Civil Engineering > GATE 2026 SET-1 > Thermal Stress
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 mm2, 60 × 103 MPa, and 12 × 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).
Correct : 90.0
Rod AB has length L = 3000 mm, A = 500 mm², E = 60,000 MPa = 60,000 N/mm², α = 12×10⁻⁶/°C, ΔT = 100°C. Spring BC has stiffness k = 25 N/mm.
Free thermal expansion of rod AB:
δT = α × ΔT × L = 12×10⁻⁶ × 100 × 3000 = 3.6 mm
Axial stiffness of rod AB:
krod = AE/L = (500 × 60,000)/3000 = 10,000 N/mm
Compatibility condition:
Since both ends A and C are fixed, the free thermal expansion is completely resisted by the rod compression and spring compression. The same internal force P acts throughout:
δT = δrod + δspring
3.6 = P/krod + P/k
3.6 = P × (1/10,000 + 1/25)
3.6 = P × (10⁻⁴ + 0.04)
3.6 = P × 0.0401
P = 3.6/0.0401 = 89.8 ≈ 90.0 kN
The internal force developed in spring BC = 90.0 kN ✓
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