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

# Define flexibility and stiffness

Theory of Structures II · Chapter 2: Force method

- Asked in: 2070 Chaitra (old course), 2+2 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/define-flexibility-stiffness-old
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Define the terms flexibility and stiffness.

## Short answer

Flexibility is displacement per unit force; coefficient f_ij is displacement at i due to unit force at j, with {Δ}=[F]{P}. Stiffness is force per unit displacement; k_ij is force at i for unit displacement at j, with {P}=[K]{Δ}, and [K]=[F]^-1.

## Answer

#### Flexibility
Flexibility is the displacement produced by a unit force. The **flexibility coefficient** $f_{ij}$ is the displacement at coordinate $i$ due to a unit force at coordinate $j$ (all other coordinates free of load). It is a measure of how easily a structure deforms. The flexibility matrix $[F]$ gives $\{\Delta\} = [F]\{P\}$. Its unit is m/kN (or rad/kNm). Example: for a cantilever of length $L$ with a tip load, $f_{11} = \dfrac{L^3}{3EI}$.

#### Stiffness
Stiffness is the force required to produce a unit displacement. The **stiffness coefficient** $k_{ij}$ is the force needed at coordinate $i$ to produce a unit displacement at coordinate $j$ while all other coordinates are held fixed. The stiffness matrix $[K]$ gives $\{P\} = [K]\{\Delta\}$. Its unit is kN/m (or kNm/rad). Example: for a beam of length $L$ fixed at both ends, a force of $\dfrac{12EI}{L^3}$ is needed to move one end through a unit deflection while both ends are prevented from rotating. The two matrices are inverses of each other, $[K] = [F]^{-1}$.

## Check yourself

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

1. What is the flexibility coefficient f_ij?
   A. Force at i to produce unit displacement at j
   B. Displacement at coordinate i due to unit force at j
   C. Energy stored per unit force
   D. Reciprocal of the load
   Answer: B. Flexibility is displacement produced by a unit force. (Key point: Flexibility definition)

2. What is the stiffness coefficient k_ij?
   A. Displacement at i per unit force at j
   B. Strain energy per unit displacement
   C. Force needed at i to produce unit displacement at j while other coordinates are held fixed
   D. Force at j when all coordinates are free
   Answer: C. Stiffness is force per unit displacement, other coordinates restrained. (Key point: Stiffness definition)

3. How are [F] and [K] related?
   A. [K] = [F]^T
   B. [K] = [F]
   C. [K] = [F]^2
   D. [K] = [F]^-1
   Answer: D. {Δ} = [F]{P} and {P} = [K]{Δ}, so the matrices are inverses. (Key point: Relation between matrices)

## Related in Theory of Structures II

- [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
- [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; 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
- [Moment distribution principle; frame stiffness matrix](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aprinciple-moment-dist-stiffness-matrix-u1v1) — Theory of Structures II, chapter 3

## Related in other subjects

- [Stiffness, flexibility and reciprocity theorem](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Amatrix-equations-flexibility-reciprocity) — Theory of Mechanism and Machine II, chapter 8
- [Flexible vs rigid; radius of relative stiffness, k](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-rigid-differences-relative-stiffness-k) — Transportation Engineering II, chapter 2