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The linear stiffness of an ideal cylindrical helical spring depends on wire shear modulus, wire diameter, mean coil diameter and active turns.
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Formula
k in N/m; G in Pa; d and D in m; N dimensionless
| Symbol | Meaning | Unit |
|---|---|---|
| Axial spring rate | N/m | |
| Shear modulus of the wire | Pa | |
| Wire diameter | m | |
| Mean coil diameter | m | |
| Number of active turns | dimensionless |
k=\frac{Gd^4}{8D^3N}The standard relation follows from torsion of the spring wire under axial spring load. It is idealized for round wire, small elastic deflection and a cylindrical spring; end turns, curvature correction, buckling, fatigue and allowable stress need separate consideration. N counts active turns, not necessarily every visible turn. Stiffness scales with d4 and inversely with D3N.
The standard relation follows from torsion of the spring wire under axial spring load. It is idealized for round wire, small elastic deflection and a cylindrical spring; end turns, curvature correction, buckling, fatigue and allowable stress need separate consideration. N counts active turns, not necessarily every visible turn. Stiffness scales with d4 and inversely with D3N.
The linear stiffness of an ideal cylindrical helical spring depends on wire shear modulus, wire diameter, mean coil diameter and active turns.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Rate of a cylindrical helical spring” and check its unit and magnitude.
For G=80 GPa, d=4 mm, D=32 mm and N=10, k≈7.81 kN/m.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: k in N/m; G in Pa; d and D in m; N dimensionless.
Formula cluster
The linear stiffness of an ideal cylindrical helical spring depends on wire shear modulus, wire diameter, mean coil diameter and active turns.
For G=80 GPa, d=4 mm, D=32 mm and N=10, k≈7.81 kN/m.
Estimate force–deflection behavior of helical springs in their linear range.
Here is how to work through a typical Rate of a cylindrical helical spring (k = Gd⁴/(8D³N)) task step by step:
Task
Work through the numerical example for “Rate of a cylindrical helical spring” and check its unit and magnitude.
Solution path
For G=80 GPa, d=4 mm, D=32 mm and N=10, k≈7.81 kN/m.
Wire diameter has a strong effect because it sets the polar section moment of the round wire. A small increase can therefore make the spring much stiffer. The d4 scaling assumes G, D and N are held fixed. Stress in the wire does not necessarily scale in the same simple way. The formula idealizes a cylindrical spring with round wire and elastic shear response. D is mean coil diameter, not outside or inside diameter; N counts active turns. Curvature and real-end corrections can change design stiffness and stress. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Active turns deform elastically and contribute to spring motion. Closed or ground end regions may be partly or fully inactive and should not be counted automatically. The effective number depends on end construction. Real springs require manufacturer geometry or a more detailed spring-design model. The formula idealizes a cylindrical spring with round wire and elastic shear response. D is mean coil diameter, not outside or inside diameter; N counts active turns. Curvature and real-end corrections can change design stiffness and stress. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
No. k describes the elastic force–deflection slope. Wire shear stress, fatigue strength, solid height, buckling and allowable travel need separate checks. A spring can have the desired rate yet yield or fatigue at its intended load. The teaching relation is not a complete component design. The formula idealizes a cylindrical spring with round wire and elastic shear response. D is mean coil diameter, not outside or inside diameter; N counts active turns. Curvature and real-end corrections can change design stiffness and stress. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: k in N/m; G in Pa; d and D in m; N dimensionless. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For G=80 GPa, d=4 mm, D=32 mm and N=10, k≈7.81 kN/m. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The formula idealizes a cylindrical spring with round wire and elastic shear response. D is mean coil diameter, not outside or inside diameter; N counts active turns. Curvature and real-end corrections can change design stiffness and stress. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: The standard relation follows from torsion of the spring wire under axial spring load. It is idealized for. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The formula idealizes a cylindrical spring with round wire and elastic shear response. D is mean coil diameter, not outside or inside diameter; N counts active turns. Curvature and real-end corrections can change design stiffness and stress. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Quanta exam set
Curated exam set for "Rate of a cylindrical helical spring":
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Which formula describes Rate of a cylindrical helical spring?
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How do you rearrange Rate of a cylindrical helical spring for Base equation for the target quantity?
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Which common mistake happens with Rate of a cylindrical helical spring?
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