GATE 2024 CE (CE1) – Question 39
The beam shown in the figure is subjected to a uniformly distributed downward load of intensity $q$ between supports A and B.
[Figure: a beam A-hinge-B-C of three equal spans $l$. A is a pin support, B and C are rollers, and there is a hinge in the beam at the end of the first span. The load $q$ acts over the first two spans, from A to B.]
Considering the upward reactions as positive, the support reactions are
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Correct answer: (A) $R_A = \frac{ql}{2}$; $R_B = \frac{5ql}{2}$; $R_C = -ql$
Explanation
The total load is $2ql$, so $R_A + R_B + R_C = 2ql$. The bending moment at the hinge is zero. For the part from A to the hinge (length $l$): $R_Al - q l\frac{l}{2} = 0$, so $R_A = \frac{ql}{2}$. For the part from the hinge to C, take moments about the hinge: $R_Bl + R_C(2l) - ql\frac{l}{2} = 0$, so $R_B + 2R_C = \frac{ql}{2}$. With $R_B + R_C = \frac{3ql}{2}$ this gives $R_C = -ql$ and $R_B = \frac{5ql}{2}$.