Constant-Force Springs: The Same Pull the Whole Way

The trick: re-curling flat strip instead of straining a coil

An ordinary spring obeys Hooke's law, F = k·x: pull it twice as far and it pulls back twice as hard, because you keep straining more and more of the same elastic body. A constant-force spring sidesteps this entirely. It is a strip of thin flat metal that has been pre-stressed — rolled and heat-set so its unloaded, relaxed shape is a tight coil of nearly constant radius, wound on a drum.

When you pull the free end, you are not stretching the metal lengthwise. You are taking a section of strip that is currently curved at the coil's small radius, straightening it flat as it leaves the drum, and letting it re-curl at the same radius as it winds back on. At every instant of the stroke the same geometry is happening: one short zone of strip is transitioning between flat and its natural curl. Because that transition — and the bending energy it costs — is identical whether you have pulled out 10 mm or 500 mm, the restoring force barely changes. The spring is, in effect, a constant-radius bending machine, and constant bending means constant force.

The governing equation and where the force comes from

The output load of a flat-strip (Neg'ator-type) constant-force spring is captured by a compact design relation:

F = E·b·t³ ÷ (26.4·Rn²)

where E is the material's elastic modulus (≈ 193 GPa for Type 301 stainless), b is strip width, t is strip thickness, and Rn is the spring's natural radius — the radius the strip curls to when relaxed. The constant 26.4 is an empirical factor that accounts for the cross-curvature (anticlastic curl) the strip picks up in the working zone. The formula is good to roughly ±10% for standard geometries.

The physics behind it is elastic bending. Straightening the strip from radius Rn imposes a peak surface bending stress σ = E·t ÷ (2·Rn). The energy to do that, released again as the strip re-curls, is what you feel as the pull. The dominant lever is thickness: because force scales with t³, doubling strip thickness multiplies the load by eight. That is why thickness tolerance — not width — is the parameter that determines whether a part hits or misses its rated force.

A worked example

Consider a Type 301 stainless strip: E ≈ 193 GPa, width b = 12.7 mm (0.5 in), thickness t = 0.20 mm, natural radius Rn = 8 mm.

• Force: F = (193×10⁹ × 0.0127 × (0.0002)³) ÷ (26.4 × 0.008²) ≈ 11.6 N — about 1.2 kgf, held flat across the whole draw.
• Peak bending stress: σ = E·t ÷ (2·Rn) = 193×10⁹ × 0.0002 ÷ 0.016 ≈ 2,410 MPa.

That stress looks alarmingly high, and it is exactly why constant-force springs live and die by their stress ratio. Type 301 in the full-hard, high-yield temper has an ultimate tensile strength around 1,300–1,900 MPa. A stress above yield means each cycle plastically works the same small strip zone — so real designs push Rn up (or t down) until working stress sits at ~40–60% of yield, trading force for the millions of cycles a tape measure or battery contact demands.

Design trade-offs and the failure mode that matters

The defining weakness is reverse-bending fatigue in one localized zone. Unlike a helical spring that spreads strain over its whole length, a constant-force spring flexes the same short arc of strip on every stroke. That zone is the crack initiation site, and it sets the life. Manufacturers therefore bin these springs by rated life — Lee Spring's stock ranges are 2,500 / 4,000 / 13,000 / 25,000 cycles, while carefully de-rated designs exceed 1,000,000. Getting long life costs force, because the only levers (bigger Rn, thinner t) both push the load down.

Other real trade-offs: the ratio of outside coil diameter to natural diameter is kept near D₀/Dn ≈ 1.2 and the aspect ratio near b/t ≈ 100 for well-behaved coiling; the force is only truly constant after the strip has been pulled out roughly 1.25× its coil diameter, so the first fraction of travel ramps up. And the spring resists uncoiling, so it must be mounted the correct way — the relaxed coil sits on the idler drum and the free end anchors to the load, not the reverse.

Constant torque, spring motors, and where they show up

Reverse the roles — anchor the strip's free end and let the drum rotate — and the same part becomes a constant-torque spring. Wind strip from a storage drum onto a larger output drum and you get a spring motor (Neg'ator motor) that delivers steady torque through many turns, used in retractable cords, tool balancers, and window-shade counterweights.

Real-world uses are everywhere once you look:

• Tape measures — the coil that snaps the blade home.
• Cordless power tools — the spring that presses battery terminals against contacts with steady force regardless of pack seating.
• Automotive — seat-belt retractors and window-regulator counterbalances.
• Cable/hose reels, IV-pole and monitor counterbalances, gym cable stacks, and self-retracting fall-arrest lanyards.
• Aerospace/space mechanisms where a predictable deployment force over long travel matters.

A misconception and a subtle pitfall

Misconception: that the force is perfectly constant. It is not — it typically holds to ≈ ±10%, and it deliberately dips at the very end of retraction (where the last coils sit at slightly different radius) and ramps at the start of extension. If a design needs the force flat across the first few millimetres too, you either pre-tension past the ramp zone or add active bias.

The subtle pitfall is hysteresis and set. Real strip loses a little force between the pull-out and let-back curves (friction plus microplasticity), and a spring left fully extended for a long time can take a permanent set, lowering its rated load. Designers counter this by staying well under yield, specifying the low-hysteresis Type 301 high-yield temper, and — critically — never storing the product with the spring fully drawn out. A second trap: because F ∝ t³, a strip that is even 5% under nominal thickness delivers about 15% low force, so incoming material thickness must be inspected, not assumed.

Constant-force (flat-strip) spring vs a conventional helical extension spring

Property	Constant-force spring	Helical extension spring
Force vs. extension	Nearly flat (≈ ±10%)	Linear rise, F = k·x
Force ratio, end vs start	≈ 1.1 : 1	Often 5:1 or more over travel
Useful stroke	Very long (many × its coil diameter)	Limited to ~2× free length
Energy stored per gram	Modest; single flexure zone	Higher; whole coil strained
Reverse-flex fatigue	One bend/unbend zone — the life limiter	Distributed, no single hotspot
Best at	Retraction, counterbalance, long travel	Return force, short precise stroke