Spring Rate Calculator

Hard springs make race cars feel every bump in the road while soft springs allow a passenger car to ride smoothly over the same bumps. The spring constant is the value that determines these two differences. This calculator offers two different methods of calculation; one method starts with force and resulting deflection, and another method calculates directly from dimensions of wire, number of turns, and material of a coil spring.

What is a spring constant?

The spring constant, also known as the stiffness of a spring, is denoted by "k" and gives the force required to compress or stretch a spring from its rest position by a certain length. If the spring constant is 100 N/mm, then a force of 100 Newtons will be needed to compress the spring one millimeter. Doubling the spring constant doubles the force required for the same compression.

A spring with a high spring constant is stiff and moves little under load. A spring with a low spring constant is flexible, compresses easily, and has a large deflection. A linear spring will have the same spring constant over its entire operating range. Progressive springs are often used in high performance suspensions and mountain bike forks; they become stiffer as they compress. They are soft on small bumps and offer firm resistance to larger jolts.

How to use this calculator tool:

Select the method that matches your known information. If you have already measured the spring or know its nominal values, select "Force and deflection". Enter two of three values (spring constant, force or deflection), and the calculator will calculate the remaining value. If you are designing or checking a coil spring, select "Spring properties". Enter wire diameter, outer diameter, effective number of turns, and material to calculate the spring constant from geometric dimensions. Metric and imperial units can be freely mixed, and all values will be converted into a uniform base unit before calculation begins.

Hooke's law (force and displacement)

According to Hooke's law, the force exerted by a spring is proportional to its displacement from equilibrium. The relationship is given by:

In this case, k is the spring constant, F is the force and delta-x is the displacement. By rearranging the equation you can also get the forms: F = k multiplied by the displacement and Displacement = F divided by k. This explains why the calculator can find a value when one of the input fields are left blank.

For example, let's say a coil spring with a nominal constant of 80 N/mm is compressed by 30 mm. The restoring force will be 2,400 Newtons, which is the product of 80 and 30, or about 245 Kilogram-force. This is enough force to support the wheel of a small car. When the spring is compressed, it also stores energy. This calculator shows both the force and stored energy.

Spring design based on dimensions

If one does not opt for a commercially available spring but constructs one oneself, the geometric dimensions and material are determined by the spring constant.

G is the wire's shear modulus, d is the diameter of the wire, D is the mean coil diameter and n is the effective number of turns. The mean coil diameter is calculated by subtracting one wire thickness from the outside diameter, so D = OD minus d. A calculation tool will determine the result. Since the wire diameter enters to the fourth power, it has the greatest influence, even a slight change in wire diameter results in a large change in value. The larger the mean coil diameter and the higher the number of turns, the softer the spring.

Here is an example of a real part: A spring made from stainless steel wire with a shear modulus of about 69 GPa, thickness of 0.072 inch, outside diameter of 0.75 inch and effective number of turns of 5.5. The mean diameter is 0.678 inch and the formula gives a value of about 3.4 N/mm which is approximately 19.5 pounds per inch. If slightly thicker wire were used the value would be considerably higher.

The shear modulus is material dependent and different from the commonly used elastic modulus (Young's modulus). If the elastic modulus E and Poisson's ratio v are known, then the shear modulus can be calculated using the formula G = E divided by 2*(1+v). A list of typical values is available in the Material menu below so that a separate search is usually not necessary.

Spring arrangements: series and parallel connection

When two springs are combined they act like a single spring with a different value. This feature allows any two springs to be combined together. When the springs are placed in parallel they share the deformation so their values add up resulting in a stiffer combination.

If the springs are in series (i.e. one end to another) they share the load so their values add up giving a softer combination than using just one spring.

Stacking springs is a common method of adjusting the effective spring constant without having to buy new parts. Two identical springs stacked in parallel will double the spring constant, while two identical springs stacked in series will halve it.

Areas of application for spring constants:

Suspension tuning is one of the most common applications for spring constants. Road vehicles typically use springs with a spring constant between 25 and 150 N/mm. Softer springs offer more comfort, while stiffer ones provide better handling in corners. The spring constants for mountain bike forks and shocks are usually between 30 and 90 pounds per inch and are chosen according to the weight of the rider. The same values also determine the spring components that often go unnoticed, such as garage door openers, industrial valves, retractable ballpoint pens or trampoline mats. At extremely low spring constants, these can be found in clock mechanisms or surgical instruments at just a few millinewtons per millimeter.

Tips for accurate results:

Measure the spring rate on a workbench with the length of the spring in its unloaded free state as your reference point. As springs that are already installed have some preload they can be difficult to measure accurately. When replacing springs on a vehicle you should also check the shock absorbers at the same time. The harder the spring, the more damping force is required to control bottoming out. With progressive springs manufacturers often give a curve or range rather than an individual number so pick the value that corresponds with your actual travel.