Mallor et al 2020b - Revision history

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 [1] Cervello S. Fatigue properties of railway axles: New results of full-scale specimens from Euraxles project. Int J Fatigue, 86:2–12, 2016. [https://doi.org/10.1016/j.ijfatigue.2015.11.028. https://doi.org/10.1016/j.ijfatigue.2015.11.028.]
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-[15] Bea JA, Doblaré M, Gracia L. Fiabilidad de elementos metálicos en crecimiento de grieta por fatiga aleatoria mediante elementos finitos probabilistas y modelos B. Rev Int Métod Numér Para Cálculo Diseño En Ing 1999;15(1):85–112. [https://www.scipedia.com/public/Bea_et_al_1999a. https://www.scipedia.com/public/Bea_et_al_1999a.]
+[15] Bea J.A., Doblaré M., Gracia L. Fiabilidad de elementos metálicos en crecimiento de grieta por fatiga aleatoria mediante elementos finitos probabilistas y modelos B. Rev Int Métod Numér para Cálculo Diseño en Ing, 15(1):85–112, 1999. [https://www.scipedia.com/public/Bea_et_al_1999a. https://www.scipedia.com/public/Bea_et_al_1999a.]
 [16] Hillmansen S, Smith RA. The management of fatigue crack growth in railway axles. Proc Inst Mech Eng Part F J Rail Rapid Transit 2004;218:327–336. [https://doi.org/10.1243/0954409043125879. https://doi.org/10.1243/0954409043125879.]

@@ Line 308: / Line 308: @@
 The variety of solutions differ in the values of the coefficients <math display="inline">a</math>, <math display="inline">{b}_{0}</math>, <math display="inline">{b}_{1}</math> and <math display="inline">{b}_{2}</math>. These solutions are classified into different Pearson distribution types depending on the quantities <math display="inline">{\beta }_{1}</math> and <math display="inline">{\beta }_{2}</math>. In other words, the type of distribution to which the data belong is entirely determined by the square of the skewness <math display="inline">{\beta }_{1}</math> presented in Eq. <span id='cite-_Ref30513951'></span>[[#_Ref30513951|(2)]] and the kurtosis <math display="inline">{\beta }_{2}</math> expressed in Eq. <span id='cite-_Ref30514082'></span>[[#_Ref30514082|(3)]]. The Pearson distribution types correspond to common probability distributions. The following types and their common distribution associated arise: type I (beta distribution), type II (symmetrical beta distribution), type III (gamma distribution), type IV (Cauchy distribution), type V (inverse-gamma distribution), type VI (beta prime distribution), type VII (Student's tdistribution) and the limit of type I, III, IV, V (normal distribution).
-Once the Pearson distribution type is determined, the two, three or four parameters of the particular distribution type can be calculated as a function of the expected value, variance, skewness and kurtosis, i.e. from the first four moments. The formulas to calculate the parameters for each type of Pearson distribution are given in [34]. As an example, the formulas for the Pearson distribution type I that corresponds to the beta distribution are included here. A random variable <math display="inline">x</math> following a beta distribution <math display="inline">Beta\left( \alpha ,\beta \right)</math>  with the shape parameters <math display="inline">\alpha</math>  and <math display="inline">\beta</math> , can be further parametrized as <math display="inline">\frac{x-\lambda }{{a}_{2}-{a}_{1}}</math> with the scale parameter <math display="inline">\left( {a}_{2}-\right. </math><math>\left. {a}_{1}\right)</math>  and with a location parameter <math display="inline">\lambda</math> . The shape parameters <math display="inline">\alpha \,</math> and <math display="inline">\beta</math>  are calculated according to the Pearson distribution family as expressed in Eq. <span id='cite-_Ref31624338'></span>[[#_Ref31624338|(11)]]
