Probabilistic S–N curves using exponential and power laws equations

Composites: Part B 56 (2014) 582–590

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Composites: Part B

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Probabilistic S–N curves using exponential and power laws equations

1359-8368/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesb.2013.08.036

⇑ Corresponding author. Tel.: +55 8432153740.E-mail address: [email protected] (R.C.S. Freire Júnior).

Raimundo Carlos Silverio Freire Júnior ⇑, Adriano Silva BelísioUniversidade Federal do Rio Grande do Norte, Technology Center, Mechanical Engineering Postgraduate Course, Lagoa Nova, CEP 59072-970 Natal, Rio Grande do Norte, Brazil

a r t i c l e i n f o

Article history:Received 6 February 2013Received in revised form 3 July 2013Accepted 12 August 2013Available online 20 August 2013

Keywords:A. LaminatesB. FatigueStatistical properties/methods

a b s t r a c t

The current study aims to associate the Weibull probability equations to the ones commonly used whenmodeling the S–N curves of composite materials, which are the exponential equation and the power lawand its generalizations. The association among them will be called as the ‘‘Probabilistic S–N Curves’’. Inorder to evaluate and compare the results of such associations, a database containing three compositematerials was used. Such materials are known as the following abbreviations: DD16, IM7/977 andT800/5245. The results obtained from the probabilistic S–N curves were compared to the probabilisticoutcomes of each stress level that had been analyzed. The results indicate that the probabilistic S–Ncurves based on the generalized power law reproduce the physical phenomenon in a better way.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

When designing machines and structures made from compositematerials, which suffer the action of cyclic loading, one of the fac-tors that must be considered is the possibility that these elementsundergo fatigue failure. The fatigue behavior of a material is ini-tially evaluated by S–N curves and these curves are modeled bydeterministic equations [1–9]. However, due to the large disper-sion of such values, a probabilistic analysis of those curves mustbe done. Therefore, it is necessary to develop a ‘‘Probabilistic S–NCurve’’ in order to evaluate both the deterministic and probabilisticaspects of the fatigue behavior of the composite.

In order to develop a probabilistic S–N curve, there are basicallythree methods that are demonstrated in the literature, the first [10]one considers that the data show a linear behavior in a log–lin (re-stricted exponential equation) or log–log graph (restricted powerlaw), and, probabilistically, the fatigue behavior follows a log–nor-mal distribution. Among the methods, this is the simplest one, butit has some disadvantages, such as considering that the variance isthe same for all stress levels analyzed (homoscedasticity) and theneed for a larger database. Moreover, most researchers [11–14]consider that the probabilistic behavior of fatigue failure followsa Weibull distribution.

The second method [15,16], and which shall serve as a basis forthe study presented in this paper, considers that the data show alinear behavior in a log–log graph (restricted power law) and fol-low a Weibull distribution. The advantage of this method is to beless dependent on the amount of samples than the previous one,

as well as its relative simplicity. However, as shown in the litera-ture [16,17], there are few studies that evaluate its quality andnone of the cited papers compares its results with the individualprobability of failure of each stress level. Furthermore, in all studiesthat use such method, the parameters of the Weibull equation areobtained by the maximum likelihood estimator, but this method[16] is very sensitive to its database, and its application is only rec-ommended to a data set that is higher than 20.

The third method [18] converts all data of fatigue in the equiv-alent static stress values and the construction of Weibull distribu-tion is developed from these results. In addition, this methodrequires an iterative process for obtaining the parameters relatedto their equationing. According to the literature [17], this methodcan be troublesome and it does not produce satisfactory results,especially if one does not have knowledge of the material staticstress.

