Team
physics

Spring Calculator

Springs convert mechanical energy through elastic deformation—a principle vital to mechanical engineering, automotive design, and precision instruments. Whether you're working with compression springs in suspension systems, tension springs in door hinges, or torsion springs in watch mechanisms, understanding the relationship between applied force and displacement is essential. This calculator handles both linear springs (compression and expansion) and rotational springs (torsion), providing you with instant calculations of spring constant, force, and deformation based on material properties and geometry.

Last updated:

Creators Davide Borchia, PhD
7,304 people find this calculator helpful

What is a Spring?

A spring is an elastic device that stores and releases energy by changing shape under load. When force is removed, it returns to its original form—the defining characteristic of elastic materials.

Springs fall into three main categories:

  • Compression springs resist being squeezed, commonly found in shock absorbers and mattresses.
  • Expansion springs resist being pulled apart, used in gate hinges and retractable mechanisms.
  • Torsion springs resist rotational forces, prevalent in door closers and clock mechanisms.

Each type follows predictable mathematical relationships between applied force (or torque) and resulting deformation, allowing engineers to specify springs with precision for any application.

Spring Force and Hooke's Law

The foundation of spring behaviour is Hooke's Law, which states that the restoring force is directly proportional to displacement. For linear springs:

F = −k × Δx

where the spring constant is calculated from material and geometric properties:

k = (G × d⁴) ÷ (8 × D³ × Nₐ)

  • F — Restoring force in Newtons (positive for expansion, negative for compression)
  • k — Spring constant in N/m, indicating stiffness
  • Δx — Displacement from natural length in metres (negative for compression, positive for extension)
  • G — Shear modulus (modulus of rigidity) in Pa, a material property
  • d — Wire diameter in metres
  • D — Mean coil diameter in metres
  • Nₐ — Number of active coils resisting deformation

Designing a Linear Spring

Spring design requires balancing manufacturability, cost, and performance. Three parameters dominate this process:

  • Spring index (C) = D ÷ d. A ratio between 5 and 10 offers optimal balance between ease of manufacturing and material utilisation. Lower ratios (tighter coils) are harder to wind; higher ratios create uneven stress distribution.
  • Free length (L₀) is the unloaded spring length. Depending on end type, it varies: plain ends (Lₚ = Nₐ × p + d), ground ends (Lₑ = Nₐ × p + 2d), squared ends (Lₛ = Nₐ × p + 3d), or squared and ground (Lₛₘ = Nₐ × p + 2d).
  • Pitch (p) is the axial distance between successive coils, controlling how tightly wound the spring is.

Material selection (steel alloys, stainless steel, titanium) determines the shear modulus and maximum operating stress, directly affecting spring constant and lifespan.

Torsion Spring Torque

Torsion springs twist rather than compress, storing rotational energy. The relationship between applied torque and twist angle mirrors linear Hooke's Law:

M = k_torsion × α

where the torsional spring constant is:

k_torsion = (d⁴ × E) ÷ (64 × D × Nₐₜ)

  • M — Applied torque in Newton-metres
  • k_torsion — Torsional spring constant in N·m per radian
  • α — Twist angle in radians
  • d — Wire diameter in metres
  • E — Elasticity modulus (Young's modulus) in Pa
  • D — Mean coil diameter in metres
  • Nₐₜ — Number of active coils resisting torsion, including end contributions

Practical Design and Measurement Tips

Avoid common pitfalls when specifying or testing springs.

  1. Sign convention matters — Hooke's Law includes a negative sign: compression (negative Δx) produces positive restoring force. When using the calculator, compress springs with negative displacement values and extend them with positive values. Consistency prevents sign errors that invalidate designs.
  2. Measure free length carefully — Free length is always measured with zero load. Remove any preload or mounting stress before measuring. Even small errors (±1 mm) significantly affect pitch calculations and active coil counts, especially in multi-turn designs.
  3. Account for end conditions — End type (plain, ground, squared) changes free length by 1–3 wire diameters. Ignoring this causes incorrect load-displacement predictions. Document the end condition when specifying springs to suppliers.
  4. Temperature affects shear modulus — Shear modulus decreases roughly 0.03% per °C above room temperature for steel. If your spring operates hot (automotive, industrial), use temperature-corrected modulus values or the spring will feel softer than calculated.

Frequently Asked Questions

How do I experimentally determine a spring constant?

Place the spring vertically. Add a calibrated weight (e.g., 1 kg = 9.81 N) and measure the resulting compression or extension from the natural length. Divide the force by the displacement: k = F ÷ Δx. Repeat with different weights to verify linearity. For precise work, plot multiple points; Hooke's Law breaks down beyond the elastic limit, typically 5–20% strain depending on material.

What's the difference between compression and extension spring constants?

Mathematically, the spring constant is identical for the same coil geometry and material. The difference is functional design: compression springs have closely spaced coils for stability under axial load, while extension springs have open coils and hooks or loops at the ends. The calculator treats both using the same formulas; the distinction is mechanical, not mathematical.

Why does spring index matter?

Spring index (C = D ÷ d) controls stress distribution and manufacturability. A low index (tight coils, C < 5) creates uneven bending stress and is difficult to wind precisely. A high index (C > 12) has lower material strength due to stress concentration at the inner surface. The sweet spot, C = 5–10, minimises both manufacturing difficulty and stress variation, reducing failure risk and cost.

How do I account for preload in spring design?

Preload is initial compression applied at assembly. It reduces the effective displacement range and shifts the operating window. When specifying, state both preload force and operating force. The calculator assumes zero preload; subtract preload displacement from measured compression to get the true working Δx.

What's the practical limit on spring compression?

Solid height (coils touching) occurs when compression reduces length to Nₐ × d. Attempting further compression damages the coils and renders the spring useless. Design safely at 50–75% of available travel; this preserves spring life and prevents nonlinear behaviour near solid height.

How do torsion springs differ from linear springs in real applications?

Torsion springs develop torque (rotational force), not linear force. They're used in door hinges, clipboards, and mechanical watches. Measurement is angular (degrees or radians) rather than linear. The calculator computes both angular displacement and equivalent force at a lever arm distance, letting you convert between rotational and linear thinking.

More physics calculators (see all)

Orifice Flow Calculator Series Inductors Calculator Drake Equation Calculator De Broglie Wavelength Calculator Magnetic Force Between Wires Calculator Darcy's Law Calculator Boyle's Law Calculator Wet-Bulb Calculator