Stress-Life (S-N) Approach (2024)

The safety factor (SF) based on the mean stress correction applied is given by the following equations.

Mean Stress = Constant

1. Goodman or Soderberg

   When SN curve is of the Stress Ratio R = -1

   (20) $$SF=\frac{s}{{\sigma }_a}=\frac{s_e}{{\sigma }_{a_0}}$$

   $$s_e$$ = Target stress amplitude against the target life from the modified SN curve

   $${\sigma }_{a_0}$$ = Stress amplitude after mean stress correction

   
   
   Figure 19.

   When SN curve is of the Stress Ratio R != -1

   
   
   Figure 20.

   $${\sigma }_a$$ = Stress Amplitude

   $${\sigma }_m$$ = Mean Stress

   $$S_{e-R}$$ = Endurance limit obtained from SN curve with R ratio

   $$S_{e-m}$$ = Mean Stress corresponding to $$S_{e-R}$$

   If $${}_{R>-1}$$, $s_e=\frac{S_{e-R}}{1-\frac{s_{m-R}}{UTS}}$

   (21) $s_{m-R}=S_{e-R}.\frac{1+R}{1-R}$

   If $${}_{R<-1}$$, $S_e=S_{e-R}$

   See Also

   Ermüdungsanalyse auf der Grundlage der S-N-Kurve: Lösen grundlegender Probleme | MachineMFGWhat is Fatigue Life – S-N Curve - Woehler Curve - Definition | Material PropertiesWöhlerkurve - Theorie und Anwendung | Ingenieurbüro Andreas Hanke

   If $${\sigma }_m{}_{>0}$$, $s_a=\frac{{\sigma }_a}{1-\frac{{\sigma }_m}{UTS}}$

   If $${\sigma }_m{}_{\le 0}$$, $s_a={\sigma }_a$

   (22) $SF=\frac{S_e}{S_a}$
2. Gerber

   (23) $$SF=\frac{s}{{\sigma }_a}=\frac{s_e}{{\sigma }_{a_0}}$$

   
   
   Figure 21.

   When SN curve is of the Stress Ratio R != -1

   
   
   Figure 22.

   (24) $S_a={\sigma }_a\cdot \left(1-{\left(\frac{{\sigma }_m}{UTS}\right)}^2\right)$ (25) $S_e=S_{e-R}\cdot \left(1-{\left(\frac{s_{m-R}}{UTS}\right)}^2\right)$ (26) $s_{m-R}=S_{e-R}.\frac{1+R}{1-R}$ (27) $SF=\frac{S_e}{S_a}$
3. Gerber2
   1. (28) $$\begin{array}{l}{\sigma }_m>0:\\ SF=\frac{s}{{\sigma }_a}=\frac{s_e}{{\sigma }_{a_0}}\end{array}$$
   2. (29) $$\begin{array}{l}{\sigma }_m\le 0:\\ SF=\frac{s}{{\sigma }_a}\end{array}$$

   When SN curve is of the Stress Ratio R != -1

   If $$R>-1$$

   (30) $S_e=S_{e-R}\cdot \left(1-{\left(\frac{s_{m-R}}{UTS}\right)}^2\right)$ (31) $s_{m-R}=S_{e-R}.\frac{1+R}{1-R}$

   If $$R<-1$$, $S_e=S_{e-R}$

   If $${\sigma }_m{}_{>0}$$, $S_a={\sigma }_a\left(1-{\left(\frac{{\sigma }_m}{UTS}\right)}^2\right)$

   If $${\sigma }_m{}_{\le 0}$$, $s_a={\sigma }_a$

   (32) $SF=\frac{S_e}{S_a}$
4. FKM
   (33) $$SF=\frac{s{\text{'}}_e}{{\sigma }_a}$$

   1. (34) $$\begin{array}{l}{\sigma }_m<\frac{-s_e}{1-m_2}\\ s{\text{'}}_e=-m,\left({\sigma }_m+\frac{s_e}{1-m_2}\right)+\frac{s_e}{1-m_2}\end{array}$$
   2. (35) $$\begin{array}{l}\frac{-s_e}{1-m_2}\le {\sigma }_m<\frac{s_e}{1+m_2}\\ s{\text{'}}_e=-m_2{\sigma }_m+s_e\end{array}$$
   3. (36) $$\begin{array}{l}\frac{s_e}{1+m_2}\le {\sigma }_m<\frac{3(1+m_3)}{1+3m_3}·\frac{s_e}{1+m_2}\\ s{\text{'}}_e=-m_3\left({\sigma }_m-\frac{s_e}{1+m_2}\right)+\frac{s_e}{1+m_2}\end{array}$$
   4. (37) $$\begin{array}{l}\frac{3(1+m_3)}{1+3m_3}·\frac{s_e}{1+m_2}\le {\sigma }_m\\ s{\text{'}}_e=-m_4\left({\sigma }_m-\frac{3(1+m_3)}{1+3m_3}·\frac{s_e}{1+m_2}\right)+\frac{1}{3}\left(\frac{3(1+m_2)}{1+3m_3}·\frac{s_e}{1+m_2}\right)\end{array}$$

   
   
   Figure 23.

5. No Mean Stress Correction

   (38) $$SF=\frac{s_e}{{\sigma }_a}$$

6. Interpolate

Safety Factor with Multi-Mean

To calculate safety factor, HyperLife creates an internal Haigh diagram for the target life using multi-mean SN curve by finding stress amplitude-mean stress pairs at the target life. Using the internally created Haigh diagram, HyperLife calculates safety factor as described in section Safety Factor in Chapter Haigh diagram. The number of data points of the Haigh diagram is the number of curves. Thus the more number of curves, the better result. When Haigh diagram is not available in mean stress ranges, OptiStruct extrapolates the Haigh diagram.

Figure 24. Conversion of Multi-Mean Curve to Haigh Diagram

Safety Factor with Multi-Ratio

To calculate safety factor, HyperLife create an internal Haigh diagram for the target life using multi-mean SN curve by finding stress amplitude-mean stress pairs at the target life. The number of data points of the Haigh diagram is the number of curves. Thus, the more number of curves, the better result. When Haigh diagram is not available in mean stress ranges, HyperLife extrapolates the Haigh diagram.

Figure 25. Conversion of Multi-Mean Curve to Haigh Diagram

Safety Factor with Haigh

Safety factor (SF) is calculated in the following manner in Figure 26.

Figure 26.

If Haigh diagram for a target life is not defined by user, OptiStruct creates Haigh diagram for the target life. In Figure 26, if target life is 10000, and Haigh diagram for N=10000 is not defined, OptiStruct will created dashed curve to calculate Safety factor.