## Q. 14.3

Design of a Hard-Drawn Wire Compression Spring

A helical compression coil spring made of hard-drawn round wire with squared and ground ends (Figure 14.7d) has spring rate $k$, diameter $d$, and spring index $C$. The allowable force associated with a solid length is $P_{all}$.

Find: The wire diameter and the mean coil diameter for the case in which the spring is compressed solid.

Given: $C$=9, $P_{all}$=45 N.

Assumptions: Static loading conditions will be considered. Factor of safety based on yielding is $n$ = 1.8.

## Verified Solution

The direct shear factor, from Equation (14.7), is $K_s$=1+(0.615/9)=1.068. The ultimate strength is estimated using Equation (14.12) and Table 14.2 as

$K_s=1+\frac{0.61 .5}{C}$       (14.7)

$S_{u s}=A d^b$        (14.12)

$S_u=A d^b=1.51\left(10^9\right) d^{-0.201}$

in which $d$ is in millimeters. Expressing $d$ in meters, the foregoing becomes

$S_u=1.51\left(10^9\right)(1000)^{-0.201} d^{-0.201}=376.7\left(10^6\right) d^{-0.201}$

The yield strength in shear, referring to Table 14.3, is then

$S_u=0.42 S_u=158.2\left(10^6\right) d^{-0.201}$       (a)

Substitution of the given numerical values into Equation (14.6) together with $\tau_{ all } / n$, the maximum design shear stress is expressed as

$\tau_t=K_s \frac{8 P D}{\pi d^3}=K_s \frac{8 P C}{\pi d^2}$         (14.6)

\begin{aligned} \tau_{\text {all }} & =\frac{8 n K_s C P_{\text {all }}}{\pi d^2} \\ & =\frac{8(1.8)(1.068 \times 9)(45)}{\pi d^2}=\frac{1982.6}{d^2} \end{aligned}      (b)

Finally, equating Equations (a) and (b) results in

$158.2(10)^6 d^{-0.201}=1982.6 d^{-2}$

from which

$d=0.00188 m =1.88 mm$

Thus, the mean coil diameter equals

$D=C d=9(1.88)=16.92 mm$

Comment: A standard 1.9 mm diameter hard-drawn wire should be used.

 TABLE 14.2 Coefficients and Exponents for Equation (14.12) $A$ Material ASTM No. $b$ MPa ksi Hard-drawn wire A227 −0.201 1510 237 Music wire A228 −0.163 2060 186 Oil-tempered wire A229 −0 193 1610 146 Chrome-vanadium wire A232 −0.155 1790 173 Chrome-silicon wire A401 −0 091 1960 218 Source: Associated Spring-Barnes Group, Design Handbook, Associated Spring-Barnes Group, Bristol, CN, 1987.

 TABLE 14.3 Approximate Strength Ratios of Some Common Spring Materials Material $S_{y s} / S_u$ $S_{e s}^{\prime} / S_u$ Hard-drawn wire 0.42 0.21 Music wire 0.40 0.23 Oil-tempered wire 0.45 0.22 Chrome-vanadium wire 0.52 0.20 Chrome-silicon wire 0.52 0.20 Source: Associated Spring-Barnes Group, Design Handbook, Associated Spring-Barnes Group, Bristol, CN, 1987. Notes: $S_{y s}$, yield strength in shear; $S_u,$ ultimate strength in tension; $S_{e s}^{\prime}$, endurance limit (or strength) in shear.