Extension Spring Calculator
Spring rate, initial tension, hook stress (Kb), Wahl body stress, Goodman fatigue, max extension — SMI / IS 7906-4.
📖 What is an Extension Spring Calculator?
An extension spring calculator applies the standard helical tension spring equations — as codified by the Spring Manufacturers Institute (SMI), IS 7906 Part 4, and EN 13906-2 — to determine the complete mechanical performance of a helical extension spring from its geometry, hook type, and material. Engineers use it during spring design to verify spring rate, initial tension, hook bending stress, body Wahl shear stress, fatigue safety, and maximum safe extension before ordering or manufacturing.
Extension springs differ fundamentally from compression springs in three key ways. First, they operate in tension: the two end hooks transfer load into the spring body. Second, they carry initial tension — a built-in pre-stress from the coiling process that must be overcome before the spring begins to extend. Third, the hook geometry is the primary failure site: the sharp curvature at the hook bend creates bending stresses that far exceed the body torsional stress at the same load level. This calculator addresses all three phenomena explicitly.
The Wahl correction factor Kw corrects the body torsional stress for wire curvature and direct shear, exactly as in compression spring design. The hook bending correction factor Kb is a separate, higher correction that accounts for the even tighter curvature at the hook inner radius. SMI data consistently shows that the hook bend is the critical stress location — over 90% of extension spring fatigue failures initiate at the inner surface of the hook.
The calculator supports five hook types: machine (full) loop, half loop, extended hook, cross-centre loop, and side-centre loop. Each has a different stress correction factor. Extended hooks have the lowest Kb and are preferred for high-cycle fatigue applications. Machine loops are the most common and economical. The hook type also affects free length — the calculator computes the correct free length including hook contributions for each hook geometry.
Ten materials are covered: hard-drawn steel, music wire, chrome-vanadium, chrome-silicon, stainless steel 302 and 316L, 17-7 PH precipitation-hardened stainless, phosphor bronze, beryllium copper, and Inconel 718. For each material, shear modulus G, Young's modulus E, density, allowable body stress fraction of UTS (per SMI), allowable hook bending stress fraction of UTS, and fatigue endurance limit fraction of UTS are applied automatically from validated SMI and IS 7906 data tables.
Four interactive uPlot charts visualise the design: force vs extension (showing the initial tension intercept and F₁/F₂ operating points), body stress vs extension (with Wahl-corrected shear and allowable limit), the modified Goodman fatigue diagram, and hook bending stress vs extension (with Kb correction and allowable bending limit). All charts are expandable to full-screen for detailed inspection.
This calculator is designed for preliminary engineering design and education. For safety-critical applications — garage door counterbalance springs, door closure mechanisms, actuator return springs, high-cycle machinery — always validate with the applicable design code and consult a qualified mechanical engineer.
📝 Extension Spring Formulas
Spring Index:
C = D / d Valid design range: 4 ≤ C ≤ 12
Wahl Correction Factor (body torsion):
Kw = (4C − 1) / (4C − 4) + 0.615 / C
Hook Bending Stress Correction Factor:
Kb = (4C − 1) / (4C − 4) [evaluated at hook curvature radius; Kb > Kw always]
Hook Torsion Stress Correction Factor:
Kt = (4C + 2) / (4C + 4)
Active Coils (closely-wound body):
Na = Lb / d (Lb = body length; coils are touching in free state)
Spring Rate:
k = (G × d⁴) / (8 × D³ × Na) [N/mm; G in MPa, d and D in mm]
Initial Torsional Pre-stress (SMI empirical chart fit):
τᵢ_mid = 990 / C^1.1 [MPa; empirical fit to SMI Fig. 5-1 mid-band, converted from psi]
τᵢ_low = 0.60 × τᵢ_mid τᵢ_mid = 1.00 × τᵢ_mid τᵢ_high = 1.40 × τᵢ_mid
