This paper presents an assessment of the fragility of a reinforced concrete (RC) element subjected to avalanche loads, and more generally to dynamic pressure fields applied orthogonally to a wall, within a reliability framework. In order to obtain accurate numerical results with supportable computation times, a light and efficient Single-Degree-of-Freedom (SDOF) model describing the mechanical response of the RC element is proposed. The model represents its dynamic mechanical response up to failure. Material non-linearity is taken into account by a moment–curvature approach, which describes the overall bending response. The SDOF model is validated under quasi-static and dynamic loading conditions by comparing its results to alternative approaches based on finite element analysis and the yield line theory. Following this, the deterministic SDOF model is embedded within a reliability framework to evaluate the failure probability as a function of the maximal avalanche pressure reached during the loading. Several reliability methods are implemented and compared, suggesting that non-parametric methods provide significant results at a moderate level of computational burden. The sensitivity to material properties, such as tensile and compressive strengths, steel reinforcement ratio, and wall geometry is investigated. The effect of the avalanche loading rate is also underlined and discussed. Finally, the obtained fragility curves are compared with respect to the few proposals available in the snow avalanche engineering field. This approach is systematic and will prove useful in refining formal and practical risk assessments. It could be applied to other similar natural hazards, which induce dynamic pressure fields onto the element at risk (e.g., mudflows, floods) and where potential inertial effects are expected and for which fragility curves are also lacking.

The hazard posed by avalanches threatens human communities in mountainous areas. Fatalities due to snow avalanches result from the practice of mountaineering or from avalanches reaching dwellings (e.g., an avalanche in 1999 killed 12 people in their homes in French Chamonix-Montroc village) or holiday accommodations (e.g., an avalanche in Val-d'Isère French valley in 1970 destroyed a vacation resort, where 39 people died; an avalanche in 2017 in the Italian Abruzzo region affected a hotel killing 29 people inside the buildings).

For formal risk assessment, fragility and/or vulnerability curves are
required to evaluate individual risks

In earthquake engineering, fragility curves have been widely studied and
methodologies to determine these have traditionally been categorized as
empirical, numerical, judgmental, or hybrid

Numerical fragility curves are mainly derived using the well-established
framework of reliability analysis

In reliability analysis, very time-consuming models are generally discarded
in favor of time-effective ones. Such alternative models, also called
meta-models, are often built on a statistical basis, e.g., using polynomial
chaos expansion (

As a response to the important issue of obtaining accurate numerical results
with reasonable computation times, this paper presents a light and efficient
SDOF model and uses it to refine the assessment of physical fragility
regarding snow avalanches to elements at risk, such as residential RC
buildings. Even if several kinds of constructive technologies are used in
snow avalanche engineering (e.g., masonry, reinforced concrete, or metallic
structures), for the sake of simplicity only the most common type of
structure found in avalanche prone areas in the Alps is considered:
reinforced concrete. In order to justify the assumptions made for the
avalanche loading, Sect.

Avalanches can be defined as the release of a snow volume that propagates
down a slope under the action of gravity. Snow avalanches can be classified
according to several criteria (e.g., snow type, release zone, weather
conditions). Two main types of avalanches are distinguished: (i) powder snow
avalanches composed of diluted dry snow, due to air incorporation,
characterized by a mean flow velocity that can reach 100 m s

Up to now, measured peak pressures span from 6.6 kPa at the Lautaret
experimental site

However, it must be stressed that such direct measurements, related back
calculations, and numerical calculations of avalanches' pressure impacts and
loading rates still suffer from large uncertainties and lack of information.
In addition, dwellings and buildings are commonly located at the bottom of
avalanche paths, in the so-called avalanche runout areas, where magnitudes of
peak pressures and loading rates are lower than those recorded in the middle
of avalanche paths, which adds further uncertainty to the analysis. This
means that engineering studies, as with this study, cannot currently rely on very
specific inputs to specify impact pressures and loading rate values. Hence,
in most of what follows, because the RC wall is supposed to be located within
the runout zone of the avalanche, a rough loading rate value of
0.1 kPa s

In order to perform the fragility analysis of an RC wall impacted by an
avalanche, the pressure field should be described in time and space. For the
sake of simplicity, based on previous research, it seems reasonable
to use the following assumptions for the modeling of the avalanche loading. A
uniform spatial distribution of the pressure field (Fig.

Simply supported RC wall loaded by a uniform pressure field

We consider a simply supported wall with length

Stress-strain relations for
concrete

The concrete and steel behavior laws are described by piece-wise linear
relationships that describe the evolution of stress

Under compression regime, the stress

For steel, the behavior law is assumed to be elastic perfectly
plastic (Fig.

Cross-section of the RC beam

Simply supported beam

The SDOF model corresponds to a dynamic mass-spring system loaded by a force
time evolution deduced from the uniform pressure field applied to the RC wall
(Fig.

