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```CHARLES DARWIN UNIVERSITY
BACHELOR OF ENGINEERING
Slope stability analysis of
tailings dam embankments
Zoe Knight
Student number: 233922
29 May 2015
Supervisor: Professor Charlie Fairfield
Co-supervisor: Professor David Lilley
ABSTRACT
Keywords: Slope stability, tailings dams, limit equilibrium methods, reliability analysis,
Monte Carlo simulation
The stability of an embankment is a vital consideration in tailings dam design,
construction and management. A general overview of dam construction methods, potential
failure mechanisms for tailings dam embankments, and different methods of analysis
including deterministic and probabilistic analyses are included in this thesis.
Deterministic analysis includes a discussion of the stability of tailings dam
embankments by analysing the factors of safety using unique values of each soil, and
geometric, property. The factor of safety is calculated using stability charts (Taylor (1948),
Bishop and Morgenstern (1960), and Spencer (1967)), and limit equilibrium methods such as
those by Fellenius (1936), Janbu (1954), Bishop (1955), Morgenstern and Price (1965), and
Spencer (1967). A parametric study and sensitivity analysis were performed to elucidate the
effects of varying the shear strength parameters and embankment geometry on the stability of
an embankment. Probabilistic methods of analysis use random variables in place of
deterministic values to find a range of factors of safety that describes the inherent variability
in both the soil and geometry in the tailings dam. A first order, second moment, (FOSM)
analysis and Monte Carlo simulations were performed to describe the factors of safety in a
risk analysis-based context.
The deterministic and probabilistic methods were applied to simple baseline cases and
a case study of an existing tailings dam using hand calculations performed in Microsoft
Excel™, and GeoSlope SLOPE/W 2012 software. The embankments were analysed under
both short- and long-term conditions (i.e. in terms of both total and effective stress
parameters), with the effects of pore water pressure and horizontal seepage considered.
This research found that there were advantages and disadvantages to both the
deterministic and probabilistic methods of analysis. The deterministic methods were generally
easy to use but the input variables were single values which may not accurately reflect the
anisotropy or inhomogeneity within the soil in the tailings dam embankment. This could
produce factors of safety that are not representative of the entire embankment. On the other
hand, probabilistic analyses took into account such anisotropy and inhomogeneity, but
required a more detailed understanding and analysis thereof. It was found that deterministic
methods of analysis were suitable in calculating factors of safety, however it is also beneficial
to perform probabilistic analyses to find the factor of safety and refined safety margins when
the variability from uncertainties is included.
ii
List of abbreviations ................................................................................................................. vii
List of tables ............................................................................................................................viii
List of figures ............................................................................................................................. x
List of symbols ........................................................................................................................ xiv
1
2
Introduction ......................................................................................................................... 1
1.1
Stability of tailings dams ............................................................................................. 1
1.2
Problem definition ....................................................................................................... 2
1.3
Thesis outline ............................................................................................................... 3
1.4
Thesis outcomes ........................................................................................................... 4
Tailings dam failures ........................................................................................................... 5
2.1
Failures over the last century ....................................................................................... 5
2.2
Types of failure .......................................................................................................... 10
2.2.1
Rotational sliding................................................................................................ 10
2.2.2
Translational sliding ........................................................................................... 12
2.2.3
Foundation failure .............................................................................................. 13
2.2.4
Overtopping ........................................................................................................ 13
2.2.5
Piping .................................................................................................................. 14
2.2.6
Liquefaction ........................................................................................................ 15
2.2.7
Seepage ............................................................................................................... 15
2.3
3
Slope stability parameters ................................................................................................. 19
3.1
3.1.1
Undrained condition ........................................................................................... 20
3.1.2
Drained condition ............................................................................................... 21
3.2
4
ANCOLD consequence category............................................................................... 17
Factor of safety .......................................................................................................... 23
3.2.1
ANCOLD recommendations .............................................................................. 24
3.2.2
Tailings dam legislation ..................................................................................... 25
Deterministic methods of analysis .................................................................................... 26
4.1
Stability charts ........................................................................................................... 26
4.1.1
Taylor’s stability charts ...................................................................................... 26
4.1.2
Bishop and Morgenstern 1960............................................................................ 29
4.1.3
Spencer 1967 ...................................................................................................... 32
4.1.1
Janbu 1968 .......................................................................................................... 33
iii
4.2
Summary of slope stability charts .............................................................................. 35
4.3
Limit equilibrium methods ........................................................................................ 36
In terms of total stress ....................................................................................................... 37
In terms of effective stress ................................................................................................ 38
5
4.3.1
Fellenius’ method 1936 ...................................................................................... 38
4.3.2
Bishop’s simplified method 1955 ....................................................................... 40
4.3.3
Morgenstern and Price 1965 ............................................................................... 41
4.3.4
Spencer’s method 1967 ...................................................................................... 42
4.3.5
Janbu’s simplified method 1968 ......................................................................... 43
4.3.6
General Limit Equilibrium method 1977 ........................................................... 46
4.4
Summary of limit equilibrium methods ..................................................................... 47
4.5
Limitations of limit equilibrium methods .................................................................. 48
4.6
Computer modelling .................................................................................................. 48
4.6.1
GeoStudio™ SLOPE/W ..................................................................................... 48
4.6.2
Microsoft Excel™ .............................................................................................. 48
4.6.3
Comparison of deterministic methods............................................................................... 50
5.1
Baseline cases ............................................................................................................ 50
5.2
Case 1: One layer soil with no pore water effects ..................................................... 51
5.2.1
Stability charts results ......................................................................................... 52
5.2.2
Undrained analysis ............................................................................................. 52
5.2.3
Drained analysis ................................................................................................. 54
5.3
Case 2: One layer soil with phreatic surface.............................................................. 55
5.3.1
Stability charts results ......................................................................................... 56
5.3.2
Undrained analysis ............................................................................................. 56
5.3.3
Drained analysis ................................................................................................. 57
5.4
Case 3: Two layer soils with no phreatic surface ...................................................... 58
5.4.1
Stability charts results ......................................................................................... 59
5.4.2
Undrained analysis ............................................................................................. 60
5.4.3
Drained analysis ................................................................................................. 61
5.5
Case 4: Two layer soils with phreatic surface ........................................................... 62
5.5.1
Stability charts results ......................................................................................... 62
5.5.2
Undrained analysis ............................................................................................. 63
5.5.3
Drained analysis ................................................................................................. 64
5.6
Summary .................................................................................................................... 64
iv
6
Parametric study ................................................................................................................ 66
6.1
Effect of varying undrained cohesion (cu) ................................................................. 66
6.2
Effect of varying undrained cohesion (cu) and slope angle (β) .................................. 69
6.3
Effect of varying drained angle of shearing resistance .............................................. 72
6.4
Effect of varying bulk unit weight (γ) ........................................................................ 74
6.4.1
Undrained conditions.......................................................................................... 74
6.4.2
Drained conditions .............................................................................................. 75
6.5
Effect of varying pore water pressure (u) .................................................................. 78
6.6
Effect of varying effective shear strength parameters ............................................... 79
7
Deterministic analysis of a case study............................................................................... 81
7.1
Downstream slope...................................................................................................... 82
7.2
Upstream slope .......................................................................................................... 84
7.3
Discussion .................................................................................................................. 85
8
Probabilistic methods of analysis ...................................................................................... 87
8.1
Uncertainties in soil properties .................................................................................. 88
8.2
Random variables and probability density functions ................................................. 89
8.3
Probability of failure .................................................................................................. 92
8.4
Acceptable probabilities of failure ............................................................................. 93
8.5
Types of probabilistic analysis .................................................................................. 97
8.6
Level I analysis .......................................................................................................... 97
8.6.1
Event tree ............................................................................................................ 97
8.6.2
Fault tree ............................................................................................................. 98
8.7
8.7.1
Drained analysis ............................................................................................... 101
8.7.2
Undrained analysis ........................................................................................... 103
8.8
Level III analysis ..................................................................................................... 106
8.8.1
Monte Carlo simulation .................................................................................... 106
8.8.2
Number of simulations ..................................................................................... 106
8.8.3
Drained analysis ............................................................................................... 108
8.8.4
Undrained analysis ........................................................................................... 110
8.9
9
Level II analysis ......................................................................................................... 99
Probabilistic analysis of the case study.................................................................... 112
Comparative analysis ...................................................................................................... 115
10
Conclusion ................................................................................................................... 122
11
References ................................................................................................................... 124
12
Bibliography ................................................................................................................ 130
v
Appendix A ............................................................................................................................ 131
Appendix B ............................................................................................................................. 133
Appendix C ............................................................................................................................. 135
Appendix D ............................................................................................................................ 136
Appendix E ............................................................................................................................. 137
Appendix F ............................................................................................................................. 141
Appendix G ............................................................................................................................ 142
vi
LIST OF ABBREVIATIONS
ANCOLD
Australian National Committee on Large Dams
DITR
Department of Industry, Tourism and Resources
C.O.V
Coefficient of Variation
ICOLD
International Commission on Large Dams
LPSDP
UNEP
United Nations Environmental Protection
US ACE
United States Army Corps of Engineers
USCOLD
United States Commission on Large Dams
US EPA
United States Environmental Protection Agency
WISE
World Information Service of Energy
vii
LIST OF TABLES
Table 2.1: ANCOLD consequence categories (ANCOLD, 2012) ........................................... 18
Table 3.1: ANCOLD recommended factors of safety for tailings dams (ANCOLD, 2012). ... 24
Table 4.1: Summary of slope stability charts by Abramson (2002) ......................................... 35
Table 4.2: Limit equilibrium methods ...................................................................................... 47
Table 5.1: Soil properties ......................................................................................................... 51
Table 5.2: Case 1 factor of safety from stability charts ............................................................ 52
Table 5.3: Case 1 factor of safety from Case 1 Spreadsheet and SLOPE/W for undrained
conditions ................................................................................................................................. 52
Table 5.4: Case 1 factor of safety from Case 1 Spreadsheet and SLOPE/W for drained
conditions ................................................................................................................................. 54
Table 5.5: Case 2 factor of safety from stability charts ............................................................ 56
Table 5.6: Factor of safety from Spreadsheet and SLOPE/W for undrained conditions.......... 56
Table 5.7: Factor of safety from Spreadsheet and SLOPE/W for drained conditions.............. 57
Table 5.8: Case 2 factors of safety when negative pore pressures are ignored ........................ 58
Table 5.9: Case 3 and Case 4 properties ................................................................................... 59
Table 5.10: Case 3 factor of safety from stability charts .......................................................... 59
Table 5.11: Case 3 factors of safety from spreadsheet and SLOPE/W for undrained conditions
.................................................................................................................................................. 60
Table 5.12: Case 3 factors of safety from spreadsheet and SLOPE/W for drained conditions 61
Table 5.13: Factor of safety from stability charts ..................................................................... 62
Table 5.14: Case 4 factors of safety from spreadsheets and SLOPE/W for undrained
conditions ................................................................................................................................. 63
Table 5.15: Case 4 factors of safety from spreadsheets and SLOPE/W for drained conditions
.................................................................................................................................................. 64
Table 5.16: Summary of results from stability charts and limit equilibrium methods ............. 65
Table 6.1: Undrained shear strength classification (BS 5930-1999) ........................................ 66
Table 6.2: Undrained shear strength classifications (AS1726-1993) ....................................... 67
Table 6.3: Values of angle of shearing resistance for different sand densities ........................ 73
Table 7.1: Case study soil properties. ....................................................................................... 81
Table 7.2: Factors of safety and failure type for a typical tailings dam downstream slope using
varying assumptions on the foundation properties and apparent cohesion. ............................. 83
Table 7.3: Case study upstream slope factors of safety for 5 kPa and zero cohesion .............. 84
viii
Table 8.1: Coefficients of Variation (from Duncan, 2000) ...................................................... 92
Table 8.2: Expected performance levels (US ACE, 1997) ....................................................... 94
Table 8.3: Random variable expected values and standard deviation .................................... 102
Table 8.4: Reliability analysis results for Case 1 in terms of effective stresses ..................... 102
Table 8.5: Random variable expected values and standard deviation .................................... 104
Table 8.6: Reliability analysis results for Case 1 in terms of total stresses ............................ 104
Table 8.7: Normal standard deviate values for levels of confidence (Abramson, 2002) ....... 107
Table 8.8: Number of trials required for levels of confidence ............................................... 108
Table 8.9: Soil properties for Baseline Case 1 Monte Carlo simulation ................................ 108
Table 8.10: Output data from @RISK and SLOPE/W ........................................................... 109
Table 8.11: Soil properties for Baseline Case 1 Monte Carlo simulation undrained ............. 110
Table 8.12: Output data from @RISK and SLOPE/W ........................................................... 111
Table 8.13: Random variable properties................................................................................. 113
Table 8.14: Case study probabilistic results from SLOPE/W ................................................ 114
Table 9.1: Comparison of results from deterministic and probabilistic analyses ................... 118
ix
LIST OF FIGURES
Figure 2.1: Failure events over time (Azam and Li, 2010) ........................................................ 5
Figure 2.2: Tailings dam failure distribution by region (Azam and Li, 2010) ........................... 6
Figure 2.3: Failure distribution by cause (Azam and Li, 2010) ................................................. 7
Figure 2.4: Tailings dam incident cause comparison with dam type ......................................... 8
Figure 2.5: Socio-economic impact of failure (Azam and Li, 2010) ......................................... 9
Figure 2.6: Failure surface ........................................................................................................ 10
Figure 2.7: Potential critical failure surfaces ............................................................................ 11
Figure 2.8: Example of rotational sliding ................................................................................. 11
Figure 2.9: Translational sliding failure ................................................................................... 12
Figure 2.10: Compound failure ................................................................................................ 12
Figure 2.11: Example of translational sliding (Swiss Confederation, 2005) ........................... 12
Figure 2.12: Example of an embankment overtopping in ........................................................ 13
Figure 2.13: Schematic of piping in an embankment (Goodarzi, Ziaei, & Shui, 2013)........... 14
Figure 2.14: Embankment failure by piping along the outlet pipe (USDA, n.d.) .................... 15
Figure 2.15: Seepage through an embankment (TADS, n.d.) .................................................. 16
Figure 2.16: Example of seepage through an embankment (USDA, n.d.) ............................... 16
