Helical Spring

How a helical spring works

The wire in a helical spring isn't bending the way a beam bends. When you push down on a coil spring, every short segment of wire experiences a torque that twists it about its own long axis. The coil shape converts that wire-level twist into bulk axial deflection. Unwind the spring mentally into a straight bar and the same axial force becomes a torque arm of length D/2 — and the rate formula falls out of the torsion-of-a-round-bar formula scaled by the number of turns.

This is why wire diameter dominates. Torsional stiffness of a round bar scales with the polar second moment J = πd⁴/32. Drop that into the deflection-per-unit-load math and you get the canonical rate equation:

k = G·d⁴ / (8·D³·n)

where
  k = spring rate (force per unit deflection, N/mm)
  G = shear modulus of the wire material (MPa)
  d = wire diameter (mm)
  D = mean coil diameter (mm)
  n = number of active coils

Mean coil diameter D is measured to the centerline of the wire, not the outside. Active coils n excludes the closed end coils, which don't deflect. For a typical compression spring with squared-and-ground ends, n = total turns − 2.

Worked example: a typical valve spring

Take an automotive engine valve spring with reasonable numbers and run them through the formula:

Material: chrome-silicon spring wire, G = 80 GPa = 80,000 MPa = 80,000 N/mm²
Wire diameter:     d = 4.0 mm
Mean coil diam:    D = 25 mm   (so spring index C = D/d = 6.25)
Active coils:      n = 6

k = (80,000 × 4⁴) / (8 × 25³ × 6)
  = (80,000 × 256) / (8 × 15,625 × 6)
  = 20,480,000 / 750,000
  = 27.3 N/mm

That's 27 N per mm of compression. A 10 mm valve lift at the spring delivers 273 N of seat-load increase on top of installed preload — typical of a naturally aspirated street engine. Now bump wire diameter from 4.0 to 4.5 mm and rerun:

k = (80,000 × 4.5⁴) / (8 × 25³ × 6)
  = (80,000 × 410.0625) / 750,000
  ≈ 43.7 N/mm

A 12.5% increase in wire diameter gives a 60% increase in rate — exactly the d⁴ effect at work. This is the lever spring designers reach for first when they need stiffness, before ever changing coil diameter or active turns.

Real-world ranges

Application	Typical rate	Wire d	Notes
Ballpoint pen detent	0.05 to 0.2 N/mm	0.2 to 0.4 mm	Low force, billions of cycles, music wire
Mattress innerspring	0.5 to 2 N/mm	1.5 to 2.0 mm	Hourglass coil, high-carbon steel
Engine valve spring	20 to 60 N/mm	3.5 to 5.0 mm	Chrome-silicon, shot-peened, often two nested springs
Passenger car suspension	20 to 40 N/mm	11 to 14 mm	Hot-coiled silicon-chrome, painted or powder-coated
Truck suspension auxiliary	80 to 200 N/mm	16 to 22 mm	Hot-coiled, often progressive-rate (variable pitch)
Garage door torsion	~10 N·mm/° (torque rate)	5 to 8 mm	Torsion variant, rated by IPPT (inch-pounds per turn)
Industrial die spring	200 to 5,000 N/mm	up to 30 mm	Rectangular wire for higher rate per volume

Helical vs other spring types

	Helical (coil)	Leaf	Torsion bar	Gas (pneumatic)	Disc (Belleville)	Air spring (bellows)
Stress mode in element	Wire torsion	Beam bending	Bar torsion	Gas compression	Conical bending	Gas + diaphragm
Force-deflection curve	Linear	Linear, can be progressive	Linear	Highly non-linear (PV=const)	S-shaped, can be flat-top	Tunable, near-constant rate
Rate per kg of metal	High	Low	Highest	Very high (no metal in the spring)	Very high (compact)	High (most mass is air)
Damping (built-in)	None	Inter-leaf friction	None	Orifice damping	Friction between stacked discs	Adjustable via valving
Packaging	Vertical, bulky	Long, flat, doubles as locator	Slender bar, end-fixed	Cylinder + piston	Stack disc	Bellows, requires compressor
Cost (relative)	1×	0.7×	1.2×	2 to 4×	1.5×	4 to 8× (system)
Typical home	Cars, valves, pens	Trucks, trailers	SUVs, race cars	Office chairs, hatches	Bolted joints, clutches	Buses, semi-trailers

Variants: compression, extension, torsion

The same coil geometry serves three different load directions, and the differences matter for design and life.

