Why are extension spring hooks a critical stress point?

Extension spring hooks are the most common failure point. The hook transitions from the coil body to the straight leg, creating a stress concentration where both bending and torsion stresses combine. Hook bending stress at the bend of the hook can be 1.5-2x higher than the spring body shear stress. Proper hook design (using machine hooks, cross-centre hooks, or side hooks depending on load) reduces stress concentration. Always verify hook stress against material yield strength, not just body stress.

Extension Spring Calculator

Q: 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.

Q: 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 stress in addition to the direct axial load. The bending stress correction factor Kb = (4C−1)/(4C−4) is always larger than 1.0 and larger than the body Wahl factor for the same spring index - 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/distortion under overload.

Q: 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 with inner radius D/2. High Kb (high hook stress). Very common, low cost. Half loop: only half a coil is bent to form the hook. The loop inner radius is smaller so the curvature correction is larger, but the overall geometry reduces the lever arm. Used in lighter-duty springs. Extended hook: the last coil is drawn out straight, then bent at a right angle, creating a hook that extends beyond the body. The turning radius at the elbow is larger, significantly reducing Kb. Preferred 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 axis.

Q: How do I calculate the free length of an extension spring?

Free length Lf = body length Lb + hook contribution from both ends. For machine (full) loops: each hook adds approximately D/2 to the spring length, so Lf = Lb + D. For half loops: each adds D/4, so Lf = Lb + D/2. For extended hooks: each adds roughly D, so Lf = Lb + 2D. Body length Lb = Na × d when coils are closely wound (which is the normal free state for extension springs - unlike compression springs that have a defined pitch). The installed length and extended length then are Lf + x1 and Lf + x2 respectively.

Q: How is maximum safe extension determined?

Maximum safe extension xMax is the extension at which the body shear stress reaches the allowable stress limit (stressFraction × UTS). The corresponding maximum force FAllow = τ_allow × π d³ / (8 D Kw). Since extension force F = Fi + k×x, it follows that xMax = (FAllow − Fi) / k. If xMax is less than the required working extension x2, the spring is over-stressed and the design must be changed - increase wire diameter d, decrease mean diameter D, or select a stronger material.

Q: 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 is inconsistent. Above 12, the spring body is slender, prone to lateral vibration, and manufacturing tolerances on initial tension become very wide. The sweet spot for fatigue-critical extension springs (e.g., garage door counterbalance springs, return springs on machinery) is C = 6–8, which balances body stress, hook stress, initial tension control, and manufacturing economy.

Q: What is initial tension in an extension spring?

Initial tension (Fi) is the preload built into a close-coiled extension spring during manufacturing. It is the force required to just start separating the coils before any extension occurs. Below the initial tension, the spring produces no force. Above it, force increases linearly with extension: F = Fi + k x x. Initial tension allows extension springs to be installed with minimal sag under light loads. It is caused by residual torsional stress introduced during coiling.

Spring rate, initial tension, hook stress (Kb), Wahl body stress, Goodman fatigue, max extension - SMI / IS 7906-4.

Wire Diameter d (mm)

Mean Coil Diameter D (mm)

Body Length Lb (mm) coil body only, excl. hooks

Initial Tension Level

Hook Type

Preload Extension x₁ (mm) installed, 0 if static

Working Extension x₂ (mm) max service extension

Material

📖 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 vs Compression Spring vs Torsion Spring

Property	Extension Spring	Compression Spring	Torsion Spring
Load direction	Tensile - spring is pulled apart	Compressive - spring is pushed together	Angular - spring is twisted
Pre-load	Initial tension Fi (coiling pre-stress)	None (F = 0 at free length)	None unless preset
Critical failure point	Hook bend (90%+ of all failures)	Wire body (coil clash at solid height)	Wire inner surface at leg-body junction
Design limit	Max safe extension x_max	Solid height + clash allowance	Mandrel clearance + max angle
Free-state coils	Closed (touching) - zero pitch	Open - defined pitch gap	Close-wound body
Stress correction	Kw (body) + Kb (hook)	Kw or Bergsträsser Kb	KB curvature-bending
Typical applications	Garage doors, door closers, trampoline, luggage	Valves, suspension, pens, keyboards	Hinges, latches, clothespins, clock springs

