---
title: "Displacement method vs force method"
subject: "Theory of Structures II"
chapter: 3
key: "semester_5/theory_of_structures_ii:displacement-method-unique"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/displacement-method-unique
site: BCT Engineering
---

# Displacement method vs force method

Theory of Structures II · Chapter 3: Displacement method

- Asked in: 2075 Asoj, 4 marks
- Video: [What is Difference b/w Force method and Displacement method ?](https://www.youtube.com/watch?v=MiD57ToPm8g) — Decor Engineer
- 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/displacement-method-unique
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-3

## Question

"Displacement method is unique in comparison to force method". Justify the statement giving suitable example.

## Short answer

The displacement method is unique because the unknowns are the fixed independent joint displacements and the restrained structure is uniquely defined, unlike the force method where the analyst must choose redundants. For example, a fixed beam has zero kinematic indeterminacy, giving fixed-end moments directly, while a continuous beam’s rotations are predetermined geometric unknowns.

## Answer

The statement means that, in the displacement (stiffness) method, **the set of unknowns and the primary structure are unique** for a given structure, while in the force (flexibility) method they depend on the analyst's choice.

#### Why the displacement method is unique
1. **Unknowns are fixed.** The unknowns are the independent joint displacements (rotations and translations) that are free to occur. The degree of kinematic indeterminacy is a single, definite number for a structure, whatever the loading.
2. **The restrained structure is fixed.** Locking all the joints gives a single restrained structure made of fixed-ended members whose fixed-end moments are available in tables. No choice is involved.
3. **No choice of redundants.** In the force method one may remove different reactions or release different members (several primary structures are possible), and the work involved changes with the choice; a poor choice can even give an unstable primary structure.
4. **Systematic and general.** The same member stiffness relations ($4EI/L$, $2EI/L$, $6EI/L^2$, $12EI/L^3$) and the same assembly rule are used for every problem, so the method is easy to program. This is why most computer packages use the displacement method.

#### Example: fixed-ended beam under a point load
The fixed beam AB (both ends fixed) is statically indeterminate to degree 2: ignoring axial effects there are four reactions ($V_A$, $M_A$, $V_B$, $M_B$) but only two equilibrium equations. In the force method one must choose the redundants (for example $M_A$ and $M_B$, or $V_B$ and $M_B$) and compute several flexibility coefficients.

In the displacement method the same beam has **zero** kinematic indeterminacy, because the ends cannot rotate or translate. The end moments are directly the fixed-end moments:
$$
M_{AB}=-\frac{Pab^2}{L^2},\qquad M_{BA}=+\frac{Pa^2b}{L^2}
$$
and no equation has to be solved.

For a two-span continuous beam ABC on simple supports, the force method has one redundant, which the analyst may choose as the middle reaction or the support moment at B. In the displacement method the unknowns are the joint rotations $\theta_A$, $\theta_B$ and $\theta_C$ (or only $\theta_B$ if the modified stiffness $3EI/L$ is used for the spans with hinged far ends). Either way the unknowns are fixed by the geometry of the structure, not chosen by the analyst. The number of unknowns may be larger in either method depending on the structure (a multi-storey frame has many redundants and many joint displacements), but in the displacement method each unknown is a uniquely identified joint displacement.

## Check yourself

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

1. In what sense is the displacement method unique?
   A. It always gives smaller moments
   B. The unknowns and restrained structure are fixed for a structure, unlike the force method's choice of redundants
   C. It needs no equations
   D. It works only for beams
   Answer: B. The kinematic unknowns are definite; the force method allows different redundants. (Key point: Meaning of unique)

2. In the force method:
   A. Only one primary structure exists
   B. Joints are locked
   C. Different redundants can be chosen and the work changes with the choice
   D. Redundants are always rotations
   Answer: C. Several primary structures are possible, and a poor choice may even be unstable. (Key point: Force method choice)

3. The fixed-ended beam has kinematic indeterminacy:
   A. Two
   B. One
   C. Three
   D. Zero, so end moments equal the fixed-end moments
   Answer: D. Both ends are restrained so no displacements are unknown. (Key point: Fixed beam example)

4. Why do computer packages mostly use the displacement method?
   A. It is systematic with the same member stiffness relations and assembly rule for all problems
   B. It needs less memory always
   C. It ignores loads
   D. It gives exact answers only
   Answer: A. A uniform procedure is easy to program. (Key point: Computers)

## Related in Theory of Structures II

- [Differences between force and displacement methods](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aforce-vs-displacement-method-differences) — Theory of Structures II, chapter 2

## Related in other subjects

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