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