What is the spring rate formula for a compression spring?

A common compression spring rate formula is k = d^4 G / (8 D^3 Na), where d is wire diameter, D is mean diameter, G is shear modulus, and Na is number of active coils.

What is spring index?

Spring index is the ratio of mean coil diameter to wire diameter, C = D/d. It is used as a practical indicator for manufacturability and stress concentration.

What is the Wahl factor?

The Wahl factor is a curvature correction factor used to estimate spring body shear stress more realistically in helical compression springs.

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Compression Spring Formulas and Definitions

This page combines the most useful compression spring formulas and compression spring definitions into one reference page. It is designed for both general internet users and engineers.

Use it when you want to understand spring terms, check symbols, compare end types, review buckling constants, or see the main equations used in compression spring design, static stress checks, fatigue checks, and stability checks.

Definitions End types Core formulas Buckling constants Fatigue formulas Symbols

Compression spring definitions

Compression spring

A helical compression spring resists axial compression and stores energy when it is shortened under load.

Axial load, F

A load acting parallel to the spring axis. In normal use, the load is applied concentrically along the axis.

Free length, L_f

The overall spring length in the unloaded condition.

Solid height, L_s

The spring length when the coils are fully closed. This is often close to the highest-load condition.

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.

Inner diameter, ID

The inside diameter of the spring.

Spring rate, k

The force required per unit deflection. A larger spring rate means a stiffer spring.

Deflection

The change in spring length caused by the applied load.

Pitch, p

The axial spacing between corresponding points on neighboring coils in the free condition.

Active coils, N_a

The coils that actually twist and contribute to spring deflection.

Total coils, N_t

The full number of coils, including inactive or end coils.

Spring index, C

The ratio of mean diameter to wire diameter, C = D/d. It helps indicate manufacturability and stress concentration.

Wahl factor, K_w

A stress-correction factor that includes curvature effects in the spring body shear stress.

Buckling

Lateral instability of a slender spring under compression. Long springs often need a stability check.

Common compression spring end types

End type changes active-coil count, total-coil count, solid height, and pitch relations.

Plain end

No pitch change at the end. Simple and economical, but often needs a proper seat for stability.

Plain and ground end

The end is ground flatter to improve seating and load transfer.

Closed end

The end coils are brought closer together. This changes inactive-coil behavior and the dimensional formulas.

Closed and ground end

A common practical end form that improves seating between flat surfaces.

Parameter	Plain	Plain and ground	Closed	Closed and ground
Total coils, N_t	N_a	N_a + 1	N_a + 2	N_a + 2
Free length, L_f	pN_a + d	p(N_a + 1)	pN_a + 3d	pN_a + 2d
Solid height, L_s	d(N_t + 1)	dN_t	d(N_t + 1)	dN_t
Pitch, p	(L_f - d) / N_a	L_f / (N_a + 1)	(L_f - 3d) / N_a	(L_f - 2d) / N_a

Core compression spring formulas

Parameter	Formula	Practical meaning
Outer diameter, OD	OD = D + d	Outside spring diameter.
Inner diameter, ID	ID = D - d	Inside spring diameter.
Spring index, C	C = D / d	Indicates curvature severity and manufacturability.
Wahl factor, K_w	K_w = (4C - 1)/(4C - 4) + 0.615/C	Stress correction for curved wire.
Curvature-only factor, K_s	K_s = (2C + 1)/(2C)	Often used for preset or prestressed spring-body checks.
Spring rate, k	k = d^4 G / (8 D^3 N_a)	Stiffness of the spring.
Shear stress with Wahl factor	τ = K_w · 8FD / (π d^3)	Used for unprestressed spring-body stress checks.
Shear stress with curvature-only factor	τ = K_s · 8FD / (π d^3)	Often used for preset spring checks.
OD at solid height	OD_s = √(D^2 + (p^2 - d^2)/π^2) + d	Useful when the spring must fit inside a bore at solid condition.
Hooke’s law	ΔF = k Δx	Connects change in force to change in deflection.

Allowable torsional stress for static compression spring design

The table below gives commonly cited allowable torsional stress levels for helical compression springs in static applications, expressed as a percentage of tensile strength. These values are practical design guidance, not a substitute for full validation.

“Unprestressed” refers to springs before set removal and typically includes K_w or K_b. “Prestressed” refers to springs after set removal and typically includes K_s.

Material	Unprestressed (before set removed)	Prestressed (after set removed)
Music wire and cold-drawn carbon steel	45% of tensile strength	60–70% of tensile strength
Hardened and tempered carbon and low-alloy steel	50% of tensile strength	65–75% of tensile strength
Austenitic stainless steels	35% of tensile strength	55–65% of tensile strength
Nonferrous alloys	35% of tensile strength	55–65% of tensile strength

Meaning of factors

K_w: Wahl factor
K_b: Bergsträsser factor
K_s: shear-stress correction factor

Source guidance: Shigley’s Mechanical Engineering Design and the Standard Handbook of Machine Design.

Buckling and stability constants

The classic compression spring stability relation can be written as:

L_f < (πD / α) · √( 2(E - G) / (2G + E) )

Here, α depends on the end condition of the spring support.

End condition	Constant α
Spring supported between flat parallel surfaces (fixed ends)	0.5
One end fixed, other end pivoted	0.707
Both ends pivoted	1
One end clamped, other end free	2

Ends supported by flat surfaces should generally be squared and ground for good seating.

Compression spring fatigue formulas

Parameter	Formula	Meaning
Force amplitude, F_a	F_a = (F_max - F_min) / 2	Alternating part of the load.
Midrange force, F_m	F_m = (F_max + F_min) / 2	Average part of the load.
Shear stress amplitude, τ_a	τ_a = K_w · 8F_aD / (π d^3)	Alternating stress in the spring body.
Midrange shear stress, τ_m	τ_m = K_w · 8F_mD / (π d^3)	Average stress in the spring body.
Torsional rupture strength, S_su	S_su = 0.67 S_ut	Approximate torsional rupture strength.
Slope line, r	r = τ_a / τ_m	Useful in the fatigue-limit relations.
Zimmerli shot-peened amplitude constant	S_sa = 398 MPa	Infinite-life amplitude component for shot-peened springs.
Zimmerli shot-peened midrange constant	S_sm = 534 MPa	Infinite-life midrange component for shot-peened springs.
Zimmerli unpeened amplitude constant	S_sa = 241 MPa	Infinite-life amplitude component for unpeened springs.
Zimmerli unpeened midrange constant	S_sm = 379 MPa	Infinite-life midrange component for unpeened springs.
Gerber-type endurance limit	S_se = S_sa / (1 - (S_sm / S_su)^2)	Shear endurance relation using a Gerber form.
Goodman-type endurance limit	S_se = S_sa / (1 - S_sm / S_su)	Shear endurance relation using a Goodman form.
Fatigue factor of safety	fos_f = S_sa,lim / τ_a	Margin against fatigue failure for the chosen criterion.

Symbol glossary

Symbol	Definition
OD	Spring outer diameter
ID	Spring inner diameter
D	Spring mean diameter
d	Wire diameter
p	Pitch
L_f	Spring free length
L_s	Spring solid height
F	Axial force
F_s	Force at solid length
ΔF	Force difference
Δx	Deflection change
G	Shear modulus
E	Elastic modulus
N_a	Number of active coils
N_t	Number of total coils
C	Spring index
K_w	Wahl factor
K_s	Curvature-only stress factor
α	End-condition constant for buckling/stability
S_ut	Ultimate tensile strength
S_su	Torsional rupture strength