ZDENĚK KALA
Department of Structural Mechanics
Brno University of Technology, Faculty of Civil Engineering
Veveří Str. 95, Brno
CZECH REPUBLIC
kala.z@fce.vutbr.cz; http://www.vutbr.cz
Abstract:  For steel structures and bridges subjected to fatigue loading, it is possible to determine the probabilities for basic phenomena that are related to the growth of fatigue cracks. Emerging new methods of probabilistic reliability assessment consider the effects of possible defects in the form of initiation cracks, which are the main cause of the propagation of fatigue cracks. The strongly nonlinear dependence between the initial crack size and fatigue resistance can lead to unrealistic probabilistic models if the types of probability density functions are selected inappropriately. The article discusses the uncertainties present in determining the variables in the calculation, which must be logically related to the probabilistic model of fatigue resistance. The aim of the present paper is to provide a methodology of inverse stochastic analysis, which is suitable for the verification of probabilistic models of fatigue crack propagation.
KeyWords:  Fatigue, fracture, mechanics, ParisErdogan, crack, stochastic, skewness, steel, bridge, structure
1 Introduction
In Eurocode 3, the widespread method of design and assessment of fatigue is based on the detail category specified Wöhler curve (SN curve) and the Palmgren–Miner cumulative damage rule [1]. Wöhler curves permits a limited lifetime to failure, which is problematically determined, based on constant amplitude and the expected number of load cycles. The methodology has been gradually developed into procedures that describe real conditions and facilitate the work of designers; however, it is not sufficiently universal [2]. Wöhler curves are only available for selected structural details given by the classification tables in design codes, such as Eurocode 3 [3], BS 5400 [4] and AASHTO [5].
The traditional Wöhler (SN) method cannot be used to determine the effect of a specific defect on the fatigue life. Linear elastic fracture mechanics presents a tool for the analysis of fatigue crack propagation of numerous cracked structural details [6, 7]. Approaches based on linear elastic fracture mechanics provide information on crack size and the growth rate of cracks under actual service loads [2].
An important input quantity for the analysis of fatigue degradation is stress history, which can be generated using deformation measurements in combination with FE models [8, 9]. The fatigue crack propagation lifespan of each structural detail and critical connection can be predicted using the standard ParisErdogan crack growth model [10]. The prediction of the lifetime of fatigue cracks requires stochastic models that consider the uncertainty of all parameters, which by their nature are random variables, see e.g. [11, 12]. Monte Carlo numerical simulation methods are effective, but are not the only tools for the analysis of fatigue degradation and lifetime of structural steel constructions and steel bridges, see e.g. [13, 14]. Results of probabilistic studies are mainly used to determine inspection times and to analyse their results, which in the absence of cracks, lead to the conditional probability of their occurrence.
2 Linear Fracture Mechanics
Linear fracture mechanics has been the subject of research for many years, especially in the field of mechanical engineering and is gradually being applied and modified for the design of load bearing building structures. Commonly applied linear elastic fracture mechanics analyses the propagation of an initial crack of magnitude a in dependence on the number of fatigue cycles N. Fatigue crack growth is generally described by Paris’s rule which is expressed by Paris and Erdogan [10].
(1)
where m and C are ParisErdogan (materialrelated) law parameters and the range of stress intensity factor ΔK can be determined by [15].
(2)
where F(a) is the geometric factor (calibration function) describing the course of crack propagation with respect to the geometry of the sample and Δs is the quasi–constant stress range.
(3)
where N_{F} is the total number of cycles at crack growth from a_{0} to a_{cr}. The quasi–constant stress range Ds = 50 MPa is considered. C, m are material constants according (6)
(4)
where c_{1}, c_{2} can be considered for steel grade S235 as c_{1} = 11.141 and c_{2} = 0.507 [16]. F(a) is the calibration function evaluated for pure bending in the form [17]:
where a is crack length and W is specimen width in the direction of crack propagation.
Fig.1: Fatigue resistance N_{F} vs a_{0}, W=400, a_{cr}=175
An example of the dependence between N_{F} and a_{0} is shown in Fig.1. With regards to the strongly nonlinear dependence between N_{F} and a_{0}, it is more practical to work with the logarithms of these variables. An example of the dependence between logarithms N_{F} and a_{0} is shown in Fig.2.
Fig.2: ln(N_{F}) vs ln(a_{0}) for W=400, a_{cr}=175
3 Probabilistic Analysis
The input random variables of the probabilistic model are listed in the Table 1. Initial crack size a_{0} has a lognormal probability density function (pdf), the other random variables have Gauss pdf. The Latin Hypercube Sampling (LHS) method [18, 19] based on repeated random sampling is used to obtain the numerical results.
