A composite column is formed by placing a steel bar, 20 mm in diameter and 200 mm long, inside an alloy cylinder of the same length whose internal and external diameters are 20 and 25 mm, respectively. The column is then subjected to an axial load of 50 kN. If E for steel is 200 000 N/mm^2 and E

Question

A composite column is formed by placing a steel bar, 20 mm in diameter and 200 mm long, inside an alloy cylinder of the same length whose internal and external diameters are 20 and 25 mm, respectively. The column is then subjected to an axial load of 50 kN. If E for steel is 200 000 N/mm² and E for the alloy is 70 000 N/mm² , calculate the stress in the cylinder and in the bar, the shortening of the column and the strain energy stored in the column.

Accepted Answer

Eqs (7.38) may be used directly to determine the stresses in the steel bar and alloy cylinder.
The areas of cross section are

[latex]\sigma_{ S }=\frac{E_{ S }}{A_{ C } E_{ C }+A_{ S } E_{ S }} P \quad \sigma_{ C }=\frac{E_{ C }}{A_{ C } E_{ C }+A_{ S } E_{ S }} P[/latex] (7.38)

[latex] A_{ st }=\frac{\pi imes 20^{2}}{4}=314.16 mm ^{2}, A_{ all }=\frac{\pi\left(25^{2}-20^{2} ight)}{4}=176.71 mm ^{2} [/latex]

Then

[latex] \sigma_{s t}=\frac{50 imes 10^{3} imes 200000}{(314.16 imes 200000+176.71 imes 70000)} [/latex]

i.e.

[latex] \sigma_{ ext {stel }}=132.9 N / mm ^{2} [/latex]

Similarly

[latex] \sigma_{ ext {all }}=\frac{50 imes 10^{3} imes 70000}{(314.16 imes 200000+176.71 imes 70000)} [/latex]

i.e.

[latex] \sigma_{ ext {alloy }}=46.5 N / mm ^{2} [/latex]

From Eq. (7.37) the shortening of the column is

[latex] \delta=\frac{P L}{A_{ C } E_{ C }+A_{ S } E_{ S }} [/latex] (7.37)

[latex] \delta=\frac{50 imes 10^{3} imes 200}{(314.16 imes 200000+176.71 imes 70000)} [/latex]

i.e.

[latex] \delta=0.13 mm [/latex]

From Eq. (7.30) the strain energy stored in the column is

[latex] U=\frac{\sigma^{2}}{2 E} imes A L_{0} [/latex] (7.30)

[latex] U=\frac{200\left[\left(\frac{132.9^{2} imes 314.16}{200000} ight)+\left(\frac{46.5^{2} imes 176.71}{70000} ight) ight]}{2} [/latex]

i.e.

[latex] U=3320.3 Nmm =3.3 Nm . [/latex]

Question 7.14: A composite column is formed by placing a steel bar, 20 mm i...

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