1st Middle European Conference on Landfill Technology
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.
1 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.
2 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.
1st Middle European Conference on Landfill Technology
Table 1. Shear strength parameters of untreated MSW (collected from accessible literature)
Reference
Landva et al.
(1984)
Van Impe (1998)
Eid et al. (2000)
Mulabdic et al.
(1998)
Kovacevic et al.
(2002)
Vukelic et al.
(2004)
Cowland et al.
(1991)
Kavazanjian (2001)
Mazzucato et al.
(1999)
Zhu et al. (2003)
Kölsch (1993)
Greco and Oggeri
(1993)
Hossain (2002)
Age of waste
samples
fresh
1 year old
fresh
/
/
9 month old
fresh
/
/
/
/
/
/
/
/
/
/
/
Cohesion
(kN/m2)
16
16
23
10
7
7
28
10
15
10
15,7
23,5
0
0
0
0
0
10
Friction
angle (0)
40
33
24
33,6
38
42
26,5
23
15
25
21
22
41
25
28
30
34
18-43 (30,5)
/
/
/
20
0
20
0
38
30
/
40
35
/
large ring
shear
apparatus
/
28-52 (38)
12-20 (17)
/
15-43 (29)
19-26 (22)
Testing device
LDS
/
simple shear
/
/
back analysis
direct shear
back analysis
in situ ; LDS
back analysis
Additional comments
/
/
/
/
/
/
/
suggested values
suggested values
/
balled waste, low density
balled waste, high density
/
at normal stresses ≈ 180 kPa
at normal stresses ≈ 110 kPa
at normal stresses ≈ 45 kPa
at normal stresses ≈ 60 kPa
14 kPa < normal stresses < 38 kPa
at normal stresses <= 20 kPa
20 < normal stresses <= 60 kPa
at normal stresses > 60 kPa
estimated values based on the
research of various authors
saturated waste
dry waste
/
55
33
/
estimated values according to the
literature and measurements
/
10,5 – 11,4
(11)
13,2 – 14,4
(13,8)
back analysis
/
/
4
56
back analysis
/
/
43
26
direct shear
/
at displacement of 25 mm,
bioreactor landfill
lower values
/
/
0
39
/
/
17
19
33
28
/
24
18
/
/
22
6,4 -31,4
17
40,4 - 49,6
fresh
0
26,4
old
fresh, loose
fresh
degraded
degraded
degraded
10 – 15
years old
0
<5
30-50 (40)
5-15 (10)
16-32 (24)
0-10 (5)
17,7
38-42 (40)
38-40 (39)
23-27 (25)
19-24 (21,5)
17-23 (20)
/
/
/
/
/
disturbed sample
/
lower values of shear strength
parameters, hardening model with
fibrous component
/
/
/
/
/
17
33
/
/
simple shear
back analysis
in situ ; LDS
direct shear
direct shear
undisturbed sample
2.1 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:
1st Middle European Conference on Landfill Technology
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
mean
Variance
Standard
deviation
Characteristic
cohesion value
Characteristic
friction angle value
Cohesion
23,4
521,7
22,8
12,01
/
Friction
angle
24,3
157,7
12,6
/
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
Reference
Jones and
Dixon (2003)
Kavazanjian
(2001)
Van Impe
(1998)
Cohesion
(kPa)
Shear Friction
Angle (0)
Additional comments
5
25
at all normal stresses
24
0
20
0
20
0
33
0
38
30
for normal stresses <= 30 kPa
for normal stresses > 30 kPa
for 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.
2.2 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.
1st Middle European Conference on Landfill Technology
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:
f ( ' ) 
1
 2
e
1  '  
 
2   
2
(2)
where σ is standard deviation, μ is mean, e is the base of the natural logarithm, π is the constant Pi and φ’ 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:
f (c ' )  e c
'
(3)
where λ is exponential function parameter (λ = 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).
1st Middle European Conference on Landfill Technology
Figure 3. Negative linear correlation between shear friction angle and cohesion
3 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 + φ'/2 (Rankine´s 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.
Factor of safety
Failure surface
1.8
1.6
1.4
1 2
0
20
40
60
80
1 2 3 4 5 6 7 8
3
4 5 6 7 8
100 120 140 160 180 200 220 240
Distance
Figure 4. The analyzed landfill section with polygonal failure surfaces
1st Middle European Conference on Landfill Technology
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 geomembraneclay residual interface strength.
The following referent values are used in the analyses:
waste:   13 kN / m 3 ; c'  19 kPa ; ' 24 0
bottom liner:   20 kN / m 3 ;
c'  0 ; ' 12 0
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 S
Mean
F of S
Standard
deviation
Min
F of S
Max
F of S
No. of
trials
P (Failure)
(%)
Reliability
Index
1.43
1.54
0.321
0.82
2.65
2000
3.43
1.673
Probability Distribution Func tion
100
Probability (%)
80
P (F of S < x)
60
40
20
0
0.000
P (Failure)
1.825
3.651
Factor of Safety
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 β = 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 β, for ultimate limit state design and for the most
common design situation, should be 4.2. Target failure rate associated with recommended β 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.
