GATE 2021 AE – Question 61
Cantilever beams AB and DC contact at their free ends through a roller. Both have 50×50 mm square cross sections and E=70 GPa. AB is 400 mm long under 20 kN/m uniform load; DC is 250 mm long. Roller compression is ___ kN (one decimal).
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Correct answer: 2.39 to 2.41
Explanation
The two cantilevers touch at their free ends through a roller, so the end deflections must be compatible: the downward deflection of the tip of AB (which carries the uniform load) is reduced by the roller force $P$ pushing up on it, and the same force $P$ pushes DC's tip down.
**Deflection of AB's tip** (length $L_1$) under UDL $w$ and the roller force $P$:
$$\delta_1=\frac{wL_1^4}{8EI}-\frac{PL_1^3}{3EI}.$$
**Deflection of DC's tip** (length $L_2$) under $P$:
$$\delta_2=\frac{PL_2^3}{3EI}.$$
**Compatibility** ($\delta_1=\delta_2$, same $E$ and $I$):
$$\frac{wL_1^4}{8EI}=\frac{P\,(L_1^3+L_2^3)}{3EI}\;\Rightarrow\;P=\frac{3wL_1^4}{8\,(L_1^3+L_2^3)}.$$
**Numbers.** $w=20\,000$ N/m, $L_1=0.4$ m, $L_2=0.25$ m:
$$P=\frac{3\times20\,000\times0.0256}{8\times(0.064+0.015625)}=\frac{1536}{0.637}=2411\text{ N}=\mathbf{2.4\ kN}.$$