📝 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]

Extension Spring Formula Quick Reference

Parameter	Symbol	Formula	Units / Notes
Spring index	C	D / d	Target 5–9
Active coils	Na	Lb / d	Lb = body length; coils touching
Spring rate	k	G d⁴ / (8 D³ Na)	N/mm
Initial tension	Fi	(τᵢ × π × d³) / (8 × D × Kw)	N; must overcome before extending
Force at extension x	F	Fi + k × x	N
Wahl factor (body)	Kw	(4C − 1)/(4C − 4) + 0.615/C	Body shear correction
Hook bending factor	Kb	(4C − 1) / (4C − 4)	Always Kb > Kw - hook governs
Body shear stress	τ_body	8 F D Kw / (π d³)	MPa
Hook bending stress	σ_b	Kb × 32 F D / (π d³)	MPa; check vs allowable separately
Max safe extension	x_max	(F_allow − Fi) / k	mm; x₂ must be < x_max
Goodman SF (body)	SF_body	1 / (τ_alt / Sₑ + τ_mean / S_us)	≥ 1.3; check hook SF separately

Hook Type Comparison

Hook Type	Relative Kb	Hook stress	Free length add (per end)	Cost	Best for
Machine (Full) Loop	Highest	Highest	≈ D/2	Lowest	General-purpose, low cycle
Half Loop	High	High	≈ D/4	Low	Compact, light-duty
Extended Hook	Low	Low	≈ D	Medium	Fatigue-critical applications
Cross-Centre Loop	Low	Low	≈ D/2	High	Precision instruments
Side-Centre Loop	Low	Low	≈ D/2	High	Offset-axis precision mechanisms

Spring Index (C = D/d) Guide for Extension Springs

C range	Initial tension control	Hook Kb	Body stress	Manufacturing	Verdict
< 4	Very high, inconsistent	Very high >1.4	Very high	Difficult	Avoid
4 – 5	High	High 1.3–1.4	High	Marginal	Use extended hook; heavy-duty only
5 – 9	Good, consistent	Moderate 1.15–1.3	Moderate	Easy	Optimal - target this range
9 – 12	Low - loose coiling	Low 1.10–1.15	Low	Acceptable	Acceptable; watch lateral vibration
> 12	Very low, erratic	Low <1.10	Low	Tangling risk	Avoid - unstable

🔧 Spring Wire Materials - Properties and Selection Guide

Material selection is particularly critical for extension springs because the hook bending stress - which is always higher than the body shear stress - is governed by the material's allowable bending stress fraction of UTS. Lower-UTS materials produce hooks that fail earlier relative to the body. The initial tension is also material-dependent: higher-G materials allow tighter coiling and higher initial tension for the same geometry. All ten materials below follow SMI, IS 7906-4, and ASTM spring wire standards.

Material Properties Reference Table (per SMI, IS 4454 / IS 3431 / IS 6603, EN 10270, ASTM A228 / A227 / A401 / A313)

Material	G (MPa)	E (MPa)	UTS range (MPa)	Density (kg/m³)	Max temp (°C)	Body τ allow / UTS	Hook σ allow / UTS	Standards
Hard-drawn Steel	79,300	200,000	1380 – 1650	7,850	120	0.45	0.75	IS 4454, ASTM A227
Music Wire (Patented)	81,500	210,000	1650 – 2200	7,850	120	0.45	0.75	IS 4454 Gr.2, ASTM A228
Chrome-Vanadium	80,000	208,000	1550 – 1900	7,840	220	0.52	0.80	IS 3431, ASTM A401
Chrome-Silicon (SAE 9254)	80,700	207,000	1700 – 2050	7,830	250	0.52	0.80	SAE 9254, DIN 17223-2
Stainless Steel 302	68,900	193,000	1150 – 1450	7,920	260	0.35	0.60	IS 6603, ASTM A313 Gr.302
Stainless Steel 316L	68,000	193,000	1050 – 1350	7,980	315	0.32	0.56	ASTM A313 Gr.316
Stainless 17-7 PH	71,700	204,000	1450 – 1750	7,780	370	0.42	0.70	ASTM A313 Gr.631
Phosphor Bronze	41,400	103,000	700 – 1000	8,860	95	0.30	0.50	IS 7811, ASTM B197
Beryllium Copper	48,300	124,000	1000 – 1380	8,250	200	0.38	0.62	ASTM B197, CDA 172
Inconel 718	77,200	200,000	1200 – 1450	8,220	650	0.35	0.58	AMS 5596, ASTM B637

