Development of Fragility Curves for Piping and Slope Stability of River Levees

water Article Development of Fragility Curves for Piping and Slope Stability of River Levees Nicola Rossi *, Mario Baˇcić , Meho Saša Kovaˇcevićand Lovorka Librić Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia; [email protected] (M.B.); [email protected] (M.S.K.); [email protected] (L.L.) * Correspondence: [email protected]; Tel.: +385-1-4639-491 Abstract: The design code Eurocode 7 relies on semi-probabilistic calculation procedures, through uti- lization of the soil parameters obtained by in situ and laboratory tests, or by the means of transformation models. To reach a prescribed safety margin, the inherent soil parameter variability is accounted for through the application of partial factors to either soil parameters directly or to the resistance. However, considering several sources of geotechnical uncertainty, including the inherent soil variability, measurement error and transformation uncertainty, full probabilistic analyses should be implemented to directly consider the site-specific variability. This paper presents the procedure of developing fragility curves for levee slope stability and piping as failure mechanisms that lead to larger breaches, where a direct influence of the flood event intensity on the probability of failure is calculated. A range of fragility curve sets is presented, considering the variability of levee material properties and varying durations of the flood event, thus providing crucial insight into the vulnerability of the levee exposed to rising water levels. The procedure is applied to the River Drava levee, a site which has shown a continuous trend of increased water levels in recent years. Citation: Rossi, N.; Baˇcić,M.; Keywords: levee; slope stability; piping; overtopping; fragility curves; Monte Carlo simulation Kovaˇcević,M.S.; Librić,L. Development of Fragility Curves for Piping and Slope Stability of River Levees. Water 2021, 13, 738. 1. Introduction https://doi.org/10.3390/w13050738 As earthen structures constructed for the purpose of flood defence, the levees should Academic Editors: Jihn-Sung Lai be verified for several potential failure modes. According to Wolff [1], these include over- and Miguel Á. Toledo topping, slope stability, external erosion, underseepage and through-seepage, with the latter two being considered as internal erosion mechanisms. These failure modes are Received: 7 January 2021 conditioned by the levee’s geometrical configuration, its material properties, and overall Accepted: 3 March 2021 hydraulic conditions of the site. Based on the examined breach characteristics of hundreds Published: 9 March 2021 of failures, Özer et al. [2] identified the external erosion as the most frequent for levees, while failures due to internal erosion and instability are less frequent but lead to larger Publisher’s Note: MDPI stays neutral breaches, and as such are emphasized within this study. Of all the internal erosion mech- with regard to jurisdictional claims in anisms, backward erosion piping is considered to be the primary failure mechanism for published maps and institutional affil- levees [3], and even accounts for one-third of all piping failures that occurred in the last cen- iations. tury [4]. Various design situations such as rainfall, high water level, seismic peak ground acceleration, etc., can be triggering factors for one or more failure mechanisms, directly or indirectly. Extensive studies have been conducted with various approaches regarding slope stability, Figure1a, with respect to rainfall [ 5,6], high water levels [7,8], and peak Copyright: © 2021 by the authors. ground accelerations [9,10], as well as combinations of various events [11]. Regarding the Licensee MDPI, Basel, Switzerland. piping failure, Figure1b, and depending on the mechanism of soil particle removal (e.g., This article is an open access article removal of particles by water forces, chemical dispersion of clays, migration of fine material distributed under the terms and through coarse matrix, etc.), various modes are identified, all pertaining to internal erosion conditions of the Creative Commons under or through the levee [12–15]. Attribution (CC BY) license (https:// The analysis of different levee failure mechanisms within the Eurocode 7 design creativecommons.org/licenses/by/ code [16] is based on the use of recommended, singular values of partial safety factors 4.0/). Water 2021, 13, 738. https://doi.org/10.3390/w13050738 https://www.mdpi.com/journal/water Water 2021, 13, 738 2 of 19 (PSFs), with a defined combination of PSFs for action and resistivity (material), depending on the adopted calculation approach for a specific design situation. The code, however, prescribes constant values of PSFs for limit states, with no variation depending on the