Question

What is the strain energy stored in a body due to bending moment? *

Answer 1

Let's consider a small segment of beam of length $ds$ subjected to bending moment. One cross section rotates about another cross section by a small amount $d\theta$ and we can write:

$$d\theta=\dfrac{1}{R}ds=\dfrac{M}{EI}ds, (1)$$

here, $R$ is the radius of curvature of the bent beam, $M$ is the bending moment, $EI$ is the flexural rigidity of the beam.

Let's write the work done by the bending moment while the segment of beam rotating through angle $d\theta$:

$$dU=\dfrac{1}{2}Md\theta (2)$$

Substituting formula (1) into the formula (2), we get:

$$dU=\dfrac{1}{2}\dfrac{M^2}{EI}ds (3)$$

Finally, we can find the strain energy stored in the beam of span $L$ by integrating equation (3) over $L$:

$$U=\displaystyle\intop_{0}^L \dfrac{M^2}{2EI}ds.$$

Answer:

$U=\displaystyle\intop_{0}^L \dfrac{M^2}{2EI}ds.$