GATE 2026 AE – Question 60

Structures · Strength of materials: stress and strain, stress-strain curves, trusses, bars, beams and shafts, determinate and indeterminate · 2 marks · Numerical answer

A stepped cantilever beam, made of a material having Young's modulus E = 200 GPa, is shown in the figure below. The length and the moment of inertia of the beam from point A to B are L$_1$ = 100 mm and I$_1$ = 100 mm$^4$, respectively. The length and the moment of inertia of the beam from point B to C are L$_2$ = 100 mm and I$_2$ = 700 mm$^4$, respectively. A shear force P = 30 N is applied at point A of the beam. The magnitude of the deflection of the beam at point A is ________ mm (rounded off to 1 decimal place).

A cantilever fixed at C on the left. The part CB has moment of inertia $I_2$ and length $L_2$, and the part BA has $I_1$ and length $L_1$. The load P acts downwards at the free end A.

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Correct answer: 0.99 to 1.01

Explanation

With $E = 2 \times 10^5$ N/mm$^2$: the segment AB bends as a cantilever fixed at B, giving $\frac{PL_1^3}{3EI_1} = \frac{30 \times 10^6}{3 \times 2 \times 10^5 \times 100} = 0.5$ mm. Segment BC carries the force $P$ and the moment $PL_1$. Its tip deflection is $\frac{PL_2^3}{3EI_2} + \frac{PL_1 L_2^2}{2EI_2} = 0.0714 + 0.1071 = 0.1786$ mm, and the tip rotation is $\frac{PL_2^2}{2EI_2} + \frac{PL_1 L_2}{EI_2} = 0.001071 + 0.002143 = 0.003214$ rad. The deflection at A is $0.5 + 0.1786 + 0.003214 \times 100 = 1.0$ mm.