Helical compression springs: rate, stress and closing solid

A torsion bar wound into a coil

Press on a coil spring and it looks as if the wire bends, but mostly it twists. Cut through the wire anywhere and the force on the spring acts at the coil's radius from the cut, so every section of wire carries a torque of F·D/2, where D is the mean coil diameter. A compression spring is a torsion bar that has been wound up to fit in a small space.

Two numbers describe its shape. The wire diameter d and the mean diameter D give the spring index C = D/d, and springs are easiest to make and use with C between about 4 and 12. Below 4 the wire is too stiff to wind easily; above 12 the spring is floppy, tangles in the box and buckles easily.

Rate: the fourth power of the wire

The spring rate, the force per millimetre of deflection, is k = d⁴·G/(8·D³·N_a), where G is the shear modulus of the wire and N_a is the number of active coils. The powers are what matter. Double the wire diameter and the spring is sixteen times stiffer; double the coil diameter and it is eight times softer.

Coils add like springs in series: every active coil deflects by the same amount under the same force, so twice as many coils give twice the deflection and half the rate. Only the coils that are free to move count. The closed coils at each end sit on each other and do nothing but seat the spring.

The inside of the coil works hardest

The torque gives a shear stress of 8·F·D/(π·d³) at the surface of the wire, as it would in a straight bar. A coil is not straight, though. The fibres on the inside of each turn are shorter than those on the outside, so the same twist strains them more, and the direct shear from the force adds on that side too. The peak stress sits on the inner surface of the coil, and that is where spring fatigue cracks start.

A correction factor carries both effects: τ = K_B·8·F·D/(π·d³), with the Bergsträsser factor K_B = (4C + 2)/(4C − 3). For a spring index of 8 it is 1.17, a 17% increase; for a tightly wound spring with C = 4 it is 1.38. The Wahl factor, an older formula, gives almost the same values.

Thin wire is stronger wire

Spring wire is drawn cold through a series of dies, and the drawing lines up and hardens the steel. Thin wire has been drawn further, so it is stronger: music wire 1 mm across has an ultimate strength over 2200 MPa, while 6 mm wire of the same steel is nearer 1700 MPa. Designers use a fitted rule for each wire, S_ut = A/d^m, from spring makers' test data.

Springs are twisted, not pulled, so the limit that matters is the torsional yield strength, about 45% of the ultimate strength for cold-drawn carbon steel. Makers raise it by set removal: the spring is coiled a little long and then pressed solid, which yields the inner surface and leaves residual stresses that oppose the working stress, so the finished spring can take a higher load without taking a permanent set.

Closing solid

Keep pressing and eventually the coils touch. That is the solid height, and it depends on how the ends are finished: squared ends are closed flat against the next coil, and ground ends are also ground down so the spring stands square. For squared and ground ends the solid height is the wire diameter times the total number of coils.

In real machines springs get pressed solid, during assembly, by a jammed mechanism or by a careless hand. A spring whose wire stress at solid height is still below yield is called solid-safe; it can be closed completely and come back to its full length. Spring designers aim for a safety factor of at least 1.2 at solid, and leave at least 15% of the travel unused beyond the maximum working force, because the coils do not all close at once and the rate stops being linear near solid.

Fatigue, buckling and surge

Valve springs, clutch springs and pump springs cycle between a preload and a maximum force millions of times. In the 1950s the spring engineer F. P. Zimmerli found that for steel spring wire under 10 mm, the fatigue strength is almost the same whatever the steel and whatever its size, because failure is ruled by the surface. Shot peening, blasting the finished spring with small steel balls, compresses that surface and raises the fatigue strength by more than half.

A long, slender spring can also buckle sideways like a column. For steel springs between flat, parallel plates it is safe from buckling if its free length is less than about 5.3 times its mean diameter; a spring longer than that needs a rod through it or a bore around it. And a spring has a mass of its own, so it has a natural frequency. If the machine drives it near that frequency, waves of compression run along the coils and the spring surges, which is why engine valve springs are designed with a natural frequency many times their operating speed.

How engineers learned this

In 1676 Robert Hooke published a Latin anagram, ceiiinosssttuv, and two years later revealed its meaning: ut tensio, sic vis, as the extension, so the force. His lectures on the subject were titled Of Spring, and his interest was practical; Hooke and Christiaan Huygens both worked on balance springs that made pocket watches keep time.

The spring's curvature took longer. In the late 1920s Arthur Wahl, at the Westinghouse research laboratories, showed that the stress on the inside of a heavy coil was well above the straight-bar value and derived the correction that still carries his name. Later work by spring makers, including Zimmerli's fatigue tests, turned spring design from rules of thumb into the calculation used today.

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