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## Q. 7.3

A cantilever of length 2 metre fails when a load of 2 kN is applied at the free end. If the section of the beam is 40 mm × 60 mm, find the stress at the failure.

## Verified Solution

Given :
Length,   L = 2 m = 2 × 10³ mm
Load,       W = 2 kN = 2000 N
Section of beam is 40 mm × 60 mm.
∴ Width of beam, b = 40 mm
Depth of beam,    d = 60 mm

Fig. 7.10 (a) shows the section of the beam.
Section modulus of a rectangular section is given by equation (7.7).

∴                      $Z=\frac{b d^2}{6}=\frac{40 \times 60^2}{6}=24000 mm ^3$

Maximum bending moment for a cantilever shown in Fig. 7.10 is at the fixed end.

∴              $M=W \times L=2000 \times 2 \times 10^3=4 \times 10^6 Nmm$

Let           $σ_{\max}$ = Stress at the failure

Using equation (7.6), we get

$M=\sigma_{\max } . Z$

∴                $\sigma_{\max }=\frac{M}{Z}=\frac{4 \times 10^6}{24000}= \pmb{1 6 6 . 6 7 N / m m ^2.}$  