## Q. 14.8

Design of a Nine-Leaf Cantilever Spring

A steel 0.9 m long cantilever spring has 80 mm wide nine leaves. The spring is subjected to a concentrated load $P$ at its free end.

Find: The depth of the leaves and the largest bending stress.

Given: $b$=80 mm, $L$=0.9 m, $P$=2.5 kN, $n=9 E$=200 GPa, $\nu$=0.3.

Assumption: Maximum vertical deflection caused by the load will be limited to 50 mm.

## Verified Solution

Equation (14.43) may be rearranged into the form

$\delta=\left(1- \nu ^2\right) \frac{6 P L}{E n b}\left(\frac{L}{h}\right)^3$      (14.43)

$h^3=\left(1- \nu ^2\right) \frac{6 P L}{E n \delta}$     (d)

Inserting the given data, we have

$h^3=\left(1-0.3^2\right) \frac{6(2500)(0.9)^3}{\left(200 \times 10^9\right)(9)(0.08)(0.05)}=1.382\left(10^{-6}\right)$

or

$h=0.0111 m =11.1 mm$

Equation (14.42) results in the maximum stress as

$\sigma=\frac{6 P L}{n b h^2}$      (14.42)

$\sigma_{\max }=\frac{6 P L}{n b h^2}=\frac{6(2500)(0.9)}{9(0.08)(0.0111)^2}=152.2 MPa$

The Goodman criterion may be used in the design of leaf springs subject to cyclic loading, as illustrated in the solution of the following numerical problem