• Compression springs are the default. Loaded along the axis with coils being pushed together. End conditions vary: plain ends (cheapest, can lean), squared (last coil flattened), squared and ground (most stable, used wherever the spring sits in a cup or pocket), and plain ground. The active coils don't include the inactive end turns, so a 6-turn squared-and-ground spring has only 4 active.
• Extension springs are stretched. They have hooks or loops at the ends, and they're typically wound with initial tension — the coils are forced together during winding, so a finite force is needed before the spring even starts deflecting. Failure usually starts at the hook bend, not the body, because the bend stress concentration can be twice the body stress.
• Torsion springs are loaded by twisting around the coil axis. They store energy by bending the wire (not twisting it as in compression). Garage doors, mousetraps, and binder-clip springs are torsion springs. The rate is M/θ = Ed⁴/(64Dn), where bending modulus E replaces shear modulus G.

Two more useful sub-variants: conical springs taper from large to small diameter so the large coils close first, giving a progressive (rising) rate; variable-pitch springs have closer-spaced coils at one end that bottom out first to do the same thing without changing diameter. Both are common in suspension and where bottom-out behavior matters.

When to use a helical spring

• Linear, predictable rate over a wide range of deflection — the load-deflection curve is straight up to about 80% of solid height.
• Compact axial packaging — high stored energy per unit volume of metal compared with leaf or bar.
• Light weight — wire is the only metal, no fixturing leaves or sliding shoes.
• High cycle count — engine valve springs survive billions of cycles when properly shot-peened.
• Cheap and standardized — catalog springs cover most rates within a factor of two with off-the-shelf parts.

Use something else when you need built-in damping (leaf, gas, or air), load-leveling (air), extreme energy density in tiny travel (Belleville), or locating function in addition to springing (leaf, where the spring eye doubles as the axle pivot).

Common failure modes and pitfalls

• Inner-fiber fatigue. When wire is coiled, the inner fiber of each coil is shorter than the outer fiber and sees ~15 to 25% higher shear stress for the same axial load. The Wahl correction factor K_w = (4C-1)/(4C-4) + 0.615/C captures this. Fatigue cracks initiate on the inside of the coil and propagate outward. Shot peening introduces compressive residual stress at the surface that roughly doubles fatigue life.
• Setting (permanent loss of free length). If a spring is compressed past the wire's elastic limit, it returns shorter than it started. This can happen on the very first cycle in service if the design stress exceeds yield, or gradually under sustained high preload (relaxation).
• Buckling. An axially loaded compression spring with slenderness ratio (free length / mean diameter) above ~4 will buckle sideways under load like a column. Solutions: shorter springs, springs in cups or sleeves, or two springs nested with opposite hand-of-wind.
• Surge. A spring has its own natural frequency from the wave equation along the wire. When a cam or impulse drives the spring at that frequency, traveling stress waves cause coil clash and accelerated fatigue. Engine valve springs are designed so their natural frequency is well above maximum cam-event frequency, often using two nested springs of different rates to spread the resonance.
• Hydrogen embrittlement. Plating or pickling spring wire without proper baking traps atomic hydrogen at grain boundaries, leading to sudden brittle fracture days or weeks after manufacture. Spec sheets always call out a post-plate bake at ~190 °C for 8+ hours.
• Hook fracture (extension springs). The bend at the hook is a stress concentrator. Catalog hooks come in several shapes (machine, side, swivel, German); side and swivel hooks have lower bend stress.

Quick design checklist

1. Pick wire material — music wire below 6 mm, chrome-silicon for valve and high-cycle, stainless for corrosion.
2. Choose spring index C in the 6 to 12 range. Below 6 you fight stress concentration; above 12 you fight buckling.
3. Use the rate formula k = Gd⁴/(8D³n) and iterate d, D, n until k matches the target. Wire d gives the biggest leverage.
4. Check shear stress at maximum load with the Wahl-corrected formula; it must stay below the wire's allowable shear stress (typically 40 to 50% of ultimate tensile for infinite life).
5. Verify free length / D ≤ 4 for unguided compression springs, or plan to guide them.
6. Confirm natural frequency is at least 13× the highest forcing frequency for high-cycle applications.
7. Specify shot peening for any spring expected to see more than 10⁴ stress cycles.