## Chapter 14

## 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.

## Step-by-Step

## 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