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 can be defined in the Assign Material dialog, where you assign them to each part.

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 in the Assign Material dialog and is assigned to parts or sets.

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.

Temperature Influence

The fatigue strength of a material reduces with an increase in temperature. Temperature influence can be accounted by applying the temperature factor C_temp to modify the fatigue endurance limit FL.

C_temp can either by assigned directly, or isothermal temperature across the part/element set can be defined to calculate C_temp as referred by FKM guidelines for elevated temperatures. The temperature defined must be in degree Celsius.

C_temp at normal temperature = 1

C_temp at elevated temperature defined as per FKM guidelines for the following materials is highlighted in the table below.

C_temp user-defined accepts a value between 0 < C_temp <= 1

C_temp set to NONE = 1


Type	Temp. Condition	C_temp Factor
None** this is for materials other than the ones below	-	= 1
Fine Grain Structural Steel	60℃ < T < 500℃	=1 - [10^-3 x (T/℃)]
Other Steels (other than stainless steel)**	100℃ < T < 500℃	=1 - [1.4*10^-3 x (T/℃-100)]
GS (Cast steel and heat treatable cast steel)	100℃ < T < 500℃	=1 - [1.2*10^-3 x (T/℃-100)]
GJS (Nodular Cast Iron) GJM (Malleable Cast Iron) GJL (Cast iron with lamellar graphite)	100℃ < T < 500℃	=1 - a_T,D x (10^-3 * T/℃)²
Aluminum materials	50℃ < T < 200℃	=1 - [1.2*10^-3 x (T/℃-50)]


Material Group	GJS	GJM	GJL
a_T,D	1.6	1.3	1.0

If both C_temp and K_f are specified, the fatigue endurance limit FL will be modified as: FL' = FL ⋅ C_temp / K_f

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 deviation 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 in Materials and Standard Error in the Material DB and My Material tabs of the sample. The SN material data input in the Material DB and My Material tabs 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. In the Assign Material dialog, 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 in the Assign Material dialog.

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 in the Assign Material dialog. Since a value of 50% survival certainty may not be sufficient for all applications, HyperLife 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 in Assign Material).
Certainty of Survival required for this analysis (Certainty of Survival in the Fatigue Module context).

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) in the Assign Material dialog).

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 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 Materials)
$S E$: Standard error (SE on Materials)

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.

Adjustment of Single SN Curves

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

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.
Figure 5.; $S R I 1 ’ = S R I 1 x C r$; $F L ’ = F L x C r$; $C r = 10^{z x b 1 x S E}$; Where z is a 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 1000 cycles in the following manner.; $F L ’ = F L * C s / K f$
Figure 6.
Static Failure: If static failure check is activated, static failure is reported when the maximum stress is higher than UTS or corrected stress amplitude is more than UTS x (1- R)/2, where R is a stress ratio that the SN curve is based on. The SN curve is modified so the program can report damage value 1.0 when stress amplitude is UTS x (1- R)/2 if UTS x (1-R)/2 is smaller than SRI1. Thus stress amplitude higher than S1000 reports a damage value different from the user-defined SN curve due to the modified b0 slope in Figure 7.
Figure 7.
Overall SN Curve Modification: Figure 8.

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.

Adjustment of Multiple SN Curves

The following adjustment is applied to multi-mean stress SN curves, multi-stress ratio SN curves and Haigh diagram.

Certainty of Survival

Uncertainty of fatigue strength of material can be taken into consideration by means of the standard error of log(stress) and certainty of survival.

For example, if the standard error of log(stress) is 0.2, and certainty of survival has to be 99.7%, HyperLife adjusts the multiple SN curves as follows:

log(fatigue strength) = log(user defined fatigue strength) – 3 x 0.2
Fatigue strength = (user defined fatigue strength ) x 10(-3 x 0.2) .

In the example, user defined fatigue strength is reduced by 3 standard error which corresponds to 99.7% in normalized Gaussian distribution.

Surface Condition and Fatigue Strength Reduction Factor

A factor for surface condition (Cs) and fatigue strength reduction factor (Kf) are applied to fatigue strength in the following manner:

Fatigue strength = (user defined fatigue strength ) * K’

Where,

K’ = 1.0 for N <= 1000

K’ = Cs/Kf for N > Nc1

log(K’) = log(Cs/Kf) x (3-logN) / (3-logNc1) for 1000 < N < Nc1

Nc1 : transition point

References

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