Thus, this work has the purpose of making a further analysis onthe method originally submitted by Whitney [15,16], in whichother equations, besides the restricted power law ones, will beused to model the S–N curve, and, especially, it will be checkedhow their results approach the individual probabilistic behavior(for each stress level). Differently from previous papers, in a data-base that is lower than or equal to 20, the method for obtaining theparameters of the Weibull equation will be the method of Thiel–Cacciari [19], which is much more suitable for this case. Finally,it should also be noted that this analysis is performed for compos-ites made from carbon and glass fiber, in which the S–N curveswere obtained from fatigue ratio values in all four regions of load-ing (tension, tension–compression, compression-tension and com-pression) and from various probabilities of failure (1%, 5% and10%); such matters have not yet been analyzed by the aforemen-tioned studies.

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2. Materials extracted from literature

For the analysis of the probabilistic S–N curves, a database witha relatively large amount of tests has been sought. Therefore, theresults of fatigue tests of three composite materials known in theliterature by DD16, IM7/977 and T800/5245 were used. TheDD16 is made of fiberglass and the others (IM7/977 and T800/5245) are made of carbon fiber.

The glass fiber-reinforced DD16 plastic obtained has a fiber vol-ume content of 36%, and it was manufactured by the resin transfermolding process, which matrix is made of ortho-polyester and thestacking sequence is [90/0/±45/0]s. In the layers at 0� and 90�, itcontains D155 fiberglass fabric (527 g/m2), and at ±45, it containsDB120 fabric (393 g/m2) [20].

The IM7/977 and T800/5245 materials are laminates made ofcarbon fiber. The T800/5245 uses the highly resistant Toray fiberand the Narmco epoxy matrix. The IM7/977 uses the highly resis-tant Hercules fiber and the 977 epoxy polyester matrix. Both lam-inates are configured [(±45.02)2]s with 65% fiber volume andmanufactured by the closed mold process (autoclave) [21,22].

For a better understanding of the stacking sequence and manu-facturing process, among others, see the following literature [20–23].

3. Mathematical equation

3.1. The equations utilized for modeling the S–N curves

As stated earlier, in order to model the S–N curves, the exponen-tial equation and the power law are used. Their generalized formsare shown by Eqs. (1) and (2), respectively.

ra ¼ A� B � logðNÞ� �c ð1Þ

logðraÞ ¼ A� B � logðNÞ� �c ð2Þ

In the Eqs. (1) and (2), A, B and C are constants that must be ob-tained while fitting the experimental data through the minimumsquare method, ra is the alternating stress applied (maximumstress minus minimum stress divided by two) and N is the averagenumber of cycles where the material failure occurs.When C = 1, theEqs. (3) and (4) are obtained, which are their restricted forms; theyare commonly employed by most researchers in the field[6,20,21,24–26].

ra ¼ A� B � logðNÞ ð3Þ

logðraÞ ¼ A� B � logðNÞ ð4Þ

The Tables 1–3 show the constants obtained for the materialsDD16, IM7/977 and T800/5245 of each equation presented above,and for each stress ratio examined as well as the total number ofsamples used for obtaining these constants.

3.2. The application of Weibull distribution for fatigue failure

The Eq. (5) shows the Weibull cumulative distribution func-tion with two parameters [27]. This equation is particularly usefulwhen it is desired to assess the probability of failure of a struc-ture or material whatsoever. In order to use such equation onthe fatigue behavior of a material, it is necessary to obtain theconstants a and b, which are the shape and scale parametersrespectively and, thereby, the failure probability (P) for a numberof failure cycles (N) is obtained, considering, of course, that otherparameters such as the stress ratio (R) and stress amplitude (ra)are constant.

PðNÞ ¼ 1� e � Nbð Þ

a� �

ð5Þ

In this paper, the probability of failure for each stress level offive or more samples is studied. The method for obtaining theparameters a and b will be the method of Thiel-Cacciari [19], whichis a non-parametric method that has the ability to estimate the val-ues of these constants in a robust way with reduced data samples(from 5 to 20).

The method consists in, from Eq. (6), obtaining the values of dis-crete cumulative probability and creating data pairs, between thenumber of cycles and its probability of failure, represented by{Ni, Pi}.

