# Other Factors Affecting Fatigue

## Surface Condition (Finish and Treatment)

Surface condition is an extremely important factor influencing fatigue strength, as fatigue failures nucleate at the surface. Surface finish and treatment factors are considered to correct the fatigue analysis results.

^{1}

Surface treatment can improve the fatigue strength of components. NITRIDED, SHOT-PEENED, and COLD-ROLLED are considered for surface treatment correction. It is also possible to input a value to specify the surface treatment factor ${C}_{treat}$ .

In general cases, the total correction factor is ${C}_{sur}={C}_{treat}\text{\hspace{0.17em}}\xb7\text{}\text{\hspace{0.17em}}{C}_{finish}$

If treatment type is NITRIDED, then the total correction is ${C}_{sur}=2.0\text{\hspace{0.17em}}\xb7\text{}\text{\hspace{0.17em}}{C}_{finish}\left({C}_{treat}=2.0\right)$ .

If treatment type is SHOT-PEENED or COLD-ROLLED, then the total correction is ${C}_{sur}$ = 1.0. It means you will ignore the effect of surface finish.

The fatigue endurance limit FL will be modified by ${C}_{sur}$ as: $FL\text{'}=FL*{C}_{sur}$ . For two segment S-N curve, the stress at the transition point is also modified by multiplying by ${C}_{sur}$ .

Surface conditions may be defined on a PFAT Bulk Data Entry. Surface conditions are then associated with sections of the model through the FATDEF Bulk Data Entry.

## Fatigue Strength Reduction Factor

In addition to the factors mentioned above, there are various other factors that could affect the fatigue strength of a structure, that is, notch effect, size effect, loading type. Fatigue strength reduction factor ${K}_{f}$ is introduced to account for the combined effect of all such corrections. The fatigue endurance limit FL will be modified by ${K}_{f}$ as: $FL\text{'}=FL/{K}_{f}$

The fatigue strength reduction factor may be defined on a PFAT Bulk Data Entry. It may then be associated with sections of the model through the FATDEF Bulk Data Entry.

If both ${C}_{sur}$ and ${K}_{f}$ are specified, the fatigue endurance limit FL will be modified as: $FL\text{'}=FL\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{C}_{sur}/{K}_{f}$

${C}_{sur}$
and
${K}_{f}$
have similar influences on the E-N formula through
its elastic part as on the S-N formula. In the elastic part of the E-N formula, a
nominal fatigue endurance limit FL is calculated internally from the reversal limit
of endurance N_{c}. FL will be corrected if
${C}_{sur}$
and
${K}_{f}$
are presented. The elastic part will be modified as
well with the updated nominal fatigue limit.

## Scatter in Fatigue Material Data

S | 2000.0 | 2000.0 | 2000.0 | 2000.0 | 2000.0 | 2000.0 |

Log (S) | 3.3 | 3.3 | 3.3 | 3.3 | 3.3 | 3.3 |

Log (N) | 3.9 | 3.7 | 3.75 | 3.79 | 3.87 | 3.9 |

The experimental scatter exists in both Stress Range and Life data. On the
MATFAT entry, the Standard Error of the scatter of log(N) is
required as input (`SE` field for S-N curve). The sample mean is
provided by the S-N curve as
$\mathrm{log}({N}_{i}^{50\%})$
, whereas, the standard error is input via the
`SE` field of the
MATFAT entry.

- Standard Error of log(N) normal distribution (
`SE`on MATFAT). - Certainty of Survival required for this analysis (
`SURVCERT`on FATPARM).

A normal distribution or gaussian distribution is a probability density function which implies that the total area under the curve is always equal to 1.0.

The user-defined SN curve data is assumed as a normal distribution, which is typically characterized by the following Probability Density Function:

- ${x}_{s}$
- The data value ( $\mathrm{log}({N}_{i})$ ) in the sample.
- ${\mu}_{s}$
- The sample mean $\mathrm{log}({N}_{i}^{sm})$ .
- ${\sigma}_{s}$
- The standard deviation of the sample (which is unknown, as you input only Standard Error (SE) on MATFAT).

