Question

A steel bar 35 mm x 35 mm in section and 100 mm in length is acted upon by a loud of 180 KN along its longitudinal axis and 400 KN and 300 KN along the axes of the lateral surface Determine

i) Change in the dimension of the bar

ii) Change in volume Take E = 205 Gpa and poissons ratio 0.3​. .

Answer 1

The stresses in the direction of X, Y and Z axis,

Along X-axis,

${P}_{x} = \frac{{F}_{x}}{A} = \frac{180∗10³}{35∗35}(N/mm∗mm)$

= 146.94 N/mm²

Along Y axis,

${P}_{y} = \frac{{F}_{y}}{A} = \frac{400∗10³}{100∗35} (N/mm∗mm)$

= 114.29 N/mm²

Along Z axis,

${P}_{z} = \frac{{F}_{y}}{A}= \frac{300∗10³}{100∗35}(N/mm∗mm)$

= 85.714 N/mm²

Now, the strain along the three principal directions are, due to stresses, ${P}_{x},]{P}_{y},{P}_{z},$

${e}_{x} = \frac{{P}_{x}}{E}−\frac{{μP}_{y}}{E}−\frac{{μP}_{z}}{E}$

${e}_{x} = \frac{146.94}{205∗10³} − \frac{0.3∗114.29}{205∗10³} - \frac{0.3* 85.714 }{205*10³}$

= 0.000424

Now,

${e}_{y} = \frac{{P}_{y}}{E} − \frac{{μP}_{z}}{E} − \frac{{μP}_{x}}{E}$

${e}_{y}=\frac{114.29}{205∗10³} - \frac{0.3*85.714}{205*10³}-\frac{0.3∗146.94}{205∗10³}$

= 0.000217

${e}_{z} = \frac{{P}_{z}}{E} −\frac{{μP}_{y}}{E}- \frac{{μP}_{x}}{E}$

ez=$\frac{85.714}{0.3*10³} - \frac{0.3∗114.29}{205∗10³} -\frac{0.3∗146.94}{205∗10³}$

= 0.000036

Volumetric sign = ex+ey+ez

= 0.000424 + 0.000217 + 0.000036 ev

= 0.000677 ev

Also, volumetric strain = $\frac{change in volume}{original volume}$

=> 0.000677 = $\frac{change in volume}{100∗35∗35}$

=> change in volume = 82.9325 mm³

Now, ${e}_{x} = \frac{∆L}{L}$

=> $0.000424 = frac{∆L}{100}$

=> ∆L = 0.0424 mm

${e}_{y} = \frac{∆W}{W}$

=> $0.000217 = frac{∆W}{35}$

=> ΔW= 0.007595 mm

${e}_{z} = \frac{∆t}{t}$

=> $0.000036 = frac{∆t}{100}$

Δt = 0.00126 mm

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Answer 2

Explanation:

A steel bar 35 mm x 35 mm in section and 100 mm in length is acted upon by a loud of 180 KN along its longitudinal axis and 400 KN and 300 KN along the axes of the lateral surface Determine

i) Change in the dimension of the bar

ii) Change in volume Take E = 205 Gpa and poissons ratio 0.3. .