A steel block 200 mm X 20 mm x 20 mm is subjected to a tensile force of 40 kN in the direction of its length. Determine the change in volume, if E is 205 kN/mm2 and Poisson's ratio = 0.3.

E=205 kN/mm²= 205•10⁹ N/m²,

μ = 0.3

σ=E•ε(longitudinal)
F/A = E•ε(longitudinal)
ε=ε(longitudinal) = F/A•E=
=40000/(20•10⁻³)²•205•10⁹=4.9•10⁻⁴.
If a=0.2 m, b=c=0.002 m, ε=4.9•10⁻⁴,

V=abc = 0.2•0.002•0.002 =8•10⁻⁷ m³
The deformed volume is
V₁ =(1+ ε)a•(1- μ ε)b•(1- μ ε)c .
Neglecting powers of ε, the deformed volume
V₁ =(1+ ε - 2•μ•ε)V .
The change in volume is
ΔV = ε(1- 2•μ)V =
=4.9•10⁻⁴(1 - 2•0.3)•8•10⁻⁷ =
=1.57•10⁻¹º m³

To determine the change in volume of the steel block under the given conditions, we need to use the formula for volumetric strain.

The formula for volumetric strain (εv) is given by:

εv = εx + εy + εz

Where:
εx = axial strain
εy = lateral strain in the y-direction
εz = lateral strain in the z-direction

To find the axial strain (εx), we can use Hooke's law for axial strain:

εx = σx / E

Where:
σx = axial stress
E = Young's modulus

Given that the tensile force (F) acting on the block is 40 kN, and the dimensions of the block are 200 mm x 20 mm x 20 mm, we can calculate the axial stress (σx):

σx = F / A

Where:
A = cross-sectional area

The cross-sectional area (A) can be calculated by multiplying the width (20 mm) with the height (20 mm):

A = 20 mm * 20 mm

Now we have all the values needed to calculate εx:

εx = (F / A) / E

To find the lateral strain in the y-direction (εy) and z-direction (εz), we can use the relationship between axial strain and lateral strain, given by Poisson's ratio (ν):

ν = -εy / εx = εz / εx

Rearranging the equation, we can find εy and εz:

εy = -ν * εx
εz = ν * εx

Given that Poisson's ratio (ν) is 0.3, we can substitute this value into the equations above to find εy and εz.

Finally, we can calculate the change in volume (ΔV) using the formula:

ΔV = V * εv

Where:
V = initial volume

The initial volume (V) can be calculated by multiplying the length (200 mm) with the width (20 mm) and height (20 mm).

Plug in the values into the respective formulas and perform the calculations to determine the change in volume.