**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.

Step-by-Step

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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) |
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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 |
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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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