Probabilistic seismic slope stability assessment of geostructures Yiannis Tsompanakis*1, Nikos D. Lagaros2, Prodromos N. Psarropoulos1 and Evaggelos C. Georgopoulos1 1
Department of Applied Sciences, Technical University of Crete, Greece
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Institute of Structural Analysis and Seismic Research, National Technical University of Athens, Greece
Dates of submission: 11 Nov 2007, revision: 23 June 2008 and acceptance: 10 July 2008
* To whom correspondence should be addressed: Yiannis Tsompanakis Department of Applied Sciences, Technical University of Crete, University Campus, Chania 73100, Greece e-mail:
[email protected] Tel : +30-28210-37634 Fax : +30-28210-37843
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Abstract Typically, seismic analysis of large-scale geostructures, such as embankments, is performed by means of deterministic pseudostatic slope stability methods where a safety factor based approach is adopted. However, probabilistic seismic fragility analysis can be a more efficient and realistic approach for interpreting more accurately the seismic performance and the vulnerability assessment of an earth structure. There exist two major approaches for performing vulnerability analysis: either approximately assuming that the demand values follow the lognormal distribution, or numerically most often via Monte Carlo simulation (MCS) method, where the probability of exceedance for every limit-state is obtained performing MCS analyses for various intensity levels. The MCS technique is considered as the most consistent reliability analysis method, having no limitations regarding its applicability range. The objective of this work is to present the efficiency of the MCS-based numerical approach versus the commonly used lognormal empirical approach for developing fragility curves of embankments.
Keywords: Probabilistic analysis, slope stability, fragility curves, geostructures, Monte Carlo simulation.
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1.
INTRODUCTION
Deterministic analysis of any structural system requires certain assumptions regarding the geometry, the material properties and other structural attributes that may affect the overall capacity of the structure. Moreover, additional simplifying assumptions have to be taken into account with respect to loads, and especially those related to seismic demand. However, in real world engineering practice there exist uncertainties associated with both randomness (aleatory uncertainty) and imperfect knowledge (epistemic uncertainty) of the problems. The aforementioned uncertainties play a crucial role, especially when they are integrated in the framework of performance-based earthquake engineering. For instance, the majority of geotechnical
earthquake engineering applications
are highly stochastic problems.
Nevertheless, geotechnical seismic design compromises with the use of deterministic simplifications due to their low computational cost and their minimal complexity, while on the other hand, probabilistic methods are constantly gaining popularity due to the advances in computational resources and the significant developments of stochastic analysis methods. In general, embankments constitute large-scale geostructures of great importance (e.g., dams, solid waste landfills), the safety and serviceability of which are directly related to environmental and social-economical issues. This kind of structures became the subject of systematic research following the 1989 Loma Prieta (Seed et al. 1990), the 1994 Northridge (Stewart et al. 1994) and the 1995 Kobe (Bertero et al. 1995) earthquakes, after which extended investigations took place to examine failures that occurred on embankments due to seismic actions. In geotechnical engineering practice the slope stability of an embankment is most frequently evaluated utilizing the deterministic pseudostatic method, in which the horizontal and vertical pseudostatic inertial forces are included in the safety factor calculations. 3
Conversely, structural reliability methods have been improved considerably during the last twenty years, as discussed in several studies (see Schueller 1997, Wen et al. 2003, Schueller 2005, among others). Structural reliability analysis can be performed either with simulation methods, such as the Monte Carlo simulation (MCS) method, or with approximation methods (i.e., first or second order reliability methods (FORM, SORM), response surface methods (RSM), etc). However, MCS appear to be the only approach capable of achieving accurate solutions for complex problems that involve nonlinearities, a great number of random variables, large variations of the uncertain parameters, etc. The major advantage of MCS is that accurate solutions can be obtained almost for every problem, while its major deficiency is the increased computational cost which can be substantially reduced via efficient metamodels (Papadrakakis et al. 1996). In earthquake engineering applications, any reliability problem can be defined with two separate sets of variables, representing the demand and the capacity. A popular way of representing the probabilistic nature of the response of a structural system is via fragility curves. In this work limit-state probabilities are obtained for a wide range of intensity levels in order to construct the fragility curves of typical embankments. In order to further exploit the findings of the study, the fragility curves obtained by MCS are compared to those based on the assumption that the demand values follow the lognormal distribution, and it is shown that this assumption may lead to curves that differ considerably from those of the more rigorous MCS-based methodology. The present investigation involves the consideration of random variability of the mechanical properties of the soil, the geometry of the geostructure, as well as the seismic intensity levels (in terms of pseudostatic horizontal acceleration). The results demonstrate the efficiency of the proposed methodology for treating large-scale problems in geotechnical earthquake engineering.
