Question

A simply supported beam of span L carries a concentrated load W at its mid-span. If the width b of the beam is constant throughout the span, then, with permissible bending stress as $\sigma$, the depth of the beam at mid-span will be

• A. $\frac{3WL}{2b\sigma }$
• B. $\sqrt{\frac{3WL}{2b\sigma }}$
• C. $\frac{6WL}{b\sigma }$
• D. $\sqrt{\frac{6WL}{b\sigma }}$

Answer 1

The correct option is B $\sqrt{\frac{3WL}{2b\sigma }}$


BM at any section X distant x from the support is given by

$M=\frac{w}{2}x$

Let depth at the mid span be d
According to BM formula

$\therefore \frac{M}{I}=\frac{E}{R}=\frac{\sigma }{y}$

$\Rightarrow \frac{W}{2}x=z.\sigma$

Let depth at section X distance x from support be $d_x$

$\therefore \frac{W}{2}x=\frac{1}{6}\sigma b{d}_x^2$

$\Rightarrow d_x=\sqrt{\frac{3wx}{\sigma b}}$

$\therefore$ At mid span,

$x=\frac{L}{2}$

$\therefore d=\sqrt{\frac{3wL}{2b\sigma }}$