---
title: "Flexibility, stiffness coefficient; suitable method"
subject: "Theory of Structures II"
chapter: 1
key: "semester_5/theory_of_structures_ii:flex-stiff-coefficient-frame-method"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/flex-stiff-coefficient-frame-method
site: BCT Engineering
---

# Flexibility, stiffness coefficient; suitable method

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2078 Kartik, 2+3 marks
- Video: [Derivation of Stiffness for Propped Cantilever AB Fixed at A & Propped at B | Structural Analysis](https://www.youtube.com/watch?v=eW9nybh0kVo) — Ekeeda GATE & ESE
- 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-coefficient-frame-method
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Define flexibility and stiffness coefficient. For the given frame, which of the methods, force or displacement, is suitable? Give your reasons. [Figure: single-storey frame with a middle column that is hinged at its base.]

## Short answer

Flexibility coefficient f_ij is displacement at i due to unit force at j; stiffness coefficient k_ij is force at i for unit displacement at j with others fixed. Displacement method is more suitable: the frame has fewer unknowns (4 vs 5 redundants) after modified stiffness for the hinged column, well-suited for slope-deflection.

## Answer

#### Flexibility coefficient
The flexibility coefficient $f_{ij}$ is the displacement at coordinate $i$ due to a unit force applied at coordinate $j$ (all other coordinates unloaded). Its unit is m/kN (or rad/kNm). The flexibility matrix $[F]$ relates displacements and forces: $\{\Delta\} = [F]\{P\}$.

#### Stiffness coefficient
The stiffness coefficient $k_{ij}$ is the force required at coordinate $i$ to produce a unit displacement at coordinate $j$ while all other coordinates are held fixed. The stiffness matrix gives $\{P\} = [K]\{\Delta\}$, and $[K] = [F]^{-1}$.

#### Suitable method
Read the figure as a single-storey, two-bay frame with fixed outer bases and a hinged base under the middle column: $j = 6,\ m = 5,\ r = 3 + 2 + 3 = 8$.

$D_s = 3(5) + 8 - 3(6) = 5$

$D_k$ (inextensible) $= 3(6) - 8 - 5 = 5$ (three top-joint rotations, one base-hinge rotation, one sway).

Because the base of the middle column is hinged, its far-end rotation can be removed with the modified stiffness $3EI/L$. The effective unknowns of the displacement method fall to **4**, compared with **5** redundants in the force method.

**Displacement method is more suitable** because
1. it has fewer unknowns (4 against 5);
2. the frame is a regular framed system with several rotating joints and sway, which suits slope-deflection or moment distribution well;
3. the force method needs a stable released structure and many moment diagrams for 5 redundants, which is long hand work.

## Check yourself

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

1. The flexibility coefficient f_ij is:
   A. Force at i due to unit displacement at j
   B. Displacement at i due to unit force at j
   C. Displacement at j due to unit force at j only
   D. Force at j due to unit force at i
   Answer: B. f_ij is the displacement at coordinate i caused by a unit force at coordinate j, other coordinates unloaded. (Key point: Flexibility coefficient)

2. In defining the stiffness coefficient k_ij, the other coordinates are:
   A. Left free
   B. Loaded with unit forces
   C. Removed
   D. Held fixed
   Answer: D. k_ij is the force at i for a unit displacement at j while all other coordinates are restrained. (Key point: Stiffness coefficient)

3. For the hinged base of the middle column, the far-end rotation is eliminated using the stiffness:
   A. 4EI/L
   B. 2EI/L
   C. 3EI/L
   D. 6EI/L
   Answer: C. With a hinged far end the modified stiffness is 3EI/L, removing one unknown. (Key point: Modified stiffness)

4. Which method is chosen and why?
   A. Force method, 4 unknowns against 5
   B. Displacement method, 4 effective unknowns against 5 redundants
   C. Force method, because bases are hinged
   D. Displacement method, because it has more unknowns
   Answer: B. Effective displacement unknowns fall to 4 versus 5 redundants, and the frame suits slope-deflection or moment distribution. (Key point: Choice of method)

## Related in Theory of Structures II

- [Force method: frame with hinged support](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aforce-method-frame-30knm-50kn-hinged) — Theory of Structures II, chapter 2
- [Force method: frame reactions at hinged support](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aforce-method-frame-50knm-hinged-d) — Theory of Structures II, chapter 2
- [Flexibility matrix: frame with 48 kNm moment](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflexibility-frame-48knm-hinged-a) — Theory of Structures II, chapter 2
- [Stiffness method: frame, flexural deformations only](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astiffness-frame-10kn-40knm-flexural) — Theory of Structures II, chapter 3
- [Stiffness method: final end moments, frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astiffness-frame-20knm-8knm) — Theory of Structures II, chapter 3
- [Force method: frame, SFD and BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aforce-method-frame-5knm-20kn-sfd) — Theory of Structures II, chapter 2

## Related in other subjects

- [Frame: roller displacement and hinge rotation](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-roller-hinge-rotation-2071b) — Theory of Structures I, chapter 3
- [Frame AFD, SFD, BMD (inclined column, triangular)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-30knm-20kn) — Strength of Materials, chapter 1
- [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