Torsion Spring Calculator
Angular spring rate, KB bending stress, coil diameter change, Goodman fatigue, leg geometry - SMI / IS 7906-3.
📖 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 vs Compression Spring vs Extension Spring
| Property | Torsion Spring | Compression Spring | Extension Spring |
|---|---|---|---|
| Output | Torque (N·mm or N·m) | Force (N) | Force (N) |
| Input | Angular deflection (degrees) | Linear deflection (mm) | Linear extension (mm) |
| Rate formula uses | Young's modulus E | Shear modulus G | Shear modulus G |
| Primary wire stress | Bending stress (σ) | Torsional shear stress (τ) | Torsional shear + hook bending |
| Stress correction factor | KB (curvature-bending) | Kw or Bergsträsser Kb | Kw (body) + Kb (hook) |
| Geometry change under load | Coil diameter decreases - check mandrel | Length decreases - check solid height | Length increases - check x_max |
| Endurance limit fraction | S_e ≈ 0.56 × UTS (bending) | S_e ≈ 0.40 × UTS (torsion) | S_e ≈ 0.40 × UTS (torsion) |
| Typical applications | Hinges, latches, clothespins, clock springs | Valves, suspension, keyboards, pens | Garage doors, door closers, trampolines |
📝 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]
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]
Torsion Spring Formula Quick Reference
| Parameter | Symbol | Formula | Units / Notes |
|---|---|---|---|
| Spring index | C | D / d | Target 4–12; optimal 6–9 |
| KB correction factor | KB | (4C² − C − 1) / (4C × (C − 1)) | Inner-surface bending correction |
| Effective active coils | Na | N_body + (L₁ + L₂) / (3π D) | Includes leg compliance |
| Angular rate (radians) | k_θ_rad | E d⁴ / (10.8 D Na) | N·mm/rad |
| Angular rate (degrees) | k_θ_deg | k_θ_rad × π / 180 | N·mm/° |
| Torque at angle θ | T | k_θ_deg × θ_deg | N·mm |
| Bending stress (KB corrected) | σ | KB × 32 T / (π d³) | MPa |
| Coil diameter under load | D_loaded | D × N_body / (N_body + θ/360) | mm; ID_loaded must > mandrel |
| Body length under load | L_body | (N_body + θ/360) × d | mm; check for interference |
| Goodman safety factor | SF | 1 / (σ_alt / S_e + σ_mean / UTS) | ≥ 1.3; S_e ≈ 0.56 × UTS |
KB Factor vs Spring Index - Torsion Spring
| Spring Index C | KB factor | Stress increase vs uncorrected | Manufacturing difficulty | Diameter change per 90° |
|---|---|---|---|---|
| 4 | 1.38 | +38% | Difficult | Large (tight coil) |
| 5 | 1.28 | +28% | Marginal | Moderate |
| 6 | 1.20 | +20% | Standard | Moderate |
| 8 | 1.11 | +11% | Easy | Small |
| 10 | 1.07 | +7% | Easy | Small |
| 12 | 1.06 | +6% | Easy; buckling risk rises | Very small |
🔧 Spring Wire Materials - Properties and Selection Guide
Torsion spring material selection differs from compression and extension springs in one key respect: the spring rate is governed by Young's modulus E (not shear modulus G) because the wire is in bending. A higher E material gives a stiffer spring for the same geometry. The allowable stress criterion is also different - the bending endurance limit (≈ 0.56 × UTS for steel, higher than the torsional endurance limit) applies to Goodman assessment. All ten materials below follow SMI, IS 7906-3, and ASTM / AMS standards.
Material Properties Reference Table (per SMI, IS 4454 / IS 3431 / IS 6603, EN 10270, ASTM A228 / A227 / A401 / A313)
E is listed first because it is the modulus that governs torsion spring rate (k_θ = E d⁴ / (10.8 D Na)). Bend allow / UTS = maximum allowable bending stress / UTS for static loading. S_e / UTS = bending endurance limit fraction used in modified Goodman assessment. UTS values are typical for 2–4 mm wire; finer wire has higher UTS. E and G are at room temperature; both decrease at elevated service temperature.
