Extension Spring Formulas and Definitions

This page combines the main extension spring formulas for static and fatigue design into one practical reference. Extension springs are also called tension springs, and their design usually requires checking both the spring body and the hook or loop ends.

Use this page with the extension spring calculators when you need to understand symbols, spring index values, correction factors, end stresses, body stresses, and fatigue safety factor equations.

Definitions Static formulas Fatigue formulas Symbols Related calculators

Extension spring definitions

Extension spring

A spring that resists pulling force. It stores energy when stretched and returns force as it tries to shorten.

Tension spring

Another common name for an extension spring.

Wire diameter, d

The diameter of the spring wire.

Mean diameter, D

The coil diameter measured from wire centerline to wire centerline.

Outer diameter, OD

The outside diameter of the spring body.

Inner diameter, ID

The inside diameter of the spring body.

Spring index, C

The ratio C = D/d. It is used for body stress and Wahl factor calculations.

Hook index, C₁ and C₂

Index values based on hook or loop radii. They are used for hook stress correction factors.

Initial tension / preload

The force that must be overcome before the extension spring starts to open or extend.

Spring body stress

The torsional shear stress in the coiled body of the spring.

Hook stress

The stress at the spring end. Extension spring hooks often control the final safety factor.

Fatigue loading

Repeated loading and unloading. Fatigue checks are needed when the spring works through many cycles.

Static extension spring formulas

Static design checks the spring body and the end hooks or loops under a maximum load. Extension spring ends are often critical, so both point A and point B should be reviewed.

Parameter	Formula	Meaning
Outer diameter	OD = D + d	Outside diameter of the spring body.
Inner diameter	ID = D - d	Inside diameter of the spring body.
Spring index	C = D / d	Used for body stress correction.
Hook index at point A	C₁ = 2R₁ / d	Index based on hook radius R₁.
Hook index at point B	C₂ = 2R₂ / d	Index based on hook radius R₂. A common recommendation is C₂ > 4.
Curvature correction at point A	K_A = (4C₁² - C₁ - 1) / [4C₁(C₁ - 1)]	Correction factor for tensile stress at point A of the hook.
Stress correction at point B	K_B = (4C₂ - 1) / (4C₂ - 4)	Correction factor for stress at point B of the hook.
Wahl factor	K_w = (4C - 1)/(4C - 4) + 0.615/C	Correction factor for spring body shear stress.
Tensile stress at point A	σ_A = F · [K_A · 16D/(πd³) + 4/(πd²)]	Combined bending and direct tensile stress at hook point A.
Shear stress at point B	τ_B = K_B · 8FD/(πd³)	Corrected hook stress at point B.
Spring body shear stress	τ_body = K_w · 8FD/(πd³)	Corrected shear stress in the coiled body.
Static factor of safety	n = allowable stress / calculated stress	Used separately for point A, point B, and the spring body.

Extension spring fatigue formulas

Fatigue design uses maximum and minimum cyclic force to calculate alternating and midrange stresses. The hook and body stresses should be checked separately because the lowest factor of safety controls the design.

Parameter	Formula	Meaning
Force amplitude	F_a = (F_max - F_min) / 2	Alternating part of the spring force.
Midrange force	F_m = (F_max + F_min) / 2	Average force during the load cycle.
Torsional rupture strength	S_su = 0.67S_ut	Approximate torsional rupture strength from ultimate tensile strength.
Tensile stress amplitude at point A	σ_Aa = F_a · [K_A · 16D/(πd³) + 4/(πd²)]	Alternating tensile stress at hook point A.
Midrange tensile stress at point A	σ_Am = F_m · [K_A · 16D/(πd³) + 4/(πd²)]	Midrange tensile stress at hook point A.
Shear stress amplitude at point B	τ_Ba = K_B · 8F_aD/(πd³)	Alternating hook shear stress at point B.
Midrange shear stress at point B	τ_Bm = K_B · 8F_mD/(πd³)	Midrange hook shear stress at point B.
Spring body stress amplitude	τ_body,a = K_w · 8F_aD/(πd³)	Alternating shear stress in the spring body.
Spring body midrange stress	τ_body,m = K_w · 8F_mD/(πd³)	Midrange shear stress in the spring body.
Zimmerli unpeened amplitude component	S_sa = 241 MPa	Amplitude endurance component used in the legacy fatigue formulas.
Zimmerli unpeened midrange component	S_sm = 379 MPa	Midrange endurance component used in the legacy fatigue formulas.
Gerber shear endurance relation	S_se,gerber = S_sa / [1 - (S_sm/S_su)²]	Gerber-type endurance limit for shear stress.
Goodman shear endurance relation	S_se,goodman = S_sa / [1 - (S_sm/S_su)]	Goodman-type endurance limit for shear stress.
Gerber tensile endurance relation	S_e,gerber = (S_r/2) / [1 - (S_r/(2S_ut))²]	Gerber-type endurance relation for hook tensile stress.
Goodman tensile endurance relation	S_e,goodman = (S_r/2) / [1 - S_r/(2S_ut)]	Goodman-type endurance relation for hook tensile stress.
Goodman factor of safety	n = S_eS_u / (stress_aS_u + stress_mS_e)	Use matching tensile or shear endurance and strength terms for point A, point B, or body.

Symbol glossary

Symbol	Definition
F	Axial force on the spring
F_max	Maximum cyclic force
F_min	Minimum cyclic force or preload force during the cycle
F_a	Force amplitude
F_m	Midrange force
d	Wire diameter
D	Spring mean diameter
OD	Spring outer diameter
ID	Spring inner diameter
C	Spring body index, D/d
C₁	Hook index based on radius R₁
C₂	Hook index based on radius R₂
K_A	Curvature correction factor for point A
K_B	Stress correction factor for point B
K_w	Wahl factor for spring body stress
S_ut	Ultimate tensile strength
S_su	Torsional rupture strength
S_r	Allowable bending strength for cycling loading at the spring end
σ	Normal tensile stress
τ	Shear stress