P ¼ i� 0:3TN þ 0:4

ð6Þ

In Eq. (6), i is the position of the checked element and TN is the totalnumber of elements. When obtaining the data pairs, it is possible toobtain a set of a values, from the Eq. (7). This set of a values is ob-tained by a combination, without repetition, among the compo-nents of the vector {Ni, Pi}, of total size M, wherein M is given byM = TN(TN � 1)/2.

am ¼ln ln 1

1�Pi

� �h i� ln lnð 1

1�PjÞ

h ilnðNiÞ � lnðNjÞ

ð7Þ

Sorting the a values in ascending order, one must choose thelargest integer k, so that k < p*(M + 1) (where p* is the pivotal per-centile which depends on the total number of samples and is de-scribed in Table 4) and from this position it is possible to findthe value of a with the following equation:

a ¼ ak þ ðakþ1 � akÞ �pkþ1 � p�

pkþ1 � pk

� �ð8Þ

The pk percentile formulation is shown by the followingequation:

pk ¼k

M þ 1ð9Þ

From the value of a, the bi values of each analyzed element areobtained, from the Eq. (10), and with this result, the average ofthese values is calculated through the Eq. (11), yielding the param-eter b.

bi ¼Ni

�lnð1� PiÞ½ �ð1=aÞð10Þ

b ¼PTN

i¼1bi

TNð11Þ

Tables 5–7 show the constant a and b for each analyzed le-vel of stress amplitude (ra) and the total number of test sam-ples (TN) of the materials DD16, IM7/977 and T800/5245respectively.

3.3. The probabilistic S–N curves

As mentioned in the introduction, the Weibull probability equa-tion must be combined to the S–N curves equation in order to de-crease the amount of testing and thereby obtain a probabilistic S–Ncurve.

To do so, it is necessary to consider the constant values of a andb for each S–N curve. For this, one should normalize the dataset bydividing the number of cycles of failure obtained experimentally(N) by the average number of cycles (N

�) obtained by Eqs. (1)–(4),

for each sample, obtaining a normalized value of number of cyclesNnor, as shown by the following equation:

Table 1Constants A, B and C, aR and bR for the exponential and power law probabilistic S–N curves (restricted and generalized), and the total number of test samples for each stress ratio(R) for DD16.

R Equation A B C aR bR Number of specimen

2 Generalized Exponencial 100.90 2.166 1.531 0.640 2.685 21Restricted Exponencial 104.51 5.970 � 0.459 3.688Generalized Power Law 2.006 0.009 1.695 0.538 3.094Restricted Power Law 2.019 0.030 � 0.379 4.00

10 Generalized Exponencial 183.79 19.185 0.858 1.240 1.568 49Restricted Exponencial 176.44 13.743 � 1.230 1.554Generalized Power Law 2.270 0.045 1.037 1.283 1.559Restricted Power Law 2.275 0.049 � 1.269 1.531

�2 Generalized Exponencial 320.22 11.884 1.520 1.160 1.593 32Restricted Exponencial 351.87 34.16 � 1.512 1.545Generalized Power Law 2.505 0.011 1.920 1.194 1.586Restricted Power Law 2.562 0.061 � 0.912 1.660

�1 Generalized Exponencial 407.06 34.543 1.194 1.540 1.411 27Restricted Exponencial 429.44 51.336 � 1.548 1.457Generalized Power Law 2.606 25 1.704 1.933 1.3131Restricted Power Law 2.661 0.079 � 0.951 1.529

�0.5 Generalized Exponencial 478.98 123.580 0.650 1.447 1.412 28Restricted Exponencial 409.92 57.690 � 1.184 1.364Generalized Power Law 2.690 0.108 1.050 1.974 1.312Restricted Power Law 2.691 0.115 � 1.714 1.238

0.1 Generalized Exponencial 288.80 59.524 0.697 1.312 1.468 93Restricted Exponencial 248.98 28.623 � 1.290 1.460Generalized Power Law 2.471 0.090 1.000 1.398 1.461Restricted Power Law 2.470 0.089 � 1.381 1.411

0.5 Generalized Exponencial 161.68 21.428 0.884 1.340 1.537 66Restricted Exponencial 154.67 16.349 � 1.325 1.522Generalized Power Law 1.988 0.023 1652 1.357 1.553Restricted Power Law 2.244 0.076 � 1.180 1.504




Table 2Constants A, B and C, aR and bR for the exponential and power law probabilistic S–N curves (restricted and generalized), and total number of test samples for each stress ratio (R)for IM7/977.


