Introduction
This guide explains the basic calculations for compression, extension and torsion springs. Maktel evaluates material, geometry, force, stress, cycle life, temperature, corrosion and installation conditions together. The formulas are for preliminary design; the final design should be verified by engineering assessment against real working conditions.
Basic symbols and spring index
d: wire diameter · Dₒ: outer diameter · Dᵢ: inner diameter · Dₘ: mean coil diameter (mm) · n: number of active coils · k: spring rate (N/mm) · G: shear modulus (N/mm²) · F: force (N) · s: spring travel (mm) · C: spring index.
Dₘ = Dₒ − d = Dᵢ + d = (Dₒ + Dᵢ) / 2
C = Dₘ / d
The spring index affects manufacturability, stress distribution and working behaviour. At low indices, forming and inner-region stresses are evaluated separately; at high indices, geometric stability is also considered.
Compression spring calculation
The spring rate expresses the change in force relative to the compression travel. Basic relationships for a helical compression spring:
Spring rate: k = G × d⁴ / (8 × Dₘ³ × n)
Force in the linear region: F = k × s
Theoretical deflection: s = 8 × F × Dₘ³ × n / (G × d⁴)
Approximate number of active coils: n = G × d⁴ / (8 × Dₘ³ × k)
Because wire diameter acts to the fourth power, small changes can significantly affect the force. Increasing the number of active coils generally makes the spring more flexible. Solid height is the minimum length at which the coils touch; it depends on wire diameter, total number of coils, end geometry and whether the ends are ground. Theoretical travel alone does not indicate a safe cycle life.
Shear stress and the Wahl factor
In a helical compression spring, the axial load creates shear stress in the wire. A correction factor is used for the increased stress on the inner side of the coil:
Ideal stress: τ = 8 × F × Dₘ / (π × d³)
Corrected stress: τ = K × 8 × F × Dₘ / (π × d³)
Wahl factor: K = (4 × C − 1) / (4 × C − 4) + 0.615 / C
K is the stress correction factor. For applications requiring dynamic loading and high cycling, Maktel evaluates stress together with force against the required working life.
Extension and torsion spring calculation
In an extension spring, the initial tension F₀ is the starting force that must be overcome before extension begins: F = F₀ + k × s. Hook geometry and stresses, body length, maximum extension and installation conditions are also examined.
A torsion spring generates moment against rotation: M = F × R, where M is the moment and R is the effective arm length of the force relative to the spring axis. Arm geometry, free and working angles, winding direction and bending stress are evaluated together with the actual installation.
Material and technical quotation information
The shear modulus G describes a material’s elastic torsional behaviour. Depending on the project, Maktel uses EN 10270-1 spring wire, oil-tempered wire, EN 10270-2 Cr-Si wire, EN ISO 6931-1 stainless wire and special alloyed materials.
Note: the formulas in this guide are for preliminary design and do not guarantee final product performance. Contact Maktel’s engineering team for project-specific calculation and verification.