Axially loaded bar (gradual loading): A bar of length l and cross-sectional area A is subjected to a gradually applied tensile load W. What is the strain energy stored?

Introduction / Context:
Strain energy is the elastic energy stored in a member due to deformation. For axial members, it is a fundamental quantity used in energy methods (Castigliano’s theorem) and in estimating resilience and impact effects.

Given Data / Assumptions:

• Prismatic bar: length l, area A.
• Linearly elastic material with Young’s modulus E.
• Load W is applied gradually from 0 to W (not suddenly or with impact).

Concept / Approach:
Under axial loading, extension δ = W l / (A E). For a gradually applied load, the work done (and stored as strain energy) equals the area under the load–deflection curve, which is a triangle: U = (1/2) * W * δ. Substituting δ gives U = (1/2) * W * (W l / (A E)) = W^2 l / (2 A E).

Step-by-Step Solution:
Compute axial extension: δ = W l / (A E).Use gradual loading energy: U = (1/2) W δ.Substitute δ ⇒ U = (1/2) W * (W l / (A E)) = W^2 l / (2 A E).

Verification / Alternative check:
Compare with sudden loading (no impact): energy is U = W^2 l / (2 A E) for gradual loading, while the maximum deflection under sudden loading is twice the gradual case, changing energy relations accordingly. This confirms the role of the loading path.

Why Other Options Are Wrong:

• W l / (A E) is the deflection, not energy.
• W^2 l / (A E) misses the 1/2 factor from the triangular load–deflection relationship.
• W^2 / (2 A E l) has incorrect dimensionality.
• (1/2) A E l is unrelated to the applied load W.

Common Pitfalls:

• Confusing gradual and sudden loading; energy depends on the load application path.

Final Answer:
U = W^2 l / (2 A E).