Table 1: Input random variables
Random variables 
Mean value_{} 
Standard Deviation 

Initial crack size 
a_{0} _{} 
0.526 mm 
0.504 mm 
Critical crack size 
a_{cr} _{} 
175 mm 
14 mm 
Specimen width 
W 
400 mm 
20 mm 
Parameter 
m 
3 
0.03 
The fatigue resistance N_{F} is the output random variable, whose statistical characteristics and pdf are examined. The mean value of N_{F} is m_{NF}=16.71E6 and standard deviation is m_{NF}=7.62E6, which are the statistical results, obtained using one million runs of the LHS method. The Chisquare goodnessoffit test does not reject the hypothesis that N_{F} has a lognormal pdf. Practically, it means that the data fit the lognormal pdf very well, but it does not necessarily imply a hundred percent fit. If N_{F} really has a lognormal pdf, then we can use this pdf (with parameters m_{NF},m_{NF}) to simulate the random realizations of N_{F} and subsequently use inverse analysis to obtain random variable a_{0}, which has statistical characteristics listed in Table 1. It may be added that the abovedescribed statistical model of initial crack size a_{0} has skewness of 3.7, which is a parameter that will also be monitored.
Let us try to study a_{0} using inverse analysis. Let us consider N_{F} as a random variable whose mean value and standard deviation are listed above with theoretical consideration of several different types of parametric pdfs.
The aim of the study is to perform an inverse analysis and obtain the histogram of a_{0}, whose pdf will be subsequently examined. The theoretically presumed pdfs introduced for N_{F} are listed in Table 2 and are also depicted in Fig.3 and Fig.4. The second chosen pdf type is the Hermite pdf, which has four parameters [20]. The third and fourth parameters of this pdf are skewness and kurtosis, which are considered to have values of 0.6 and 3.
Table 2: Variants for N_{F} random pdf
Var. 

Mean Value_{} 
Standard Deviation 
1 
Lognormal 
16.71E6 
7.62E6 
2 
Hermite 
16.71E6 
7.62E6 
3 
Truncated Gauss 
16.71E6 
7.62E6 
4 
Decreasing Triangular 
16.71E6 
7.62E6 
5 
Growing Triangular 
21.71E6 
7.62E6 
Statistical analysis of a_{0} is evaluated using ten thousand simulation runs of the LHS method. The results are shown in Fig. 5 to Fig. 14.
Fig.3: Pdfs of N_{F}  variants 1, 2, 3
Fig.4: Pdfs of N_{F}  variants 4, 5
Fig.5: Observations of a_{0} for Variant 1
Fig.6: Histogram of a_{0} for Variant 1
Fig.7: Observations of a_{0} for Variant 2
Fig.8: Histogram of a_{0} for Variant 2
Fig.9: Observations of a_{0} for Variant 3
Fig.10: Histogram of a_{0} for Variant 3
Fig.11: Observations of a_{0} for Variant 4
Fig.12: Histogram of a_{0} for Variant 4
Fig.13: Observations of a_{0} for Variant 5
Fig.14: Histogram of a_{0} for Variant 5
From the graphs shown above, it can be observed that the selection of inappropriate types of probability density functions of fatigue resistance lead to remote observations of the size of the initial crack. Observations of the initial crack must be proportionate to the frequency of the occurrence of real failures, which are observed during inspections of steel structures or bridges subjected to cyclical loads, see e.g. [12, 21, 22].
4 Conclusion
Inverse analysis a_{0} is an important part of the verification of stochastic models because it provides information on crack propagation, which is needed to plan regular inspections. The inverse analysis performed in this article showed that if the fatigue resistance N_{F} has a lognormal pdf then a_{0} has a lognormal pdf. However, the statistical characteristics a_{0} correspond to the original values listed in Table 1 only approximately. Even so, the lognormal pdf is, out of all the pdf types introduced for N_{F}, the most suitable distribution that can be accepted with a high probability in stochastic models of structural elements subjected to fatigue damage. The unsuitable pdfs include Hermite, Truncated Gauss and Growing Triangular pdfs, which lead to the occurrence of large (unreal) frequencies of observations of high values of a_{0}. The Decreasing Triangular pdf, whose random realization maximum of a_{0} determined in this article has a value of 4.8mm, is also worth considering in probabilistic studies. Practically, it is necessary to adopt for a_{0} such a pdf, whose probability density function is zero in the vicinity of zero values of a_{0} and at the same time decreases very rapidly when observing higher values. The lognormal pdf satisfies these requirements, but is not necessarily the only suitable pdf.
Acknowledgement
The article was elaborated within the framework of project GAČR 1701589S.
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