10 Generalized Exponencial 380.010 8.648 1.367 0.582 2.394 15Restricted Exponencial 388.390 17.985 — 0.532 2.792Generalized Power Law 257 0.0045 1.88 0.635 1.801Restricted Power Law 2.592 0.024 — 0.426 3.363

�1.5 Generalized Exponencial 625.310 1.474 2.629 0.649 1.543 16Restricted Exponencial 688.780 37.454 — 0.405 10.547Generalized Power Law 2.81 0.00067 3 0.741 1.184Restricted Power Law 2.844 0.029 1.000 0.359 9.922

�1 Generalized Exponencial 890.500 48.336 1.140 0.730 1.973 22Restricted Exponencial 910.790 64.691 — 0.723 2.032Generalized Power Law 2.95 0.020 1.36 0.764 1.957Restricted Power Law 2.971 0.040 — 0.680 2.049

�0.3 Generalized Exponencial 878.920 0.017 5.444 0.968 2.091 27Restricted Exponencial 1018 56.491 — 0.403 4.968Generalized Power Law 2.99 0.00089 3 0.968 2.207Restricted Power Law 3.002 0.027 — 0.318 5.74

584 R.C.S. Freire Júnior, A.S. Belísio / Composites: Part B 56 (2014) 582–590

Table 3Constants A, B and C, aR and bR for the exponential and power law probabilistic S–N curves (restricted and generalized), and total number of test samples for each stress ratio (R)for T800/5245.


10 Generalized Exponencial 394.200 11.896 1.371 0.785 2.092 26Restricted Exponencial 412.520 25.408 — 0.669 2.417Generalized Power Law 2.60 0.010 1.62 0.793 2.122Restricted Power Law 2.621 0.033 — 0.564 2.598

�0.3 Generalized Exponencial 1083.3 60.247 1.166 1.492 1.470 25Restricted Exponencial 1111.2 84.183 — 1.370 1.564Generalized Power Law 3.03 0.017 1.5 1.477 1.466Restricted Power Law 3.056 0.043 — 1.041 1.723

0.1 Generalized Exponencial 743.65 44.480 0.993 0.817 1.834 33Restricted Exponencial 742.60 43.736 — 0.818 1.830Generalized Power Law 2.864 0.017 1.31 0.804 1.832Restricted Power Law 2.885 0.034 — 0.744 1.971

0.5 Generalized Exponencial 415.980 14.461 0.867 0.414 4.503 17Restricted Exponencial 412.880 10.970 — 0.424 5.471Generalized Power Law 2.62 0.014 0.97 0.419 4.505Restricted Power Law 2.617 0.013 — 0.424 4.339

Table 4Percentile values of p� pivotal shape parameter a for different TNvalues.

TN M p�

5 10 0.45106 15 0.45097 21 0.45298 28 0.45519 36 0.4562

10 45 0.457111 55 0.458912 66 0.459613 78 0.461014 91 0.461615 105 0.462316 120 0.462517 136 0.463518 153 0.464519 171 0.464820 190 0.4652

Table 5Constants a and b for DD16 for each stress amplitude ra and the number of usedsamples.

