A steel rod with a 150 mm diameter is inserted into a copper rod of 300 mm diameter. The internal diameter of the copper rod is 150 mm. The axial load acting on the rod is 75 kN an... A steel rod with a 150 mm diameter is inserted into a copper rod of 300 mm diameter. The internal diameter of the copper rod is 150 mm. The axial load acting on the rod 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.
Understand the Problem
The question asks for the calculation of stresses in a steel rod inserted in a copper rod under a given axial load. It provides specific dimensions and material properties, which will be used to apply stress formulas to determine the resulting stresses in both materials.
Answer
Stress in Steel: $$ \sigma_{steel} $$, Stress in Copper: $$ \sigma_{copper} $$
Answer for screen readers
- Stress in Steel: $$ \sigma_{steel} $$
- Stress in Copper: $$ \sigma_{copper} $$
Steps to Solve
- Identify Given Data
Start by determining the properties of the materials and the dimensions provided.
- Young's modulus for steel ($E_{steel}$) = 200 GPa
- Young's modulus for copper ($E_{copper}$) = 110 GPa
- Axial load ($P$) applied = Given value
- Lengths and cross-sectional areas ($A_{steel}$, $A_{copper}$) of both rods
- Calculate Stress in Each Material
Stress is defined as force per unit area. Use the formula:
$$ \sigma = \frac{P}{A} $$
For both materials:
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For steel: $$ \sigma_{steel} = \frac{P}{A_{steel}} $$
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For copper: $$ \sigma_{copper} = \frac{P}{A_{copper}} $$
- Calculate Strain Using Young's Modulus
Using Young's modulus ($E$), we can calculate the strain ($\varepsilon$) in each material with the formula:
$$ E = \frac{\sigma}{\varepsilon} $$
Rearranging gives us:
$$ \varepsilon = \frac{\sigma}{E} $$
For both materials:
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For steel: $$ \varepsilon_{steel} = \frac{\sigma_{steel}}{E_{steel}} $$
-
For copper: $$ \varepsilon_{copper} = \frac{\sigma_{copper}}{E_{copper}} $$
- Check Compatibility of Strains
Since both materials are connected, the total elongation must be the same. Set the elongations equal to each other based on the individual strains and lengths:
$$ \varepsilon_{steel} L_{steel} + \varepsilon_{copper} L_{copper} = 0 $$
- Solve the System of Equations
Use the two strain equations to create a system of equations. Substitute the values from the stress calculations and solve for the unknown stresses or elongations.
- Conclude with Final Stress Values
Substitute back to find the final stress values in both materials, ensuring to keep track of signs based on tension or compression.
- Stress in Steel: $$ \sigma_{steel} $$
- Stress in Copper: $$ \sigma_{copper} $$
More Information
The results will reflect the stresses developed in both the steel and copper rods due to the axial load. Steel usually has a higher stress due to its greater modulus, resulting in smaller elongation compared to copper.
Tips
- Not using the correct units when calculating stress (make sure to convert GPa to N/m² if necessary).
- Forgetting the relation between strain components in connected materials, leading to incorrect assumptions about the system.
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