A helical spring is a spiral wound wire with a constant coil diameter and uniform pitch.   The most common form of helical spring is the compression spring but tension springs are also widely used. .   Helical springs are generally made from round wire... it is comparatively rare for springs to be made from square or rectangular sections.  The strength of the steel used is one of the most important criteria to consider in designing springs.  Most helical springs are mass produced by specialists organisations.  It is not recommended that springs are made specifically for applications if off-the-shelf springs can be obtained to the job.

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

The spring index (C) for helical springs in a measure of coil curvature ..

For most helical springs C is between 3 and 12

Generally springs are designed to have a deflection proportional to the applied load (or torque -for torsion springs).   The "Spring Rate" is the Load per unit deflection.... Rate (N/mm) = F(N) / δ _e(deflection=mm)

For General purpose springs a maximum stress value of 40% of the steel tensile stress may be used. However the stress levels are related to the duty and material condition (ref to relevant Code/standard). Reference Webpage Spring Materials

a)   Stress

A typical compression spring is shown below

Consider a compression spring under an axial force F.   If a section through a single wire is taken it can be seen that, to maintain equilibrium of forces, the wire is transmits a pure shear load F and also to a torque of Fr.  

The stress in the wire due to the applied load =

This equation is simplified by using a traverse shear distribution factor K _d = (C+0,5)/C.... The above equation now becomes.

The curvature of the helical spring actually results in higher shear stresses on the inner surfaces of the spring than indicated by the formula above.  A curvature correction factor has been determined ( attributed to A.M.Wahl). This (Wahl) factor K _w is shown as follows.

This factor includes the traverse shear distribution factor K _d.. The formula for maximum shear stress now becomes.

A table relating K_W to C is provided below

The spring axial deflection is obtained as follows.

The force deflection relationship is most conventiently obtained using Castigliano's theorem. Which is stated as ... When forces act on elastic systems subject to small displacements, the displacement corresponding to any force collinear with the force is equal to the partial derivative to the total strain energy with respect to that force.

For the helical spring the strain energy includes that due to shear and that due to torsion.
Referring to notes on strain energy Strain Energy

Replacing T= FD/2, l = πDn, A = πd² /4 The formula becomes.

Using Castiglianos theorem to find the total strain energy....

Substituting the spring index C for D/d The formula becomes....

In practice the term (1 + 0,5/C²) which approximates to 1 can be ignored

c)  Spring Rate

The spring rate = Axial Force /Axial deflection

In practice the term (C² /(C² + 0,5)) which approximates to 1 can be ignored

The figure below shows various end designs with different handing.   Each end design can be associated with any end design.  The plain ends are not desirable for springs which are highly loaded or for precise duties.

The table below shows some equations affected by the end designs...

Note: The results from these equations is not necessarily integers and the equations are not accurate.   The springmaking process involves a degree of variation...

The formulae provided for the compression springs generally also apply to extension springs.

An important design consideration for helical extensions springs is the shape of the ends which transfers the load to the the spring body.  These must be designed to transfer the load with minimum local stress concentration values caused by sharp bends.   The figures below show some end designs.. The third design C) design has relatively low stress concentration factors.

An Extension spring is sometimes tightly wound such that it is prestressed with an initial stress τ _i . This results in the spring having a property of an initial tension which must be exceeded before any deflection can take place.   When the load exceeds the initial tension the spring behaves according the the formulae above.  This relationship is illustrated in the figure below

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The initial tension load can be calculated from the formula.... T _i = π τ _i d ³/ ( 8 D)

Best range of of Initial Stress (τ _i) for a spring related to the Spring Index C = (D/d)

If the coils in a tension spring are not tightly wound, there is no initial tension and the relevant equations are identical to those for the spring under compression as identified above.

The equations for tension springs with initial tension are provided below

Spring Rate and Stress

These are helical springs with coils progressively change in diameter to give increasing stiffness with increasing load.  This type of spring has the advantage that its compressed height can be relatively small.  A major user of conical springs is the upholstery industry for beds and settees.

Allowable Force on Spring...
F_a = allowable force (N)..τ = allowable shear stress (N/m²)