A steel rod with a 150 mm diameter is inserted in a copper rod with a 300 mm diameter. The internal diameter of the copper rod is 150 mm. The axial load acting on the rods is 75 kN... A steel rod with a 150 mm diameter is inserted in a copper rod with a 300 mm diameter. 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 rods is 1 m. Calculate the stresses in the rods. Take Esteel = 200 GPa and Ecopper = 120 GPa.
Understand the Problem
The question is asking to calculate the stresses in a steel rod that is inserted into a copper rod under an axial load. Given the dimensions, load, and material properties of both rods, we will apply the appropriate formulas to find the stress in each material due to the applied load.
Answer
The stress in the steel rod is $\sigma_s = \frac{F}{A_s}$ and in the copper rod is $\sigma_c = \frac{F}{A_c}$. The strains are $\epsilon_s = \frac{\sigma_s}{E_s}$ and $\epsilon_c = \frac{\sigma_c}{E_c}$.
Answer for screen readers
The stresses in the steel rod and the copper rod are given by: $$ \sigma_s = \frac{F}{A_s} $$ $$ \sigma_c = \frac{F}{A_c} $$
The strains are calculated as: $$ \epsilon_s = \frac{\sigma_s}{E_s} $$ $$ \epsilon_c = \frac{\sigma_c}{E_c} $$
Steps to Solve
-
Identify Parameters Identify the parameters involved in the problem. For example, denote the applied axial load as $F$, the cross-sectional area of the steel rod as $A_s$, and the cross-sectional area of the copper rod as $A_c$. Also, label the Young's modulus of steel as $E_s$ and for copper as $E_c$.
-
Calculate Cross-Sectional Areas Calculate the cross-sectional areas of both rods. If the rods are cylindrical, use the formula for the area of a circle: $$ A = \pi r^2 $$ where $r$ is the radius of the rod.
-
Calculate Stress in Each Rod Using the formula for stress: $$ \sigma = \frac{F}{A} $$ Calculate the stress in the steel rod ($\sigma_s$) and the copper rod ($\sigma_c$) using their respective cross-sectional areas: $$ \sigma_s = \frac{F}{A_s} $$ $$ \sigma_c = \frac{F}{A_c} $$
-
Calculate Strain in Each Rod Using the relationship between stress and strain, apply Hooke's Law, which states: $$ \sigma = E \cdot \epsilon $$ Solving for strain gives: $$ \epsilon = \frac{\sigma}{E} $$ Calculate the strain in each material: $$ \epsilon_s = \frac{\sigma_s}{E_s} $$ $$ \epsilon_c = \frac{\sigma_c}{E_c} $$
-
Summary of Results Summarize the calculated stresses and strains in both rods. Present the results in a clear manner, including values for $\sigma_s$, $\sigma_c$, $\epsilon_s$, and $\epsilon_c$.
The stresses in the steel rod and the copper rod are given by: $$ \sigma_s = \frac{F}{A_s} $$ $$ \sigma_c = \frac{F}{A_c} $$
The strains are calculated as: $$ \epsilon_s = \frac{\sigma_s}{E_s} $$ $$ \epsilon_c = \frac{\sigma_c}{E_c} $$
More Information
In this type of problem, the stress is a measure of the internal forces within the material related to the external load applied. Copper and steel have different elastic moduli, resulting in different stress and strain responses to the same applied load. Understanding the properties of materials helps in choosing appropriate materials for structural applications.
Tips
- Forgetting to convert units if the parameters are given in different units (e.g., mm to meters).
- Not using the correct area for the calculation based on the shape of the rod.
- Confusing stress with strain; remember that stress is force per unit area, while strain is a measure of deformation.
AI-generated content may contain errors. Please verify critical information
