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For the springs in this discussion, Hooke's Law is typically assumed to hold,

We can expand the spring constant k as a function of the material properties of the spring. Doing so and solving for the spring displacement gives,

where G is the material shear modulus, na is the number of active coils, and D and d are defined in the drawing. The number of active coils is equal to the total number of coils nt minus the number of end coils n* that do not help carry the load,

The value for n* depends on the ends of the spring. See the following illustration for different n* values:

Geometrical Factors
The spring index, C, can be used to express the deflection,

The useful range for C is about 4 to 12, with an optimum value of approximately 9. The wire diameter, d, should conform to a standard size if at all possible.

The active wire length La can also be used to form an expression for the deflection,

Shear Stress in the Spring
The maximum shear stress tmax in a helical spring occurs on the inner face of the spring coils and is equal to,

where W is the Wahl Correction Factor which accounts for shear stress resulting from spring curvature,