---
title: "Physical significance of flexibility and stiffness matrices"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:flex-stiff-significance"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/flex-stiff-significance
site: BCT Engineering
---

# Physical significance of flexibility and stiffness matrices

Theory of Structures II · Chapter 2: Force method

- Asked in: 2079 Baishakh, 4 marks
- Video: [Difference between Stiffness and Flexibility with example](https://www.youtube.com/watch?v=8SjAhf4vB3k) — Civil Engineering
- 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/flex-stiff-significance
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Explain the physical significance of flexibility and stiffness matrices with suitable examples.

## Short answer

Flexibility matrix [F] gives displacements {Δ} = [F]{P}; element f_ij is displacement at i from a unit force at j, measuring softness. Stiffness matrix [K] gives forces {P} = [K]{Δ}; k_ij is force at i for unit displacement at j with other coordinates fixed, measuring stiffness. They are symmetric and [K] = [F]^{-1}.

## Answer

#### Flexibility matrix $[F]$
Each element $f_{ij}$ is the displacement at coordinate $i$ caused by a unit force at coordinate $j$. The relation is $\{\Delta\} = [F]\{P\}$. It measures how *soft* the structure is: a large $f_{ij}$ means a small force gives a large displacement. Column $j$ is the deflected shape produced by a unit load at $j$. Diagonal terms are always positive, and the matrix is symmetric ($f_{ij} = f_{ji}$, Maxwell's theorem). It exists only for a stable, restrained structure and is the matrix used in the force method.

#### Stiffness matrix $[K]$
Each element $k_{ij}$ is the force needed at coordinate $i$ to give a unit displacement at coordinate $j$ while all other coordinates are held fixed. The relation is $\{P\} = [K]\{\Delta\}$. It measures how *stiff* the structure is: column $j$ is the set of restraint forces that holds the structure in the shape of a unit displacement at $j$. Diagonal terms are positive, and the matrix is symmetric. It is used in the displacement method.

The two are inverses: $[K] = [F]^{-1}$.

#### Example: cantilever of length $L$
Coordinates: 1 = vertical deflection at the free end, 2 = rotation at the free end.

$$
[F] = \begin{bmatrix} \dfrac{L^3}{3EI} & \dfrac{L^2}{2EI} \\[2mm] \dfrac{L^2}{2EI} & \dfrac{L}{EI} \end{bmatrix}, \qquad
[K] = [F]^{-1} = \begin{bmatrix} \dfrac{12EI}{L^3} & -\dfrac{6EI}{L^2} \\[2mm] -\dfrac{6EI}{L^2} & \dfrac{4EI}{L} \end{bmatrix}
$$

- $f_{11} = L^3/3EI$ is the tip deflection due to a unit tip load.
- $k_{22} = 4EI/L$ is the moment needed at the tip to give a unit rotation when the tip deflection is held zero.
- Note $k_{11} = 12EI/L^3 \neq 1/f_{11} = 3EI/L^3$. The stiffness is not the reciprocal of the flexibility for a matrix, because the stiffness holds the other coordinate fixed while the flexibility leaves it free.

## Check yourself

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

1. A large flexibility coefficient means:
   A. A small force gives a large displacement
   B. The structure is stiff
   C. The force is zero
   D. The matrix is singular
   Answer: A. Flexibility measures how soft the structure is. (Key point: Flexibility meaning)

2. The relation between [F] and [K] is:
   A. [K] = [F]^-1
   B. [K] = [F]
   C. [K] = [F]^T
   D. [K] x [F] = 0
   Answer: A. They are matrix inverses. (Key point: Matrix relation)

3. Why is k11 (12EI/L^3) not 1/f11 (3EI/L^3)?
   A. Stiffness holds the other coordinate fixed, flexibility leaves it free
   B. Arithmetic error
   C. Units differ
   D. L differs
   Answer: A. Individual coefficients are not reciprocals; the matrices are inverses. (Key point: Why not reciprocal)

4. Flexibility matrix is used in the ___ method and stiffness matrix in the ___ method.
   A. displacement, force
   B. force, displacement
   C. force, force
   D. displacement, displacement
   Answer: B. Flexibility goes with the force method, stiffness with the displacement method. (Key point: Usage)

## Related in Theory of Structures II

- [Properties of stiffness and flexibility matrices](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astiffness-flexibility-matrix-properties) — Theory of Structures II, chapter 2
- [Flexibility, stiffness coefficient; suitable method](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflex-stiff-coefficient-frame-method) — Theory of Structures II, chapter 1
- [Define flexibility and stiffness](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-flexibility-stiffness-old) — Theory of Structures II, chapter 2
- [Flexibility, stiffness; properties of flexibility matrix](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflexibility-stiffness-properties-flexibility-matrix) — Theory of Structures II, chapter 2
- [Stiffness matrix: element matrices, member forces, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astiffness-element-matrices-600kn-bmd) — Theory of Structures II, chapter 3

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