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2.
SEISMIC SLOPE STABILITY
Since the failure of embankments is directly related to slope instabilities (either of the embankment mass or its foundation), seismic slope stability analysis is certainly considered as a critical component of the geotechnical seismic design process. In engineering practice it is based on three main categories of methods; namely: stress deformation analysis, permanent deformation analysis and pseudostatic analysis. Stress deformation analyses are mainly performed utilizing the finite element method with the application of complicated constitutive models to describe the potential nonlinear material behaviour. However, the parameters required for the implementation of the models cannot be easily or accurately quantified in the laboratory or in situ. It is evident that in the case of waste landfills this deficiency of stress deformation analysis is critical. In contrast, permanent deformation analyses are based on the calculation of seismic deformations through the simple sliding block approach proposed by Newmark (1965). Due to their complexity or their inherent uncertainties, the two aforementioned methods are usually excluded from the seismic design of embankments. Most frequently, the assessment of seismic slope stability is obtained via pseudostatic analyses. Based on the limit equilibrium methods of static slope stability analysis, and including the horizontal and vertical inertial forces, the results are provided in terms of the minimum factor of safety (FoS). The basic limitation of this method is the selection of the proper value of the seismic coefficient, which controls the inertial forces on the soil masses. Contemporary seismic norms (e.g., Eurocode 8 (EC8) Greek Seismic Code (EAK)), suggest the use of pseudostatic slope stability analysis utilizing a proper value for the related seismic coefficient (the so-called pseudostatic horizontal acceleration or PHA), equal to a specific portion of the design peak ground acceleration (PGA) at the site of interest. However, this approach does not take into account the actual dynamic response of the structure and the corresponding derformations, resulting to 5
incapability of predicting the actual response of the geostructure during a more severe seismic event. Therefore, in special cases it is advisable to adopt more accurate dynamic analysis procedures, e.g., for embankments characterized by high non-symmetric geometry or potentially problematic foundation, and/or when local site conditions (stratigraphy, topography, geomorphology) may play an important role (Zania et al. 2008a). In this study, the typical trapezoid embankment shown in Figure 1 is examined. The probabilistic calculations involved for the construction of the fragility curves of the geostructure (especially when dynamic analyses had to be performed) are extremely timeconsuming. Thus, in the current investigation pseudostatic analyses were conducted in which the required slope stability analyses were performed using a computer code developed by the authors (Zania et al. 2008b) that utilizes the simplified Bishop’s method for dry conditions and the assumption of circular failure surfaces within the body of the embankment. In addition, it is capable of randomly modifying the mechanical and geometrical characteristics of the model, as well as the acting pseudostatic horizontal acceleration values throughout the probabilistic analyses via the MCS method.
3.
PROBABILISTIC SEISMIC ANALYSIS OF EMBANKMENTS
Despite their very low probabilities of occurrence, severe earthquake events may produce extensive damages to engineering systems. Therefore, it is essential to establish a reliable procedure for assessing the seismic risk of real world structures and infrastructures. These procedures can only be created within the framework of probabilistic safety analysis (PSA), since it provides a rational framework for taking into account the various sources of uncertainty that may influence the structural performance. Seismic fragility analysis is considered as the core of PSA, which provides a measure of the safety margins of every engineering system under any specified hazard levels. 6
Nowadays, it is possible to predict via deterministic methodologies the level of ground shaking which is necessary to achieve a target level of response and/or damage state for a given structure. The material properties and certain other parameters that affect the overall capacity of the system can be determined in a similar manner. This type of deterministic assessment requires that certain assumptions should also be made about the ground motion and local site conditions, since both of them affect seismic demand. Nevertheless, the values of these parameters are not exact; they undoubtedly posses various types of randomness and uncertainty. An increasingly popular way of characterizing the probabilistic nature of seismic phenomena is through the use of the so-called fragility curves. A fragility curve provides the failure probability of a system as a function of certain seismic intensity measure. In the case of an embankment, fragility curves can provide the failure probability of the slopes as a function of the imposed pseudostatic acceleration.
