Spring Design: Method

Helical Spring Stresses, Deflection, Stability and Fatigue

Scope and source

The tool follows Chapter 10 of Budynas and Nisbett, Shigley's Mechanical Engineering Design: helical compression springs of round wire (Sections 10-1 to 10-10), extension springs with hook ends (Section 10-11) and torsion springs with straight ends (Section 10-12). Every equation and table number quoted on the results page is that book's. The default inputs are its Example 10-1 (No. 16 music wire, 7/16 in outside diameter, 12.5 squared coils), whose printed torsional yield strength, yield load, rate, deflection, solid and free lengths, buckling check and pitch the page reproduces; an assumed working load of 5 lbf is added for the working-stress and fatigue sections.

Wire strength and moduli

Spring wire is stronger the smaller it is drawn, so the tensile strength is a function of diameter: S_ut = A/d^m (Eq. 10-14) with the constants A and m of Table 10-4 for music wire A228, hard-drawn A227, oil-tempered A229, chrome-vanadium A232, chrome-silicon A401, 302 stainless A313 (three diameter bands) and phosphor bronze B159 (three bands). The constants are given in kpsi·in^m and in MPa·mm^m; the tool uses the set of the chosen unit system, so SI results match the SI column of the handbook rather than a converted USCS value (the two differ by a few tenths of a percent because the printed SI constants are rounded). E and G come from Table 10-5, which lists them by diameter band for music wire and hard-drawn wire and as single values for the other materials; oil-tempered A229 wire takes the oil-tempered row of that table (28.5 Mpsi, 11.2 Mpsi).

Allowable strengths are fractions of S_ut by material class: torsional yield for compression springs from Table 10-6 (45 % for music and cold-drawn carbon wire, 50 % for hardened-and-tempered carbon and low-alloy wire, 35 % for austenitic stainless and nonferrous wire before set removal; 65, 70 and 60 % after set removal, the middle of the handbook's ranges), the extension-spring body, hook-torsion and hook-bending allowables of Table 10-7, the torsion-spring bending yield of Table 10-9 (78, 85 and 61 % before set removal) and the cyclic allowables of Tables 10-8 (extension springs) and 10-10 (torsion springs).

Compression springs

Mean coil diameter D = OD − d, spring index C = D/d, recommended between 4 and 12. The shear stress in the wire is τ = K_B·8FD/(πd³) with the Bergsträsser factor K_B = (4C + 2)/(4C − 3), which the handbook uses for both static and fatigue stresses; Wahl's factor is shown for comparison and differs by less than 1.4 %. The rate is k = d⁴G/(8D³N_a), with the active coils, solid length, free length and pitch related by the end-condition table (Table 10-1: plain, plain and ground, squared, squared and ground). The static check finds the load that brings the wire to S_sy; when no free length is entered the spring is taken as wound to L₀ = L_s + F_y/k, the length at which it is just solid-safe (Example 10-1 f). With a free length, the load and stress at solid height follow and the factor of safety there should be at least 1.2 (Eq. 10-21); the clash allowance ξ = F_s/F_max − 1 should be at least 0.15 (Eq. 10-17, 10-20).

Buckling: with the end-condition constant α of Table 10-2, the effective slenderness is λ_eff = αL₀/D and the critical deflection y_cr = L₀C′₁[1 − (1 − C′₂/λ_eff²)^½] (Eq. 10-11), C′₁ = E/[2(E − G)], C′₂ = 2π²(E − G)/(2G + E). The spring is stable at every deflection when L₀ < (πD/α)[2(E − G)/(2G + E)]^½; for steel the handbook rounds the coefficient to 2.63 (Eq. 10-13). Both are printed; the margin uses the exact form with the material's own E and G, which for music wire gives a coefficient of 2.54 rather than 2.63, so a spring that the rounded rule calls just stable may show a small negative buckling margin.

Surge: the weight of the active coils is W = π²d²DN_aγ/4 and the fundamental frequency of a spring between flat parallel plates f = ½(kg/W)^½ (Eq. 10-25, 10-26; one quarter when one end is free). The handbook asks for a surge frequency 15 to 20 times the forcing frequency; the margin is taken against 15.

Fatigue: Zimmerli found that unpeened spring steels in torsion have an endurance limit that does not depend on size, alloy or strength, S_sa = 35 kpsi at S_sm = 55 kpsi (57.5 and 77.5 kpsi peened), with S_su = 0.67 S_ut. The alternating and midrange stresses are formed from F_a = (F₂ − F₁)/2 and F_m = (F₂ + F₁)/2 with K_B on both, and the factors of safety are the Gerber (S_se = S_sa/[1 − (S_sm/S_su)²], Eq. 10-28), Goodman (S_se = S_sa/(1 − S_sm/S_su), Eq. 10-29a) and Sines (n = S_sa/τ_a) criteria of Section 10-7. The Gerber value sets the fatigue margin, as the handbook's examples do.

