Safety Factor

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

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

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

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

   
   
   

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

   
   
   

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

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

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

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

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

   $$SF= \frac{S_e}{S_a}$$

2. Gerber
   $$SF=\frac{s}{{\sigma }_a}=\frac{s_e}{{\sigma }_{a_0}}$$

   
   
   

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

   
   
   

   $$S_a={\sigma }_a\cdot \left(1-{\left(\frac{{\sigma }_m}{UTS}\right)}^2\right)$$

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

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

   $$SF= \frac{S_e}{S_a}$$
3. Gerber2
   1. $$\begin{array}{l}{\sigma }_m>0:\\ SF=\frac{s}{{\sigma }_a}=\frac{s_e}{{\sigma }_{a_0}}\end{array}$$
   2. $$\begin{array}{l}{\sigma }_m\le 0:\\ SF=\frac{s}{{\sigma }_a}\end{array}$$

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

   If $R>-1$

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

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

   $$SF= \frac{S_e}{S_a}$$

4. FKM
   $$SF=\frac{s{\text{'}}_e}{{\sigma }_a}$$

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

   
   
   

5. No Mean Stress Correction
   $$SF=\frac{s_e}{{\sigma }_a}$$

Stress Ratio = Constant

1. Goodman

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

   $$SF=\frac{OB}{OA}=\frac{1}{\left(\frac{{\sigma }_a}{s_e}+\frac{{\sigma }_m}{UTS}\right)}$$

   
   
   

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

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

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

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

   If ${\sigma }_m{}_{>0}$ , $SF=\frac{1}{\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{UTS}}$

   If ${\sigma }_m{}_{\le 0}$ , $SF= \frac{S_e}{{\sigma }_a}$

2. Gerber

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

   1. $$\begin{array}{l}\text{If}{\sigma }_m=0:\\ SF=\frac{s_e}{{\sigma }_a}\end{array}$$
   2. $$\begin{array}{l}\text{If}{\sigma }_m\ne 0:\\ SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma }_m}\right)}^2·\frac{{\sigma }_a}{s_e}\left[-1+\sqrt{1+{\left(\frac{2s_e{\sigma }_m}{UTS\cdot {\sigma }_a}\right)}^2}\right]\end{array}$$

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

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

   If ${\sigma }_m{}_{\ne 0}$ , $SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma }_m}\right)}^2\cdot \frac{{\sigma }_e}{S_e}\cdot \left(-1+\sqrt{1+{\left(\frac{2{\sigma }_m{S}_e}{UTS{\sigma }_a}\right)}^2}\right)$

   If ${\sigma }_m{}_{=0}$ , $SF= \frac{S_e}{{\sigma }_a}$

3. Gerber2
   1. $$\begin{array}{l}\text{If }{\sigma }_m\le 0:\\ SF=\frac{s_e}{{\sigma }_a}\end{array}$$
   2. $$\begin{array}{l}\text{If}{\sigma }_m\ge 0:\\ SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma }_m}\right)}^2·\frac{{\sigma }_a}{s_e}\left[-1+\sqrt{1+{\left(\frac{2s_e{\sigma }_m}{UTS\cdot {\sigma }_a}\right)}^2}\right]\end{array}$$

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

   If $R>-1$

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

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

   If ${\sigma }_m{}_{>0}$ , $SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma }_m}\right)}^2\cdot \frac{{\sigma }_a}{S_e}\cdot \left(-1+\sqrt{1+{\left(\frac{2{\sigma }_m{S}_e}{UTS{\sigma }_a}\right)}^2}\right)$

   If ${\sigma }_m{}_{\le 0}$ , $SF= \frac{S_e}{{\sigma }_a}$

4. FKM
   $$SF=\frac{s_e}{{\sigma }_{a_0}}$$

   ${\sigma }_{a_0}$ = Corrected Stress Amplitude in Constant R mean stress correction

5. No Mean Stress Correction
   $$SF=\frac{s_e}{s_a}$$

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, HyperLife extrapolates the 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.

   
   
   

   Safety Factor with Haigh

   Safety factor (SF) is calculated in the following manner in Figure 9.

   
   
   

   When target life is 100000:

   • Constant R : SF = OB/OA
   • Constant mean : SF = OD/OC

   If Haigh diagram for a target life is not defined, HyperLife creates Haigh diagram for the target life. In Figure 9, if target life is 10000, and Haigh diagram for N=10000 is not defined, HyperLife will created dashed curve to calculate Safety factor.