Question

A simply supported beam is subjected to a uniformly distributed load. Which one of the following statements is true?

• A. Maximum or minimum shear force occurs where the curvature is zero.
• B. Maximum or minimum bending moment occurs where the shear force is zero.
• C. Maximum or minimum bending moment occurs where the curvature is zero.
• D. Maximum bending moment and maximum shear force occur at the same section.

Answer 1

The correct option is B Maximum or minimum bending moment occurs where the shear force is zero.


For section AB

$Forequilibrium,\sum M_0=0$

$M+V\Delta x+W_x{\Delta }_x\frac{\Delta X}{2}-(M+\Delta M)=0$

$\Delta M=V\Delta x+W_x\frac{(\Delta x{)}^2}{2}$

${lim}_{\Delta x\to 0}\frac{\Delta M}{\Delta X}={lim}_{\Delta x\to 0}(V+W_x\frac{\left(\Delta x\right)}{2})$

$\Rightarrow \frac{dM}{dx}=V....\left(i\right)$

and we know that for any function to be maximum or minimum it's differential should be equal to zero.

Hence from equation (i) for bending moment (M) to be maximum or minimum
$\Rightarrow \frac{dM}{dx}=0$

$Hence,\frac{dM}{dx}=0\Rightarrow V=0$