The plane frame has a hinge and a roller support, and is loaded as shown in the figure. Both the columns have same height.
(Figure not to scale)
What is the absolute value of the maximum bending moment (in kN-m) in the frame?
Show Hint
Find the support reactions first, then work out the bending moment at the top of each column and at the point-load location; the frame's moment comes out to zero at the roller-side column top.
Step 1: Split the loading into two simpler cases and use superposition.
Since the frame is linear elastic, the total bending moment under both loads together equals the sum of the moments from each load acting alone. Break the problem into Case (a): only the 50 kN horizontal load acts, and Case (b): only the 90 kN vertical load acts.
Step 2: Solve Case (a), the 50 kN load alone.
With only the 50 kN horizontal force at A(0,3) (using the same coordinates as before, P at origin, R at (4,0)): moments about P give $V_R' \times 4 = 50\times3=150$, so $V_R'=37.5$ kN, and vertical equilibrium (no vertical load in this case) gives $V_P'=-37.5$ kN. Horizontal equilibrium gives $H_P'=50$ kN.
Moment at A: $M_A' = H_P'\times3 = 150$ kN-m. Moment at B: $M_B' = H_P'\times3 + V_P'\times2 = 150 + (-37.5)(2) = 150-75=75$ kN-m.
Step 3: Solve Case (b), the 90 kN load alone.
With only the 90 kN downward force at B(2,3): moments about P give $V_R''\times4=90\times2=180$, so $V_R''=45$ kN, and vertical equilibrium gives $V_P''=90-45=45$ kN. Since there is no horizontal load in this case, $H_P''=0$.
Moment at A: $M_A''=H_P''\times3=0$. Moment at B: $M_B''=H_P''\times3+V_P''\times2=0+45\times2=90$ kN-m.
Step 4: Add the two cases (superposition) to get the actual moments.
$M_A = M_A'+M_A'' = 150+0=150$ kN-m.
$M_B = M_B'+M_B''=75+90=165$ kN-m.
This matches the combined-loading result directly, confirming the answer through an entirely separate route (solving two simpler one-load problems and adding them) instead of solving the combined loading in one pass.
Step 5: Confirm this is the overall maximum.
Going further along the beam from B to C, the 90 kN load's own moment contribution works in the opposite sense and brings the total moment back down to zero at C (the top of the right column, which carries only the vertical roller reaction and so no bending moment at all). So among A (150), B (165), and C (0), the biggest is at B.
Final Answer:
The maximum absolute bending moment is 165 kN-m, at the point where the 90 kN load is applied.
\[ |M|_{max} = 165 \text{ kN-m} \]
