Compute the thermal stresses in steel:
Consider the compound bar shown in below figure consisting of a steel bolt of diameter 18 mm, surrounded through a copper tube of outer and inner diameters 30 mm and 18 mm respectively. The assembly is just snug at 15oC. The material properties are given as:
Young's modulus of steel, ES = 200 kN/mm2
Young's modulus of copper, EC = 120 kN/mm2.
Coefficient of linear thermal expansion of steel, αs = 12 × 10-6 m/m/ o C
Coefficient of linear thermal expansion of copper, αc = 18 × 10-6 m/m/ o C
Compute the thermal stresses in steel and copper when the temperature of the assembly is raised to 45oC.
In this case, when the temperature of the assembly is raised, copper will expand more than steel and hence, will exert a thrust on the steel nut and bolt head thus producing tension in the steel. Since there is no other external force this tension has to be kept in equilibrium by the compressive force induced in copper. Hence by formulating a compatibility condition of total strains (thermal + elastic) the problem could be solved.
Now let the forces induced in steel and copper are designated as Ps and Pc respectively in kN units.
Thermal strain in steel, εst = αs × ΔT
= 12 × 10-6 × (40 - 15)
= 3 × 10-4 m/m
Thermal strain in copper, αc × ΔT
= 18 × 10-6 × (40 - 15)
= 4.5 × 10-4 m/m
Elastic strain in steel, δse = Ps /As Es
= Ps/ (π/4) × 182 × 200
= 1.965 × 10-5 Ps
Elastic strain in copper, δce = Pc /Ac Ec
= Pc/ (π/4) × (302 - 182) 120
= 1.8421 × 10-5 Pc