Advanced Structural Analysis/Part I - Theory/Failure Modes/Fatigue/Crack Initiation/Loading/Influence of Mean Stress/Algebraic Models

The effect of mean stress, $$\sigma_m$$,can be modeled by the following relations:


 * Soderberg:   $$ \sigma_a = \sigma_{a, \sigma_m = 0} (1 - \frac{\sigma_m} {\sigma_y}) $$; which is conservative for most engineering alloys
 * Modified Goodman:    $$ \sigma_a = \sigma_{a, \sigma_m = 0} (1 - \frac{\sigma_m}{\sigma_u}) $$; which is nonconservative when $$ \sigma_m < 0 $$.  It is however, a good approximation for brittle materials and conservative for ductile alloys, when $$ \sigma_m \leq 0 $$
 * Gerber:    $$ \sigma_a = \sigma_{a, \sigma_m = 0} (1 - {\frac{\sigma_m}{\sigma_u}}^2) $$; which is only valid when $$\sigma_m \geq 0 $$, as a result of the quadratic term.  It is a good approximation for ductile alloys