---
title: "Compatibility conditions; Maxwell's theorem"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:compatibility-conditions-maxwell"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/compatibility-conditions-maxwell
site: BCT Engineering
---

# Compatibility conditions; Maxwell's theorem

Theory of Structures II · Chapter 2: Force method

- Asked in: 2070 Chaitra (old course), 2+2 marks
- Video: [Betti's Law and Maxwell's Reciprocal Theorem [ HINDI ] | structural analysis-1](https://www.youtube.com/watch?v=wZs_q0AaTTw) — Ganesh Shegar
- 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/compatibility-conditions-maxwell
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Explain compatibility conditions. Also describe Maxwell's reciprocal theorem.

## Short answer

Compatibility conditions require continuous displacements that agree with support conditions, supplying extra equations for indeterminate structures. Maxwell's reciprocal theorem states δ_ij=δ_ji for linearly elastic structures, so flexibility and stiffness matrices are symmetric.

## Answer

#### Compatibility conditions
Compatibility conditions are the geometric requirements that the deformed structure must satisfy: displacements must be continuous inside members and at the joints, and must agree with the support conditions (for example zero deflection at a rigid support, zero rotation at a fixed end, equal rotation of connected members at a rigid joint). They are independent of the material and loading.

In an indeterminate structure, equilibrium alone cannot give all the forces. The extra equations come from compatibility. In the force method, the redundants $X_j$ are found from

$$
\Delta_{i0} + \sum_j f_{ij}X_j = \delta_i
$$

Example: for a propped cantilever with prop reaction $R_B$, the compatibility condition is that the deflection at the prop is zero, $\Delta_{B0} - f_{BB}R_B = 0$, so $R_B = \dfrac{wL^4/8EI}{L^3/3EI} = \dfrac{3wL}{8}$.

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

$$
\delta_{AB} = \delta_{BA}, \qquad f_{ij} = f_{ji}
$$

It follows from Betti's law (or from the independence of strain energy from the order of loading). It makes the flexibility and stiffness matrices symmetric, which reduces the number of coefficients to be found in the force method and gives a check on the computation. Example: for a cantilever of length $L$, the deflection at mid-span due to a unit tip load is $\dfrac{5L^3}{48EI}$, which equals the tip deflection due to a unit load at mid-span.

## Check yourself

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

1. What do compatibility conditions require?
   A. Forces balance at each joint
   B. Displacements are continuous and agree with support conditions
   C. The material is linear
   D. Loads are symmetric
   Answer: B. They are geometric requirements on the deformed shape, independent of material and load. (Key point: Compatibility conditions)

2. For a propped cantilever under UDL w, which compatibility condition gives R_B = 3wL/8?
   A. Rotation at A is zero and deflection at B is wL/8
   B. Moment at B is zero
   C. Deflection at the prop B is zero
   D. Shear at A is zero
   Answer: C. Setting the deflection at the prop to zero: wL^4/8EI = R_B L^3/3EI, so R_B = 3wL/8. (Key point: Propped cantilever example)

3. What does Maxwell's reciprocal theorem state and give?
   A. f_ij = -f_ji; antisymmetric matrix
   B. The deflection under a load is maximum at midspan
   C. Load and deflection are always proportional
   D. f_ij = f_ji in a linearly elastic structure; symmetric flexibility matrix
   Answer: D. Displacement at A due to unit load at B equals that at B due to unit load at A, making [F] symmetric and giving a check. (Key point: Maxwell reciprocal theorem)

## Related in Theory of Structures II

- [Betti's law and Maxwell's reciprocal theorem](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Abetti-maxwell-theorems) — Theory of Structures II, chapter 2
- [State and prove Maxwell's reciprocal theorem](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amaxwell-reciprocal-theorem) — Theory of Structures II, chapter 2

## Related in other subjects

- [Strain compatibility and displacement theory](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Acompatibility-small-displacement-check) — Mechanics of Solids, chapter 3
- [Maxwell's equations and continuity equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-continuity-equation) — Engineering Physics, chapter 9, asked 2×
- [Reciprocal levelling: RL of A, B at 1160.18 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-rl-b-1160-18) — Surveying I, chapter 5, asked 2×
- [Reciprocal levelling: RL of A, B at 1260.18 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-rl-b-1260-18) — Surveying I, chapter 5, asked 2×