The loading rate (e.g., from 0.1 to 6 kPa s

Bending moment–curvature relation

The characteristic load–displacement curve (

The

The curvature is defined as

From Newton's second law, the dynamic mechanical balance of the SDOF produces
the following ordinary differential equations. For the elastic phase, where

If needed, non-uniform spatial distributions of the pressure field can be
described by the SDOF model

To validate the SDOF model, a finite element simulation of the RC wall
response to an avalanche load was undertaken using the computation software
Cast3M

Cross section discretization of the beam multi fiber finite element. The diameter sizes of the steel reinforcements are knowingly exaggerated.

Under quasi-static loading conditions, the ultimate resistance of RC
slabs under uniformly distributed pressure can be derived from classical
yield line theory

For a simply supported one-way slab, the only collapse mechanism that can
arise is depicted in Fig.

Comparisons of

Table

Results demonstrated that both models are in very good agreement under either
quasi-static (Fig.

Parameter values for models comparison. The following notations are adopted: Ult. is an abbreviation of ultimate, S signifies steel, and C signifies concrete.

Ultimate displacement, ultimate pressure and computation time provided by the three approaches considering quasi-static pushover test.

The quantification of failure probability is carried out through the
reliability analysis of the physical model

Two classes of inputs are considered random variables, i.e., geometrical (

Marginal distributions of input
parameters. “determ.” means deterministic, which corresponds to a
coefficient of variation (COV) equal to zero. In the case of independent
variables, normal distributions are used (

To describe geometrical uncertainties, normal distributions are largely
assumed

Regarding both compressive and tensile strength parameters, in a first
approximation, normal distributions with a COV of

No data is available regarding the reinforcement ratio's COV. As

Statistical distributions of

The JCSS

For the yield strength of steel (

Four reliability methods are used, two non-parametric ones and two parametric ones. Non-parametric approaches consist of a direct estimate derived from the fragility curve with no assumptions regarding the output function. Parametric approaches assume the shape of the output probability density function via functional relationships and estimates of their constitutive parameters. The four considered methods are as follows.

A direct Monte Carlo (MC) approximation of the cumulative distribution function to build the empirical cumulative distribution function (ECDF).

A Gaussian kernel smoothing approximation using the Monte Carlo samples (MCKS).

A method based on parametric distribution definitions of the CDF, with parameters deduced following a Taylor expansion of the first two statistical moments of the resistance (TECDF).

Fitting a parametric distribution to the Monte Carlo samples via the maximum likelihood estimation method (MLECDF).

The extensive reliability methods library of the OpenTURNS software, which is
dedicated to the treatment of uncertainty, risk, and statistics, was used to
build the fragility curves from these four methods

Fragility curves can be assessed by using the output samples of direct Monte
Carlo simulations such as

Direct MC simulations of input variables can provide a discrete PDF of the
model's output. However, the resulting curve is a piecewise linear function.
The Gaussian kernel smoothing method allows the output PDF to be estimated
considering a normal, i.e., Gaussian, kernel function

Hereafter,

If a functional shape of the fragility curve is postulated, e.g., normal or
log-normal CDF, the parameters can be deduced from the first
(

From the MC sampling, the output CDF can also be fitted assuming the
functional shape of the fragility curve. The maximum likelihood estimation
(MLE) allows estimators

This section is divided in three sub-sections:
Sect.

Reliability method comparisons
between empirical cumulative distribution functions (ECDF) with set

The comparison between each method, presented in Sect.

We defined the fragility range as the interval between the 2.5 % and
97.5 %
quantile of the limit pressure CDF, i.e., the pressure range in which the
fragility increases from

In the case of the TECDF method, the approximation of the first statistical
moments and the second centered statistical moments combined with normal or
log-normal CDF needs only 15 simulations at the first order of the Taylor
expansion. One simulation allows the mean to be estimated at the
first order and 14 simulations allow the variance to be estimated at the
first order. The second order mean estimate needs 113 simulations. For
the TECDF method, the approximation of the fragility curve exhibits slight
differences compared to the ECDF fragility curve regardless of the assumed
output CDF (Fig.

The efficiency and drawbacks of each method are summed up in the scheme of
Fig.

Advantages and drawbacks of each method to derive fragility curves.

Independent input PDFs give similar fragility curves when they are centered
around the same mean input values (Fig.

Statistical distributions effects on
fragility curves (built with 300 data using the Gaussian Kernel Smoothing method)
considering

The 2.5,%, 50 %, and 97.5 % quantiles (in kPa) of the fragility curve according to the input PDF reference set.

Four combinations are considered to investigate the effect of the number and
the class of random variables, i.e., (i) the deterministic case,
set

The 2.5 %, 50 %,
97.5 % quantiles (in kPa), and the fragility range ratio

Effects on fragility curves (built with 300 data using the Gaussian Kernel Smoothing method) of the mean values of

The ultimate pressure value (

The influence of the reinforcement ratio is explored for several typical
values. The lower the reinforcement ratio, the lower the ultimate pressure
(Fig.

As the reinforcement ratio plays an important role in the failure mode
of the structure, a high density reinforcement ratio is tested, such as

Depending on the RC wall mechanical properties and on the avalanche loading
time evolution, inertial effects can develop and modify the structural
response through time. In order to assess the effect of the avalanche loading
rate (

Effects on fragility curves of the avalanche loading rate. Fragility curves have been computed using the ECDF reliability method.