+Once the Pearson distribution type is determined, the two, three or four parameters of the particular distribution type can be calculated as a function of the expected value, variance, skewness and kurtosis, i.e. from the first four moments. The formulas to calculate the parameters for each type of Pearson distribution are given in [34]. As an example, the formulas for the Pearson distribution type I that corresponds to the beta distribution are included here. A random variable <math display="inline">x</math> following a beta distribution <math display="inline">Beta\left( \alpha ,\beta \right)</math>  with the shape parameters <math display="inline">\alpha</math>  and <math display="inline">\beta</math>, can be further parametrized as <math display="inline">\frac{x-\lambda }{{a}_{2}-{a}_{1}}</math> with the scale parameter <math display="inline">\left( {a}_{2}-\right. </math><math>\left. {a}_{1}\right)</math>  and with a location parameter <math display="inline">\lambda</math>. The shape parameters <math display="inline">\alpha \,</math> and <math display="inline">\beta</math>  are calculated according to the Pearson distribution family as expressed in Eq. <span id='cite-_Ref31624338'></span>[[#_Ref31624338|(11)]]
 {| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
@@ Line 542: / Line 542: @@
 In case (ii), the probability density function for the lifespan <math display="inline">N</math> is assumed to be log-normally distributed, and therefore it is derived from the standardized form for the log-normal probability density function <math display="inline">f\left( x;\sigma \right)</math>  with <math display="inline">\sigma</math>  as a shape parameter. Commonly, the probability density function is further parameterized in terms of a location parameter <math display="inline">\lambda</math>  and a scale parameter <math display="inline">{e}^{\mu }</math>. The location parameter <math display="inline">\lambda</math>  was set equal to zero, hence only two parameters remain to be estimated. The physical meaning of the location parameter is realistic since it indicates that crack growth occurs after any given cycle and so the support of the distribution is nonnegative. The shape <math display="inline">\sigma</math>  and the scale <math display="inline">{e}^{\mu }</math> parameters of the log-normal are functions of the expected value and the variance of the fatigue life <math display="inline">N</math>. As a consequence, the lifespan <math display="inline">N</math> distributed log-normally is described by the probability density function <math display="inline">{f}_{N}\left( N;\sigma ,\lambda ,{e}^{\mu }\right)</math>  where <math display="inline">\sigma</math>, <math display="inline">\lambda</math>  and <math display="inline">{e}^{\mu }</math> are fixed parameters.
-In case (iii), the Pearson distribution type was automatically determined based on the quantities <math display="inline">{{\beta }_{1}}_{N}=</math><math>{{\gamma }_{1}^{2}}_{N}</math> and <math display="inline">{{\beta }_{2}}_{N}</math> as it is described in Section <span id='cite-_Ref31636819'></span>[[#_Ref31636819|2.2]]. In this example, the procedure leads to the Pearson type I that corresponds to the beta distribution. Afterwards, the parameters of the beta distribution were estimated making the statistical moments of the constructed lifespan distribution match the first four prescribed moments predicted by the Pr. Eq. In this case, the probability density function for the lifespan <math display="inline">N</math> beta-distributed is derived from the standardized form for the beta probability density function <math display="inline">f\left( x;\alpha ,\beta \right)</math>  in Eq. <span id='cite-_Ref31636896'></span>[[#_Ref31636896|(14)]] with <math display="inline">\alpha</math> , <math display="inline">\beta</math>  as shape parameters and where <math display="inline">\, \Gamma</math>  is the gamma function <math display="inline">\Gamma \left( x\right) =</math><math>\left( x-1\right) !