Stochastic Inverse Analysis of Fatigue Cracks Based on Linear Fracture Mechanics

Keywords: Fatigue, fracture, mechanics, Paris-Erdogan, 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 (S-N 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 (S-N) 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 life-span of each structural detail and critical connection can be predicted using the standard Paris-Erdogan 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 Paris-Erdogan (material-related) 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 a₀ to a_cr. The quasi–constant stress range Ds = 50 MPa is considered. C, m are material constants according (6) (4) where c₁, c₂ can be considered for steel grade S235 as c₁ = -11.141 and c₂ = -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 a₀, W=400, a_cr=175 An example of the dependence between N_F and a₀ is shown in Fig.1. With regards to the strongly non-linear dependence between N_F and a₀, it is more practical to work with the logarithms of these variables. An example of the dependence between logarithms N_F and a₀ is shown in Fig.2. Fig.2: ln(N_F) vs ln(a₀) 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 a₀ has a log-normal probability density function (