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## Q. 10.16

A weight 200 N falls through a height h = 300 mm onto the middle of a simply supported beam of length 3 m. Calculate the required cross-sectional area A and the maximum bending is not to exceed 7 MPa and E = 10 GPa for the beam material.

## Verified Solution

We note the static deflection, $\delta_{ st }$ of the simply supported beam due to 200 N load at midpoint is $\delta_{\text {st }}=W L^3 / 48 E I$ and this assumed to be much less than 300 mm. Therefore, from the results of the previous problem

$\sigma_{\max }=\sqrt{\frac{18 E h W}{A L}}$

Therefore, the cross-sectional area is

$A=\frac{18 W E h}{\sigma_{\max }^2 L}=18 \frac{W}{\sigma_{\max }} \frac{E}{\sigma_{\max }} \frac{h}{L}$

$=\frac{(18)(200)}{(7)\left(10^6\right)} \frac{10\left(10^3\right)}{7} \frac{300}{3000} m ^2$

or          A = 0.07347 m² ⇒ A = 734.7 cm²

The required cross-sectional area of the beam is 734.7 cm².