1st Middle European Conference on Landfill Technology
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 level
High
Good
Above average
Below average
Poor
Unsatisfactory
Hazardous
Reliability
index (β)
5.0
4.0
3.0
2.5
2.0
1.5
1.0
Probability of
failure (pf)
0.0000003
0.00003
0.001
0.006
0.023
0.07
0.16
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 KovacevicZelic 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.
4 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
5 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.
6 References
Cowland J.W., Tang K.Y., Gabay J. 1993. Density and strength properties of Hong Kong refuse, Proceedings Sardinia 93,
Fourth International Landfill Symposium, S. Margherita di Pula, Cagliari (Italy), 1433 - 1446
Eid H.T., Stark T.D., Evans W.D., Sherry P.E. 2000. Municipal solid waste slope failure. I: Waste and foundations soil
properties, Journal of Geotechnical and Geoenvironmental Engineering, 126(5), 397-407
Gabr A., Valero S.N. 1995. Geotechnical properties of municipal solid waste, Geotechnical Testing Journal, GTJODJ, 18(2),
241-251
Greco O.Del., Oggeri C. 1993. Geotechnical parameters of sanitary wastes, Proceedings Sardinia 93, Fourth International
Landfill Symposium, S. Margherita di Pula, Cagliari (Italy), Vol.2, 1421-1431
1st Middle European Conference on Landfill Technology
Ivsic T., Petrovic I., Veric F. 2004. Overview of parameters for stability analysis on waste disposal sites, Gradevinar 56(11),
665-674 (in Croatian)
Jones D.R.V., Dixon N. 2003. Stability of Landfill Lining Systems: Report No. 1 – Literature Review (R&D Technical Report P1385/TR1), Environment Agency, Rio House, Waterside Drive, Aztec West, Almondsbury, Bristol, BS32 4UD
Kavazanjian Jr.E. 2001. Mechanical properties of municipal solid waste, Proceedings Sardinia 01, 8 th International Waste
Management and Landfill Symposium, S. Margherita di Pula, Cagliari (Italy), Vol.3, 415-424
Kovacevic-Zelic B., Kvasnicka P., Domitrovic D. 2002. Stability Analysis for the Landfill Jakusevec, Proceedings of the 12 th
Danube-European Conference, Passau (Germany), 503-506
Kölsch F. 1993. The bearing behaviour of domestic waste and related consequences for stability, Proceedings Sardinia 93,
Fourth International Landfill Symposium, S. Margherita di Pula, Cagliari (Italy), Vol.2, 1393-1410
Krahn J. 2004. Stability Modeling with SLOPE/W, Calgary: GEO-SLOPE/W International Ltd
Landva A.O., Clark J.I., Weisner W.R., Burwash W.J. 1984. Geotechnical Engineering and Refuse Landfills, 6th National
Conference On Waste Management in Canada, Vancouver (British Columbia)
Mazzucato A., Simonini P., Colombo S. 1999. Analysis of block slide in a MSW landfill, Proceedings Sardinia 99, Seventh
International Waste MAnagement and Landfill Simposium, S. Margherita di Pula, Cagliari (Italy)
Mulabdic M., Sesar S., Zgaga V. 1998. Ispitivanje parametara čvrstoće za analize stabilnosti odlagališta otpada Jakuševec, V.
Međunarodni simpozij Gospodarenje otpadom, Zagreb (Croatia), 413-420.
Petrovic I., Ivsic T., Veric F., Kovacic D. 2006. Sensitivity analyses of global stability in sanitary landfills, Sixth European
Conference on Numerical Methods in Geotechnical Engineering, September 2006, Graz (Austria), 501 - 506
Probabilistic model code, Part 1 – Basis of Design. Joint Committee on Structural Safety, JCSS-OSTL/DIA/VROU/-10-112000, 12th Draft
Schneider H.R., 1997. Definition and determination of characteristic soil properties, Proc. 14th Int. Conf. on Soil Mechanics
and Geotechnical Engineering, Hamburg, 2271-2274
Singh S., Murphy J. 1990. Evaluation of the stability of sanitary landfills, Geotechnics of Waste Fills – Theory and Practice,
ASTM STP 1070, Arvid Landva, G. David Knowles, editors, American Society for Testing and Materials, Philadelphia, 240
– 258
Spencer E. 1967. A method of analysis of the stability of embankments assuming parallel interslice forces, Geotechnique,
London, Vol. XVII, No. 1, 11-26
U.S. Army Corps of Engineers 1999. Risk-based analysis in geotechnical engineering for support of planning studies,
engineering and design, Rep. No. 20314-1000, Dep. of Army, Washington, D.C.
Van Impe W.F. 1998. Environmental geotechnics – ITC5 – reports and future goals, Geotechnical Hazards – proceedings of
the XIth Danube – European conference on soil mechanics and geotechnical engineering, Porec (Croatia), 127-155.
Vukelic A., Kovacevic-Zelic B., Drnjevic B. 2004 Landslide at the Jakusevec landfill, Proceedings of the Third european
Geosynthetic Conference, Munich (Germany), 87-92
Zhu X., Jin J., Fang P. 2003. Geotechnical behaviour of the MSW in Tianziling landfill, Journal of Zhejiang University, Vol.4,
No.3, 324-330
The paper may be considered for
(Please indicate your choice by putting  in the appropriate box)
1. Oral Presentation
2. Poster Session
