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

# Properties of stiffness and flexibility matrices

Theory of Structures II · Chapter 2: Force method

- Asked in: 2071 Chaitra, 6 marks
- Video: [Properties Of Stiffness Matrix in FEM | Structural Analysis | FEM for Mechanical & Civil Engineering](https://www.youtube.com/watch?v=hiymSIY0GDk) — Mahesh Gadwantikar
- 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/stiffness-flexibility-matrix-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

List the properties of stiffness and flexibility matrices for a given system. Draw a neat sketch of a system and explain.

## Short answer

Flexibility matrix is square, symmetric with positive diagonal entries, gives displacements from forces; column j is the displaced shape for a unit force at coordinate j. Stiffness matrix is square, symmetric, positive definite when rigid-body motion is prevented, gives forces from displacements; column j lists holding forces for unit displacement at j. They are inverse.

## Answer

A system is described by $n$ coordinates (points and directions of force or displacement). The flexibility matrix $[F]$ gives displacements from forces, $\{\Delta\} = [F]\{P\}$, and the stiffness matrix $[K]$ gives forces from displacements, $\{P\} = [K]\{\Delta\}$.

#### Properties of the flexibility matrix $[F]$
1. It is a square matrix of order $n$ (number of coordinates).
2. It is symmetric, $f_{ij} = f_{ji}$ (Maxwell's reciprocal theorem).
3. The diagonal terms $f_{ii}$ are always positive (a force does positive work through its own displacement).
4. Off-diagonal terms may be positive, negative or zero.
5. It exists only for a stable, adequately supported system; a mechanism has no flexibility matrix.
6. It is positive definite, and $[F] = [K]^{-1}$.
7. Column $j$ is the deformed shape caused by a unit force at coordinate $j$.

#### Properties of the stiffness matrix $[K]$
1. It is a square, symmetric matrix of order $n$: $k_{ij} = k_{ji}$.
2. The diagonal terms $k_{ii}$ are always positive.
3. Off-diagonal terms may be positive, negative or zero.
4. It is positive definite when the rigid-body motion is prevented. If the system is free, $[K]$ is singular (it has zero determinant, because rigid-body motions need no force).
5. Column $j$ is the set of forces needed to hold the system in the shape of a unit displacement at $j$ (all others zero).
6. $[K] = [F]^{-1}$, and for any coordinate $k_{ii} \ge 1/f_{ii}$.

#### Illustration: cantilever of length $L$ with two coordinates
Coordinate 1 is the vertical deflection at the free end and coordinate 2 is the rotation at the free end.

```text
  ||==========================o  -> (1) deflection
  fixed                       |  -> (2) rotation
```

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

Both are symmetric with positive diagonals. Also $k_{11} = 12EI/L^3 > 1/f_{11} = 3EI/L^3$, because $k_{11}$ holds the rotation at zero while $f_{11}$ leaves it free.

## Check yourself

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

1. Why is the flexibility matrix symmetric?
   A. Because the diagonal terms are positive
   B. Because of Maxwell's reciprocal theorem, f_ij = f_ji
   C. Because it is a square matrix
   D. Because [K] is singular
   Answer: B. Maxwell's theorem equates the displacement at i due to a unit load at j with that at j due to a unit load at i. (Key point: Symmetry)

2. Which statement is correct?
   A. [K] = [F] always, even free
   B. [F] exists for a mechanism
   C. [K] is always positive definite
   D. [K] = [F]^-1, and [K] of a free system with rigid-body motion is singular
   Answer: D. A free system moves as a rigid body without force, so [K] has zero determinant; supports make it positive definite. (Key point: Inverse relation and singularity)

3. What does column j of [K] represent?
   A. Displacements due to a unit force at j
   B. Forces needed to hold the shape of a unit displacement at j with all others zero
   C. Total strain energy
   D. The reactions of the structure
   Answer: B. k_ij is the force at i for unit displacement at j with the other displacements held at zero. (Key point: Meaning of a column)

## Related in Theory of Structures II

- [Physical significance of flexibility and stiffness matrices](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflex-stiff-significance) — 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
- [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
- [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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