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Spring Rate Calculator

The spring rate calculator can help you obtain the spring rate (spring constant) for any given spring.

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Combined rate12 N/mma single spring
Force480 N12 N/mm × 40 mm
Deflection40 mm
Rate in other units12,000 N/m68.522 lbf/in, 1.2237 kgf/mm
Energy stored9.6 J½kx² — it goes as the square of deflection, so the last millimetre stores far more than the first
Energy in the first half of the travel2.4 Ja quarter of the total, not half
Force at twice the deflection960 Nlinear in x, unlike the energy
Energy at twice the deflection38.4 Jfour times as much

The formula

F = kx; E = ½kx²

Force is linear, energy is not

Hooke's law says the force a spring exerts is proportional to how far it is deflected, so doubling the deflection doubles the force. The energy stored, however, is ½kx² and so goes as the square: doubling the deflection stores four times the energy. The consequence is that most of a spring's energy lives in the last part of its travel — the first half of the deflection holds only a quarter of the total.

Combining springs works opposite to the way most people guess. Side by side, springs share the load and each deflects the same amount, so the rates add and the assembly is stiffer. End to end, each carries the full load and they all stretch, so the assembly is softer — two identical springs in series give half the rate of one.

Hooke's law holds only up to the elastic limit. Beyond it a spring takes a permanent set and no longer returns to its free length, at which point none of these figures apply.