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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWG0bGaamOCaiaadwgacaWGHbGaamiDaaqabaaaaa@3BA3@ .

In general cases, the total correction factor is C s u r = C t r e a t · C f i n i s h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaOGaeyypa0Jaam4qamaaBaaa leaacaWG0bGaamOCaiaadwgacaWGHbGaamiDaaqabaGccaaMe8UaeS 4JPFMaaGzaVlaaysW7caWGdbWaaSbaaSqaaiaadAgacaWGPbGaamOB aiaadMgacaWGZbGaamiAaaqabaaaaa@4E41@

If treatment type is NITRIDED, then the total correction is C s u r = 2.0 · C f i n i s h ( C t r e a t = 2.0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaOGaeyypa0JaaGOmaiaac6ca caaIWaGaaGjbVlabl+y6NjaaygW7caaMe8Uaam4qamaaBaaaleaaca WGMbGaamyAaiaad6gacaWGPbGaam4CaiaadIgaaeqaaOWaaeWaaeaa caWGdbWaaSbaaSqaaiaadshacaWGYbGaamyzaiaadggacaWG0baabe aakiabg2da9iaaikdacaGGUaGaaGimaaGaayjkaiaawMcaaaaa@552A@ .

If treatment type is SHOT-PEENED or COLD-ROLLED, then the total correction is C s u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaaaa@39D3@ = 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaaaa@39D3@ as: F L ' = F L * C s u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaadY eacaGGNaGaeyypa0JaamOraiaadYeacaGGQaGaam4qamaaBaaaleaa caWGZbGaamyDaiaadkhaaeqaaaaa@3F6A@ . For two segment S-N curve, the stress at the transition point is also modified by multiplying by C s u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaaaa@39D3@ .

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGMbaabeaaaaa@37DD@ is introduced to account for the combined effect of all such corrections. The fatigue endurance limit FL will be modified by K f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGMbaabeaaaaa@37DD@ as: F L ' = F L / K f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaadY eacaGGNaGaeyypa0JaamOraiaadYeacaGGVaGaam4samaaBaaaleaa caWGMbaabeaaaaa@3D79@

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaaaa@39D3@ and K f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGMbaabeaaaaa@37DD@ are specified, the fatigue endurance limit FL will be modified as: F L ' = F L · C s u r / K f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaadY eacaGGNaGaeyypa0JaamOraiaadYeacaaMe8UaeS4JPFMaaGjbVlaa doeadaWgaaWcbaGaam4CaiaadwhacaWGYbaabeaakiaac+cacaWGlb WaaSbaaSqaaiaadAgaaeqaaaaa@46EA@

C s u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaaaa@39D3@ and K f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGMbaabeaaaaa@37DD@ 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 Nc. FL will be corrected if C s u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaamyDaiaadkhaaeqaaaaa@39D3@ and K f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGMbaabeaaaaa@37DD@ 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 Ctemp to modify the fatigue endurance limit FL.

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

Ctemp at normal temperature = 1

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

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

Ctemp set to NONE = 1

Type Temp. Condition Ctemp 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 - aT,D x (10-3 * T/℃)2
Aluminum materials 50℃ < T < 200℃ =1 - [1.2*10-3 x (T/℃-50)]
Material Group GJS GJM GJL
aT,D 1.6 1.3 1.0

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

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 modify the curves according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).


Figure 2. S-N Curve with Scatter Data
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. The material data input in the Material DB and My Material tabs is based on the mean of the scatter.


Figure 4. Probability Function of the Log(N) Distribution for S-N Scatter

The experimental scatter exists in both Stress Range and Life data. In the Assign Material dialog, the Standard Deviation (or Standard Error) of the scatter of log(N) is required as input (SE field for S-N curve). Any normal distribution is fully defined by specifying the mean and its standard deviation, in the case of S-N fatigue analysis in HyperLife, the mean is provided by the S-N curve as log ( N i 50 % ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaiikaiaad6eadaqhaaWcbaGaamyAaaqaaiaaiwdacaaI WaGaaiyjaaaakiaacMcaaaa@3E3A@ , whereas, the standard deviation is input via the SE field in the Assign Material dialog.

If the specified S-N curve is directly utilized, without any perturbation, then the mean values of the normal distributions at each data point is assumed to be used, leading to a certainty of survival of 50%, by default. This is a consequence of S-N material data that is available for the mean values of log(N). 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.
  1. Standard Deviation (Standard Error) of log(N) normal distribution (SE in Assign Material).
  2. 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. log(N) values which are higher than the mean are always more conservative. Therefore, for certainty of survival calculations, any value greater than 50% implies that the log(N) values lower than the mean are being considered. The total area of the curve starting from the right end of the Normal curve (which is typically positive infinity for a PDF) is directly equal to the certainty of survival (or probability of survival).

A typical normal distribution is characterized by the following Probability Density Function:(1)
P(x)= 1 2π σ 2 e ( xμ ) 2 2 σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI cacaWG4bGaaiykaiabg2da9maalaaabaGaaGymaaqaamaakaaabaGa aGOmaiabec8aWjabeo8aZnaaCaaaleqabaGaaGOmaaaaaeqaaaaaki aadwgadaahaaWcbeqaaiabgkHiTmaalaaabaWaaeWaaeaacaWG4bGa eyOeI0IaeqiVd0gacaGLOaGaayzkaaWaaWbaaWqabeaacaaIYaaaaa WcbaGaaGOmaiabeo8aZnaaCaaameqabaGaaGOmaaaaaaaaaaaa@4BCD@
Where,
x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@
The sample value ( log ( N i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaiikaiaad6eadaWgaaWcbaGaamyAaaqabaGccaGGPaaa aa@3C17@ ) for which the probability of occurrence is calculated in a particular range (based on the specified certainty of survival).
μ
The mean log( N i 50% ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaiikaiaad6eadaqhaaWcbaGaamyAaaqaaiaaiwdacaaI WaGaaiyjaaaakiaacMcaaaa@3E3A@
σ
The standard deviation (or standard error, given by SE field ).
The area of the curve from the right end can be understood to be equal to the certainty of survival specified in the Certainty of Survival field. This integral equation can be solved directly to generate the value of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ (or log ( N i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaiikaiaad6eadaWgaaWcbaGaamyAaaqabaGccaGGPaaa aa@3C17@ ) which will allow a perturbation of the S-N curve. However, the general practice is to convert the 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 convert the certainty of survival values into corresponding log ( N i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaiikaiaad6eadaWgaaWcbaGaamyAaaqabaGccaGGPaaa aa@3C17@ 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.
The equation used to convert the normal distribution to a standard normal distribution is:(2)
z= xμ σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2 da9maalaaabaGaamiEaiabgkHiTiabeY7aTbqaaiabeo8aZbaaaaa@3D6F@
This transformation leads to a corresponding standard normal distribution with a mean of 0.0 and standard error of 1.0. Replacing the values with fatigue data, and rewriting the equation.(3)
log( N i )=log( N i 50% )+z*SE MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaiikaiaad6eadaWgaaWcbaGaamyAaaqabaGccaGGPaGa eyypa0JaciiBaiaac+gacaGGNbGaaiikaiaad6eadaqhaaWcbaGaam yAaaqaaiaaiwdacaaIWaGaaiyjaaaakiaacMcacqGHRaWkcaWG6bGa aiOkaiaadofacaWGfbaaaa@4991@
The value of z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@ is procured from the standard normal distribution Z-tables based on the input value of the certainty of survival (which is the area under the curve from the right end of the curve up to the corresponding negative Z-value). 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.

References

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