The GATE Grind

GATE 2025 AE – Question 29

Structures · Strength of materials: Strain energy, Castigliano's principles, three-dimensional Hooke's law, plane stress and strain, Euler buckling · 1 mark · Multiple choice

A uniform symmetric cross-section cantilever beam of length $L$ is subjected to a transverse force $P$ at the free end, as shown in the figure. The Young's modulus of the material is E and the moment of inertia is I. Ignoring the contributions due to transverse shear, the strain energy stored in the beam is _____.

A cantilever of length L fixed at the left, with a downward force P at the free end.
  1. $\frac{P^2L^3}{6EI}$
  2. $\frac{PL^3}{3EI}$
  3. $\frac{PL^3}{6EI}$
  4. $\frac{P^2L^3}{3EI}$

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Correct answer: (A) $\frac{P^2L^3}{6EI}$

Explanation

The bending moment at distance $x$ from the free end is $M = Px$, so the strain energy is $U = \int_0^L \frac{M^2}{2EI}dx = \frac{P^2}{2EI}\cdot\frac{L^3}{3} = \frac{P^2L^3}{6EI}$.