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.

Surface finish correction factor

C_{f i n i s h}

is used to characterize the roughness of the surface. It is presented on diagrams that categorize finish by means of qualitative terms such as polished, machined or forged. ^¹

Figure 1. Surface Finish Correction Factor for Steels

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 $C_{t r e a t}$ .

In general cases, the total correction factor is $C_{s u r} = C_{t r e a t} \cdot C_{f i n i s h}$

If treatment type is NITRIDED, then the total correction is $C_{s u r} = 2.0 \cdot C_{f i n i s h} (C_{t r e a t} = 2.0)$ .

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

The fatigue endurance limit FL will be modified by $C_{s u r}$ as: $F L' = F L * C_{s u r}$ . For two segment S-N curve, the stress at the transition point is also modified by multiplying by $C_{s u r}$ .

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: $F L' = F L / K_{f}$

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 $C_{s u r}$ and $K_{f}$ are specified, the fatigue endurance limit FL will be modified as: $F L' = F L \cdot C_{s u r} / K_{f}$

$C_{s u r}$ 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 $C_{s u r}$ and $K_{f}$ are presented. The elastic part will be modified as well with the updated nominal fatigue limit.

Scatter in Fatigue Material Data

The S-N and E-N curves (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 estimate the worst mean log(N) according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).

To understand these parameters, let us consider the S-N curve as an example. When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude Sa or range SR 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 3. One Segment S-N Curve in log-log Scale

Consider the situation where S-N scatter leads to variations in the possible S-N curves for the same material and same sample specimen. Due to natural variations, the results for full reversed rotating bending tests typically lead to variations in data points for both Stress Range (S) and Life (N). Looking at the Log scale, there will be variations in Log(S) and Log(N). Specifically, looking at the variation in life for the same Stress Range applied, you may see a set of data points which look like this.


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

As with many processes, the distribution of Log(N) is assumed to be a Normal Distribution. There is a full population of possible values of log(N) for a particular value of log(S). The mean of this full population set is the true population mean and is unknown. Therefore, statistically estimate the worst true population mean of log(N) based on the input sample mean (SN curve on MATFAT) and Standard Error (SE on MATFAT) of the sample. The SN material data input on the MATFAT entry is based on the mean of the normal distribution of the scatter in the particular user sample used to generate the data.

Figure 4. Probability Function of the Log(N) Normal Distribution for S-N Scatter. of a particular user-defined sample data

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 $\log (N_{i}^{50 %})$ , whereas, the standard error is input via the SE field of the MATFAT entry.

If the specified S-N curve is directly utilized, without any perturbation, the sample mean is directly used, leading to a certainty of survival of 50%. This implies that OptiStruct does not perturb the sample mean provided on the MATFAT entry. Since a value of 50% survival certainty may not be sufficient for all applications, OptiStruct can internally perturb the S-N material data to the required certainty of survival defined by you. To accomplish this, the following data is required.

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:

P (x_{s}) = \frac{1}{\sqrt{2 π σ_{s}^{2}}} e^{- \frac{{(x_{s} - μ_{s})}^{2}}{2 σ_{s}^{2}}}

Where,

$x_{s}$: The data value ( $\log (N_{i})$ ) in the sample.
$μ_{s}$: The sample mean $\log (N_{i}^{s m})$ .
$σ_{s}$: 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 $S E = (\frac{σ_{s}}{\sqrt{n_{s}}})$ , where $σ_{s}$ 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.

The general practice is to convert a normal distribution function into a standard normal distribution curve (which is a normal distribution with mean=0.0 and standard error=1.0). This allows us to directly use the certainty of survival values via Z-tables.

Note: The certainty of survival is equal to the area of the curve under a probability density function between the required sample points of interest. It is possible to calculate the area of the normal distribution curve directly (without transformation to standard normal curve), however, this is computationally intensive compared to a standard lookup Z-table. Therefore, the generally utilized procedure is to first convert the current normal distribution to a standard normal distribution and then use Z-tables to parameterize the input survival certainty.

