---
title: "Indeterminacy and stability of truss-frames"
subject: "Theory of Structures II"
chapter: 1
key: "semester_5/theory_of_structures_ii:indeterminacy-stability-two-structures-2067"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/indeterminacy-stability-two-structures-2067
site: BCT Engineering
---

# Indeterminacy and stability of truss-frames

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2067 Asar, 5 marks
- Video: [Structural Theory | Determinacy, Indeterminacy and Stability of Trusses](https://www.youtube.com/watch?v=CZMHRSB7GX0) — the Engineer's Insight
- 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/indeterminacy-stability-two-structures-2067
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Determine the external/internal static indeterminacy and kinematic indeterminacy of the structures shown in figure below. Are they stable or unstable? [Figure: (i) a truss-like frame with X-bracing in two panels, hinged at one support and roller at the other end, plus a cantilever arm; (ii) a braced frame with diagonals and three supports (fixed, hinged, roller).]

## Short answer

X-braced truss with cantilever: static indeterminacy 2, kinematic 11. Braced single-storey frame: static 5, kinematic 5 (inextensible). Both are stable.

## Answer

#### (i) Truss with X-bracing in two panels and a cantilever arm
Reading: top and bottom chords with three verticals and two X-braced panels, hinge at one end and roller at the other, plus a cantilever arm of two bars carrying a joint.

$j = 6 + 1 = 7,\ m = 4 + 3 + 4 + 2 = 13,\ r = 3$

- $D_s = m + r - 2j = 13 + 3 - 14 = 2$ (internal 2, one per X-panel; external 0)
- $D_k = 2j - r = 14 - 3 = 11$
- Stable: triangulated panels, the cantilever arm is attached to the rigid truss by two bars, and the reactions are non-concurrent.

#### (ii) Braced single-storey frame
Reading: three columns (fixed, hinged, roller), two beams and two pin-ended diagonals.

$j = 6,\ m = 7,\ r = 3 + 2 + 1 = 6,\ c = 4\ \text{(bar ends)}$

- $D_s = 3(7) + 6 - 18 - 4 = 5$ (external 3, internal 2 from the two closed braced bays)
- $D_k = 18 - 6 - 7 = 5$ (inextensible)
- Stable: braced panels are rigid and reactions prevent all rigid-body motion.

**Answer: (i) $D_s = 2,\ D_k = 11$; (ii) $D_s = 5,\ D_k = 5$. Both stable.**

## Check yourself

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

1. For j=7, m=13, r=3, Ds is:
   A. 2
   B. 3
   C. 0
   D. 4
   Answer: A. 13 + 3 - 14 = 2, one per X-panel. (Key point: Truss with cantilever arm)

2. Dk = 2j - r =
   A. 14
   B. 11
   C. 13
   D. 17
   Answer: B. 14 - 3 = 11. (Key point: Dk of truss)

3. Braced frame: 3m + r - 3j - c = 21 + 6 - 18 - 4 =
   A. 5
   B. 9
   C. 3
   D. 7
   Answer: A. c = 4 from the bar ends, so Ds = 5. (Key point: Braced frame)

4. The cantilever arm is stable because:
   A. It is attached to the rigid truss by two non-collinear bars
   B. It is loaded
   C. It is hinged
   D. It has a roller
   Answer: A. Two bars attach the extra joint to the rigid truss. (Key point: Cantilever arm stability)

## Related in Theory of Structures II

- [External/internal static and kinematic indeterminacy of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-asar-2070) — Theory of Structures II, chapter 1
- [Indeterminacy and stability of frame and truss](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-frame-and-truss-baishakh) — Theory of Structures II, chapter 1
- [Indeterminacy and stability of truss and frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-truss-and-frame-bhadra) — Theory of Structures II, chapter 1
- [Static and kinematic indeterminacy of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-old-2070) — Theory of Structures II, chapter 1
- [Static and kinematic indeterminacy of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-bhadra) — Theory of Structures II, chapter 1
- [Indeterminacy of frame with internal hinges](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-trapezoid-frame) — Theory of Structures II, chapter 1

## Related in other subjects

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- [Truss indeterminacy; forces in BC, BG, HG, GD](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Atruss-indeterminacy-bc-bg-hg-gd) — Applied Mechanics, chapter 7
- [Frame horizontal deflection at roller support](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-virtual-work-roller-2076) — Theory of Structures I, chapter 3