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No. 16 music-wire compression spring: torsional yield strength, yield load, rate, solid and free lengths, buckling and pitch (Shigley Example 10-1)

Tool	Spring Design
Source	Budynas, R. G., and Nisbett, J. K. (2015). Shigley's Mechanical Engineering Design, 10th ed. McGraw-Hill Education, New York. Example 10-1 (Tables 10-1, 10-2, 10-4, 10-5, 10-6; Eqs. 10-1, 10-5, 10-7, 10-9, 10-13, 10-14).
Level	Undergraduate

Problem

A helical compression spring is made of No. 16 music wire (d = 0.037 in). The outside coil diameter is 7/16 in (mean diameter 0.400 in), the ends are squared and there are 12.5 total turns. Estimate the torsional yield strength of the wire, the static load corresponding to it, the spring rate, the deflection under that load, the solid length, the free length that leaves no permanent set after compression to solid, whether buckling is possible at that length, and the pitch of the body coils.

Inputs to Enter

Spring type helical compression, material music wire A228, set not removed, unpeened
Wire diameter 0.037 in, mean coil diameter 0.400 in, 12.5 total coils, squared (closed) ends, free length blank (solid-safe at yield), both ends fixed
Working loads 3 to 5 lbf (assumed, not part of the example), n_s 1.2, clash allowance 0.15, density 0.284 lb/in³

Open the Spring Design tool (loaded with this example)

Answers

S_ut = A/d^m = 201/0.037^0.145 = 324.2 kpsi (book 324); S_sy = 0.45 S_ut = 145.9 kpsi (book 145.8)
C = 10.81, K_B = 1.124; load at yield F = π d³ S_sy/(8 K_B D) = 6.45 lbf (book 6.45)
N_a = 10.5, G = 11.85 Mpsi: k = d⁴G/(8D³N_a) = 4.13 lbf/in (book 4.13); y = F/k = 1.56 in (book 1.56)
L_s = (N_t + 1)d = 0.500 in; L_0 = y + L_s = 2.06 in (book 2.06); p = (L_0 − 3d)/N_a = 0.186 in (book 0.186)
Buckling: Shigley's 2.63 D/α = 2.10 in > L_0 (book: unlikely); the exact Eq. 10-12 with E 29.0, G 11.85 Mpsi gives 2.03 in, so the tool's margin is −0.09 and it flags the case as borderline

Notes

The tool's default inputs are this spring; press Submit. The book's S_ut, S_sy, F, k, y, L_s, L_0 and p are reproduced to the printed digits (test_spring_design.php, 451 checks, with 11th-edition solutions-manual problems and a numpy cross-check in test-scripts/output/spring_check.py). The buckling verdict differs only because the book uses the rounded steel coefficient 2.63 where the tool evaluates the coefficient from the wire's own moduli.