Conical Spring Calculator
Estimate deflection, stiffness, solid length, minimum free geometry, and maximum nominal shear stress for a round-wire conical compression spring.
Engineering screening tool—verify inputs and results independently. Use at your own risk.
Spring inputs
Enter mean coil diameters; the large mean diameter must exceed the small mean diameter.
Calculated response
The free length and pitch are equality-based geometric minima at the entered load. Add operating clearance and verify buckling, coil contact, stress correction, fatigue, and manufacturing tolerances separately.
How to use
- Enter round-wire diameter d, large mean coil diameter 2r₁, and small mean coil diameter 2r₀ in mm; require 2r₁ > 2r₀ > 0.
- Enter the positive active-coil count n, shear modulus G in GPa, and positive compressive force magnitude F in N. This calculator has one compression-spring mode.
- Select Calculate, or edit a field for an immediate update. Select Reset to restore the 2 mm, 18 mm, 10 mm, 30-coil, 79.3 GPa, and 100 N example.
- Read deflection, maximum nominal shear stress, secant stiffness, closed length, minimum free length, and minimum pitch in the displayed units. All outputs are rounded to two decimals.
- Correct any warning before interpreting results. Do not use this simplified elastic model for nonlinear coil contact, buckling, surge, fatigue-life certification, plasticity, rectangular wire, nonuniform pitch, or safety-critical final design.
Worked example
1. For d = 2 mm, 2r₁ = 18 mm, 2r₀ = 10 mm, n = 30, G = 79.3 GPa, and F = 100 N, the results are f = 56.14 mm, τmax = 572.96 MPa, k = 1.78 kN/m, Lᶜ = 60.00 mm, L₀,min = 116.14 mm, and smin = 3.87 mm/coil. 2. The high nominal stress means the geometry requires an allowable-stress and fatigue review; the equality-based minimum free dimensions also need practical clearance.
Equations used
Closed length: Lᶜ = nd
Axial deflection: f = 16n(r₁ + r₀)(r₁² + r₀²)F/(d⁴G)
Minimum free length: L₀,min = Lᶜ + f
Minimum pitch: smin = L₀,min/n
Maximum nominal shear stress: τmax = 16Fr₁/(πd³)
Secant stiffness: k = F/f
d is wire diameter; r₁ and r₀ are the large- and small-end mean coil radii; n is active-coil count; G is shear modulus; F is the positive compression magnitude; f is axial deflection; Lᶜ is coil-bound length; L₀,min and smin are equality-based minimum free length and pitch; τmax is the nominal large-end shear stress; and k is secant stiffness. Lengths use compatible units, G and τ use compatible stress units, and force uses N. The calculator converts mm and GPa to SI internally.
Theory and method
The model treats the mean coil radius as varying continuously between r₀ and r₁ over n active round-wire turns. The deflection relation combines torsional compliance across that taper. The maximum nominal shear stress is evaluated at the large-radius end, where the force moment is greatest. Closed length is nd; adding calculated deflection gives the minimum unloaded length that reaches the closed geometry at the entered force.
Assumptions and limits: homogeneous isotropic material, linear shear elasticity, round wire, concentric axial compression, evenly distributed active turns, small deformation before coil contact, and no initial preload. Curvature/Wahl stress correction, direct shear, end-coil details, pitch variation, nesting clearance, buckling, surge, temperature relaxation, yielding, and fatigue are excluded. Real springs require a free length and pitch above the reported equality-based minima.
Reference: A. van Beek, Advanced Engineering Design: Lifetime Performance and Reliability, conical compression-spring calculation collection; equations reproduced above. Validate production designs against an applicable spring standard and supplier fatigue/material data.
Continue in TriboSolver to refine stress, material, contact, nonlinear response, and service loading.
