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 $m + r - 2j$.

Answer 2

Rather than deriving the formula from scratch again, we can verify which option is correct by testing all four formulas against a truss whose degree of static indeterminacy we already know from basic experience: a simple, stable, statically determinate triangular truss.

Take the most basic pin-jointed truss possible — a single triangle with 3 joints (\(j = 3\)), 3 members (\(m = 3\)), and 3 independent reaction components at its supports (\(r = 3\), e.g. a pin and a roller). This truss is well known to be exactly statically determinate, meaning its degree of static indeterminacy must equal 0.

1. m + r + 2j: Substituting gives \(3 + 3 + 2(3) = 12\), nowhere close to 0, so this formula is inconsistent with a known determinate truss.
2. m - r + 2j: Substituting gives \(3 - 3 + 2(3) = 6\), also not 0, so this formula fails the check as well.
3. m + r - 2j: Substituting gives \(3 + 3 - 2(3) = 6 - 6 = 0\), exactly matching the known determinate result for this basic truss.
4. m + r - 3j: Substituting gives \(3 + 3 - 3(3) = 6 - 9 = -3\), which is not 0 and does not represent a valid non-negative indeterminacy for a determinate structure.

Only the formula \(m + r - 2j\) correctly reduces to zero for this simple, known statically determinate truss, confirming it as the general expression for degree of static indeterminacy.

Therefore, the correct answer is m + r - 2j.