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

Question

A beam is a structural member which is subjected to

Answer 1

Step 1: Each joint of a plane pin-jointed frame gives 2 independent equilibrium equations ($\sum F_x = 0, \sum F_y = 0$), so $j$ joints give $2j$ equations total.
Step 2: The total unknowns in the structure are the m member forces plus the r reaction components, i.e. $m + r$ unknowns.
Step 3: Degree of static indeterminacy is the excess of unknowns over the available equilibrium equations.
\[ \boxed{D_s = (m+r) - 2j} \]

Answer 2

A third way to confirm the formula is to test it on a truss that we know is indeterminate to degree 1, by adding exactly one redundant member to the basic determinate triangular truss used before.

Starting from the same determinate triangle ($j = 3$, $m = 3$, $r = 3$, giving $D_s = 0$), suppose we add one extra diagonal member somewhere without adding any new joint. Now $m = 4$ while $j = 3$ and $r = 3$ stay the same. Physically, adding one extra member without adding a joint makes the truss indeterminate to the first degree, so we expect $D_s = 1$ for this modified truss.

m + r + 2j: Substituting the new values gives $4 + 3 + 2(3) = 13$, far from the expected value of 1, so this formula is inconsistent.
m - r + 2j: Substituting gives $4 - 3 + 2(3) = 7$, also inconsistent with the expected value of 1.
m + r - 2j: Substituting gives $4 + 3 - 2(3) = 7 - 6 = 1$, exactly matching the expected first-degree indeterminacy caused by the one extra member.
m + r - 3j: Substituting gives $4 + 3 - 3(3) = 7 - 9 = -2$, which is negative and does not match the expected value of 1.

Testing the formula on a truss with one deliberately added redundant member confirms it correctly picks up the expected increase in indeterminacy.

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

Answer 3

A third way to confirm the formula is to test it on a truss that we know is indeterminate to degree 1, by adding exactly one redundant member to the basic determinate triangular truss used before.

Starting from the same determinate triangle ($j = 3$, $m = 3$, $r = 3$, giving $D_s = 0$), suppose we add one extra diagonal member somewhere without adding any new joint. Now $m = 4$ while $j = 3$ and $r = 3$ stay the same. Physically, adding one extra member without adding a joint makes the truss indeterminate to the first degree, so we expect $D_s = 1$ for this modified truss.

m + r + 2j: Substituting the new values gives $4 + 3 + 2(3) = 13$, far from the expected value of 1, so this formula is inconsistent.
m - r + 2j: Substituting gives $4 - 3 + 2(3) = 7$, also inconsistent with the expected value of 1.
m + r - 2j: Substituting gives $4 + 3 - 2(3) = 7 - 6 = 1$, exactly matching the expected first-degree indeterminacy caused by the one extra member.
m + r - 3j: Substituting gives $4 + 3 - 3(3) = 7 - 9 = -2$, which is negative and does not match the expected value of 1.

Testing the formula on a truss with one deliberately added redundant member confirms it correctly picks up the expected increase in indeterminacy.

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

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

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The Correct Option is C

Approach Solution - 1

Approach Solution -2

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