Figure 2.17: Construction of a tailings dam with lining ......................................................... 17
Figure 3.1: Soil element at failure ............................................................................................ 19
Figure 3.2: Mohr failure circles and Mohr failure envelope .................................................... 19
Figure 3.3: Mohr-Coulomb failure criterion ............................................................................ 19
Figure 3.4: Mohr-Coulomb failure criterion for undrained conditions .................................... 20
Figure 3.5: Mohr-Coulomb failure criterion for drained conditions ........................................ 22
Figure 3.6: Mohr-Coulomb failure criterion for drained conditions with apparent cohesion .. 22
Figure 3.7: Failure surface ........................................................................................................ 23
Figure 3.8: Schematic of embankment after slip circle failure................................................. 24
Figure 4.1: Critical slip circles: (a) toe circle (b) slope circle (c) midpoint circle .................. 26
Figure 4.2: Taylor’s stability chart for undrained clay slopes (Steward et al. ., 2011) ............ 27
Figure 4.3: Taylor’s stability chart for drained soil slopes (Steward et al. ., 2011) ................. 28
Figure 4.4: Bishop and Morgenstern’s charts for stability coefficients m and n for 𝑐′𝛾 𝐻 = 0
(Murthy, 2002) ......................................................................................................................... 29
Figure 4.5: Bishop and Morgenstern’s charts for stability coefficients m and n for 𝑐′𝛾 𝐻 =
0.025 and 𝑛𝑑 = 1.00 (Murthy, 2002) ...................................................................................... 30
x
Figure 4.6: Bishop and Morgenstern’s charts for stability coefficients m and n for 𝑐′𝛾 𝐻 =
0.025 and 𝑛𝑑 = 1.25 (Murthy, 2002) ...................................................................................... 30
Figure 4.7: Bishop and Morgenstern’s charts for stability coefficients m and n for 𝑐′𝛾 𝐻 =
0.05 and 𝑛𝑑 = 1.00 (Murthy, 2002) ........................................................................................ 31
Figure 4.8: Bishop and Morgenstern’s charts for stability coefficients m and n for 𝑐′𝛾 𝐻 =
0.05 and 𝑛𝑑 = 1.25 (Murthy, 2002) ........................................................................................ 31
Figure 4.9: Bishop and Morgenstern’s charts for stability coefficients m and n for 𝑐′𝛾 𝐻 =
0.05 and 𝑛𝑑 = 1.50 (Murthy, 2002) ........................................................................................ 32
Figure 4.10: Spencer’s stability charts for different pore pressure ratios (Abramson, 2002) .. 33
Figure 4.11: Janbu stability chart for 𝜙 = 0 soils (Abramson, 2002) ...................................... 34
Figure 4.12: Method of slices ................................................................................................... 36
Figure 4.13: Free body diagram of ith slice- Fellenius method................................................ 39
Figure 4.14: Free body diagram of ith slice- Bishop's simplified method ............................... 40
Figure 4.15: Free body diagram of ith slice- Morgenstern and Price method .......................... 41
Figure 4.16: Free body diagram of ith slice- Spencer’s method .............................................. 42
Figure 4.17: Free body diagram of ith slice- Janbu's simplified method ................................. 44
Figure 4.18: Correction Factor (f0) (Duncan & Wright, 2005) ................................................ 45
Figure 5.1: Baseline cases ........................................................................................................ 50
Figure 5.2: Case 1 embankment geometry ............................................................................... 51
Figure 5.3: Case 1 most critical slip circle in undrained SLOPE/W analysis with 20 slices ... 53
Figure 5.4 : Distribution of failure circles analysed (GEO-SLOPE, 2007) .............................. 53
Figure 5.5: Case 1 most critical slip circle in drained SLOPE/W analysis with 20 slices ....... 54
Figure 5.6: Case 2 embankment geometry with phreatic surface............................................. 55
Figure 5.7: Most critical slip circle in undrained SLOPE/W analysis with 20 slices............... 56
Figure 5.8: Most critical slip circle in drained SLOPE/W analysis with 20 slices................... 57
Figure 5.9: Case 3 embankment geometry showing two soils and no phreatic surface. .......... 59
Figure 5.10: Most critical slip circle in undrained SLOPE/W analysis with 20 slices............. 60
Figure 5.11: Most critical slip circle in drained SLOPE/W analysis with 20 slices................. 61
Figure 5.12: Case 4 embankment geometry showing two soils and phreatic surface as entered
into SLOPE/W .......................................................................................................................... 62
Figure 5.13: Most critical slip circle in undrained SLOPE/W analysis with 20 slices............. 63
Figure 5.14: Most critical slip circle in drained SLOPE/W analysis with 20 slices................. 64
Figure 6.1: Most critical slip circles for undrained cohesion 1 kPa in SLOPE/W .................. 67
Figure 6.2: Most critical slip circles for cu = 300 kPa in SLOPE/W ....................................... 68
Figure 6.3: Change in the factor of safety arising from changes in the undrained cohesion ... 69
xi
Figure 6.4: Changes in undrained cohesion for different slope angles .................................... 70
Figure 6.5: Sensitivity plot ....................................................................................................... 71
Figure 6.6: Effect of varying drained angle of shearing resistance (ϕ′) ................................... 72
Figure 6.7: Effect of varying the tangent of angle of shearing resistance ................................ 73
Figure 6.8: Effect of varying bulk unit weight and angle of shearing resistance in drained
conditions ................................................................................................................................. 74
Figure 6.9: Varying bulk unit weight in undrained conditions................................................. 75
Figure 6.10: Varying bulk unit weight in drained conditions................................................... 76
Figure 6.11: Varying bulk unit weight for drained conditions with phreatic surface .............. 77
Figure 6.12: Location of phreatic surfaces ............................................................................... 78
Figure 6.13: Factors of safety when varying angle of phreatic surface.................................... 78
Figure 6.14: Failure surface when phreatic surface angle <10° ............................................... 79
Figure 6.15: Failure surface when phreatic surface angle >10° ............................................... 79
Figure 6.16: Sensitivity plot (GEO-SLOPE, 2014) .................................................................. 80
Figure 7.1: Typical tailings dam cross section ......................................................................... 81
Figure 7.2: Case study embankment in SLOPE/W (2007)....................................................... 81
Figure 7.3: Critical failure surface with sandstone foundation and zero effective cohesion.... 82
Figure 7.4: Critical failure surface of upstream slope with no apparent cohesion and at current
tailings pond level ..................................................................................................................... 84
Figure 8.1: Soil uncertainties flow chart (after Lacasse and Nadim, 1996) ............................. 88
Figure 8.2: Deterministic and statistical descriptions of a soil property .................................. 90
Figure 8.3: Normal and lognormal distribution curves (US ACE, 1997) ................................ 90
Figure 8.4: Normal probability distribution function (US ACE, 2006) ................................... 91
Figure 8.5: Area under the standard normal distribution curve ............................................... 92
Figure 8.6: Comparison of two situations with different means and distributions of factor of
safety (Christian, Ladd, Baecher, 1994) ................................................................................... 93
Figure 8.7: ANCOLD (1994) societal risk guideline (McDonald, 2008) ................................ 95
Figure 8.8: A portion of an event tree ...................................................................................... 98
Figure 8.9: Fault tree analysis vs event tree analysis (Taguchi, 2014). .................................... 98
Figure 8.10: Fault tree for slope failure of a uranium mine tailings dam. ................................ 99
Figure 8.11: Lognormal probability density function of factor of safety .............................. 100
Figure 8.12: Transformed probability density function.......................................................... 101
Figure 8.13: Baseline case 1 in drained conditions ................................................................ 101
Figure 8.14: Baseline case 1 in undrained conditions ............................................................ 104
Figure 8.15: Probability of failure versus number of trials .................................................... 107
xii
Figure 8.16: Probability distribution curves for drained conditions ....................................... 109
Figure 8.17: Probability distribution curves for undrained conditions ................................... 111
Figure 8.18: Case study histogram of factors of safety from SLOPE/W ............................... 114
Figure 9.1: Capacities of deterministic and probabilistic design approaches ........................ 117
Figure A.1: Upstream construction method ........................................................................... 131
Figure A.2: Downstream construction method....................................................................... 132
Figure A.3: Centreline construction method .......................................................................... 132
Figure B.1: Factor of safety vs number of slices .................................................................... 133
xiii
LIST OF SYMBOLS
α
angle (°)
𝛽
reliability index
𝛾
bulk unit weight (kN/m3)
unit weight of water (kN/m3)
𝛽
slope angle (°)
𝛾𝑑
dry unit weight (kN/m3)
𝛾𝑤
𝜎
standard deviation
σ
total stress (kPa)
σ′
effective stress (kPa)
ε
level of confidence
θ
interslice force inclination angle (°)
λ
scaling factor
μ
mean
τ
shear stress (kPa)
τf
shear stress (kPa)
τmob
mobilised shear stress (kPa)
ϕu
undrained angle of shearing
developed friction angle (°)
ϕ′
effective angle of shearing
effective cohesion (kPa)
cu
undrained cohesion (kPa)
normal standard deviate
E
interslice force
factor of safety from moment
FS
factor of safety
Fϕ′
factor of safety with respect to angle
h
depth of point in soil mass below
ground surface (m)
cu.mob mobilised undrained cohesion (kPa)
d
factor of safety from force
of friction
circle and centre of slice
c′
factor of safety with respect to
equilibrium
distance between centre of slip
slice width (m)
Fc′
Fm
mobilised angle of shearing
b
factor of safety
equilibrium
resistance (°)
b
F
Ff
resistance (°)
ϕmob
expected mean value
cohesion
resistance (°)
ϕd
E(x)
xiv
H
height of slope (m)
L
length of base of slice (m)
m
number of variables
m
stability coefficient
Md
driving moment
Mr
resisting moment
N
normal force of slice
N
number of trials
n
stability coefficient
𝑛𝑑
depth factor
No
stability number
Ns
stability number
r
𝑟𝑢
pore pressure ratio
u
pore water pressure (kPa)
W
self-weight of slice
X
interslice shear force
z
pore pressure depth
1
INTRODUCTION
Tailings dams are above-ground storage facilities used at mine sites to store the waste (the
tailings) from mineral processing. The tailings are a combination of residual mine material
and mine water and can contain material that is hazardous to the environment and surrounding
infrastructure. There have been a considerable number of highly publicised failures of tailings
dams over the last 30 years, and the trend is continuing with approximately two to five major
tailings dam failures per year, worldwide (Davies et al., 2002). An increase in the social and
corporate awareness of the risks and liabilities associated with tailings dams, coupled with
increasing regulatory attention and public scrutiny, have led to an increase in ensuring tailing
dam embankments are stable. The Australian Government Leading Practice Sustainable
Development Program (LPSDP) (Department of Industry, Tourism and Resources DITR,
2007) states that “…there are a number of historical mine sites in Australia that carry a
negative legacy of environmental and social impacts… from tailings storage facilities”. A
modern tailings dam is an engineered structure with the “…basic requirement to provide safe,
stable and economical storage of tailings presenting negligible public health and safety risks
and acceptably low social and environmental impacts during operation and post-closure”
(DITR, 2007).
1.1
Stability of tailings dams
The stability of a tailings dam is controlled by the slope angle, soil material properties, the
nature and strength of the foundations, the degree of compaction therein, and the hydraulic
regime in and around the dam (Lottermoser, 2007). The design of tailings dams is highly sitespecific as they are dependent on the climatic, geologic, topographic, hydrogeological and
geotechnical characteristics on site, as well as regulatory requirements for safety and
environmental performance (US EPA, 194). The design of tailings dam embankments must
ensure that the structure is able to withstand the potential loading conditions expected over the
life of the dam (ANCOLD, 2012). One method of analysis of the stability of an embankment
involves the calculation of a factor of safety to ensure that the risk of failure is acceptably
low. Factors of safety can be calculated for design purposes, or back-analysed for existing
dams, to determine their likely stability. There are different methods used to calculate factors
of safety including the use of stability charts and limit equilibrium methods. In such
deterministic methods, the input variables are treated as constants although parametric studies
varying them are often undertaken, time, and budget, permitting.
1
Soils are inherently variable: even within an engineered, constructed, embankment, the
material is potentially anisotropic and inhomogeneous. This is especially true for tailings
dams where the embankments are generally constructed, in stages, from borrowed fill to
increase the available storage as the level of the impoundment rises. In contrast to a
deterministic analysis, probabilistic analyses treat the input parameters as variables that
change according to an assigned probability distribution function. This sets the factors of
safety in a risk-based context to represent the variability in the soil profile.
Here, the factors of safety were calculated using both deterministic and probabilistic methods
of analysis: they were then used to determine the most suitable prediction of the stability of
tailings dams embankments.
1.2
Problem definition
There is extensive literature available that relates to slope stability and includes coverage of
both limit equilibrium, and probabilistic, methods. In the past, deterministic analyses allowed
the design process to be simplified by neglecting uncertainties in the model, compromising on
various critical and realistic features of the embankment that would affect the stability. Since
potentially threatening uncertainties are not considered, many embankments have failed that
were designed and deemed ‘safe’ according to deterministic methods of analysis. Probabilistic
methods of analysis have been difficult to adapt into routine analysis due to their complexity,
and the presupposed knowledge, time, and effort required to perform the analysis and
interpret the results. There is an ominous need for further understanding of probabilistic
modelling to design and analyse the stability of embankments to help militate against failures
in the future.
The task of analysing stability has become relatively easy due to the increased knowledge
gleaned from the lessons of previous tailings dam failures and the increased availability of
easy-to-use software (e.g. @RISK and SLOPE/W). However, there is a downside to making
the process easy; the geotechnical fundamentals can often be overlooked, which could lead to
misinterpretation of the results with unknown consequences for the ultimate safety of an
embankment. This can ultimately put lives and the environment at risk of significant damage
as a result of a tailings dam failure.
The research in this thesis aims to identify whether the factor of safety determined using
deterministic approaches gives a suitable prediction for the stability of tailings dams
embankments. A comparative analysis of the factors of safety calculated using deterministic
2
methods (no uncertainty included) and probabilistic methods (including uncertainty) is
performed to demonstrate the differences in the approaches. The outcomes of this thesis can
provide the engineering community with the necessary tools to understand the fundamental
geomechanics governing the stability of tailings dams and an insight into analysing and
evaluating stability assessments.
1.3
Thesis outline
Chapter 2 investigates the different modes of failures of tailings dam embankments in the last
century. The different failure types are then discussed with examples of each failure type. A
national consequence category is introduced which is used in practice to determine various
design and operational requirements depending on the severity of failure of the dam.
Chapter 3 introduces the loading conditions (drained or undrained) and shear strength
parameters (cohesion and angle of shearing resistance) used in slope stability analyses. The
term ‘factor of safety’ is then introduced which forms the basis of how stability is represented
in an analysis. National recommendations for factors of safety and stability are discussed.
The deterministic methods of analysis, including stability charts and limit equilibrium
methods (Fellenius’ method (1936), Bishop’s simplified method (1955), Morgenstern and
Price’s method (1965), and Janbu’s simplified method (1968)) are discussed in Chapter 4.
These methods are then applied to four baseline cases in Chapter 5 to compare stability chart
output with limit equilibrium methods: all cases were analysed by calculations done with
Microsoft Excel™ spreadsheets developed by the author and GeoStudio™ SLOPE/W (2012).
The resulting factors of safety were compared and analysed to confirm the validity and
assumptions inherent in the use of these limit equilibrium methods.
A parametric study was performed in Chapter 6 by varying the embankment geometry (slope
angle β), shear strength parameters for undrained conditions (cohesion cu) and drained
conditions (cohesion cʹ and angle of shearing resistance ϕʹ), bulk unit weight (γ) and
groundwater conditions (location of phreatic surface zw, i.e. its depth measured vertically from
the ground surface). Combinations of parameters were analysed, including an assessment of
the effects of variations in: the undrained cohesion and slope angle, the drained angle of
shearing resistance and the bulk unit weight of the soil, and the effective shear strength
parameters and the soil bulk unit weight. Sensitivity analyses were performed, alongside the
parametric study, to evaluate the effects of changes to each on the resultant factor of safety.
3
In Chapter 7, the deterministic methods of analysis were applied to a case study involving a
typical tailings dam embankment, currently in service in the Northern Territory, to determine
the stability of its upstream and downstream slopes.
In Chapter 8, this research then considered the inherent variability manifest in a given soil
profile and analysed the sources of uncertainty in the subsequent geotechnical analysis
thereof. The baseline cases were re-analysed, now using the reliability index First Order
Second Moment method (US Army Corps of Engineers, 2006) to highlight the effect of using
a range of values for the input soil properties instead of a single, deterministic, value. A
Monte Carlo simulation was performed on the baseline case using Palisade’s @RISK
software (2014) and SLOPE/W (2012) to compare the methods and confirm the accuracy of
SLOPE/W as this software was then used to analyse the case study in terms of probability of
failure. A comparative analysis was undertaken in Chapter 9, with conclusions and
recommendations for future research provided in Chapter 10.
1.4
Thesis outcomes
This research found that there were advantages and disadvantages to both the deterministic
and probabilistic methods of analysis. The deterministic methods, including stability charts
and limit equilibrium methods, were generally easy to use in the determination of a factor of
safety. The input variables were single values that may not have accurately reflected the
anisotropy or inhomogeneity within the soil in the embankment (tailings dam) under
investigation. This could produce factors of safety that were not representative of the entire
embankment. On the other hand, probabilistic analyses took into account such anisotropy and
inhomogeneity, but required a more detailed understanding thereof.
The baseline cases and case study analysed showed that there was little difference (less than
2%) between the factors of safety calculated using limit equilibrium methods and Monte
Carlo simulations. Nowadays, due to the increases in technological capability it is not difficult
to perform a Monte Carlo simulation; however, assessing the results requires a more detailed
understanding of statistics than is generally taught at a tertiary level. Based on the literature
review and analyses in this thesis, setting regulatory guidelines for probabilities of failure is
recommended. The recommended minimum guidelines would ideally take into account the
expected performance level of a tailings dam and the consequences of failure so that each dam
is analysed individually, as opposed to setting a standard minimum probability of failure that
all tailings dams must achieve.
4
2
TAILINGS DAM FAILURES
2.1
Failures over the last century
Azam and Li (2010) compiled a review of tailings dam failures over the last one hundred
years from available data from primary data bases:
i.
United Nations Environmental Protection (UNEP)
ii.
International Commission on Large Dams (ICOLD)
iii.
World Information Service of Energy (WISE)
iv.
United States Commission On Large Dams (USCOLD)
v.
United States Environmental Protection Agency (US EPA)
They found that eight to nine tailings dams failed per decade in the 1940s and 1950s but
during the 1960s, 1970s, and 1980s the number of failures per decade rose to around 50 (the
accurate, definitive, quantification of such phenomena on a global basis remains elusive:
many smaller failures go unreported, and indeed may not be viewed as failures especially if
they arise during construction where instant remedial measure may be taken and designs
changed on an ad hoc basis). The higher failure rate during the later decades may be attributed
to an increase in the global demand for mined materials after World War II. Failures were
significantly reduced in 1990s and 2000s with approximately 20 failures per decade. This
improvement was due to “…sufficient engineering experience, implementation of tougher
safety criteria and improved construction technology” (Azam and Li, 2010).
