#### Question

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .

Find the point at which M is maximum in a given case.

#### Solution

\[\text { Given }: \hspace{0.167em} M = \frac{Wx}{3} - \frac{W x^3}{3 L^2}\]

\[ \Rightarrow \frac{dM}{dx} = \frac{W}{3} - 3 \times \frac{W x^2}{3 L^2}\]

\[ \Rightarrow \frac{dM}{dx} = \frac{W}{3} - \frac{W x^2}{L^2}\]

\[\text { For maximum or minimum values of M, we must have }\]

\[\frac{dM}{dx} = 0\]

\[ \Rightarrow \frac{W}{3} - \frac{W x^2}{L^2} = 0\]

\[ \Rightarrow \frac{W}{3} = \frac{W x^2}{L^2}\]

\[ \Rightarrow x = \frac{L}{\sqrt{3}}\]

\[\text { Now }, \]

\[\frac{d^2 M}{d x^2} = - \frac{2Wx}{L^2} < 0\]

\[\text { So, M is maximum at } x = \frac{L}{\sqrt{3}} .\]