Helical springs are formed by winding spring material into a helix. The most common forms are compression springs, which resist axial compression, and extension springs, which resist axial extension.
This page provides engineering equations and reference information for cylindrical helical springs, including spring index, spring rate, shear stress, deflection, compression-spring end arrangements, extension-spring initial tension, rectangular-section springs and conical springs.
A cylindrical helical spring is normally formed from wire or bar wound at a substantially constant coil diameter. Round wire is the most common section, although square, rectangular and other sections are used for particular applications.
Where a suitable stock spring can satisfy the required load, travel, available space, service life and environmental conditions, it will often be more economical than specifying a special-purpose spring. A custom spring becomes appropriate where the application requirements cannot reasonably be met by an available standard component.
Compression Springs
Extension Springs
| Symbol | Description |
|---|---|
| C | Spring index = D/d |
| d | Wire diameter |
| D | Mean coil diameter = (Di + Do)/2 |
| Di | Spring inside diameter |
| Do | Spring outside diameter |
| Dil | Spring inside diameter under load |
| E | Young's modulus |
| F | Axial force |
| Fi | Initial axial force for a close-coiled extension spring |
| G | Shear modulus / modulus of rigidity |
| Kd | Direct-shear correction factor |
| KW | Wahl stress correction factor |
| L0 | Free length |
| Ls | Solid length |
| n | Number of active coils |
| nt | Total number of coils |
| p | Pitch |
| δ | Axial deflection |
| τ | Shear stress |
| τi | Initial shear stress in an extension spring |
| τmax | Maximum corrected shear stress |
The equations below may be used with any consistent system of units. For practical spring calculations, force in N, dimensions in mm and modulus in N/mm2 are convenient and produce stresses in N/mm2.
The spring index C is the ratio of mean coil diameter to wire diameter:
C = D / d
Spring index is an important geometrical parameter because it affects coil curvature, stress concentration and manufacturability.
The existing RoyMech design information commonly considers spring indices in approximately the 3 to 12 range. This should not be regarded as a universal design limit. The practical range depends on spring size, material, manufacturing process and the applicable specification.
Within the approximately linear working region of a conventional cylindrical helical spring, spring rate is the change in axial force divided by the corresponding change in deflection:
k = F / δ
where k is normally expressed in N/mm when force is in N and deflection is in mm.
Where a spring operates between two loaded positions, spring rate may be determined from:
k = (F2 - F1) / (δ2 - δ1)
This form is particularly useful for extension springs having initial tension, because their force-deflection relationship does not pass through the origin.
The allowable working stress of a spring should be based on the actual material, material condition, manufacturing process, operating temperature, surface condition and loading duty.
Use material data and allowable stress appropriate to the spring material, condition and applicable design standard. For cyclic applications also see Fatigue in Springs.
Consider a cylindrical helical compression spring subjected to an axial force F.
A section through the spring wire is subjected principally to torsion caused by the axial force acting at the mean coil radius, together with direct shear.
For mean coil diameter D, the torque in the wire is:
T = FD / 2
The torsional shear stress for a round wire is therefore:
τt = 8FD / (πd3)
The direct shear stress caused by the axial force is:
τd = 4F / (πd2)
Combining torsional and direct shear gives:
τ = 8FD / (πd3) + 4F / (πd2)
Using the spring index C = D/d, this may be written:
τ = [8FD / (πd3)] (1 + 1/(2C))
The direct-shear correction factor is therefore:
Kd = (C + 0.5) / C
and:
τ = Kd 8FD / (πd3)
Coil curvature causes the maximum stress at the inside of the coil to be greater than predicted by the simple straight-bar torsion calculation. The commonly used Wahl factor combines the effects of curvature and direct shear:
KW = (4C - 1) / (4C - 4) + 0.615 / C
The corrected maximum shear stress is then:
τmax = KW 8FD / (πd3)
| C | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| KW | 1.58 | 1.40 | 1.31 | 1.25 | 1.21 | 1.18 | 1.16 | 1.14 | 1.13 | 1.12 | 1.11 | 1.10 | 1.10 | 1.09 |
The force-deflection relationship can be obtained from strain energy. RoyMech's original derivation uses Castigliano's theorem and includes the strain energy associated with both torsion and direct shear.
For an elastic system subjected to a force F, Castigliano's theorem gives the displacement in the direction of that force as:
δ = ∂U / ∂F
where U is the total strain energy. See also Strain Energy.
