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

# Indeterminacy and stability of frame and truss

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2068 Baishakh, 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-frame-and-truss-baishakh
- 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 the figure below. Are they geometrically stable? [Figure: (i) a two-bay frame with internal hinges and diagonal bracing members, fixed bases and a pinned support; (ii) a multi-panel truss with double diagonals, supports as drawn.]

## Short answer

Braced two-bay frame: static indeterminacy 5, kinematic 6 (inextensible) or 12 (extensible). Double-diagonal truss: static indeterminacy 4, kinematic 17. Both structures are geometrically stable.

## Answer

#### (i) Braced two-bay frame
Reading: one storey, three columns (two fixed, one pinned), two beams, one pin-ended diagonal in each bay, and two internal hinges.

$j = 6,\ m = 5 + 2 = 7,\ r = 3 + 3 + 2 = 8,\ c = 2\ \text{hinges} + 4\ \text{bar ends} = 6$

- $D_s = 3(7) + 8 - 3(6) - 6 = 5$. External: $r - 3 = 8 - 3 = 5$; internal: each brace adds 1 and each hinge releases 1, so $2 - 2 = 0$. Check: the frame without braces gives $15 + 8 - 18 - 2 = 3$, and each brace adds 1, giving 5.
- Kinematic (inextensible members): the columns stop vertical movement of the top joints, the beams tie them together, and the braces stop sway, so no joint translation remains. The unknowns are the rotations of the three top joints (3), the rotation of the pinned base (1) and the extra rotations at the two hinges (2): $D_k = 6$. (The shortcut $3j - r + c' - m = 5$ undercounts here, because the two braces give two constraints against only one sway.) With axial deformation, $D_k = 3j - r + c' = 18 - 8 + 2 = 12$.
- Stability: reactions are non-concurrent and non-parallel, and the bays are closed by braces. The frame is **geometrically stable**.

#### (ii) Double-diagonal truss
Reading: four panels, chords, five verticals, both diagonals in each panel, hinge and roller.

$j = 10,\ m = 4 + 4 + 5 + 8 = 21,\ r = 3$

- $D_s = m + r - 2j = 21 + 3 - 20 = 4$ (internal)
- $D_k = 2j - r = 17$
- Stability: $m + r = 24 > 2j = 20$, all panels are braced triangles, and the three reactions are neither parallel nor concurrent. The truss is **stable**.

**Answer: (i) $D_s = 5$, $D_k = 6$ for inextensible members (12 if extensible), stable; (ii) $D_s = 4$, $D_k = 17$, stable.**

## Check yourself

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

1. A pin-ended diagonal brace contributes how many hinge conditions to c?
   A. Two, one at each end
   B. One
   C. Zero
   D. Four
   Answer: A. Each hinged bar end gives one equation of condition, so a brace adds 2 (four bar ends in two braces). (Key point: Equations of condition for braces)

2. Why is Dk of the braced frame small (5)?
   A. The braces prevent sway
   B. Supports are fixed
   C. Joints are hinged
   D. There are no columns
   Answer: A. Inextensible braces stop sway, leaving only rotations. (Key point: Braced frame Dk)

3. Double-diagonal truss: m=21, r=3, j=10 gives Ds:
   A. 4
   B. 3
   C. 24
   D. 0
   Answer: A. 21 + 3 - 20 = 4, all internal. (Key point: Truss Ds)

4. Why is the truss stable?
   A. m + r exceeds 2j, panels are braced triangles and reactions are neither parallel nor concurrent
   B. It has more verticals than diagonals
   C. It has a roller
   D. It has no redundants
   Answer: A. Count and arrangement conditions are both satisfied. (Key point: 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 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
- [Degrees of indeterminacy of frame with hinges](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-hinges-1) — 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
- [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
- [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

## 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
- [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