StressLife (SN) Approach
SN Curve
The SN curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure.
for segment 1
Where, $S$ is the nominal stress range, ${N}_{f}$ are the fatigue cycles to failure, ${b}_{l}$ is the first fatigue strength exponent, and $SI$ is the fatigue strength coefficient.
The SN approach is based on elastic cyclic loading, inferring that the SN curve should be confined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant plasticity is occurring. This is commonly referred to as highcycle fatigue.
SN curve data is provided for a given material using the Materials module.
Multiple SN Curves
 Multimean SN curves: group of SN curves defined at different mean stress.
 Multiratio SN curves: group of SN curves defined at different stress ratio R.
 MultiHaigh Diagram: group of Haigh curves defined at different Number of Cycles.
Rainflow Cycle Counting
Cycle counting is used to extract discrete simple "equivalent" constant amplitude cycles from a random loading sequence.
One way to understand "cycle counting" is as a changing stressstrain versus time signal. Cycle counting will count the number of stressstrain hysteresis loops and keep track of their range/mean or maximum/minimum values.
Rainflow cycle counting is the most widely used cycle counting method. It requires that the stress time history be rearranged so that it contains only the peaks and valleys and it starts either with the highest peak or the lowest valley (whichever is greater in absolute magnitude). Then, three consecutive stress points (1, 2, and 3) will define two consecutive ranges as $\text{\Delta}{S}_{12}=\left{S}_{1}{S}_{2}\right$ and $\text{\Delta}{S}_{23}=\left{S}_{2}{S}_{3}\right$ . A cycle from 1 to 2 is only extracted if $\text{\Delta}{S}_{12}\le \text{\Delta}{S}_{23}$ . Once a cycle is extracted, the two points forming the cycle are discarded and the remaining points are connected to each other. This procedure is repeated until the remaining data points are exhausted.
Simple Load History
Next, pick the first three stress values (1, 2, and 3) and determine if a cycle is present.
In this case, $\text{\Delta}{S}_{14}=\text{\Delta}{S}_{45}$ , so another cycle is extracted from point 1 to 4. After these two points are also discarded, only point 5 remains; therefore, the rainflow counting process is completed.
Two cycles (2→3 and 1→4) have been extracted from this load history. One of the main reasons for choosing the highest peak/valley and rearranging the load history is to guarantee that the largest cycle is always extracted (in this case, it is 1→4). If you observe the load history prior to rearrangement, and conduct the same rainflow counting process on it, then clearly, the 1→4 cycle is not extracted.
Complex Load History
The load history is rearranged such that all points including and after the highest load are moved to the beginning of the load history and are removed from the end of the load history.
 Complex Load History
Equivalent Nominal Stress
Since SN theory deals with uniaxial stress, the stress components need to be resolved into one combined value for each calculation point, at each time step, and then used as equivalent nominal stress applied on the SN curve.
Various stress combination types are available with the default being "Absolute maximum principle stress". "Absolute maximum principle stress" is recommended for brittle materials, while "Signed von Mises stress" is recommended for ductile material. The sign on the signed parameters is taken from the sign of the Maximum Absolute Principal value.
"Critical plane stress" is also available as a stress combination for uniaxial calculations (stress life and strain life).
For example, if number of planes requested is 20, then stress is calculated every 10 degrees.
By default, HyperLife also calculates at $\theta $ = 45 and 135degree planes in addition to the requested number of planes. This is to include the worst possible damage if occurring on these planes.
Mean Stress Correction
Generally, SN curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fullyreversed, and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to take into account the effect of nonzero mean stresses.
The Gerber parabola and the Goodman line in Haigh's coordinates are widely used when considering mean stress influence, and can be expressed as:
Gerber
Goodman
When SN curve is of the Stress Ratio R = 1
Gerber2
Improves the Gerber method by ignoring the effect of negative mean stress.
When SN curve is of the Stress Ratio R != 1
If ${S}_{m}>0$ , same as Gerber
If ${S}_{m}\le 0$ , ${S}_{aR}={S}_{a}$
Soderberg
Is slightly different from GOODMAN; the mean stress is normalized by yield stress instead of ultimate tensile stress.
