---
title: "Theorems on displacement"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:theorems-on-displacement"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/theorems-on-displacement
site: BCT Engineering
---

# Theorems on displacement

Theory of Structures II · Chapter 2: Force method

- Asked in: 2076 Chaitra, 4 marks
- Video: [Moment Area Theorems Explained - Structural Analysis](https://www.youtube.com/watch?v=4W47rgTE09M) — structurefree
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/theorems-on-displacement
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Explain theorems on displacement with suitable illustration.

## Short answer

Maxwell's theorem states deflection at A due to a unit load at B equals deflection at B due to a unit load at A. Betti's law generalizes this to work reciprocity. Castigliano's second theorem gives displacement as the partial derivative of strain energy, and the unit load method uses the integral of M m over EI.

## Answer

Displacements of an elastic structure can be found, and checked, with the following theorems.

#### 1. Maxwell's reciprocal displacement theorem
In a linearly elastic structure, the deflection at point A due to a unit load at point B equals the deflection at B due to a unit load at A:

$$
\delta_{AB} = \delta_{BA}
$$

**Illustration.** Take a cantilever of length $L$ with A at mid-span and B at the free end.

- Unit load at B: deflection at A $= \dfrac{5L^3}{48EI}$.
- Unit load at A: deflection at B $= \dfrac{(L/2)^2\left(3L - L/2\right)}{6EI} = \dfrac{5L^3}{48EI}$.

The two are equal. The same reciprocity holds for rotations, and for mixed pairs: the rotation at A due to a unit load at B equals the deflection at B due to a unit couple at A.

#### 2. Betti's law
The work done by a first system of loads $P_1$ through the displacements caused by a second system $P_2$ equals the work of $P_2$ through the displacements caused by $P_1$: $\sum P_1\Delta_{12} = \sum P_2\Delta_{21}$. Maxwell's theorem is the special case of two unit loads.

#### 3. Castigliano's second theorem
The displacement at a point in the direction of a load equals the partial derivative of the strain energy with respect to that load: $\Delta = \dfrac{\partial U}{\partial P}$. For a beam, $\Delta = \displaystyle\int \frac{M}{EI}\frac{\partial M}{\partial P}dx$.

#### 4. Unit load (virtual work) theorem
The displacement at a point is $\Delta = \displaystyle\int \frac{M\,m}{EI}\,dx$, where $M$ is the moment due to the real loads and $m$ is the moment due to a unit load at the point and in the direction of the required displacement.

These theorems give the flexibility coefficients and are used to apply the compatibility condition in the force method.

## Check yourself

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

1. What does Maxwell's reciprocal theorem state?
   A. The deflection at A due to a unit load at B equals the deflection at B due to a unit load at A
   B. The deflection at A equals the deflection at B under the same load
   C. The strain energy equals the work of the loads
   D. The deflection at A is proportional to the load at B squared
   Answer: A. delta_AB = delta_BA for a linear elastic structure; it also holds for rotations and for force-couple pairs. (Key point: Maxwell's theorem)

2. How are Betti's law and Maxwell's theorem related?
   A. They are unrelated theorems
   B. Betti's law is the general statement for two load systems; Maxwell's theorem is the special case of two unit loads
   C. Maxwell's theorem is general and Betti's law covers only unit loads
   D. Betti's law covers only rotations
   Answer: B. Betti: sum P1 Delta_12 = sum P2 Delta_21. Taking each system as a single unit load gives Maxwell. (Key point: Betti's law)

3. In Delta = integral of M m/EI dx, what is m?
   A. The moment due to the real loads
   B. The moment due to a unit load at the support
   C. The moment due to a unit load at the point and in the direction of the required displacement
   D. The slope of the real loads' moment diagram
   Answer: C. M is from the real loads and m from a unit virtual load placed where the displacement is wanted. (Key point: Unit load method)