R ra a ba Number of test specimen

2 69.00 0.670 712.1 986.25 1.08 36.51 5

10 93.60 1.718 1690.6 7109.35 0.961 40.756 5124.65 1.568 8.544 7

�2 155.25 2.587 518.63 10180.75 1.912 131.63 7207.00 7.641 23.91 5258.75 1.680 1.601 5

�1 179.00 4.802 46.225 5241.00 4.241 6.253 5276.00 2.570 1.533 6310.00 0.887 0.514 5

�0.5 155.25 2.177 16.517 5207.00 2.371 1.962 5258.75 1.570 0.441 5

0.1 92.70 1.731 696.05 793.15 1.204 379.87 14108.00 1.444 130.25 5108.45 1.640 104.26 13147.60 1.840 2.047 11186.30 1.638 0.401 17

0.5 60.25 1.008 889.78 1581.50 2.067 34.972 781.75 1.368 42.752 682.00 0.964 22.196 9103.5 1.543 1.847 8

0.7 41.40 3.161 266.26 551.75 1.688 26.668 562.10 1.378 5.698 5

0.8 27.60 1.224 1275.7 532.80 1.442 214.02 537.90 1.958 60.452 544.80 2.188 10.685 5

0.9 18.95 2.517 539.93 622.40 1.100 66.712 524.15 1.076 10.027 525.85 0.828 0.300 5

a The values given must be multiplied by 103.


Nnor ¼NN

ð12Þ

This way, one can rewrite Eq. (5) as shown by Eq. (13) where aR

and bR are the shape and scale parameters of each S–N curve,respectively (constant stress ratio). For this case, the values of aR

and bR are calculated using the method of Thiel–Cacciari [19] whenthe number of samples is equal to or less than 20 and for a numberof samples above 20 the method of maximum likelihood estima-tion will be used.

P ¼ 1� e� Nnor

bR

� �aRh i

ð13Þ

The method of maximum likelihood estimation of the parame-ters obtain the shape (a) and scale (b) parameters through the Eqs.(14) and (15), where Ni is the number of cycles that each samplewas submitted until its disruption.PTN

i¼1Nai � lnðNiÞPTN

i¼1Nai

� 1TN�XTN

i¼1

lnðNiÞ �1a¼ 0 ð14Þ

b ¼ 1TN�XTN

i¼1

Nai

" # 1að Þ

ð15Þ
By those findings, one can now replace Eq. (13) in Eqs. (1)–(4),thereby obtaining their probabilistic S–N curve, which is shownby the following equations:

Table 6Constants a and b for IM7/977 for each stress amplitude ra and the number of usedsamples.

R ra a ba Number of Test specimen

10 272.73 0.568 3007.3 5318.18 0.764 157.50 5

�1.5 498.74 0.727 474.12 5580.81 0.307 13.766 5

�1 595.96 0.845 92.38 6696.97 0.972 5.589 5

�0.3 650.00 1.714 546.44 5715.70 2.431 371.61 5778.05 1.194 113.73 6843.7 1.610 82.53 6


Table 7Constants a and b for T800/5245 for each stress amplitude ra and the number of usedsamples.

R ra a ba Number of test specimen

10 249.75 1.266 1535.6 6273.15 0.690 146.97 7317.25 1.201 32.67 8337.50 0.797 6.818 5

�0.3 715.39 0.817 101.20 5840.16 3.739 4.316 5

0.1 493.92 1.049 815.87 7540.23 0.981 182.08 7582.17 0.833 7.738 6627.08 1.145 0.754 5

0.5 348.34 0.550 949.94 5360.84 0.438 375.43 5373.98 0.431 48.244 5


100 101 102 103 104 105 106 107 1080

100

200

300

400

500

600

700

Stre

ss A

mpl

itude

(MPa

)

Number of Cycles

Experimental Data 1% S-N Curve of Probability 5% S-N Curve of Probability 1% Probability of Failure for each Stress Level 5% Probability of Failure for each Stress Level

Fig. 1. Comparison of the probabilistic S–N curves to the individual results at 1%and 5% probability of failure for T800/5245 with R = 0.1. The S–N curve used hereinis based on the restricted exponential equation (Eq. (18)).

