'The deflection caused by any external force is equal to the partial derivative of the strain energy with respect to that force' — this statement is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 2005
Detailed Explanation
Castigliano gave two related theorems. Textbooks differ in how they number them, so it helps to know both statements.
First theorem: \( P_i = \dfrac{\partial U}{\partial \delta_i} \) (U in terms of displacements). Second theorem: \( \delta_i = \dfrac{\partial U}{\partial P_i} \) (U in terms of forces).
Why “Castigliano's second theorem” Is Correct
Deflection = ∂U/∂P is Castigliano's second theorem in the standard numbering, which is the book's key. Some Indian textbooks call this the "first theorem" and call ∂U/∂R = 0 (the least work principle) the second, so read the options carefully.
| Theorem | Statement |
|---|---|
| Castigliano I | Force = ∂U/∂(displacement) |
| Castigliano II | Displacement = ∂U/∂(force) |
| Least work (minimum strain energy) | ∂U/∂R = 0 for a redundant reaction R |
| Maxwell's reciprocal | δ_ij = δ_ji |
Why the Other Options Are Wrong
- A. Castigliano's first theorem: In the standard numbering the first theorem gives force as ∂U/∂δ (some books swap the names).
- C. the theorem of minimum strain energy: Minimum strain energy (least work) gives ∂U/∂R = 0 for redundants.
- D. Maxwell's theorem: Maxwell's theorem is the reciprocal deflection theorem.
Key Point
δ = ∂U/∂P → Castigliano's second theorem (standard numbering).
Key Concepts for Students
- Least work is a special case: the redundant's displacement is zero.
- Use a fictitious load to find deflection where no load acts.
- Betti–Maxwell reciprocal theorem: δ_AB = δ_BA.
Practice more: this is question 43 in Slope and Deflection MCQs – Part 3. Try the whole set and check every answer instantly.