3.1. Seismic fragility analysis of geostructures Fragility analysis is currently considered as one of the most useful computational tools for determining the dynamic behavior of any engineering system over a large range of seismic intensity levels. Specifically, a fragility curve of an earth structure provides the probability that its slope exceeds a given damage state for a certain seismic hazard level. In general, there exist two approaches to perform fragility analysis: the first is based on the assumption that the demand values follow the lognormal distribution, while the second is based on the Monte Carlo simulation technique performing reliability analysis of the slope for each intensity level (Tantalla et al. 2001). The main objective of the present investigation is to perform probabilistic slope stability analysis of a characteristic geostructure utilizing the pseudostatic method and to develop fragility curves to assess the vulnerability state of the examined earth structure. In the current 7
study, both the approximate and the numerical approaches have been implemented. The following general assumption has been made: the empirical fragility curves can be expressed in the form of the previously described lognormal distribution function and can be developed as a function of the pseudostatic horizontal acceleration (PHA) in order to represent the intensity of the seismic ground motion. The use of PHA is reasonable for this purpose, since this work adopts the pseudostatic slope stability approach, recommended by most of the modern seismic norms.
Similarly, in order to develop fragility curves using reliability
analysis methods (such as MCS), the embankment has to be assessed over a number of different PHA values. Therefore, seven different hazard levels (see Table 1) ranging from minor (PHA=0.01g) to very severe (PHA=0.50g) seismic intensity were studied, in an effort to cover a sufficient range of the seismic demand. As shown in Table 2, the Damage States (DS) considered are defined in terms of safety factor for the embankment slope stability and cover the whole range of its potential Vulnerability States (VS) and the relative Safety Margins (SM). To obtain discrete damage states, a properly selected range of the geostructural damage index must be specified. Similar correlation is used for buildings via proper damage indices, such as the interstory drift. The correlation of specific FoS values with the vulnerability assessment of the slopes is a crucial factor for the construction of the fragility curves. Therefore, the association presented in Table 2 is used in the present study, which follows the guidelines of the geotechnical design norm Eurocode 7 ((ΕC7) [18]) and the seismic norms (Eurocode 8 (ΕC8) [7], Greek Seismic Code (EAK) [11]), where the adopted characteristic damage index (FoS) values are described as: - FoS = 1.0 is the acceptable factor of safety for the stability of a slope under pseudostatic conditions (EC8, EAK), - FoS = 1.25 is the acceptable factor of safety for the stability of a slope for static conditions when considering the existence of water (EC7), 8
- FoS = 1.4 is the acceptable factor of safety under dry static conditions (EC7). In addition, a rather extreme upper value of FoS = 2.0 is considered as the indicator for very high safety margins. It has to be noted that the present implementation is a preliminary simplifying approach to deal with this important and complex problem. The above classification of vulnerability states with the aforementioned values of FoS does not take into account the permanent seismic deformations of the geostructure, which may be marginal even for the case of “pseudostatic failure” (for FoS < 1.0). Furthermore, the vulnerability and the corresponding DS of a geostructure, in terms of deformation and instability, may differ substantially from the values of Table 2 in the case of “problematic” materials (like soft clay, loose sand, organic material, waste, etc). In such special cases each vulnerability state should be determined more elaborately on a case-by-case basis. Nevertheless, there exist cases in which not even marginal deformations are allowed, and thus FoS should be much greater that unity. It has to be stressed that even the seismic norms use the “pseudostatic failure” value of FoS = 1.0 only as a lower bound and not as an absolute limit. Moreover, due to various simplifications used in the pseudostatic slope stability method, it is realistic to try to have a more clear perception of the embankment’s safety margins via its fragility curves constructed using values of FoS > 1.0. The employed procedure for the fragility curves generation that was employed in the present investigation can be summarized as follows: a. model the uncertainties of the geostructure b. use different levels of pseudostatic horizontal acceleration to perform pseudostatic slope stability analyses c. construct the fragility curves 9
Brief descriptions regarding the construction of fragility curves, both empirically and numerically, are presented in the subsequent sections.
