Survival analysis of a liquefiable embankment subjected to sequential earthquakes
Introduction
Seismic hazard analysis involves the quantitative estimation of the ground motion (GM) hazards of a specific area. It requires the knowledge of the geologic evidence, the fault activity, the magnitude and the historical seismicity of the studied region [6,30]. The Probabilistic Seismic Hazard Analysis (PSHA) considers the uncertainties in the earthquake size, location and occurrence time. It estimates the mean frequency of exceedance of any spectral acceleration at the site [5]. The level of shaking produced from this analysis comes from the contribution of the magnitude , the source-to-site distance R and often the deviation of the GM from the predicted value (ε) [5]. Given the aforementioned information, the Ground Motion Prediction Equations (GMPEs) create the relationship between the magnitude, distance, and other model parameters and the Intensity Measures (IM). In this context, the study of the non linear behavior of the structures needs a recall to a large number of acceleration time histories. In addition and for particular scenarios, available data resources are sometimes inadequate to characterize the models due to several problems (i.e. ground motions from very large magnitude earthquakes, near-fault ground motions, basin effects) [37,50,54,57,60, among others]. For this reason, artificial or synthetic earthquakes could be used. They are conducted based on several methods (i.e. stochastic ground motion model, the composite source method, among others) and are useful when real motions are not available.
In practice, structures are designed to resist the first damaging earthquake scenario [23]. But during their service life, the structures are not only exposed to a single seismic event but also to multiple or repeated earthquake shocks. Previous works in this context have been conducted on various structures like buildings or bridges [14,17,18,20,24,25,41,[47], [48], [49],63, among others]. As a consequence of the later, structural damage accumulation by consecutive earthquake loading will be produced. The damage accumulation according to Iervolino et al. [25], is mainly due to two phenomena: i) continuous deterioration of the material which is called “aging” or ii) cumulative damage due to repeated load, also known as “sequential earthquakes”. The cumulative damage of the structure during its working life, is known as the Life Cycle of the structure [25,33,46,49,51,52,62, among others]. In another context, the life cycle of the structure can take lots of definitions. It can be considered as the cycle needed for a structure to be constructed, maintained and economically valued (i.e. LCSA [26]). It can also be considered as the time length of the structure until the occurrence of an event of interest (i.e. equipment failure, damage, complex system), or in other words, the time-to-event study. The later is known as the Survival Analysis. It is generally defined as a set of statistical methods to analyze data that has the time of occurrence of an event of interest as the outcome. Such analysis is not a new subject in medicine precisely [9,13,21]. For example, it is used to validate the impact of a certain disease on different types of patients, or the occurrence of specific symptoms after a drug. Reflecting this analysis in the geotechnical field, it is, to the knowledge of the authors, still a new topic [12,15,39].
Otherwise, the behavior of the structure (e.g. reinforced concrete buildings) under seismic sequence loading is assessed based on the Incremental Dynamic Analysis (IDA). It consists in subjecting the structural model to multiple ground motion records each scaled to different intensities [59]. Then, a limit state is considered in which the structure reaches failure when it exceeds the limits. On the other hand, previous studies in structural analysis have shown that, for mid-rise buildings, ten to twenty records are enough to have an estimation of the seismic demand [55,59, among others]. IDA in this case, is easily applied since it does not have a large set of earthquake scenarios to draw fragility curves. Whereas in earthquake geotechnical engineering and particularly in liquefaction related problems, this approach is not enough to represent the overall response of the geo-structure due to i) the multi-physical aspects of the soil (solid, water and air), ii) its history of loading that will affect its future behavior [36,56] and iii) the correlation of the soil response with several intensity measures of the real seismic motions (i.e. Arias intensity, number of cycles) [11].
The present work aims to quantify numerically the liquefaction-induced damage on an embankment due to sequential earthquake loading. Following the Performance Based Earthquake Engineering (PBEE) approach, a PSHA should be conducted in which the seismicity of the site and the occurrence rate of earthquake are identified. In this work, the site of concern is located in Mygdonia, Greece. The reference to the fully probabilistic hazard analysis in this study are based on the work of Aristizábal et al. [2]. A large number of time histories was generated using stochastic simulations from synthetic ground motion models (e.g. Rezaeian and Der Kiureghian [45] and Boore [7]). Nevertheless any other stochastic models are also suitable to be used under the proposed methodology. At the beginning of this work, the induced damage was quantified based on a set of GM records without sequences similarly to a site-specific seismic analysis. Concerning the sequential analysis, the methodology adopted in this study is shown in Fig. 1. Assuming that the working life of the embankment () is 100 years, and according to the PSHA and the catalog GM constructed for this site, the event rate of the mainshocks () is 0.44 events/year. Thus, 44 acceleration time histories () should occur during this period. Then the sequential loading is obtained by a random permutation of the obtained number of mainshocks. In order to calculate the survival function (), a threshold damage should be identified. Hence, the lifetime distribution of the embankment can be estimated as well as its Mean Time To Failure (MTTF, the expected time to failure for a non-repairable system). In this study, the survival analysis is computed based on a non parametric statistical method [27]. The main advantage behind this method is that it does not require the assumptions of a particular probability distribution (i.e. Weibull, exponential,log-logistic) of the structure's survival function. Also in this work, a numerical parametric analysis is performed in order to quantify the impact of considering (or not) the loading history and the recovery time between each ground motion on the obtained MTTF of the embankment. This study points out the importance of the history of loading since it affects the overall performance of the embankment. Finally, two synthetic ground motions models are assessed in order to generalize, to a certain extent, this work. The 2D finite element calculations were performed using the GEFDyn code [3]. For the soil behavior, an elastoplastic multi-mechanism model that takes into consideration the history of loading was used.
