Two metal beams are joined together by four rivets, as the drawing indicates. Each rivet has a radius of 5.0 10-3 m and is to be exposed to a shearing stress of no more than 4.6 108 Pa. What is the maximum tension T that can be applied to each beam, assuming that each rivet carries one-fourth of the total load?
4* (single rivet area)*(maximum shear stress)
= maximum tension
how do you get the single rivet area?
To find the maximum tension T that can be applied to each beam, we can start by considering the shearing stress on each rivet.
1. First, let's consider the total load on each rivet. Since there are four rivets and each carries one-fourth of the total load, the load on each rivet can be calculated as follows:
Load on each rivet = Total load / Number of rivets
= T / 4
2. Now, let's calculate the cross-sectional area of the rivet. The cross-sectional area of a cylinder (rivet) can be found using the formula:
Area = π * (radius)^2
Area = π * (5.0 * 10^(-3))^2
Area = π * (25 * 10^(-6))
Area = 7.85 * 10^(-5) m^2
3. The shearing stress on each rivet can be calculated using the formula:
Shearing stress = Load / Area
4.6 * 10^8 Pa = (T/4) / (7.85 * 10^(-5) m^2)
4. Rearranging the equation to solve for T, we get:
T/4 = (4.6 * 10^8 Pa) * (7.85 * 10^(-5) m^2)
T/4 = 3.61 * 10^4 N
5. Multiplying both sides of the equation by 4, we find:
T = 4 * (3.61 * 10^4 N)
T = 1.44 * 10^5 N
Therefore, the maximum tension T that can be applied to each beam is 1.44 * 10^5 N.
To determine the maximum tension that can be applied to each beam, we need to consider the shearing stress acting on each rivet.
Given:
- Radius of each rivet (r) = 5.0 × 10^(-3) m
- Maximum shearing stress allowable (σ_max) = 4.6 × 10^8 Pa
First, let's find the area of one rivet:
Area of a circular rivet = π × (radius of the rivet)^2
Area of one rivet = π × (5.0 × 10^(-3))^2
Area of one rivet = 3.14 × (5.0 × 10^(-3))^2
Area of one rivet ≈ 7.85 × 10^(-5) m^2
Since there are 4 rivets, the total area resisting the shearing force is:
Total area = 4 × Area of one rivet
Total area = 4 × 7.85 × 10^(-5) m^2
Total area ≈ 3.14 × 10^(-4) m^2
Next, we can calculate the maximum force (F_max) that each rivet can withstand:
F_max = σ_max × Total area
F_max = 4.6 × 10^8 Pa × 3.14 × 10^(-4) m^2
F_max = 1.4444 × 10^5 N
Since each rivet carries one-fourth of the total load, the maximum tension (T) that can be applied to each beam is:
T = F_max / 4
T = (1.4444 × 10^5 N) / 4
T ≈ 3.61 × 10^4 N
Therefore, the maximum tension T that can be applied to each beam is approximately 3.61 × 10^4 N.