 ${S}_{e}$
 Equivalent stress amplitude
 ${S}_{a}$
 Stress amplitude
 ${S}_{m}$
 Mean stress
 ${S}_{y}$
 Yield stress
FKM
There are two available options for FKM correction in HyperLife. They are activated by setting FKM MSS to 1 slope/4 slopes in the Assign Material dialog.
 Regime 1 (R > 1.0)
 ${S}_{e}^{A}={S}_{a}\left(1M\right)$
 Regime 2 (∞ ≤ R ≤ 0.0)
 ${S}_{e}^{A}={S}_{a}+M*{S}_{m}$
 Regime 3 (0.0 < R < 0.5)
 ${S}_{e}^{A}=\left(1+M\right)\frac{{S}_{a}+\left(\raisebox{1ex}{$M$}\!\left/ \!\raisebox{1ex}{$3$}\right.\right){S}_{m}}{1+\raisebox{1ex}{$M$}\!\left/ \!\raisebox{1ex}{$3$}\right.}$
 Regime 4 (R ≥ 0.5)
 ${S}_{e}^{A}=\frac{3{S}_{a}{\left(1+M\right)}^{2}}{3+M}$
 ${S}_{e}^{A}$
 Stress amplitude after mean stress correction (Endurance stress)
 ${S}_{m}$
 Mean stress
 ${S}_{a}$
 Stress amplitude
 $M$
 Slope entered for region 2
If all four slopes are specified for mean stress correction, the corresponding Mean Stress Sensitivity values are slopes for controlling all four regimes. Based on FKMGuidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( $R={S}_{\mathrm{min}}/{S}_{\mathrm{max}}$ ) values. The corrected value is then used to choose the SN curve for the damage and life calculation stage.
 Regime 1 (R > 1.0)
 ${S}_{e}^{}=\left({S}_{a}+{M}_{1}{S}_{m}\right)\left(\left(1{M}_{2}\right)/\left(1{M}_{1}\right)\right)$
 Regime 2 (∞ ≤ R ≤ 0.0)
 ${S}_{e}^{}={S}_{a}+{M}_{2}{S}_{m}$
 Regime 3 (0.0 < R < 0.5)
 ${S}_{e}^{}=\left(1+{M}_{2}\right)\frac{{S}_{a}+{M}_{3}{S}_{m}}{1+{M}_{3}}$
 Regime 4 (R ≥ 0.5)
 ${S}_{e}=\left(\left(1+3{M}_{3}\right){S}_{a}{M}_{4}\left(1+3{M}_{3}\right){S}_{m}\right)\left(\left(1+{M}_{2}\right)/\left(\left(13{M}_{4}\right)\left(1+{M}_{3}\right)\right)\right)$
 ${S}_{e}$
 Fully reversed fatigue strength (Endurance stress)
 ${S}_{m}$
 Mean stress
 ${S}_{a}$
 Stress amplitude
 ${M}_{i}$
 Slopes at each region
Interpolate
 MultiMean SN Curves
 Life is usually determined by interpolation of two SN curves with respect to mean stress. A log function mentioned below is a 10 base log function.
 Case A
 If a cycle has a mean stress of 150MPa at point A, HyperLife locates point 1 and point 2 in Figure 15. Then HyperLife linearly interpolates logN1 and logN2 with respect to mean stress in order to determine logN_A at mean stress 150MPa. Once logN_A is determined, life (N_A) and corresponding damage can be determined.
 Case B
 If the cycle has a mean stress greater than the maximum mean stress of
the curve set (180MPa in this case), HyperLife offers two options to choose its behavior.
 Option 1 , Curve Extrapolation = False
 Use an SN curve of the maximum mean stress (the SN curve of mean stress 180 MPa in this case). In the example in HyperLife, N1 is the life HyperLife will report.
 Option 2 , Curve Extrapolation = True
 Extrapolate log(N) of the two SN curves with the highest mean stress values. In the example in Figure 15, log(N) will be extrapolated from log(N1) and log(N2) with respect to mean stress.
 Case C
 If the cycle has a mean stress less than the minimum mean stress of the curve set (90MPa in this case), HyperLife will use the SN curve of the minimum mean stress to determine life. In the example in Figure 15, life will be N2.
 MultiStress Ratio SN Curve
 Life is usually determined by interpolation of 2 SN curves with respect to mean stress. When multistress ratio SN curves are used, HyperLife assumes that you will not define SN curves with stress ratio greater than or equal to 1, which are SN curves with compressive stress or zero stress amplitude. A log function mentioned below is a 10 base log function. R denotes a stress ratio.
