Compression springs are mechanical springs that deform under axial compressive load and revert when the load is removed. In mechanical systems, they can be used to either resist compressive forces, absorb shocks and vibrations, keep components apart, or store and release elastic energy as needed.

This article tackles the different compression spring formulas in order to guide designers in the design of systems that use this component.

## Compression Spring Rate Formula

Inherently, the behavior of a compression spring is governed by its spring rate or spring constant. This is what characterizes the deformation of a compression spring relative to the force applied. In essence, it measures the difficulty of compressing the spring.

Hence, for a spring with a linear behavior, the spring rate can be approximated using the formula:

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Where:

- k = spring rate [N/mm]
- F = force applied on the spring [N]
- x = deflection from the equilibrium position [mm]

## Compression Spring Working Principle

Before learning about the other spring calculations in addition to the spring rate, it is important to understand how a compression spring works.

A compression spring works as a device that stores elastic potential energy when compressed and releases the same energy when restored to its original shape. This ability is made possible by the spring material’s elastic property.

As the coils are pushed tighter against each other by a compressive load, its atomic bonds are strained and the equilibrium distance between the atoms— that is, the distance such that the overall energy of the body is minimum— is disturbed. This allows the spring material to absorb energy as it is deformed.

Once the compressive load is removed, the tendency of the body is to revert back to the state of minimum energy. Hence, the distance between the atoms goes back to equilibrium, thereby releasing the stored energy.

## Compression Spring Calculations

There are different types of compression springs, configured into different shapes and made of different materials. The most common type is the straight helical compression spring, mainly because it is easy to manufacture and design, yet also very reliable in terms of performance and functionality.

For simplicity, this article considers a straight helical compression spring to discuss how to calculate the different spring parameters.

### Spring Diameters

A straight helical compression spring is made from a wire with a constant diameter, formed into a cylindrical helix, as shown in the diagram below.

From the cross-sectional diagram above, the following equations can be used to relate the diametrical parameters of a compression spring:

Where:

- d = diameter of the wire [mm]
- D
_{i}= inner diameter of the spring [mm] - D
_{o}= outer diameter of the spring [mm] - D = mean diameter of the spring [mm]

### Spring Index

Using the values of the wire diameter and the mean diameter, the spring index can be calculated using the formula:

Where:

- C = spring index [unitless]

The spring index is a measure of the relative tightness of the coils, which also indicates how easily a spring can be manufactured. In general, springs with an index between 4 and 12 are considered manufacturable, however, the range between 6-9 is preferred for low-tolerance springs subjected to cyclic loading.

### Spring Pitch Formula

The pitch is defined as the distance between two successive coils, measured from the midpoint of the spring material at the resting position. It should be noted that this value is not the same as the length of the gap between the coils; it is larger since it also includes the radii of the two adjacent coils.

The formula for calculating pitch depends on the configuration of the end coils. The four most common configurations for the end coils are closed and square ends, closed and ground ends, double closed ends, and open ends, as shown in the diagram below.

These end coils are configured depending on the application. For example, some end coils are closed or left without any gap to allow stability and more support if the spring needs to stand upright.

For closed and square ends, the pitch can be calculated using the formula:

Where:

- p = pitch of the spring [mm]
- L
_{f}= free length [mm] - d = wire diameter [mm]
- N
_{a}= number of active coils

For closed and ground ends:

For double closed ends:

For open ends:

The free length of the spring is its total axial length at resting position, that is, at zero compressive loads.The number of active coils can be obtained by counting the number of open-wound coils— those that have gaps in between them.

Active coils are the coils that allow the spring to be elastic. When a compressive force is applied against the spring, they are the ones that push back.

It is also important to note that some springs have a variable pitch, as shown in the diagram below. Spring manufacturers sometimes employ this feature to control spring rate at constant coil diameter.

It is also used to prevent spring surge, which occurs when the spring’s natural frequency matches the frequency of the applied force. This causes the spring to oscillate erratically, which can shorten the life span of the spring or, worse, cause permanent damage.

### Slenderness Factor

The slenderness factor is defined as the ratio between the free length of the spring and the mean coil diameter. Together with the ratio between the deflection and the free length, this value is an indication of the spring’s tendency to buckle.

For example, the curves below show the boundaries between the stable and unstable states of parallel and non-parallel compression springs based on the slenderness factor and the ratio between the deflection and free length.

### Spring Constant Formula

As stated above, spring constant is the inherent characteristic of a compression spring that governs how it deflects in relation to a force applied. Influenced by its geometry and material composition, the spring constant can be obtained using the formula:

Where:

- k = spring rate [N/mm]
- G = shear modulus of elasticity of the wire material [N/mm
^{2}]

The shear modulus of elasticity of some of the most common spring materials are listed in the table below:

The inverse of the spring rate is called compliance. If a spring, for example, has a spring rate equal to 5 N/mm, then the compliance would be 0.2 mm/N.