Figure 2.1: Failure events over time (Azam and Li, 2010)
5
The number of failures per region (Figure 2.2) shows that Australia has suffered almost 20%
of those dam failures occurring before 2000, but has had no recorded failures since. It is
important to note that these graphs only show the failures that have been recorded. There is
likely to be a much larger number of failed tailings dams, however the extent of the failure
may determine whether it is recorded globally, or just locally. The ICOLD Committee on
Tailings Dams and Waste Lagoons (1994) in their review of tailings dams failures (Bulletin
121) found a reluctance amongst the owners of locally recorded information in publishing
failure rates. As well as not recording all tailings dam incidents there are also the missed
opportunities of publishing lessons learned from the unrecorded failures. There have been
tailings dam incidents (such as leaks, releases of water during and after high rainfall events,
and overtopping) in Australia since 2000, yet none of these have been recorded as a failure
and are not, therefore, included in the failure distributions presented here. The decrease in
dam failures in the last decade may also represent the effect of introducing better legislation
and guidelines.
Figure 2.2: Tailings dam failure distribution by region (Azam and Li, 2010)
Azam and Li (2010) also analysed the causes of failure for all recorded tailings dam incidents.
They found that the main causes of failure were unusual weather (38%) and seepage (31%).
6
Figure 2.3: Failure distribution by cause (Azam and Li, 2010)
Rico et al. (2008) define managerial causes of failure as: inappropriate dam construction
procedures, improper maintenance of drainage structures, and inadequate long-term
monitoring programmes. Unusual weather and management failures have a bearing on the
failure mechanism. For example, unusual weather, especially in terms of rainfall, can increase
the chances of a seepage failure or overtopping due to the additional water present: it could
therefore be difficult to differentiate the exact initial cause of failure. Azam and Li (2010)
state that the 20% increase in managerial causes of failure from pre-2000 to post-2000 could
indicate that the “…rush for natural resource exploitation has compromised on engineering
standards”. The effects of climate change could also be a reason why the distribution of
unusual weather failures has increased. It is interesting to note that the number and rate of
failures caused by foundation subsidence, overtopping, seepage, and structural defects has
decreased, but the number and rate initially caused by slope instability have increased
(although most types of failure result in some form of slope instability). This shows that,
although improved engineering practices have decreased the number, and proportion, of many
types of failure mechanisms, slope instability itself remains a failure mechanism that still
causes an excessive number of incidents. ICOLD Bulletin 121 provided a summary of failure
types separated by tailings dam type (upstream, downstream, centreline) for dam failures prior
to 2000. The graph shows that upstream dams have failed more often than downstream or
7
centreline constructed dams. This could also be due to the number of dams constructed using
the upstream method is far greater than the number of downstream or centreline dams. For
information on the methods of construction, see Appendix A.
Figure 2.4: Tailings dam incident cause comparison with dam type (ICOLD Bulletin 121, 2001)
Azam and Li (2010) assigned a parameter that best described each failure, or incident, to
analyse the socio-economic impacts as a result of tailings dam failures.
8
Figure 2.5: Socio-economic impact of failure (Azam and Li, 2010)
The failure distribution (and number) of failure cases due to environmental pollution and
infrastructure damage have decreased from 52% to 35% and from 20% to 15%, respectively.
This decrease could be attributed to the majority of failures occurring in small to intermediate
sized facilities where the release of tailings can be better managed and regulated by
authorities. The distribution (and number of cases) of public health impacts rose from 2% to
20%, and the failure distribution affecting loss of lives increased from 26% to 30%. These
increases could be as a result of large catastrophic failures such as the fluorite mine failure in
Stava, Italy in 1985 (WISE, 2009) and the Harmony Gold Mine in Merriespruit, South Africa
in 1994, (Tailings.info, 2014).
The question still exists as to what constitutes a failure. Davies et al. (2000) refer to Leonards
(1982) (the Terzaghi lecture) where failure is defined as “…an unacceptable difference
between expected and observed performance”. They believe this terminology captures what
failure means in the context of tailings dams. The failure does not have to be a catastrophic
event, it can also include far less dramatic events where there are still valuable lessons to be
learned.
It is difficult to quantify the number of tailings dams failures compared to the total number of
tailings dams constructed, since there is no single database that records each and every failure
or storage facility. Davies (2002) states that there are approximately 3500 tailings dams
worldwide and in the past 30 years there have been approximately two to five major tailings
9
dam failure incidents per year. This equates to an annual probability of failure between 1 in
700 (0.14%) and 1 in 1750 (0.06%).
2.2
Types of failure
An analysis of potential failure methods, shapes, and locations is required to determine the
stability of an embankment. The Australian Government LPSDP (2007) cites a summary of
the main causes of failure from ICOLD Bulletin 121 as “…a lack of control of the water
balance, lack of control of construction and a general lack of understanding of the features
that control safe operations”. The most common types of failure are “…slope instability,
This thesis focuses on analysing slope instability failure mechanisms (predominantly
rotational sliding), however it is important to note that there other causes of failure that can
contribute to embankment instability. Davies (2002) states that “each and every failure is
entirely predictable in hindsight”. Hence, the importance of understanding the different failure
types, and what causes a tailings dam embankment to fail, is demonstrated.
2.2.1
Rotational sliding
In two-dimensional analysis, the failure surface can be approximated by a circular arc, often
called a slip circle. There are an infinite number of possible failure arcs (shown in Figure 2.7),
however, failure will occur at the most critical slip circle with the lowest factor of safety
(Horn, 1960). This occurs when the shear stress (𝜏𝑚𝑜𝑏 ) along the failure circle is equal to, or
greater than, the shear strength of the soil (𝜏𝑓 ), i.e. when the factor of safety is less than 1. The
shear strength resisting slope movement must be greater than the shear stress created by the
self-weight (W) of the sliding mass for the slope to be stable (Sivakugan, 2007).
W
𝝉𝒎𝒎𝒎
𝝉𝒇
Figure 2.6: Failure surface
10
Figure 2.7: Potential critical failure surfaces
The failure is rotational as the failure mass is in the shape of an arc rotating about an
instantaneous centre of rotation. It is most commonly caused by changes to the phreatic
surface or material permeability, settlement of the foundation, ground vibration, or a steep
slope angle (US EPA, 1994).
There are many methods for determining the factor of safety of potential slip circles
including: Bishop’s simplified, Fellenius (1936), Janbu (1954), and Morgenstern and Price’s
(1965) methods. These methods are explained in Section 4.3.
Figure 2.8: Example of rotational sliding in an embankment adjacent to a water main in Sheoaks, Victoria (CCMA,
2008)
11
2.2.2
Translational sliding
Translational sliding is similar to rotational sliding; however the failure surface is straight
rather than arcuate, as shown in Figure 2.9. It occurs when a weaker soil overlies a stiff
stratum, causing the weaker soil to slide downwards (Sivakugan, 2007).
Stiff stratum
Stiff stratum
Figure 2.9: Translational sliding failure
Figure 2.10: Compound failure
A combination of rotational and translational failure (called a compound failure) occurs when
the slip surface is circular but is intercepted by a stiff stratum causing a straight failure surface
along the interface between strata.
Figure 2.11: Example of translational sliding (Swiss Confederation, 2005)
Translational and compound failure surfaces are known as non-circular failure surfaces.
Methods due to Morgenstern and Price (1965), Spencer (1967), and Janbu (1954) can be used
to calculate factors of safety for non-circular failure surfaces.
12
2.2.3
Foundation failure
The foundation properties are important in the design and construction of a tailings dam
embankment. Potential planes of weaknesses are inherent at the interface between the
foundation and embankment. Sliding and bearing failure are possible failure types as well as
compound or translational failures arising from the slope itself.
Foundation failure can occur during staged construction as the existing embankments are
treated as foundations for subsequent dyke placement and compaction As the dam wall height
increases, the added weight of the structure increases the pore water pressures in the existing
dyke. The excess pore water pressures dissipate over time, which in turn increases the
effective stresses and the strength of the foundation, improving the factor of safety (Duncan
and Wright, 2005). The strength of the existing foundation is important as it may be sufficient
for a particular dam height, but even with the increased strength caused by the increase in
self-weight; the foundation may be insufficient to maintain overall stability as the height is
increased (ICOLD Bulletin 106). Embankment construction should be timed so that the
material has had sufficient time for consolidation before a new dyke is constructed.
2.2.4
Overtopping
Overtopping can occur when there is a rise in water levels, due to high rainfall, flood waters,
or mismanagement of tailings entering the pond. If overtopping occurs, the embankment wall
will be breached and possible erosion of the downward slope can lead to complete failure of
the dam (Lottermoser, 2007). Effective diversion of surface water and run off is a key
component in the design of tailings dams to prevent excess storm water entering the pond (US
EPA, 1994).
Figure 2.12: Example of an embankment overtopping in Googong, New South Wales 1976 (EHA, 2008)
13
One of the basic requirements in the design of a tailings dam is the maintenance of the
phreatic surface within the embankment (US EPA, 1994). The phreatic surface should be as
low as possible for embankment stability. The phreatic surface in a tailings dam is the level
surface at which full saturation is reached in the impoundment, i.e. where the pore water
pressures are zero. Management of impoundment levels and the rate of filling an
impoundment are important considerations in controlling the phreatic surface.
Overtopping can occur when the mine’s environmental approvals, such as dewatering
licences, constrain the ability to control the phreatic surface while continuing to increase the
level of tailings in the impoundment. St Barbara’s Gold Ridge tailings dam on the Solomon
Islands is reportedly close to collapse from overtopping as its water levels continue to rise
(Mining Australia, 2015). The mine was hit by Cyclone Ita in 2014 which brought with it high
rainfall, raising the water level in the tailings dam to critical levels. The ABC (2015) reports
that St Barbara wants to release untreated water from the tailings dam into a nearby river (to
reduce the water levels), but the government is insisting it must be treated first. This is an
example that shows that overtopping and controlling the phreatic surface is a real factor that
can affect the stability of an embankment and how environmental approvals must be
considered in the design to ensure the severity of situations currently being experienced at the
Gold Ridge tailings dams can be reduced.
2.2.5
Piping
Piping is internal erosion which can lead to rapid failure of a tailings embankment. It occurs
when there is seepage in, or below, the embankment which creates a low pressure region
causing water or liquid tailings to flow to this area (US EPA, 1994). This is when the critical
hydraulic gradient of the soil is exceeded (i.e. when the resultant body force per unit volume
becomes zero upon the upwards seepage force matching the self-weight).
Figure 2.13: Schematic of piping in an embankment (Goodarzi, Ziaei, & Shui, 2013)
14
The water erodes the embankment and with sufficient velocity, the eroded channel can
propagate upstream to the source of the seepage. Failure occurs when the water reaches the
upstream source allowing streams of tailings to pass through the embankment. Piping is more
likely to occur at defects in the dam, such as in areas of poor compaction or around tensile
cracking patterns arising from differential settlement.
Figure 2.14: Embankment failure by piping along the outlet pipe (USDA, n.d.)
2.2.6
Liquefaction
If soil particles in the embankment are unconsolidated and similarly sized they are susceptible
to failure from liquefaction (US EPA, 1994). Liquefaction is the process of soil particles
suspended in the tailings liquids becoming viscous (as a result of earthquakes or moving
machinery) and being able to flow through weaknesses in the soil. Liquefaction can lead to
rapid failure of the embankment with substantial releases of tailings thereafter.
2.2.7
Seepage
The US EPA (1994) defines seepage as the “…movement of water (contaminated or
uncontaminated) through and around the dam and impoundment”. It states that the primary
factors affecting the volume of seepage are “…depth to the ground water table and infiltration
capacities of the unsaturated zone and tailings” (US EPA, 1994).
15
Figure 2.15: Seepage through an embankment (TADS, n.d.)
Figure 2.16: Example of seepage through an embankment (USDA, n.d.)
Contaminated water from the tailings impoundment can seep through the foundation and
potentially contaminate groundwater (or aquifers), leading to public health risks and
environmental damage (DITR, 2007). The US EPA (1994) states that “historically, controlled
seepage through embankments was encouraged to lower the phreatic surface and increase
stability”, however, with increasing knowledge of the environmental impacts of contaminated
seepage, the need for seepage control and collection was increased. The effective control of
seepage, by the use of liners, filter wells, drainage, or decanting systems is now an important
consideration in the design of embankments (US EPA, 1994). McWhorter & Nelson (1979)
state that “Tailings impoundments are frequently constructed utilising liners to minimise
seepage losses, particularly if the tailings contain toxic or hazardous materials.”
16
Figure 2.17: Construction of a tailings dam with lining in Perth, Western Australia (Cape Crushing, 2013)
2.3
ANCOLD consequence category
ANCOLD have developed a Dam Failure Consequence Category used to determine various
design and operational requirements depending on the severity of failure of the dam. The
consequence category is determined by “…evaluating the consequences of dam failure with
release of water and tailings through a risk assessment process”. The assessment includes
damage to business, society, the environment, and its impact on human health. Each of the
aforementioned failure methods would have different consequences throughout the life of a
dam; for example, “erosion would be readily repaired during operation but could become a
potential mechanism for large-scale failure post-closure when limited maintenance is likely”
(ANCOLD, 2012). It is therefore necessary to analyse such dams to assess the likelihood of
each type, or mechanism, of damage arising: the use of risk assessment and statistical
methods at each stage of a dam’s life, including post-closure, is recommended.
17
Table 2.1: ANCOLD consequence categories (ANCOLD, 2012)
TYPE OF
DAMAGE
MINOR
MODERATE
MAJOR
CATASTROPHIC
Infrastructure
(dam, houses,
commerce,
farms,
community)
importance
< \$10 M
\$10 M to \$100 M
\$100 M to \$1 B
> \$1 B
Some restrictions
Significant impacts
Severe to crippling
bankruptcy
Public health
< 100 people
affected
100 to 1000 people
affected
Social
dislocation
< 100 person or
months
> 10,000 people
affected for over one
year
> 10,000 person
months or numerous
Affected area
< 1 km2
100 to 1000 person
months or 20 to
months
< 5 km2
< 1000 people
affected for more
than one month
> 1000 person
months or > 200
< 20 km2
> 20 km2
Impact duration
< 1 year
< 5 years
< 20 years
> 20 years
Impact on the
environment
Damage limited to
items of low
conservation value
cleared land,
ephemeral streams,
non-endangered
flora and fauna).
Possible
remediation.
Significant effects
on rural land and
local flora and
fauna.
Limited effects on:
A. Item(s) of
local and state
natural
heritage
B. Native flora
and fauna
within
forestry,
aquatic and
conservation
reserves, or
recognised
habitat
corridors,
wetlands or
fish breeding
areas.
Extensive rural
effects.
Significant effects
on river system
and areas A & B.
Limited effects on:
C. Item(s) of
National or
World natural
heritage.
D. Native flora
and fauna
within
national parks,
recognised
wilderness
areas,
RAMSAR,
wetlands and
national
protected
aquatic
reserves.
Remediation
difficult.
Extensively affects
areas A & B.
Significantly affects
areas C & D.
Remediation involves
significantly altered
ecosystems.
18
3
SLOPE STABILITY PARAMETERS
3.1
Coulomb (1776) observed that the cohesive component of the shearing strength remained
constant for a given soil and is independent of the applied stress. The second component of
the shear strength is the frictional resistance: the angle of shearing resistance, which was
further investigated by Mohr in 1900. Mohr (1900) produced circles - Mohr circles - using the
principal stresses at failure (𝜎1 and 𝜎3 ) to represent the state of stress at a point. A line drawn
tangential to a series of Mohr circles shows the Mohr failure envelope which identifies that, at
failure, the shear stress is equal to the shearing strength. The Mohr failure hypothesis thus
identifies the point of tangency of the Mohr failure envelope as the inclination of the failure
plane, i.e. the angle of shearing resistance (Ranjan & Rao, 2007).
𝜏
𝜎1
𝜎3
𝜎𝑓
𝜏𝑓
𝜎3
𝜏𝑓
𝜎3
𝜎1
Figure 3.1: Soil element at failure
𝜎3
𝜎1 𝜎𝑓
𝜎
𝜎1
Figure 3.2: Mohr failure circles and Mohr failure envelope
The Mohr-Coulomb failure criterion incorporates both shear strength parameters: cohesion (c)
and angle of shearing resistance (𝜙) in an equation for the shear strength at failure (𝜏𝑓 ):
𝜏
c
𝜏𝑓 = 𝑐 + 𝜎 tan 𝜙
𝜙
3.1
𝜎
Figure 3.3: Mohr-Coulomb failure criterion
19
The type of loading condition (undrained or drained) determines what shear strength
parameters (total or effective) to use for stability calculations. Each loading condition
represents a different physical condition that needs to be analysed with specific types of
material strengths (ANCOLD, 2012).
3.1.1
Undrained condition
The undrained condition occurs in the short-term when there has been insufficient time
allowed for the excess pore water pressures to dissipate.
