Stress-Life (S-N) Approach

Typically, the S-N curve (and other fatigue properties) of a material is obtained from experiment; through fully reversed rotating bending tests. Due to the large amount of scatter that usually accompanies test results, statistical characterization of the data should also be provided (certainty of survival is used to modify the S-N curve according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).

Figure 1. S-N Data from Testing

When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude $$S_a$$ or range $$S_R$$ versus cycles to failure $$N$$, the relationship between $$S$$ and $$N$$ can be described by straight line segments. Normally, a one or two segment idealization is used.

Figure 2. One Segment S-N Curves in Log-Log Scale

(1) $S=S1{\left(N_f\right)}^{b1}$

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 S-N approach is based on elastic cyclic loading, inferring that the S-N 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 high-cycle fatigue.

S-N curve data is provided for a given material using the Materials module.

One way to understand "cycle counting" is as a changing stress-strain versus time signal. Cycle counting will count the number of stress-strain 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 $$\text{Δ}{S}_{12}=\left|S_1-S_2\right|$$ and $\text{Δ}{S}_{23}=\left|S_2-S_3\right|$. A cycle from 1 to 2 is only extracted if $\text{Δ}{S}_{12}\le \text{Δ}{S}_{23}$. 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

Figure 6. Continuous Load History

Since this load history is continuous, it is converted into a load history consisting of peaks and valleys only.

Figure 7. Peaks and Valleys for Rainflow Counting. 1, 2, 3, and 4 are the four peaks and valleys

It is clear that point 4 is the peak stress in the load history, and it will be moved to the front during rearrangement (Figure 8). After rearrangement, the peaks and valleys are renumbered for convenience.

Figure 8. Load History after Rearrangement and Renumbering

Next, pick the first three stress values (1, 2, and 3) and determine if a cycle is present.

$\text{Δ}{S}_{23}\le \text{Δ}{S}_{34}$, hence a cycle is extracted from point 2 to 3. Now that a cycle has been extracted, the two points are deleted from the graph.

Figure 9. Delete and Reconnect Remaining Points

In this case, $\text{Δ}{S}_{14}=\text{Δ}{S}_{45}$, 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 rainflow counting process is the same regardless of the number of load history points. However, depending on the location of the highest peak/valley used for rearrangement, it may not be obvious how the rearrangement process is conducted.Figure 10 shows just the rearrangement process for a more complex load history. The subsequent rainflow counting is just an extrapolation of the process mentioned in the simple example above, and is not repeated here.

Figure 10. Continuous Load History

Since this load history is continuous, it is converted into a load history consisting of peaks and valleys only:

Figure 11. Peaks and Valleys for Rainflow Counting

Clearly, load point 11 is the highest valued load and therefore, the load history is now rearranged and renumbered.

Figure 12. Load History After Rearrangement and Renumbering

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.

Since S-N 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 S-N 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 135-degree planes in addition to the requested number of planes. This is to include the worst possible damage if occurring on these planes.

Generally, S-N curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully-reversed, 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 non-zero 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 $S_m\le 0$, $S_{a-R}=S_a$

Soderberg

Is slightly different from GOODMAN; the mean stress is normalized by yield stress instead of ultimate tensile stress.

FKM

If only one slope field is specified for mean stress correction, the corresponding Mean Stress Sensitivity value ($M$) for Mean Stress Correction is set equal to Slope in Regime 2 (Figure 14). Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ($$R=S_{\min }/S_{\max }$$) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.

Figure 14.

Note: The FKM equations below illustrate the calculation of Corrected Stress Amplitude ($S_e^A$). The actual value of stress used in the Damage calculations is the Corrected stress range (which is $2\cdot S_e^A$). These equations apply for SN curves input by the user (by default, any user-defined SN curve is expected to be input for a stress ratio of R=1.0).

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.

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 FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ($$R=S_{\min }/S_{\max }$$) values. The corrected value is then used to choose the S-N curve for the damage and life calculation stage.

Interpolate

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.

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

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

Stress Ratio = Constant

1. Goodman

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

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

   
   
   Figure 24.

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

   If $$R>-1$$, $s_e=\frac{S_{e-R}}{1-\frac{s_{m-R}}{UTS}}$(40) $s_{m-R}=S_{e-R}. \frac{1+R}{1-R}$

   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. (41) $$\begin{array}{l}\text{If}{\sigma }_m=0:\\ SF=\frac{s_e}{{\sigma }_a}\end{array}$$
   2. (42) $$\begin{array}{l}\text{If}{\sigma }_m\ne 0:\\ SF=\frac{1}{2}{\left(\frac{UTS}{{\sigma }_m}\right)}^2·\frac{{\sigma }_a}{s_e}\left[-1+\sqrt{1+{\left(\frac{2s_e{\sigma }_m}{UTS\cdot {\sigma }_a}\right)}^2}\right]\end{array}$$
   When SN curve is of the Stress Ratio R != -1(43) $S_e=S_{e-R}\left(1-{\left(\frac{S_{m-R}}{UTS}\right)}^2\right)$ (44) $s_{m-R}=S_{e-R}. \frac{1+R}{1-R}$

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

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

   If $$R>-1$$(47) $S_e=S_{e-R}\cdot \left(1-{\left(\frac{s_{m-R}}{UTS}\right)}^2\right)$ (48) $s_{m-R}=S_{e-R}. \frac{1+R}{1-R}$

   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
   (49) $$SF=\frac{s_e}{{\sigma }_{a_0}}$$

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

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

   
   
   Figure 25. 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 26. Conversion of Multi-Mean Curve to Haigh Diagram

   Safety Factor with Haigh

   Safety factor (SF) is calculated in the following manner in Figure 27.

   
   
   Figure 27.

   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 27, if target life is 10000, and Haigh diagram for N=10000 is not defined, HyperLife will created dashed curve to calculate Safety factor.

Safety factor (SF) is calculated in the following manner in Figure 30.

Figure 30.

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.