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.