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 2D trusses: Stable and determinate → \( m + r = 2j \) Unstable → \( m + r<2j \) Redundant → \( m + r>2j \)

Answer 1

The Correct Option is C

Step 1: Understanding the stability condition of a 2D truss.
For a two-dimensional truss structure, the basic equation relating the number of members, joints, and reactions for a stable and statically determinate structure is: \[ m + r = 2j \] This equation ensures that the structure has just enough constraints to remain stable without being redundant.
Step 2: Identifying the condition for instability.
If the total number of unknowns (members and reactions) is less than the number of equilibrium equations available, the structure becomes unstable. Mathematically, this condition is expressed as: \[ m + r<2j \] In this case, the structure does not have sufficient members or reactions to maintain equilibrium, leading to instability.
Step 3: Analysis of the given options.
(A) \( m + r = 2j \): This represents a statically determinate and stable truss, not an unstable one.
(B) \( m - r = 2j \): This equation does not correspond to standard truss stability criteria.
(C) \( m + r<2j \): Correct — this indicates fewer constraints than required, resulting in an unstable truss.
(D) \( m - r<2j \): This condition is not used for determining 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. Therefore, the correct condition for instability is \( m + r<2j \).