Typical range: C=5 → ~98–228 MPa; C=7 → ~70–162 MPa; C=10 → ~47–111 MPa
Initial Tension Force:
Fᵢ = (τᵢ × π × d³) / (8 × D × Kw) [N]
Force at Extension x:
F = Fᵢ + k × x (spring only extends when F > Fᵢ)
Body Shear Stress (Wahl corrected):
τ_body = (8 × F × D) / (π × d³) × Kw [MPa]
Allowable: τ_allow = stressFraction × UTS (0.30–0.52 depending on material)
Hook Bending Stress (SMI 2.4.3):
σ_b = Kb × (32 × F × D) / (π × d³) [MPa]
Allowable: σ_allow = hookStressFraction × UTS (0.56–0.80 depending on material)
Hook Torsion Stress:
τ_hook = Kt × (16 × F × D) / (π × d³) [MPa]
Free Length (Machine Loop hooks):
Lf = Lb + D (each machine loop adds D/2 per end)
Length at Extension x:
L(x) = Lf + x
Maximum Safe Extension:
F_allow = (τ_allow × π × d³) / (8 × D × Kw)
x_max = (F_allow − Fᵢ) / k
Energy Stored:
W = 0.5 × k × (x₂² − x₁²) + Fᵢ × (x₂ − x₁) [N·mm = mJ]
Modified Goodman Fatigue Safety Factor (body):
τ_mean = (τ_body2 + τ_body1) / 2 τ_alt = (τ_body2 − τ_body1) / 2
SF_body = 1 / (τ_alt / Sₑ + τ_mean / S_us)
Sₑ ≈ 0.40 × UTS (torsional endurance limit) S_us ≈ 0.65 × UTS
Hook Fatigue Safety Factor (bending Goodman):
σ_mean = (σ_b2 + σ_b1) / 2 σ_alt = (σ_b2 − σ_b1) / 2
Sₑ_bend ≈ Sₑ / 0.577 (bending endurance converted from torsional via von Mises)
SF_hook = 1 / (σ_alt / Sₑ_bend + σ_mean / σ_allow_hook)
Note: Hook fatigue typically governs — extension springs fail at hooks 90%+ of the time.
Cycle Life Estimation (S-N power law — SAE HS-1 / Shigley):
N = (A / τ_alt)^(1/b) where A ≈ 0.9 × UTS, b ≈ 0.11
Stress-ratio shortcut: τ_alt / UTS < 0.30 → infinite life; 0.30–0.45 → ~1M cycles;
0.45–0.60 → ~100k; 0.60–0.75 → ~10k; > 0.75 → < 1k cycles
Same S-N applied independently to hook bending stress for hook life estimate.
Safe Operating Frequency (surge prevention):
f_operating ≤ fn / 20 (operating frequency must stay well below natural frequency)
Exceeding fn / 13 causes resonance (surge) and rapid fatigue failure.
Natural Frequency:
fn = (d / (2π × D² × Na)) × √(G / (2ρ)) × 1000 [Hz; G in MPa, ρ in kg/m³]
C = D / d Valid design range: 4 ≤ C ≤ 12
Wahl Correction Factor (body torsion):
Kw = (4C − 1) / (4C − 4) + 0.615 / C
Hook Bending Stress Correction Factor:
Kb = (4C − 1) / (4C − 4) [evaluated at hook curvature radius; Kb > Kw always]
Hook Torsion Stress Correction Factor:
Kt = (4C + 2) / (4C + 4)
Active Coils (closely-wound body):
Na = Lb / d (Lb = body length; coils are touching in free state)
Spring Rate:
k = (G × d⁴) / (8 × D³ × Na) [N/mm; G in MPa, d and D in mm]
Initial Torsional Pre-stress (SMI empirical chart fit):
τᵢ_mid = 990 / C^1.1 [MPa; empirical fit to SMI Fig. 5-1 mid-band, converted from psi]
τᵢ_low = 0.60 × τᵢ_mid τᵢ_mid = 1.00 × τᵢ_mid τᵢ_high = 1.40 × τᵢ_mid
Typical range: C=5 → ~98–228 MPa; C=7 → ~70–162 MPa; C=10 → ~47–111 MPa
Initial Tension Force:
Fᵢ = (τᵢ × π × d³) / (8 × D × Kw) [N]
Force at Extension x:
F = Fᵢ + k × x (spring only extends when F > Fᵢ)
Body Shear Stress (Wahl corrected):
τ_body = (8 × F × D) / (π × d³) × Kw [MPa]
Allowable: τ_allow = stressFraction × UTS (0.30–0.52 depending on material)
Hook Bending Stress (SMI 2.4.3):
σ_b = Kb × (32 × F × D) / (π × d³) [MPa]
Allowable: σ_allow = hookStressFraction × UTS (0.56–0.80 depending on material)
Hook Torsion Stress:
τ_hook = Kt × (16 × F × D) / (π × d³) [MPa]
Free Length (Machine Loop hooks):
Lf = Lb + D (each machine loop adds D/2 per end)
Length at Extension x:
L(x) = Lf + x
Maximum Safe Extension:
F_allow = (τ_allow × π × d³) / (8 × D × Kw)
x_max = (F_allow − Fᵢ) / k
Energy Stored:
W = 0.5 × k × (x₂² − x₁²) + Fᵢ × (x₂ − x₁) [N·mm = mJ]
Modified Goodman Fatigue Safety Factor (body):
τ_mean = (τ_body2 + τ_body1) / 2 τ_alt = (τ_body2 − τ_body1) / 2
SF_body = 1 / (τ_alt / Sₑ + τ_mean / S_us)
Sₑ ≈ 0.40 × UTS (torsional endurance limit) S_us ≈ 0.65 × UTS
Hook Fatigue Safety Factor (bending Goodman):
σ_mean = (σ_b2 + σ_b1) / 2 σ_alt = (σ_b2 − σ_b1) / 2
Sₑ_bend ≈ Sₑ / 0.577 (bending endurance converted from torsional via von Mises)
SF_hook = 1 / (σ_alt / Sₑ_bend + σ_mean / σ_allow_hook)
Note: Hook fatigue typically governs — extension springs fail at hooks 90%+ of the time.