100 101 102 103 104 105 106 107 1080

100

200

300

400

Stre

ss A

mpl

itude

(MPa

)

Number of Cycles

Experimental Data 1% S-N Curve of Probability 5% S-N Curve of Probability 1% Probability for each Stress Level 5% Probability for each Stress Level

Fig. 2. Comparison of the probabilistic S–N curves to individual results at 1% and 5%probability of failure for DD16 with R = �0.5. The S–N curve used herein is based onthe generalized exponential equation (Eq. (16)).

100 101 102 103 104 105 106 107 1080

100

200

300

400St

ress

Am

plitu

de (M

Pa)

Number of Cycles

1% Probability for each Stress Level 5% Probability for each Stress Level

1% S-N Curve of Probability 5% S-N Curve of Probability

Experimental Data

Fig. 3. Comparison of the probabilistic S–N curves to individual results at 1% and 5%probability of failure for DD16 with R = �0.5. The S–N curve used herein is based onthe restricted exponential equation (Eq. (18)).


ra ¼ A� B � logN

bR � �lnð1� PÞ½ �ð1=aRÞ

!" #c

ð16Þ

logðraÞ ¼ A� B � logN


!" #c

ð17Þ

ra ¼ A� B � logN


!ð18Þ

logðraÞ ¼ A� B � logN


!ð19Þ

It is important to mention that it is possible to incorporate theconstant bR to the A in Eqs. (18) and (19) (the restricted equations)thereby reducing the number of unknowns and causing theseequations to become similar to those obtained in the literature[15,16,24]. However, as this procedure cannot be performed forEqs. (16) and (17) (generalized equations) it was decided to keepsuch value in every analysis checked.

The values of A, B, C, aR e bR for each material and eachequation used for all the fatigue ratio studied are found inTables 1–3 for materials DD16, IM7/977 and T800/5245,respectively.

Table 8Root mean square error for 1%, 5% and 10% failure probability of generalized and restricted exponential equations for DD16, IM7/977 and T800/5245.

DD16a IM7-977a T800-5245a

1% 5% 10% 1% 5% 10% 1% 5% 10%

Generalized Exponential 41 20 13 69 69 43 49 35 33Restricted Exponential 51 30 20 520 280 175 62 33 27

a The values given must be multiplied by 10–4.

Fig. 4. Root mean square error for 1%, 5% and 10% probability of failure of bothgeneralized and restricted exponential equations for DD16, IM7/977 and T800/5245.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Prob

abilit

y S-

N C

urve

(log

(N))

Values obtained by equation 5 (log(N))

Fig. 5. Dispersion of the results obtained from the S–N curve (generalizedexponential) and the probabilities of each stress level for 1% of failure of DD16.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Prob

abilit

y S-

N C

urve

(log

(N))


Fig. 6. Dispersion of the results obtained from the S–N curve (generalizedexponential) and the probabilities of each stress level for 10% of failure of DD16.


4. Results and discussion

4.1. Probabilistic S–N curves based on exponential equations

Analyzing the results obtained from Eqs. (16) and (18), andcomparing them to the results of the individual probability of eachstress level, it was realized that the probabilistic S–N curves show,in most cases, conservative results for both generalized and re-stricted equations. An example can be seen in Fig. 1 for the mate-rial T800/5245 with R = 0.1 and analyzed with 1% and 5% ofprobability.

Despite the conservative results, a problem encountered in theconstruction of the probabilistic S–N curves with the exponentialequation was to obtain the stress amplitude values lower than orequal to zero for values of elevated numbers of cycles. Such resultdemonstrates that the exponential equations (both generalizedand restricted) cannot satisfactorily represent the fatigue behaviorof the composite materials for those cases. An example can be seen

in Figs. 2 and 3 for DD16 with R = �0.5 by applying the generalizedequation (Eq. (16)) and the restricted equation (Eq. (18)),respectively.

By the comparison of the results obtained for the two exponen-tial equations to the number of cycles obtained for each individualstress level, considering the failure probability of 1%, 5% and 10%(shown in Table 8 and Fig. 4), it is noticed that there is an increaseof the root mean square (RMS) error, with the decrease in the per-centage of the failure probability. This result demonstrates thatthese equations may not satisfactorily represent the resultantprobabilistic behavior of the material fatigue for low values ofprobability of failure, even though they show conservative results,as previously stated. The root mean square (RMS) error employedherein was calculated from Eq. (20) by using logarithm and nor-malized values.

