---
title: "Structural idealization; stability condition of truss"
subject: "Theory of Structures II"
chapter: 1
key: "semester_5/theory_of_structures_ii:structural-idealization-truss-stability"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/structural-idealization-truss-stability
site: BCT Engineering
---

# Structural idealization; stability condition of truss

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2069 Chaitra, 5 marks
- Video: [STABILITY OF TRUSSES USING STATIC INDETERMINACY EQUATION](https://www.youtube.com/watch?v=KIBnzOFV4pY) — Structural Funda by:- KheNiSiR
- 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/structural-idealization-truss-stability
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

What is structural idealization? Explain the necessary and sufficient condition for stability of a truss.

## Short answer

Structural idealization replaces a real structure with a simplified analytical model. A plane truss is stable when m + r ≥ 2j and it is built from properly arranged triangles with supports that prevent rigid-body motion.

## Answer

#### Structural idealization
It is the replacement of an actual structure by a simple analytical model (line members, ideal supports, pinned or rigid joints, idealized loads and linear elastic material) that can be analysed and still represents the real behaviour with acceptable accuracy. A roof truss is idealized as pin-jointed bars with loads at the joints.

#### Stability of a plane truss
A truss is **stable** when it keeps its shape and position under any load. Two conditions are checked.

**1. Necessary condition (count)**
- Internally, the members must satisfy $m \ge 2j - 3$ (for a stable, internally rigid truss).
- Including supports, $m + r \ge 2j$.

If $m + r < 2j$ the truss is unstable (a mechanism). If $m + r = 2j$ it is determinate if stable, and if $m + r > 2j$ it is indeterminate.

**2. Sufficient condition (arrangement)**
The count is not enough, so the arrangement must also be right:
- Each part must be a rigid unit, normally a chain of triangles. A panel without a diagonal is a mechanism.
- Reactions must be at least 3 and must not be all parallel or all concurrent at a point, otherwise the truss can move as a rigid body.
- No joint may be held by only two collinear members, and no part may form a collapsible chain of links; such arrangements are geometrically unstable even when the count is satisfied.

A truss is stable when $m + r \ge 2j$ **and** it is built from properly arranged triangles with supports preventing rigid-body motion. Example: a triangle with hinge and roller has $j=3,\ m=3,\ r=3$, and $m + r = 6 = 2j$, so it is stable and determinate.

## Check yourself

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

1. Necessary count condition for a stable truss including supports is:
   A. m + r >= 2j
   B. m + r <= 2j
   C. m = j
   D. r = 2j
   Answer: A. If m + r < 2j the truss is a mechanism. (Key point: Count condition)

2. Why is the count not sufficient?
   A. Members may be too long
   B. The arrangement may be wrong, such as a panel without a diagonal
   C. Loads may be too large
   D. Joints may be rigid
   Answer: B. The arrangement must form rigid triangles and the supports must prevent rigid-body motion. (Key point: Sufficient condition)

3. The reactions of a stable truss must be at least 3 and:
   A. All parallel
   B. All concurrent
   C. Neither all parallel nor all concurrent
   D. All vertical
   Answer: C. Parallel or concurrent reactions allow rigid-body motion. (Key point: Reaction arrangement)

4. If m + r = 2j and the truss is stable, it is:
   A. Indeterminate
   B. Determinate
   C. A mechanism
   D. Over-braced
   Answer: B. Equality means exactly enough members; if stable, determinate. (Key point: Determinate case)

## Related in Theory of Structures II

- [Structural idealization and formulation of problems](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astructural-idealization-formulation-steps) — 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 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-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

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