 Case A
 If a cycle has R = 0.2 at point A, HyperLife locates point 1 and point 2 in Figure 16. Then HyperLife linearly interpolates logN1 and logN2 with respect to mean stress in order to determine logN_A at R = 0.2. Once R value and stress amplitude of the cycle are given, we can always calculate mean stress of the cycle. Once logN_A is determined, life (N_A) and corresponding damage can be determined. It is worthwhile to mention that HyperLife does not use stress ratio for interpolation because R can be an infinite value when maximum stress is zero.
 Case B
 If the cycle has R greater than the maximum R of the curve set (R=0 in
this case), HyperLife offers two options to
choose its behavior.
 Option 1, Curve Extrapolation = False
 Use an SN curve of the maximum R (the SN curve of R= 0 in this case). In the example in Figure 16, N1 is the life HyperLife will report.
 Option 2, Curve Extrapolation = True
 Extrapolate log(N) of the two SN curves with the highest R values. In the example in Figure 16, log(N) will be extrapolated from log(N1) and log(N2) with respect to mean stress.
 Case C
 If the cycle has R less than the minimum R of the curve set (R= 1 in this case), HyperLife will use the SN curve of the minimum R to determine life. In the example in Figure 16, life will be N2.
 Constant Life Haigh Diagram
 Life is usually determined by interpolation of two Haigh diagrams with
respect to stress amplitude. A log function mentioned below is a 10 base
log function.
 Interpolation on a Constant Mean Stress Line
 If you choose constant mean stress line for linear
interpolation of Haigh diagram, HyperLife interpolates two Haigh
diagrams on a constant mean stress line as described in the following.
 Case A
 If a cycle has a mean stress and stress amplitude at point A, HyperLife locates point 1 and point 2 in Figure 17. Life of point A should be between 1000 and 100000. HyperLife linearly interpolates log(1000) and log(100000) with respect to stress amplitude along Sm_A constant mean stress line in order to determine logN_A at point A. Once logN_A is determined, life (N_A) and corresponding damage can be determined.
 Case B
 If a point (mean stress, stress amplitude) is located above or below all the Haigh diagrams, life of the point is calculated by extrapolation of the two highest or two lowest curves. In the example in Figure 17, log(1000) and log(100000) will be extrapolated with respect to stress amplitude along Sm_B constant mean stress line.
 Case C
 In this case, stress amplitude at point 5 and point 6 may be calculated from extrapolation. Once stress amplitudes become available at the 2 points, a procedure described in case A is applied.
 Interpolation on a Constant Stress Ratio Line
 If you choose constant stress ration line for linear
interpolation of Haigh diagram, HyperLife interpolates two Haigh
diagrams on a constant stress ratio line as described in the following.
 Case A
 If a cycle has a mean stress and stress amplitude at point A, HyperLife locates point 1 and point 2 in Figure 18. Life of point A should be between 1000 and 100000. HyperLife linearly interpolates log(1000) and log(100000) with respect to stress amplitude along RA constant stress ratio line in order to determine logN_A at point A. Once logN_A is determined, life (N_A) and corresponding damage can be determined.
 Case B
 If a point (mean stress, stress amplitude) is located above or below all the Haigh diagrams, life of the point is calculated by extrapolation of the two highest or two lowest curves. In the example in Figure 18, log(1000) and log(100000) will be extrapolated with respect to stress amplitude along R=RB constant stress ratio line.
 Case C
 In this case, stress amplitude at point 5 and point 6 may be calculated from extrapolation. Once stress amplitudes become available at the 2 points, a procedure described in case A is applied on constant stress ratio line R=RC.
Damage Accumulation Model
 ${N}_{if}$
 Materials fatigue life (number of cycles to failure) from its SN curve at a combination of stress amplitude and means stress level $i$ .
 ${n}_{i}$
 Number of stress cycles at load level $i$ .
 ${D}_{i}$
 Cumulative damage under ${n}_{i}$ load cycle.
The linear damage summation rule does not take into account the effect of the load sequence on the accumulation of damage, due to cyclic fatigue loading. However, it has been proved to work well for many applications.
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.
