Structural Analysis

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: 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: In such a case, the truss lacks sufficient constraints and becomes unstable.
Step 3: Evaluation of the options.
(A) : This represents a stable and determinate structure, not instability.
(B) : This is not a valid criterion for truss stability.
(C) : Correct — this indicates insufficient members or reactions, leading to instability.
(D) : 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 .

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: 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: 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) : This represents a statically determinate and stable truss, not an unstable one.
(B) : This equation does not correspond to standard truss stability criteria.
(C) : Correct — this indicates fewer constraints than required, resulting in an unstable truss.
(D) : 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 .