ࡱ> ~5@bjbj22 XXXHMNNN:^^^8^_:ycNdNdNdddh5i<qi 3555MhhR$cRvNxch"hxxvNddd3xlNdNdd3x32",Nkddc Crh^. I0y?6N?\kbTk?N0itn.3qs3iiivv::dL^::^Probabilistic Stability Analysis of Sanitary Landfill Jakusevec I. Petrovic University of Zagreb, Faculty of Geotechnical Engineering, Varazdin, Croatia Keywords: landfill, shear strength, stability, probability, waste ABSTRACT: Fifty pairs of municipal solid waste strength parameters were obtained through extensive literature review. Central tendency and dispersion of collected data were obtained. Chi-Square goodness of fit test revealed that shear friction angle obey normal probability density function while cohesion obey exponential probability density function. The parameters are negatively correlated which reflects the tendency of a larger value of cohesion to occur with a smaller value of shear friction angle. Obtained statistical data are used as input parameters for probabilistic stability analysis of sanitary landfill Jakusevec in Zagreb. The analysis is conducted in order to establish the level of confidence that can be placed in design of sanitary landfill Jakusevec. Introduction Global stability of the sanitary landfill is one of the dominant problems in the landfill design from the geotechnical point of view. In order to ensure the stability of landfill the design considerations should include the stability of waste, bottom liner and drainage layers, final cover system, as well as the stability of foundation soil. The stability of landfill is affected by the wide range of parameters. Similar to common geotechnical design for embankment stability, the design situations for landfill consider the geometry of the design section, the strength parameters of main materials and the influence of possible pore pressures (leachate in waste). Probabilistic stability analysis, as a tool for estimation of safety level in geotechnical problems, became in the last few years a quite popular method. Moreover, in the case of landfills where uncertainties of waste strength parameters are very high, the use of probabilistic methods is fully justified. In order to conduct the probabilistic stability analysis it is important to know basic statistical data of parameters which are included in the analysis. Therefore, 50 pairs of MSW strength parameters were collected to determine central tendency, dispersion and probability density function of mentioned parameters. One of the most useful figures which can be obtained with probabilistic stability analysis is the reliability index of analysed structure. The reliability index value expresses the distance of the mean margin of safety M (M is defined as the difference between the resistance and the load) from its critical value. It can be also considered as a way of normalizing the factor of safety with respect to its uncertainty. From practical point of view the reliability index presents the expected performance level of analysed structure. Probability of failure (probability of obtaining factor of safety less then 1.0), which is usually presented through factor of safety probability distribution function, is also a very common figure used in evaluation of performance level of analysed structure. Both values are obtained for the proposed design of sanitary landfill Jakusevec, the main waste dump in Zagreb, the capital of the Republic of Croatia. The dump has been used since 1960s for disposal of various kinds of waste while the rehabilitation of landfill started in the mid 90-ties. Obtained values represent the current level of confidence that can be placed in design of Jakusevec landfill. Collected data and basic statistical indicators Fifty pairs of shear strength parameters were collected from the available literature. They are shown in Table 1. It should be mentioned that presented data do not include parameters obtained with triaxial testing device since these parameters are not real strength parameters, but only virtual strength parameters defined at arbitrary chosen level of deformation. Table 1. Shear strength parameters of untreated MSW (collected from accessible literature) ReferenceAge of waste samplesCohesion (kN/m2) Friction angle (0)Testing deviceAdditional commentsLandva et al. (1984)fresh1640LDS/1 year old1633/fresh2324//1033,6/Van Impe (1998)/738//9 month old742simple shear/fresh2826,5//1023/suggested values/1515/suggested values/1025back analysis//15,721direct shearballed waste, low density/23,522balled waste, high density/041//025back analysisat normal stresses H" 180 kPa/028at normal stresses H" 110 kPa/030at normal stresses H" 45 kPa/034at normal stresses H" 60 kPa/1018-43 (30,5)in situ ; LDS14 kPa < normal stresses < 38 kPa/200back analysisat normal stresses <= 20 kPa/03820 < normal stresses <= 60 kPa/2030at normal stresses > 60 kPaEid et al. (2000)/4035/estimated values based on the research of various authorsMulabdic et al. (1998)/28-52 (38)12-20 (17)large ring shear apparatussaturated waste/15-43 (29)19-26 (22)dry wasteKovacevic et al. (2002)/5533/estimated values according to the literature and measurementsVukelic et al. (2004)/10,5 11,4 (11)13,2 14,4 (13,8)back analysis/Cowland et al. (1991)/456back analysis/Kavazanjian (2001)/4326direct shear//039at displacement of 25 mm, bioreactor landfill/1733simple shearlower values/1928back analysis/Mazzucato et al. (1999)/2418in situ ; LDSundisturbed sample/2217disturbed sampleZhu et al. (2003)/6,4 -31,440,4 - 49,6direct shear/ Klsch (1993)fresh026,4direct shearlower values of shear strength parameters, hardening model with fibrous componentold017,7Greco and Oggeri (1993)fresh, loose< 538-42 (40)//fresh30-50 (40)38-40 (39)//degraded5-15 (10)23-27 (25)//degraded16-32 (24)19-24 (21,5)//degraded0-10 (5)17-23 (20)//Hossain (2002)10 15 years old1733//Measures of central tendency, dispersion and characteristic values of collected data Even though the origin of data, with respect to the testing method, is