| Material | E (MPa) governs rate | G (MPa) | UTS range (MPa) | Density (kg/m³) | Max temp (°C) | Bend allow / UTS | S_e / UTS (bending) | Standards |
|---|---|---|---|---|---|---|---|---|
| Hard-drawn Steel | 200,000 | 79,300 | 1380 – 1650 | 7,850 | 120 | 0.68 | 0.50 | IS 4454, ASTM A227 |
| Music Wire (Patented) | 210,000 | 81,500 | 1650 – 2200 | 7,850 | 120 | 0.78 | 0.56 | IS 4454 Gr.2, ASTM A228 |
| Chrome-Vanadium | 208,000 | 80,000 | 1550 – 1900 | 7,840 | 220 | 0.75 | 0.56 | IS 3431, ASTM A401 |
| Chrome-Silicon (SAE 9254) | 207,000 | 80,700 | 1700 – 2050 | 7,830 | 250 | 0.75 | 0.56 | SAE 9254, DIN 17223-2 |
| Stainless Steel 302 | 193,000 | 68,900 | 1150 – 1450 | 7,920 | 260 | 0.68 | 0.50 | IS 6603, ASTM A313 Gr.302 |
| Stainless Steel 316L | 193,000 | 68,000 | 1050 – 1350 | 7,980 | 315 | 0.68 | 0.48 | ASTM A313 Gr.316 |
| Stainless 17-7 PH | 204,000 | 71,700 | 1450 – 1750 | 7,780 | 370 | 0.72 | 0.54 | ASTM A313 Gr.631 |
| Phosphor Bronze | 103,000 | 41,400 | 700 – 1000 | 8,860 | 95 | 0.56 | 0.40 | IS 7811, ASTM B197 |
| Beryllium Copper | 124,000 | 48,300 | 1000 – 1380 | 8,250 | 200 | 0.62 | 0.46 | ASTM B197, CDA 172 |
| Inconel 718 | 200,000 | 77,200 | 1200 – 1450 | 8,220 | 650 | 0.68 | 0.50 | AMS 5596, ASTM B637 |
E is listed first because it is the modulus that governs torsion spring rate (k_θ = E d⁴ / (10.8 D Na)). Bend allow / UTS = maximum allowable bending stress / UTS for static loading. S_e / UTS = bending endurance limit fraction used in modified Goodman assessment. UTS values are typical for 2–4 mm wire; finer wire has higher UTS. E and G are at room temperature; both decrease at elevated service temperature.
Material Selection Guide - Torsion Springs
1
Hard-drawn Steel (IS 4454, ASTM A227) - Economy choice for static and low-cycle torsion. Lowest cost. Use for torsion springs that actuate infrequently - door-latch return springs, furniture fold mechanisms, cable retractors. E = 200,000 MPa gives a moderate angular rate. Bending allowable is 0.68 × UTS. Surface quality limits fatigue life - do not use for applications cycling above 10⁵ times or in corrosive environments. Max 120 °C.
2
Music Wire (IS 4454 Gr.2, ASTM A228) - Best ambient-temperature torsion spring for high cycle life. Highest E (210,000 MPa) of any wire in the table - gives the stiffest spring per unit geometry, making it ideal for compact, high-torque torsion springs. Bending allowable 0.78 × UTS and S_e = 0.56 × UTS provide excellent Goodman safety factors. Standard choice for precision instrument springs, mouse traps, clock springs, and any torsion application cycling above 10⁵ cycles at ambient temperature. Wire range 0.1–6 mm.
3
Chrome-Vanadium (IS 3431, ASTM A401) - Elevated temperature and industrial-duty torsion springs. E = 208,000 MPa (close to music wire), with service to 220 °C. Lower notch sensitivity than plain carbon steel makes it more tolerant of surface damage at the critical leg-to-body junction - the stress concentration at the leg bend is analogous to the hook in extension springs, and notch insensitivity is valuable here. Use for automotive interior torsion springs (door hinges, sun visor), oven door springs, and any spring requiring rated torque above 120 °C. Most common Indian engineering choice per IS 3431.
4
Chrome-Silicon / SAE 9254 - Highest-torque alloy steel torsion spring. Maximum UTS (up to 2050 MPa in fine wire) and service to 250 °C. Use when the torsion spring must deliver the maximum torque possible from the smallest cross-section - compact automotive latches, heavy-duty industrial pivots, high-temperature actuation mechanisms. E = 207,000 MPa is nearly identical to chrome-vanadium, so spring rate is similar. Bending allowable 0.75 × UTS. Premium material; use only when chrome-vanadium cannot meet the torque or temperature requirement.
