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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .

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

Answer 1

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

\[ \Rightarrow \frac{dM}{dx} = \frac{WL}{2} - 2 \times \frac{Wx}{2}\]

\[ \Rightarrow \frac{dM}{dx} = \frac{WL}{2} - Wx\]

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

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

\[ \Rightarrow \frac{WL}{2} - Wx = 0\]

\[ \Rightarrow \frac{WL}{2} = Wx\]

\[ \Rightarrow x = \frac{L}{2}\]

\[\text { Now,} \]

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

\[\text { So,M is maximum at }x = \frac{L}{2} . \]