GATE 2022 AE – Question 57
The tip deflection and tip slope of a tip-loaded cantilever of length L are $NL^3/(3EI)$ and $NL^2/(2EI)$, where N is tip force. A rectangular cantilever PQ carries transverse load F at its midpoint. End Q is free in Case I and simply supported in Case II. The ratio of maximum bending-stress magnitudes at P in Case I to Case II is (one decimal place).
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Correct answer: 2.69 to 2.71
Explanation
Let the cantilever PQ have total length $2l$ and the load $F$ act at its midpoint. P is the fixed end.
**Case I (Q free).** The bending moment at P is $M_I=F\,l$.
**Case II (Q simply supported).** The beam is statically indeterminate, with an unknown support reaction $R$ at Q. Use compatibility: the deflection at Q must be zero.
- Deflection at Q due to $F$ at the midpoint: the midpoint deflects $\dfrac{Fl^3}{3EI}$ and its slope is $\dfrac{Fl^2}{2EI}$, and the beam beyond the load stays straight, so
$$\delta_Q=\frac{Fl^3}{3EI}+\frac{Fl^2}{2EI}\cdot l=\frac{5Fl^3}{6EI}.$$
- Deflection at Q due to $R$ (a tip load on a cantilever of length $2l$): $\dfrac{R(2l)^3}{3EI}=\dfrac{8Rl^3}{3EI}$.
Setting the sum to zero in magnitude: $\dfrac{8R}{3}=\dfrac{5F}{6}\Rightarrow R=\dfrac{5F}{16}$.
**Moment at P in Case II:**
$$M_{II}=Fl-R(2l)=Fl-\frac{5Fl}{8}=\frac{3Fl}{8}.$$
**Ratio of maximum bending stresses** (same section, so the ratio of moments):
$$\frac{M_I}{M_{II}}=\frac{Fl}{3Fl/8}=\frac83=\mathbf{2.7}.$$