Answer:
Minimum factor of safety for infinite life fatigue: 4.5
Maximum stress in the shaft: 80 MPa
Number of cycles to failure (estimated): less than infinite
Step-by-step explanation:
Solution for infinite life fatigue factor of safety:
Step 1: Calculate the alternating stress
The alternating stress is the half of the stress range, which is the difference between the maximum and minimum stress in the shaft.
σ_a = (σ_max - σ_min) / 2
For a rotating shaft with a force applied perpendicular to the axis, the maximum stress is at the surface of the shaft and is given by:
σ_max = F / A
Where:
F is the applied force
A is the area of the shaft
The minimum stress is zero, since the shaft is rotating and the force is always changing direction.
σ_min = 0
Therefore, the alternating stress is:
σ_a = F / (2A)
Step 2: Determine the yield strength
The yield strength is the maximum stress that a material can withstand before it starts to deform permanently. The yield strength of AISI 1020 CD steel is 345 MPa.
Step 3: Calculate the fatigue strength
The fatigue strength is the maximum alternating stress that a material can withstand for an infinite number of cycles without failing. The fatigue strength of AISI 1020 CD steel is 180 MPa.
Step 4: Calculate the factor of safety
The factor of safety is the ratio of the fatigue strength to the alternating stress. A factor of safety of at least 2 is generally recommended for infinite life fatigue design.
FS = σ_f / σ_a
Where:
σ_f is the fatigue strength
σ_a is the alternating stress
FS = 180 MPa / (F / (2A))
Step 5: Check for yielding
The maximum stress in the shaft must also be less than the yield strength to prevent yielding.
σ_max < σ_y
Where:
σ_y is the yield strength
F / A < 345 MPa
Results:
The minimum factor of safety for infinite life fatigue is:
FS = 180 MPa / (F / (2A)) = 180 MPa / (6 kN / (2 * π * 25 mm^2)) = 4.5
The maximum stress in the shaft is:
σ_max = F / A = 6 kN / (π * 25^2 mm^2) = 80 MPa
Since the maximum stress in the shaft is less than the yield strength, the shaft will not yield.
If the life is not infinite:
The number of cycles to failure for a rotating shaft can be estimated using the following equation:
N = (σ_f / σ_a)^m
Where:
N is the number of cycles to failure
σ_f is the fatigue strength
σ_a is the alternating stress
m is the fatigue exponent, which is typically between 3 and 10 for carbon steel
For AISI 1020 CD steel, a fatigue exponent of 5 is typically used.
N = (180 MPa / (F / (2A)))^5
If the alternating stress is equal to the fatigue strength, the shaft will fail after an infinite number of cycles. Therefore, the number of cycles to failure for the given shaft will be less than infinite.