Question

If there are $m$ unknown member forces, $r$ unknown reaction components and $j$ number of joints, then the degree of static indeterminacy of a pin-jointed plane frame is given by

• A. $m + r + 2j$
• B. $m - r + 2j$
• C. $m + r - 2j$
• D. $m + r - 3j$

Hint:

If degree of static indeterminacy is zero, the truss is statically determinate; if positive, it is statically indeterminate.

Answer 1

The Correct Option is C

Step 1: Recall equilibrium equations for a plane pin-jointed frame.
Each joint in a plane truss provides two independent equilibrium equations:
\[ \sum F_x = 0, \quad \sum F_y = 0 \]
Step 2: Total available equations.
For $j$ joints, total equilibrium equations available are:
\[ 2j \]
Step 3: Count total unknowns.
Total unknowns in the structure are:
\[ m + r \]
Step 4: Degree of static indeterminacy.
Degree of static indeterminacy is defined as:
\[ \text{DSI} = \text{Unknowns} - \text{Equations} = (m + r) - 2j \]
Step 5: Conclusion.
Hence, the correct expression is $m + r - 2j$.