Goodman and Soderberg lines for shafts

In a fatigue diagram the mean stress $\sigma_m$ is plotted on the horizontal axis and the alternating stress $\sigma_a$ on the vertical axis. A line is drawn to show the combinations of stress that the material can bear for infinite life.

 sigma_a
    ^
 Se *
    |\ ' .
    | \     ' .      Goodman line
    |  \        ' .
    |   \ Soderberg   ' .
    |    \  line          ' .
    +-----*-------------------*----> sigma_m
          Sy                  Sut

(Soderberg line: from $S_e$ on the vertical axis to $S_y$ on the horizontal axis. Goodman line: from $S_e$ to $S_{ut}$. The Soderberg line is the lower one.)

Equations

Soderberg line (based on yield strength):

$$\frac{\sigma_m}{S_y} + \frac{\sigma_a}{S_e} = \frac{1}{n}$$

Modified Goodman line (based on ultimate strength):

$$\frac{\sigma_m}{S_{ut}} + \frac{\sigma_a}{S_e} = \frac{1}{n}$$

where $n$ is the factor of safety and $S_e$ is the corrected endurance limit. The stress concentration factor $K_f$ is applied to the alternating component (and, for brittle materials, also to the mean component).

Comparison

For a shaft with bending and torsion, the stresses are combined using the distortion energy theory: the equivalent mean and alternating stresses are found and substituted in the line equation.