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

# Indeterminacy and stability of truss and frame

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2066 Bhadra, 10 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-truss-and-frame-bhadra
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Compute static indeterminacy, kinematic indeterminacy and stability of the structures shown in figure given below. Axial deformation of the members are neglected. [Figure: (i) a multi-panel truss with double diagonals and hinged supports; (ii) a two-bay, multi-storey frame with an internal hinge, X-braced storeys and three base supports.]

## Short answer

With inextensible members: double-diagonal truss on hinged supports has static indeterminacy 5, kinematic 0; two-bay braced frame with hinge has static 16, kinematic 9. Both structures are stable.

## Answer

Axial deformation is neglected, so every member is axially rigid.

#### (i) Double-diagonal truss with hinged supports
Reading: four panels, chords, five verticals, X-diagonals, both supports hinged.

$j = 10,\ m = 21,\ r = 4$

- Static: $D_s = m + r - 2j = 21 + 4 - 20 = 5$ (external 1, internal 4)
- Kinematic: with rigid bars and a stable arrangement no joint can move, so $D_k = 0$.
- Stability: every panel is triangulated and four reactions (non-concurrent) prevent rigid-body motion. The truss is **stable**.

#### (ii) Two-bay braced frame with an internal hinge
Reading: two storeys, three column lines, X-bracing in all four panels, one internal hinge, three base supports (fixed, hinged, roller).

$j = 9,\ m_{frame} = 10,\ \text{braces} = 8,\ r = 6,\ c = 1\ \text{(hinge)} + 16\ \text{(bar ends)}$

- Static: $D_s = 3(18) + 6 - 3(9) - 17 = 16$ (frame alone 8, plus 8 braces)
- Kinematic: rigid X-braces stop all joint translations, so only rotations remain: six upper joints (6), hinged base (1), roller base (1), and the extra rotation at the hinge (1). $D_k = 9$.
- Stability: braced panels and well-placed reactions give a **stable** frame.

**Answer: (i) $D_s = 5$, $D_k = 0$, stable; (ii) $D_s = 16$, $D_k = 9$, stable.**

## Check yourself

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

1. For the truss with hinged supports (m=21, r=4, j=10), Ds =
   A. 5
   B. 4
   C. 3
   D. 1
   Answer: A. 21 + 4 - 20 = 5 (external 1, internal 4). (Key point: Truss Ds with 4 reactions)

2. Why is Dk = 0 for the truss?
   A. No joints exist
   B. Rigid bars in a stable arrangement allow no joint movement
   C. Supports are rollers
   D. Ds is zero
   Answer: B. With axially rigid bars and a stable triangulated arrangement no joint can move. (Key point: Kinematic of rigid truss)

3. Why do only rotations remain in the braced frame?
   A. Rigid X-braces stop all joint translations
   B. Hinges stop translations
   C. The roller stops them
   D. The columns are short
   Answer: A. Axially rigid braces prevent translations, leaving rotations (9 in total). (Key point: Frame Dk with rigid braces)

4. How is c obtained for the braced frame?
   A. 1 hinge + 16 brace-end hinges = 17
   B. Only 1
   C. 8
   D. 16 only
   Answer: A. Eight braces have two hinged ends each, plus the one internal hinge. (Key point: Bar-end count)

## Related in Theory of Structures II

- [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: frame with hinge and truss](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-and-truss) — 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
- [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
- [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 and stability of truss-frames](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-two-structures-2067) — Theory of Structures II, chapter 1

## Related in other subjects

- [Static determinacy and stability of a truss](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astatic-determinacy-stability-truss) — Strength of Materials, chapter 1