Reference · Methodology

Compression Spring Design

Shigley's 11th Ed. Ch. 10 · EN 13906-1 · Associated Spring Design Handbook

Complete methodology for calculating spring rate, corrected shear stress, deflection, solid length, and buckling for helical compression springs.

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[01]Nomenclature

Symbol	Description	Unit (SI)	Unit (Imperial)
d	Wire diameter	mm	in
D	Mean coil diameter	mm	in
C	Spring index D/d	—	—
N_a	Active coils	—	—
k	Spring rate	N/mm	lbf/in
δ	Deflection	mm	in
τ	Shear stress	MPa	psi
K_W	Wahl correction factor	—	—
G	Shear modulus	GPa	psi
F	Applied force	N	lbf
L_free	Free length	mm	in
L_solid	Solid length	mm	in

[02]Equations

Spring Index

Ratio of mean coil diameter to wire diameter. Practical range is 4 to 12. Below 4, the spring is difficult to manufacture. Above 12, the spring is prone to tangling and buckling.

C = \frac{D}{d}

Spring Rate

The spring rate (stiffness) relates applied force to deflection: F = kδ. G is the shear modulus of the wire material.

k = \frac{Gd^4}{8D^3 N_a}

Wahl Correction Factor

Accounts for the non-uniform stress distribution due to curvature and direct shear in the coil. For C < 4, the correction becomes large and the spring is impractical.

K_W = \frac{4C - 1}{4C - 4} + \frac{0.615}{C}

Corrected Shear Stress

Maximum shear stress at the inner surface of the coil, corrected by the Wahl factor. Compare against the allowable shear stress for the wire material.

\tau = K_W \frac{8FD}{\pi d^3}

Deflection

Spring deflection under applied force F. The first form derives from Castigliano's theorem; the second follows from the spring rate definition.

\delta = \frac{8FD^3 N_a}{Gd^4} = \frac{F}{k}

Solid Length

Minimum physical length when all coils are touching (for closed-ground ends). Other end conditions: closed only = (N_a + 2)d, plain = (N_a + 1)d, plain-ground = N_a × d.

L_{\text{solid}} = (N_a + 2) \times d

Buckling Ratio

Free length to mean diameter ratio. Springs with L_free/D > 4 are susceptible to buckling under compression and may need a guide rod or housing.

\frac{L_{\text{free}}}{D}

[03]Worked Example

Music wire, d = 2 mm, D = 20 mm, N_a = 8, F = 100 N, G = 81.7 GPa, closed-ground ends, L_free = 50 mm.

Step 1: C = 20/2 = 10.0 (within ideal range 4–12)

Step 2: k = (81,700 × 2&sup4;) / (8 × 20³ × 8) = 2.553 N/mm

Step 3: K_W = (4×10−1) / (4×10−4) + 0.615/10 = 1.145

Step 4: τ = 1.145 × (8 × 100 × 20) / (π × 2³) = 729.7 MPa

Step 5: δ = 100 / 2.553 = 39.17 mm

Step 6: L_solid = (8+2) × 2 = 20 mm

Step 7: Buckling ratio = 50/20 = 2.5 (≤ 4, stable)

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[04]Assumptions & Limitations

Assumes linear elastic behavior within the proportional limit of the wire material
Wahl factor is valid for spring indices C ≥ 4; below C = 4, manufacturing is impractical
End condition formulas assume standard commercial tolerances
Buckling check is for springs without guides; guided springs have higher buckling resistance
Does not account for fatigue life, creep, or stress relaxation under sustained load
Shear modulus G varies with temperature; values assume room temperature (20°C)

[05]References

[1]Budynas, R.G. & Nisbett, J.K. — Shigley's Mechanical Engineering Design, 11th Ed., Ch. 10

[2]EN 13906-1:2013 — Cylindrical Helical Springs Made from Round Wire — Calculation and Design

[3]Associated Spring — Engineering Guide to Spring Design

[4]Wahl, A.M. — Mechanical Springs, 2nd Ed.

Related References

Bolt Torque Reference→Shaft Deflection Reference→

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