Safety Factor

Safety factor is calculated based on the endurance limit or target stress (at target life) against the stress amplitude from the working stress history.

HyperLife calculates this ratio via two criteria:

Mean Stress = Constant
Stress Ratio = Constant

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

Mean Stress = Constant

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

Figure 1.

$S F = \frac{s}{σ_{a}} = \frac{s_{e}}{σ_{a_{0}}}$

$s_{e}$ = Target stress amplitude against the target life from the modified SN curve
$σ_{a_{0}}$ = Stress amplitude after mean stress correction

Figure 2.

When SN curve is of the Stress Ratio R != -1
Figure 3.

$σ_{a}$ = Stress Amplitude
$σ_{m}$ = 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 $_{R > - 1}$ , $s_{e} = \frac{S_{e - R}}{1 - \frac{s_{m - R}}{U T S}}$
Figure 4.

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

If $_{R < - 1}$ , $S_{e} = S_{e - R}$
If $σ_{m}_{> 0}$ , $s_{a} = \frac{σ_{a}}{1 - \frac{σ_{m}}{U T S}}$
If $σ_{m}_{\leq 0}$ , $s_{a} = σ_{a}$
Figure 5.

$S F = \frac{S_{e}}{S_{a}}$
Gerber

Figure 6.

$S F = \frac{s}{σ_{a}} = \frac{s_{e}}{σ_{a_{0}}}$

Figure 7.

When SN curve is of the Stress Ratio R != -1
Figure 8.

Figure 9.

$S_{a} = σ_{a} \cdot (1 - {(\frac{σ_{m}}{U T S})}^{2})$

Figure 10.

$S_{e} = S_{e - R} \cdot (1 - {(\frac{s_{m - R}}{U T S})}^{2})$

Figure 11.

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

Figure 12.

$S F = \frac{S_{e}}{S_{a}}$
Gerber2
1. Figure 13.
  
  $\begin{array}{l} σ_{m} > 0 : \\ S F = \frac{s}{σ_{a}} = \frac{s_{e}}{σ_{a_{0}}} \end{array}$
2. Figure 14.
  
  $\begin{array}{l} σ_{m} \leq 0 : \\ S F = \frac{s}{σ_{a}} \end{array}$
When SN curve is of the Stress Ratio R != -1
If $R > - 1$
Figure 15.

$S_{e} = S_{e - R} \cdot (1 - {(\frac{s_{m - R}}{U T S})}^{2})$

Figure 16.

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

If $R < - 1$ , $S_{e} = S_{e - R}$
If $σ_{m}_{> 0}$ , $S_{a} = σ_{a} (1 - {(\frac{σ_{m}}{U T S})}^{2})$
If $σ_{m}_{\leq 0}$ , $s_{a} = σ_{a}$
Figure 17.

$S F = \frac{S_{e}}{S_{a}}$
FKM
Figure 18.

$S F = \frac{s'_{e}}{σ_{a}}$
1. Figure 19.
  
  $\begin{array}{l} σ_{m} < \frac{- s_{e}}{1 - m_{2}} \\ s'_{e} = - m, (σ_{m} + \frac{s_{e}}{1 - m_{2}}) + \frac{s_{e}}{1 - m_{2}} \end{array}$
2. Figure 20.
  
  $\begin{array}{l} \frac{- s_{e}}{1 - m_{2}} \leq σ_{m} < \frac{s_{e}}{1 + m_{2}} \\ s'_{e} = - m_{2} σ_{m} + s_{e} \end{array}$
3. Figure 21.
  
  $\begin{array}{l} \frac{s_{e}}{1 + m_{2}} \leq σ_{m} < \frac{3 (1 + m_{3})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}} \\ s'_{e} = - m_{3} (σ_{m} - \frac{s_{e}}{1 + m_{2}}) + \frac{s_{e}}{1 + m_{2}} \end{array}$
4. Figure 22.
  
  $\begin{array}{l} \frac{3 (1 + m_{3})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}} \leq σ_{m} \\ s'_{e} = - m_{4} (σ_{m} - \frac{3 (1 + m_{3})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}}) + \frac{1}{3} (\frac{3 (1 + m_{2})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}}) \end{array}$
Figure 23.
No Mean Stress Correction

Figure 24.

