---
title: "Castigliano's theorem and unit load method"
subject: "Mechanics of Solids"
chapter: 5
key: "semester_5/mechanics_of_solids:castigliano-unit-load-theorems"
source: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/castigliano-unit-load-theorems
site: BCT Engineering
---

# Castigliano's theorem and unit load method

Mechanics of Solids · Chapter 5: Deflections and Slope of Statically Determinate and Indeterminate Structures

- Asked in: year not given, 6 marks
- Video: [CASTIGLIANO'S THEOREM in Just Over 10 Minutes!](https://www.youtube.com/watch?v=JH6iOhOe8h0) — Less Boring Lectures
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/castigliano-unit-load-theorems
- Whole chapter: https://www.bctengineer.com/md/semester_5/mechanics_of_solids/chapter-5

## Question

State Castigliano's second theorem and prove it for a linearly elastic body. Explain the unit (dummy) load method for finding the deflection of a beam or frame and state where each method is used in engineering.

## Short answer

Castigliano's second theorem states that the partial derivative of strain energy with respect to a load gives the corresponding displacement. The unit load method applies a unit load at the point of interest and computes deflection as ∫(M m / EI) dx. Both methods are used for deflections in beams, frames, trusses, and for solving statically indeterminate structures.

## Answer

#### Strain energy expressions

For a linear elastic member (energy stored equals work done by gradually applied loads):

| Action | Strain energy |
|---|---|
| Axial | $U = \displaystyle\int \frac{N^2}{2AE}\,dx$ |
| Bending | $U = \displaystyle\int \frac{M^2}{2EI}\,dx$ |
| Torsion | $U = \displaystyle\int \frac{T^2}{2GJ}\,dx$ |
| Shear | $U = \displaystyle\int \frac{\alpha V^2}{2GA}\,dx$ ($\alpha$ = shear form factor) |

For slender beams the bending term dominates; the others are usually neglected.

#### Castigliano's second theorem

**Statement:** For a linearly elastic structure in equilibrium, the partial derivative of the total strain energy (written in terms of the loads) with respect to a load $P_i$ equals the displacement $\delta_i$ of the point of application of that load, in the direction of the load.

$$
\delta_i = \frac{\partial U}{\partial P_i},\qquad \theta_i = \frac{\partial U}{\partial M_i}
$$

**Proof:** Let loads $P_1,\dots,P_n$ act and produce energy $U$. Increase $P_i$ by $dP_i$; the energy becomes $U + \dfrac{\partial U}{\partial P_i}dP_i$.

Now apply the loads in the reverse order: first $dP_i$ alone, then $P_1\ldots P_n$. The energy is

$$
\tfrac12\,dP_i\,d\delta_i + U + dP_i\,\delta_i
$$

The first term is the work of $dP_i$ while it is applied alone. The last term is the work of $dP_i$ (already acting at its full value) as the other loads are applied and move its point of application through $\delta_i$. The final state, and so the stored energy, does not depend on the order of loading. Equating the two expressions and neglecting the second-order term $dP_i\,d\delta_i$:

$$
\frac{\partial U}{\partial P_i}\,dP_i = \delta_i\,dP_i \quad\Rightarrow\quad \delta_i = \frac{\partial U}{\partial P_i}
$$

If no load exists at the point of interest, apply a fictitious load $Q$ (or moment $M_0$), differentiate, then set $Q = 0$.

For beams, differentiating under the integral sign:

$$
\delta = \int \frac{M}{EI}\frac{\partial M}{\partial P}\,dx
$$

#### Unit (dummy) load method

Apply a unit load (or unit moment for slope) at the point and in the direction of the required deflection, and let $m$ be the bending moment due to it. By the principle of virtual work (external work $1\cdot\delta$ = internal work):

$$
1\cdot\delta = \int \frac{M\,m}{EI}\,dx
$$

For trusses: $\delta = \sum \dfrac{F\,f\,L}{AE}$, where $F$ is the member force from the real load and $f$ from the unit load. Because the derivation uses only virtual work, the method can also be used for non-linear material by replacing $M/EI$ by the actual curvature.

#### Use

- Deflections of beams, bent frames, hooks, curved bars and trusses, where double integration is lengthy.
- Deflection at a point where no load acts, or when the loading is discontinuous.
- Solution of **statically indeterminate** structures: choose the redundant $R$ and impose a compatibility condition, e.g. $\partial U/\partial R = 0$ when the support does not move.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does Castigliano's second theorem state?
   A. Displacement equals total strain energy
   B. Load equals dU/d(delta) only for non-linear bodies
   C. The partial derivative of U with respect to load P_i equals the displacement at P_i in its direction
   D. Strain energy equals force times deflection
   Answer: C. delta_i = dU/dP_i for a linearly elastic structure; theta_i = dU/dM_i for moments. (Key point: Castigliano's second theorem)

2. How is the deflection found at a point with no load?
   A. Apply a fictitious load Q there, differentiate, then set Q = 0
   B. It cannot be found
   C. Use the nearest load
   D. Apply Q and keep Q in the answer
   Answer: A. A dummy load or moment is added, dU/dQ is found, then Q is set to zero. (Key point: Fictitious load)

3. What is the unit load expression for beam deflection?
   A. delta = integral of M^2/(2EI) dx
   B. delta = integral of M m/(EI) dx
   C. delta = integral of m/(EI) dx
   D. delta = M L/(EI)
   Answer: B. By virtual work, 1 x delta = integral of M m/(EI) dx, where m is the moment due to the unit load. (Key point: Unit load method)

4. How is Castigliano's theorem used for a statically indeterminate structure with an unyielding support?
   A. Set the redundant to zero
   B. Neglect compatibility
   C. Choose a redundant R and impose dU/dR = 0
   D. Use only equilibrium equations
   Answer: C. The support does not move, so the displacement in the direction of the redundant, dU/dR, is zero. (Key point: Use of the methods)

## Related in Mechanics of Solids

- [Bent bracket deflection by unit load method](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Abracket-deflection-unit-load) — Mechanics of Solids, chapter 5
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