Question

A cylindrical sample of cross-sectional area \(A\), length \(L\), and Young's modulus \(E\) is subjected to a constant uniaxial load \(P\). Within the elastic limit, the expression for the total strain energy \((U)\) stored in the sample is

• A. \(U=\dfrac{PL^{2}}{2AE}\)
• B. \(U=\dfrac{PL}{4AE}\)
• C. \(U=\dfrac{PLA^{2}}{E}\)
• D. \(U=\dfrac{P^{2}L}{2AE}\)

Hint:

For an axially loaded bar, \[ \boxed{ U=\frac{P^2L}{2AE} =\frac12 P\delta. } \]

Answer 1

The Correct Option is D

Step 1: Recall the expression for strain energy. For a member subjected to an axial load, \[ U=\frac{1}{2}P\delta, \] where \(\delta\) is the axial deformation.

Step 2: Substitute the deformation. The axial deformation is \[ \delta=\frac{PL}{AE}. \] Hence, \[ U = \frac12 P \left(\frac{PL}{AE}\right) = \frac{P^2L}{2AE}. \] Therefore, \[ \boxed{ U=\frac{P^2L}{2AE} } \] Hence, \[ \boxed{(D)} \] is the correct answer.