5
Stainless Steel 302 (IS 6603, ASTM A313) - Standard corrosion-resistant torsion spring. E = 193,000 MPa - approximately 8% lower than alloy steel. This means the same geometry gives 8% less angular stiffness than a chrome-vanadium spring; compensate by slightly reducing coil diameter or active coil count. Bending allowable 0.68 × UTS. Use for torsion springs in food machinery, laboratory equipment, medical devices, and outdoor electrical enclosures. The most cost-effective stainless choice for general corrosion-resistant applications.
6
Stainless Steel 316L (ASTM A313 Gr.316) - Chloride and chemical-resistant torsion springs. E and G identical to 302, with the critical addition of molybdenum for chloride-pitting resistance. Use over 302 wherever chloride exposure is possible - coastal outdoor structures, marine hardware, pharmaceutical processing equipment, chemical plant actuation. Slightly lower UTS and S_e than 302; design with conservative stress utilisation. For torsion springs in harsh outdoor or sea-spray environments where a standard stainless would pit and fatigue within months, 316L is the minimum acceptable grade.
7
Stainless 17-7 PH (ASTM A313 Gr.631) - High-performance stainless for aerospace and precision instruments. E = 204,000 MPa - close to alloy steel - giving near-steel stiffness with stainless corrosion resistance. UTS up to 1750 MPa. Use for torsion springs in aerospace mechanisms, navigation instruments, satellite deployment springs, and high-precision medical devices where low mass, high torque, and corrosion resistance must coexist. Bending allowable 0.72 × UTS. Service to 370 °C. Significantly more expensive than austenitic grades - specify only when 302/316L cannot meet the torque or temperature requirement.
8
Phosphor Bronze (IS 7811, ASTM B197) - Non-magnetic, electrically conductive torsion springs. E = 103,000 MPa - only half of steel - so a phosphor bronze torsion spring will be approximately half as stiff as the same geometry in steel. Either the coil diameter must be reduced or the active coil count decreased to match a steel spring rate. Bending allowable 0.56 × UTS. Use for relay and switch return springs requiring non-magnetic material, saltwater-resistant connector springs, and intrinsically safe environments. Max 95 °C. Cannot be used as a direct steel substitute without re-dimensioning.
9
Beryllium Copper (ASTM B197, CDA 172) - Best non-ferrous torsion spring material for demanding applications. E = 124,000 MPa - higher than phosphor bronze but still about 60% of steel. Highest UTS of any copper alloy (up to 1380 MPa). Non-magnetic, non-sparking, excellent fatigue life for a non-ferrous material. Use for torsion springs in explosive environments, precision analytical instruments, high-reliability aerospace connectors, and military equipment where both non-magnetic and high-torque requirements coexist. Handle with care - beryllium fumes are carcinogenic. Max 200 °C. Expensive; justify by requirement, not preference.
10
Inconel 718 (AMS 5596, ASTM B637) - Torsion springs for extreme temperature service (above 370 °C). E = 200,000 MPa at room temperature (comparable to alloy steel), decreasing to approximately 170,000 MPa at 650 °C - account for this 15% rate reduction at maximum service temperature when verifying torque accuracy. Retains useful creep resistance and spring rate at temperatures that would cause all steel torsion springs to set and relax permanently. Use for jet engine actuation springs, gas turbine compressor variable-vane springs, exhaust-system pivot springs, and nuclear instrumentation. Very expensive; specify only where thermal extremes exclude all steel options.
Torsion Spring: Material Impact on Key Parameters
How material choice changes the design - at identical geometry (same d, D, Na):
Angular spring rate k_θ ∝ E: Music wire (E=210,000) gives 5% higher rate than chrome-vanadium (E=208,000), 9% higher than 302 stainless (E=193,000), 70% higher than phosphor bronze (E=103,000).
Bending stress - identical for same torque and geometry regardless of material (σ = KB × 32T / πd³). Material affects only the ALLOWABLE, not the actual stress.
Coil diameter change under load - D_loaded = D × N_body / (N_body + θ/360) - independent of material. Mandrel clearance check is geometry-only.
Body length increase - L_loaded = (N_body + θ/360) × d - d is the only material-linked variable. Denser packing (smaller d in non-ferrous) shortens the body but requires more coils for the same rate.
Winding direction and material residual stress: All spring wire has residual coiling stress from manufacturing. For steel wire, residual stress is favourable (compressive on the outer surface) when loaded in the winding-tighter direction - the standard design orientation. Loading in the winding-open direction reverses this, dramatically reducing fatigue life for ALL materials. This is a geometry constraint, not a material choice, but choosing a material with high S_e / UTS (music wire, chrome-vanadium) provides a wider safety margin in the Goodman diagram regardless of residual stress state.
