EngineeringMechanical EngineeringPart 12 of 15
Helical Spring Design and Selection
Spring stress is independent of the number of coils. That single fact separates the two halves of spring design: stress fixes the wire and diameter, coil count fixes the rate.
- Rate and pre-load
- Wahl factor
- Active coils
- Buckling check
Executive summary
Springs are classified by the load they carry: tension, compression, torsion or bending. This page addresses helical cylindrical tension and compression springs made from round spring steel wire — by a wide margin the most common in mechanical design.
It is normally more cost-effective and faster to select a stock spring than to design one, and replacements remain readily available. Custom design is warranted only when no standard spring meets the requirement — but the design method is worth knowing regardless, because it is what tells you whether a stock spring is being used within its limits.
Rate, pre-load and working deflection
A spring deforms elastically, so Hooke's law applies and the force–deflection line is straight through the origin. Its slope is the spring rate.
- k
- spring rate or spring constant, N/mm
- F
- total force, or the change in force between two positions, N
- x
- total deflection, or the corresponding change in deflection, mm
Springs are almost always fitted with a pre-load — a force present when the working deflection is zero. An engine valve spring holds the valve shut with a pre-load force; as the valve opens, the spring deflects further and the force rises to its maximum.
Worked example: rate from two operating points
A valve spring exerts 200 N closed and 250 N open, with an 8 mm working deflection.
Spring index and allowable stress
- C
- spring index
- D
- mean coil diameter, mm — note that catalogues normally specify springs by outside diameter
- d
- wire diameter, mm
Most engineering springs have an index between about 4 and 15, rising as the spring grows.
| Size | Mean diameter D | Wire diameter d | Spring index C |
|---|---|---|---|
| Small | Below 8 mm | Below 1 mm | 4 to 8 |
| Medium | 8 to 24 mm | 1 to 4 mm | 8 to 12 |
| Large | Above 24 mm | Above 4 mm | 12 to 15 |
Allowable stress
The maximum allowable stress at design load depends on three things: the wire material, the wire diameter (smaller wire tolerates higher stress), and the service duty.
| Duty | Number of cycles | Type of load |
|---|---|---|
| Light | Below 104 | Static or gradually applied. |
| Average | 104 to 106 | Gradually applied through to light shock. |
| Heavy | Above 106 | Light through to heavy shock. |
Allowable stress curves for hard-drawn spring steel are published against wire diameter for each duty class. To provide a factor of safety, the maximum calculated stress should not exceed about 85 per cent of the value read from the curve — for both compression and extension springs.
Stress and the Wahl factor
A helical spring is stressed in torsional shear plus bending. Starting from the torsional shear formula and substituting the torque produced by the applied force acting at the mean radius gives the shear stress; the Wahl factor then corrects for the additional bending component.
- f
- combined stress in the spring, MPa
- K
- Wahl factor, always greater than 1
- F
- the applied load — not the change in load
- C
- spring index
- In these expressions F is the load itself, not the change in load. Using the working force range here is a common and serious error.
- Spring stress is independent of the number of coils. Two springs of identical wire and diameter carry the same stress at the same load regardless of coil count.
- Do not confuse K, the Wahl factor, with k, the spring rate.
Worked example
A 300 N load acts on a compression spring of 4 mm wire and 40 mm mean diameter, giving a spring index of 10.
Coils, free length and buckling
With stress settling the wire and diameter, the coil count settles the rate.
- n
- number of active coils
- G
- modulus of rigidity of the wire — usually taken as 78.6 GPa for spring steel
- d
- wire diameter, mm
- C
- spring index
- k
- required spring rate, N/mm
| Spring type | End condition | Total coils N |
|---|---|---|
| Extension | All coils active; loops or hooks formed at each end. | N = n |
| Compression | Two end coils squared and ground, or squared only for fine wire. These sit flat and do not deflect. | N = n + 2 |
Fractional coils are not manufactured in practice; round up to the next half coil. A calculated 5.23 coils becomes 5.5; a calculated 6.74 becomes 7.
Free length
- Ca
- clash allowance — the proportion by which design deflection is increased so the spring cannot go solid under load. At least 0.2; commercial springs commonly carry 0.3 to 0.4.
- x2
- total deflection from the zero-load position
An extension spring at rest has its coils wound tightly together, so the solid stack is simply N·d plus the ends. A compression spring must have space between coils to deflect — and must never be allowed to go solid, or the load path bypasses the spring entirely.
Buckling
L/D above 10
The spring will buckle under essentially any load or deflection. It must be guided — over a rod or inside a tube — or redesigned.
L/D below 10
Buckling depends on the deflection ratio. Compare maximum deflection divided by free length against the published buckling curve for the end conditions.
Design procedure and worked selection
- Establish the two operating pointsForce and position at each, giving the working deflection and the required rate.
- Determine dutyCycle count and load character, giving the service duty class.
- Trial a wire diameter and spring indexUsing the size guidance for the envelope available.
- Calculate the Wahl factor and stress at maximum loadComparing against 85 per cent of the allowable stress for that wire diameter and duty.
- Adjust wire or diameter until stress passesStress is fixed by wire, diameter and load only — coil count will not help here.
- Calculate active coils for the required rateThen add end coils and round to the next half coil.
- Calculate free length with clash allowanceConfirming it fits the installed envelope.
- Check bucklingAnd provide a guide rod or tube if required.
- Search the stock range firstIf a standard spring meets rate, maximum deflection and envelope, specify it instead of a custom part.
For a spring required to rise from 250 N to 350 N over a 10 mm lift, the rate is 10 N/mm and the maximum force 350 N. Scanning a stock range for the closest rate gives a candidate; its maximum usable deflection is then free length minus minimum recommended length, and that figure — not the free length — is what must exceed the required total deflection.
Design checklist
- Both operating points defined, and pre-load distinguished from working load.
- Required rate calculated from the change in force over the working deflection.
- Service duty class established from cycle count and load character.
- Spring index within a sensible range for the spring size.
- Wahl factor applied and stress calculated at maximum load, not at load range.
- Calculated stress within 85 per cent of the allowable value for that wire diameter and duty.
- Active coil count calculated, end coils added and total rounded to the next half coil.
- Clash allowance of at least 20 per cent included in compression spring free length.
- Buckling checked, with a guide rod or tube provided where required.
- Installed and working lengths both confirmed against the available envelope.
- Stock spring range searched before committing to a custom design.