Spring Wire Stress (Wahl), Solid Height, and Buckling
The three checks the spring-rate calculation leaves out. Wire shear stress tau = Kw x 8 F D / (pi d^3), where the Wahl factor Kw = (4C - 1)/(4C - 4) + 0.615/C corrects the plain torsional stress for wire curvature and direct shear; solid height Nt d with ground ends, (Nt + 1) d unground, and the travel left before the spring stacks solid; and a slenderness screen L0/D against the 5.26 limit for squared-and-ground ends on flat plates. A 0.080 in wire on a 0.75 in coil at 5 lb runs 18,651 psi uncorrected but 21,545 psi with Kw = 1.155, and a tight index bites hard: Kw is 1.40 at C = 4 against 1.12 at C = 12. The allowable stress by material and duty is the spring maker's. Companion to helical-spring-rate.
Formula and source
C = D/d; Kw = (4C - 1)/(4C - 4) + 0.615/C; tau_uncorrected = 8 F D / (pi d^3); tau = Kw x tau_uncorrected; Ls = Nt d (ground ends) or (Nt + 1) d (unground); travel = L0 - Ls; slenderness = L0/D, stable under 5.26 for squared-and-ground ends on parallel flat plates.
The standard round-wire helical-compression-spring wire shear stress with the Wahl correction factor, the Shigley end-condition solid-height table, and the absolute-stability slenderness limit, per Machinery's Handbook and Shigley's Mechanical Engineering Design, by name.
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