Worked Example: Mechanical Springs Design

This worked example demonstrates how to determine the required number of active coils for a given stiffness and calculate the maximum shear stress in the spring.

1. Problem Statement

A helical compression spring is designed to exert a force of $500 \text{ N}$ when deflected by $20 \text{ mm}$ . The spring is made of steel wire with a shear modulus $G = 79.3 \text{ GPa}$ . The mean coil diameter is $D = 30 \text{ mm}$ and the wire diameter is $d = 5 \text{ mm}$ . Calculate the required number of active coils and the maximum shear stress.

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2. Given Data

Axial Force (F)

Deflection (y)

Wire Diameter (d)

Mean Coil Diameter (D)

Shear Modulus (G)

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3. Formula and Calculation

Calculate the required spring rate ( $k$ ): $k = \frac{F}{y}$ For initial values: $k = \frac{500}{0.020} = 25,000 \text{ N/m}$

Calculate the required number of active coils ( $N_a$ ):

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The required number of active coils is approximately $9.18$ coils.

Calculate the spring index ( $C$ ) and the correction factor ( $K_s$ ): $C = \frac{D}{d}$ $K_s = 1 + \frac{0.5}{C}$ For initial values: $C = \frac{30}{5} = 6$ and $K_s = 1 + \frac{0.5}{6} = 1.0833$

Calculate the maximum shear stress ( $\tau$ ): $\tau = K_s \frac{8 F D}{\pi d^3}$ For initial values: $\tau = 1.0833 \times \frac{8 \times 500 \times 0.030}{\pi (0.005)^3} = 3.31 \times 10^8 \text{ Pa} = 331 \text{ MPa}$

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