Reliability-based design of rockfall passive systems height
Introduction
Among the natural hazards, rockfalls constitute a serious threat to life, properties and infrastructures, due to their spatial and temporal unpredictability and high energy involved.1,2 This kind of events can severely affect civil structures and infrastructures as hamlets, roads, railways,3, 4, 5 but has also industrial implications related, e.g., to quarries and open pit mines, also linked with the optimization of the orebody exploitation process.6, 7, 8, 9 As a consequence, rockfall risk reduction can represent one of the most significant aspects in various engineering and human activities. Concern about passive mitigation measures has become a central issue in rockfall risk management10,11 and, as a result, much research in the last decades has focused on the development of new technologies. Among these structures, rockfall net fences and embankments represent the most effective solutions in case of events characterized by large energy and/or high trajectories.12 The design of these protection measures is still under debate and a standardized procedure is not yet available, due both to the complexity of the problem and to the several technologies adopted.13, 14, 15, 16, 17 Nowadays, following the CE marking procedure,18 the design of the net fences is oriented towards a performance-based design approach, evaluating energy absorption capacity and height. On the contrary, no codified procedure exists either for CE making and the design of rockfall embankments.19,20
The understanding and the modelling of the rockfall phenomenon is difficult to achiev21, 22e21, 22 as appropriate assumptions, e.g. considering the source zone location,23 the initial released volume and the impacting volume,24 have to be made. Simplifications concerning the direction of the impact on the barrier or the precise position of the system along the slope are generally introduced. Few papers analyze these aspects25, 26, 27, 28 and the different resisting mechanisms that can arise in passive protection systems according to the impacting block volume29,30 and the impacted element.31 Although complex numerical models have been realized,32, 33, 34 the coupling between the impacting block and a retaining structure is tricky, and difficulties often arise when merging the results of trajectory analyses and the model of a net fence or an embankment.35
Mimicking the performance-based design approach, i.e. starting from the assumption that the failure of the system can occur for block kinetic energies or trajectory heights greater than the interception capacity of the system, and the current practice based on the Eurocode 0,36 the present work aims at identifying a compelling solution for the design of these retaining systems. This procedure is based on a time-dependent reliability-based approach introduced in De Biagi et al.,37 discussed in Marchelli et al.,38 and here enhanced (Sec. 2), which considers all the possible distributions of the velocity, mass and height of the block impacting against the protective system located in an arbitrary position along the slope. With the aim of merging it with the ultimate limit state procedure suggested in the Eurocodes,36,39 which is based on safety factors applied to resistance and actions, equivalent partial safety factors for the impacting block energy and height are here derived, with particular focus on the intercepting height of the systems.
Different combinations are analyzed, considering or not a mutual independence between the two failure modes (Sec. 3.1). Sensitivity analyses are performed to investigate the parameters which mostly affect the adopted partial safety factors (Sec. 4). As already illustrated in the previously cited papers by the Authors, the values of the factors largely depend upon many variables. With the aim of providing a profitable tool for the design of such structures, a shallow neural network was built in order to create two input-output relationships that can be used to evaluate the partial safety factors for the height and the energy, given a failure probability (Sec. 5). At the end, conclusions and future perspective are suggested (Sec. 6).
Section snippets
Time-dependent reliability approach
This section provides the fundamental principles of the reliability approach adopted for rockfall protection structures. The proposed method constitutes an improvement of the method proposed in De Biagi et al. 37 and discussed in Marchelli et al. 38 Basically, rockfall passive mitigation measures as net fences and embankments have to intercept a falling block and to withstand its dynamic impact without exceedingly deforming, breaking, or collapsing. As a result, the possible failure modes of
Equivalent partial safety coefficient: different combinations
The reliability level of a structure is related to a specific failure probability. The proposed approach can be alternatively considered for the design or for the calibration of the partial safety factors to be adopted in the current semi-probabilistic design framework.48 The partial safety factors, namely γ in the design codes, say,36 are coefficients (always larger than one) that serve for computing the design value of the variable, usually denoted with subscript d, from its characteristic
Influence of the variables on
The purpose of this section is to deeply investigate the influence of the input variables on the value of the partial safety factor . The adopted variables are within a range that encompasses the majority of the scenarios that can occur in a real slope.38 Although an annual failure probability equal to 10−4 is considered for the study, it has to be remembered that has to be properly chosen according to a risk analysis. Following Authors’ judgment, the chosen is realistic and can be
Function fitting with a shallow neural network
In the previous section the influence of , , , λ, α, N, and on , given a , was assessed. Similarly, in De Biagi et al. 37 the dependency of to , α, N, and was highlighted. The purpose of this section is to present a possible fitting for straightforwardly estimating and without performing the integrals reported in Sec. 2.
With this purpose 85000 simulations were performed applying the proposed time-dependent reliability method, with a Monte Carlo sampling
Conclusions
The time-dependent reliability approach introduced for the design of rockfall passive measures37 was enhanced accounting for the variability of the impacting block mass on the intercepting height of the system. Starting from the assumption that the failure of the system can occur for a kinetic energy or a trajectory height of the block greater than the interception capacity of the system, the proposed reliability-based approach takes into account the variability in time of the block mass and
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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