A steel rod with a diameter of 150 mm is inserted in a copper rod with a diameter of 300 mm. The internal diameter of the copper rod is 150 mm. The axial load acting on the rods is... A steel rod with a diameter of 150 mm is inserted in a copper rod with a diameter of 300 mm. The internal diameter of the copper rod is 150 mm. The axial load acting on the rods is 75 kN, and the length of the rod is 1 m. Calculate the stresses in the rod. Take E_steel = 200 GPa, E_copper = 120 GPa. Read more

Steps to Solve

1. Identify the given parameters

We need the following parameters to find the stress in each rod:

• Diameter of steel rod ($D_s$)
• Diameter of copper rod ($D_c$)
• Length of steel rod ($L_s$)
• Length of copper rod ($L_c$)
• Axial load ($P$)
• Elastic modulus of steel ($E_s$)
• Elastic modulus of copper ($E_c$)

2. Calculate the cross-sectional areas

The cross-sectional area for each rod can be calculated using the formula for the area of a circle $A = \pi \left(\frac{D}{2}\right)^2$.

For steel:

$$ A_s = \pi \left(\frac{D_s}{2}\right)^2 $$

For copper:

$$ A_c = \pi \left(\frac{D_c}{2}\right)^2 $$

3. Compute stress

Stress is defined as the force per unit area. The stress in each rod can be calculated using the formula $\sigma = \frac{P}{A}$.

For steel:

$$ \sigma_s = \frac{P}{A_s} $$

For copper:

$$ \sigma_c = \frac{P}{A_c} $$

4. Adjust for different materials

Since the rods are made of different materials, we should also consider their elastic properties. The stresses can be affected by the ratio of their elastic moduli:

$$ \sigma_s = \frac{E_s}{E_c + E_s} \cdot P $$

$$ \sigma_c = \frac{E_c}{E_c + E_s} \cdot P $$

5. Final calculations

Substituting the areas and re-evaluating the stresses based on the provided dimensions and the axial load will give the final longitudinal stresses in each rod.