The handy spring calculator expresses the relationship between the force or torque applied to a spring and its deformation. This article will show you how to use this spring compression tool. Also, we will cover what the spring compression formula is and how to calculate the spring constant for different configurations. Keep reading to understand this amazing mechanism that has been used for a long time:
- What is a spring?
- How to calculate spring force in a compression / expansion spring?
- How to design an expansion / compression spring?
- How do you calculate coil spring momentum in a torsion spring?
- How to calculate the spring constant and design one? – a real-life example.
- FAQs.
What is a spring?
A spring is a mechanical device that stores energy while deforming its shape when a force or torque is applied. If you want to know specifically which type of energy, check the elastic potential energy.
There are several types of springs; however, in this spring calculator, we will cover the three main types of springs. Its categorization depends on the direction of the external force or whether torque is being applied. Such types are:
- Compression springs: springs that are designed to work under a compression force.
- Expansion (extension) springs: springs that are designed to work under a traction force.
- Torsion springs: springs that experience momentum due to a force applied outside of the center of gravity of the spring, specifically on one of the spring legs. Such force would make the spring rotate if we did not fix the other leg. Since the spring does not rotate, it deforms because of the torsional force, and it stores energy like the other springs.
For compression and expansion springs, the spring compression formula applies but in different directions, as you will see in the next sections. Both follow the same behaviour as the one explained by Hooke's law.
How to calculate spring force in a compression / expansion spring?
Although any spring follows the same principle as the energy conversion, expansion/compression springs and torsion springs have different formulas. In this section, we will cover the equation for the force of a spring:
F = −ktc × Δx
- F — Force – represents the force that reacts to the compression or expansion of a spring, measured in newtons (N);
- ktc — Spring constant for traction or compression – indicates the linear relation between the change in spring length and the force applied. It is measured in newtons/meter (N/m); and
- Δx — Spring change in length due to the force applied, measured in meters (positive for elongation and negative for compression).
The minus sign tells us the force is a restoring force: it always points opposite to the deformation, pushing or pulling the spring back to its rest position.
How to design an expansion / compression spring?
If we want to buy or build a spring, we need to specify more detailed information. You might have already guessed that the spring constant should also be related to its shape. Indeed, compressing a spring that is thicker than another is much more complicated if we assume we are dealing with the same material.
The following equation relates material properties and geometrical shape; it is called the spring rate formula:
ktc = G × d⁴ / (8 × D³ × Na)
- G — Shear modulus – also known as the modulus of rigidity, it measures the elastic shear stiffness of a material. You will typically find its values expressed in GPa.
- d — Wire diameter – represents the wire thickness.
- D — Mean diameter – indicates the size between the outer and inner diameter.
- Na — Number of active coils – indicates the number of coils that are working when the spring is under load.
Once you know the spring rate ktc, you can pick a target deflection Δx and check the force the spring will deliver with F = ktc × Δx, iterating on the wire diameter, mean diameter or number of coils until the force and stress suit your design.
How do you calculate coil spring momentum in a torsion spring?
A torsion spring works mainly in bending, not in pure torsion. The spring rate (torque per revolution) of a round-wire helical torsion spring is:
k = E × d⁴ / (10.8 × D × n)
- E — Young's modulus of the wire material;
- d — Wire diameter;
- D — Mean coil diameter; and
- n — Number of active coils.
The factor 10.8 (instead of the theoretical 10.2) accounts for friction between the coils. The torque (bending moment) at a deflection angle α — expressed in revolutions, where one full turn = 360° — is simply:
M = k × α
And the maximum bending stress in the wire is:
σ = 32 × M / (π × d³)
How to calculate the spring constant and design one? – Real life example
Consider a steel compression spring (G ≈ 79,300 MPa) with a wire diameter d = 2 mm, a mean diameter D = 20 mm and Na = 5 active coils, compressed by Δx = 10 mm.
- Spring rate: k = 79,300 × 2⁴ / (8 × 20³ × 5) ≈ 1.98 N/mm.
- Force: F = k × Δx = 1.98 × 10 ≈ 19.8 N.
- Elastic potential energy: U = ½ × k × Δx² = ½ × 1.98 × 10² ≈ 99 N·mm ≈ 0.099 J.
If 19.8 N is not enough force, increase the wire diameter (k grows with d⁴), reduce the mean diameter (k grows as 1/D³) or use fewer active coils — and re-check the result with the calculator above.
How to use this spring calculator
- Pick the spring type: compression, expansion (extension) or torsion.
- Choose the unit system: metric (mm, MPa, N) or American/Imperial (in, psi, lbf).
- Select the spring material to auto-fill the shear modulus G (or Young's modulus E for torsion springs), or choose Custom to type your own value.
- For compression/expansion springs you may type a known spring constant; otherwise enter the geometry (wire diameter, mean diameter, number of active coils) and the tool computes it for you.
- Enter the deflection — a length for compression/expansion springs, or an angle in degrees for torsion springs.
- Read the spring force (or torque), spring constant and stored energy in the results panel.
FAQs
What is the spring constant?
The spring constant k is the ratio between the force applied to a spring and the deformation it produces: k = F / Δx. A stiffer spring has a larger spring constant, so it needs more force to be deformed by the same amount.
How do I calculate the force of a spring?
Multiply the spring constant by the deformation: F = k × Δx. The minus sign in Hooke's law (F = −kΔx) only tells you the force is a restoring force that opposes the deformation.
What is the difference between a compression and an extension spring?
A compression spring is designed to shorten under load (it pushes back), while an extension — or expansion — spring is designed to stretch under load (it pulls back). Both obey Hooke's law and share the same spring rate formula; only the direction of the force changes.
Why does a thicker wire make a stiffer spring?
Because the spring rate grows with the fourth power of the wire diameter: k ∝ d⁴. Doubling the wire diameter makes the spring 16 times stiffer (if everything else stays the same), whereas the mean diameter has the opposite effect since k ∝ 1/D³.
What value of shear modulus should I use?
Typical shear modulus values: music wire / steel ≈ 79.3 GPa (11.5 × 10⁶ psi), stainless steel 302 ≈ 69 GPa (10 × 10⁶ psi), phosphor bronze ≈ 41.4 GPa (6 × 10⁶ psi), beryllium copper ≈ 48.3 GPa (7 × 10⁶ psi). For torsion springs use Young's modulus E instead.