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

# Flexibility, stiffness; properties of flexibility matrix

Theory of Structures II · Chapter 2: Force method

- Asked in: 2069 Asar, 2+3 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/flexibility-stiffness-properties-flexibility-matrix
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Define flexibility and stiffness. What are the properties of the flexibility matrix?

## Short answer

Flexibility f_ij is displacement at i due to unit force at j; stiffness k_ij is force at i for unit displacement at j. Flexibility matrix is square, symmetric, positive definite, has positive diagonal terms, equals the inverse of stiffness matrix, and depends only on geometry and material, not on loads.

## Answer

#### Flexibility
The flexibility $f_{ij}$ is the **displacement at coordinate i caused by a unit force acting at coordinate j** (all other coordinates carry no force). For a beam fixed at one end, the tip deflection due to a unit tip load is $f=L^3/3EI$. Its unit is m/kN (displacement per unit force). The flexibility matrix $[F]$ collects all $f_{ij}$, so that
$$
\{\Delta\}=[F]\{P\}
$$

#### Stiffness
The stiffness $k_{ij}$ is the **force required at coordinate i to produce a unit displacement at coordinate j** while all other coordinates are held fixed (zero displacement). For the same cantilever the tip stiffness is $k=3EI/L^3$. Its unit is kN/m. The stiffness matrix $[K]$ gives
$$
\{P\}=[K]\{\Delta\}
$$
For a stable structure with the same coordinates, $[K]=[F]^{-1}$.

#### Properties of the flexibility matrix
1. **Square:** its order equals the number of coordinates (redundants) chosen.
2. **Symmetric:** $f_{ij}=f_{ji}$ (Maxwell's reciprocal theorem).
3. **Diagonal terms are positive:** $f_{ii}>0$, because a force always moves its own point in its own direction (work done is positive).
4. **Positive definite:** the strain energy $U=\tfrac12\{P\}^T[F]\{P\}$ is always positive, so the determinant is positive.
5. **Inverse is the stiffness matrix:** $[F]=[K]^{-1}$.
6. **Depends on the chosen released structure and coordinates**, but the final answer does not.
7. **Depends only on geometry and material** (L, E, I, A), not on the loads. Off-diagonal terms may be positive, negative or zero.

## Check yourself

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

1. What does the flexibility coefficient f_ij represent?
   A. Force at i needed to produce a unit displacement at j
   B. Displacement at coordinate i caused by a unit force at coordinate j
   C. Strain energy stored at i due to a unit load
   D. Moment at j caused by a unit rotation at i
   Answer: B. Flexibility is displacement per unit force: the displacement at i due to a unit force at j with no other forces acting. (Key point: Definition of flexibility)

2. When defining the stiffness k_ij, what condition is applied to the other coordinates?
   A. They carry no force
   B. They carry the same unit force
   C. They are held fixed (zero displacement)
   D. They are free to move
   Answer: C. Stiffness is the force at i for a unit displacement at j while all other coordinates are held at zero displacement. (Key point: Definition of stiffness)

3. Which theorem makes the flexibility matrix symmetric, f_ij = f_ji?
   A. Castigliano's first theorem
   B. Clapeyron's theorem
   C. Maxwell's reciprocal theorem
   D. Mohr's theorem
   Answer: C. Maxwell's reciprocal theorem states the displacement at i due to a unit load at j equals that at j due to a unit load at i. (Key point: Symmetry of the flexibility matrix)

4. For a stable structure with the same coordinates, how are [F] and [K] related, and what about the diagonal terms of [F]?
   A. [K] = [F]^T and f_ii can be negative
   B. [K] = [F]^-1 and f_ii > 0
   C. [K] = [F]^-1 and f_ii < 0
   D. [K] = [F] and f_ii = 0
   Answer: B. [F] and [K] are inverses, and f_ii is always positive because a force does positive work moving its own point in its own direction. (Key point: Relation between [F] and [K])

## 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
- [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
- [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 matrix method: portal frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflexibility-matrix-portal-12000) — Theory of Structures II, chapter 2
- [Flexibility matrix: frame reactions at D, SFD, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflexibility-matrix-frame-50kn-30knm-d) — Theory of Structures II, chapter 2

## Related in other subjects

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- [Plate stiffness matrix with two elements](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Astiffness-two-element-plate) — Computational Techniques in Civil Engineering, chapter 4
- [Bar stiffness matrix and 2D truss element](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Abar-stiffness-truss-rotation) — Computational Techniques in Civil Engineering, chapter 4