A steel block 200mm*20mm*20mm is subjected to a tensile force of 40kn in the direction of its length.Determine the change in volume,if Eis 205kn/mm square and poisson's ratio=0.3.
Unit volume expansion, e
=εx+εy+εz
=(1-2ν)(σx+σy+σz)/E
Using
σx=40*10^3 N
σy=σz=0
E=205 GPa
ν=0.3
Volume change
=0.2*0.02*0.02*e
=0.0008e
E=205 kN/mm²= 205•10⁹ N/m².
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,
μ = 0.3
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
Well, isn't this a "force"-ful question? Let's calculate the change in volume, shall we?
First, we need to find out the original volume of the steel block. Since it's a rectangular solid, we can calculate it using the formula: original volume = length * width * height.
Plugging in the values, we get: original volume = 200mm * 20mm * 20mm. Let's convert this to meters for simplicity. That gives us: original volume = 0.2m * 0.02m * 0.02m.
Now we can find the change in volume using the formula: change in volume = original volume * (tensile stress / E) * (1 - 2 * Poisson's ratio).
Substituting the given values, we get: change in volume = 0.2m * 0.02m * 0.02m * (40kN / 205kN/mm^2) * (1 - 2 * 0.3).
After a little bit of math, we can calculate the change in volume for this tugging steel block. Let's just hope it doesn't go through an identity crisis!
To determine the change in volume, we can make use of the equation relating the change in volume to the longitudinal strain and the Poisson's ratio.
The given dimensions of the steel block are:
Length (L) = 200mm
Width (W) = 20mm
Height (H) = 20mm
The tensile force acting on the block is 40kN. To find the change in volume, follow these steps:
Step 1: Convert the given force from kN to N:
Force (F) = 40kN = 40,000N
Step 2: Calculate the cross-sectional area of the block:
Cross-sectional area (A) = Width (W) * Height (H)
A = 20mm * 20mm
A = 400mm^2
Step 3: Convert the area from mm^2 to m^2:
A = 400mm^2 = 400 * 10^-6 m^2
A = 0.0004m^2
Step 4: Calculate the longitudinal strain (ε) using the formula:
ε = (Force * Length) / (Young's modulus * Area)
Given Young's modulus (E) = 205kN/mm^2 = 205 * 10^3 N/mm^2
Given Area (A) = 0.0004m^2
ε = (40,000N * 200mm) / (205 * 10^3 N/mm^2 * 0.0004m^2)
ε = (8,000,000mm N) / (82,000N)
ε ≈ 0.0975
Step 5: Calculate the change in volume (ΔV) using the formula:
ΔV = -2 * (Poisson's ratio) * ε * (Length * Width * Height)
Given Poisson's ratio (ν) = 0.3
Given Length (L) = 200mm
Given Width (W) = 20mm
Given Height (H) = 20mm
ΔV = -2 * 0.3 * 0.0975 * (200mm * 20mm * 20mm)
ΔV = -0.04875 * (800,000mm^3)
ΔV ≈ -38,750mm^3
Therefore, the change in volume of the steel block is approximately -38,750mm^3.
To determine the change in volume of the steel block, we can use the elastic modulus (E), the Poisson's ratio (ν), and the magnitude of the applied tensile force (F). The formula we will use is:
ΔV/V = 3ε
Where ΔV/V represents the change in volume (as a fraction or ratio of the initial volume), and ε represents the strain.
To calculate the strain (ε), we need to find the stress (σ) first. Stress is defined as the force per unit area, and in this case, it can be calculated using the formula:
σ = F / A
Where σ is the stress, F is the force, and A is the cross-sectional area of the block.
Given:
Length (L) = 200 mm
Width (W) = 20 mm
Height (H) = 20 mm
Force (F) = 40 kN = 40,000 N
Elastic Modulus (E) = 205 kN/mm^2 = 205 N/mm^2
Poisson's Ratio (ν) = 0.3
First, let's calculate the cross-sectional area (A) of the block:
A = W * H
A = 20 mm * 20 mm
A = 400 mm^2
Now, let's calculate the stress (σ):
σ = F / A
σ = 40,000 N / 400 mm^2
σ = 100 N/mm^2
Next, let's calculate the strain (ε) using Poisson's ratio (ν):
ε = σ / E * [(1 - ν) / (1 + ν)]
ε = 100 N/mm^2 / 205 N/mm^2 * [(1 - 0.3) / (1 + 0.3)]
ε = 100/205 * [0.7/1.3]
ε = 0.34146
Now, we can find the change in volume (ΔV/V):
ΔV/V = 3ε
ΔV/V = 3 * 0.34146
ΔV/V ≈ 1.02438
Therefore, the change in volume is approximately 1.02438 times the initial volume.
Please note that the units have been cancelled in this calculation, assuming consistent units are used throughout the formula.