Question 1.24: A steel bar 50 mm x 50 mm x 300 mm is under compressive load...
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.
\sigma_x, \sigma_y and \sigma_z are compressive and therefore taken -ve in Eq. (1.23),
\varepsilon_x=\frac{1}{E}\left[-\sigma_x+\mu\left(\sigma_y+\sigma_z\right)\right]\varepsilon_y=\frac{1}{E}\left[-\sigma_y+\mu\left(\sigma_z+\sigma_x\right)\right]=0
\varepsilon_z=\frac{1}{E}\left[-\sigma_z+\mu\left(\sigma_y+\sigma_x\right)\right]=0
By symmetry, \sigma_y=\sigma_z=\sigma
or, -\sigma+0.3\left(\sigma_x+\sigma\right)=0
or, \sigma=\frac{0.3 \sigma_x}{0.7}=\frac{0.3 \times 100000}{0.7 \times 50 \times 50}=17.14 N / mm ^2
which gives, \varepsilon_x =\frac{1}{200000}(-40+0.3 \times 2 \times 17.14) =-0.00014858 mm / mm
or, \Delta L=-0.00014858 \times 300=-0.044574 mm
Apparent modulus of elasticity,
E_{m, x}=\frac{\sigma_x}{\varepsilon_x}=\frac{40}{0.00014858}=269215 N / mm ^2Change in volume =\Delta L \times A=-0.0446 \times 50 \times 50 =111.5 mm ^3