Cycle Life Estimation (S-N power law — SAE HS-1 / Shigley):
N = (A / τ_alt)^(1/b) where A ≈ 0.9 × UTS, b ≈ 0.11
Stress-ratio shortcut: τ_alt / UTS < 0.30 → infinite life; 0.30–0.45 → ~1M cycles;
0.45–0.60 → ~100k; 0.60–0.75 → ~10k; > 0.75 → < 1k cycles
Same S-N applied independently to hook bending stress for hook life estimate.
Safe Operating Frequency (surge prevention):
f_operating ≤ fn / 20 (operating frequency must stay well below natural frequency)
Exceeding fn / 13 causes resonance (surge) and rapid fatigue failure.
Natural Frequency:
fn = (d / (2π × D² × Na)) × √(G / (2ρ)) × 1000 [Hz; G in MPa, ρ in kg/m³]
✍️ How to Use This Calculator
1
Enter wire diameter d (cross-section of wire) and mean coil diameter D (average of OD and ID: D = OD − d = ID + d). The spring index C = D/d will be computed — aim for 5–9.
2
Enter the body length Lb — the length of the coil body only, excluding hooks. Active coils Na = Lb / d (coils touch in the free state for extension springs).
3
Select the initial tension level. Low = lightly wound (small threshold force); Medium = typical production spring; High = tightly wound (large threshold force). This sets the pre-stress τᵢ per SMI data.
4
Select the hook type. Machine loop is the most common and affordable. Extended hook gives the lowest hook bending stress and is preferred for fatigue-critical applications. Hook type also affects free length Lf.
5
Enter preload extension x₁ (installed stretch from free length; use 0 if spring starts at rest) and working extension x₂ (maximum service extension from free length).
6
Select the material. Shear modulus G, Young's modulus E, density, allowable stresses, and fatigue properties are applied automatically from SMI / IS 7906 material tables.
7
Click Calculate Extension Spring. Check both body stress PASS/FAIL and hook bending stress PASS/FAIL separately — hook stress is often the governing limit. Verify fatigue SF > 1.3 and working extension x₂ < x_max.
8
Review the four charts: Force vs Extension (shows initial tension intercept), Body Stress vs Extension, Goodman Fatigue Diagram, and Hook Bending Stress vs Extension.