The figure of merit fom = −(relative cost)π²d²N_tD/4 (Eq. 10-22) compares candidate springs of the same function by material cost, less negative being cheaper. For the entered wire, material, maximum load, clash allowance and n_s, the design index of Eq. (10-23) is also printed: it is the index at which a spring of this wire would be exactly solid-safe, and the handbook's design procedure (Section 10-8, Examples 10-2 and 10-3) sets C to it and reads the rest of the spring from k and the deflection.

Extension springs

Extension springs are wound with their coils pressed together by an initial tension F_i, which the hooks must overcome before the coils separate; the load-deflection line is F = F_i + ky. The preferred initial-tension stress for a given index is Eq. (10-41), τ_i = 33 500/exp(0.105C) ± 1000[4 − (C − 3)/6.5] psi (231/exp(0.105C) ± 6.9[4 − (C − 3)/6.5] MPa), and the tool checks the entered F_i against that band or, when none is entered, takes its lower end. The hooks add to the deflection, N_a = N_b + G/E (Eq. 10-40), and the free length inside full twisted-loop hooks is L₀ = 2(D − d) + (N_b + 1)d (Eq. 10-39).

The body is checked in torsion with K_B against the body allowable. The hook is checked at two points of Fig. 10-6: A, where the wire bends through the mean radius r₁ (bending plus the axial stress, σ_A = F[(K)_A16D/(πd³) + 4/(πd²)] with (K)_A = (4C₁² − C₁ − 1)/[4C₁(C₁ − 1)], C₁ = 2r₁/d, Eq. 10-34, 10-35), and B, the side bend of radius r₂ in torsion (τ_B = (K)_B8FD/(πd³), (K)_B = (4C₂ − 1)/(4C₂ − 4), C₂ = 2r₂/d, Eq. 10-36, 10-37). A hook formed on the coil has r₁ = D/2; r₂ should not be below 2d. Fatigue of the body uses Zimmerli's data as for a compression spring; the hooks use the repeated-stress allowables of Table 10-8 at the chosen life (body torsion, hook torsion and hook bending as percentages of S_ut, printed for A228 and 302 stainless and applied here to every material) turned into a Gerber intercept S_e = (S_r/2)/[1 − (S_r/(2S_u))²] (Eq. 10-42). A second body yield check places the preload on the load line through the initial-tension stress, as the handbook does in its extension-spring example.

Torsion springs

A torsion spring is loaded by a moment that winds the coils and stresses the wire in bending. The inner fibre carries the larger stress, σ = K_i32M/(πd³) with K_i = (4C² − C − 1)/[4C(C − 1)] (Eq. 10-43, 10-44). Straight ends of lengths l₁ and l₂ bend as well and add (l₁ + l₂)/(3πD) turns to the active count (Eq. 10-49); the rate per turn is k′ = d⁴E/(10.8DN_a) (Eq. 10-51), the constant 10.8 replacing the theoretical 10.2 to allow for friction between the coils and between the coils and the pin. The rotation under a moment is θ = M/k′ turns, and because the coils wind tighter the mean diameter under load is D′ = N_bD/(N_b + θ) (Eq. 10-52), so the inside diameter D′ − d must stay clear of the pin the spring works over. Static strength is the bending yield of Table 10-9; fatigue uses the repeated allowable S_r of Table 10-10 for the material class, life and surface condition, the Gerber intercept of Eq. (10-54) and the Gerber factor of Eq. (10-55). Moments must close the coils (wind the spring up); a moment that opens them is not covered.

Reading the results

Margins of safety are allowable over applied minus one: at solid height, at the maximum working load (or moment), for buckling (critical deflection over the deflection to solid), for surge (frequency ratio over 15) and for fatigue (the Gerber factor minus one); hooks carry their own static and fatigue margins. Notes above the results flag an index outside 4 to 12, fewer than 3 or more than 15 active coils, a clash allowance below the required value, a solid-height factor below the required value, a free length above the critical length, an initial tension outside the preferred band, a coil that closes onto its pin, and a wire diameter outside the tabulated range of its material. The drawing is a schematic with the coil and wire proportions as entered; the chart shows the load-deflection (or moment-rotation) line with the working points, the solid height and the yield level.

Not covered

Conical, barrel and variable-pitch springs, rectangular wire, Belleville washers, the second and higher surge modes, temperature effects on the moduli, relaxation at elevated temperature, stress-relief and presetting process details beyond the tabulated set-removed fractions, and the mode of hook failure by pull-out of a half-loop.