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 \( m + r<2j \).

Answer 2

For a two-dimensional truss, static determinacy and stability are governed by comparing the total number of unknowns (members plus support reactions, \( m+r \)) against the number of independent equilibrium equations available (two per joint, giving \( 2j \)). Each option is checked against this comparison.

1. Option \( m + r = 2j \): Here the unknowns exactly match the available equations, meaning every unknown can, in principle, be solved for uniquely; this describes a statically determinate and (typically) stable truss, not an unstable one, so it does not describe instability.
2. Option \( m - r = 2j \): Subtracting the number of reactions from the number of members does not correspond to any recognized structural-analysis balance between unknowns and equations; this combination has no standard physical meaning in truss stability theory.
3. Option \( m + r<2j \): Here the number of available unknowns (members plus reactions) is smaller than the number of equilibrium equations that must be satisfied at every joint; there simply are not enough members or supports to hold every joint in equilibrium under arbitrary loading, which is exactly the condition that leaves the structure under-constrained and hence unstable (a mechanism).
4. Option \( m - r<2j \): Since \( r \) is subtracted here rather than added, this expression does not track the true total count of unknowns in the structure and does not correspond to the recognized instability criterion.

Comparing the true count of unknowns, \( m+r \), against the number of equilibrium equations, \( 2j \), shows that having fewer unknowns than equations is precisely what makes a truss unstable.

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