Two structural members are connected by an internal hinge and are loaded externally as shown in the figure. A and B are roller supports and C is a pin support. Neglecting the mass of the members, find the magnitude of the vertical reaction force at C, in kN.
Show Hint
Split the structure at the hinge first, since it carries no moment, then work out the reactions piece by piece.
Step 1: Write the overall equilibrium equations for the full structure.
Taking $A$ as the origin, with $B$ at 1.2 m, the 25 kN load at 1.4 m, and $C$ at 2.0 m, vertical balance of the whole structure gives $R_A + R_B + R_C = 10 + 25 = 35$ kN.
Taking moments of the whole structure about $A$: $R_B(1.2) + R_C(2.0) - 10(0.5) - 25(1.4) = 0$, which simplifies to $1.2R_B + 2.0R_C = 40$.
Step 2: Bring in the hinge condition as a third equation.
Since the hinge at 1.0 m from $A$ transmits no moment, the moment of everything to its left about the hinge must be zero on its own: $R_A(1.0) - 10(0.5) = 0$, giving $R_A = 5$ kN.
This extra condition is what makes an otherwise indeterminate three reaction beam solvable.
Step 3: Substitute $R_A$ into the vertical balance.
$R_B + R_C = 35 - 5 = 30$, so $R_B = 30 - R_C$.
Step 4: Substitute into the moment equation and solve for $R_C$.
$1.2(30 - R_C) + 2.0R_C = 40$
$36 - 1.2R_C + 2.0R_C = 40$
$0.8R_C = 4$, so $R_C = 5$ kN.
Final Answer:
Solving the whole structure together with the hinge condition gives the same reaction as splitting it into two pieces.
\[ \boxed{R_C = 5 \text{ kN}} \]
