Question

A beam of cross section area $A$ is made of a material of young’s modulus $Y$. The beam is bent into the arc of a circle of radius $R$. The bending moment is proportional to:
(A) $\dfrac{Y}{R}$
(B) $\dfrac{Y}{{RA}}$
(C) $\dfrac{R}{Y}$
(D) $YR$

Answer 1

Hint: The bending moment of the beam can be determined by the bending equation of the material. The bending moment depends on the moment of inertia, young’s modulus and the radius of the gyration of the bending material.

Complete step by step answer:
The bending moment of the beam is given by,
$\dfrac{M}{I} = \dfrac{E}{R}$
Where, $M$ is the bending moment of the beam, $I$ is the moment of the inertia of the beam, $E$ is the young’s modulus of the beam and $R$ is the radius of the gyration of the bending beam.

Given that,
The beam of the cross sectional area, $A$,
The young’s modulus of the beam is, $Y$,
The beam is bent into the arc of a circle of radius is, $R$,
Now,
The bending moment of the beam is given by,
$\dfrac{M}{I} = \dfrac{E}{R}\,..................\left( 1 \right)$
By substituting the young’s modulus in the above equation (1), then the above equation (1) is written as,
$\dfrac{M}{I} = \dfrac{Y}{R}$
By keeping the bending moment in the above equation in one side and the other terms in the other side, then the above equation is written as,
$M = \dfrac{Y}{R} \times I$
By assuming the moment of inertia for the beam is constant, then the above equation is said to be, the bending moment is directly proportional to the $\dfrac{Y}{R}$.

Hence, the option (A) is the correct answer.

Note:The bending moment is directly proportional to the young’s modulus of the beam and the bending moment is inversely proportional to the bending radius or the radius of the gyration. The young’s modulus is of different value for different material and the radius of bending is increased, then the bending moment decreases.