Fatigue Failure Criterion

Fluctuating stresses in a component are characterized by the mean stress and the stress amplitude. Presence of mean stress shifts the σ_e − N curve downward. Thus, a machine part under a higher mean stress will have a shorter fatigue life for the desired stress level.

In fatigue diagrams, the mean stress (σ_a) is plotted on the abscissa and stress amplitude (σ_v) is plotted on the ordinate. When σ_v is zero, the load is purely static and the criterion of failure is the ultimate strength (σ_ut) for brittle materials or yield strength (σ_yt) for ductile materials. When mean stress is zero, the stress is completely reversing, and the criterion of failure is the endurance limit (σ_e).

Figure 5.12: Fatigue failure criteria.

When the component is subjected to both σ_a and σ_v, the actual failure occurs at different scattered points. There exists a border dividing the safe region from the unsafe region for various combinations of σ_a and σ_v.

Important Criteria for the Borderline

1. Gerber Parabola

A parabolic curve joining σ_e on the ordinate and σ_ut on the abscissa is called the Gerber parabola of fatigue failure. This criterion fits the failure points of experimental data in the best manner.

Equation: (σ_a / σ_ut)² + (σ_v / σ_e) = 1

With factor of safety N: (N × σ_a / σ_ut)² + (N × σ_v / σ_e) = 1

2. Soderberg Line

A straight line joining σ_e on the ordinate and σ_y on the abscissa is called the Soderberg line of fatigue failure.

Equation: σ_a / σ_y + σ_v / σ_e = 1

With factor of safety N: σ_a / σ_y + σ_v / σ_e = 1 / N

The factor of safety shifts the criterion’s line towards the origin.

3. Goodman Line

A straight line joining σ_e on the ordinate and σ_ut on the abscissa is called the Goodman line of fatigue failure.

Equation: σ_a / σ_ut + σ_v / σ_e = 1

With factor of safety N: σ_a / σ_ut + σ_v / σ_e = 1 / N

The Goodman line is a safer option for design consideration because it lies completely inside the Gerber parabola. The Soderberg line is a more conservative failure criterion.

Figure 5.13: Modified Goodman lines for fluctuating axial/bending stresses (σ) and fluctuating torsional shear stresses (τ).