E3S Web of Conferences 415, 07010 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341507010

A time-independent reliability based design approach for debris
flow flexible barriers
Maddalena Marchelli1* and Chiara Deangeli1
1DIATI, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy

Abstract. The design of debris flow protective structures has represented a challenge since the last decades.
The design process might consider several variables related to actions and resistances, mainly linked toimpact pressure and dynamic pressure resistance. Debris flow events do not have always the same
magnitude, and small events can occur with a high probability. Therefore, an events frequency-magnituderelationship might be considered in the design of protective structures. In the Eurocodes framework, thefixed partial safety factors design approach not considering the intrinsic site variability of these phenomena,
does not allow obtaining a specific probability of failure. Reliability-based approaches reveal to becompelling solutions. Focusing on flexible barriers, the paper presents a novel time-independent reliability
approach, which considers all the possible debris flow events and provides the failure probability of a barrier
in a given temporal interval.

1 Introduction
Debris flows are one of the most hazardous natural
phenomena, due to their unpredictability, and highvelocities. These processes, constitute a serious threat tolife, properties and infrastructures [1]. Consequently,
debris flow risk reduction represents a significant aspectin risk management policies.
Debris flow flexible barriers are effective protective
measures, due both to the ease of transport andinstallation [2]. These barriers mainly involve a
combination of steel ropes and nets consisting in hightensile wire in steel, whose assembling and components
technologies are constantly evolving [3] (Fig. 1).
As structural works made as kits, theirperformances have to be assessed in relation to theiressential characteristic, i.e. the dynamic impact pressure
resistance, the height and the maximum deformation. A
European guideline introducing codified methods forassessing such devices has been developed in 2016, i.e.
EAD 340020000106 [4]. In the framework of Eurocodes[5, 6], it should be verified that:𝐸𝑑 ≤ 𝑅𝑑(1)where 𝐸𝑑 , 𝑅𝑑 are the design values of the effects of theactions and of the resistances, respectively. Providingthat Eqn. (1) is satisfied, the current design procedure
consists in choosing a suitable product, and, thus,
resistances coincide with performances, while actions
can be obtained by both codified tests or advanced
numerical modelling. The considered actions are thoserelated to the impact pressure exerted by the flow, whose
design value should be accurately evaluated throughinvestigations on past events and trustworthy analyticalor numerical debris-flow dynamic analyses [7].
Nevertheless, modelling these unsteady phenomena isvery difficult, as they are characterized by entrainmentand deposition during their path, exhibiting also grainsize segregation, excess of water pressure, whose extent
is generally difficult to measure. Moreover, transition
from sliding to flow might occur during the motion 811. Consequently, the generally adopted models adopt
several simplifications. Additionally, in the same site
events can differ one from another and smaller eventsoccur with a higher frequency.
In the framework of limit state design with partial
safety factors approach, no specifications on bothcharacteristic values and partial safety factors arepresent in the Eurocodes [12] referring to debris flows
barrier. Due to the large variability in the statistics
related to actions, the choice of a unique, i.e. fixed, value
for each safety factors is not suitable, as fixed factors notallow obtaining a specific probability of failure. Specificparametric studies devoted to define a possible non-sitespecific range of factors for obtaining a target failure
probability in a given time are still missing.
Consequently, assuming that the failure of thesystem occurs for debris flow pressure exceeding theresistance of the barrier, the present work aims atidentifying a solution for the design of such system,
based on the reliability approach (Eurocode 0 [5]). The
profitability of such kind of approach was highlighted
by [12], in which a general framework is provided.
Nevertheless, the time-variant reliability issues have notbeen tackled yet, e.g. magnitude-frequency relationshipof the possible events and their consequences have notbeen considered. Addressing these issues, following the* Corresponding author: maddalena.marchelli@polito.it© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 07010 (2023)
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general suggestions provided by ISO 2394 [13], theAuthors propose a methodology, mimicking a method
defined for rockfall flexible barriers [14-16],
considering that:(i) the probability of failure, i.e. the probability ofhaving actions larger than the resistances, should bedefined for a specific time period;(ii) actions can vary during the event and, mostimportantly, their maxima vary from one event toanother. A frequency-magnitude relationship for themaximum action should be evaluated and anintegration over time of the actions should beconsidered in the design process.
The proposed methodology is delineated in Sec. 2,
while its limitations and advantages are discussed inSec. 3. Finally, conclusions and future perspective areoutlined.

