GATE 2025 CE (CE1) – Question 50

Structural Analysis · Deflections of determinate beams, frames and trusses · 2 marks · Numerical answer

For the beam and loading shown in the figure, the second derivative of the deflection curve of the beam at the mid-point of AC is given by $\frac{\alpha M_0}{8EI}$. The value of $\alpha$ is ___________________ (*rounded off to the nearest integer*).

A simply supported beam ABC of total length 2L, with a pin support at A and a roller support at B. The part from A to C, of length L, has rigidity 2EI and the part from C to B, of length L, has rigidity EI. A couple $M_0$ acts at A.

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Correct answer: 3

Explanation

For a simply supported beam of span $2L$ with a couple $M_0$ at one end, the bending moment falls linearly from $M_0$ at A to zero at B, so $M(x) = M_0\left(1 - \frac{x}{2L}\right)$. At the mid-point of AC, $x = \frac{L}{2}$, so $M = 0.75M_0$. The second derivative of the deflection is $\frac{M}{EI_{local}}$, and the rigidity there is $2EI$, so it is $\frac{0.75M_0}{2EI} = \frac{3M_0}{8EI}$. So $\alpha = 3$.