The strain energy of a structure due to bending is given by

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BPSC AE Paper 5 (Civil) 11 Nov 2022 Official Paper
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  1. \(\frac{1}{3} \int \frac{M^2 d x}{E I}\)
  2. \(\frac{1}{2} \int \frac{M^2 d x}{E I} \)
  3. \( \int \frac{M^2 d x}{E I}\)
  4. \(2 \int \frac{M^2 d x}{E I}\)

Answer (Detailed Solution Below)

Option 2 : \(\frac{1}{2} \int \frac{M^2 d x}{E I} \)
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Concepts:

The strain energy is the energy stored in a body due to its elastic deformation.

The most general formula for strain energy is:

\(U = {\frac{1}{2}} \times F \times \delta \)

Where, F is the applied force and δ is deformation

However, When stress σ is proportional to strain, ϵ, the strain energy formula is: 

   \(U = {\frac{1}{2}} \times V \times \sigma \times \epsilon\); V is the volume of material.

Based on the above formals, the strain energy stored in the material due to different types of forces/moments is given as:

1. Strain energy due to bending is:

   \(U = \int\frac{M^{2}}{2EI}dx\)

2. Strain energy due to Torsion is :

    \(U = \int\frac{T^2L}{2GJ}dx\)

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