3.2.
Lognormal assumption
A number of methodologies which are based on the lognormal assumption have been proposed for developing fragility curves, most of which have been applied to reinforced concrete framed structures (UTCB 2006, Kappos and Panagopoulos 2008) and bridges (Mander et. al 2007, Kwon and Elnashai 2008). In the present study, an attempt has been made to implement a similar methodology for developing approximate fragility curves for geotechnical earthquake engineering applications. In brief, the empirical approach is based on the assumption that the probability of reaching or exceeding a given DS can be modeled as a cumulative lognormal distribution. Recently, new proposals have been presented, in which other types of functions are used in the empirical approach (Leon and Atanasiu 2007). For a specific DS of the geostructure, which can be related to a certain damaxe index (i.e., FoS value), the probability of reaching or exceeding a DS is modeled as (Shinozuka et al. 2000):
1 PHA F DS | PHA ln tot PHA DS
(1)
where: F denotes the probability that the DS related to the specific hazard level PHA (i.e., the horizontal acceleration used in the pseudostatic simulations) will be equal or exceed the target range of damage index (see Table 1), Φ the standard normal cumulative distribution function, βtot the standard deviation of the natural logarithm of FoS for each DS , and PHADS the mean value of PHA at which the geostructure reaches the threshold of the specific damage level. The basic parameter in Eq. (1) is βtot, which is modeled as the combination of three contributors to damage variability: βC, βD and βDS, as described in the following equation: 10
β tot βC2 β 2D β 2DS
(2)
where βtot denotes the lognormal standard deviation that describes the total variability for each DS, βC the lognormal standard deviation parameter that describes the variability of the capacity, βD the lognormal standard deviation parameter that represents the variability of the seismic demand, βDS the lognormal standard deviation parameter that describes the uncertainty the estimation of the median value of the threshold of each DS. The proper range of value of βtot was initially examined by Dutta and Mander (1998), who used a theoretical approach which was subsequently confirmed via real damage data from recent California earthquakes. When there is no accurate data to determine the specific value of βtot, it usually takes values in the range of 0.30 to 0.70. A typical value of βtot for bridges is 0.60. The lower value (0.30) is used for nuclear structures due to the greater uncertainties that exist and mostly the increased safety demands that are imposed for such structures (NRC 2006), while the upper limit (0.70) is used for existing reinforced concrete structures. In the present study, a parametrical analysis with several values of βtot in the aforementioned range has been performed. Since no significant sensitivity of the fragility analysis results with respect to βtot was observed, the value of βtot = 0.30 has been selected as the most appropriate for this type of problems.
3.3. Monte Carlo simulation based fragility analysis Theory and methods of structural reliability have been developed substantially during the last twenty years and recent developments in various engineering fields are well documented in a large number of publications. Despite the improvements in the efficiency of the reliability analysis techniques, they still require disproportionate computational effort for treating realworld problems. Structural reliability analysis can be performed either with simulation methods, such as the Monte Carlo Simulation (MCS), or with approximation methods. For 11
instance, first and second order approximation methods (FORM and SORM) lead to formulations that require prior knowledge of the means and variances of the random variables and the definition of a differentiable failure function. In contrast, MCS appears to be the only universal method that can provide accurate solutions for problems regardless the complexity and/or the dimensions of the application. The major advantage of MCS is that accurate solutions can be obtained for any problem, while the only disadvantage is that it is generally more time-consuming. Fragility analysis based on MCS technique requires the repeated solution of the reliability problem for each set of random variables examined. It is obvious that the computational cost for developing fragility curves via MCS is great, especially when earthquake loading is considered in a probabilistic manner, since a vast number of analyses have to be performed for each hazard level. In this study the MCS method with Latin Hypercube Sampling (LHS) reduction technique is employed for performing the risk assessment analysis of large-scale embankments in order to accurately calculate each damage level probability. Typically, MCS uses a random number generator to select the value of each random variable using its probability density function. In general, in geotechnical engineering practice there are four probability density functions that are most commonly used: uniform distribution, triangular distribution, normal distribution, and lognormal distribution. Other probability density functions could be used provided that there were test data that matched those functions. More frequently and also in this study, the normal distribution is used, especially for the basic parameters encountered in pseudostatic slope stability analyses, such as unit weight (γ), friction angle (φ), cohesion (c). Details about the most important probabilistic parameters that are used in geotechnical engineering and the corresponding probability density functions can by found in Lacasse and Nadim (1996) and USACE (2006).
