GATE 2026 CE (CE1) – Question 51

Solid Mechanics · Simple stress and strain relationships · 2 marks · Numerical answer

A homogenous, linearly elastic rod AB is connected to a linearly elastic spring BC in between the fixed supports at A and C, as shown in the figure. The cross-sectional area, modulus of elasticity, and the coefficient of thermal expansion of the rod AB are 500 mm$^2$, $60 \times 10^3$ MPa, and $12 \times 10^{-6}$ per °C, respectively. The stiffness ($k$) of spring BC is 2500 N/mm.

The internal force (in kN) that will develop in the spring BC when the temperature of rod AB is increased by 100 °C is _________ (*rounded off to one decimal place*).

A rod AB of length 3 m fixed at A, joined at B to a spring BC of stiffness $k = 2500$ N/mm, whose other end is fixed at C.

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Correct answer: 7.16 to 7.24

Explanation

The free thermal expansion of the rod is $\alpha\Delta T L = 12 \times 10^{-6} \times 100 \times 3000 = 3.6$ mm. The axial stiffness of the rod is $\frac{AE}{L} = \frac{500 \times 60000}{3000} = 10000$ N/mm. The rod and the spring are in series, so the force is $F = \frac{3.6}{\frac{1}{10000} + \frac{1}{2500}} = \frac{3.6}{0.0005} = 7200$ N, which is 7.2 kN.