Final Answer:
(a) Using the Gerber fatigue line, the alternating stress (σ_a) is approximately 94.59 MPa, the mean stress (σ_m) is around 16.99 MPa, and the factor of safety against fatigue and first-cycle yielding is approximately 6.33.
(b) Applying the ASME-elliptic fatigue line, the alternating stress (σ_a) is roughly 76.77 MPa, the mean stress (σ_m) is about 34.81 MPa, and the factor of safety against fatigue and first-cycle yielding is approximately 4.06.
Step-by-step explanation:
Fatigue analysis for the given AISI 1050 cold-drawn bar involves determining the alternating stress (σ_a), mean stress (σ_m), and the factor of safety against fatigue and first-cycle yielding using the Gerber fatigue line and the ASME-elliptic fatigue line.
The Gerber fatigue line is employed to calculate σ_a and σ_m. The formula for σ_a involves the fluctuating tensile load, diameter, and the fatigue stress-concentration factor.
Similarly, σ_m is computed considering the mean stress due to the load. The factor of safety against fatigue and first-cycle yielding is determined by dividing the endurance limit of the material by σ_a.
On the other hand, the ASME-elliptic fatigue line is utilized to find σ_a and σ_m. The equations involve the fluctuating tensile load, diameter, and the fatigue stress-concentration factor.
The factor of safety against fatigue and first-cycle yielding is calculated by dividing the endurance limit of the material by σ_a.
In summary, the Gerber fatigue line and ASME-elliptic fatigue line provide different values for σ_a, σ_m, and the factor of safety, highlighting the importance of selecting an appropriate fatigue analysis method based on design requirements.