A steel cable of length 10 m and cross-sectional area 1200 mm2 is loaded with an object of weight 30 kN. Given that the cable is not loaded beyond its proportional limit and Young's modulus of steel is 210 GPa, draw a stress-strain curve for the loaded cable and shade in the area which indicates the strain energy stored per unit volume, and calculate the strain energy stored by the cable.
σ=Eε,
W/A= Eε,
ε=W/AE,
Energy=E ε²/2 = E W²/2A²E²=
= W²/2A²E=
=(30000)²/2•(1200•10⁻⁶)²•210•10⁹=
=1.49•10³ J.
To draw the stress-strain curve for the loaded cable, we need to calculate the stress and strain first.
1. Calculate the Stress:
Stress (σ) is defined as the force per unit area. In this case, the force is the weight of the object, and the area is the cross-sectional area of the cable.
Stress (σ) = Force / Area
= 30 kN / (1200 mm^2) [Converting mm^2 to m^2 by dividing by 1000^2]
= (30 × 10^3 N) / (1200 × 10^(-6) m^2) [Converting kN to N]
Stress (σ) = 25 × 10^6 N/m^2
2. Calculate the Strain:
Strain (ε) is defined as the change in length per unit original length. In this case, the change in length is the elongation of the cable, and the original length is 10 m.
Strain (ε) = Elongation / Original Length
= Change in Length / Original Length
To find the change in length, we can use Hooke's Law, which states that the stress is directly proportional to the strain as long as the material is not loaded beyond its proportional limit.
Hooke's Law: Stress (σ) = Young's Modulus (E) × Strain (ε)
Substituting the values, the equation becomes:
25 × 10^6 N/m^2 = (210 × 10^9 N/m^2) × Strain (ε)
Solving for Strain (ε):
Strain (ε) = (25 × 10^6 N/m^2) / (210 × 10^9 N/m^2)
= 1.19 × 10^(-4)
Now we have the stress and strain values. We can plot these points on a graph to draw the stress-strain curve.
Next, we need to calculate the strain energy stored per unit volume and shade the area on the graph representing it.
3. Calculate the Strain Energy:
Strain energy (U) is defined as the work done on a material to deform it. In this case, the work done is equal to the area under the stress-strain curve.
The strain energy per unit volume (u) can be calculated by dividing the strain energy by the volume of the cable.
Strain energy per unit volume (u) = Strain energy (U) / Volume
To find the volume, we can multiply the cross-sectional area of the cable by its length.
Volume = Cross-sectional Area × Length
= (1200 mm^2) × (10 m) [Converting mm^2 to m^2 by dividing by 1000^2]
Volume = 0.12 m^2
Now, divide the strain energy by the volume to get the strain energy per unit volume.
Strain energy per unit volume (u) = Strain energy (U) / Volume
Finally, calculate the strain energy stored by the cable by multiplying the strain energy per unit volume by the volume of the cable.
Strain Energy (U) = Strain energy per unit volume (u) × Volume
Shading the area on the graph representing strain energy stored per unit volume is not possible without actual numerical values. However, the strain energy stored per unit volume represents the area under the stress-strain curve up to the given strain value (ε).
To draw the stress-strain curve for the loaded cable and calculate the strain energy stored by the cable, we need to follow a few steps:
Step 1: Calculate the stress
Stress (σ) is defined as the force (F) applied per unit area (A).
Given that the weight of the object is 30 kN and the cross-sectional area of the cable is 1200 mm², we need to convert the units:
30 kN = 30,000 N
1 mm² = 1 x 10⁻⁶ m²
Stress (σ) = Force (F) / Area (A)
σ = 30,000 N / (1200 mm² x 1 x 10⁻⁶ m²/mm²)
σ = 30,000 N / (1.2 x 10⁻³ m²)
σ = 25 x 10⁶ Pa
Step 2: Calculate the strain
Strain (ε) is defined as the change in length (ΔL) divided by the original length (L).
Given that the original length of the cable is 10 m, we need to calculate the change in length.
Strain (ε) = Change in Length (ΔL) / Original Length (L)
From Hooke's law, we know that stress is proportional to strain within the proportional limit. Hence, we can use a known formula:
σ = E x ε
where E is Young's modulus.
Rearranging the formula, we get:
ε = σ / E
ε = (25 x 10⁶) / (210 x 10⁹)
ε = 0.119 x 10⁻³
Step 3: Draw the stress-strain curve
To draw the graph, the stress (σ) will be on the y-axis, and the strain (ε) will be on the x-axis. Start from the origin (0, 0), and plot the point (ε, σ). Then connect the points to form the curve.
Step 4: Shade the area indicating strain energy stored per unit volume
To shade the area indicating strain energy stored per unit volume, we need to calculate the area under the stress-strain curve up to the point of interest.
The formula for strain energy per unit volume (U) is given by:
U = (1/2) x σ x ε
Calculate U using the values we've obtained:
U = (1/2) x (25 x 10⁶ Pa) x (0.119 x 10⁻³)
U = 1.4875 J/m³
This is the strain energy stored per unit volume.
Step 5: Calculate the strain energy stored by the cable
To calculate the total strain energy stored by the cable, we need to multiply the strain energy per unit volume (U) by the volume (V) of the cable.
Given that the cable's length is 10 m and the cross-sectional area is 1200 mm², we can calculate the volume:
V = Length (L) x Area (A)
V = 10 m x 1200 mm² x 1 x 10⁻⁶ m²/mm²
V = 12 x 10⁻³ m³
Now we can calculate the total strain energy stored:
Total Strain Energy (E) = U x V
E = 1.4875 J/m³ x 12 x 10⁻³ m³
E = 0.01785 J
Therefore, the strain energy stored by the cable is 0.01785 Joules.