The above distribution is the distribution of the user-defined sample, and not the
full population space. Since the true population mean is unknown, the estimated
range of the true population mean from the sample mean and the sample SE and
subsequently use the user-defined Certainty of Survival (`SURVCERT`) to perturb the sample mean.

Standard Error is the standard deviation of the normal distribution created by all the sample means of samples drawn from the full population. From a single sample distribution data, the Standard Error is typically estimated as $SE=\left({\scriptscriptstyle \raisebox{1ex}{${\sigma}_{s}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{{n}_{s}}$}\right.}\right)$ , where ${\sigma}_{s}$ is the standard deviation of the sample, and ${n}_{s}$ is the number of data values in the sample. The mean of this distribution of all the sample means is actually the same as the true population mean. The certainty of survival is applied on this distribution of all the sample means.

For the normal distribution of all the sample means, the mean of this distribution is the same as the true population mean $\mu $ , the range of which is what you want to estimate.

Statistically, you can estimate the range of true population mean as:

That is,

Since the value on the left hand side is more conservative, use the following equation to perturb the SN curve:

- $\mathrm{log}({N}_{i}^{m})$
- Perturbed value
- $\mathrm{log}({N}_{i}^{sm})$
- User-defined sample mean (SN curve on MATFAT)
- $SE$
- Standard error (SE on MATFAT)

**Z-Values (Calculated)****Certainty of Survival (Input)**- 0.0
- 50.0
- 0.5
- 69.0
- 1.0
- 84.0
- 1.5
- 93.0
- 2.0
- 97.7
- 3.0
- 99.9

Based on the above example (S-N), you can see how the S-N curve is modified to the required certainty of survival and standard error input. This technique allows you to handle Fatigue material data scatter using statistical methods and predict data for the required survival probability values.

## SN Curve Modification

- Certainty of Survival
- If the certainty of survival is not 0.5 and standard error (SE) is not
0.0, an SN curve is modified by shifting SRI1 and FL.$$SRI1\text{'}=SRI1*Cr$$$$FL\text{'}=FL*Cr$$$$Cr={10}^{z*b1*SE}$$Where z is the z-value in standard normal distribution that corresponds to the certainty of survival.
- Surface Condition and Fatigue Strength Reduction Factor
- A factor for surface condition (
$Cs$
) and fatigue strength reduction factor (
$Kf$
) are applied to fatigue limit to modify
slope of the SN curve after
`Nc_stat`cycles in the following manner:$$FL\text{'}=FL*\frac{Cs}{Kf}$$Where`Nc_stat`is the number of cycles at static failure transition. - Static Failure
- When a specimen fails at less than or equal to a certain low number of
cycles, the failure is not considered as a fatigue failure but
considered as a static failure. Once the failure is considered as a
static failure, the low number of cycles is defined as the number of
static failure transition cycles (
`Nc_stat`). In fatigue analysis, stress amplitudes are supposed to be less than the stress amplitude (Sc_stat) corresponding to`Nc_stat`in an SN curve in order to apply fatigue failure theories. If static failure check is enabled, OptiStruct checks whether stress or stress amplitudes in stress history is greater than Sc_stat, and reports a warning when a stress or stress amplitude exceeds Sc_stat.`Nc_stat`can be specified directly on MATFAT.- Sc_stat is defined via
`ALPHA`field on MATFAT which is a scaling factor to the`UTS`to determine the static failure stress threshold.

- Overall SN Curve Modification
- Combining factors from certainty of survival, surface condition, fatigue strength reduction factor, and static failure, the final SN curve that is used in damage calculation is depicted in Figure 8

## References

^{1}Yung-Li Lee, Jwo. Pan, Richard B. Hathaway and Mark E. Barekey. Fatigue testing and analysis: Theory and practice, Elsevier, 2005