A square steel bar is 50 mm on a side and 250 mm long. It is loaded by an axial tensile force of 200 kN. If E = 200 GPa and v = 0.3, determine the change of volume per unit volume.
Options:
1) 0.000823
2) 0.0234
3) 0.0002
4) 0.00016
Answer (2)
To determine the change of volume per unit volume, also known as volumetric strain, we can use the relationship involving longitudinal and lateral strains in materials.
Given:
Length of steel bar, L = 250 mm
Side of square base, a = 50 mm
Axial tensile force, F = 200 kN = 200 , 000 N
Young's modulus, E = 200 GPa = 200 , 000 MPa
Poisson's ratio, v = 0.3
First, let's calculate the axial or longitudinal strain using: Longitudinal strain ( ε ) = E σ where σ is the axial stress given by: σ = A F = 50 × 50 mm 2 200 , 000 N = 80 MPa
Thus, the longitudinal strain is: ε = 200 , 000 MPa 80 MPa = 0.0004
Next, we calculate the lateral strain which is due to Poisson's effect: Lateral strain ( ε l ) = − v × ε = − 0.3 × 0.0004 = − 0.00012
Now, we calculate the change of volume per unit volume (volumetric strain) using the formula: Volumetric strain = ε + 2 × ε l = 0.0004 + 2 ( − 0.00012 ) = 0.0004 − 0.00024 = 0.00016
Therefore, the change of volume per unit volume is represented by option 4) 0.00016 .
The change of volume per unit volume, or volumetric strain, is calculated to be 0.00016. This is determined using the longitudinal strain and lateral strain with the given parameters of the steel bar. Hence, the correct option is 4) 0.00016.
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