---
title: "Define stiffness coefficient"
subject: "Theory of Structures II"
chapter: 3
key: "semester_5/theory_of_structures_ii:define-stiffness-coefficient"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/define-stiffness-coefficient
site: BCT Engineering
---

# Define stiffness coefficient

Theory of Structures II · Chapter 3: Displacement method

- Asked in: 2073 Shrawan, 1 marks
- Video: [Structural Dynamics, Lesson 1d: Fundamentals, Stiffness Coefficients of Frame Elements](https://www.youtube.com/watch?v=E0nDyB3Vioc) — Silvia's Brainery on YouTube
- 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-stiffness-coefficient
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-3

## Question

Define and explain the term stiffness coefficient.

## Short answer

A stiffness coefficient k_ij is the force (or moment) at coordinate i needed to produce a unit displacement at coordinate j while all other coordinates are fixed. It defines the structural stiffness matrix; typical values are 4EI/L for end rotation and 12EI/L³ for transverse translation.

## Answer

**Stiffness coefficient** $k_{ij}$ is the **force (or moment) required at coordinate $i$ to produce a unit displacement (or rotation) at coordinate $j$**, while the displacement at every other coordinate is held at zero. It is the basic quantity of the stiffness (displacement) method; its unit is force per unit displacement (kN/m for a translation, kNm/rad for a rotation).

#### Explanation
- Each column of the structure stiffness matrix $[K]$ is the set of forces needed to hold the structure in the shape produced by a unit displacement at one coordinate. Thus $k_{ij}$ is the force at $i$ when coordinate $j$ is displaced by one unit.
- The equation $\{P\}=[K]\{\Delta\}$ follows by superposition.
- By Betti's law $k_{ij}=k_{ji}$, and $k_{ii}>0$.
- Examples for a prismatic member of length $L$ and rigidity $EI$:
  - unit rotation at one end, with the far end fixed and no translation: $k=4EI/L$ (moment at the same end) and $2EI/L$ (moment carried to the far end)
  - unit relative transverse translation of the ends, with both ends held against rotation: $k=12EI/L^3$ (shear) and $6EI/L^2$ (end moment)
  - unit axial extension: $k=AE/L$.

## Check yourself

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

1. What is the stiffness coefficient k_ij?
   A. Force at coordinate i due to unit displacement at j, others held zero
   B. Displacement at i due to unit force at j
   C. Force at i due to unit force at j
   D. Strain energy per unit length
   Answer: A. Stiffness is force per unit displacement, with all other coordinates restrained. (Key point: Definition)

2. Which statement about k_ij is correct?
   A. k_ij has unit of metres per kN
   B. k_ij = k_ji and k_ii > 0
   C. k_ij is always negative
   D. k_ii may be zero
   Answer: B. By Betti's law the stiffness matrix is symmetric and diagonal terms are positive. (Key point: Properties)

3. For a prismatic member with far end fixed, the moment at the rotated end for unit rotation is?
   A. 3EI/L
   B. 12EI/L^3
   C. 4EI/L
   D. 2EI/L
   Answer: C. 4EI/L at the same end and 2EI/L carried over to the far end. (Key point: Member values)

## Related in Theory of Structures II

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