Question

The elastic strain energy stored in a rectangular cantilever beam of length $L$, subjected to a bending moment $M$ applied at the end of it is:

• A. $\dfrac{M^2 L}{2EI}$
• B. $\dfrac{ML^2}{2AE}$
• C. $\dfrac{ML^2}{3EI}$
• D. $\dfrac{ML^2}{16EI}$

Hint:

For cantilever beams with a moment at the end, remember that bending moment is constant throughout the length. Use $\dfrac{M^2 L}{2EI}$ for stored strain energy.

Answer 1

The Correct Option is A

The elastic strain energy $U$ stored in a beam due to bending is given by: \[ U = \int_0^L \frac{M(x)^2}{2EI} dx \] For a cantilever beam of length $L$ with a moment $M$ applied at the free end, the moment throughout the beam is constant and equal to $M$.
Substitute this into the equation: \[ U = \int_0^L \frac{M^2}{2EI} dx = \frac{M^2}{2EI} \int_0^L dx = \frac{M^2 L}{2EI} \] Hence, the strain energy stored is $\dfrac{M^2 L}{2EI}$.