The Mohr-Coulomb failure criterion may be expressed by Eq. 3.2 in terms of the total stresses
acting, using undrained shear strength parameters 𝑐𝑢 and φu. (Sivakugan & Das, 2009):
𝜏𝑓 = 𝑐𝑢 + 𝜎 tan 𝜙𝑢
3.2
where 𝜏𝑓 is the shear strength at failure, cu is the undrained cohesion, 𝜎 is the total stress and
𝜙𝑢 is the undrained angle of shearing resistance.
The angle of shearing resistance is equal to zero for the undrained condition resulting in the
shear strength (𝜏𝑓 ) being equal to the cohesion (c) (Skempton, 1948). There is negligible
change in excess pore water pressure in the short-term, so the undrained cohesion controls the
stability during construction (Ladd, 1991), and:
𝜏𝑓 = 𝑐𝑢
𝜏
3.3
cu
𝜎3
𝜎1
𝜎
Figure 3.4: Mohr-Coulomb failure criterion for undrained conditions
ANCOLD (2012) does not recommend the use of total shear strength parameters since they
“…do not directly account for pore water pressures, and for tailings dams the use of total
strength parameters may give misleading results”. However, total stress conditions should still
be analysed to determine the stability in short term conditions, as “…it represents the critical
20
condition for loading problems since the factor of safety increases with time due to
It is possible that external environmental factors could constrain an increase in factor of safety
over time, in some situations. This would happen if the environmental factors caused a
decrease at a rate faster than consolidation increases the factor of safety. Examples of this
could include recurring seismic activity (from earth tremors or machinery) or continued heavy
rainfall greater than what the dam is designed for, or combinations of environmental factors
that were not analysed simultaneously. This thesis analyses short- and long-term conditions as
instantaneous cases assuming that the effects of environmental influences between each
3.1.2
Drained condition
The drained condition assumes long-term stability under steady state conditions, with no rapid
change in phreatic surface or geometry (ANCOLD, 2012). It assumes that excess pore
pressures caused by loading have dissipated at a slow rate since construction. At this point,
the shear induced pore pressures are equal to zero (u = 0) (Ladd, 1991).
A drained condition analysis is conducted in terms of effective stresses and uses the drained
shear strength parameters 𝑐′ and 𝜙′ (Sivakugan & Das, 2009). The Mohr-Coulomb failure
criterion for shear strength at failure in terms of effective stresses is given by:
𝜏𝑓 = 𝑐 ′ + 𝜎 ′ 𝑡𝑎𝑛𝜙 ′
3.4
Where the effective stress (𝜎′) is given by the total stress (𝜎) minus the excess pore water
pressure (u) (Terzaghi, 1936):
𝜎′ = 𝜎 − 𝑢
3.5
Permeability is an important factor in the rate of drainage of excess pore pressures which
determines the change from short- to long-term conditions. In soils of low permeability with
100% saturation and zero air voids ratio, drainage will be slow and a long period of time is
required to achieve drained conditions, whereas in soils with a high permeability, drainage
will be rapid (Craig, 2004). After sufficient time for consolidation, the effective cohesion is
zero, as the total vertical stresses are then carried by the soil skeleton through inter-granular
contact.
21
𝜏
𝜙′
𝜎′
Figure 3.5: Mohr-Coulomb failure criterion for drained conditions
When the failure envelope does not pass through the origin the τ-axis intercept is known as
the apparent cohesion. This is a result of capillary pressure in overconsolidated soils which
makes cohesionless materials temporarily acquire the characteristics of cohesive materials
(Terzaghi, Peck, & Mesri 1996). The apparent cohesion increases the strength, however, the
extra strength and cohesive characteristics would be lost if the soil dries out or becomes
saturated or submerged (Venkatramaiah, 2006). Values of apparent cohesion can reach 10 to
14 kPa (Yong, 1981) but are unreliable for design purposes.
𝜏
Apparent
cohesion
c′
𝜙′
𝜎′
Figure 3.6: Mohr-Coulomb failure criterion for drained conditions with apparent cohesion
22
3.2
Factor of safety
The factor of safety (F) is the ratio of the shear strength (𝜏𝑓 ) divided by the mobilised shear
stress (𝜏𝑚𝑜𝑏 ) resisting failure (Sivakugan & Das, 2009) as shown in Eq. 3.6. A factor of
safety greater than unity means that the stresses resisting failure are greater than the stresses
driving the failure, and hence, the embankment is considered stable.
𝐹=
𝜏𝑓
𝜏𝑚𝑜𝑏
3.6
W
𝝉𝒎𝒎𝒎
𝝉𝒇
Figure 3.7: Failure surface
It is important to note that the factors of safety are instantaneous. The factor of safety
calculated is only true for the unique soil properties immediately preceding ultimate failure.
The factor of safety changes during failure and will also change over time due to a number of
factors including consolidation, further vertical construction, level of tailings impoundment,
etc. At a value of one, a critical failure surface has been established where the resisting forces
are equal to the destabilising forces. As soon as the destabilising forces are greater than
resisting forces the embankment fails, and when this happens the failure surface and, hence,
factor of safety change significantly as a result of the movement of the soil mass. The soil
mass will stop moving when the driving forces then become resisting forces as they pile up at
the toe of the embankment. An example of what an embankment looks like after the soil mass
has moved (i.e. after the embankment has failed) is shown in Figure 3.8.
23
Figure 3.8: Schematic of embankment after slip circle failure
Factors of safety of tailings dam embankments can be calculated from stability charts, limit
equilibrium methods, probabilistic analysis methods, finite element analysis methods, or a
combination thereof. The first three methods are discussed in this thesis.
3.2.1
ANCOLD recommendations
ANCOLD (2012) state that factors of safety depend on the consequences of failure, material
properties, and subsurface conditions: there are no ‘rules’ governing what constitutes an
acceptable factor of safety. Recommended minimum factors of safety for tailings dams under
Table 3.1: ANCOLD recommended factors of safety for tailings dams (ANCOLD, 2012).
Recommended minimum
factor of safety
Shear strength to be used
for evaluation
Long-term drained
1.5
Effective
Short-term undrained
(potential loss of containment)
1.5
Consolidated undrained
Short-term undrained
(no potential loss of containment)
1.3
Consolidated undrained
When there is ‘no potential loss of containment’ in short-term undrained conditions, a lower
factor of safety is recommended. This is not recommended for those cases analysed in this
thesis since the factor of safety assumes a ‘safer’ embankment with no loss of containment. It
cannot be determined that a tailings dam constructed to a factor of safety of 1.3 will never
have any loss of containment. Therefore, the minimum factor of safety, in this thesis, has been
calculated assuming the worst conditions (i.e. including the potential loss of containment) as
this is what better represents a real embankment failure.
24
3.2.2
Tailings dam legislation
ANCOLD (2012) state that the regulation of tailings dams in Australia comes under State, or
Territory, Government legislation with each state, or territory, having its own unique body of
legislation, regulations, and guidelines. The ANCOLD Guidelines on Tailings Dams (2012) is
often referred to in state legislation, with some states making compliance mandatory
(ANCOLD, 2012). Tailings dams in the Northen Territory must comply with the Waste
Management and Pollution Control Act 1998 and the Mining Management Act 2001. For the
purposes of design criteria, monitoring, and closing procedures, the ANCOLD Guidelines
should be adopted as mandatory requirements in the Northern Territory. This would ensure a
certain standard of tailings dams which, in the future, may help to mitigate some of the
current issues experienced with ‘legacy’ dams.
The Australian Government DITR (2007) published a tailings management manual entitled
“Leading Practice Sustainable Development Program for the Mining Industry”, which
incorporates the key issues affecting sustainable development into a risk-based approach for
tailings management. These key issues include: planning for construction, operation and
closure, seepage control, and water management (DITR, 2007).
The first guidelines, entitled “Guidelines on Tailings Dam Design, Construction and
Operation”, by ANCOLD were published in 1999 (ANCOLD, 1999). The publication of the
LPSDP in 2007 led to a new edition of the ANCOLD guidelines being published with the aim
of “… providing a single document that supports the DITR.” (ANCOLD, 2012). The new
release was also attributed to a considerable increase in the recognition of environmental
responsibilities since 1999, and the considerable body of new information available pertaining
to the importance of design for closure and the use of risk assessment techniques in design
and management (ANCOLD, 2012). These types of developments are vital to continually
improve the design, and as-built standard, of tailings dams, to say nothing of their long-term
stability.
25
4
DETERMINISTIC METHODS OF ANALYSIS
Deterministic methods, also referred to as traditional methods, calculate a single factor of
safety which is unique to the set of soil properties used to analyse the slope. They do not take
into account uncertainties in the input parameters; it is assumed that the input values are
representative of all soil properties. This chapter will introduce traditional methods of
analysing slope stability including site investigation, stability charts, and limit equilibrium
methods. Chapter 5 applies these methods, along with a comparative (parametric) analysis, to
a baseline case and an industry-specific case study.
4.1
Stability charts
4.1.1
Taylor’s stability charts
Taylor’s stability charts for homogenous undrained (ϕu = 0) and drained (c′ = 0) soil slopes
are a “…simple and straightforward” method of determining factors of safety (Steward et al.,
2011). Taylor’s stability charts are based on three types of failure circles: toe circle, slope
circle and midpoint circle (Taylor, 1937, 1948), as shown in Figure 4.1.
Figure 4.1: Critical slip circles: (a) toe circle (b) slope circle (c) midpoint circle (Steward et al. ., 2011)
Taylor (1937) proposed a stability number (𝑁𝑆 ), to be used in stability charts, as:
𝑁𝑆 =
𝛾𝐻
𝑐𝑢,𝑚𝑜𝑏
4.1
where 𝛾 is the bulk unit weight, H is the height of the slope and 𝑐𝑢,𝑚𝑜𝑏 is the shear strength
mobilised along the failure arc. The interrelationship between 𝑛𝑑 , 𝛽 and 𝑁𝑆 can be identified
in Taylor’s stability charts for undrained clays and drained soils.
26
Taylor’s stability charts are based on the general assumptions that the analysis is considering
only a two-dimensional limit equilibrium, the slopes are simple and homogenous, and that the
slip surfaces are circular (Steward et al., 2011). The exact location of the slip circle is not
defined using the stability charts but the type of circle can be identified. The stability charts
developed by Taylor do not consider the pore water pressures.
Figure 4.2 shows Taylor’s stability chart for undrained clays which is used to determine the
stability number when 𝑛𝑑 and 𝛽 are known. Equation 4.1 is then used to determine the
mobilised shear strength. The factor of safety can then be calculated using Equation 3.6,
assuming 𝜏𝑓 = 𝑐𝑢 and 𝜏𝑚𝑜𝑏 = 𝑐𝑢,𝑚𝑜𝑏 .
Figure 4.2: Taylor’s stability chart for undrained clay slopes (Steward et al. ., 2011)
Therefore, using Equation 3.6, the factor of safety can be defined as:
𝐹=
𝜏𝑓
𝜏𝑚𝑜𝑏
c ′ + σ′ tan𝜙 ′
= ′
𝑐 𝑚𝑜𝑏 + 𝜎 ′ 𝑡𝑎𝑛𝜙 ′ 𝑚𝑜𝑏
4.2
The factors of safety with respect to cohesion (𝐹𝑐′ ) and angle of shearing resistance (𝐹𝜙′ ) can
be defined as (Liu & Evett, 2004):
27
𝐹𝑐′ =
𝐹𝜙′ =
𝑐′
𝑐 ′ 𝑚𝑜𝑏
𝑡𝑎𝑛𝜙 ′
𝑡𝑎𝑛𝜙 ′ 𝑚𝑜𝑏
4.3
4.4
Liu and Evett (2004) assume that the degree of mobilisation is the same for the cohesion and
angle of shearing resistance, resulting in the same factors of safety.
𝐹 = 𝐹𝑐′ = 𝐹𝜙′
4.5
𝑐 ′ 𝑚𝑜𝑏 and 𝜙 ′ 𝑚𝑜𝑏 can be determined from Taylor’s stability chart (Figure 4.3) along with an
iterative process to calculate F when 𝐹 = 𝐹𝑐′ = 𝐹𝜙′ .
Figure 4.3: Taylor’s stability chart for drained soil slopes (Steward et al. ., 2011)
28
4.1.2
Bishop and Morgenstern 1960
Bishop and Morgenstern (1960) developed stability charts to include the effects of pore water
pressure. They proposed the pore pressure ratio (𝑟𝑢 ) :
𝑟𝑢 =
𝑢
𝛾ℎ
4.6
where u is the pore water pressure at any point on the assumed surface and h is the depth of
point in the soil mass below the ground surface. This method is based on the assumption that
the pore pressure ratio is constant throughout the cross-section (Bishop & Morgenstern,
1960).
4.7
The factor of safety (𝐹𝑠 ) is defined as:
𝐹𝑠 = 𝑚 − 𝑛 𝑟𝑢
where m and n are stability coefficients. The values of the stability coefficients are determined
by stability charts for
where 𝑛𝑑 =
𝑐′
𝛾𝐻
values of 0.000, 0.025, and 0.05 with different depth factors (𝑛𝑑 )
𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝑓𝑖𝑟𝑚 𝑠𝑡𝑟𝑎𝑡𝑢𝑚 𝑓𝑟𝑜𝑚 𝑡𝑜𝑝 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒
intermediate values.
𝐻𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑙𝑜𝑝𝑒
. The charts can be interpolated for
Figure 4.4: Bishop and Morgenstern’s charts for stability coefficients m and n for
𝒄′
𝜸𝑯
= 𝟎 (Murthy, 2002)
29
Figure 4.5: Bishop and Morgenstern’s charts for stability coefficients m and n for
(Murthy, 2002)
Figure 4.6: Bishop and Morgenstern’s charts for stability coefficients m and n for
(Murthy, 2002)
𝒄′
= 𝟎. 𝟎𝟐𝟓 and 𝒏𝒅 = 𝟏. 𝟎𝟎
𝒄′
= 𝟎. 𝟎𝟐𝟓 and 𝒏𝒅 = 𝟏. 𝟐𝟓
𝜸𝑯
𝜸𝑯
30
Figure 4.7: Bishop and Morgenstern’s charts for stability coefficients m and n for
(Murthy, 2002)
Figure 4.8: Bishop and Morgenstern’s charts for stability coefficients m and n for
(Murthy, 2002)
𝒄′
= 𝟎. 𝟎𝟓 and 𝒏𝒅 = 𝟏. 𝟎𝟎
𝒄′
= 𝟎. 𝟎𝟓 and 𝒏𝒅 = 𝟏. 𝟐𝟓
𝜸𝑯
𝜸𝑯
31
Figure 4.9: Bishop and Morgenstern’s charts for stability coefficients m and n for
(Murthy, 2002)
4.1.3
𝒄′
𝜸𝑯
= 𝟎. 𝟎𝟓 and 𝒏𝒅 = 𝟏. 𝟓𝟎
Spencer 1967
Spencer’s stability charts are based on the assumption of both force and moment equilibrium
(Spencer, 1967). The charts use three different pore pressure ratios (𝑟𝑢 ): 0, 0.25, and 0.5.
Spencer proposed an equation for the developed friction angle (𝜙𝑑 ) as:
𝜙𝑑 = tan −1 �
tan 𝜙
�
𝐹𝑠
4.8
32
Figure 4.10: Spencer’s stability charts for different pore pressure ratios (Abramson, 2002)
An iterative process is required to determine the factor of safety. The first iteration assumes a
factor of safety and calculates
𝑐′
𝐹𝛾𝐻
, then using the stability charts, the mobilised friction
tan 𝜙
angle can be found. The factor of safety is then equal to tan 𝜙 , if this differs from the assumed
factor of safety then further iterations are required.