UTS values typical for 2–4 mm wire. UTS increases significantly for smaller diameters (e.g. music wire at 0.5 mm can reach 2700+ MPa). Hook σ allow / UTS = allowable bending stress fraction at hook inner radius (higher than body because bending endurance limit > torsion endurance limit). Extension springs have no solid height limit - hook stress and x_max govern design.

Material Selection Guide - Extension Springs

Hard-drawn Steel (IS 4454, ASTM A227) - Economy choice for static and low-cycle applications. Adequate for springs cycled fewer than 10⁵ times in non-corrosive environments. Hook stress governs - choose an extended hook type (lower Kb) to keep hook bending stress within the 0.75 × UTS allowable. Do not use where fatigue failure would be hazardous. Typical uses: household appliances, agricultural implements, general machinery guards. Max 120 °C.

Music Wire (IS 4454 Gr.2, ASTM A228) - Best high-cycle choice at ambient temperature. Highest UTS and finest surface finish of all plain carbon steels. For extension springs cycling above 10⁵ times - door closers, precision actuators, counterbalance mechanisms - this is the standard recommendation. High G (81,500 MPa) also enables tighter coiling and higher initial tension in compact bodies. Always pair with extended or half-loop hooks to keep hook bending stress in check at high cycle counts. Wire range 0.1–6 mm.

Chrome-Vanadium (IS 3431, ASTM A401) - Best elevated-temperature or heavy-duty choice. Highest body allowable (0.52 × UTS) and hook allowable (0.80 × UTS) of all materials in the table - critical for extension springs where hook stress is the governing limit. Use when operating temperatures reach 150–220 °C (industrial ovens, engine bay return springs) or when maximum force per unit volume is the design objective. Lower notch sensitivity than plain carbon wire improves tolerance to hook manufacturing variation.

Chrome-Silicon / SAE 9254 - Maximum stress density at elevated temperature. Highest UTS alloy steel in the table. Use when the spring envelope is tightly constrained and maximum initial tension plus working force are both needed. Also preferred for springs in exhaust-adjacent locations (up to 250 °C). Hook allowable is 0.80 × UTS - same as chrome-vanadium. Premium material; justify by showing chrome-vanadium cannot meet the stress or temperature requirement.

Stainless Steel 302 (IS 6603, ASTM A313) - Corrosion-resistant standard for most wet environments. The workhorse stainless choice for extension springs in food machinery, outdoor equipment, medical devices, and laboratory instruments. Lower G (68,900 MPa) means initial tension is lower for the same geometry - account for this when specifying preload. Hook allowable is 0.60 × UTS. Use extended hooks with 302 in fatigue-critical corrosive applications to keep hook stress well within the lower allowable.

Stainless Steel 316L (ASTM A313 Gr.316) - For chloride, marine, and chemical environments. Molybdenum addition (2–3%) prevents chloride pitting that would cause premature hook fatigue in salt spray or chemical splash. Use over 302 whenever the environment contains chlorides - marine deck equipment, coastal outdoor fittings, chemical process spring latches. Hook allowable is slightly lower (0.56 × UTS) than 302 - design accordingly. Lower strength means a longer or larger spring body; verify that the free length fits the space envelope.

Stainless 17-7 PH (ASTM A313 Gr.631) - High strength stainless for aerospace and precision applications. Precipitation hardening gives UTS up to 1750 MPa with corrosion resistance close to 302. Use for aerospace actuator return springs, precision medical instrument springs, and defence equipment where both corrosion resistance and maximum force per unit weight are mandatory. Hook allowable is 0.70 × UTS - better than standard austenitic grades. Much more expensive than 302/316L; specify only when lower-grade stainless cannot meet the stress requirement.