📄 Example Calculations
Example 1 — General-purpose hard-drawn steel extension spring (machine loop, default inputs)
1
Inputs: d = 2.0 mm, D = 14 mm, Lb = 60 mm, Low initial tension, Machine loop hooks, x₁ = 2 mm, x₂ = 8 mm, Hard-drawn steel (G = 79,300 MPa, UTS = 1480 MPa)
2
Spring geometry: C = 14 / 2 = 7.0 ✓ | Na = 60 / 2 = 30 coils | Kw = (27/24) + 0.615/7 = 1.213 | Kb = (27/24) = 1.125
3
Spring rate: k = (79,300 × 2⁴) / (8 × 14³ × 30) = 1,268,800 / 659,520 = 1.924 N/mm
4
Initial tension (SMI empirical): τᵢ_mid = 990 / 7^1.1 = 990 / 8.50 = 116.4 MPa | Low → τᵢ = 0.60 × 116.4 = 69.8 MPa | Fᵢ = (69.8 × π × 8) / (8 × 14 × 1.213) = 12.8 N
5
Forces: F₁ = 12.8 + 1.924 × 2 = 16.6 N | F₂ = 12.8 + 1.924 × 8 = 28.2 N
6
Body stress at x₂: τ_body = (8 × 28.2 × 14) / (π × 8) × 1.213 = 152 MPa | Allowable = 0.45 × 1480 = 666 MPa → PASS (23% utilisation)
7
Hook bending stress: σ_b2 = 1.125 × (32 × 28.2 × 14) / (π × 8) = 282 MPa | Allowable = 0.75 × 1480 = 1110 MPa → PASS (25% utilisation)
8
Fatigue — body: τ_body1 = 127 MPa | τ_mean = 140 MPa | τ_alt = 13 MPa | Sₑ = 0.40 × 1480 = 592 MPa | SF_body = 1/(13/592 + 140/962) = 5.8 ✓ | τ_alt/UTS = 0.9% < 30% → Infinite body life
9
Fatigue — hook: σ_b1 = 1.125 × (32 × 16.6 × 14)/(π × 8) = 166 MPa | σ_mean = (282+166)/2 = 224 MPa | σ_alt = 58 MPa | Sₑ_bend = 592/0.577 = 1026 MPa | SF_hook = 1/(58/1026 + 224/1110) = 3.0 ✓ | σ_alt/UTS = 3.9% < 30% → Infinite hook life
Free length Lf = 60 + 14 = 74 mm | x_max ≈ 358 mm | Energy at x₂ = 107 mJ | Spring Status: ✔ SAFE — infinite life expected
Example 2 — Stainless 302 spring with extended hooks (fatigue-critical application)
1
Inputs: d = 1.6 mm, D = 10 mm, Lb = 30 mm, Low initial tension, Extended hooks, x₁ = 3 mm, x₂ = 12 mm, Stainless 302 (G = 68,900 MPa, UTS = 1300 MPa)
2
Spring geometry: C = 10 / 1.6 = 6.25 ✓ | Na = 30 / 1.6 = 18.75 coils | Kw = (24/21) + 0.615/6.25 = 1.241 | Kb = (24/21) = 1.143
3
Spring rate: k = (68,900 × 1.6⁴) / (8 × 10³ × 18.75) = 452,071 / 15,000,000 = 0.452 N/mm
4
Forces: Fᵢ (low) ≈ 2.1 N | F₁ = 2.1 + 0.452 × 3 = 3.46 N | F₂ = 2.1 + 0.452 × 12 = 7.52 N
5
Body stress: τ_body2 = 198 MPa | Allowable = 0.35 × 1300 = 455 MPa → PASS (44%)
6
Hook bending (extended hook — lower Kb): σ_b = 346 MPa | Allowable = 0.60 × 1300 = 780 MPa → PASS (44%) — extended hook significantly reduces hook stress vs machine loop
7
Goodman fatigue: τ_mean = (198 + 91) / 2 = 144.5 MPa | τ_alt = 53.5 MPa | Sₑ = 0.35 × 1300 = 455 MPa | S_us = 0.60 × 1300 = 780 MPa | SF = 1 / (0.118 + 0.185) = 3.30 — Excellent fatigue life
Free length Lf = 30 + 20 = 50 mm (extended hooks add D each end) | x_max = 212 mm | Energy at x₂ = 50 mJ
❓ Frequently Asked Questions
What is initial tension in an extension spring and how is it controlled?
Initial tension is a pre-stress built into the spring during the coiling process. The coiling machine winds the wire tighter than the natural pitch, creating a compressive fit between adjacent coils. This means a threshold force Fi must be applied before the spring begins to extend. Fi is set by the winding tightness and is controlled by adjusting coiler speed and pitch tooling. SMI provides charts relating initial torsional stress (τi) to spring index C. For most designs, τi = 0.01–0.045 × G. Initial tension is not present in compression springs and must always be included in extension spring force calculations.
Why is hook stress the critical failure point in extension springs?
The hook is where the wire transitions from the helical body into the end loop. At this bend, the wire experiences combined bending stress and torsional shear in addition to the direct axial load. The bending stress correction factor Kb = (4C−1)/(4C−4) is always larger than 1.0 and typically larger than the body Wahl factor for the same C — meaning the hook bend is always the highest-stressed location in the spring. SMI data shows that 90%+ of extension spring failures originate at the hook, either by fatigue cracking from the inner surface of the hook bend or by straightening under sustained high load.
What is the difference between machine loop, half loop, and extended hook?
