Energy Methods MCQ with answer and solution

Concept

Strain energy derivative for structural displacement

Formula

Δ_i = ∂ U / ∂ P_i

Step-by-step solution

1. Castigliano's theorem applies to linearly elastic structures under conservative load systems. 2. Total strain energy U is expressed in terms of external applied forces P_i. 3. Differentiating U partially with respect to force P_i yields the corresponding displacement Δ_i at that load point: Δ_i = dU / dP_i. 4. Therefore, displacement is found using the partial derivative of strain energy with respect to force. Answer: A

Explanation

Castigliano's theorem states that the partial derivative of total strain energy stored in an elastic structure with respect to an applied concentrated force equals the deflection of the structure at the point of application along the line of action of that force.

Why the other options are wrong

B) Virtual work is an independent variational method not requiring energy differentiation. C) Equilibrium equations solve internal forces from statics directly. D) Compatibility conditions enforce geometric continuity rather than evaluating energy derivatives.

Shortcut

Derivative w.r.t Force = Displacement; Derivative w.r.t Displacement = Force.

Common mistake

Confusing differentiation with respect to load (yields displacement) with differentiation with respect to displacement (yields load).