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

# Define flexibility coefficient

Theory of Structures II · Chapter 2: Force method

- Asked in: 2073 Shrawan, 1 marks
- Video: [Flexibility influence coefficients](https://www.youtube.com/watch?v=i-u0LSI9m4g) — Viswanathan R
- 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-coefficient
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Define and explain the term flexibility coefficient.

## Short answer

Flexibility coefficient f_ij is the displacement at coordinate i due to a unit force at coordinate j. For flexural members f_ij = ∫(m_i m_j/EI)dx, for trusses f_ij = Σ(n_i n_j L/AE). The flexibility matrix is symmetric (f_ij = f_ji) by Maxwell's reciprocal theorem.

## Answer

The **flexibility coefficient** $f_{ij}$ is the displacement at coordinate $i$ (in the direction of the redundant $X_i$) caused by a unit force acting at coordinate $j$, with all other redundants and loads absent.

$$
f_{ij} = \int \frac{m_i\,m_j}{EI}\,dx
$$

for flexural members (for trusses, $f_{ij} = \sum n_in_jL/AE$). Its unit is m/kN for a force coordinate or rad/kNm for a moment coordinate. The coefficients form the flexibility matrix $[F]$, which is symmetric ($f_{ij} = f_{ji}$, Maxwell). Example: for a cantilever of length $L$, the tip deflection due to a unit tip load is $f_{11} = L^3/3EI$.

## 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 due to unit displacement at j
   B. Displacement at i due to a unit force at j
   C. Stiffness of member
   D. Slope due to load
   Answer: B. f_ij is the displacement at coordinate i caused by a unit force at coordinate j. (Key point: Definition)

2. What is f_ij for flexural members?
   A. ∫ m_i m_j /EI dx
   B. ∫ M0 m_i dx
   C. Σ n L
   D. EI/L
   Answer: A. f_ij = ∫ m_i m_j /EI dx; for trusses it is Σ n_i n_j L/AE. (Key point: Formulae)

3. What property does the flexibility matrix have?
   A. Diagonal only
   B. Symmetric, f_ij = f_ji (Maxwell)
   C. Always singular
   D. Negative terms only
   Answer: B. By Maxwell's reciprocal theorem, f_ij = f_ji. (Key point: Matrix property)

## 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
- [Define stiffness coefficient](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-stiffness-coefficient) — Theory of Structures II, chapter 3