Machine (full) loop: the hook is formed by bending the last full coil into a circular loop. High Kb (highest hook stress). Very common, low cost. Half loop: only half a coil is bent to form the hook. Slightly different curvature but still high hook stress. Extended hook: the last coil is drawn out straight then bent at a right angle, creating an elbow with a larger turning radius. This significantly reduces Kb and hook bending stress — the preferred choice for fatigue-critical applications. Cross-centre and side-centre loops are special forms used in precision instruments where the hook axis intersects or is offset from the spring centreline.
How do I calculate the free length of an extension spring?
Free length Lf = body length Lb + hook contributions from both ends. For machine (full) loops: each hook adds approximately D/2, so Lf = Lb + D. For half loops: each adds D/4, so Lf = Lb + D/2. For extended hooks: each adds approximately D, so Lf = Lb + 2D. Body length Lb = Na × d when coils are closely wound — the normal free state for extension springs. Once Lf is known, the installed length is Lf + x₁ and the fully extended working length is Lf + x₂.
How is maximum safe extension determined for an extension spring?
Maximum safe extension x_max is the extension at which the body shear stress reaches the material's allowable stress limit (stressFraction × UTS). The corresponding maximum force is F_allow = τ_allow × π d³ / (8 D Kw). Since the total spring force is F = Fi + k × x, it follows that x_max = (F_allow − Fi) / k. Unlike compression springs, there is no solid height — the spring can in theory extend indefinitely until it yields. If your required working extension x₂ exceeds x_max, increase wire diameter d, decrease mean coil diameter D, or choose a stronger material.
What spring index should I target for extension springs?
Extension springs should target C = D/d in the range 5–9. Below 4, initial tension is very high and coiling consistency is poor. Above 12, the spring body is slender and prone to lateral vibration (surging), and manufacturing tolerances on initial tension become very wide. For fatigue-critical extension springs — garage door counterbalance springs, return springs, door closure mechanisms — the sweet spot is C = 6–8, which balances body stress, hook stress, initial tension control, and manufacturing economy. The hook type should be selected to keep hook bending stress within the allowable limit regardless of C.
How is extension spring cycle life estimated?
Cycle life is estimated using the S-N power law from SAE HS-1 and Shigley's Mechanical Engineering Design: N = (A / τ_alt)^(1/b), where A ≈ 0.9 × UTS and b ≈ 0.11 for steel spring wire. The alternating torsional stress τ_alt = (τ_max − τ_min) / 2 drives fatigue; the mean stress τ_mean is accounted for via the modified Goodman line. A practical shortcut used widely in spring manufacturing: if τ_alt / UTS < 30%, life is effectively infinite (> 10⁶ cycles); at 45% of UTS expect ~1M cycles; at 60% expect ~100k cycles; at 75% only ~10k cycles remain. Critically, hook cycle life must be evaluated separately from body life — hook bending stress is higher and hook fatigue typically governs, often giving 5–10× fewer cycles than the body at the same load. Always check both body SF and hook SF, and choose a hook type that reduces the hook stress concentration factor Kb.
What is the safe operating frequency for an extension spring?
The operating frequency of a spring-driven mechanism must stay well below the spring's natural frequency fn to avoid resonance (surge). The accepted engineering rule is: operating frequency ≤ fn / 20. At fn / 13 or higher, resonance begins — the spring develops standing waves, coils clash violently, and fatigue life collapses dramatically. The natural frequency fn = (d / (2π D² Na)) × √(G / 2ρ), where d and D are in metres and G is in Pa. For extension springs clamped at one end, effective fn is doubled. In practice, if your mechanism cycles faster than fn / 20, you must either shorten the spring (reduce Na), increase wire diameter d, or reduce coil diameter D — all of which raise fn. Surge is a common cause of unexpected premature fatigue failure in high-speed extension spring applications.
📌 Quick Tips
💡Extension spring index C = D/d should be 4–12. Tight coiling (C < 4) causes excessive initial tension and coiling stress during manufacture.
💡Hook failure is the most common mode of extension spring failure. Always verify hook bending stress (σ_b) — it is often the governing limit, not body shear stress.
💡Initial tension is the pre-stress wound into the spring during coiling. It gives the spring a threshold force below which the spring does not extend — important for maintaining contact in mechanisms.
💡Unlike compression springs, extension springs have no solid height limit. The design constraint is the allowable stress at full extension. Always check maximum safe extension before specifying travel.
💡For dynamic applications, choose a hook type that minimises stress concentration. Machine (full) loops have the highest Kb; extended hooks and side-centre loops have significantly lower hook stress.