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GATE 2024 ME – Question 58

Engineering Mechanics · Plane trusses and frames · 2 marks · Numerical answer

A three-hinge arch ABC in the form of a semi-circle is shown in the figure. The arch is in static equilibrium under vertical loads of $P = 100$ kN and $Q = 50$ kN. Neglect friction at all the hinges. The magnitude of the horizontal reaction at B is ______ kN (*rounded off to 1 decimal place*).

A semi-circular arch of radius 6 m with hinges at the supports A and C and at the crown B. The span is 12 m. Load P acts 3 m to the left of B and load Q acts 3 m to the right of B.

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Correct answer: 37.31 to 37.69

Explanation

Take A at $x = 0$, B (the crown) at $x = 6$ and C at $x = 12$ m, so P acts at $x = 3$ and Q at $x = 9$. Moments about C give $12A_y = 100(9) + 50(3) = 1050$, so $A_y = 87.5$ kN. The crown hinge carries no moment, so for the left half, taking moments about B (6 m above the supports): $A_y(6) - 100(3) - H_A(6) = 0$, so $H_A = \frac{525 - 300}{6} = 37.5$ kN. The horizontal force on the left half must balance, so the horizontal reaction at the hinge B is also 37.5 kN.