4.1.1
𝑑
Janbu 1968
Janbu (1968) developed stability charts for different cases including stratified soils, cases
where surcharge or tension cracks are present, undrained short-term total stress behaviour,
drained long-term effective stress behaviour, and different water levels outside and inside the
slope (Briaud, 2013). The stability charts for undrained soils are shown here:
33
Figure 4.11: Janbu stability chart for 𝝓 = 𝟎 soils (Abramson, 2002)
34
4.2
Summary of slope stability charts
Further developments of stability charts have been summarised by Abramson (2002):
Table 4.1: Summary of slope stability charts by Abramson (2002)
Method
Taylor (1948)
Bishop and
Morgenstern
(1960)
Gibson and
Morgenstern
(1962)
Spencer (1967)
Parameters
Slope
inclination 𝜷
(degrees)
Analytical
methods
𝜙=0
𝑐𝑢 , 𝑐, 𝜙
0 – 90
𝑐, 𝜙, 𝑟𝑢
11 – 26.5
𝑐𝑢
0 – 90
𝜙=0
𝑐, 𝜙, 𝑟𝑢
0 – 34
Spencer
𝜙=0
Janbu
generalised
Friction circle
Bishop
Janbu (1968)
𝑐𝑢 , 𝑐, 𝜙, 𝑟𝑢
0 – 90
Hunter and
Schuster (1968)
𝑐𝑢
0 – 90
𝜙=0
Chen and Giger
(1971)
𝑐, 𝜙
20 – 90
Limit analysis
11 – 26
Bishop
O’Connor and
Mitchell (1977)
𝑐, 𝜙, 𝑟𝑢
Friction circle
Hoek and Bray
(1977)
Cousins (1978)
Charles and
Soares (1984)
Barnes (1991)
Notes
Undrained analysis
Dry slopes only
First to include water
effects
Undrained analysis with
𝑐𝑢 increasing linearly
with depth; zero strength
at ground level
Toe circles only
Extensive series of charts
for seepage and tension
crack effects
Undrained analysis with
𝑐𝑢 increasing linearly
with depth; finite strength
at ground level
Extended Bishop and
Morgenstern (1960) to
include 𝑁𝑐 = 0.1
Includes groundwater and
tension cracks
𝑐, 𝜙
0 – 90
𝑐, 𝜙, 𝑟𝑢
0 – 45
Friction circle
𝜙
26 – 63
Bishop
Nonlinear MohrCoulomb failure envelope
𝑐, 𝜙, 𝑟𝑢
11 – 63
Bishop
Extension of Bishop and
Morgenstern (1960);
wider range of slope
angles
Wedge
Three-dimensional
analysis of wedge block
Extension of Taylor
(1948)
35
4.3
Limit equilibrium methods
Limit equilibrium methods calculate the factor of safety by equilibrium of force or
equilibrium of moment, or both, using methods involving finite slices and the analysis
thereof. For force (or moment) equilibrium, the factor of safety is found by the sum of the
resisting forces (or moments) divided by the sum of the destabilising force (or moment).
Figure 4.12 shows a slope embankment with a potential failure circle split into vertical slices
for use in the method of slices. The following derivations show how the factor of safety can
be found using method of slices for total and effective stress conditions.
b
crest
α
r
W
self-weight of individual slice
N
normal force
𝜏𝑓
shear strength resisting slide
b
distance between centre of
slip circle and centre of slice
W
L
toe
N
𝝉𝒇
r
α
angle between horizontal and
α
base of slice
L
length of the base of the slice
Figure 4.12: Method of slices
36
Moment equilibrium about centre of slip circle for driving moment (𝑀𝑑 ):
𝑀𝑑 = � 𝑊 𝑏
Where:
𝑏 = 𝑟 𝑠𝑖𝑛𝛼
Substituting b into Equation 4.9:
𝑀𝑑 = � 𝑊 𝑟 𝑠𝑖𝑛𝛼
4.9
4.10
4.11
Moment equilibrium about centre of slip circle for resisting moment (𝑀𝑟 ):
𝑀𝑟 = � 𝑟 𝜏𝑓 = 𝑟 � 𝜏𝑓
4.12
Assuming unit thickness, the area of slice is equal to L.
The shear force (𝝉𝒇 ) is equal to the shear stress (𝝉𝒎𝒎𝒎 ) multiplied by the area of the slice:
Substituting S into Equation 4.12:
𝜏𝑓 = 𝜏𝑚𝑜𝑏 𝐿
𝑴𝒓 = 𝒓 � 𝝉𝒎𝒎𝒎 𝑳
4.13
4.14
Using Equation 3.6, the resisting moment can be written in terms of F:
𝑴𝒓 = 𝒓 �
𝝉𝒇 𝑳
𝑭
4.15
By equation and rearranging Equations 4.11 and 4.15 for limit equilibrium conditions, the
factor of safety can then be written as:
𝑭=
∑ 𝝉𝒇 𝑳
∑ 𝑾 𝒔𝒊𝒏𝜶
4.16
In terms of total stress
The shear strength (𝜏𝑓 ) can be expressed by Mohr-Coulomb’s formula in terms of
total stress, as shown in Equation 3.2:
37
𝜏𝑓 = 𝑐 + 𝜎 𝑡𝑎𝑛𝜙
4.17
Substituting 𝜏𝑓 into Equation 4.16:
𝐹=
𝑁
∑ (𝑐 + 𝜎 𝑡𝑎𝑛𝜙) 𝐿
∑ 𝑊 𝑠𝑖𝑛𝛼
4.18
Since 𝜎 = 𝐿 , the equation for factor of safety in terms of total stress is:
𝐹=
∑ 𝑐 𝐿 + 𝑁 𝑡𝑎𝑛𝜙
∑ 𝑊 𝑠𝑖𝑛𝛼
4.19
In terms of effective stress
The shear strength (𝜏𝑓 ) can be expressed by Mohr-Coulomb’s formula in terms of
effective stress, as shown in Equation 3.4:
𝜏𝑓 = 𝑐′ + 𝜎 ′ 𝑡𝑎𝑛𝜙′
4.20
Substituting 𝜏𝑓 into Equation 4.16:
Since 𝜎 ′ =
𝑁
𝐿
𝐹=
4.21
− 𝑢, the equation for factor of safety in terms of effective stress is:
𝐹=
4.3.1
∑ (𝑐′ + 𝜎 ′ 𝑡𝑎𝑛𝜙′) 𝐿
∑ 𝑊 𝑠𝑖𝑛𝛼
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) 𝑡𝑎𝑛𝜙 ′ )
∑ 𝑊 𝑠𝑖𝑛𝛼
4.22
Fellenius’ method 1936
The Fellenius method, also known as the Swedish or Ordinary method of Slices, is the first
and most simple method of slices recorded in literature (Sivakugan & Das, 2009). The method
assumes that the interslice forces are ignored and satisfies moment equilibrium only. Figure
4.13 shows the forces on an individual slice with the forces shown in red assumed negligible.
38
W
S
α
N
Figure 4.13: Free body diagram of ith slice- Fellenius method
Factor of safety in terms of total stress:
𝐹=
∑ (𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙)
∑ 𝑊 sin 𝛼
4.23
Factor of safety in terms of effective stress:
∑ (𝑐 ′ 𝐿 + (𝑊 cos 𝛼 − 𝑢 𝐿) tan 𝜙′)
𝐹=
∑ 𝑊 sin 𝛼
4.24
The factor of safety can be hand calculated due to its simplicity. The GEO-SLOPE Stability
Modelling (2008) guide recommends that the Fellenius method “should not be used in
practice, due to potential unrealistic factors of safety”, this is because the method
underestimates factors of safety as it ignores the effects of any interslice forces. The Fellenius
method has been used in this thesis for comparative purposes only and the baseline case
analyses demonstrate the differences (often significant) between Fellenius’ method and other
methods that do take into account interslice forces.
39
4.3.2
Bishop’s simplified method 1955
Bishop’s simplified method assumes that there are no interslice shear forces, only interslice
normal forces acting horizontally on the slice (Bishop, 1955).
Ei+1
Ei
W
𝝉𝒇
α
N
Figure 4.14: Free body diagram of ith slice- Bishop's simplified method
The derivation for factor of safety using Bishop’s simplified method is shown below
(Sivakugan and Das (2009); Duncan and Wright (2005)).
Equilibrium of forces in the vertical direction (positive is upwards) and rearranging for N:
𝑁=
𝑊 − 𝜏𝑓 sin 𝛼
cos 𝛼
4.25
The effective normal stress can be written as:
𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan 𝜙 ′
𝜏𝑓 =
𝐹
4.26
Substituting 4.26 into 4.25 gives:
𝑐 ′ 𝐿 sin 𝛼 𝑢 𝐿 sin 𝛼 tan 𝜙 ′
−
𝐹
𝐹
𝑁=
4.27
sin 𝛼 tan 𝜙 ′
cos 𝛼 +
𝐹
Substituting Equation 4.27 into the equation for factor of safety in terms of effective stress
𝑊−
4.22 and rearranging gives:
𝑭=
∑�
Alternatively written as:
𝒄′ 𝑳 𝐜𝐨𝐬 𝜶 + (𝑾 − 𝒖 𝑳 𝐜𝐨𝐬 𝜶) 𝐭𝐚𝐧 𝝓′
�
𝐬𝐢𝐧 𝜶 𝐭𝐚𝐧 𝝓′
𝐜𝐨𝐬 𝜶 +
𝑭
∑ 𝑾 𝐬𝐢𝐧 𝜶
4.28
40
𝑭=
𝐬𝐞𝐜𝜶
∑ �(𝒄′ 𝒎 + (𝑾 − 𝒖𝒎) 𝐭𝐚𝐧 𝝓′) �
��
𝐭𝐚𝐧𝜶 𝐭𝐚𝐧𝝓′
𝟏+
𝑭
4.29
∑ 𝑾 𝐬𝐢𝐧𝜶
The formula for factor of safety (F) is non-linear as it has F on both sides of the equation,
so an iterative process is required to solve for a factor of safety that lies within acceptable
limits. An initial estimate of the factor of safety is required for the first iteration and iterations
continue until the factor of safety is within a specified tolerance.
Duncan and Wright (2005) state that Bishop’s simplified method is more accurate than the
Fellenius method, especially for effective stress analyses with high pore water pressures. This
is because Bishop’s method takes into account the interslice normal forces whereas Fellenius’
method does not. Wright, Kulhawy & Duncan (1973) conclude that the values of the factor of
safety calculated using Bishop’s simplified method are generally similar (from 0% to 6%) to
those calculated using methods that satisfy both force and moment equilibrium (evident in
Section 6).
4.3.3
Morgenstern and Price 1965
The Morgenstern and Price (1965) method considers limit equilibrium of both force and
moment for each slice in circular and non-circular slip surfaces. The method assumes a
relationship between the interslice forces (X and E) with a function (𝑓(𝑥)) that varies
continuously across the failure surface and an unknown scaling factor (𝜆):
𝑋 = 𝜆 × 𝑓(𝑥) × 𝐸
4.30
The force function can be constant (same as Spencer’s method), half-sine, trapezoidal or datapoint specified (GEO-SLOPE International Ltd, 2008).
Xi+1
Ei+1
θ
θ
Ei
Xi
W
S
α
N
Figure 4.15: Free body diagram of ith sliceMorgenstern and Price method
41
This equation has too many unknowns so alternatively two factors of safety are calculated,
one from moment equilibrium, and the other from force equilibrium.
Factor of safety (force equilibrium):
𝐹𝑓 =
∑[(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan 𝜙 ′ ) sec 𝛼]
∑(𝑊 − (𝑋𝑖+1 − 𝑋𝑖 )) tan 𝛼 + ∑(𝐸𝑖+1 − 𝐸𝑖 )
4.31
Factor of safety (moment equilibrium):
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan 𝜙 ′ )
𝐹𝑚 =
∑ 𝑊 sin 𝛼
4.3.4
4.32
Spencer’s method 1967
Spencer’s method assumes that the interslice forces, at whatever inclination, are parallel
(Spencer, 1967). The method satisfies equilibrium of both force and momentum for circular
and non-circular failure surfaces The relationship between the interslice shear forces (X) and
interslice forces (E) is constant. Calculations for Spencer’s method are generally performed
using computer analysis, since it requires an iterative process to calculate a factor of safety
(F) and interslice force inclination angle (θ).
Xi+1
Ei+1
θ
θ
Ei
Xi
W
𝝉𝒇
α
N
Figure 4.16: Free body diagram of ith slice- Spencer’s method
The normal force (N) can be derived by equilibrium of the forces perpendicular to the
interslice forces:
𝑁=
𝑐 ′ 𝐿 𝑠𝑖𝑛𝛼 𝑢 𝐿 𝑠𝑖𝑛𝛼 𝑡𝑎𝑛𝜙 ′
−
𝐹
𝐹
𝑠𝑖𝑛𝛼 𝑡𝑎𝑛𝜙 ′
𝑐𝑜𝑠𝛼 +
𝐹
𝑊 − (𝐸𝑖+1 − 𝐸𝑖 ) 𝑡𝑎𝑛𝜃 −
4.33
42
Similar to Morgenstern and Price’s method, two factors of safety are calculated, one from
moment equilibrium, and the other from force equilibrium. At a certain interslice inclination,
the moment equilibrium factor of safety and the force equilibrium are equal (Spencer, 1967).
Factor of safety (moment equilibrium):
𝐹𝑚 =
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) 𝑡𝑎𝑛𝜙 ′ )
∑ 𝑊 𝑠𝑖𝑛𝛼
4.34
Equilibrium of forces in horizontal direction:
𝜏𝑓 𝑐𝑜𝑠𝛼 − 𝑊 𝑠𝑖𝑛𝛼 + 𝐸𝑖 − 𝐸𝑖+1 = 0
4.35
Equation for effective normal stress (Equation 4.26):
𝜏𝑓 =
(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) 𝑡𝑎𝑛𝜙 ′ )
𝐹
4.36
Substituting 4.36 into 4.35 gives:
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) 𝑡𝑎𝑛𝜙 ′ ) 𝑐𝑜𝑠𝛼
− 𝑊 𝑠𝑖𝑛𝛼 + 𝐸𝑖 − 𝐸𝑖+1 = 0
𝐹𝑓
4.37
Factor of safety (force equilibrium):
∴ 𝐹𝑓 =
4.3.5
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) 𝑡𝑎𝑛𝜙 ′ ) 𝑐𝑜𝑠𝛼
𝑊 𝑠𝑖𝑛𝛼 − (𝐸𝑖 − 𝐸𝑖+1 )
4.38
Janbu’s simplified method 1968
Janbu’s simplified method uses horizontal force equilibrium to identify a factor of safety for
circular or non-circular failure surfaces (Janbu, 1968). It is similar to Bishop’s simplified
method as it assumes the interslice shear forces are zero, and the interslice normal forces (E)
act horizontally, but Janbu’s simplified method only satisfies horizontal force equilibrium (as
opposed to moment equilibrium). Bishop’s simplified method is essentially the factor of
safety with respect to moment equilibrium, whereas Janbu’s simplified method gives the
factor of safety in terms of force equilibrium.
43
Ei+1
Ei
W
S
α
N
Figure 4.17: Free body diagram of ith slice- Janbu's simplified method
The equation for normal force as derived in Bishop’s simplified method:
𝑁=
𝑊−
𝑐 ′ 𝐿 sin 𝛼 𝑢 𝐿 sin 𝛼 tan 𝜙 ′
−
𝐹
𝐹
sin 𝛼 tan 𝜙 ′
cos 𝛼 +
𝐹
4.39
Horizontal force equilibrium gives factor of safety (𝐹0 ):
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) 𝑡𝑎𝑛𝜙 ′ )
𝐹0 =
∑ 𝑊 𝑠𝑖𝑛𝛼
4.40
The method takes into account the iterative processes of Bishop’s simplified method but also
accounts for inaccuracies with a correction factor (𝑓0 ):
𝐹 = 𝑓0 × 𝐹0
4.41
The correction factor (𝑓0 ) can be found graphically as shown in Figure 4.18. The factor of
safety (F) can then be calculated using Equation 4.41 and the correction factor (𝑓0 ) from
Figure 4.18. This method requires the failure surface depth to be known to be able to calculate
a factor of safety using the correction factor. It would be time consuming to find the most
critical failure surface using this method by hand.
44
Figure 4.18: Correction Factor (f0) (Duncan & Wright, 2005)
The need for a correction factor in the use of Janbu’s simplified method raises doubt in the
level of confidence of using Bishop’s simplified method as the latter does not have a
correction factor to account for the interslice forces. The correction factor aims to account for
the effect of the interslice shear forces, however even with the correction factor the factors of
safety calculated are not similar to those from Bishop’s simplified method. Section 5
compares Janbu’s simplified method to other methods discussed in this section, and the
factors of safety calculated were lower (between 2 and 9%) than those of Bishop’s simplified
method, Morgenstern and Price’s method, Sarma’s method and the General Limit Equilibrium
method. The factor of safety calculated using Janbu’s simplified method was often more
similar to the factor of safety found using Fellenius’ method, which, as aforementioned,
should not be used in practice. The doubt in level of confidence is then placed on Janbu’s
correctional factor rather than on Bishop’s simplified method. Duncan and Wright (2005) say
that the assumption of horizontal interslice forces “…almost always produces factors of safety
that are smaller than those obtained by more rigorous procedures that satisfy complete
equilibrium”.
45
4.3.6
General Limit Equilibrium method 1977
Fredlund and Krahn (1977) developed a simpler derivation of Morgenstern and Price’s
method, known as the General Limit Equilibrium method (GLE). The GLE method can use
different interslice force functions such as constant, half-sine, or trapezoidal (GEO-SLOPE
International Ltd, 2008). Two factors of safety equations exist: one for moment equilibrium
the other for force equilibrium.
