(constant and rate)

Spring coefficient is variously named; spring constant, spring rate, stiffness, etc. but all of these essentially mean the same thing. The amount of deflection (in any direction) one can expect from a given load.

This property is important for all structural and mechanical elements as it will identify expected deflections under design loads that may need to be modified, which can be achieved by altering the element’s sectional or material properties (e.g. I or E respectively) or a combination of both.

The spring coefficient is also very useful for identifying the natural frequency of an element

(see Mode Shapes, Vortex Shedding and Vibration Damping).

Fig 1. Spring Rate

This calculator is not limited to plane (simple) sections, if the material (E) from which a structure is constructed is constant throughout, and its sectional properties (I, J, A, etc.) are known and consistent, this calculator could be used to determine the spring coefficient of, for example, a bridge, a crane boom or a fabricated beam.

The simplest way to understand this property (spring coefficient) is to imagine a spring (Fig 1) compressed (or stretched) by an axial load (F) that produces a deflection (y).

A unit length (1.0 m, in, etc.) of this deflection would be generated by the force; F÷y, in other words a load of F÷y would produce a deflection in the spring of 1.0m (or 1.0in).

This load (F÷y) is the spring coefficient of the helical spring concerned.

The same property can be established for *any* fully elastic structure, i.e. one that returns to the same pre-loaded condition after releasing the applied load (F).

In order for this property to be regarded as a coefficient both force and deflection must be linearly related, i.e. double the load and the deflection doubles, treble the load and the deflection trebles, etc.

Exactly the same argument applies to a beam, shaft, column, etc., in that a lateral load (F) applied a given distance from one end will induce a deflection commensurate with its magnitude, and providing the deflection is small relative to its length (<5%), it will vary linearly with the load.

If you have more than one spring acting together you will need to compound them, however, the way you do this will depend upon their arrangement.

They may be arranged in ...

**Series**; each one in-front of another

or

**Parallel**; each one along-side another

As this property is similar in concept and behaviour as stiffness they can be added together in similar manner

You may use any units you like, but you must be consistent.

Metric and Imperial examples are provided in its help pages for guidance.

The output units for all spring coefficients are in the form of force per unit length (N/m, lbf/in, etc.), except for a torsional shaft;

The spring coefficient of a shaft under torsion is expressed in 'N.m' ...

This is because it is providing a spring coefficient of torque per degree or radian of twist (N.m/degree or radian) and comes from the following relationship:

T/J = G.θ/L

from which the spring coefficient (T/θ) is derived as follows:

T/θ = G.J/L (i.e. N.m {/degree or radian})

**Input Data:**

E (Young’s modulus): the elastic modulus of the material which must be the same throughout the cross-section of any element for the spring coefficient (k) to be accurate

I (second moment of area): the sectional property for resistance to bending which must be the same throughout the element’s length for the spring coefficient (k) to be accurate

J (polar moment of inertia): the sectional property for resistance to torsion which must be the same throughout the element’s length for the spring coefficient (k) to be accurate, but the element does not need to be circular. For non-circular elements, simply add up any two second moments of area at right angles to each other (J = Iˣˣ + Iʸʸ)

(see **Units** above)

L (length): the total length of a beam, shaft, spring or cord

ℓ (part length or distance): distance from and end or edge to the load (where the spring coefficient is required)

ν (Poisson’s ratio): the three-dimensional elastic property of the material which must be the same throughout the cross-section of any element for the spring coefficient (k) to be accurate. The most brittle material will have a ν ≤ 0.5 a material with zero elasticity will have ν ≥ 0

For example:

steel: 0.3

Rubber: 0.5

Plastic: <0.25

(see CalQlata’s Metal Properties)

A (area): the cross-sectional area which must be the same throughout the element’s length for the spring coefficient (k) to be accurate

Ø (diameter): the outside diameter of a circular plate that must be supported all around the edge for the spring coefficient (k) to be accurate

r (radius): the approximate (or average) radius of the applied load

t (thickness): the thickness of the plate which must be the same throughout for the spring coefficient (k) to be accurate

R (radius): the mean radius of the helical spring, i.e. to the centre of the wire

d (diameter): the diameter of the helical spring wire which must be the same throughout for the spring coefficient (k) to be accurate

T (tension): the tensile (axial) force applied to the cord

**Output Data:**

k (spring coefficient): The calculated spring coefficient is for the location where the load is applied

Whilst beams, shafts, columns, etc. normally refer to single elements (tube, H-beam, channel, etc.), they also apply to any complex structure of consistent sectional modulus and material such as a crane boom or simple bridge. Spring Coefficients can be used for any and all such structures as long as the correct sectional modulus and material properties are entered.

The accuracy of Spring Coefficient is dependent upon the accuracy of the input data. If all input data is correct and accurate there is no expected error margin in the results.

You will find further reading on this subject in reference publications^{(1, 3 & 4)}