</math>. Once again, the probability density function is further parameterized by introducing two parameters representing the location and the scale. The shape parameters <math display="inline">\alpha</math>  and <math display="inline">\beta</math>, the scale and the location parameter <math display="inline">\lambda</math>  were calculated by using the closedform expressions derived in Section <span id='cite-_Ref31636819'></span>[[#_Ref31636819|2.2]] that express the parameters as function of the first four moments of the fatigue life <math display="inline">N</math> distribution. Recall that in these expressions the expected value <math display="inline">{\mu }_{N}</math> or first raw moment is denoted as <math display="inline">{\mu '}_{1}</math>. As a result, the lifespan <math display="inline">N</math> following a beta distribution is characterized by the probability density function <math display="inline">{f}_{N}\left( N;\alpha ,\beta ,\lambda ,{a}_{2}-\right. </math><math>\left. {a}_{1}\right)</math>  given in Eq. <span id='cite-_Ref39844746'></span>[[#_Ref39844746|(15)]]
+In case (iii), the Pearson distribution type was automatically determined based on the quantities <math display="inline">{{\beta }_{1}}_{N}=</math><math>{{\gamma }_{1}^{2}}_{N}</math> and <math display="inline">{{\beta }_{2}}_{N}</math> as it is described in Section <span id='cite-_Ref31636819'></span>[[#_Ref31636819|2.2]]. In this example, the procedure leads to the Pearson type I that corresponds to the beta distribution. Afterwards, the parameters of the beta distribution were estimated making the statistical moments of the constructed lifespan distribution match the first four prescribed moments predicted by the Pr. Eq. In this case, the probability density function for the lifespan <math display="inline">N</math> beta-distributed is derived from the standardized form for the beta probability density function <math display="inline">f\left( x;\alpha ,\beta \right)</math>  in Eq. <span id='cite-_Ref31636896'></span>[[#_Ref31636896|(14)]] with <math display="inline">\alpha</math>, <math display="inline">\beta</math>  as shape parameters and where <math display="inline">\, \Gamma</math>  is the gamma function <math display="inline">\Gamma \left( x\right) =</math><math>\left( x-1\right) !</math>. Once again, the probability density function is further parameterized by introducing two parameters representing the location and the scale. The shape parameters <math display="inline">\alpha</math>  and <math display="inline">\beta</math>, the scale and the location parameter <math display="inline">\lambda</math>  were calculated by using the closedform expressions derived in Section <span id='cite-_Ref31636819'></span>[[#_Ref31636819|2.2]] that express the parameters as function of the first four moments of the fatigue life <math display="inline">N</math> distribution. Recall that in these expressions the expected value <math display="inline">{\mu }_{N}</math> or first raw moment is denoted as <math display="inline">{\mu '}_{1}</math>. As a result, the lifespan <math display="inline">N</math> following a beta distribution is characterized by the probability density function <math display="inline">{f}_{N}\left( N;\alpha ,\beta ,\lambda ,{a}_{2}-\right. </math><math>\left. {a}_{1}\right)</math>  given in Eq. <span id='cite-_Ref39844746'></span>[[#_Ref39844746|(15)]]
 {| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
@@ Line 634: / Line 634: @@
 The authors acknowledge the Spanish Ministry of Economy, Industry and Competitive through the National Programme for Research Aimed at the Challenges of Society that financially supported the project RTC-2016-4813-4.
 ==References==
-[1] Cervello S. Fatigue properties of railway axles: New results of full-scale specimens from Euraxles project. Int J Fatigue 2016;86:2–12. [https://doi.org/10.1016/j.ijfatigue.2015.11.028. https://doi.org/10.1016/j.ijfatigue.2015.11.028.]
+[1] Cervello S. Fatigue properties of railway axles: New results of full-scale specimens from Euraxles project. Int J Fatigue, 86:2–12, 2016. [https://doi.org/10.1016/j.ijfatigue.2015.11.028. https://doi.org/10.1016/j.ijfatigue.2015.11.028.]
 [2] Novosad M, Fajkoš R, Řeha B, Řezníček R. Fatigue tests of railway axles. Procedia Eng 2010;2:2259–68. [https://doi.org/10.1016/j.proeng.2010.03.242. https://doi.org/10.1016/j.proeng.2010.03.242.]
@@ Line 710: / Line 710: @@
 [36] Haldar A, Mahadevan S. Probability, reliability, and statistical methods in engineering design. John Wiley; 2000.