The normal force is derived from the vertical equilibrium of force either in terms of the
interslice force (E) or the interslice shear force (X):
𝑁=
𝑐 ′ 𝐿 sin𝛼 𝑢 𝐿 sin𝛼 tan𝜙 ′
−
𝐹
𝐹
′
sin𝛼 tan𝜙
cos𝛼 +
𝐹
𝑊 − (𝐸𝑖+1 − 𝐸𝑖 ) tan 𝜃 −
𝑁=
𝑐 ′ 𝐿 sin𝛼 𝑢 𝐿 sin𝛼 tan𝜙 ′
−
𝐹
𝐹
sin𝛼 tan𝜙 ′
cos𝛼 +
𝐹
𝑊 − (𝑋𝑖+1 − 𝑋𝑖 ) −
4.42
4.43
Factor of safety for moment equilibrium for circular slip circle:
𝐹𝑚 =
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan𝜑 ′ )
∑ 𝑊 sin𝛼
4.44
Factor of safety for force equilibrium:
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan𝜙 ′ ) cos𝛼
𝐹𝑓 =
∑ 𝑁 sin𝛼
4.45
An iterative process is then used to calculate the value of scaling factor (𝜆) when the factor of
safety for force equilibrium is equal to the factor of safety for moment equilibrium. The first
iteration assumes the vertical shear forces (Xi and Xi+1) are zero, so the horizontal interslice
forces (Ei and Ei+1) can be calculated. The vertical shear forces are then calculated using the
assumed values of the scaling factor (𝜆) and function (𝑓(𝑥)) in Equation 4.30, which then
allows the normal force (N) and subsequently the factors of safety to be calculated
(Albataineh, 2006). The SLOPE/W Engineering Book (2008) states that the GLE method is
“very useful for understanding what is happening behind the scenes and understanding the
reasons for differences between various methods”, however it is not necessarily a method for
routine analysis in practice, due to its inherent costs, time taken, and difficulty in subsequent
interpretation (GEO-SLOPE International Ltd, 2008).
46
4.4
Summary of limit equilibrium methods
Further developments of limit equilibrium methods were performed by Lowe and Karafiath
(1960), US Army Corps of Engineers (1970), Janbu (generalised method) (1973) and Sarma
(1973).


Vertical







Sarma

Lowe and
Karafiath

US ACE
Fellenius
Morgenstern
and Price

Janbu’s
generalised

Vertical
Janbu’s
simplified
Satisfies force
equilibrium
Spencer
Method
Bishop’s
simplified
Table 4.2: Limit equilibrium methods
Satisfies moment
equilibrium





Circular slip
surfaces









Non-circular slip
Surfaces









Interslice normal
force (E)









Interslice shear
forces (X)









by slice
Many reports including GEO-SLOPE International Ltd. (2008) (software manual), ANCOLD
(2012) and Duncan & Wright (2005) recommend that limit equilibrium analysis used in
practice should use methods that satisfy both force and moment equilibria as a minimum
factor of safety. Methods that have interslice forces include Bishop’s simplified (1955),
Morgenstern and Price (1965), and Spencer (1967). Fellenius’ (1936) and Janbu’s simplified
(1968) methods also provide a factor of safety; however do not consider interslice forces so
the result can be underestimated.
47
4.5
Limitations of limit equilibrium methods
Wright, Kulhawy, & Duncan (1973) state that limit equilibrium methods are “…subject to
criticism on theoretical grounds” since assumptions are employed that result in the normal
stress calculated using only conditions of static equilibrium, without consideration of the
stress-strain characteristics of the soil. The GEO-SLOPE SLOPE/W Stability Modelling
(2008) document states that “…this has two consequences, one is that local variations in
safety factors cannot be considered, the other is that the calculated stress distributions are
often unrealistic”. To overcome this issue, stresses can be found using the upper and lower
bound theorem methods, or by finite element analysis then incorporated in the stability
analysis to compute a factor of safety. Alternatively, the factor of safety can be analysed using
probabilistic analyses to account for local variations in soil properties.
Wright, Kulhawy & Duncan (1973) also state that most limit equilibrium methods assume
that the factor of safety for each slice is the same however this would only be true at failure
when each slice has a factor of safety of one.
4.6
Computer modelling
4.6.1
GeoStudio™ SLOPE/W
GeoStudio™ SLOPE/W (2012) is a computer modelling program which uses limit
equilibrium methods to determine a factor of safety of soil embankments. SLOPE/W was
used for the deterministic analyses to calculate factors of safety and also used for the
probabilistic analysis to determine the probability of failure using Monte Carlo simulations.
The GEO-SLOPE Slope Modelling (2008) document states “…the SLOPE/W results must
always be judged in context of what can exist in reality”.
4.6.2
Microsoft Excel™
Microsoft Excel™ spreadsheets can be used for simple limit equilibrium methods to obtain
factors of safety. Spreadsheets have been developed by the author for Bishop’s simplified
method and Fellenius’ method for undrained and drained conditions. Both methods have
limitations in that the width, height, and angle of the base of each slice are unknown and need
to be specified beforehand. The number of slices and iterations also has to be manually
entered.
The spreadsheets are not able to determine the location of the slip circle. This is a significant
limitation as each potential failure surface will need to be analysed separately to determine
48
which failure surface produces the lowest factor of safety. The accuracy of the factor of safety
calculated would then depend on how many failure circles have been analysed and whether or
not the most critical surface had been identified. This process would be time-consuming and
uneconomical in a professional environment but provides a useful check for Bishop’s and
Fellenius’ methods used in SLOPE/W in this thesis. SLOPE/W provides the width, midheight, and angle of the base for each slice which can then be entered into the spreadsheets to
confirm the factor of safety. The spreadsheets are also useful in understanding the steps
involved in using Bishop’s or Fellenius’ methods and in helping to identify where errors may
arise.
Other programs like MATLAB™ could be used: MS-Excel™ was chosen due to its ease of
use and simplicity.
4.6.3
A student version of the @RISK software suite, developed by Palisade Corporation (2014),
was purchased and used for the probabilistic analysis of embankments. It is an add-in to MSExcel™ which can be applied to any existing spreadsheet which has inputs, a calculation, and
an output. The process of performing a Monte Carlo simulation is described in Section 8.8.1.
49
5
COMPARISON OF DETERMINISTIC METHODS
5.1
Baseline cases
Four soil embankments were analysed using the aforementioned stability charts and limit
equilibrium methods (calculated by the developed MS-Excel™ spreadsheets and by
SLOPE/W software). This set of four solutions was used to represent a baseline set of cases
against which the effects of changing the input parameters upon the output factors of safety
and failure mechanisms could be compared. The resulting factors of safety were compared for
the following cases:
Case 1: One soil layer, no phreatic surface
Case 2: One soil layer with a phreatic surface
Case 3: Two soil layers, no phreatic surface
Case 4: Two soil layers with a phreatic surface
Case 1
Case 2
Case 3
Case 4
Figure 5.1: Baseline cases
50
5.2
Case 1: One layer soil with no pore water effects
Case 1 is a homogenous, isotropic soil layer with the phreatic surface assumed to be low
enough to not have an effect on the embankment (shown in Figure 5.2). The entry range
(where the slip circle enters the embankment) was defined as being from (4, 15) to (10, 15)
and the exit range was set to (28, 6) to (34, 5); these ranges were kept constant for each
baseline case.
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.2: Case 1 embankment geometry
The embankment was tested under drained and undrained conditions using effective and total
stress parameters, respectively.
Table 5.1: Soil properties
cu (kPa)
20
c′ (kPa)
0
Total stress parameters (undrained)
𝜙u (°)
𝛾𝑏 (kN/m3)
𝛽 (°)
Effective stress parameters (drained)
𝛾𝑏 (kN/m3)
𝜙′ (°)
𝛽 (°)
0
35
18
18
26.6
26.6
The angle of repose (𝛽) of 26.6° represents an embankment profiled to a gradient of 1V:2H.
The failure slip circle was tested with five slices and 20 slices, separately. These parameters
were used for each baseline case. Appendix B shows a brief analysis of the number of slices
used in the analysis versus the factor of safety which concludes that they give an acceptable
upper and lower bound to the number of slices required for analysis.
51
The factor of safety was first found using stability charts developed by Taylor (1937), Bishop
& Morgenstern (1960), Spencer (1967), and Janbu (1968), then compared to factors of safety
calculated using the author’s Case 1 Spreadsheet and SLOPE/W.
5.2.1
Stability charts results
Table 5.2: Case 1 factor of safety from stability charts
Stability charts
Analysis
Undrained
Taylor
Bishop &
Morgenstern
Spencer
Janbu
0.656
n/a
n/a
0.656
n/a
1.9
1.502
n/a
Drained
The drained analysis using Bishop and Morgenstern’s and Spencer’s stability charts gave
different (20%) factors of safety. Since the effects of pore water pressure were ignored, the
pore water ratio (𝑟𝑢 ) was zero for both Bishop and Morgenstern’s and Spencer’s stability
charts. This resulted in Bishop and Morgenstern’s factor of safety to be an approximate
reading for stability coefficient m from their stability charts, since 𝐹𝑠 = 𝑚 − 𝑛 𝑟𝑢 . The
difference between the drained factors of safety was attributed to Spencer’s method
comparing the internal angle of shearing resistance (ϕ) with a mobilised friction angle (𝜙𝑑 )
whereas Bishop and Morgenstern’s method depends more on the pore water pressures.
5.2.2
Undrained analysis
Table 5.3: Case 1 factor of safety from Case 1 Spreadsheet and SLOPE/W for undrained conditions
No. of
slices
5
20
Method
Bishop
simplified
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
0.784
0.784
n/a
n/a
n/a
n/a
SLOPE/W
0.792
0.792
0.680
0.792
0.792
0.796
0.729
0.729
n/a
n/a
n/a
n/a
SLOPE/W
0.735
0.735
0.719
0.734
0.734
0.732
52
0.735
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.3: Case 1 most critical slip circle in undrained SLOPE/W analysis with 20 slices
The range of failure surfaces is shown in Figure 5.4 with the red arcs showing the lower
factors of safety.
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.4 : Distribution of failure circles analysed (GEO-SLOPE, 2007)
The factors of safety calculated in the spreadsheet and in SLOPE/W were all less than one
which suggested that it would not be possible to construct this embankment due to its inherent
instability. Figure 5.3 shows that the undrained failure surface was a deep-seated failure
resembling a mid-point circle, as predicted by Taylor’s and Janbu’s stability charts. The
failure surface contained the largest possible mass within the defined entry/exit points and
elevation which is a result of the undrained cohesion of 20 kPa corresponding to a very soft to
soft soil (AS1726-1993; BS 5930-1999). On comparison, the undrained values from stability
charts were approximately 10% less than those calculated using limit equilibrium methods in
Table 5.3. This difference was attributed to the simplicity of the stability charts as a method.
53
The factors of safety were the same for Fellenius’ and Bishop’s simplified method. This was
found to be true for undrained conditions for each case analysed in this thesis. Appendix C
shows calculations that confirm Fellenius’ method is equal to Bishop’s simplified method for
undrained conditions.
5.2.3
Drained analysis
Table 5.4: Case 1 factor of safety from Case 1 Spreadsheet and SLOPE/W for drained conditions
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
1.479
1.449
n/a
n/a
n/a
n/a
SLOPE/W
1.479
1.448
1.447
1.479
1.479
1.479
1.477
1.448
n/a
n/a
n/a
n/a
SLOPE/W
1.477
1.445
1.445
1.477
1.477
1.477
No. of
slices
5
20
Method
Bishop
simplified
1.477
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.5: Case 1 most critical slip circle in drained SLOPE/W analysis with 20 slices
Table 6.4 shows that methods that satisfy the same equilibrium conditions produce the same
factor of safety (i.e. Morgenstern and Price’s, Spencer’s, and General Limit Equilibrium).
Janbu’s simplified method calculated similarly close values to Fellenius’ method which could
have been the result of Janbu’s simplified method not satisfying moment equilibrium and
Fellenius’ method not taking into account the interslice normal forces when Bishop’s
simplified method did.
54
The factors of safety were 3.5% below the minimum recommended by ANCOLD (2012) in
Table 3.1. Further investigation of the consequences of failure would be required to determine
whether this factor of safety was acceptable for a tailings dam built with these material
properties, or not. In reality, an embankment would not have homogenous, isotropic soil
properties; there would be variations within the embankment resulting in different local
factors of safety. The values used in this analysis were idealised assumptions deemed to
represent the soil properties for the entire embankment.
Bishop and Morgenstern’s factor of safety from their stability chart was approximately 30%
higher than the values found using limit equilibrium methods in Table 5.4, while Spencer’s
was within 4%.. This showed that Spencer’s stability chart gave a suitable factor of safety for
drained conditions.
5.3
Case 2: One layer soil with phreatic surface
Case 2 analysed the same downstream embankment as Case 1 but a phreatic surface was
included to determine the changes in behaviour of each method of analysis and their factors of
safety. The phreatic surface was assumed to occur three metres below the top of the
embankment at x = 0, y = 12; this was where the imagined level to which the retained pond
was allowed to rise. It was assumed that a drain would be placed at the toe of the embankment
so the phreatic surface ends at the toe (Fell et al., 2005).
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.6: Case 2 embankment geometry with phreatic surface
To ensure the same phreatic surface location was used in SLOPE/W and the spreadsheets, the
pore pressure depth (z) for each slice was entered into the spreadsheet using the pore water
pressures (from SLOPE/W) divided by the acceleration due to gravity (9.81 m/s2).
55
5.3.1
Stability charts results
Table 5.5: Case 2 factor of safety from stability charts
Analysis
Undrained
Taylor
Bishop &
Morgenstern
Spencer
Janbu
0.656
n/a
n/a
0.656
n/a
1.266
1.038
n/a
Drained
For undrained analysis, Taylor and Janbu’s stability charts gave the same factors of safety as
in Case 1 (i.e. the change in phreatic surface did not affect the factor of safety). Bishop and
Morgenstern’s and Spencer’s stability charts gave different factors of safety for the drained
condition compared to Case 1, since the pore water pressures were now included. The pore
pressure ratio (𝑟𝑢 ) calculated for Bishop and Morgenstern’s stability charts was 0.33, however
for use in Spencer’s method 𝑟𝑢 = 0.25 which accounts for the differences between the two
factors of safety.
5.3.2
Undrained analysis
Table 5.6: Factor of safety from Case 2 Spreadsheet and SLOPE/W for undrained conditions
Method
Bishop
simplified
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
0.755
0.755
n/a
n/a
n/a
n/a
SLOPE/W
0.748
0.748
0.730
0.757
0.757
0.756
0.735
0.735
n/a
n/a
n/a
n/a
SLOPE/W
0.730
0.730
0.719
0.735
0.735
0.730
No. of
slices
5
20
0.730
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.7: Most critical slip circle in undrained SLOPE/W analysis with 20 slices
56
The short-term analysis, in the presence of a phreatic surface, gave slightly lower (0.3%)
factors of safety compared to Case 1 when there was no phreatic surface present. The phreatic
surface did not have much effect on this embankment under undrained conditions since the
cohesive component of the strength of the soil was already very soft. A phreatic surface in a
soil with a greater cohesive strength has been investigated in the parametric study in Section
6.
5.3.3
Drained analysis
Table 5.7: Factor of safety from Case 2 Spreadsheet and SLOPE/W for drained conditions
No. of
slices
5
20
Method
Bishop
simplified
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
1.718
1.668
n/a
n/a
n/a
n/a
SLOPE/W
1.432
1.240
1.294
1.437
1.437
1.437
1.647
1.578
n/a
n/a
n/a
n/a
SLOPE/W
1.387
1.189
1.258
1.391
1.391
1.391
Figure 5.8: Most critical slip circle in drained SLOPE/W analysis with 20 slices
The factors of safety calculated using the spreadsheets wee much greater than SLOPE/W
(between 20 and 40% higher). The difference arose in the way in which negative pore water
pressures were treated in SLOPE/W and the Case 2 Spreadsheet.
57
Negative pore pressures, or matric suction, exist when the base of the slice is above the
phreatic surface (Fredlund, 1987). The effect of including the negative pore water pressures
factor of safety (GEO-SLOPE International Ltd, 2008). The slices that experienced matric
suction are highlighted in Figure 5.8. The Case 2 (and Case 4) Spreadsheets initially included
the effects of negative pore water pressure, but were removed to calculate more realistic
factors of safety.
If soil suction was not considered in Bishop’s simplified method, and this calculation was
performed on site, the factor of safety would be overestimated by approximately 20% which
could lead to errors in determining the critical failure surface. Fredlund (1987) says ignoring
water pressures above the phreatic surface is a “reasonable assumption”.
