| name | leaf-spring |
| description | Leaf spring design — semi-elliptic, quarter-elliptic, multi-leaf, stress and deflection formulas, progressive rate, interleaf friction, fatigue, SAE standards. |
| metadata | {"priority":7,"promptSignals":{"phrases":["leaf spring","semi-elliptic spring","parabolic spring","multi-leaf spring","quarter elliptic"],"minScore":3}} |
Leaf Spring Design — Complete Skill
Types
- Semi-elliptic: both ends pinned; load at center; most common (vehicle suspension)
- Quarter-elliptic (cantilever): one end fixed; load at free end
- Parabolic (tapered leaf): variable section; uniform stress distribution
- Multi-leaf: several leaves of same or different length with interleaf friction
Single-Leaf (Constant Section) — Semi-Elliptic
Length = 2L (total); span = 2L; load F at center
Deflection: δ = FL³ / (3EI) × 2 = FL³ / (4EI) [for semi-elliptic with two cantilever halves]
More precisely: δ = FL³ / (4Ebh³/12) = 3FL³ / (Ebh³) [where b=width, h=thickness]
Max bending stress: σ = 3FL / (bh²) [at center, top/bottom fiber]
Spring rate: k = Ebh³ / (4L³) [N/mm]
Multi-Leaf (Equal-Length Leaves)
n leaves, each width b, thickness h
Stiffness: k = Enb h³ / (4L³) [simply scales with n]
Stress: σ = 3FL / (nbh²) [distributed across n leaves]
SAE multi-leaf: leaves graduated in length; master leaf longest
Nipping: leaves pre-stressed by bending before assembly to equalize stress
Graduated-Length (Triangular Plate) Equivalent
Equivalent uniform-stress cantilever:
k = Ewb h³ / (6L³) [for triangular plate or equivalent graduated-leaf]
σ = 6FL / (wbh²)
Parabolic Spring (Uniform-Stress Design)
Leaf width tapers as b(x) = b₀(1-x/L)² → uniform σ along length
σ_max = constant = 6FL / (b₀h²)
δ = 6FL³ / (Eb₀h³) [more deflection than constant section]
Benefits: lighter, no interleaf friction, better fatigue life
Interleaf Friction
In multi-leaf springs: friction between leaves creates hysteresis
Friction coefficient μ ≈ 0.15–0.25 (dry steel-steel); ~0.07 with lubricant
Hysteresis band: ΔF ≈ 2μ × (contact force) × tanθ
Design consideration: friction improves damping but reduces precision and fatigue life
Eye and Mounting
- Shackle end allows length change as spring deflects
- Fixed eye provides pivot; shackle eye allows swing
- Wrap angle and bushing material critical for durability
Fatigue Design
Critical location: center clamp (stress concentration K_t ≈ 1.5–2.0)
At bolt holes: K_t up to 3.0
Shot peening: increases endurance limit 20–50%
SAE spring steel: S_u = 1400–1650 MPa (52100, 5160, EN45A)
Endurance limit (unpeened): σ_e ≈ 0.35 S_u; (shot peened): σ_e ≈ 0.50 S_u
SAE Materials (Automotive)
| Grade | S_u [MPa] | Use |
|---|
| 5160-H | 1480 | Automotive main spring |
| 9260 | 1550 | High strength |
| EN45A | 1400 | European standard |
| 52100 | 1600 | Truck heavy duty |
Heat treatment: Oil quench + temper 400–450°C → ~HRC 46-48
Design Procedure
- Determine load F, span 2L, required rate k and δ
- Select b and material; solve for h from k equation
- Check σ_max ≤ σ_allow (typically 600–900 MPa depending on material)
- Verify fatigue safety at center clamp with K_t
- Check eye geometry and shackle angle at max load
- Add nipping if multi-leaf to equalize stress
Progressive Rate
Achieved by:
- Auxiliary (helper) leaf engaging at bump stops
- Variable-pitch spacing between leaves
- Air bag assist
Progressive rate described by: F(δ) = k₁δ + k₂δ² for piecewise linear engagement
Output
Provide: leaf dimensions b × h [mm], L [mm], n (number of leaves), k [N/mm], σ_max [MPa] vs. σ_allow, δ at rated load [mm], fatigue life estimate, material and heat treatment.
