## Q. 14.2

Allowable Load of a Helical Compression Spring

A helical compression spring for mechanical device is subjected to an axial load $P$. Determine:

a. The yield strength in the shear of the wire.

b. The allowable load $P$ corresponding to yielding.

Design Decisions: Use a 0.0625 in. music wire. The mean diameter of the helix is $D$=0.5 in. A safety factor of 1.5 is applied due to uncertainty about the yielding.

## Verified Solution

The spring index is $C=D / d=0.5 / 0.0625=8 .$

a. Through the use of Equation (14.12) and Table 14.2, we have

$S_{u s}=A d^b$          (14.12)

$S_u=A d^b=186\left(0.0625^{-0.163}\right)=292 ksi$

Then, by Table 14.3, $S_{y s}=0.4(292)=117 ksi$.

b. The allowable load is obtained by applying Equation (14.6) as

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

$P_{\text {all }}=\frac{\tau_{\text {all }} \pi d^2}{8 K_s C}$

where

$\tau_{ all }=\frac{S_{y s}}{n}=\frac{117}{1.5}=78 ksi$

$K_s=1+\frac{0.615}{8}=1.077$       (from Equation (14.7))

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

Hence,

$P_{\text {all }}=\frac{\pi(78,000)(0.0625)^2}{8(1.077)(8)}=13.9 lb$

 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.