Energy Methods MCQ with answer and solution

Concept

Castigliano's second theorem is a fundamental energy method used to calculate displacements and slopes in both statically determinate and indeterminate structures.

Formula

δ_i = dU / dP_i

Step-by-step solution

1. Understand Castigliano's second theorem: Deflection = partial derivative of total strain energy (U) with respect to the applied load (P). 2. Recognize that strain energy U depends on internal moments, shears, or axial forces, which can be determined for both determinate structures (via equilibrium) and indeterminate structures (via compatibility along with equilibrium). 3. Therefore, the method is valid for statically determinate as well as indeterminate beams, frames, and trusses.

Explanation

Castigliano's second theorem states that the partial derivative of the total strain energy with respect to any external load (force or moment) gives the displacement (or rotation) in the direction of that load. Because strain energy can be formulated for any elastic structure regardless of whether its internal forces can be found using equations of equilibrium alone, this theorem is universally applicable to both statically determinate and indeterminate systems (beams, frames, and trusses).

Why the other options are wrong

A: Incorrectly restricts the theorem to determinate beams and frames only. B: Incorrectly restricts the theorem to determinate trusses only. C: Leaves out indeterminate structures, which can also be analyzed using Castigliano's theorems.

Shortcut

Castigliano's second theorem applies to all linear elastic structures (determinate and indeterminate).

Common mistake

Thinking energy methods like Castigliano's theorem are restricted only to determinate structures.