where 𝜈 is the mean expected annual frequency of adebris flow of any size, while 𝑝𝑓,𝑎 is the probability of
failure if a debris flow impacts against the barrier, i.e.
considering the certain occurrence of debris flow events.
As other natural phenomena, small events can occur
with a higher probability than huge ones [14].
This means that the effects of the actions are timedependent, i.e. for a given return period T, a pressurep(T) can be defined. The design method herein proposedcomputes the failure of the barrier considering all thepossible events that can occur at any time, and, for thisreason, can be considered as “time-independent”.
Another consideration relates to the pressure, which,during each event, is not a deterministic value but canbe described as a probabilistic function, whosereference, i.e. characteristic value, is 𝑝𝑘 .
Consequently, 𝑝𝑓,𝑎 , can be obtained through:∞𝑝𝑓,𝑎 = ∫0 𝑝𝑓 |(𝑝𝑘 = 𝜋)𝑓𝑝𝑘 (𝜋)𝑑𝜋(3)being 𝑝𝑓 |(𝑝𝑘 = 𝜋) the probability of failure if thepressure 𝑝 has its characteristic value 𝑝𝑘 equal to 𝜋, i.e.
a random variable through which the characteristicvalue 𝑝𝑘 is defined, and 𝑓𝑝𝑘 (𝜋) the probability density
function related to each pressure distribution, whosecharacteristic value is 𝜋. The integral is calculated over
all the possible impact pressures, to provide the timeindependency of the method. It should be stated that theprobability 𝑓𝑝𝑘 (𝜋) is difficult to be obtained. Providing∞that ∫−∞ 𝑓𝑝𝑘 (𝜋) 𝑑𝜋 is equal to 1, Hong et al. [17], basedon a large set of measurements on real phenomena,suggest to assume a Weibull distribution. It means that:

Fig. 1 Debris flow flexible barrier (Courtesy of Geobrugg
Italia)𝑘 𝜋 𝑘−1

𝑓𝑝𝑘 (𝜋) = ( )𝜇 𝜇2 Methodology𝑘𝑒 −(𝜋/𝜇)(4)where 𝑘 and 𝜇 are coefficient that should be properly
calibrated (Fig. 2a). Obviously, Eqn. (4) holds for𝜋 ≥ 0. The conditional failure probability 𝑝𝑓 |(𝑝𝑘 = 𝜋)is studied through a state function, which describes bothsafe and unsafe barrier conditions, accounting for the
resistance 𝑝𝐵 .The state function 𝐺(𝐩) can be defined as:

This section provides the fundamental principles of thetime-independent reliability approach adopted for debris
flow protection structures. Debris flow flexible barriers
have to retain the solid phase of the debris flow mixture,
resisting the dynamic pressure exerted by the flow,
without exceedingly deforming, breaking, or collapsing.
The height of the flow is generally assumed to becomparable with the channel height. Therefore, thefailure mode of such barriers can be simplified into a
failure process related to exceeding pressure, when thedynamic pressure resistance 𝑝𝐵 , is lower than the flowpressure. The resistance 𝑝𝐵 is evaluated with theassessment procedure of [4], i.e. with either
standardized field test or numerical analysis. The
boundary between failure and safety can bemathematically described through a limit state functionthat allows obtaining the design values (subscript d) ofthe variables. Vice-versa, if the effects of the actions areknown, the limit state function enables to assess thereliability of the protection structure.
Going deeply into the reliability approach, theprobability of failure 𝑝𝑓 , in a given period 𝜏 (in years),
can be computed as:
𝑝𝑓 (𝜏) = 1 − 𝑒 −𝜈𝜏𝑝𝑓,𝑎(2)𝑝𝐺(𝐩) = 𝐺 (𝑝 ) = 𝑝𝐵 − 𝑝𝐵

(5)where 𝑝 is the flow pressure, described as a probabilisticvariable, as stated before (Fig. 2b).
The failure probability, for each event, is computed as:𝑝𝑓 |(𝑝𝑘 = 𝜋) = 𝑃(𝐺(𝐩) ≤ 0) =∬𝐺(𝐩)≤0 𝑓𝐺 |(𝑝𝑘 = 𝜋)(𝐩)𝑑𝑝𝐵 𝑑𝑝(6)where 𝑓𝐺 is the probability density function of thedifference between barrier resistance and debris-flow
impact pressure, provided that 𝑝𝑘 is equal to 𝜋.

