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: For a 2D truss, the number of unknowns is the number of member forces plus reaction forces, \( m+r \), and the number of independent equilibrium equations is \( 2j \) (two per joint).
Step 2: When \( m+r = 2j \), the truss is exactly determinate; when \( m+r>2j \), it is statically indeterminate (redundant).
Step 3: When the unknowns fall short of the equations needed, \( m+r<2j \), the structure cannot satisfy equilibrium at every joint under general loading and becomes a mechanism. \[ \boxed{\text{Instability condition: } m + r<2j} \]

Answer 2

Another way to see this is through the idea of degrees of freedom: an unstable truss is one that still has at least one uncontrolled degree of freedom (a possible rigid-body-like motion) left over even after accounting for all members and supports.

1. Option \( m + r = 2j \): This balance leaves zero net degrees of freedom beyond what is needed for equilibrium, corresponding to a stable, determinate structure with no unconstrained motion.
2. Option \( m - r = 2j \): There's no structural meaning attached to subtracting reactions from members and equating that difference to \( 2j \); it does not track any physical degree-of-freedom count.
3. Option \( m + r<2j \): Since \( 2j \) represents the number of equilibrium constraints needed and \( m+r \) represents how many constraint-providing elements (members and supports) actually exist, a shortfall (\( m+r<2j \)) means there are leftover, unconstrained degrees of freedom, i.e. the structure can move without any member changing length. This is precisely instability.
4. Option \( m - r<2j \): This subtracts rather than adds the reaction count, which does not correspond to the actual number of restraining elements present, so it fails to track true degrees of freedom.

Thinking in terms of leftover unconstrained motion rather than a simple equation count leads to the same conclusion about which comparison signals instability.

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