Phosphor Bronze (IS 7811, ASTM B197) - Non-magnetic, non-sparking, electrically conductive. The material of choice for extension springs in electronic equipment (relay return springs, connector springs), intrinsically safe environments, and saltwater without steel-corrosion tolerance. Very low G (41,400 MPa) means a much lower initial tension and spring rate for a given geometry - the spring will be significantly larger or weaker than a steel equivalent. Hook allowable is 0.50 × UTS. Never use in high-stress structural applications. Max 95 °C.

Beryllium Copper (ASTM B197, CDA 172) - Premium non-ferrous for high-stress, high-reliability service. Highest-performing copper alloy. UTS up to 1380 MPa, non-magnetic, non-sparking, excellent fatigue resistance. For extension springs in explosive environments, aerospace connectors, and high-precision medical or scientific instruments where steel cannot be tolerated. Hook allowable 0.62 × UTS. Handle with extreme care during forming - beryllium dust is a severe carcinogen. Use only when no alternative meets both the electrical/safety and the mechanical requirements.

Inconel 718 (AMS 5596, ASTM B637) - High-temperature service above 370 °C. Use for extension springs in jet engines, gas turbines, exhaust manifold retaining springs, and high-temperature process equipment where all steel alternatives would relax or fail. G comparable to steel (77,200 MPa), so initial tension and spring rate behave similarly to alloy steel at room temperature. Hook allowable is 0.58 × UTS. Note that G decreases ~15% at 650 °C - account for this in angular rate calculations at elevated service temperature. Very expensive; reserve for thermal extremes only.

Hook Type and Material Interaction

Hook bending stress correction Kb depends on hook geometry, not material. Material determines the allowable:

Machine (full) loop: Kb = (4C − 1) / (4C − 4) - highest hook stress, lowest cost
Half loop: Kb slightly lower than machine loop
Extended hook: Kb significantly lower (larger turning radius) - preferred for fatigue
Cross-centre / Side-centre: Kb similar to extended - specialist use

Material × Hook pairing rules:
Hard-drawn + machine loop → highest hook stress risk → use only for low-cycle static springs
Music wire + extended hook → lowest stress + highest UTS → optimal for high-cycle fatigue
Stainless 302/316L + extended hook → corrosion + fatigue → use when both environments and cycles matter
Chrome-vanadium + any hook → 0.80 × UTS hook allowable → most tolerant of machine loop hooks
Phosphor bronze + any hook → hook allowable only 0.50 × UTS → must keep extension modest; prefer extended hook

Rule of thumb: If hook stress / hook allowable > 0.80, switch to extended hook or higher-UTS material before any other change.

✍️ How to Use This Calculator

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.

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).

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.

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.

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).

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.

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.

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)

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)

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

Spring rate: k = (79,300 × 2⁴) / (8 × 14³ × 30) = 1,268,800 / 659,520 = 1.924 N/mm

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

Forces: F₁ = 12.8 + 1.924 × 2 = 16.6 N | F₂ = 12.8 + 1.924 × 8 = 28.2 N

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)

Hook bending stress: σ_b2 = 1.125 × (32 × 28.2 × 14) / (π × 8) = 282 MPa | Allowable = 0.75 × 1480 = 1110 MPa → PASS (25% utilisation)

Free length Lf = 60 + 14 = 74 mm | x_max ≈ 358 mm | Energy at x₂ = 107 mJ | Spring Status: ✔ SAFE - infinite life expected

Try this example →

Example 2 - Stainless 302 spring with extended hooks (fatigue-critical application)

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)

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

Spring rate: k = (68,900 × 1.6⁴) / (8 × 10³ × 18.75) = 452,071 / 15,000,000 = 0.452 N/mm

Forces: Fᵢ (low) ≈ 2.1 N | F₁ = 2.1 + 0.452 × 3 = 3.46 N | F₂ = 2.1 + 0.452 × 12 = 7.52 N

Body stress: τ_body2 = 198 MPa | Allowable = 0.35 × 1300 = 455 MPa → PASS (44%)

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

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

Try this example →

❓ 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.

🔗 Related Calculators

📌 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.