@@ Line 583: / Line 583: @@
 |  style="text-align: center;"|<span style="text-align: center; ">394 964 </span> <math display="inline">\left( = {e}^{\mu }\right)</math>
 |-style="text-align:center"
-|  rowspan="2" style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; ">Pearson type I (Beta)</span>
+|  rowspan="2" style="text-align: center;"|<span style="text-align: center; ">Pearson type I (Beta)</span>
 |  style="text-align: center;"|<span style="text-align: center; ">8.49 </span> <math display="inline">\left( = \alpha \right)</math><span style="text-align: center; ">,</span>
 |  style="text-align: center;"|<span style="text-align: center; ">232 076 </span>
@@ Line 597: / Line 597: @@
 <div id="_Ref31637565"></div>
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
 |style="padding:10px;"| [[File:Review_912708961870_3760_FIG5.png|600px|frameless|center|]]

@@ Line 433: / Line 433: @@
 | <span style="text-align: center; ">-6.47%</span>
 |-style="text-align:center"
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">50</span>
+| <span style="text-align: center; ">50</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">0.689</span>
+|  <span style="text-align: center; ">0.689</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">0.646</span>
+|  <span style="text-align: center; ">0.646</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">-6.24%</span>
+| <span style="text-align: center; ">-6.24%</span>
 |}
@@ Line 471: / Line 471: @@
 |  [%]
 |-style="text-align:center"
-|  style="border-top: 1pt solid black;"|<span style="text-align: center; ">5</span>
+|  <span style="text-align: center; ">5</span>
-|  style="border-top: 1pt solid black;"|<span style="text-align: center; ">4.33</span>
+|  <span style="text-align: center; ">4.33</span>
-|  style="border-top: 1pt solid black;"|<span style="text-align: center; ">3.84</span>
+|  <span style="text-align: center; ">3.84</span>
-|  style="border-top: 1pt solid black;"|<span style="text-align: center; ">-11.33%</span>
+|  <span style="text-align: center; ">-11.33%</span>
 |-style="text-align:center"
 | <span style="text-align: center; ">10</span>
@@ Line 496: / Line 496: @@
 | <span style="text-align: center; ">-7.47%</span>
 |-style="text-align:center"
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">50</span>
+|  <span style="text-align: center; ">50</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">3.88</span>
+|  <span style="text-align: center; ">3.88</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">3.60</span>
+|  <span style="text-align: center; ">3.60</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">-7.22%</span>
+|  <span style="text-align: center; ">-7.22%</span>
 |}
@@ Line 520: / Line 520: @@
 | <math>{{\beta }_{2}}_{N}</math>
 |-style="text-align:center"
-|  style="border-bottom: 1pt solid black;"|[mm]
+|  [mm]
-|  style="border-bottom: 1pt solid black;"|[Cycles]
+|  [Cycles]
-|  style="border-bottom: 1pt solid black;"|[Cycles]
+| [Cycles]
-|  style="border-bottom: 1pt solid black;"|[dimensionless]
+|  [dimensionless]
-|  style="border-bottom: 1pt solid black;"|[dimensionless]
+| [dimensionless]
 |-style="text-align:center"
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center;">50</span>
+|  <span style="text-align: center;">50</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; >398 853</span>
+|  <span style="text-align: center; >398 853</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">56 103</span>
+|  <span style="text-align: center; ">56 103</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">0.646</span>
+|  <span style="text-align: center; ">0.646</span>
-|  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">3.60</span>
+| <span style="text-align: center; ">3.60</span>
 |}
@@ Line 573: / Line 573: @@
 ! Probability Distr. !! Shape !! Location !! Scale
 |-style="text-align:center"
-|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; ">Normal</span>
+|  text-align: center;"|<span style="text-align: center; ">Normal</span>
-|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; ">-</span>
+|  text-align: center;"|<span style="text-align: center; ">-</span>
-|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; ">398 853 </span> <math display="inline">\left( = {\mu }_{N}\right)</math>
+|  text-align: center;"|<span style="text-align: center; ">398 853 </span> <math display="inline">\left( = {\mu }_{N}\right)</math>
-|  style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; ">56 103 </span> <math display="inline">\left( {=\sigma }_{N}\right)</math>
+|  style="text-align: center;"|<span style="text-align: center; ">56 103 </span> <math display="inline">\left( {=\sigma }_{N}\right)</math>
 |-style="text-align:center"
 |  style="text-align: center;"|<span style="text-align: center; ">Log-normal</span>
@@ Line 588: / Line 588: @@
 |  style="text-align: center;"|<span style="text-align: center; ">4 825 029 </span>
 |-style="text-align:center"
-|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; ">237.33 </span> <math display="inline">\left( = \beta \right)</math>
+|  style="text-align: center;"|<span style="text-align: center; ">237.33 </span> <math display="inline">\left( = \beta \right)</math>
-|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; "> </span> <math display="inline">\left( = \lambda \right)</math>
+|  style="text-align: center;"|<span style="text-align: center; "> </span> <math display="inline">\left( = \lambda \right)</math>
-|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; "> </span> <math display="inline">\left( {=a}_{2}-{a}_{1}\right)</math>
+|  style="text-align: center;"|<span style="text-align: center; "> </span> <math display="inline">\left( {=a}_{2}-{a}_{1}\right)</math>
 |}

@@ Line 512: / Line 512: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-! Pr. Eq. !! Pr. Eq. !! Pr. Eq. !! Pr. Eq.