3 Discussion
This section deals with (i) the probability distributions
in time of the relevant variables and (ii) the capability ofthe method to be used for deriving reliable partial safety2

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factors that can, thus, be adopted in a semi-probabilistic
design approach. In this last case, a debris flow eventwith a given return period is considered for the design
actions.

should account for those variables, together with theirjoint probability density functions.
Taking the hydrodynamic formulation as an
example, Eqn. (5) turns thus into:𝜌𝐺(𝐩) = 𝐺 ( 𝑣 ) = 𝑝𝐵 − 𝜌𝑣 2 𝛼𝑝𝐵(9)and Eqn. (3) into:∞
𝑝𝑓,𝑎 = ∫0 𝑝𝑓 |(𝜌𝑘 = 𝛿 and 𝑣𝑘 = 𝜔 )𝑓𝜌𝑘𝑣𝑘 (𝛿, 𝜔)𝑑𝜋 (10)being 𝑓𝜌𝑘 𝑣𝑘 the joint probability density function of thecharacteristic values of debris density and velocity,defined through the random variables 𝛿 and 𝜔,
respectively (Fig. 3). It is worth mentioning that timedependent probabilistic distributions of each variable
are very difficult to achieve, as well as the assessmentof the goodness of the existing predicting impactmodels, as shown in [12].
Another important issue concerns the distribution of
the dynamic pressure resistance of the barrier and its
characteristic value. Unfortunately, this information canbe obtained only for products verified with advanced
numerical tests [21]. As the design of such systems isbased on the choice of proper flexible kits among thoseavailable on the market, a Dirac-𝛿 distribution, at themaximum tested value according to [4], can be used.

Fig. 2 Probability density function 𝒇𝒑𝒌 (𝝅) related to any
possible pressure distribution, whose characteristic value is π,described with a Weibull distribution (a); and probability
density function of the pressure for different events (e.g. i andj, whose characteristic values are 𝝅𝒊 and 𝝅𝒋 , respectively) withboth safe and unsafe conditions (b). The capacity of the barrier
is assumed to be described by a Dirac-𝜹 distribution.

3.1 Uncertainties
approachrelatedtothereliability

The proposed equations account for the most generalsituation, assuming to have a magnitude-frequency
relationship based, e.g., on catalogue of recorded pastevents. The equations do not consider the method ofcalculation of the impact pressure 𝑝 exerted by the flow.
Several models have been developed to estimate a peakpressure value [18] based on a hydrodynamic approachor on a hydro-static approach [19,20]. In thehydrodynamic approach the impact pressure iscomputed through:𝑝 = 𝜌𝑣 2 𝛼

Fig. 3 Probability density functions of 𝝆 and 𝒗 for different
events (e.g. i and j, whose characteristic values are 𝜹𝒊, 𝝎𝒊 and𝜹𝒋, 𝝎𝒋 , respectively) with both safe and unsafe conditions.

(7)

3.2 Capabilities of the method: derivation of thepartial safety factors

where 𝜌 is the density of the debris flow mixture, , 𝑣 isthe velocity of the fluid, and 𝛼 a dynamic coefficient,which depends on the grain size distribution, and, thus,
related to 𝜌 itself. In the hydrostatic approach, theimpact pressure can be evaluated using:𝑝 = 𝑘𝑠 𝜌𝑔ℎ

The proposed time-independent reliability approachallows obtaining a failure probability for a given
protection system in a given period. Moreover, thismethod can be used to derive site-specific partial safety
factors which, if adopted in a more straightforwardsemi-probabilistic design framework [5,6], guarantee toobtain a given failure probability. Unlike structuralengineering, in geotechnical engineering the actions arevery site-specific and, consequently, the use of a fixedvalue for each factor entails a non-unique value offailure probability. In the partial safety factor approach,(8)where 𝑘𝑠 is an empirical coefficient, 𝑔 the gravity, and ℎ
the flow depth.
The impact pressure is thus described through othervariables, i.e. 𝜌, 𝑣, ℎ, each of which can be defined by aprobability distribution. In this case, the state function