various, all data are treated equally. Two main reasons were found for such approach. First, there are not enough data to make a separate analysis for each type of tests. And secondly, no matter what approach different authors used to obtain strength parameters, they are all, at the end, used for calculation of shear strength of waste according to the Mohr-Coulomb failure criterion. According to these observations, basic statistical indicators of shear strength parameters are calculated. It should be noted that shear strength parameters collected by Singh and Murphy (1990) are also included. In addition, by using Schneider (1997) equation: Xk = Xm 0,5SD (1) where Xk is the characteristic value of soil property, Xm is the arithmetic mean value and SD is the standard deviation, it is possible to obtain characteristic cohesion and shear friction angle values. The results are shown in Table 2. Table 2. Measures of central tendency, dispersion and characteristic values of collected data Arithmetic meanVarianceStandard deviationCharacteristic cohesion valueCharacteristic friction angle valueCohesion23,4521,722,812,01/Friction angle24,3157,712,6/18,05 From the results presented in Table 2 it can be noticed that arithmetic mean value for cohesion is in good correlation with recommendations from Kavazanjian (2001) and Van Impe (1998) (at some but not all stress levels), while shear friction angle is in good correlation with recommendation from Jones and Dixon (2003). The summary of recommendations from referred authors is presented in Table 3. Table 3. Shear strength parameters recommendations according to various authors ReferenceCohesion (kPa)Shear Friction Angle (0)Additional commentsJones and Dixon (2003)525at all normal stressesKavazanjian (2001)24 00 33for normal stresses <= 30 kPa for normal stresses > 30 kPaVan Impe (1998)20 0 200 38 30for normal stresses <= 20 kPa for n. s. between 20 - 60 kPa for normal stresses > 60 kPa Characteristic values presented in Table 2 reveal that recommendations of referred authors are not conservative as it may seem at first impression. It should be also noticed that standard deviation and variance shows that shear friction angle have a much less dispersion of data than cohesion. The reasons for such high standard deviation in cohesion value lays in fact that fibrous component of waste have a great influence on cohesive behavior of untreated MSW. Chi-square goodness of fit test and correlation Chi-Square test was conducted in order to find theoretical probability density function which empirical frequencies of obtained data obey. Both variables are tested to several standard theoretical probability density function, namely normal, exponential, gamma, log-normal and Chi-Square distribution. The significance level of 0.05 was selected for all tests. From selection of continuous distributions it is obvious that cohesion and shear friction angle are treated as continuous variables since both parameters can adopt any value. The testing was carried out with use of Statistica, the software package for data analysis. Obtained results have shown that shear friction angle obey normal probability density function while cohesion obey exponential probability density function. The results are presented in Figure 1 and Figure 2.  Figure 1. The goodness of fit of the shear friction angle frequencies to the normal distribution Figure 1 shows that empirical frequencies of shear friction angle are in very good agreement with normal distribution which is presented with equation:  EMBED Equation.3  (2) where  is standard deviation,  is mean, e is the base of the natural logarithm,  is the constant Pi and  is shear friction angle.  Figure 2. The goodness of fit of the cohesion frequencies to the exponential distribution Figure 2 shows that empirical frequencies of cohesion are in good agreement with exponential distribution which is presented with equation:  EMBED Equation.3  (3) where  is exponential function parameter ( = 0,04266873545384), e is the base of the natural logarithm and c is cohesion. Figure 3 presents correlation between these two variables. Coefficient of correlation reveals that shear friction angle and cohesion are correlated to medium extent. The parameters are negatively correlated which reflects the tendency of a larger value of shear friction angle to occur with a smaller value of cohesion. Interestingly, the obtained coefficient of correlation for waste strength parameters is in good agreement with numerous laboratory tests on soils which shows that soil shear friction angle and cohesion is also often negatively correlated with correlation coefficient in range from -0.72 to 0.35 (Krahn, 2004).  Figure 3. Negative linear correlation between shear friction angle and cohesion Probabilistic stability analysis The design stability of landfills is usually verified by common limit equilibrium methods. Although the applicability of these methods for waste landfill is often questioned, their routine use in many geotechnical applications makes them still a popular tool for common design. Actually, the main goal in most geotechnical design analyses is not to exactly replicate all physical conditions, but to simulate crude stability mechanisms and apply an adequate margin of safety against unfavorable event. The limit equilibrium methods are based on rigid body static principles. They do not consider the deformations of the real body and stress-strain compatibilities. As a representative for limit equilibrium method the Spencer (1967) method is chosen, which is often routinely used for the polygonal sliding surfaces. The analyses are made using program package Slope/W and critical sliding surfaces are obtained. The basic data for the analyses are obtained from design documentation of waste landfill Jakusevec in the City of Zagreb in Croatia. The analyzed sliding surfaces were chosen as polygonal surfaces with bottom part passing through bottom liner and upper part passing through waste. The slope of the waste part of sliding surface is inclined to horizontal at an angle 45 + '/2 (Rankines active state plane). The referent geometry of analyzed landfill cross section is shown in Figure 4. The position of sliding surface with minimal factor of safety with respect to the Spencer method (No. 3) is also shown.  