$S F = \frac{s_{e}}{σ_{a}}$

Stress Ratio = Constant

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

Figure 25.

$S F = \frac{O B}{O A} = \frac{1}{(\frac{σ_{a}}{s_{e}} + \frac{σ_{m}}{U T S})}$

Figure 26.

When SN curve is of the Stress Ratio R != -1
If $R > - 1$ , $s_{e} = \frac{S_{e - R}}{1 - \frac{s_{m - R}}{U T S}}$
Figure 27.

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

If $R < - 1$ , $s_{e} = S_{e - R}$
If $σ_{m}_{> 0}$ , $S F = \frac{1}{\frac{σ_{a}}{S_{e}} + \frac{σ_{m}}{U T S}}$
If $σ_{m}_{\leq 0}$ , $S F = \frac{S_{e}}{σ_{a}}$
Gerber
When SN curve is of the Stress Ratio R = -1
1. Figure 28.
  
  $\begin{array}{l} If σ_{m} = 0 : \\ S F = \frac{s_{e}}{σ_{a}} \end{array}$
2. Figure 29.
  
  $\begin{array}{l} If σ_{m} \neq 0 : \\ S F = \frac{1}{2} {(\frac{U T S}{σ_{m}})}^{2} \cdot \frac{σ_{a}}{s_{e}} [- 1 + \sqrt{1 + {(\frac{2 s_{e} σ_{m}}{U T S \cdot σ_{a}})}^{2}}] \end{array}$
When SN curve is of the Stress Ratio R != -1
Figure 30.

$S_{e} = S_{e - R} (1 - {(\frac{S_{m - R}}{U T S})}^{2})$

Figure 31.

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

If $σ_{m}_{\neq 0}$ , $S F = \frac{1}{2} {(\frac{U T S}{σ_{m}})}^{2} \cdot \frac{σ_{e}}{S_{e}} \cdot (- 1 + \sqrt{1 + {(\frac{2 σ_{m} S_{e}}{U T S σ_{a}})}^{2}})$
If $σ_{m}_{= 0}$ , $S F = \frac{S_{e}}{σ_{a}}$
Gerber2
1. Figure 32.
  
  $\begin{array}{l} If σ_{m} \leq 0 : \\ S F = \frac{s_{e}}{σ_{a}} \end{array}$
2. Figure 33.
  
  $\begin{array}{l} If σ_{m} \geq 0 : \\ S F = \frac{1}{2} {(\frac{U T S}{σ_{m}})}^{2} \cdot \frac{σ_{a}}{s_{e}} [- 1 + \sqrt{1 + {(\frac{2 s_{e} σ_{m}}{U T S \cdot σ_{a}})}^{2}}] \end{array}$
When SN curve is of the Stress Ratio R != -1
If $R > - 1$
Figure 34.

$S_{e} = S_{e - R} \cdot (1 - {(\frac{s_{m - R}}{U T S})}^{2})$

Figure 35.

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

If $R < - 1$ , $S_{e} = S_{e - R}$
If $σ_{m}_{> 0}$ , $S F = \frac{1}{2} {(\frac{U T S}{σ_{m}})}^{2} \cdot \frac{σ_{a}}{S_{e}} \cdot (- 1 + \sqrt{1 + {(\frac{2 σ_{m} S_{e}}{U T S σ_{a}})}^{2}})$
If $σ_{m}_{\leq 0}$ , $S F = \frac{S_{e}}{σ_{a}}$
FKM

Figure 36.

$S F = \frac{s_{e}}{σ_{a_{0}}}$

$σ_{a_{0}}$ = Corrected Stress Amplitude in Constant R mean stress correction
No Mean Stress Correction

Figure 37.

$S F = \frac{s_{e}}{s_{a}}$
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.
Figure 38. 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 39. Conversion of Multi-Mean Curve to Haigh Diagram

Safety Factor with Haigh
Safety factor (SF) is calculated in the following manner in Figure 9.
Figure 40.

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.