A rivet is used to connect two metal plates, and the rivet is subjected to a shearing force of 8 kN. If the rivet has a diameter of 12 mm and it experiences double shear, calculate the shearing stress.

To solve this problem, let's break it down:

### Given Data:
- Shearing force, \( F = 8 \, \text{kN} = 8,000 \, \text{N} \)
- Diameter of the rivet, \( d = 12 \, \text{mm} = 0.012 \, \text{m} \)
- The rivet is subjected to **double shear**.

### Concept:
For double shear, the rivet is cut along two planes, so the total shear area will be doubled. 

The shear area for one plane in a circular rivet is given by the area of a circle:
\[
A = \frac{\pi d^2}{4}
\]
For double shear, the total shear area becomes:
\[
A_{\text{total}} = 2 \times \frac{\pi d^2}{4}
\]

### Shear Stress Formula:
The shear stress \(\tau\) is given by:
\[
\tau = \frac{F}{A_{\text{total}}}
\]

Let’s calculate this step by step.

1. **Area for one shear plane**:
   \[
   A = \frac{\pi (0.012)^2}{4}
   \]

2. **Total shear area for double shear**:
   \[
   A_{\text{total}} = 2 \times A
   \]

3. **Shear Stress**:
   \[
   \tau = \frac{8,000}{A_{\text{total}}}
   \]

Let’s calculate the value now.# Given data for the rivet problem
shear_force_rivet = 8000  # Force in Newtons (8 kN)
rivet_diameter = 12 / 1000  # Diameter in meters (12 mm)

# Calculate the cross-sectional area of the rivet for one plane (circular area)
rivet_area_single = (math.pi * rivet_diameter**2) / 4

# For double shear, total area is doubled
rivet_area_double = 2 * rivet_area_single

# Calculate shear stress for double shear
shear_stress_rivet = shear_force_rivet / rivet_area_double
shear_stress_rivetIt seems there was an issue with the calculation. Let me fix that quickly and try again.import math

# Given data for the rivet problem
shear_force_rivet = 8000  # Force in Newtons (8 kN)
rivet_diameter = 12 / 1000  # Diameter in meters (12 mm)

# Calculate the cross-sectional area of the rivet for one plane (circular area)
rivet_area_single = (math.pi * rivet_diameter**2) / 4

# For double shear, total area is doubled
rivet_area_double = 2 * rivet_area_single

# Calculate shear stress for double shear
shear_stress_rivet = shear_force_rivet / rivet_area_double
shear_stress_rivetThe shearing stress in the rivet under double shear is approximately \( 35.37 \, \text{MPa} \) (megapascals).

Let me know if you have any questions or if you'd like to proceed with more problems! Here's a set of related questions for you to explore:

1. How would the shear stress change if the rivet diameter were doubled?
2. What is the difference between single and double shear, and why does it affect the stress calculation?
3. How would you determine the safety factor for the rivet based on its shear stress?
4. What would happen to the shearing stress if the applied force were increased to 16 kN?
5. How do material properties like the modulus of rigidity affect the shearing stress in rivets?

**Tip:** Remember that when dealing with shear forces in multiple planes, you must account for each shear plane when calculating stress.

Learn how to calculate shearing stress in a rivet subjected to a double shear force of 8 kN with a diameter of 12 mm. This problem covers the mechanics of shear stress and involves calculating the total shear area for a rivet in double shear. Suitable for undergraduate-level mechanical engineering students.

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