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For each simulation (embankment’s pseudostatic slope stability analysis) the factor of safety (FoS) is calculated. Each FoS is used to develop a probability density function of the earth structure vulnerability, while the number of unsatisfactory performance events is also determined. From the calculated probability density function of the FoS, or simply from the ratio of “failure simulations” (i.e., when FoS is less than a threshold value) over the total number of simulations defines the probability of geostructure’s unsatisfactory performance pexc (i.e., to exceed a certain FoS) is given by:
p exc
NH N SIM
(3)
where NH and NSIM are the number of “failure” and total simulations, respectively.
4.
NUMERICAL STUDY
In the current investigation, numerical pseudostatic slope stability analyses of the embankment shown in Figure 1 were performed, using the Bishop’s method in conjunction with the MCS technique to take into account the uncertainties of the problem. Note also that for the performed pseudostatic slope stability analyses not only the pseudostatic horizontal acceleration (PHA) was used, but the pseudostatic vertical acceleration (PVA) was also taken into account. In accordance to contemporary seismic norms [7, 11], vertical acceleration was set equal to PVA = ±0.50×PHA, to account for the vertical pseudostatic inertial force. For earthquake engineering applications, the reliability problem can be defined as a problem of two separate types of random variables representing the seismic demand and the capacity of the structure. In this work uncertainty both on capacity and demand has been considered. Uncertainty on capacity is taken into consideration through the soil mechanical properties and more specifically the unit weight (γ), friction angle (φ) and cohesion (c), as well as the embankment’s geometry. Regarding the uncertainty of the seismic demand it is 13
imposed through the seismic intensity levels. The geometry of this simple trapezoid example is determined using three parameters: height, slope width and deck width (see Figure 1). In this work two distinct cases were examined: in the first the geometry of the geostructure was considered deterministic (by simply using the mean values of the dimensions given in Table 3), while on the latter the dimensions were also considered as random variables. As aforementioned the normal distribution is used for the basic parameters encountered in pseudostatic slope stability analyses (i.e., geometry, unit weight, friction angle and cohesion), while the lognormal distribution is used for the seismic coefficient (PHA) levels. The mean values and the corresponding coefficient of variation (COV) values for the soil mechanical properties and the geometry are given in Table 3, while for the seismic demand coefficient are shown in Table 4. Though cohesion may have a greater scattering, thus greater COV value, than the other mechanical properties of the geostructure, the same COV value (10%) has been used for all geotechnical parameters for simplicity. In contrast, a greater variance (20%) has been allowed for the geometrical stochastic variables. Allowing a varying geometry is performed under the perspective of achieving an improved topology of the embankment, which is feasible provided that any possible change in the dimensions is not violating any other constructional and/or functional limitations. Two cases were examined in the present investigation: In the first, the so-called reference case, the mechanical properties of which are given in Table 3, the two approaches of fragility curves generation (empirical and MCS-based) were compared, while the effect of the randomness of geometrical dimensions was also examined. In the second set of analyses, a thorough investigation has been performed where the geometry was kept fixed and a significant number of various combinations of the embankment’s mechanical parameters (c, φ, γ) was examined, covering a wide range of soil and waste material properties. In all these analyses both approaches for fragility curves generation were implemented and compared. 14
4.1 Reference case results Initially, the reference case was examined using the data presented in Tables 2 to 4 examining both fragility analysis approaches and both types of geometry (deterministic and stochastic) of the embankment shown in Figure 1. For the generation of empirical fragility curves (via the standard lognormal type of Equation (1)), the median value of the FoS of the examined embankment was calculated for each intensity level. These values, which were used for the generation of the empirical fragility curves, are presented in Table 5. The comparison of the standard empirical approach (denoted as Lognormal in the graphs) and the numerical approach (denoted as MCS) is depicted in Figure 3, where both types of geometry (deterministic and stochastic) are shown. By comparing the plot of Figure 3a with the one in Figure 3b, it is obvious that the empirical fragility curves of the embankment with deterministic and stochastic geometry, respectively, have very small differences in their lower and upper intervals, and differ substantially in the region of moderate values of PHA and medium damage levels. More specifically, at these regions the fragility curves for the deterministic geometry have more steep inclination than the corresponding