For the helical spring, the torsional and direct-shear strain energy may be written:
U = T2l / (2GJ) + F2l / (2GA)
where, for round wire:
T = FD/2, l = πDn, J = πd4/32, A = πd2/4
Substitution gives:
U = 4F2D3n / (Gd4) + F2Dn / (2Gd2)
Applying Castigliano's theorem:
δ = 8FD3n / (Gd4) + FDn / (Gd2)
or:
δ = 8FD3n / (Gd4) [1 + 1/(8C2)]
For many practical springs the direct-shear contribution has relatively little effect. The commonly used simplified round-wire compression-spring deflection equation is:
δ ≈ 8FD3n / (Gd4)
Spring rate is force divided by the corresponding elastic deflection. Using the usual simplified round-wire cylindrical compression-spring relationship:
k ≈ Gd4 / (8D3n)
Compression springs can be manufactured with several end arrangements. The choice affects seating, free length, solid length, active coils and the way the spring transfers load into its supporting components.
Plain ends are generally less suitable where accurate seating and alignment are important. Closed and ground ends provide a more nearly square bearing surface and are commonly used for more demanding applications.
The following relationships are retained from the existing RoyMech design data as useful approximate geometrical relationships. Actual coil geometry varies with manufacturing practice, and the applicable spring specification should be used where dimensional control is important.
| Term | Plain | Plain and Ground | Closed | Closed and Ground |
|---|---|---|---|---|
| End coils ne | 0 | 1 | 2 | 2 |
| Total coils nt | n | n + 1 | n + 2 | n + 2 |
| Free length L0 | pn + d | p(n + 1) | pn + 3d | pn + 2d |
| Solid length Ls | d(nt + 1) | dnt | d(nt + 1) | dnt |
| Pitch p | (L0 - d)/n | L0/(n + 1) | (L0 - 3d)/n | (L0 - 2d)/n |
The calculated numbers of coils and associated dimensions are not necessarily exact manufacturing values. Spring manufacture involves dimensional variation, and appropriate tolerances should be specified for the application.
An extension spring is designed to resist axial separation. Many of the basic relationships used for cylindrical helical compression springs also apply to the active body coils of an extension spring, but extension springs introduce two particularly important additional considerations: initial tension and end-hook stresses.
The spring ends transfer the external load into the spring body. Hooks, loops and other end forms introduce local bending and stress concentrations and can therefore become critical parts of an extension spring, particularly under fatigue loading.
End geometry should therefore be considered as part of the spring design rather than simply as an attachment added after the body coils have been calculated.
A close-wound extension spring may be manufactured with initial tension. This produces an initial axial force that must be exceeded before the coils begin to separate.
The relationship between initial tension force and initial shear stress may be expressed as:
Fi = πτid3 / (8D)
Once the coils have separated and the spring is operating in its approximately linear range, the load may be represented approximately by:
F = Fi + kδ
The following initial-stress ranges are retained from the existing RoyMech data. The original source and material/manufacturing basis have not yet been established sufficiently to present these values as universal design limits. They should therefore be treated as historical reference information rather than allowable stresses.
| Spring Index C = D/d |
Legacy Initial Stress Range τi (N/mm2) | |
|---|---|---|
| Lower | Upper | |
| 3 | 140 | 205 |
| 4 | 120 | 185 |
| 5 | 110 | 165 |
| 6 | 95 | 150 |
| 7 | 90 | 140 |
| 8 | 80 | 125 |
| 9 | 70 | 110 |
| 10 | 60 | 100 |
| 11 | 55 | 90 |
| 12 | 45 | 85 |
| 13 | 40 | 75 |
| 14 | 35 | 65 |
| 15 | 30 | 60 |
| 16 | 25 | 55 |
Where an extension spring is manufactured without initial tension, the force-deflection relationship of the active coils approaches that of the corresponding compression-spring calculation.
Although round wire is the normal choice for helical springs, rectangular section material can be used where its geometry provides an advantage for a particular design.
The following equations and factors are retained from the existing RoyMech reference data.
k = K2 Gbt3 / (nD3)
τ = KWK1FD / (bt2)
| b/t | 1.0 | 1.5 | 1.75 | 2.0 | 2.5 | 3.0 | 4.0 | 6.0 | 8.0 | 10.0 |
|---|---|---|---|---|---|---|---|---|---|---|
| K1 | 2.41 | 2.16 | 2.09 | 2.04 | 1.94 | 1.87 | 1.77 | 1.67 | 1.63 | 1.60 |
| K2 | 0.18 | 0.25 | 0.272 | 0.292 | 0.317 | 0.335 | 0.385 | 0.381 | 0.391 | 0.399 |
A conical compression spring has a coil diameter that changes progressively along its length. Depending on its geometry and pitch, the arrangement can provide a non-linear force-deflection characteristic and can allow adjacent coils to telescope as the spring is compressed, giving a relatively small compressed height.
Conical springs are used where packaging, variable stiffness or reduced solid height makes them preferable to a conventional cylindrical spring.
The original RoyMech page contains image-based equations for allowable force and stiffness for this spring form. These equations are retained below as legacy reference information pending independent verification of the original formulation and notation.
The following standards are relevant to cylindrical helical spring design, specification, measurement and testing. The applicable standard depends on spring type, manufacturing process and the purpose of the calculation or inspection.