The formula above applies to linear springs with a constant spring rate. However, it should be noted that there are some spring configurations that have a variable spring rate. These are called progressive or variable-rate springs.

Progressive springs normally have uneven gaps between the coils, which causes the spring rate to change as the spring is compressed. This allows them to stiffen more quickly as they absorb energy. Hence, they are popularly used in heavy-duty applications such as vehicle suspensions, overhead doors, and agricultural equipment.

### Combining Spring Constants

Two or more compression springs can be connected in series or parallel to alter the overall behavior of the spring system. The group of springs can then be treated as one with a combined spring rate.

For springs connected in series, the combined spring rate can be calculated using the formula:

Where:

- k
_{comb}= combined spring rate [N/mm] - n = number of springs [unitless]

On the other hand, for springs connected in parallel, the combined spring rate can be calculated using the formula:

This means that the spring rates of springs connected in parallel add directly, while those in series combine reciprocally.

### Spring Force Formula

When a compressive load is applied to a compression spring, the spring exerts a restoring force against the load with a magnitude proportional to the spring rate and the deflection, that is, the change in its axial length relative to the equilibrium position.

This is governed by Hooke’s Law, as shown in the equation below:

Where:

- F = restoring force of the spring [N]
- x = deflection from the equilibrium position [mm]

The formula above assumes that the equilibrium position is at x = 0. Also, note that this relationship only applies within the spring’s elastic range, that is, from the equilibrium position down to its solid height.

### Spring Elastic Potential Energy

As the spring deforms due to the compressive load and the atomic bonds of the spring material are strained, it stores elastic energy within its body. The amount of this stored elastic energy is equal to the area under the load-deformation curve, as shown in the diagram below.

Hence, for a helical compression spring with a constant spring rate, the total elastic potential energy can be obtained using the formula:

Where:

- U = stored elastic potential energy [N-mm]

### Maximum Spring Compression Formula

The maximum spring compression is generally considered to occur at the spring’s solid height, which happens when there is no more gap between any of the coils. A fully compressed spring, therefore, should not be shorter than that height.

For compression springs with closed and square ends, the solid height can be calculated using the formula:

Where:

- L
_{s}= solid height [mm] - N
_{t}= total number of coils (active + inactive) [unitless]

On the other hand, for compression springs with closed and ground ends, the solid height can be calculated using the formula:

For other end configurations, the general formula below should apply:

Where:

- t
_{1}+ t_{2}= sum of the thickness of the end coils [mm]

From the solid height, it follows that the maximum spring compression or deflection from the equilibrium position can be calculated using the formula:

Where:

- x
_{max}= maximum spring compression relative to the equilibrium position [mm]

### Spring Capacity Formula

Based on the maximum spring compression, the spring capacity or the maximum load a compression spring can handle can then be calculated using the formula:

The spring capacity is also the magnitude of the load that will result to a fully compressed spring.

### Weight of a Compression Spring

The mass of a compression spring can be calculated by multiplying the density of the wire material and its total volume. This results in the following equation:

Where:

- m
_{spring}= mass of the spring [kg] - ρ = density of the wire material [kg/mm
^{3}]

To get the weight of the spring, multiply its mass by the gravitational acceleration, which is equal to 9.81 m/s^{2}.

### Maximum Shear Stress Formula

Lastly, when designing a compression spring, it is important to consider the maximum stress that the wire material may be subjected to.

Although a compression spring is expected to handle axial compressive loads, its wire is actually not subjected to axial stress. Instead, it experiences shear stresses divided into two components: a torsional shear stress induced by the twisting of the spring, and a direct shear stress due to the force applied.

This causes the maximum shear stress to occur at the inner portion of the wire material with a magnitude equal to the following equation:

Where:

- τ
_{max}= maximum shear stress [N/mm^{2}]

## Types of Compression Spring

Compression springs are produced in different shapes. Aside from straight helical springs, some of the other configurations are conical, hourglass, and barrel, as shown in the diagram below.

### Conical Spring

The coils of a conical spring are wound in the shape of a cone— that is, the mean diameter gradually tapers from one end to the other. This configuration creates a progressive spring rate, which is ideal for applications requiring low solid height and prevents surging.

### Hourglass Spring

As the name suggests, the coils of an hourglass or convex spring are wound in the shape of an hourglass— that is, the mean diameter in the middle portion is shorter than both ends. This symmetrical configuration is ideal for applications that require the spring to stay centered over a particular point.

### Barrel Spring

Barrel springs are a complete opposite of hourglass springs. Instead of having the smallest mean diameter in the middle, barrel springs have a larger diameter in the middle than on both ends. This configuration is suitable for applications that require stability and resistance to surging as the springs decompress, for example, in automotive, furniture, and toy industries.