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.
To draw a stress-strain curve for the loaded cable and calculate the strain energy stored, we need to understand a few concepts and then apply the appropriate formulas.
1. Stress and Strain:
Stress (σ) is defined as the force (F) applied per unit area (A). It is given by the formula:
σ = F / A
Strain (ε) is the ratio of the change in length (∆L) to the original length (L₀). It is given by the formula:
ε = ∆L / L₀
2. Young's Modulus (E):
Young's modulus (E) is a measure of the stiffness of a material. It relates stress to strain and is given by the formula:
E = σ / ε
3. Strain Energy (U):
Strain energy is the energy stored in a material due to deformation. For a linearly elastic material like steel, it can be calculated using the formula:
U = (1/2) * σ * ε * V
Where σ is stress, ε is strain, and V is the volume of the material.
To draw the stress-strain curve, we first need to calculate the stress and strain values for different loading conditions of the cable. Given that the cable is loaded with a weight of 30 kN (30,000 N) and has a cross-sectional area of 1200 mm² (1.2 × 10⁻³ m²), we can calculate the stress:
σ = F / A
= 30,000 N / 1.2 × 10⁻³ m²
= 25 × 10⁶ Pa
Now, for each value of stress, we can calculate the corresponding strain using Young's modulus:
ε = σ / E
= 25 × 10⁶ Pa / 210 × 10⁹ Pa
= 0.119
We can plot the stress-strain curve using these values. The x-axis represents strain, and the y-axis represents stress.
Once the stress-strain curve is drawn, we need to shade in the area that indicates the strain energy stored per unit volume. This area corresponds to the integral of the stress-strain curve.
To calculate the strain energy stored, we need to determine the volume of the cable. Since the cable is assumed to be uniform, the volume can be calculated as follows:
V = A × L
= 1.2 × 10⁻³ m² × 10 m
= 0.012 m³
Substituting the stress and strain values into the strain energy formula, we can calculate the strain energy stored:
U = (1/2) * σ * ε * V
= (1/2) * 25 × 10⁶ Pa * 0.119 * 0.012 m³
= 17,854 Joules (J)
Therefore, the strain energy stored by the cable is 17,854 J.