Question

For a 2-dimensional truss structure, if \( m \) is the number of members, \( j \) is the number of joints and \( r \) is the number of reactions, then the condition for instability of the structure is

• A. \( m + r = 2j \)
• B. \( m - r = 2j \)
• C. \( m + r<2j \)
• D. \( m - r<2j \)

Hint:

For a 2D truss: Stable and determinate if \( m + r = 2j \), Unstable if \( m + r<2j \), Statically indeterminate if \( m + r>2j \).

Answer 1

The Correct Option is C

Step 1: Total unknowns in a 2D truss are the member forces plus reaction components, giving \( m + r \); total available equilibrium equations are two per joint, giving \( 2j \).
Step 2: A determinate, stable truss satisfies \( m + r = 2j \) exactly.
Step 3: If the unknowns fall short of the equations, \( m + r<2j \), the joints cannot all be held in equilibrium without the structure deforming as a mechanism, which is the instability condition. \[ \boxed{m + r<2j} \]

Answer 2

A slightly different angle is to think of member-plus-reaction count as the "supply" of constraints and the joint equations as the "demand," then check each option against whether supply meets demand.

1. Option \( m + r = 2j \): Supply exactly equals demand, so every joint's two equilibrium conditions can be met with a unique, determinate set of forces; this is the boundary case of a stable, determinate truss.
2. Option \( m - r = 2j \): This expression effectively treats reactions as reducing rather than adding to the available constraints, which does not reflect how reactions actually function (they add restraining unknowns, they do not subtract from the count), so it has no valid structural interpretation.
3. Option \( m + r<2j \): Supply falls short of demand here, meaning some of the equilibrium requirements at the joints cannot be met by the existing members and reactions alone; the structure is then free to move under load without developing internal member forces at every joint, which is the defining behaviour of an unstable truss.
4. Option \( m - r<2j \): Because it subtracts reactions rather than counting them as part of the supply, this expression misrepresents the actual constraint count and does not match the physical instability criterion.

Framing the comparison as constraint supply versus equilibrium demand again shows that a shortfall in supply is what causes instability.

So the correct answer is \( m + r<2j \).