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Extension spring

Force and stress at the installed (preload) and working extensions, plus the fatigue safety factor for cycling between them. Both the coil body (shear) and the hook (bending) are checked — the hook almost always governs, and it can't be shot-peened, so it's flagged separately.

Extension spring — working stroke k = —

Wire diameter d mm

Mean coil diameter D mm

Active (body) coils n_a turns

Initial tension F_i N

Preload extension δ₁ mm

Working extension δ₂ mm

Material

Body finish

Engineering notes

Two operating points

Rate k = G·d⁴ / (8·D³·n_a); total force F = F_i + k·δ at preload (δ₁) and working (δ₂). Coils stay closed until F > F_i.
Body shear τ = K_W · 8·F·D / (π·d³). Hook bending σ = K_b · 16·F·D / (π·d³) + 4·F/(π·d²), K_b from the hook bend radius (taken = coil radius).
S_ut = A / d^m; static shear yield ≈ 0.45·S_ut, hook bending yield ≈ 0.75·S_ut.

Fatigue

Body: Goodman with Zimmerli shear endurance (unpeened S_sa=241, S_sm=379 MPa; peened 398/534), S_su=0.67·S_ut.
Hook: Goodman in bending with endurance ≈ 0.4·S_ut. The hook usually governs and can't be shot-peened inside — generously radius it.
n_f ≥ 1 is infinite life (>10⁶ cycles). The reported n_f is the worse of body and hook.

Result

Live