The effect of mean stress, ${\displaystyle \sigma _{m}}$,can be modeled by the following relations:
• Soderberg: ${\displaystyle \sigma _{a}=\sigma _{a,\sigma _{m}=0}(1-{\frac {\sigma _{m}}{\sigma _{y}}})}$;
• Modified Goodman: ${\displaystyle \sigma _{a}=\sigma _{a,\sigma _{m}=0}(1-{\frac {\sigma _{m}}{\sigma _{u}}})}$;
which is nonconservative when ${\displaystyle \sigma _{m}<0}$. It is however, a good approximation for brittle materials and conservative for ductile alloys, when ${\displaystyle \sigma _{m}\leq 0}$
• Gerber: ${\displaystyle \sigma _{a}=\sigma _{a,\sigma _{m}=0}(1-{\frac {\sigma _{m}}{\sigma _{u}}}^{2})}$;
which is only valid when ${\displaystyle \sigma _{m}\geq 0}$, as a result of the quadratic term. It is a good approximation for ductile alloys