nature or the duration of the design situation and no recommendation regarding the target reliability values. Du Thinh [17] notes that, during the design process, an engineer must select a set of characteristic values and the corresponding PSFs, hoping to obtain in the end a design that satisfies a prescribed reliability level. On the other hand, the design code Eurocode 0 [18] provides minimum values for the reliability index for three consequence classes, but these are only defined for buildings, not for geotechnical structures. Some other Water 2021, 13, x FOR PEER REVIEW design codes, such as those in [19], acknowledge the uncertain nature of soils, by defining2 of 19 target values of pf and equivalent reliability indices for three consequence levels, based on random finite-element analyses. (a) (b) FigureFigure 1. 1.LeveeLevee failure failure mechanisms mechanisms analysed analysed in in the the study: study: slope slope instability instability (a ()a and) and internal internal erosion erosion (b (b).). TheConcerning analysis of the different soil related levee uncertainties,failure mechanisms Phoon within and Kulhawy the Eurocode [20] identified 7 design code three [16]main is based sources onof the geotechnical use of recommended, uncertainties: singular (1) inherent values soilof partial variability, safetywhich factors describes (PSFs), withthe a variation defined combination of properties of fromPSFs for one action spatial and location resistivity to another,(material), (2) depending measurement on the er- adoptedror, which calculation implies approach the scatter for of a measurements specific design on situation. presumably The homogeneous code, however, soil pre- vol- scribesumes, constant and (3) transformation values of PSFs uncertainty,for limit stat where,es, with in no the variation process depending of model characterization, on the nature orwhich the duration includes of linking the design the on-sitesituation and and laboratory no recommendation test results regarding to the design the target parameters, reli- abilitysome values. degree ofDu uncertainty Thinh [17] isnotes introduced. that, during By implementing the design process, a Eurocode an engineer 7 semi-probabilistic must select a approach,set of characteristic which utilizes values statistical and the methods corresponding to select PSFs, characteristic hoping to values obtain of in geotechnical the end a designparameters, that satisfies both spatial a prescribed correlations reliability between level. the On same the other parameter hand, sat the different design code sampling Eu- rocodepoints 0 and [18] cross-correlations provides minimum between values different for the reliability parameters index at the for same three sampling consequence point classes,are neglected but these [21 are]. Further, only defined depending for build on theings, associated not for geotechni failure mechanismscal structures. of Some levees, otherdifferent design material codes, parameters such as those will controlin [19], theacknowledge limit states the and uncertain different modelsnature areof soils, necessary by to predict the resistance, and thus no uniform reliability level can be obtained with the defining target values of p and equivalent reliability indices for three consequence levels,load based and on resistance random finite-element factor design analyses. method [ 22]. The degree of uncertainty involved in calculationConcerning of levees the soil is especially related uncertainties, high for slope Phoon stability and [ 23Kulhawy] and piping [20] mechanismsidentified three [24 ]. mainEven sources though of the geotechnical geotechnical uncertainties: community has (1) been inherent more soil progressive variability, in the which implementation describes theof variation different probability-basedof properties from methods one spatial in analyses location of levees,to another, understanding (2) measurement of levee error, failure whichmechanisms implies isthe still scatter limited of measurements [25], while their on behaviour presumably during homogeneous critical conditions soil volumes, mostly remains uncertain. and (3) transformation uncertainty, where, in the process of model characterization, which This paper contributes to the efforts of levee vulnerability evaluations, through the includes linking the on-site and laboratory test results to the design parameters, some de- demonstration of a methodology for calculation of fragility curves for relevant failure mech- gree of uncertainty is introduced. By implementing a Eurocode 7 semi-probabilistic ap- anisms of slope stability and piping. Among the many available probabilistic methods [26], proach, which utilizes statistical methods to select characteristic values of geotechnical this study adopts

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