RMS ¼ 12 � TN

�XTN

i¼1

logðNiÞlogð107Þ

� logðNiÞlogð107Þ

" #2

ð20Þ

In order to better demonstrate the decrease in the value accu-racy of exponential probabilistic S–N curves for small probabilitiesof failure, the example of Figs. 5 and 6 may be used. These figuresrepresent the dispersion of the results obtained by the probabilisticS–N curve (generalized exponential equation) and the probabilitiesof the individual stress level (Eq. (5)) for 1 and 10% of DD16,respectively. By those figures it can be noticed that the data is moredispersed in 1% than it is in 10% of the failure probability.

Another important aspect to be noticed, in Fig. 4 and Table 8, isthat for the probabilistic S–N curve based on a generalized expo-nential equation (Eq. (16)), the root mean squared errors aremostly smaller than those obtained by the restricted equation; thisresult is important because it demonstrates that the use of a gen-eralization represents the resultant probabilistic fatigue behaviorof the composite material in a better way.

100 101 102 103 104 105 106 107 1080

100

200

300

400

500

600

700

800

900

Stre

ss A

mpl

itude

(MPa

)

Number of Cycles


Fig. 7. Comparison of the probabilistic S–N curves to the individual results at 1%and 5% probability of failure for IM7/977 with R = �0.3. The S–N curve used hereinis based on the restricted exponential equation (Eq. (18)).

100 101 102 103 104 105 106 107 1080

100

200

300

400

Stre

ss A

mpl

itude

(MPa

)

Number of Cycles


Fig. 8. Comparison of the probabilistic S–N curves to the individual results at 1%and 5% probability of failure for DD16 with R = �2. The S–N curve used herein isbased on the generalized power law (Eq. (17)).

Stre

ss A

mpl

itude

(MPa

)

100

200

300

400

1000

101

Experimental Data1% S-N Curve of Probability5% S-N Curve of Probability1% Probability for each Stress Level5% Probability for each Stress Level

102

Number of Cycles103 104 105 106 107 108

Fig. 9. Comparison of the probabilistic S–N curves to the individual results at 1%and 5% probability of failure for DD16 with R = �2. The S–N curve used herein isbased on the restricted power law (Eq. (19)).

Table 9Root mean square error for 1%, 5% and 10% failure probability of generalized and restricted Power Laws for DD16, IM7/977 and T800/5245.

DD16a IM7-977a T800-5245a

1% 5% 10% 1% 5% 10% 1% 5% 10%

Generalized power law 47 23 15 73 83 53 47 34 32Restricted power law 110 68 44 740 468 316 102 60 43

a The values given must be multiplied by 10–4.


By analyzing the IM7/977 material specifically, unsatisfactoryresults were obtained, reinforcing that the restricted equationshould be done carefully. Although this equation presents conser-vative results, with values of number of cycles that are lower thanthose obtained in the analysis of individual stress rates, it may notrepresent the experimental results, as demonstrated in Fig. 7 forIM7/977 with R = �0.3.

An explanation for not yielding reliable results using the re-stricted exponential equation in the probabilistic analysis, in someinstances, occurs due to the fact that some of the S–N curves can-not be linearized on a semi-logarithm plane (which is represented

by the restricted exponential equation) without a considerable er-ror in respect to the experimental data. This result consequently af-fects the response probability, presented at the probability S–Ncurve. This can be verified by the variation of the constant C (whichis the value that differentiates the two exponential equations), forthe general exponential equation is much higher than 1 (as it canbe seen in Table 1), causing the restricted exponential equation(Eq. (18)) to not satisfactorily represent the data analyzed in thecase.