ones resulting from the stochastic geometry. Therefore, for the same PHA value, the probability of reaching a more severe damage level is bigger in the deterministic case than in the stochastic. A similar trend is observed in the region of medium PHA and DS when comparing the numerical fragility curves of the embankment with deterministic and stochastic geometry, also shown in the graphs of Figure 3. Nevertheless, in this case due to the more accurate MCS-based calculations, the trend of obtaining smoother curves for the stochastic geometry case is also observed, apart from the medium regions, in the upper and lower intervals of the fragility curves. The comparison of the empirical and the numerical approaches for the deterministic geometry types is presented in Figure 3a, where it is verified that there is consistency between 15
the two methods, except for the regions of the upper and lower values. In contrast, the comparison of the empirical and the numerical approaches for the stochastic geometry, shown in Figure 3b, reveals that there is a significant variation between the two methods in all the regions of the curves. The numerical method provides more reliable results in the whole range of seismic intensity and damage levels. This can be attributed to the fact that the degree of uncertainty becomes greater, since the number of the probabilistic variables are increased from four (c, φ, γ, PHA) for the embankment with deterministic geometry, to eight (c, φ, γ, PHA and the four geometrical parameters) for the earth structure with stochastic geometry.
4.2 Parametric study results After the investigation of the reference case, a parametric study was performed covering a significant number of geotechnical parameter combinations of the examined model. The purpose of this parametric study was twofold: a) to further investigate and compare the two fragility analysis approaches, and b) to examine how and to what extend the shape of the fragility curves is affected by the variation of the basic parameters (the soil material properties) of the problem. As it was previously discussed in the reference case, the empirical approach has better performance when the geometry of the embankment is considered deterministic rather than stochastic, thus, the geometry of the geostructure in this set of analyses was kept fixed, as determined by the values given in Table 3. Therefore, only the soil material properties and seismic intensity level were considered as stochastic variables. The stochastic data shown in Table 4 were used for the imposed PHA levels. The examined soil material properties of the embankment were the following: cohesion c=5 and 10kPa, friction angle φ=20º and 25º, and unit weight γ=10, 13, 15, 18 and 20 kN/m3. Concerning stochastic variability of the above data, the same conventions as in the reference case were used (see Table 3), i.e., the aforementioned values were considered as mean values 16
of the stochastic parameters that follow the normal distribution with 10% standard deviation. It has to be noted that the lower values of unit weight, which are unrealistic for typical soil embankments, correspond to special “embankments”, i.e., waste landfills. In such cases, the existence of stochastic variations of the geostructure’s properties is unavoidable and even more pronounced. The results that arise from the generation of the empirical fragility curves for the aforementioned sets of geotechnical parameter combinations, are presented in Figures 4 to 7. It is obvious that even for this set of runs having deterministic geometry, there are many cases where there exist great discrepancies between the two methods, not only in the lower and upper intervals but in the medium regions of the fragility curves as well. Nevertheless, there is no clear trend with respect to the comparison of the two fragility methods, i.e., to determine under what circumstances the empirical curves approach more closely the numerical ones. From a geotechnical point of view, there is a great variation with respect to the shape of the fragility curves. For instance, the impact of cohesion (c) is the following: when it increases (from 5 to 10 kPa) it shifts the fragility curves to the right. This remark is evident by comparing the plots of Figures 4 and 6 (with same friction angle φ=20º) and the plots of Figures 5 and 7 (with equal friction angle φ=25º). In other words, the increase of cohesion value results to a less vulnerable embankment. In addition, the fragility curves become slightly smoother as cohesion becomes greater. A similar trend is observed when considering the impact of the friction angle (φ). By comparing the plots of Figures 4 and 5 (with same cohesion c=5) and the plots of Figures 6 and 7 (with equal cohesion c=10), it is clear that the increase of friction angle value results to slightly smoother fragility curves that are also moved to the right, thus for the same PHA level lead to a less severe damage state. In contrast, the opposite trend is observed with respect to the unit weight (γ). More specifically, as γ increases, the fragility curves become much steeper and also are shrunk to the left. For 17
instance, by observing the plots in Figures 4 to 7, it is obvious that the smaller value of unit weight (γ=10 kN/m3, which is typical for waste landfills) has smoother vulnerability curves than those obtained for larger unit weight values.