Table 5.8: Case 2 factors of safety when negative pore pressures are ignored ((note: the SLOPE/W results are the
same as those shown in Table 5.7).
No. of
slices
5
20
Method
Bishop’s
simplified
Fellenius
1.414
1.294
SLOPE/W
1.432
1.240
1.370
1.541
SLOPE/W
1.387
1.189
The factors of safety calculated using Fellenius’ methods remained higher (4% for five slices,
25% for 20 slices) than those calculated by SLOPE/W when the matric suction was zero.
There was a significant difference between the factors of safety from Bishop’s simplified
method and Fellenius’ method (between 8 and 14% for 5 slices, and between 11 and 15%
difference at 20 slices). This was due to the Fellenius method only satisfying moment
equilibrium and no interslice shear or normal forces being considered; this means that “…
there is nothing to counterbalance the lateral components of the base shear and normal
[stress]…” (GEO-SLOPE International Ltd, 2008). It is recommended that Fellenius’ method
not be used for stability analysis in cases where the pore water pressures are included as it
calculates unrealistic factors of safety.
5.4
Case 3: Two layer soils with no phreatic surface
Case 3 analysed the same embankment but with two layers of soil as shown in Figure 6.10.
The bottom layer was referred to as the foundation and was a denser, homogenous, isotropic
material. The soil properties were idealised representations of the properties present
58
throughout the embankment in an attempt to address the potential inhomogeneity and
anisotropy therein.
Elevation (m)
15
10
Embankment
5
Foundation
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.9: Case 3 embankment geometry showing two soils and no phreatic surface.
Table 5.9: Case 3 and Case 4 properties
Embankment
Foundation
Embankment
Foundation
5.4.1
Total stress parameters (undrained)
cu (kPa)
𝜙u (°)
𝛾𝑏 (kN/m3)
20
0
18
80
0
20
Effective stress parameters (drained)
𝛾𝑏 (kN/m3)
c′ (kPa)
𝜙′ (°)
0
35
18
0
40
20
𝛽 (°)
26.6
𝛽 (°)
26.6
Stability charts results
Table 5.10: Case 3 factor of safety from stability charts
Analysis
Undrained
Drained
Taylor
Bishop &
Morgenstern
Spencer
Janbu
0.90
n/a
n/a
0.94
n/a
1.90
1.50
n/a
The undrained factors of safety calculated from Taylor’s and Janbu’s charts were similar to
the values found using limit equilibrium methods in Table 5.11. Taylor’s charts gave a value
that was 0.7% lower and Janbu’s charts a value that was 3.6% higher. This was reassuring in
59
that values from stability charts used for short-term analysis gave quick results relevant to the
stability of an embankment.
Spencer’s charts produced a factor of safety of 1.5 which was 1.5% higher than Bishop’s
simplified method (see Table 5.12). This was consistent with the results from Case 1, where
Spencer’s method calculated a suitably similar factor of safety and the value from Bishop and
Morgenstern’s stability charts was an overestimate (in this case by 22%) since no pore water
pressures were considered).
5.4.2
Undrained analysis
Table 5.11: Case 3 factors of safety from spreadsheets and SLOPE/W for undrained conditions
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
0.921
0.921
n/a
n/a
n/a
n/a
SLOPE/W
0.921
0.921
0.864
0.921
0.921
0.921
0.908
0.907
n/a
n/a
n/a
n/a
SLOPE/W
0.906
0.906
0.879
0.906
0.906
0.906
No. of
slices
5
20
Method
Bishop
simplified
0.906
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.10: Most critical slip circle in undrained SLOPE/W analysis with 20 slices
Since the foundation had a greater cohesion and bulk unit weight (compared to the
embankment fill), the deep-seated failure only occurred in the weaker material of the
embankment, with a small section of local translational failure. The factor of safety had
improved, from the situation in Case 1, by approximately 20%, indicating that the effect of a
firm stratum as a foundation can improve the factor of safety. However since the failure
60
surface was tangential to the foundation, there was an assumed plane of weakness between the
two layers. Azami, Yacoub & Curran (2012) state that “The stability of slopes is highly
dependent on the presence and configuration of the weak planes in the materials and the shape
and possible slip surface is also influenced by the orientation of the weak planes.”. The
material in the foundation is often different from that in the embankment material, especially
for tailings dams since the embankments are constructed from coarse mine tailings. As
discussed in Section 2.2.7, liners can be used between the foundation and embankments
which can also act as a plane of weakness.
5.4.3
Drained analysis
Table 5.12: Case 3 factors of safety from spreadsheets and SLOPE/W for drained conditions
Method
Bishop
simplified
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
1.479
1.449
n/a
n/a
n/a
n/a
SLOPE/W
1.479
1.448
1.447
1.479
1.479
1.479
1.477
1.448
n/a
n/a
n/a
n/a
SLOPE/W
1.478
1.445
1.445
1.477
1.477
1.477
No. of
slices
5
20
1.477
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.11: Most critical slip circle in drained SLOPE/W analysis with 20 slices
The drained analysis for Case 3 was unaffected by the stiff foundation. In comparison to Case
1, the embankment failed the same way in which it would have if there was no foundation.
This was because the soil had a high angle of shearing resistance (35°) in the long-term and
was sufficiently strong that the strength of the embankment had no effect.
61
The factors of safety calculated by the Case 3 Spreadsheet and SLOPE/W were similar (< 1%
difference), which gave confidence in the use of the SLOPE/W suite to analyse more complex
cases in terms of drained analyses.
5.5
Case 4: Two layer soils with phreatic surface
Case 4 analysed the same embankment as Case 3 but now with a phreatic surface (the same
phreatic surface as Case 2).
Elevation (m)
15
10
Embankment
5
Foundation
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.12: Case 4 embankment geometry showing two soils and phreatic surface as entered into SLOPE/W
5.5.1
Stability charts results
Table 5.13: Factor of safety from stability charts
Analysis
Undrained
Drained
Taylor
Bishop &
Morgenstern
Spencer
Janbu
0.90
n/a
n/a
0.94
n/a
1.64
1.50
n/a
The short-term factors of safety using Taylor’s and Janbu’s stability charts were the same as
were calculated for Case 3. These values gave close (< 4% difference) representations of what
was calculated in Table 5.14 using Bishop’s simplified method.
The factor of safety using Bishop and Morgenstern’s stability charts was 1.64 (15% higher
than Bishop’s simplified method values in Table 5.15). This factor of safety depended on the
depth to the phreatic surface (zw) which varied for each slice. the resultant factor of safety may
not necessarily be the most critical since the depth to the phreatic surface (zw) was estimated
to the midpoint. Further in situ testing to determine the depth to the phreatic surface could be
undertaken, however it would be difficult to determine the phreatic surface position compared
62
to its saturation depth, potential capillary fringe, and natural moisture content in the
embankment.
The factor of safety from Spencer’s stability charts was 5% higher than that that given by
drained Bishop’s simplified method in Table 5.15. The estimates for long-term conditions are
overestimated compared to computer analyses so further calculations should be performed if
this were a real embankment to verify the minimum factor of safety.
5.5.2
Undrained analysis
Table 5.14: Case 4 factors of safety from spreadsheets and SLOPE/W for undrained conditions
Method
Bishop
simplified
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
0.911
0.911
n/a
n/a
n/a
n/a
SLOPE/W
0.910
0.910
0.870
0.910
0.910
0.910
0.908
0.907
n/a
n/a
n/a
n/a
SLOPE/W
0.906
0.906
0.879
0.906
0.906
0.906
No. of
slices
5
20
0.906
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.13: Most critical slip circle in undrained SLOPE/W analysis with 20 slices
The short-term failure surface is a deep circle with a section tangential to the base. The
section tangential to the base depended on the width of the slice; a wider slice would have a
wider section tangential to the base, therefore the number of slices used in the analysis
governs the shape of the failure surface. The factors of safety calculated using the spreadsheet
and SLOPE/W were very similar (< 0.3% difference) (again, providing confidence when
using SLOPE/W for more complex cases).
63
5.5.3
Drained analysis
Table 5.15: Case 4 factors of safety from spreadsheets and SLOPE/W for drained conditions
Method
Bishop
simplified
Fellenius
Janbu
simplified
Morgenstern
and Price
Spencer
General
limit
equilibrium
1.434
1.333
n/a
n/a
n/a
n/a
SLOPE/W
1.438
1.326
1.335
1.439
1.439
1.439
1.426
1.477
n/a
n/a
n/a
n/a
SLOPE/W
1.426
1.340
1.349
1.427
1.426
1.427
No. of
slices
5
20
1.426
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 5.14: Most critical slip circle in drained SLOPE/W analysis with 20 slices
The failure surface in Case 4 was similar to that in Case 3, however the factor of safety had
decreased by approximately 3.5% with the addition of the phreatic surface. The slight
difference in the shape of the failure surface was due to the foundation layer consisting of soil
with a greater bulk unit weight. It was found previously that the foundation layer increased
the factor of safety, however, now that the phreatic surface was also included, the factor of
safety decreased, showing that the inclusion of water in the embankment reduces stability.
5.6
Summary
The deterministic analysis of the four baseline cases assumed that the soil was homogenous
and isotropic giving the embankment constant properties. In reality, this would not be true due
to variations in the materials: the data used in any analysis would have to be idealised as sets
of values used to describe the average soil properties (namely cohesion, bulk unit weight, and
angle of shearing resistance). Nevertheless, the baseline cases were used to identify
64
differences arising from the use of stability charts and limit equilibrium methods for
undrained and drained conditions.
Table 5.16: Summary of results from stability charts and limit equilibrium methods
Case
Stability charts
Limit equilibrium methods
Case 1 (one layer soil with no phreatic surface)
Undrained
0.656
0.735
Drained
1.502
1.477
Case 2 (one layer soil with phreatic surface)
Undrained
0.656
0.730
Drained
1.266
1.387
Case 3 (two soil layers with no phreatic surface)
Undrained
0.90
0.906
Drained
1.50
1.478
Case 4 (two soil layers with phreatic surface)
Undrained
0.90
0.906
Drained
1.50
1.426
Bishop and Morgenstern’s stability charts were not suitable for calculating factors of safety
for embankments with no phreatic surface. Spencer’s stability charts (shown in Table 5.16)
gave similar factors of safety to the limit equilibrium methods, with or without a phreatic
surface being included.
Ultimately, it was found that the stability charts gave a good indication of the stability of an
embankment. For a quick analysis (perhaps during a site investigation), these methods would
be suitable for determining stability based on in situ soil properties, however further analysis
would be required to verify the factors of safety. More data observations would help to
determine soil properties that better represent their variation and distribution in the
embankment. Using averaged values from observations in a limit equilibrium analysis would
provide factors of safety to a reasonable level of confidence, however the level of confidence
would be refined if the soil variability was described using probability methods instead.
The deterministic analysis also showed that the spreadsheets made for Bishop’s simplified
and Fellenius’ method calculated similar factors of safety to those found using SLOPE/W.
This showed that the methods used in SLOPE/W had been both understood and applied
correctly, as well as their having been verified by hand-calculation, making SLOPE/W a
suitable program to analyse future slope stability problems.
65
6
PARAMETRIC STUDY
A parametric study was performed to analyse the differences in factors of safety when
different soil properties were used. The inputs in the deterministic analysis (Section 5) used
single idealised values for cohesion, angle of shearing resistance, and bulk unit weight, which
in reality would be based on observations. The variability in the soil cannot always be
captured from observations since the number of observations taken may be limited by
funding, logistics, or time. Therefore this parametric study helped to evaluate the factors of
safety for different combinations of soil properties. It was also useful in identifying factors of
safety for design purposes if the variation in the soil for construction is known, or to help
specify what type of soil can be used to achieve a certain factor of safety. The factors of safety
are found by analysing the effect of variations in:
•
undrained cohesion (cu)
•
undrained cohesion (cu) and slope angle (β)
•
drained angle of shearing resistance (ϕ′)
•
bulk unit weight (γ) in undrained and drained conditions
•
pore water pressure (u)
•
effective shear strength parameters together (including apparent cohesion).
The parametric study was somewhat analogous to the probabilistic analysis; however the
parameters were selected individually rather than assigned a probability distribution and
randomly generated. The parametric study can be time consuming, but it allows the effects of
varying different properties to be easily analysed.
6.1
Effect of varying undrained cohesion (cu)
The effects of varying the undrained condition from 1 kPa to 300 kPa were tested in
SLOPE/W. The undrained strength values were taken from Table 6.1.
Table 6.1: Undrained shear strength classification (BS 5930-1999)
Stiffness state
Undrained strength cu (kPa)
Hard
> 300
Very stiff
150 - 300
Stiff
75 - 150
Firm
40 - 75
Soft
20 - 40
Very soft
< 20
66
The Australian Standard AS1726-1993 also provides classifications (and a field guide to
consistency) for different stiffness states, shown in Table 6.2.
Table 6.2: Undrained shear strength classifications (AS1726-1993)
Stiffness state
Undrained strength
cu (kPa)
Hard
> 200
Field guide to consistency
Can be indented with difficult by thumb nail
Very stiff
100 - 200
Stiff
50 - 100
Firm
25 - 50
Can be indented by thumb nail
Can be indented by thumb
Cannot be moulded by fingers
Can be moulded by strong finger pressure
Soft
12 -25
Can be moulded by light finger pressure
Very soft
< 12
Exudes between fingers when squeezed in hand
The Australian shear strength classifications have slightly different ranges (with each range
being on the softer side) compared to the British Standards. Ultimately, the classification of
soil will not affect the stability of an embankment as the shear strength values are more usable
than the stiffness state term associated with it. The parametric study has been performed
incorporating the ‘hard’ values from the BS 5930-1999.
The analysis was performed on the embankment used in Case 1. The bulk unit weight (𝛾 = 18
kN/m3), and slope angle (𝛽 = 26.6°) remained constant while the undrained cohesion (cu) was
varied. The range of critical failure surfaces for the embankments with an undrained cohesion
of 1 kPa and 300 kPa are shown in Figure 6.1 and Figure 6.2, respectively.
0.104
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 6.1: Most critical slip circles for undrained cohesion 1 kPa in SLOPE/W (GeoStudio™, 2007)
67
At 1 kPa, it is highly unlikely that the soil is strong enough to form an embankment, so a slip
circle failure as shown in Figure 6.1 would be improbable.
10.985
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 6.2: Most critical slip circles for cu = 300 kPa in SLOPE/W (GeoStudio, 2007)
Clays with an undrained cohesion of greater than 300 kPa can be described as very weak
mudstone or as hard clay (BS 5930-1999). Figure 6.2 shows a deep seated circular failure,
however failure surfaces at this strength are unlikely to be circular slips since, even in their
saturated states, the clays would fail in a brittle manner (BS 5930-1999).
Figure 6.1 and Figure 6.2 do not show plausible failures due to the unrealistic critical failure
surface assigned to a very soft and very stiff values of undrained cohesion. It has been used
here to highlight the fact that the factor of safety increases with an increase in undrained
cohesion, but it also highlights the importance of understanding the inputs into, and results
from, a simple computer program such as SLOPE/W. It is easy enough to analyse an
embankment with 300 kPa strength and believe that the type of failure would be circular,
when in fact this is not practically admissible.
68
Factor of safety vs undrained cohesion
12
R² = 1
10
Factor of safety
8
6
4
Fellenius
Janbu Simplified
2
Morgenstern & Price
Bishop F
0
0
|
soft
|
50
firm
|
100
150
stiff
|
Undrained cohesion cu (kPa)
200
very stiff
250
300
|
Figure 6.3: Change in the factor of safety arising from changes in the undrained cohesion using Bishop’s simplified,
Janbu’s simplified, Fellenius’, and Morgenstern and Price’s methods.
Figure 6.3 shows a linear relationship (R2 = 1) between the undrained cohesion and factor of
safety. This relationship assumes the failure surfaces for all cu values would be circular which,
as discussed, is not likely to represent the true failure surfaces. The slight difference (less than
one percent) between each method is a result of rounding errors and assumptions about the
interslice forces.
The undrained cohesion is assumed to be constant throughout the embankment for this
analysis. In reality, there would be a slight increase in cohesion with depth (GEO-SLOPE
International Ltd, 2008) and variability horizontally within the soil. The effect of this would
be a shallower failure surface. Methods of analysing stability while taking into account the
variability of cohesion are discussed in the probabilistic analysis in Section 8.
6.2
Effect of varying undrained cohesion (cu) and slope angle (β)
The undrained cohesion was varied between 1 and 300 kPa for different slope angles from
1V:2H (2.9°) to 10V:1H (84.3°).