4.2. Probabilistic S–N curves based on the power law

Analyzing the results obtained by Eqs. (17) and (19) and com-paring to the results of probability of each individual stress level,it has been found that, as occurred herein for the exponential equa-tion, the S–N probability curves also show conservative results,with lower values of number of cycles than those obtained in theanalysis of individual stress rates; an example can be found forDD16 (R = �2) in Figs. 8 and 9 for the S–N curve based on the powerlaw and its general and restricted equations, respectively.

Analyzing the results, it has been found that there is an advan-tage by using Eqs. (17) and (19), because these equations do notshow stress amplitude values that are less than or equal to zero,thus, the S–N curve based on the power law better represents thefatigue behavior of the composite than the one based on the expo-nential equation.

When comparing the results of Eqs. (17) and (19), for the prob-abilities 1%, 5% and 10% (Table 9 and Fig. 10), it is noticed that thereis an increase in the root mean square error with the decrease inthe probability of failure. That also shows that these equations rep-resent higher percentages of failure probability in a better way, andthat, to lower percentages, these equations may overestimate the

Fig. 10. Root mean square error for 1%, 5% and 10% probability of failure ofgeneralized and restricted power laws for DD16, IM7/977 and T800/5245.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Prob

abilit

y S-

N C

urve

(log

(N))


Fig. 11. Dispersion of the results obtained from the S–N curve (generalized powerlaw) and the probabilities of each stress level for 1% of failure of DD16.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Prob

abilit

y S-

N C

urve

(log

(N))


Fig. 12. Dispersion of the results obtained from the S–N curve (restricted powerlaw) and the probabilities of each stress level for 1% of failure of DD16.

100 101 102 103 104 105 106 107 1080

100

200

300

400

500

600

700

800

900

1000

Stre

ss A

mpl

itude

(MPa

)

Number of Cycles


Fig. 13. Comparison of the probabilistic S–N curves to the individual results at 1%and 5% probability of failure for IM7/977 with R = �0.3. The S–N curve used hereinis based on the generalized power law (Eq. (17)).

100 101 102 103 104 105 106 107 1080

100

200

300

400

500

600

700

800

900

Stre

ss A

mpl

itude

(MPa

)

Number of Cycles


Fig. 14. Comparison of the probabilistic S–N curves to the individual results at 1%and 5% probability of failure for IM7/977 with R = �0.3. The S–N curve used hereinis based on the restricted power law (Eq. (19)).


analysis of the probabilistic fatigue behavior. The same is shown inthe exponential equation presented in the previous section.

Similarly to what happened to the exponential equation, thegeneralized power law showed higher correlation coefficients thanits restricted equation; an example of this phenomenon can be ob-served in the dispersion graphs shown in Figs. 11 and 12 for DD16

with 1% of failure for the general and restricted power laws,respectively.

It is also observed in Table 9 and Fig. 10 that, for IM7/977 mod-eled with Eq. (19) (restricted power law), unsatisfactory results arepresented, unlike what was obtained by the general power law (Eq.(17)). In order to better illustrate this result, the Figs. 13 and 14 canbe observed, which compare the S–N curve based on the general-ized power law (Eq. (17)) to its restricted equation (Eq. (19)) withthe probability for each stress level, respectively, for IM7/977 withR = �0.3.

5. Conclusions

Based on the presented results, it can be concluded that all theprobabilistic S–N curves show conservative results for the majorityof the studied cases, when compared to the probabilities of thenumber of cycles obtained for each individual stress level.

As it turned out, a representation of the probabilistic behaviorto fatigue through a linear equation on the plane log–log (powerlaw restricted) or on the plane semi-log (restricted equation expo-nential) may show an unsatisfactory response; this is particularly


true for IM7/977, where shows unsatisfactory results in the re-stricted equations relative to that, obtained for each stress.

According to the results, the probabilistic S–N curves based onthe exponential equation should be cautiously used because, as itwas previously stated, it may not represent the resultant probabi-listic fatigue behavior of the composite, presenting inconsistentvalues (negative stress amplitude).

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Probabilistic S–N curves using exponential and power laws equations

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