 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
${s}_{e}$ = 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
${S}_{eR}$ = Endurance limit obtained from SN curve with R ratio
${S}_{em}$ = Mean Stress corresponding to ${S}_{eR}$
If ${}_{R\text{}\text{}1}$ , ${s}_{e}=\frac{{S}_{eR}}{1\frac{{s}_{mR}}{UTS}}$
If ${}_{R\text{}\text{}1}$ , ${S}_{e}={S}_{eR}$
If ${\sigma}_{m}{}_{>\text{}0}$ , ${s}_{a}=\frac{{\sigma}_{a}}{1\frac{{\sigma}_{m}}{UTS}}$
If ${\sigma}_{m}{}_{\le 0}$ , ${s}_{a}={\sigma}_{a}$  GerberWhen SN curve is of the Stress Ratio R != 1
 Gerber2
When SN curve is of the Stress Ratio R != 1
If $R\text{}\text{}1$If $R\text{}\text{}1$ , ${S}_{e}={S}_{eR}$
If ${\sigma}_{m}{}_{>\text{}0}$ , ${S}_{a}={\sigma}_{a}\left(1{\left(\frac{{\sigma}_{m}}{UTS}\right)}^{2}\right)$
If ${\sigma}_{m}{}_{\le 0}$ , ${s}_{a}={\sigma}_{a}$  FKM
 No Mean Stress Correction
 Goodman or Soderberg
 Stress Ratio = Constant

 Goodman
When SN curve is of the Stress Ratio R = 1
When SN curve is of the Stress Ratio R != 1
If $R\text{}\text{}1$ , ${s}_{e}=\frac{{S}_{eR}}{1\frac{{s}_{mR}}{UTS}}$If $R\text{}\text{}1$ , ${s}_{e}={S}_{eR}$
If ${\sigma}_{m}{}_{>\text{}0}$ , $SF=\frac{1}{\frac{{\sigma}_{a}}{{S}_{e}}+\frac{{\sigma}_{m}}{UTS}}$
If ${\sigma}_{m}{}_{\le 0}$ , $SF=\frac{{S}_{e}}{{\sigma}_{a}}$
 Gerber
When SN curve is of the Stress Ratio R = 1
When SN curve is of the Stress Ratio R != 1If ${\sigma}_{m}{}_{\ne 0}$ , $SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma}_{m}}\right)}^{2}\cdot \frac{{\sigma}_{e}}{{S}_{e}}\cdot \left(1+\sqrt{1+{\left(\frac{2{\sigma}_{m}{S}_{e}}{UTS{\sigma}_{a}}\right)}^{2}}\right)$
If ${\sigma}_{m}{}_{=0}$ , $SF=\frac{{S}_{e}}{{\sigma}_{a}}$
 Gerber2
When SN curve is of the Stress Ratio R != 1
If $R\text{}\text{}1$If $R\text{}\text{}1$ , ${S}_{e}={S}_{eR}$
If ${\sigma}_{m}{}_{>\text{}0}$ , $SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma}_{m}}\right)}^{2}\cdot \frac{{\sigma}_{a}}{{S}_{e}}\cdot \left(1+\sqrt{1+{\left(\frac{2{\sigma}_{m}{S}_{e}}{UTS{\sigma}_{a}}\right)}^{2}}\right)$
If ${\sigma}_{m}{}_{\le 0}$ , $SF=\frac{{S}_{e}}{{\sigma}_{a}}$
 FKM
${\sigma}_{{a}_{0}}$ = Corrected Stress Amplitude in Constant R mean stress correction
 No Mean Stress Correction
 Interpolate
 Safety Factor with MultiMean
 To calculate safety factor, HyperLife creates an internal Haigh diagram for the target life using multimean SN curve by finding stress amplitudemean 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 MultiRatio
 To calculate safety factor, HyperLife create an internal Haigh diagram for the target life using multimean SN curve by finding stress amplitudemean 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 27.When target life is 100000:
 Constant R : SF = OB/OA
 Constant mean : SF = OD/OC
 Goodman
Safety Factor with MultiMean
Safety Factor with MultiRatio
Safety Factor with Haigh
 Constant R : SF = OB/OA
 Constant mean : SF = OD/OC
If Haigh diagram for a target life is not defined by user, HyperLife creates Haigh diagram for the target life. In Figure 30, if target life is 10000, and Haigh diagram for N=10000 is not defined, HyperLife will created dashed curve to calculate Safety factor.