5.
CONCLUSIONS
Fragility analysis offer a precise and efficient way to determine a geostructure’s performance and the evaluation of its seismic vulnerability for multiple hazard levels and multiple damage states, in the viewpoint of the state-of-the-art Performance-based Earthquake Engineering (PBEE). Under this perspective, this paper presented the two most commonly used approaches for establishing fragility curves, as the outcome of the probabilistic pseudostatic slope stability analysis, of large-scale embankments under pseudostatic seismic loading conditions. The proposed implementation involved the consideration of random variability of geometry, soil mechanical properties as well as the imposed seismic intensity levels on the geostructure. Both empirical and numerical methodologies were used and compared for the generation of the fragility curves of a typical embankment for various combinations of soil material properties and the influence of each parameter (c, φ, γ) was highlighted. The impact of the geometry type (fixed or random) was also considered, since it also affects substantially the shape of the curves; regardless of the fragility analysis method, the stochastic geometry results to smoother curves compared to the deterministic geometry. When comparing the two approaches it was found that in some occasions there is consistency between the two types of fragility analysis, usually in the medium regions of the fragility curves, when the geometry of the embankment is considered constant. However, in the majority of the deterministic geometry cases examined there are big intervals (especially in the upper and lower regions) where there are significant variations between numerical and 18
empirical curves. Moreover, this discrepancy between empirical and numerical fragility curves was more evident when the dimensions of the geostructure were also considered as random variables. Conclusively, despite its simplicity and its low computational cost, the application of empirical fragility curves has questionable accuracy compared to the more accurate (provided that the MCS sampling size is big enough) and more elaborate numerical approach. Nevertheless, the generation of the fragility curves using the numerical methodology is a very computational intensive task, since a much greater number of simulations must be executed. The implementation of efficient approximation techniques (such as neural networks), which is the next step of this ongoing research, can alleviate this deficiency of the MCS-based approach and increase the applicability range and the effectiveness of the numerical methodology.
6.
ACKNOWLEDGEMENTS
This paper is part of the 03ED454 research project, implemented within the framework of the “Reinforcement Program of Human Research Manpower” (PENED) and co-financed by National and Community Funds (75% from E.U.-European Social Fund and 25% from the Greek Ministry of Development-General Secretariat of Research and Technology).
7.
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TABLE LEGENDS
Table 1: Pseudostatic PHA intensity levels.
Table 2: Correlation of damage index (FoS) with the Vulnerability State and Safety Margins of the embankment
Table 3: Probabilistic data for the mechanical and geometrical properties of the embankment.
Table 4: Probabilistic data for the seismic coefficient (PHA).
Table 5: Reference case: median FoS values used in the empirical approach.
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TABLES Table 1: Pseudostatic PHA intensity levels. Case
Characterization
PHA value
Case I Case II Case III Case IV Case V Case VI Case VII
Minor Slight Low Low to Moderate Moderate Severe Very Severe
0.01g 0.05g 0.10g 0.20g 0.30g 0.40g 0.50g
Table 2: Correlation of damage index (FoS) with the Vulnerability State and Safety Margins of the embankment Vulnerability Safety Range of State Margins damage index Optimal Very High FoS > 2.0 Sufficient High 1.4 < FoS
