Springs: Design, Stress and Fatigue

Compression, extension, torsion and disc springs: rate, stress, Wahl factor, buckling, surge and fatigue with a worked relief-valve spring.

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Types of spring
Spring rate
Stress and the Wahl factor
Worked example: the relief valve spring
Buckling, surge and solid height
Fatigue and set
Torsion, extension and disc springs
Spring materials
FAQ

Types of spring

A spring stores energy as elastic deformation and gives it back. Its job in a machine is to supply a force or a torque (a valve seat, a clutch, a latch), to absorb shock, to keep parts in contact (a cam follower) or to isolate vibration. The helical compression spring is the commonest; others are the extension spring (hooks at the ends and an initial tension), the torsion spring (a helix loaded by a moment), the disc (Belleville) spring (a stack of dished washers for large forces in small space), the leaf spring (vehicles), wave and flat springs (compact) and the gas spring. Fatigue life and relaxation, not static strength, usually decide the design.

Spring rate

k = F ÷ y = G d⁴ ÷ (8 D³ N_a)

Stress and the Wahl factor

Twisting the wire loads it in torsion: τ = 8 F D ÷ (π d³) in a straight bar. In a coil the inside of the curve is more heavily stressed, and the direct shear adds, so the stress is multiplied by the Wahl factor K_w = (4C − 1) ÷ (4C − 4) + 0.615 ÷ C, where C = D ÷ d is the spring index (4 to 12 is good manufacturing practice: a smaller index is hard to coil, a larger one tangles and buckles).

Worked example: the relief valve spring

Worked example: the spring that holds the 160 bar relief valve of the press shut

A 6 mm poppet sees 160 bar, so the spring must push with F = p A = 16 × π ÷ 4 × 6² = 452 N at the set point (Valves and Flow Control). Music wire d = 4 mm, mean diameter D = 24 mm, 7 active coils, squared and ground ends (9 total coils).

Spring index C = 24 ÷ 4 = 6 (in the good range); Wahl factor K_w = 1.150 + 0.103 = 1.253.
Rate k = 79 300 × 256 ÷ (8 × 13 824 × 7) = 26.2 N/mm; compressing it to 452 N takes 17.2 mm.
Stress τ = K_w × 8 F D ÷ (π d³) = 1.253 × 8 × 452 × 24 ÷ (π × 64) = 541 MPa. The static limit for a preset music-wire spring is about 45 percent of the tensile strength (1 650 MPa for 4 mm wire): 743 MPa, so the factor is 1.37.
Solid height H_s = N_t d = 36 mm. Free length L₀ = H_s + working deflection + 15 percent clash allowance + a little = 36 + 17.2 + 2.6 + 3 ≈ 59 mm; the force at solid would be 598 N.
The mass of the spring is about 67 g.

Force and corrected shear stress of the relief valve spring against deflection.

Buckling, surge and solid height

A slender compression spring can buckle sideways. The limit is L₀ ÷ D below about 2.63 ÷ α for the end condition (α = 0.5 for fixed–free ends, 0.7 for fixed–pinned, 1.0 for both ends pinned, 0.5 for both ends fixed); here L₀ ÷ D = 2.5, well inside the limit of about 2.63 ÷ 0.5 = 5.3 for the worst case, so no guide is needed. Longer springs need a guide rod or sleeve (with clearance, as friction and wear appear) or two springs in series with a plate between them. Surge is a standing wave running up and down the coils when the spring is cycled near its natural frequency: the coils clash and the stress doubles. The fundamental surge frequency with both ends fixed is f = ½ √(k ÷ m_active) in hertz, with k in N/m and m_active the mass of the active coils in kg: for this spring 355 Hz (the active mass is 52 g). Keep the operating frequency at least 13 times below (for a 1 Hz relief valve cycle that is easily satisfied) or use a lower-mass, higher-frequency design. Never compress a spring to solid in service: stop at about 85 percent of the solid deflection.

Fatigue and set

Springs that cycle millions of times fail by fatigue, almost always from a surface flaw at the inside of the coil, so wire quality, shot peening and the stress range matter. The Zimmerli data for unpeened music wire give a torsional endurance (10⁷ cycles) of S_sa = 241 MPa amplitude with a mean of S_sm = 379 MPa; a modified-Goodman line through those points with the ultimate in shear S_su = 0.67 S_ut allows the safety factor for any cycle to be found. Spring set (permanent shortening) occurs if the stress is too high; presetting (compressing to solid once at the factory) removes it and lets stresses up to about 0.6 S_ut be used for static duty.

