Question

A steel tube with 2.4 cm external diameter and 1.8 cm internal diameter encloses a copper rod 1.5 cm diameter to which it is rigidly joined at each end. If at a temperature of 10°C, there is no longitudinal stress, calculate the stresses in the rod and tube when the temperature is raised to 200°C. E_S = 210,000 N/mm², α_s = 11 × 10^-6/°C, E_C = 100,000 N/mm², αC = 18 × 10^-6/°C

Answer 1

Given that:

E_S = 210,000 N/mm²

E_C = 100,000 N/mm²

α_s = 11 × 10^–6/°C,

α_C = 18 × 10^–6/°C

∆T = 200°C

Apply compatibility condition:

e_C + e_S = (α_C – α_S ) ∆t

e_C + e_S = (18 – 11) × 10^-6 × 200

e_C + e_S = 0.0014 ...(i)

From the equilibrium condition; 

Compressive force on copper = Tensile force on steel

P_B = P_C

e_C.A_C.E_C = e_S .A_S.E_S

e_C = e_S[(A_S/A_C)(E_S/E_C)]

= e_S [{(π/4)(2.4² – 1.8²)/(π/4)(1.5²)}(210/100)]

e_C = 2.35e_S ...(ii)

Substituting the value of equation(ii) in equation (i)

2.35e_S + e_S = 0.0014

e_S = 0.000418  ...(iii)

e_C = 0.000982 ...(iv)

Stress in steel tube δ_S = e_S .E_S

= 0.000418 × 210,000

σ_S = 87.7 N/mm² 

Stress in copper tube σ_C = e_C.E_C

= 0.000982 × 100,000

σ_C = 98.2 N/mm²