Compression Spring Calculator.

Designing a Helical Compression Spring

A compression spring is a torsion bar wound into a helix. Load the ends and the wire twists — which is why every stress in a coil spring is a shear stress, not a bending stress, and why the wire diameter appears to the fourth power in the rate and the third power in the stress.

Spring Rate

k = G · d⁴ / (8 · D³ · n_a)

Where G is the shear modulus (79,300 MPa for music wire), d is wire diameter, D is mean coil diameter (OD − d, not OD), and n_a is the number of active coils. Every term is sensitive:

• Increase wire diameter 10% → rate rises 46% (1.1⁴).
• Increase coil diameter 10% → rate falls 25% (1/1.1³).
• Add one active coil to eight → rate falls 11%.

Using OD instead of mean diameter is the single most common error in hand calculations, and it always makes the spring look softer than it is.

Spring Index and the Wahl Factor

The spring index C = D / d governs both manufacturability and stress. The wire on the inside of a coil is stressed more than the outside because of curvature and the direct shear component. The Wahl factor corrects for this:

K_W = (4C − 1) / (4C − 4) + 0.615 / C

At C = 4 the correction is 1.40 — the inside fibre carries 40% more stress than the uncorrected formula predicts. At C = 10 it drops to 1.14. Keep the index in the 4 to 12 range: below 4 the wire is hard to coil and cracks on the inside diameter; above 12 the spring tangles, buckles and is difficult to handle in automated assembly. The comfortable target is 6 to 9.

Solid Height and Available Travel

Total coils depend on the end treatment. For the common squared-and-ground end, total coils = active + 2, and the solid height is simply L_s = n_t × d. Travel to solid is L₀ − L_s.

Never design a spring to reach solid in service. Leave at least 15% of the working deflection as clash allowance — if the spring goes solid under a shock load, stress spikes without limit and the spring takes a permanent set or breaks. Check the stress at solid height, not just at working deflection; that is the number that determines whether an over-travel event destroys the spring.

Allowable Shear Stress

Spring wire tensile strength is not a fixed number — it rises sharply as wire gets thinner because of the drawing process. For music wire the relationship is approximately S_ut = 2211 × d^−0.145 MPa, so 1 mm wire reaches about 2211 MPa while 6 mm wire is nearer 1740 MPa.

Shot peening raises fatigue allowables by 20% or more and is standard practice on any spring expected to survive millions of cycles.

Buckling

A slender compression spring buckles sideways like a column. The stability limit depends on how the ends are held:

• Both ends on parallel ground plates — stable while L₀ < 5.26 × D
• One end fixed, one pivoted — stable while L₀ < 3.72 × D
• Both ends free to pivot — stable while L₀ < 2.63 × D

Past those limits the spring needs a guide rod or a bore to run in. Allow radial clearance of at least 10% of the coil diameter for the spring to grow — a compression spring's outer diameter increases as it is compressed, roughly by the amount the pitch closes up.

Surge Frequency

A spring has its own natural frequency. If the operating cycle approaches it, the coils resonate, the spring surges, and it fails quickly regardless of how comfortable the static stress looked. Keep the natural frequency at least 13 to 15 times the operating frequency. This is the criterion that sizes automotive valve springs.

Worked Example

Ø3 mm music wire, 25 mm OD, 8 active coils, 70 mm free length, squared and ground, compressed 20 mm.

• Mean diameter D = 25 − 3 = 22 mm, index C = 22/3 = 7.33 (good)
• Rate k = 79300 × 3⁴ / (8 × 22³ × 8) = 6,423,300 / 681,472 ≈ 9.43 N/mm
• Working load F = 9.43 × 20 ≈ 189 N
• Wahl K_W = (4×7.33 − 1)/(4×7.33 − 4) + 0.615/7.33 = 1.117 + 0.084 = 1.20
• Stress τ = 1.20 × 8 × 189 × 22 / (π × 27) = ≈ 471 MPa
• Solid height = 10 coils × 3 = 30 mm, travel to solid = 40 mm
• Buckling limit = 5.26 × 22 = 116 mm — 70 mm free length is safe

Related Tools

For fastener preload in the same assembly, see Bolt Torque. For structural members under load, use Beam Deflection. For interference-fitted spring seats see Press Fit, and for the material hardness spec see Hardness Conversion.