Figure 4. The analyzed landfill section with polygonal failure surfaces The strength parameters for waste were selected to represent the common suggested design values (Ivsic et al. 2004; Petrovic et al. 2006). The values for bottom liner are selected as a conservative choice of geomembrane-clay residual interface strength. The following referent values are used in the analyses: waste:  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3  bottom liner:  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3  Based on the selected parameters and statistical data presented in previous chapters, probabilistic stability analysis for a critical sliding surface (a surface with a minimal factor of safety value) was conducted. The results of analysis are presented in Table 4 and Figure 5. Table 4. Factor of safety, probability of failure and reliability index obtained from probabilistic analysis Spencer F of SMean F of SStandard deviationMin F of SMax F of SNo. of trialsP (Failure) (%)Reliability Index1.431.540.3210.822.6520003.431.673  Figure 5. Probability distribution function of the 2000 Monte Carlo factors of safety showing probability of failure From Table 4 it can be noticed that mean factor of safety value obtained with probabilistic method is slightly higher then the factor of safety obtained with Spencer method, which may lead to conclusion that overall stability of landfill is sufficient. However, from the same table and Figure 5 it could be also noticed that probability of failure is 3.43%. According to the Table 5, the expected level of performance for calculated probability of failure is between poor and unsatisfactory. Since there is no direct relationship between the deterministic factor of safety and probability of failure, there is another way of looking at the risk of instability known as a reliability index. Reliability index describes stability by the number of standard deviations separating the mean factor of safety from its defined failure value of 1.0. For this analysis the calculated reliability index is  = 1.673. Once again, Table 5 suggests that, for the calculated reliability index of sanitary landfill Jakusevec, expected level of performance is somewhat between unsatisfactory and poor. In addition, Probabilistic model code (2000) recommends that , for ultimate limit state design and for the most common design situation, should be 4.2. Target failure rate associated with recommended  is 10-5, which is also significantly lower value then the calculated one. In other words, both references suggest that for design of sanitary landfill Jakusevec which can be considered as safe reliability index should be much higher and probability of failure much lower then the calculated one. Table 5. Expected levels of performance in terms of probability of failure and corresponding reliability indexes (U.S. Army Corps of Engineers, 1999) Expected performance levelReliability index ()Probability of failure (pf)High5.00.0000003Good4.00.00003Above average3.00.001Below average2.50.006Poor2.00.023Unsatisfactory1.50.07Hazardous1.00.16 Based on the calculated and recommended reliability index value it can be concluded that the level of confidence that can be placed in a design of Jakusevec landfill is not too high, in spite the fact that the factor of safety value is satisfactory. The reason for such low reliability index value is most probably not connected with waste strength parameters but with low strength parameters of bottom liner, since it is well known fact that bottom liner is actually the most critical element of modern sanitary landfills with respect to stability issues. The results from this analysis are in very good agreement with results from analysis conducted by Kovacevic-Zelic et al (2002). The authors also found that proposed design of sanitary landfill Jakusevec needs to be improved with interventions in bottom liner system or perhaps with interventions in geometrical issues of landfill design. Conclusions Waste landfill is unique type of object. Proper design of waste landfills requires interdisciplinary approach of wide variety of experts. Between them, the geotechnical engineer has important role to ensure mechanical stability of waste landfill for proper functioning of all landfill components. Probabilistic stability analyses with variation of shear strength parameters of waste were performed. The probability of failure and reliability index of sanitary landfill Jakusevec was determined. The following conclusions can be drawn from the presented results: mean value for cohesion is in good correlation with recommendations from Kavazanjian (2001) and Van Impe (1998) mean value for shear friction angle is in good correlation with recommendation from Jones and Dixon (2003) cohesion obey exponential probability density function shear friction angle obey normal probability density function shear friction angle and cohesion are correlated to medium extent in spite the fact that factor of safety is satisfactory, the expected level of performance of sanitary landfill Jakusevec is between unsatisfactory and poor to increase expected level of confidence it is necessarily to make improvements in design of bottom liner or in the geometrical issues of landfill design Acknowledgements The financial support of the Ministry of Science, Education and Sports of the Republic of Croatia for the project "Characterization of municipal solid waste" (160-0831529-3031) is gratefully acknowledged. 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