+! !! Pr. Eq. !! Pr. Eq. !! Pr. Eq. !! Pr. Eq.
 |-style="text-align:center"
 | <math>a</math>

@@ Line 475: / Line 475: @@
 |  style="border-top: 1pt solid black;"|<span style="text-align: center; ">3.84</span>
 |  style="border-top: 1pt solid black;"|<span style="text-align: center; ">-11.33%</span>
-|-
+|-style="text-align:center"
 | <span style="text-align: center; ">10</span>
 | <span style="text-align: center; ">4.13</span>
 | <span style="text-align: center; ">3.73</span>
 | <span style="text-align: center; ">-9.53%</span>
-|-
+|-style="text-align:center"
 | <span style="text-align: center; ">20</span>
 | <span style="text-align: center; ">4.00</span>
 | <span style="text-align: center; ">3.66</span>
 | <span style="text-align: center; ">-8.38%</span>
-|-
+|-style="text-align:center"
 | <span style="text-align: center; ">30</span>
 | <span style="text-align: center; ">3.94</span>
 | <span style="text-align: center; ">3.63</span>
 | <span style="text-align: center; ">-7.83%</span>
-|-
+|-style="text-align:center"
 | <span style="text-align: center; ">40</span>
 | <span style="text-align: center; ">3.90</span>
 | <span style="text-align: center; ">3.61</span>
 | <span style="text-align: center; ">-7.47%</span>
-|-
+|-style="text-align:center"
 |  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">50</span>
 |  style="border-bottom: 1pt solid black;"|<span style="text-align: center; ">3.88</span>
@@ Line 505: / Line 505: @@
 Once again, the kurtosis provided by the MC simulations is considered as the framework of reference. It is observed that the larger the crack the lower kurtosis and, in both calculations, a stabilization in magnitude is reached. The error in the probabilistic equation ranges from 11% to 7%, decreasing as the crack depth increases. The maximum difference of the Pr. Eq. calculates a kurtosis about 0.5 smaller.
-At this point, the first part of the methodology presented in section <span id='cite-_Ref31790969'></span>[[#_Ref31790969|2.1]] is completed. As a result, the first to fourth moments of the underlying lifespan probability distribution are available after applying the full second-order approach. The expected value, first raw moment, and the variance, second central moment are taken from [25], whereas the skewness and the kurtosis, third and fourth central standardized moments, respectively, have been calculated above. Recapitulating, the moments of fatigue life <math display="inline">N</math> provided by the probabilistic NASGRO equations for a crack depth equal to 50 mm are enclosed in <span id='cite-_Ref31629726'></span>[[#_Ref31629726|Table 3]].
+At this point, the first part of the methodology presented in Section <span id='cite-_Ref31790969'></span>[[#_Ref31790969|2.1]] is completed. As a result, the first to fourth moments of the underlying lifespan probability distribution are available after applying the full second-order approach. The expected value, first raw moment, and the variance, second central moment are taken from [25], whereas the skewness and the kurtosis, third and fourth central standardized moments, respectively, have been calculated above. Recapitulating, the moments of fatigue life <math display="inline">N</math> provided by the probabilistic NASGRO equations for a crack depth equal to 50 mm are enclosed in <span id='cite-_Ref31629726'></span>[[#_Ref31629726|Table 3]].