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: Understanding the basic stability condition.
For a two-dimensional truss structure, the condition for a stable and statically determinate structure is given by: \[ m + r = 2j \] This equation ensures that the number of unknown forces (members and reactions) is equal to the number of available equilibrium equations.
Step 2: Identifying the instability condition.
If the number of members and reactions is less than what is required for equilibrium, the structure will not be able to maintain stability. This condition is mathematically written as: \[ m + r<2j \] In such a case, the truss lacks sufficient constraints and becomes unstable.
Step 3: Evaluation of the options.
(A) \( m + r = 2j \): This represents a stable and determinate structure, not instability.
(B) \( m - r = 2j \): This is not a valid criterion for truss stability.
(C) \( m + r<2j \): Correct — this indicates insufficient members or reactions, leading to instability.
(D) \( m - r<2j \): This condition is not used for analyzing truss instability.
Step 4: Conclusion.
The structure becomes unstable when the total number of members and reactions is less than twice the number of joints. Hence, the correct condition is \( m + r<2j \).

Answer 2

Static determinacy of a 2D truss compares the number of unknowns, \( m+r \) (member forces plus reactions), against the number of independent joint-equilibrium equations, \( 2j \); testing each option against this comparison identifies the instability condition.

1. Option \( m + r = 2j \): An exact match between unknowns and equations means the structure is just-determinate, with a unique solution existing for every member force and reaction under general loads; this is the signature of a stable, determinate truss, not an unstable one.
2. Option \( m - r = 2j \): There is no standard structural relationship that equates the difference of members and reactions to twice the joint count; this expression does not arise from the joint-equilibrium method at all.
3. Option \( m + r<2j \): When the number of unknowns is smaller than the number of equilibrium equations that must hold, some equations cannot be satisfied for arbitrary loading unless the structure deforms as a mechanism, meaning there are not enough members/reactions to prevent motion; this is exactly what defines instability.
4. Option \( m - r<2j \): Subtracting the reaction count instead of adding it does not correctly represent the true number of unknowns present in the structure, so this inequality does not correspond to any recognized stability criterion.

The comparison that correctly tracks total unknowns against total equilibrium equations shows a deficit is what causes instability.

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