COIL & BELLEVILLE SPRINGS

Helical compression springs follow F = k·δ with stiffness k = G·d⁴/(8·D³·N). The Wahl correction factor accounts for the stress concentration on the inner edge of the wire due to curvature. Belleville (disc) springs use the Almen-Laszlo equations to capture their nonlinear load-deflection curve, which can be flat or even regressive depending on the height-to-thickness ratio h/t. Stacks in series add deflections; stacks in parallel add forces.

• k = G·d⁴ / (8·D³·N)
• τ = K_W · 8·F·D / (π·d³)
• K_W = (4C-1)/(4C-4) + 0.615/C, C = D/d
• σ_BV = 4E·δ / [(1-ν²)·M·De²]·[h - δ/2 + t]
• f_n = 0.5·sqrt(k·g/W)

References & StandardsSAE HS-795, DIN 2092 (Belleville), Shigley Ch. 10

Coil springs handle linear load-deflection problems: valves, suspension, seal preload, return springs. Belleville washers handle high-load short-deflection cases: bolted joints needing constant clamp force as gaskets relax, machine-tool spindles, or thermal-compensation in fastener stacks. Verify solid-height clearance, buckling for slender coils (Lf/D > 5.26 for free ends), and fatigue life if cyclic.