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

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

\log (N_{i}^{s m}) - z * S E \leq μ \leq \log (N_{i}^{s m}) + z * S E

That is,

\log (N_{i}^{s m}) - z * S E \leq \log (N_{i}^{m}) \leq \log (N_{i}^{s m}) + z * S E

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

\log (N_{i}^{m}) = \log (N_{i}^{s m}) - z * S E

Where,

$\log (N_{i}^{m})$: Perturbed value
$\log (N_{i}^{s m})$: User-defined sample mean (SN curve on MATFAT)
$S E$: Standard error (SE on MATFAT)

The value of

z

is procured from the standard normal distribution Z-tables based on the input value of the certainty of survival. Some typical values of Z for the corresponding certainty of survival values are:

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

This section describes how a slope-based SN curve is modified in OptiStruct.

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.

S R I 1' = S R I 1 * C r

F L' = F L * C r

C r = 10^{z * b 1 * S E}

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 (

C s

) and fatigue strength reduction factor (

K f

) are applied to fatigue limit to modify slope of the SN curve after Nc_stat cycles in the following manner:

F L' = F L * \frac{C s}{K f}

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.

SN Curve Modification and Static Failure Transition:

OptiStruct modifies the user-defined SN curve when certainty of survival is not 0.5, surface treatment is applied, surface finish is applied, or fatigue strength factor is applied. In the modification due to surface treatment (Cr), surface finish (Cs), or fatigue strength factor (Kf), SN curve is only modified after Nc_stat because surface treatment, surface finish, or fatigue strength factor should not affect static failure behavior which does not follow fatigue theories.

Once static failure check is enabled, OptiStruct modifies slope (b0) near UTS so that number of cycles at UTS can be 1 as depicted in Figure 7. During rainflow cycle counting, OptiStruct issues warning messages in the following cases:

Stress is greater than Sc_stat in stress history (both single SN curve and multi SN curves)
Stress amplitude after mean stress correction is greater than Sc_stat (single SN curve).

By default, OptiStruct checks static failure and continues to evaluate damages of remaining stress history (CHK=CONT on FATPARM). You can choose an option to stop damage calculation as soon as OptiStruct detects static failure (CHK=STOP on FATPRM). You can also choose an option to disable static failure check (CHK=NOCHK on FATPARM). If static failure check is disabled, slope near UTS in SN curve is not modified.

Depending on the UTS value, there can be a special case where the calculated slope b0 becomes 0.0. In this case damage is 1.0 for all the stress amplitudes greater than or equal to Sc_stat. If calculated b0 is positive, OptiStruct errors out.

Instead of OptiStruct calculating b0 using UTS and Sc_stat, you can define b0 directly in MATFAT. If b0 is defined, b0 is honored as it is. If b0 is set to 0.0, damage at Sc_stat is 1.0, and damage at stress amplitude greater than Sc_stat is more than 1.0.

You can choose how Nc_stat is defined in FATPARM. You can directly define Nc_stat. This is the default way to define Nc_stat. The default value of Nc_stat is 1000. Another way to define Nc_stat is to specify Sc_stat. Sc_stat is specified by a fraction of UTS (using ALPHA field on MATFAT). Default Sc_stat value is 0.9*UTS. If Sc_stat is specified, OptiStruct calculates Nc_stat using the slope of the SN curve after SN curve shift due to certainty of survival.

If static failure check is activated, static failure is reported when the maximum stress is higher than Sc_stat or corrected stress range is more than Sc_stat.

If static failure check (CHK on FATPARM) is set to CONT, the SN curve is modified so that OptiStruct can report a damage value of 1.0 when stress range is 2*UTS and 2*UTS is smaller than SRI1. Thus stress range higher than Sc_stat reports a damage value different from the user-defined SN curve due to the modified b0 slope in the picture below. If static failure check is set to STOP, damage is greater than 1 when stress range is more than Sc_stat. OptiStruct errors out if 2*UTS is less than Sc_stat.

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