Chapter 14

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)
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.