---
title: "Determinacy and stability of frame"
subject: "Applied Mechanics"
chapter: 6
key: "semester_1/applied_mechanics:frame-determinacy-stability-check"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/frame-determinacy-stability-check
site: BCT Engineering
---

# Determinacy and stability of frame

Applied Mechanics · Chapter 6: Analysis of Beams and Frames

- Asked in: 2074 Asoj, 3 marks; 2067 Asoj, 3 marks (asked 2 times)
- Video: [Static Determinacy, Indeterminacy and Stability of a Plane Frame - Solved Examples](https://www.youtube.com/watch?v=gcPP8DCa2sw) — AF Math & Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/frame-determinacy-stability-check
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-6

## Question

How can you check the determinacy and stability of the frame? Explain with examples.

## Short answer

For a plane frame with r support reactions, if r <3 it is unstable; r =3 with non-parallel, non-concurrent reactions is stable and determinate; r >3 is externally indeterminate. Internal determinacy: degree = (3m + r) -3j - h (hinge releases). A closed loop adds 3 to indeterminacy.

## Answer

#### Frame
A frame is a rigid-jointed structure of beams and columns in which members carry axial force, shear force and bending moment.

#### Stability and external determinacy
Let r = number of unknown support reactions, and the number of equations of static equilibrium = 3 for a plane structure.
- If r < 3: the structure is **unstable** (a mechanism).
- If r = 3 and the reactions are not all parallel and do not meet at one point: **stable and statically determinate**.
- If r > 3: statically **indeterminate externally** to degree r − 3 (if stable).
- Add one extra equation for each internal hinge (M = 0 at the hinge). So the number of available equations is 3 + h, where h is the number of internal hinges (releases).

Geometric instability: even if r ≥ 3, the structure is unstable if the reactions are all parallel or all pass through one point.

#### Internal determinacy (closed loops)
For a plane frame with m members, j joints and r reactions, with rigid joints:
Degree of indeterminacy = (3m + r) − 3j − (extra equations of condition, e.g. hinges)
- = 0 determinate, > 0 indeterminate, < 0 unstable.
Each closed loop of rigid members adds 3 to the degree of indeterminacy.

#### Examples
- **Portal frame with a hinge and a roller** (r = 3, no loop): determinate and stable. Reactions are found by ΣFx = ΣFy = ΣM = 0.
- **Portal frame fixed at both bases** (r = 6): indeterminate to the 3rd degree.
- **Three-hinged portal frame** (hinge at each base and one at the crown): r = 4 and h = 1, so the degree = 4 − (3 + 1) = 0, determinate and stable.
- **Cantilever frame** with fixed support: r = 3, determinate.
- A frame resting on three rollers (r = 3, but all three reactions vertical and parallel) is **geometrically unstable**: it cannot resist any horizontal load.

## Check yourself

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

1. For a plane structure with r reactions (no hinges), what is the condition r < 3?
   A. Unstable (a mechanism)
   B. Stable and determinate
   C. Indeterminate
   D. Stable but indeterminate
   Answer: A. Fewer than 3 reactions cannot satisfy 3 equilibrium equations in general, so the structure is unstable. (Key point: Reaction count and stability)

2. How does an internal hinge change the available equations?
   A. Each hinge removes one equation
   B. Each hinge adds one extra equation (M = 0 at hinge)
   C. It adds three equations
   D. It has no effect
   Answer: B. Equations available = 3 + h, since the moment at a hinge is zero. (Key point: Effect of internal hinge)

3. For a three-hinged portal frame with r = 4 and h = 1, what is the degree of indeterminacy?
   A. 0, determinate and stable
   B. 1
   C. 3
   D. -1, unstable
   Answer: A. 4 - (3 + 1) = 0. (Key point: Three-hinged portal frame)

## Related in Applied Mechanics

- [Discrete vs continuum structure; stability](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Adiscrete-continuum-structure-stability-determinacy) — Applied Mechanics, chapter 6
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