Stress-Life (S-N) Approach

S-N Curve

The S-N curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure.

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 = S 1 {(N_{f})}^{b 1}

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 $S I$ 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.

Multiple SN Curves

HyperLife supports the following Multiple SN curve types:

Multi-mean S-N curves: group of S-N curves defined at different mean stress.
Multi-ratio S-N curves: group of S-N curves defined at different stress ratio R.
Multi-Haigh Diagram: group of Haigh curves defined at different Number of Cycles.

Note: Refer Mean Stress = Interpolate, to understand how life is determined when Multiple SN curves are assigned.

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 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

Δ S_{12} = | S_{1} - S_{2} |

and

Δ S_{23} = | S_{2} - S_{3} |

. A cycle from 1 to 2 is only extracted if

Δ S_{12} \leq 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.

If $S_{i}$ represents the stress value, point $_{i}$ then:(2) $Δ S_{12} = | S_{1} - S_{2} |$ (3) $Δ S_{23} = | S_{2} - S_{3} |$

As you can see from Figure 8 , $Δ S_{12} \geq Δ S_{23}$ ; therefore, no cycle is extracted from point 1 to 2. Now consider the next three points (2, 3, and 4).(4) $Δ S_{23} = | S_{2} - S_{3} |$ (5) $Δ S_{34} = | S_{3} - S_{4} |$

$Δ S_{23} \leq Δ S_{34}$ , 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

The same process is applied to the remaining points:(6) $Δ S_{14} = | S_{1} - S_{4} |$ (7) $Δ S_{45} = | S_{4} - S_{5} |$

In this case, $Δ S_{14} = Δ S_{45}$ , 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.

Equivalent Nominal Stress

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 ).

Normal Stress resolved at each plane 𝜃 is calculated by:(8)

\begin{array}{l} σ = σ_{x} (\cos^{2} θ) + σ_{y} (\sin^{2} θ) + 2 σ_{x y} (\cos θ \sin θ) \\ θ = 0, 10, 20, 30......170 degrees, \end{array}

HyperLife expects a number of planes (n) as input, which are converted to equivalent 𝜃 using the following formula.(9)

θ = \frac{180}{n - 2}

For example, if number of planes requested is 20, then stress is calculated every 10 degrees.

By default, HyperLife also calculates at 𝜃 = 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.

Mean Stress Correction

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:

Multi-mean SN curves

Life is usually determined by interpolation of 2 SN curves with respect to mean stress. Note that a log function mentioned below is a 10 base log function.

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.

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 2 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 only one slope is defined and if mean stress correction on an SN module is set to FKM:

Regime 1 (R > 1.0): $S_{e}^{A} = S_{a} (1 - M)$
Regime 2 (-∞ ≤ R ≤ 0.0): $S_{e}^{A} = S_{a} + M * S_{m}$
Regime 3 (0.0 < R < 0.5): $S_{e}^{A} = (1 + M) \frac{S_{a} + (\frac{M}{3}) S_{m}}{1 + \frac{M}{3}}$
Regime 4 (R ≥ 0.5): $S_{e}^{A} = \frac{3 S_{a} {(1 + M)}^{2}}{3 + M}$

Where,

$S_{e}^{A}$: 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 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.

If four slopes are defined and mean stress correction is set to FKM:

Regime 1 (R > 1.0): $S_{e}^{} = (S_{a} + M_{1} S_{m}) ((1 - M_{2}) / (1 - M_{1}))$
Regime 2 (-∞ ≤ R ≤ 0.0): $S_{e}^{} = S_{a} + M_{2} S_{m}$
Regime 3 (0.0 < R < 0.5): $S_{e}^{} = (1 + M_{2}) \frac{S_{a} + M_{3} S_{m}}{1 + M_{3}}$
Regime 4 (R ≥ 0.5): $S_{e} = ((1 + 3 M_{3}) S_{a} - M_{4} (1 + 3 M_{3}) S_{m}) ((1 + M_{2}) / ((1 - 3 M_{4}) (1 + M_{3})))$

Where,

$S_{e}$: Fully reversed fatigue strength (Endurance stress)
$S_{m}$: Mean stress
$S_{a}$: Stress amplitude
$M_{i}$: Slopes at each region

Gerber

When SN curve is of the Stress Ration R = -1(10)

