Issue 39

M. Krejsa et alii, Frattura ed Integrità Strutturale, 39 (2017) 143-159; DOI: 10.3221/IGF-ESIS.39.15 144 detection of cracks from measurement made during the operational period of the bridge cannot be incorporated into reliability calculations [2, 24]. The essential tools for these calculations are provided by fracture mechanics and the reliability theory used in a probabilistic framework, e.g. [11, 13]. Linear elastic fracture mechanics – LEFM, is fully sufficient for fatigue crack propagation (LEFM is demonstrated to be a powerful tool to facilitate fatigue assessment due to the fact that initial cracks in real structures are unavoidable) on the condition of small scale yielding before the crack tip [6]. The initiation position of a fatigue crack is often found at an inclusion, impurity or surface flaw, which acts as a local stress raiser resulting in small scale plastic deformation. Analysis of the fatigue life based on linear elastic fracture mechanics requires that the random nature of production, crack growth, applied load and the subsequent failure due to cracking are properly taken into account. A prerequisite, however, is sufficient database of input random variables [4, 5 and 33]. Common studies of the misalignment of failure probability P f (or the reliability index  ) over time tend to focus only on structural details, however, a comprehensive probabilistic methodology generally applicable to bridge structures is currently missing. Insight into the stochastic interactions among random factors (load, geometric and material characteristics [8, 25, 27 and 28]) affecting the reliability of steel bridges is absolutely crucial to understanding the misalignment of failure probability of steel bridges [1]. Numerous stochastic methods, which enable the determination of failure probability, respectively the reliability index, have been developed [16, 22]. A substantial part of these methods are based on the crude Monte Carlo simulation method (MC), whose disadvantage is poor efficiency due to the need of a high number of simulation steps [26]. Advanced and stratified simulation methods, for e.g. Markov chain Monte Carlo simulations (MCMC), also applied in fatigue damage prognosis, strive to increase the efficiency of these computational methods [35]. An alternative solution is the evaluation of reliability using the reliability index, which can be efficiently determined using the Latin Hypercube Sampling method (LHS). The LHS method is capable of evaluating the reliability index from a small number of simulation steps (hundreds to thousands) [12]. Stochastic methods denoted as approximation methods enable probabilistic assessment of reliability analytically. Another category of stochastic approaches for the quantification of the reliability of structures are represented by methods, which determine the failure probability on the basis of numerical integration, for e.g. the Direct Optimized Probabilistic Calculation method – DOProC, which was comprehensively published, e.g. in [9, 20]. It appears that this method is very effective for many probabilistic problems. It is also distinguished by higher accuracy than that of simulation methods, resp. approximation ones. This calculation method was applied in the solution of some engineering problems, among others, the assessment of reliability of steel bridge structures loaded by fatigue [19]. Probabilistic modelling of fatigue crack progression leads to designing a system of regular inspections of structures [21, 23]. M ODELLING OF THE FATIGUE CRACK PROPAGATION USING LINEAR FRACTURE MECHANICS hree sizes are important for the characteristics of the propagation of fatigue cracks. The first size is the initiation size of the crack that corresponds to a random failure in an element subject to random loads. Existence of the initiation cracks during the propagation should be revealed, along the measurable length of the crack , during inspections. The third important size has been referred to as far as the critical size – it is the final recorded size before a brittle fracture results in a failure. It would be advisable to use another method to specify the acceptable final size. Building structures and bridges are sized for extreme loads. Fatigue loads are investigated into only in details that are liable to fatigue cracks caused by variable operation loads [37]. If the load-bearing element is designed with a reasonable designed reliability margin for effects of the extreme load, then a crack will negatively influence the designed condition. The fatigue crack damage depends on a number of stress range cycles. This is a time factor of reliability in the course of reliability for the entire designed service life [15, 36]. It is assumed that in the course of time the failure rate increases, while the reliability drops. If the propagation of the fatigue crack is included into the failure rate, it is necessary to investigate into the fatigue crack and define the maximum acceptable weakening. The weakening depends on the acceptable crack size which comprises safety margins for the critical crack size that may occur in consequence of a brittle fracture and, more often in steel structures, in consequence of a ductile fracture. The reason for this type of degradation of a load-bearing element in the course of time is the random existence of the initiation crack and propagation of the crack in the consequence of variable load effects. The result is the weakening of the element that has been sized for extreme load effects. The crack propagates in a stable way until it reaches the acceptable size that is a limit for the required reliability [14]. When investigating into the propagation, the fatigue crack that deteriorates a certain area of the structure components is described with one dimension only a . In order to describe the propagation of the crack, the linear elastic fracture T