diameter increases rate 16×. Rate decreases with D (larger coil = more lever arm, more angular compliance) and with Na (more active coils = more twist per unit torque). Leg length contributes to Na_effective: Na_eff = N_body + (L₁ + L₂) / (3πD). For precision spring rate, leg contribution must be included, especially when legs are long relative to D.
What is the difference between right-hand and left-hand winding? +
Winding direction determines which rotation direction winds the spring tighter (increasing deflection angle closes the coils) versus which direction opens them (uncoiling). A right-hand wound spring winds tighter when rotated clockwise as viewed from the leg-1 end. For a torsion spring to work correctly, it must always be loaded in the winding-tighter direction — this keeps residual stress from the coiling process aligned with the operating stress, which gives better fatigue life. Loading in the winding-open direction reverses the residual stress relationship and dramatically reduces fatigue life. Always confirm winding direction matches the application's required rotation.
How do I include leg geometry in the spring rate calculation? +
Each straight leg deflects angularly like a cantilever beam under the applied torque, contributing to the total angular deflection. SMI defines the leg angular compliance contribution as Na_leg = (L₁ + L₂) / (3πD), where L₁ and L₂ are the straight leg lengths from the last coil tangent point to the load application point. For short legs (L < 0.5D) the contribution is small (less than 0.1 coil each). For long legs (L = 3D), each leg contributes approximately 0.32 coil. The effective active coils Na = N_body + Na_leg governs both spring rate and deflection calculations. Ignoring legs over-predicts spring rate and under-predicts angular deflection at a given torque.
What bending stress level is acceptable for fatigue-critical torsion springs? +
For a torsion spring cycling between preload angle θ₁ and working angle θ₂, the mean and alternating bending stresses are σ_mean = (σ₂ + σ₁)/2 and σ_alt = (σ₂ − σ₁)/2. The modified Goodman safety factor SF = 1 / (σ_alt / S_e + σ_mean / S_ut) where S_e ≈ 0.56 × UTS for steel wire in bending and S_ut = UTS. For general use, SF > 1.3 is acceptable. For high-cycle applications (automotive latches, appliance mechanisms, precision instruments) targeting more than 10⁷ cycles, SMI recommends SF ≥ 1.5–2.0. Shot-peening the wire (which introduces compressive residual stress at the surface) can approximately double the effective fatigue life by suppressing crack initiation at the inner-surface stress concentration.

📌 Quick Tips

💡Spring index C = D/d should be 4–12. Torsion springs are more forgiving than compression springs at low C but coiling becomes difficult below 4 and stress concentration rises sharply.
💡Torsion springs wind tighter under load — the coil mean diameter decreases and the body length increases. Always verify that the working coil OD does not contact a housing bore.
💡Unlike compression and extension springs, the primary stress in a torsion spring is bending (not torsional shear). Use the KB (Wahl curvature-bending) correction factor, not Kw.
💡For fatigue applications, always apply the preload (θ₁) first and then cycle to working angle (θ₂). Residual coiling stress and operating stress must be in the same direction (stress-relieving preload reduces fatigue life in torsion springs).
💡Leg geometry matters: a longer moment-arm leg reduces the load required for a given torque but increases the leg bending stress at the body-to-leg bend. Keep leg length to 1–3 × D for most applications.

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📌 Quick Tips

💡Spring index C = D/d should be 4–12. Torsion springs are more forgiving than compression springs at low C but coiling becomes difficult below 4 and stress concentration rises sharply.
💡Torsion springs wind tighter under load — the coil mean diameter decreases and the body length increases. Always verify that the working coil OD does not contact a housing bore.
💡Unlike compression and extension springs, the primary stress in a torsion spring is bending (not torsional shear). Use the KB (Wahl curvature-bending) correction factor, not Kw.
💡For fatigue applications, always apply the preload (θ₁) first and then cycle to working angle (θ₂). Residual coiling stress and operating stress must be in the same direction (stress-relieving preload reduces fatigue life in torsion springs).
💡Leg geometry matters: a longer moment-arm leg reduces the load required for a given torque but increases the leg bending stress at the body-to-leg bend. Keep leg length to 1–3 × D for most applications.