A short bar of copper, 25 mm in diameter, is enclosed centrally within a steel tube of external diameter 36 mm and thickness 3 mm. At 0°C the ends of the bar and tube are rigidly fastened together and the complete assembly heated to 80°C. Calculate the stress in the bar and in the tube if E for

Question

A short bar of copper, 25 mm in diameter, is enclosed centrally within a steel tube of external diameter 36 mm and thickness 3 mm. At 0°C the ends of the bar and tube are rigidly fastened together and the complete assembly heated to 80°C. Calculate the stress in the bar and in the tube if E for copper is 100 000 N/mm² , E for steel is 200 000 N/mm² and the coefficients of linear expansion of copper and steel are 0.000 01/°C and 0.000 006/°C, respectively.

Accepted Answer

The cross-sectional areas of the steel tube and copper bar are, respectively,

[latex] A_{ st }=\frac{\pi\left(36^{2}-30^{2} ight)}{4}=311.02 mm ^{2}, A_{ c }=\frac{\pi imes 25^{2}}{4}=490.9 mm ^{2} [/latex]

Then, from Eqs (7.49)

[latex]\begin{aligned}\sigma_{ C } &=\frac{P_{ C }}{A_{ C }}=\frac{\Delta T\left(\alpha_{ S }-\alpha_{ C } ight) E_{ C } A_{ S } E_{ S }}{A_{ C } E_{ C }+A_{ S } E_{ S }} \\sigma_{ S } &=\frac{P_{ S }}{A_{ S }}=\frac{\Delta T\left(\alpha_{ S }-\alpha_{ C } ight) A_{ C } E_{ C } E_{ S }}{A_{ C } E_{ C }+A_{ S } E_{ S }}\end{aligned}[/latex] (7.49)

[latex] \sigma_{s t}=\frac{80(0.000006-0.00001) imes 490.9 imes 100000 imes 200000}{(490.9 imes 100000+311.02 imes 200000)} [/latex]

i.e.

[latex] \sigma_{ ext {steel }}=-28.3 N / mm ^{2} [/latex]

The stress in this case is negative so that, unlike the steel reinforcement in the concrete column of Fig. 7.28, the steel is in tension. This is logical since the coefficient of linear expansion of copper is greater than that of steel. Again, from Eqs (7.49)

[latex] \sigma_{ c }=\frac{80(0.000006-0.00001) imes 311.02 imes 100000 imes 200000}{(490.9 imes 100000+311.02 imes 200000)} [/latex]

i.e.

[latex] \sigma_{ ext {copper }}=-17.9 N / mm ^{2} [/latex]

The concrete in the column of Fig. 7.21 is in tension; here the negative answer indicates that the copper tube is in compression; again an expected result.

Question 7.17: A short bar of copper, 25 mm in diameter, is enclosed centra...

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