Question

A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by $M=\frac{Wx}{3}-\frac{W}{3}\frac{x^3}{L^2}$. Find the point at which M is maximum in this case.

Answer 1

Given,

$M=\frac{W}{3}x-\frac{W}{3}\frac{x^3}{L^2}$

differentiating the above equation, we get,

$\frac{dM}{dx}=\frac{W}{3}-W\frac{x^2}{L^2}$

equate, $\frac{dM}{dx}=0$

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

$\frac{W}{3}=W\frac{x^2}{L^2}$

$x=\frac{L}{\sqrt{3}}$

$\frac{d^{2}M}{d{x}^2}=-2\frac{W}{L^2}x<0$

Therefore, $f\left(x\right)$ is maximum when $x=\frac{L}{\sqrt{3}}$