Strain energy definition in strength of materials Strain energy of a structural member is the internal work stored due to deformation under loading. Which description best captures this definition for members that may elongate or shorten?

Introduction / Context:
Strain energy is the elastic energy stored within a body when it is subjected to loads that cause deformation. It underpins energy methods such as Castigliano’s theorems and provides insight into how structures distribute load and deform under service conditions.

Given Data / Assumptions:

• Linear elastic range is assumed so that energy is recoverable upon unloading.
• Members may experience tension (elongation) or compression (shortening).
• Deformations are small, and work done equals area under the load–deflection curve.

Concept / Approach:

For a uniaxial bar, strain energy U = ∫ P du = ∫ σ dε · Volume. In linear elasticity with constant E and uniform stress, U = σ^2/(2E) · Volume = P^2 L/(2 A E). This same concept generalizes to bending, shear, torsion, and combined states, always representing the internal work associated with the deformation resisted by the member.

Step-by-Step Solution:

Verification / Alternative check:

In bending, U = ∫ M^2/(2 E I) dx; in torsion, U = ∫ T^2/(2 G J) dx. These expressions represent energy stored due to deformation modes other than simple tension/compression, reinforcing the definition as “work done to deform.”

Why Other Options Are Wrong (individually):

• Each of (a), (b), and (c) is only a partial statement; strain energy encompasses all these cases simultaneously.
• “None of these” is incorrect because all are true facets of the same concept.

Common Pitfalls:

• Mistaking external work for purely dissipative effects; in the elastic range, it is stored and recoverable.
• Forgetting that compression stores energy just as tension does.

Final Answer:

all the above.