3E3S Web of Conferences 415, 07010 (2023)
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only a characteristic value of the impact pressure,associated to a specific return period, should beconsidered. The reference return period T can be 100
years or more. For a given T, the characteristic pressure
becomes:
1L. Canelli, A.M. Ferrero, M. Migliazza, A.
Segalini, Nat. Hazards Earth Syst. Sci. 12, 5(2012)
3. C. Wendeler, V. Budimir, M. Denk, Debris
flow protection with flexible ring net barriers– 10 years of experience, in Proceedings of theXVI Danube - European Conference on
Geotechnical Engineering, 7-0 June 2018,Skopje, R. Macedonia (2018)
4. EAD 340020000106, Flexible kits forretaining debris flows and shallow
landslides/open hill debris flows, EOTA
(2016)
5. EN 1990:2002, Eurocode 0 - Basis of
Structural Design (2002)
6. EN 1997–1:2004, Eurocode 7 - Geotechnical
Design. Part 1: General Rules, (2004)
7. D.Tiranti, C. Deangeli, Front. Earth Sci., 3, 8(2015)
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Scenarios Using Different DTMs, inEngineering Geology for Society andTerritory, Springer, 2, 1667-1671 (2015)
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12. F. Vagnon, A.M. Ferrero, L.R. Alejano,
Landslides, 17, 49-59 (2020)
13. ISO 2394 :2015, General Principles on
Reliability for Structures (2015)
14. V. De Biagi, M. Marchelli, D. Peila, Eng.
Struct., 213, 110553, (2020)
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Rock Mech. Min. Sci., 139 104664 (2021)
16. M. Marchelli, V. De Biagi, D. Peila,Geosciences, 10, 8 (2020)
17. Y. Hong, J.P. Wang, D.Q. Li, ZJ Cao, C.W.W.
Ng, P. Cui, Eng. Geo., 187, 122-134 (2015)
18. F. Vagnon, Landslides, 17, 313-333 (2020)
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Scheidl. Debris flow impact estimation, inProceedings of the 11th international
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Li, Y. Zheng, Eng Geol 251, 81-92 (2019)22. UNI 11211–5:2019. Opere di difesa dalla
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1/𝑘𝑝𝑘 (𝑇) = −𝜇 [ln ( )]
𝜆𝑇

2.

(10)

Hence, given a specific 𝑝𝑓 (𝜏), with the pressure
probability distribution of the specific site, the minimum
required 𝑝𝐵,𝑚𝑖𝑛 is computed. The partial safety factor forthe pressure 𝛾𝑝 can thus be obtained as:
𝛾𝑝 =𝑝𝐵,𝑚𝑖𝑛𝑝𝑘 (𝑇)

.

(11)

Extending the study for different sites a possible rangeof 𝛾𝑝 can be obtained. Although there are no specificprescriptions on the allowable risk due to debris-flow, aone-year failure probability equal to value 𝑝𝑓 (1yr) =10−5 can be adopted, which is coherent with thesuggestions for rockfall barrier provided in the ItalianStandard [22].

4 Conclusions
Flexible barriers are among the most adopted passive
mitigation measures for debris flow. During the last
decade, several technologies have been developed,
resulting on very complex systems, whose essential
characteristics should be assessed according to [4]. In
this framework, the evaluation of the proper actions toconsider in the design process represents a key aspect,
nowadays still under debate. Today’s regulation does
not specifically deal with such structures. In theframework of the Eurocodes, a partial safety factors
approach with fixed factors might not be trustworthy as
it does not account for a specific failure probability, nor
for the site-specificity of the problem. The profitabilityof reliability-based approaches has recently highlighted.
Nevertheless, the variability in time of the input
variables has not been considered yet.
The present paper proposes a novel reliability
approach, which determines the failure probability of agiven product during a specific temporal interval,
accounting for all possible events, for which afrequency-magnitude relationship is defined. For thisreason, the approach is time-integrated and can bedefined as time-independent. Given a target failure
probability, this method allows also to obtain sitespecific partial safety factors to adopt in a semiprobabilistic approach.
Further developments could investigate more deeply
the distributions of the input variable or the method
could be extended also to rigid barriers [23,24] or evenembankments [25].

References
1.

M. Jakob, O. Hungr. Debris-flow hazards andrelated phenomena. Springer-Verlag, Berlin
Heidelberg, 749, (2005)

4