If j represents the number of joints and r the number of external reactions, a structure that has 2j − r members will be:
Try it first: pick an option, then read the explanation.
Correct answer: A. Stable
Asked in SSC JE 29/01/2018 (Morning)
Detailed Explanation
Each pin joint in a plane truss gives 2 equations (ΣFx = 0, ΣFy = 0), so a truss has 2j equations. The unknowns are the m member forces plus the r reactions.
\[ m + r = 2j \;\Rightarrow\; m = 2j - r \;(= 2j - 3 \text{ for 3 reactions}) \]
Why “Stable” Is Correct
With exactly this many members, the number of unknowns equals the number of equations, and the truss is a perfect truss: stable and statically determinate. (Strictly, the count is necessary; the members must also be arranged properly, e.g. in triangles.)
| Truss | Condition (j joints, m members, 3 reactions) | Behaviour |
|---|---|---|
| Perfect (determinate) | m = 2j − 3 | Stable; solvable by statics |
| Deficient | m < 2j − 3 | Unstable (a mechanism) |
| Redundant | m > 2j − 3 | Stable but statically indeterminate |
Why the Other Options Are Wrong
- B. Unstable: Unstable trusses have fewer members than 2j − r.
- C. Depends upon the structure: The exam answer is that this count gives a stable (perfect) truss.
- D. Depends upon the magnitude of load: Stability does not depend on load magnitude.
Key Point
Perfect truss: m = 2j − r (= 2j − 3).
Key Concepts for Students
- m < 2j − 3: deficient (unstable).
- m > 2j − 3: redundant (indeterminate).
- A simple truss is built by adding two members and one joint at a time to a triangle.
Practice more: this is question 4 in Beams and Trusses MCQs – Part 1. Try the whole set and check every answer instantly.