Very few snow avalanche fragility and vulnerability curves have been reported
in the literature. However, to put our results in a broader perspective, the
herein obtained fragility curves were plotted against existing curves. First,
the numerical fragility curves proposed by

Comparison of the article fragility curve to

Based on classical engineering approaches,

Many points can explain the differences in shapes and values between all these curves. Indeed, even if some similarity is expected, all curves, especially those representing the failure probability on the one hand and the sensitivity of damage as function of avalanche pressure on the other, do not necessarily have to follow the same trends. Specifically, several factors can explain the differences we highlighted.

First, the failure probability gives the probability that the structure
exceeds the ultimate damage state, whereas the sensitivity of damage gives a
deterministic value of damage ratio, which is rather different. For instance,
it can be assumed that the expert in

This paper presents the derivation of fragility curves for a reinforced concrete wall loaded by a dynamic pressure field due to a snow avalanche. Methods from the reliability framework have been implemented and combined with a simplified SDOF model, which is light and efficient. A one-way simply supported RC wall has been considered and a deterministic model based on an equivalent mass-spring system has been used to represent its mechanical behavior up to the rupture when subjected to a uniform pressure field. The ability of the SDOF model to predict the RC wall mechanical response has been validated based on comparisons with FEA and limit analyses. Using a SDOF approach significantly reduces the computation time needed to perform a single simulation and allows accounting for the physics involved up to the collapse of the structure during wall–avalanche interactions. Second, four reliability methods have been implemented to derive fragility curves. All methods gave similar results regardless of the configuration considered, at least for the core of the distribution. The advantages and drawbacks of each method have been identified, and the kernel smoothing method was selected as a reasonable compromise for further parametric and sensitivity studies. This comprehensive framework could be valuable for a wide range of reliability-based engineering applications where structural members are loaded by non-uniform pressure fields which can evolve through time.

For our specific snow avalanche case study, systematic fragility curves were derived. The results emphasize that fragility curves are very sensitive to physical parameters such as the RC wall's geometry, its reinforcement ratio or the loading features. In particular, the spread of the fragility range appeared to be strongly variable. However, as soon as the fragility range was standardized by its 50 % quantile, the relative fragility spread remained almost the same. These results supplement the few fragility and vulnerability curves already available in snow avalanche engineering literature. They will be of great value for future works that seek to refine formal risk evaluation in avalanche prone areas.

According to the scarce available measurement data, it was assumed that the response of the structure was quasi-static. The SDOF model formulation has been made within a dynamic framework. The proposed SDOF model is thus able to describe the occurrence of potential additional resisting forces (structural inertia), which are governed by the pressure time evolution. The effect of the latter was explored, underlining the increase of the apparent strength with the loading rate when triangular pressure time evolutions are considered and high loading rates are imposed onto the RC wall. As this result cannot be generalized, further research is needed to explore the influence of various pressure time evolutions on the fragility curves derivation. Moreover, our approach can be implemented for other types of structures with different technologies (e.g., other RC structure configurations, masonry, timber or metallic structures) and/or more sophisticated structure geometries. Finally, extension to other mass movements hazards such as debris flows, rockfalls or ice avalanches, for which similar gaps in engineering need to be filled, may be pursued. It should be kept in mind that for each hazard, the challenge will be to propose simplified mechanical models able to account for the main physics with a reduced computation time.

As a perspective, the main difficulty concerns the modeling of the avalanche pressure, which can vary significantly as function of meteorological conditions and especially in terms of pressure magnitude, spatial distribution and typical time of variation. Pressure magnitude is implicitly taken into account by the fragility curves but the spatial distribution and pressure variations through time can have a significant influence on the structure mechanics. The structure's mechanical features are generally better known than the avalanche loading. Thus, further research, accounting for several typical spatial distributions and time evolutions of the pressure, might be of specific interests to highlight the influence of avalanche loadings on curves, which are used in formal risk evaluation.

Data of the curves from Figs. 11 and 12 can be freely accessed on the website

DB and NE designed the research. DB and PF planned and carried out the simulations. All authors contributed to the analysis of the results and to the writing of the manuscript.

The authors declare that they have no conflict of interest.

The authors are grateful to the ANR research program MOPERA (MOdélisation Probabiliste pour l'Etude du Risque d'Avalanche), the MAP3 ALCOTRA INTERREG program, the Chilean National Commission for Scientific and Technological Research (CONICYT) under grant Redes 150119 and grant Fondecyt Postdoc 3160483, the Chilean National Research Center for Integrated Natural Disaster Management CONICYT/FONDAP/15110017 (CIGIDEN), and the ECOS-CONICYT Scientific cooperation program under project “Multi-risk assessment in Chile and France: application to seismic engineering and mountain hazards ECOS170044 and ECOS action C17U02” for financially supporting this work. Irstea is member of Labex Osug@2020. Edited by: Perry Bartelt Reviewed by: four anonymous referees