S_{e} = \frac{S_{r}}{(1 - {(\frac{S_{m}}{S_{u}})}^{2})}

When SN curve is of the Stress Ratio R != -1(11)

S_{a - R} = (\sqrt{(1 + \frac{4 \cdot S_{e}^{2} \cdot {(1 + R)}^{2}}{{(1 - R)}^{2} S_{u}^{2}})} - 1) \cdot \frac{{(1 - R)}^{2} S_{u}^{2}}{2 S_{e} \cdot {(1 + R)}^{2}}

(12)

S_{e} = \frac{S_{a}}{1 - {(S_{m} ∕ S_{u})}^{2}}

Goodman

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

(13)

S_{e} = \frac{S_{r}}{(1 - \frac{S_{m}}{S_{u}})}

When SN curve is of the Stress Ratio R != -1(14)

S_{a - R} = \frac{S_{a} \cdot S_{u}}{S_{u} - S_{m} + S_{a} (\frac{1 + R}{1 - R})}

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 $S_{m} \leq 0$ , $S_{a - R} = S_{a}$

SODERBERG

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

When SN curve is of the Stress Ratio R = -1(15)

S_{e} = \frac{S_{a}}{(1 - \frac{S_{m}}{S_{y}})}

Where,

$S_{e}$: Equivalent stress amplitude
$S_{a}$: Stress amplitude
$S_{m}$: Mean stress
$S_{y}$: Yield stress

When SN curve is of the Stress Ratio R != -1(16)

S_{a - R} = \frac{S_{a} \cdot S_{y}}{S_{y} - S_{m} + S_{a} (\frac{1 + R}{1 - R})}

FKM

Interpolate

Multi-Mean SN Curves

Life is usually determined by interpolation of 2 SN curves with respect to mean stress. Note that 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 16. 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 16, 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 16, life will be N2.

Multi-Stress Ratio SN Curve

Life is usually determined by interpolation of 2 SN curves with respect to mean stress. When multi-stress ratio SN curves are used, HyperLife assumes that user 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 17. 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 17, 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 17, 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 17, 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.

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 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 2 highest or 2 lowest curves. In the example in Figure 18, 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.

Damage Accumulation Model

Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:(17)

\sum D_{i} = \sum \frac{n_{i}}{N_{i f}} \geq 1.0

Where,

$N_{i f}$: Materials fatigue life (number of cycles to failure) from its S-N 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.

The fatigue life or damage obtained for the event specified can be scaled in HyperLife as shown below. Scaled life or scaled damage will be available as additional output from the fatigue evaluation. (18)

S c a l e d L i f e = \frac{E q u i v a l e n t l i f e u n i t s}{D a m a g e}

Life (which is 1/Damage) is scaled in equivalent units.(19)

S c a l e d D a m a g e = \frac{D a m a g e}{A l l o w a b l e M i n e r S u m}

Linearly accumulated damage can be modified by applying the Allowable Miner sum. Scaled life and scaled damage are supported for SN, EN, Transient Fatigue, Weld Fatigue, and Vibrational Fatigue.

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

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

$s_{e}$ = Target stress amplitude against the target life from the modified SN curve

$σ_{a_{0}}$ = Stress amplitude after mean stress correction

Figure 19.

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

Figure 20.

$σ_{a}$ = 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 $_{R > - 1}$ , $s_{e} = \frac{S_{e - R}}{1 - \frac{s_{m - R}}{U T S}}$
(21) $s_{m - R} = S_{e - R} . \frac{1 + R}{1 - R}$
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 $σ_{m}_{\leq 0}$ , $s_{a} = σ_{a}$ (22) $S F = \frac{S_{e}}{S_{a}}$
Gerber
(23) $S F = \frac{s}{σ_{a}} = \frac{s_{e}}{σ_{a_{0}}}$

Figure 21.
When SN curve is of the Stress Ratio R != -1

Figure 22.
(24) $S_{a} = σ_{a} \cdot (1 - {(\frac{σ_{m}}{U T S})}^{2})$ (25) $S_{e} = S_{e - R} \cdot (1 - {(\frac{s_{m - R}}{U T S})}^{2})$ (26) $s_{m - R} = S_{e - R} . \frac{1 + R}{1 - R}$ (27) $S F = \frac{S_{e}}{S_{a}}$
Gerber2
1. (28) $\begin{array}{l} σ_{m} > 0 : \\ S F = \frac{s}{σ_{a}} = \frac{s_{e}}{σ_{a_{0}}} \end{array}$
2. (29) $\begin{array}{l} σ_{m} \leq 0 : \\ S F = \frac{s}{σ_{a}} \end{array}$
When SN curve is of the Stress Ratio R != -1

