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 \( m + r<2j \).