The paper is structured as follows. It starts by introducing the theory behind the survival analysis in Section 2. The geometry and the numerical model are shown in Section 3. The development of the used synthetic ground motion model is presented in Section 5. The site-specific seismic analysis of the embankment is developed in Section 6. Then, in Section 7, the sequential and the survival analysis are presented. Finally, the different types of sequential analysis approaches are developped in Section 8, as well as the consideration of different synthetic GM models. The paper is closed with conclusions.
Section snippets
Overview of the survival analysis
The survival analysis is the analysis of time-to-event data. These data describe the length of time until the occurrence of a well-defined end point of interest [9,28,53, among others]. Survival analysis is conducted via survival (or survivor) functions or hazard functions. Let T be a non-negative random variable that represents the surviving time. Denoting the duration of each event as t, the probability density function of T is , and its cumulative distribution is . First, the
Geometry
The model's geometry is a levee of 9 m high composed of dry dense sand. The foundation is formed of 4 m loose to medium sand (LMS) on the top of a 6 m dense sand. The bedrock is located under the dense sand. The water table starts 1 m below the surface to keep the dam dry. The inclination of the levee is a slope of 1:3 (vertical: horizontal). The geometry in this work is inspired from [Rapti et al. [43], Lopez-Caballero and Khalil [34]], and is detailed in Fig. 2.
Soil constitutive model
As for the constitutive model,
Assumptions for this study
For the study of the life-cycle of a levee subjected to sequential signals, basic assumptions are made:
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The cumulative damage of the levee is due to the effect of the series of mainshocks only. The effect of aftershocks is not taken into account.
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The effect of aging is not considered. For example, there is no consideration of the rain or sun, the wind load or any other type of loads that may be caused from external uncontrolled conditions. Also aging needs a deeper study of the material
Input ground motions
The seismicity of the site requires the knowledge of the geographical location, the site characteristic and the magnitude-frequency distribution of the earthquakes. The seismic hazard analysis involves the quantitative estimation of the ground motion characteristic at a particular site with the help of deterministic or probabilistic approaches. The Probabilistic Seismic Hazard Analysis (PSHA) sets a predictive relationship for each ground motion parameter in each source. This method combines
Site-specific seismic analysis
In Section 5, the used synthetic ground motions were presented. Two models were used (Mod.R and Mod.B). In this section, the response of the embankment based on each seismic load will be developed. Since the crest settlement is the mode of failure normally studied in case of embankments, it will be the parameter for the damage quantification. It is calculated by considering each ground motion as a single event. The percentage relative crest settlement as calculated by Swaisgood [58] is the
Survival analysis of the levee
Also in the scope of the PBEE methodology, the lifetime of the structure is the length of time until failure occurs. In order to calculate it, the degradation of the structure over time should be considered. Thus, the study of its performance due to sequential loading is required. In Section 6, it was shown that the two synthetic ground motion models gave different responses and that Mod.R induced more damage. Thus, this model will be used to compute the seismic sequential loads as well as the
Influence of various parameters on the survival analysis
To this point of this article, it was represented an analysis of the relative crest settlement of an embankment after both, a single event and sequential events of mainshocks. This later considered sequences of 44 mainshocks with a recovery time between each. In addition, the survival function was calculated based on a non parametric approach. In this section, answers on the following questions will be discussed respectively:
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What if the loading history was not considered? Technically, what will
Conclusions
This paper presents the survival analysis of a liquefiable embankment subjected to sequential earthquakes. First, it started with a site-specific seismic analysis where the damage was quantified after a set of unsequenced ground motion records. To generalize this work, two synthetic ground motion models were used. They were extracted from the studies of Rezaeian and Der Kiureghian [45] and Boore [7] and were designated in this paper as Mod.R and Mod.B respectively. Then, one synthetic model
CRediT authorship contribution statement
C. Khalil: Conceptualization, Methodology, Formal analysis, Software, Writing - original draft, review & editing. F. Lopez-Caballero: Conceptualization, Methodology, Formal analysis, Software, Writing - original draft, review & editing.
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.
Acknowledgment
This work, within the ISOLATE project, benefited from French state funding managed by the National Research Agency reference under program Mobility and Sustainable Urban Systems (DS06) 2017 reference No. ANR-17-CE22-0009. The research reported in this paper has been supported in part by the SEISM Paris Saclay Research Institute.
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