If $R > - 1$ (30) $S_{e} = S_{e - R} \cdot (1 - {(\frac{s_{m - R}}{U T S})}^{2})$ (31) $s_{m - R} = S_{e - R} . \frac{1 + R}{1 - R}$

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 $σ_{m}_{\leq 0}$ , $s_{a} = σ_{a}$ (32) $S F = \frac{S_{e}}{S_{a}}$
FKM
(33) $S F = \frac{s'_{e}}{σ_{a}}$
1. (34) $\begin{array}{l} σ_{m} < \frac{- s_{e}}{1 - m_{2}} \\ s'_{e} = - m, (σ_{m} + \frac{s_{e}}{1 - m_{2}}) + \frac{s_{e}}{1 - m_{2}} \end{array}$
2. (35) $\begin{array}{l} \frac{- s_{e}}{1 - m_{2}} \leq σ_{m} < \frac{s_{e}}{1 + m_{2}} \\ s'_{e} = - m_{2} σ_{m} + s_{e} \end{array}$
3. (36) $\begin{array}{l} \frac{s_{e}}{1 + m_{2}} \leq σ_{m} < \frac{3 (1 + m_{3})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}} \\ s'_{e} = - m_{3} (σ_{m} - \frac{s_{e}}{1 + m_{2}}) + \frac{s_{e}}{1 + m_{2}} \end{array}$
4. (37) $\begin{array}{l} \frac{3 (1 + m_{3})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}} \leq σ_{m} \\ s'_{e} = - m_{4} (σ_{m} - \frac{3 (1 + m_{3})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}}) + \frac{1}{3} (\frac{3 (1 + m_{2})}{1 + 3 m_{3}} \cdot \frac{s_{e}}{1 + m_{2}}) \end{array}$
Figure 23.
No Mean Stress Correction
(38) $S F = \frac{s_{e}}{σ_{a}}$
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, OptiStruct 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 26.

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 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.

Stress Ratio = Constant

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

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

Figure 27.

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

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

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. (41) $\begin{array}{l} If σ_{m} = 0 : \\ S F = \frac{s_{e}}{σ_{a}} \end{array}$
2. (42) $\begin{array}{l} If σ_{m} \neq 0 : \\ S F = \frac{1}{2} {(\frac{U T S}{σ_{m}})}^{2} \cdot \frac{σ_{a}}{s_{e}} [- 1 + \sqrt{1 + {(\frac{2 s_{e} σ_{m}}{U T S \cdot σ_{a}})}^{2}}] \end{array}$
When SN curve is of the Stress Ratio R != -1(43) $S_{e} = S_{e - R} (1 - {(\frac{S_{m - R}}{U T S})}^{2})$ (44) $s_{m - R} = S_{e - R} . \frac{1 + R}{1 - R}$

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. (45) $\begin{array}{l} If σ_{m} \leq 0 : \\ S F = \frac{s_{e}}{σ_{a}} \end{array}$
2. (46) $\begin{array}{l} If σ_{m} \geq 0 : \\ S F = \frac{1}{2} {(\frac{U T S}{σ_{m}})}^{2} \cdot \frac{σ_{a}}{s_{e}} [- 1 + \sqrt{1 + {(\frac{2 s_{e} σ_{m}}{U T S \cdot σ_{a}})}^{2}}] \end{array}$
When SN curve is of the Stress Ratio R != -1

If $R > - 1$ (47) $S_{e} = S_{e - R} \cdot (1 - {(\frac{s_{m - R}}{U T S})}^{2})$ (48) $s_{m - R} = S_{e - R} . \frac{1 + R}{1 - R}$

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
(49) $S F = \frac{s_{e}}{σ_{a_{0}}}$

$σ_{a_{0}}$ = Corrected Stress Amplitude in Constant R mean stress correction
No Mean Stress Correction
(50) $S F = \frac{s_{e}}{s_{a}}$

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.

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 30.

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 by user, OptiStruct creates Haigh diagram for the target life. In Figure 30, if target life is 10000, and Haigh diagram for N=10000 is not defined, OptiStruct will created dashed curve to calculate Safety factor.