A steel bar 50 mm x 50 mm x 300 mm is under compressive load of 100 kN applied in the direction of its length. Find the changes in length and volume if all lateral strains are prevented. Also find the apparent modulus of elasticity if E = 200 GPa and µ = 0.3.

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Accepted Answer

[latex]\sigma_x, \sigma_y and \sigma_z[/latex] are compressive and therefore taken -ve in Eq. (1.23),

[latex]\varepsilon_x=\frac{1}{E}\left[-\sigma_x+\mu\left(\sigma_y+\sigma_z ight) ight] [/latex]

[latex]\varepsilon_y=\frac{1}{E}\left[-\sigma_y+\mu\left(\sigma_z+\sigma_x ight) ight]=0 [/latex]

[latex]\varepsilon_z=\frac{1}{E}\left[-\sigma_z+\mu\left(\sigma_y+\sigma_x ight) ight]=0 [/latex]

By symmetry, [latex]\sigma_y=\sigma_z=\sigma [/latex]

or, [latex] -\sigma+0.3\left(\sigma_x+\sigma ight)=0 [/latex]

or, [latex]\sigma=\frac{0.3 \sigma_x}{0.7}=\frac{0.3 imes 100000}{0.7 imes 50 imes 50}=17.14 N / mm ^2[/latex]

which gives, [latex]\varepsilon_x =\frac{1}{200000}(-40+0.3 imes 2 imes 17.14) =-0.00014858 mm / mm [/latex]

or, [latex]\Delta L=-0.00014858 imes 300=-0.044574 [/latex] mm

Apparent modulus of elasticity,

[latex]E_{m, x}=\frac{\sigma_x}{\varepsilon_x}=\frac{40}{0.00014858}=269215 N / mm ^2 [/latex]

Change in volume [latex]=\Delta L imes A=-0.0446 imes 50 imes 50 =111.5 mm ^3 [/latex]

Question 1.24: A steel bar 50 mm x 50 mm x 300 mm is under compressive load...

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