Shot peening: Applicable to steel and stainless torsion springs. Induces compressive residual stress on the inner surface of the coil (highest-stressed location). Effectively increases the Goodman fatigue life by approximately 50–100% for cyclic applications above 10⁶ cycles. Not economically viable for non-ferrous materials; use a higher-allowable material instead.
Angular spring rate k_θ ∝ E: Music wire (E=210,000) gives 5% higher rate than chrome-vanadium (E=208,000), 9% higher than 302 stainless (E=193,000), 70% higher than phosphor bronze (E=103,000).
Bending stress - identical for same torque and geometry regardless of material (σ = KB × 32T / πd³). Material affects only the ALLOWABLE, not the actual stress.
Coil diameter change under load - D_loaded = D × N_body / (N_body + θ/360) - independent of material. Mandrel clearance check is geometry-only.
Body length increase - L_loaded = (N_body + θ/360) × d - d is the only material-linked variable. Denser packing (smaller d in non-ferrous) shortens the body but requires more coils for the same rate.
Winding direction and material residual stress: All spring wire has residual coiling stress from manufacturing. For steel wire, residual stress is favourable (compressive on the outer surface) when loaded in the winding-tighter direction - the standard design orientation. Loading in the winding-open direction reverses this, dramatically reducing fatigue life for ALL materials. This is a geometry constraint, not a material choice, but choosing a material with high S_e / UTS (music wire, chrome-vanadium) provides a wider safety margin in the Goodman diagram regardless of residual stress state.
Shot peening: Applicable to steel and stainless torsion springs. Induces compressive residual stress on the inner surface of the coil (highest-stressed location). Effectively increases the Goodman fatigue life by approximately 50–100% for cyclic applications above 10⁶ cycles. Not economically viable for non-ferrous materials; use a higher-allowable material instead.
Material Quick-Select Table - Torsion Springs
| Material | E (MPa) | UTS range (MPa) | Bend allow / UTS | Max temp (°C) | Rate vs steel | Choose when |
|---|---|---|---|---|---|---|
| Hard-drawn Steel | 200,000 | 1380–1650 | 0.68 | 120 | Baseline | Low-cycle, low-cost static springs |
| Music Wire | 210,000 | 1650–2200 | 0.78 | 120 | +5% stiffer | High-cycle ambient - best default |
| Chrome-Vanadium | 208,000 | 1550–1900 | 0.75 | 220 | +4% stiffer | Elevated temp, automotive, heavy duty |
| Chrome-Silicon | 207,000 | 1700–2050 | 0.75 | 250 | +4% stiffer | Max torque density, exhaust adjacency |
| Stainless 302 | 193,000 | 1150–1450 | 0.68 | 260 | −4% softer | Corrosion resistance (non-chloride) |
| Stainless 316L | 193,000 | 1050–1350 | 0.68 | 315 | −4% softer | Marine / chloride environments |
| Stainless 17-7 PH | 204,000 | 1450–1750 | 0.72 | 370 | +2% stiffer | Aerospace, high-strength + corrosion |
| Phosphor Bronze | 103,000 | 700–1000 | 0.56 | 95 | −48% softer | Non-magnetic, non-sparking, relay springs |
| Beryllium Copper | 124,000 | 1000–1380 | 0.62 | 200 | −38% softer | Explosive environment, precision instruments |
| Inconel 718 | 200,000 | 1200–1450 | 0.68 | 650 | Baseline (−15% at 650°C) | Extreme temperature (>370°C), jet engines |
✍️ 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
Try this example →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
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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
Try this example →❓ 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.
What is the difference between a torsion spring and an extension spring?
A torsion spring stores energy by twisting and exerts a torque (rotational force). It is used in garage doors (counterbalance), clothespins, and hinges. An extension spring stores energy by stretching and exerts a linear force. Both obey Hooke's law (force/torque proportional to deformation), but the spring constant units differ: N/mm (linear) vs N-mm/degree or N-mm/rad (angular).
How does the Wahl correction factor affect torsion spring stress?
The Wahl correction factor (Kb) accounts for the curvature effect in coiled springs: stress is higher on the inner coil surface than simple bending theory predicts. Kb = (4c - 1)/(4c - 4) + 0.615/c, where c = D/d (spring index). For tight coils (c = 4), Kb is approximately 1.38, meaning actual stress is 38% higher than uncorrected bending stress. Always use the corrected stress for fatigue life calculations.