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To solve this problem, we need to determine the bearing stress between the collar and the plate, given the conditions provided in the diagram.

### **Step 1: Understanding the Problem**
- **Force Applied**: \( P = 50 \, \text{kN} \)
- **Collar Diameter**: \( D_{\text{collar}} = 130 \, \text{mm} \)
- **Shaft Diameter**: \( D_{\text{shaft}} = 90 \, \text{mm} \)

### **Step 2: Area Calculation**
Bearing stress (\(\sigma_b\)) is calculated using the formula:

\[
\sigma_b = \frac{P}{A_b}
\]

where \( A_b \) is the bearing area between the collar and the plate.

The bearing area \( A_b \) can be calculated as the area of the collar's cross-section minus the area of the shaft's cross-section:

\[
A_b = \frac{\pi}{4} \left( D_{\text{collar}}^2 - D_{\text{shaft}}^2 \right)
\]

Substituting the given diameters:

\[
A_b = \frac{\pi}{4} \left( (130 \, \text{mm})^2 - (90 \, \text{mm})^2 \right)
\]

### **Step 3: Calculate the Bearing Area**

\[
A_b = \frac{\pi}{4} \times (16900 \, \text{mm}^2 - 8100 \, \text{mm}^2)
\]

\[
A_b = \frac{\pi}{4} \times 8800 \, \text{mm}^2
\]

\[
A_b = \frac{3.1416}{4} \times 8800 \, \text{mm}^2
\]

\[
A_b \approx 6911.5 \, \text{mm}^2
\]

### **Step 4: Calculate the Bearing Stress**

Now, using the bearing stress formula:

\[
\sigma_b = \frac{50 \times 10^3 \, \text{N}}{6911.5 \, \text{mm}^2}
\]

\[
\sigma_b \approx 7.23 \, \text{MPa}
\]

### **Answer**:
The closest option to our calculated result is **7.23 MPa**, which would be the correct answer.

### Would you like more details or have any questions?

### Relative Questions:
1. How would the bearing stress change if the force was doubled?
2. What if the collar diameter was increased to 150 mm?
3. How does the difference between the collar and shaft diameters affect the bearing stress?
4. What would happen to the stress if the material of the shaft was different?
5. How can we reduce the bearing stress in this situation?

### Tip:
Always ensure to double-check unit conversions, especially when dealing with areas and forces, as incorrect units can lead to significant errors in stress calculations.

What is the bearing stress between the collar and the plate, assuming there is no clearance between the shaft and the plate hole and a 50 kN force is applied?

Learn how to calculate the bearing stress between a collar and a plate under a 50 kN force. This solution covers key concepts in mechanical engineering stress analysis, including the use of the bearing stress formula and area calculation for circular sections.

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