**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.

Step-by-Step

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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) |
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A |
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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 |
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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. |

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