Mean Stress Matters
Most fatigue data comes from fully reversed loading tests (mean stress = 0), but real components almost always operate with a non-zero mean stress. A bolted joint has tensile preload, a rotating shaft has bending superimposed on axial loads, and a pressure vessel cycles around a positive mean pressure. John Goodman recognized in 1899 that tensile mean stress reduces the allowable alternating stress, and his diagram provides a simple, conservative method to account for this effect.
The Goodman Line
The modified Goodman diagram plots alternating stress (σₐ) on the vertical axis and mean stress (σₘ) on the horizontal axis. A straight line from the endurance limit Se (at σₘ = 0) to the ultimate strength Sᵤ (at σₐ = 0) defines the failure boundary. Any operating point (σₘ, σₐ) inside this line is predicted to have infinite fatigue life; points outside will eventually fail. The safety factor is the ratio of the distance from origin to the Goodman line along the load line.
Comparing Failure Theories
The Goodman line is a conservative approximation. The Gerber parabola, which passes through the same endpoints but bulges outward, typically fits experimental data for ductile metals more closely. The Soderberg line, connecting Se to the yield strength Sy, is more conservative still and additionally prevents yielding. This simulation overlays all three criteria so you can compare their predictions and choose the appropriate one for your design context and required safety margin.
Design Application
In practice, the Goodman diagram is combined with stress concentration factors, surface finish corrections, and reliability factors. The modified endurance limit Se already accounts for these derating factors. Engineers plot the operating point considering the worst-case combination of alternating and mean stress, then verify that the safety factor meets the design requirement — typically n > 1.5 for general machine design and n > 3 for safety-critical applications.