Worked example: the relief valve cycles between 300 and 452 N

τ_max = 541 MPa and τ_min = 359 MPa: amplitude τ_a = 91 MPa, mean τ_m = 450 MPa.
S_su = 0.67 × 1 650 = 1 106 MPa; the endurance amplitude at zero mean is S_se = 241 ÷ (1 − 379 ÷ 1 106) = 367 MPa.
Fatigue safety factor n_f = 1 ÷ (τ_a ÷ S_se + τ_m ÷ S_su) = 1.53: acceptable for infinite life (for important springs aim at 1.2 or more, and shot-peen to raise it).
The same spring as a valve spring on an engine would cycle billions of times: use a chrome-silicon or chrome-vanadium wire, peen it and keep the stress amplitude under about 400 MPa.

Torsion, extension and disc springs

Torsion spring: torque is M = k_θ θ with k_θ = E d⁴ ÷ (64 D N) in N·mm per radian, the wire stressed in bending, not torsion (σ = K_i 32 M ÷ (π d³)). Example: E = 207 GPa, d = 2 mm, D = 15 mm, N = 6: k_θ = 575 N·mm/rad, so 1 500 N·mm turns it 149°. Load it in the direction that tightens the coil and support it on a rod.
Extension spring: wound with initial tension F_i that must be overcome before it extends; F = F_i + k y. With F_i = 12 N, k = 2.2 N/mm and a 40 mm extension: F = 100 N. The hooks are the weak point (stress concentrations): design them with a large radius and check them.
Disc (Belleville) springs are stacked: in parallel (same direction) the force adds, 3 discs of 3 000 N at 0.8 mm give 9 000 N at 0.8 mm; in series (alternating) the deflection adds, 4 discs give 3 000 N at 3.2 mm. Friction in a parallel stack gives hysteresis. They are used for bolted joints, clutches and heavy preloads.
Rubber, gas and wave springs fill gaps where space, noise or cost decide.

Spring materials

Material	Tensile strength (MPa)	Use	Max temperature
Music wire (ASTM A228)	1 600 to 2 200 (d = 4 mm: 1 650)	best fatigue, cheap	to 120 °C
Oil-tempered wire (A229)	1 200 to 1 600	general-purpose, large wire	to 150 °C
Chrome-silicon (A401)	1 700 to 2 000	valve springs, high stress and temperature	to 250 °C
Stainless 302/304 (A313)	1 100 to 1 600	corrosion resistance, lower fatigue	to 290 °C
Phosphor bronze, Be-Cu	700 to 1 300	conductive, non-magnetic	to 100 to 200 °C
Inconel X-750	1 200 to 1 400	high temperature	to 600 °C

Wire strength falls with diameter (thin wire is stronger): use the supplier’s chart. Stainless and non-ferrous springs fatigue worse than music wire at the same stress. Corrosion (hydrogen embrittlement after plating, stress corrosion of stainless) causes unexpected spring failures; see Corrosion and Protection. Safety: a compressed spring stores energy and can fly out: use a spring compressor, a guide and a cover, and never cut a loaded spring.

FAQ

What is the spring index?

The ratio D ÷ d of the mean coil diameter to the wire diameter. 4 to 12 is the manufacturing sweet spot; below 4 the stress concentration is severe, above 12 the spring tangles.

How many active coils?

Coils that deflect: the total less the dead (end) coils. For squared and ground ends, N_t = N_a + 2. The rate depends on N_a.

Why did my spring get shorter?

It took a set: the stress was above the elastic limit, or it relaxed at high temperature. Preset it, lower the stress, or choose a better material.

Can I stack two springs in parallel?

Yes, with one inside the other with opposite hand to avoid tangling; the rates add. Make sure that the inner spring does not bind and that both reach solid together.

References

Shigley, J. E., Mischke, C. R., Budynas, R. G., Mechanical Engineering Design, McGraw-Hill (springs); Wahl, A. M., Mechanical Springs, McGraw-Hill; Spring Manufacturers Institute, Handbook of Spring Design.
Machinery’s Handbook, Industrial Press (spring design and data); Associated Spring, Design Handbook; ISO 26909 (springs vocabulary), EN 13906 (cylindrical helical springs).
Zimmerli, F. P., Human Failures in Spring Applications, Mechanical Springs Handbook.

Last reviewed: October 2026. Material strengths and endurance data are typical; use the wire supplier’s values for design.