---
title: "Differences between force and displacement methods"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:force-vs-displacement-method-differences"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/force-vs-displacement-method-differences
site: BCT Engineering
---

# Differences between force and displacement methods

Theory of Structures II · Chapter 2: Force method

- Asked in: 2072 Kartik, 6 marks
- Video: [Force method and Displacement method( Comparison)](https://www.youtube.com/watch?v=qAa2dAkUcRY) — Research Made Easy through Dr. MG
- 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/force-vs-displacement-method-differences
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

List the differences between force and displacement methods. Draw a neat sketch of a system and explain.

## Short answer

Force method uses redundant forces as unknowns and satisfies compatibility, suited for structures with few redundants. Displacement method uses joint displacements as unknowns and satisfies equilibrium, is systematic and preferred for computer analysis of frames with many joints.

## Answer

The **force (flexibility) method** takes the redundant forces as unknowns and satisfies compatibility of displacements. The **displacement (stiffness) method** takes the joint displacements as unknowns and satisfies equilibrium of the joints.

| Point | Force method | Displacement method |
|---|---|---|
| Unknowns | Redundant forces or moments ($D_s$) | Joint displacements ($D_k$) |
| Governing condition | Compatibility of displacements | Equilibrium of joints |
| Matrix used | Flexibility matrix $[F]$ | Stiffness matrix $[K]$ |
| Equations | $[F]\{X\} = -\{\Delta_0\}$ | $[K]\{\Delta\} = \{P\}$ |
| Primary structure | Statically determinate (redundants removed) | Kinematically determinate (all joints fixed) |
| Best for | Structures with few redundants (low $D_s$), trusses, arches | Structures with few joint displacements (low $D_k$), multi-storey frames, continuous beams |
| Suitable for computer | Less systematic, since the primary structure is not unique | Very systematic, so it is the basis of computer programs |
| Effect of settlement/temperature | Enters through the load vector $\Delta_0$ | Enters through fixed-end forces |

#### Illustration: portal frame with fixed bases
```text
        B ______ C
        |        |
        |        |
       A///    ///D     (both fixed)
```

For the portal frame with fixed bases, $D_s = 3$ and $D_k = 3$ (rotations of B and C, one sway) if axial deformation is neglected. If one base is replaced by a hinge, $D_s = 2$, but $D_k$ becomes 4 (the rotation at the hinge is added), or stays 3 if the modified stiffness $3EI/L$ is used for that column. A two-span continuous beam with hinged ends on rigid supports has $D_s = 1$ (the middle reaction) and $D_k = 3$ (rotations at the three supports), or $D_k = 1$ (rotation at the middle support only) if the modified stiffness is used for both end spans.

**Force method steps:** remove the redundants, find $M_0$ and $m_i$, find $f_{ij}$ and $\Delta_{i0}$, solve $[F]\{X\} = -\{\Delta_0\}$, then superpose.

**Displacement method steps:** fix all joints, apply fictitious restraints, find the fixed-end forces, find the stiffness coefficients, solve $[K]\{\Delta\} = \{P\}$ for the joint displacements, then find the member end moments.

Choose the method that gives fewer unknowns: a frame with many redundant members but few joints suits the displacement method, whereas an arch or a truss with one or two redundants suits the force method.

## Check yourself

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

1. What are the unknowns in the force method and the displacement method?
   A. Joint displacements; redundant forces
   B. Redundant forces; joint displacements
   C. Both use support reactions
   D. Both use member lengths
   Answer: B. Force method: redundants (D_s). Displacement method: joint displacements (D_k). (Key point: Unknowns)

2. Which pairing is correct?
   A. Force method: equilibrium, stiffness matrix
   B. Displacement method: compatibility, flexibility matrix
   C. Force method: compatibility, flexibility matrix; displacement method: equilibrium, stiffness matrix
   D. Both: compatibility, stiffness matrix
   Answer: C. Force method satisfies compatibility with [F]{X} = -{D0}; displacement method satisfies joint equilibrium with [K]{D} = {P}. (Key point: Governing condition and matrix)

3. A structure with few joint displacements but many redundants is best analysed by:
   A. The force method
   B. Neither method
   C. A determinate analysis only
   D. The displacement method
   Answer: D. Choose the method with fewer unknowns; many redundants but few joint displacements suits the displacement method. (Key point: Choice of method)

## Related in Theory of Structures II

- [Displacement method vs force method](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adisplacement-method-unique) — Theory of Structures II, chapter 3

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