69
Changes in undrained cohesion for different
slope angles β
140
Factor of safety
120
β = 2.9° (1V:20H)
100
β = 15°
80
β = 26.6° (1V:2H)
60
β = 30°
40
β = 45°
20
β = 60°
0
β = 75°
0
50
100
150
200
250
300
Undrained cohesion cu (kPa)
β = 84.3° (10V:1H)
10
9
8
Factor of safety
7
6
5
4
3
2
1
0
0
|
soft
|
50
firm
|
100
150
200
250
stiff
|
very stiff
Undrained cohesion cu (kPa)
300
|
Figure 6.4: Changes in undrained cohesion for different slope angles
Figure 6.4 shows that at shallow slope angles (< 30°), an increase in the undrained cohesion
improved the factor of safety. The factor of safety was less sensitive to changes in undrained
cohesion at angles greater than 30°. This can be seen in the sensitivity plot in Figure 6.5.
If a factor of safety of 1.5 is considered stable according to ANCOLD (2012), then the
minimum undrained cohesion for slope 1V:20H would be approximately 3.5 kPa, whereas for
slope 10V:1H, the minimum undrained cohesion would be approximately 60 kPa. This shows
that at shallow slope angles a weaker material can be used which may provide economic
70
benefits if there is sufficient space to construct long, shallow embankments, and the cohesion
is strong enough to support the loads expected to be present. Soil with undrained cohesion of
3.5 kPa is very soft material and it would not be possible to construct an embankment with it.
Steeper slope angles require less land area for construction but require soils with greater
undrained cohesion. Undrained cohesion of 60 kPa is generally the accepted minimum
allowable strength to ensure plant equipment can drive over the embankment. If additional
strength is required in an existing embankment but there is little land area available to extend
the toe of the embankment, then the use of soil stabilisation can become expensive.
Sensitivity Data
140
120
Factor of safety
100
80
1V:20H
60
10V:1H
40
20
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Sensitivity Range
Figure 6.5: Sensitivity plot
For presentation purposes, the undrained cohesion values have been normalised to values
from -1 to 1, with -1 being the lowest value (1 kPa) and 1 being the highest value (300 kPa).
This is a default function when performing sensitivity analyses in SLOPE/W to allow
different variables to be analysed with the same x-axis.
Zero cohesion could not be analysed in SLOPE/W using undrained conditions since the angle
of shearing resistance equals zero and therefore there are no shear strength parameters used in
the analysis. Without any shear strength (𝜏𝑓 ) there would be no soil - only water). This is
represented in Coulomb’s equation for shear strength:
𝜏𝑓 = 𝑐 + 𝜎 tan 𝜙
6.1
𝜏𝑓 = 0 + 𝜎 tan 0 = 0
71
6.3
Effect of varying drained angle of shearing resistance
The angles of shearing resistance were varied from 2° to 85° to show the change in factor of
safety, as shown in Figure 6.6.
Effect of varying angle of shearing resistance
30
Factor of safety
25
20
R² = 0.9597
15
10
5
0
0
10
20
30
40
50
60
70
80
90
Drained angle of shearing resistance φ′ (°)
Figure 6.6: Effect of varying drained angle of shearing resistance (ϕ′)
The graph shows the effect of varying the angle of shearing resistance (ϕ′) when the total bulk
unit weight was 18 kN/m3. Figure 6.6 shows that a greater angle of shearing resistance gives a
greater factor of safety. An exponential trendline closely represents the spread of data (R2 =
0.96) however at the lower and upper limits an exponential trend-line failed to represent the
data. The drained angle of shearing resistance could not be tested at 0° and 90°, since the
tangents of 0° and 90° are zero and infinite, respectively. In reality, a zero angle of shearing
resistance, as well as no cohesion, means that there is no shear strength, i.e. the soil does not
exist. At the upper extent, an angle of shearing resistance of 90° represents a very dense
material (i.e. rock). The angle of shearing resistance of this type of soil (or rock) would
physically be difficult to determine using standard testing methods. As the shear strength
increases, the failure surface would be less circular and more like a wedge, due to the
increased frictional resistance between soil grains. So analysing the same failure surface for
varying angles of shearing resistance does not give accurate factors of safety since the failure
surface would be varying as the angle of shearing resistance varies.
The effect of varying tanϕ′ is shown in Figure 6.7. This is done because the tangent angle of
shearing resistance is used in the limit equilibrium methods to calculate the factor of safety.
72
Figure 6.7: Effect of varying the tangent of angle of shearing resistance
The effect of varying tan ϕ′ has a linear relationship with the factor of safety; an increase in
tan ϕ′ has a proportional linear increase in the factor of safety. The highlighted region shows
angles of shearing resistance for very loose sand (30°) to very dense sand (45°) as defined by
Meyerhof (1956) and Peck et al. (1974) in Table 6.3. This region shows that very loose sand
had a factor of safety of 1.2 under the drained conditions in Case 1 and very dense sand had a
factor of safety of 2.1. ANCOLD’s (2012) factor of safety recommendation of 1.5 requires a
tan ϕ′ value of 36.7° which corresponds to medium sand (Meyerhof, 1956).
Table 6.3: Values of angle of shearing resistance for different sand densities
Soil type
Angle of shearing resistance ϕ′ (°)
Peck et al. (1974)
Meyerhof (1956)
< 29
< 30
Loose sand
29 – 30
30 – 35
Medium sand
30 – 36
35 – 40
Dense sand
36 – 41
40 – 45
> 41
> 45
Very loose sand
Very dense sand
A change in bulk unit weight (from 1 to 25 kN/m3) caused no change in the factor of safety
when varying the drained angle of shearing resistance. The factor of safety increased when the
angle of shearing resistance was increased, but varying the bulk unit weight had no effect, as
shown in Figure 6.8. The limit equilibrium methods use the bulk unit weight in determining
73
the weight of each slice (area of slice multiplied by the specified unit weight multiplied by an
assumed 1 m thick slice normal to the plane of analysis). The sum of the weights of each slice
is in the numerator and denominator in Bishop’s simplified method and so the effects cancel
out. This means that the factor of safety is independent of the bulk unit weight for drained
conditions. Figure 6.8 shows the factors of safety for unit weights (γ) from 14 – 22 kN/m3 and
when varying the drained angle of shearing resistance (ϕ′) from 25 – 45°.
Effect of varying γ and ϕ′
2.5
Factor of safety
2
1.5
φ′ = 45
φ′ = 40
1
φ′ = 35
φ′ = 30
φ′ = 25
0.5
0
14
15
16
17
18
Unit weight γ
19
20
21
22
(kN/m3)
Figure 6.8: Effect of varying bulk unit weight and angle of shearing resistance in drained conditions
6.4
Effect of varying bulk unit weight (γ)
6.4.1
Undrained conditions
Figure 6.9 shows the factors of safety for different total weights in undrained conditions. An
increase in the bulk unit weight increases the weight of each slice and hence the normal force
at the base of the slice is also increased which leads to a greater shear resistance (𝜏𝑚𝑜𝑏 ), This
results in a lower factor of safety, evident in Figure 6.9 and Equation 6.2.
𝐹=
𝜏𝑓
𝜏𝑚𝑜𝑏
6.2
74
Varying bulk unit weight in undrained conditions
500
450
1V:20H (2.9°)
400
1V:2H (26.6°)
Factor of safety
350
10V:1H (84.3°)
300
250
200
150
100
50
0
0
5
10
Bulk unit weight γ
15
20
25
(kN/m3)
35
30
Factor of safety
25
20
15
10
5
0
14
15
16
|
|
uniform dry mixed grain dry
loose sand
loose sand
17
18
19
20
21
22
|
|
|
|
uniform dry uniform wet mixed grain wet mixed grain wet
dense sand
loose sand
loose sand
dense sand
Soil unit weight γ (kN/m3)
Figure 6.9: Varying bulk unit weight in undrained conditions
6.4.2
Drained conditions
The bulk unit weights were varied from 1 to 25 kN/m3 for slope angles 1V:20H, 1V:2H, and
10V:1H.
75
Varying bulk unit weight in drained conditions
16
14
Factor of safety
12
10
1V:20H (2.9°)
8
1V:2H (26.6°)
6
10V:1H (84.3°)
4
2
0
0
5
10
|
water
15
20
|
dense mixed grain sand
25
|
concrete
Bulk unit weight γ (kN/m3)
Figure 6.10: Varying bulk unit weight in drained conditions
A zero unit weight suggests the soil is made only of voids which is not possible. The other
end of the scale shows a bulk unit weight which is close to the bulk unit weight of concrete.
For drained conditions, the bulk unit weight does not have an effect on the factor of safety;
however a shallow slope (2.9°) gives a factor of safety that is over 130 times higher than a
factor of safety for a steep embankment (84.3°) (in this case).
Considering an embankment with weight W and slope angle β, the embankment is in
equilibrium when the sliding forces are equal to the restraining forces divided by a factor of
safety (Smith, 2014), i.e.:
𝑊 cos 𝛽 tan 𝜙′
𝐹
tan 𝜙′
𝐹=
tan 𝛽
𝑊 sin 𝛽 =
6.3
This shows that the weight of the material has no effect on the factor of safety or stability or
the embankment, however the bulk unit weight does indirectly effect the values of undrained
cohesion or angle of shearing resistance (higher values of bulk unit weight will have relative
high values of shear strength parameters).
76
Figure 6.11 shows the effect of varying bulk unit weight at slope angles of 15° and 26.6° with
a phreatic surface. These angles were used because the phreatic surface was close enough to
the failure surface to have an effect on the factor of safety. At angles greater than 30° the
phreatic surface did not affect the factor of safety at different bulk unit weights and the factors
of safety were constant. The phreatic surface could be raised to fully submerge the
embankment for the analysis however this would not resemble real life situations.
Varying bulk unit weight for drained
conditions with phreatic surface
3
Factor of safety
2.5
2
1.5
1
β = 15°
0.5
β =26.6° (1V:2H)
0
0
5
10
Bulk unit weight γ
15
20
25
(kN/m3)
Figure 6.11: Varying bulk unit weight for drained conditions with phreatic surface
Figure 6.11 shows that when the failure surface was close to the phreatic surface then the bulk
unit weight did have an effect on the factor of safety. The unit weight varies for each slice as
some slices were affected by the phreatic surface and will require use of a saturated unit
weight while other slices remain unaffected by the pore water pressures and an engineer can
continue to use the bulk unit weight (see Figure 6.15). At 26.6° the factor of safety was
asymptotic to approximately 0.2 for unit weights from 1 to 5 kN/m3, then to approximately
1.5 for unit weights greater than 20 kN/m3. A similar effect was seen for a 15° slope, where
the factor of safety was asymptotic to both 0.2 and 2.6, respectively. The values were
asymptotic due to the number of slices that were affected by the pore water pressure. The
values would vary for different embankment geometries and different locations of phreatic
surface. The effect on the factor of safety for varying the bulk unit weight in undrained and
drained conditions is also shown in the probabilistic analysis in Section 8.5.
77
6.5
Effect of varying pore water pressure (u)
The effect of varying the level of the phreatic surface was analysed to determine the change in
factor of safety. The Case 2 embankment geometry was used with the angle of the phreatic
Elevation (m)
surface varying from zero to 18.4° (10V:30H).
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
θ
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Distance (m)
Figure 6.12: Location of phreatic surfaces
Varying angle of phreatic surface
1.5
Factor of safety
1.2
0.9
0.6
0.3
0
0
2
4
6
8
10
12
14
16
18
20
Phreatic surface angle (θ)
.
Figure 6.13: Factors of safety when varying angle of phreatic surface
Figure 6.13 shows that when the phreatic surface is low, the factor of safety is higher. At
some angle (in this case 10°) the phreatic surface begins to have an effect on the failure
surface and the factor of safety is reduced. This is because the failure surface now intercepts
the phreatic surface and the pore water pressures reduce the total stresses, ultimately lowering
the factor of safety. Figure 6.14 and Figure 6.15 show the differences in the failure arc when
the phreatic surface is 9.5° and 11.3°, respectively. In Figure 6.14 the failure arc is above the
78
phreatic surface and the pore water pressures are zero. In Figure 6.15 the failure arc goes
below the phreatic surface; these slices are subject to pore water pressure.
Figure 6.14: Failure surface when phreatic surface angle <10°
Figure 6.15: Failure surface when phreatic surface angle >10°
The position of the phreatic surface which would result in a lower factor of safety would vary
for different embankments.
6.6
Effect of varying effective shear strength parameters
A sensitivity analysis was performed in SLOPE/W for drained conditions. A drained cohesion
was used which represented an apparent cohesion (for all other cases the drained cohesion
was assumed to have been zero). Brand (1982) and GEO-SLOPE (2008) say that a zero
strength condition is unrealistic and is seldom zero, however apparent cohesion should not be
79
relied upon in the design or back-analysis of the stability of an embankment. The sensitivity
data used apparent cohesion for values from 1 to 15 kPa (values used in a previous parametric
study by Tan, Lee, and Sivadass, 2008), drained angles of shearing resistance from 30° to
45°, and bulk unit weights between 14 and 22 kN/m3.
Sensitivity Data
3
2.8
Bulk unit weight γ (kN/m3)
W
Factor of Safety
2.6
Cohesion c′ (kPa)
2.4
2.2
Angle of shearing resistance ϕ′ (°)
2
1.8
1.6
0
-1
1
Sensitivity Range
Figure 6.16: Sensitivity plot (GEO-SLOPE, 2014)
The sensitivity plot shows that the factor of safety was most sensitive to changes in apparent
cohesion and angle of shearing resistance, and was less sensitive to changes in the bulk unit
weight. This was consistent with what was analysed in this parametric study (see also
subsequent probabilistic analysis).
The methods of analysis ultimately depend on the accuracy of the input parameters. If the
input parameters are incorrect then the outcome of the analysis is subject to errors as well.
The parametric study shows the effects on the factor of safety when soil properties are varied.
It is useful for deterministic analysis to understand what effect an increase or decrease, or the
variability, in a soil property will have on the stability of an embankment. The deterministic
methods of analysis were applied to a case study in the following chapter, which includes a
parametric study of the foundation properties.
80
7
DETERMINISTIC ANALYSIS OF A CASE STUDY
The deterministic methods of analysis discussed in previous sections have been applied to a
case study tailings dam embankment (Figure 7.1) to determine its stability.
Figure 7.1: Typical tailings dam cross section
Table 7.1: Case study soil properties (original table of soil properties are included in Appendix D).
Effective cohesion
c′ (kPa)
Material
Angle of shearing
resistance ϕ′ (°)
Bulk unit weight
γ (kN/m3)
Rock armour
0
40
20
Earth fill
5
35
20
Foundation
0
varies
varies
Figure 7.2 shows the embankment geometry, phreatic surface and soil regions as entered into
Elevation (m)
SLOPE/W (2012).
16
14
12
10
8
6
4
2
0
-2
Tailings
Earth fill
Foundation
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
Distance (m)
Figure 7.2: Case study embankment in SLOPE/W (2007)
81
80
The data provided was in terms of effective stresses for drained conditions. The earth fill has
an effective cohesion of 5 kPa which was assumed to be apparent cohesion. The embankment
has been analysed with apparent cohesion and with zero effective cohesion to determine what
effect the apparent cohesion has on the stability of the embankment. The exit range ends at the
toe of the embankment as it was assumed that the foundation will not fail by bearing failure.
7.1
Downstream slope
The case study does not provide detailed information on the foundation of the embankment,
except that there are regions of sandy clay (alluvium) at the trial pit, and high plasticity clay
and sandstone where bore holes have been drilled. Therefore, the foundation properties have
been varied based on the available data:
•
No foundation – with apparent cohesion
•
No foundation – with zero effective cohesion
•
2 m of clay – with apparent cohesion
•
2 m of clay – with zero effective cohesion
•
2 m of alluvium – with apparent cohesion
•
2 m of alluvium – with zero effective cohesion
•
1 m of clay on 1m of sandstone – with apparent cohesion
•
1 m of clay on 1m of sandstone – with zero effective cohesion
•
2 m of sandstone – with apparent cohesion
•
2 m of sandstone – with zero effective cohesion
Each case was analysed using Bishop’s simplified method in SLOPE/W, with the results
shown in Table 7.2 (the failure surfaces for each case are shown in Appendix E).
Elevation (m)
2.070
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure 7.3: Critical failure surface with sandstone foundation and zero effective cohesion
82
Table 7.2: Factors of safety and failure type for a typical tailings dam downstream slope using varying assumptions on
the foundation properties and apparent cohesion.
Foundation
type
No foundation
2 m clay
2 m alluvium
1 m clay and
1 m sandstone
2 m sandstone
Foundation properties
ϕ′ (°)
γ (kN/m3)
-
30
28
-
Earth fill
cʹ (kPa)
Factor of
safety
5
2.279
Shallow toe circle
0
2.082
Shallow toe circle
5
1.914
0
1.846
5
1.826
0
1.759
19
18
30
19
5
2.052
36
21
0
1.901
5
2.179
0
2.070
36
21
Failure type
Deep toe circle
through foundation
Deep toe circle
through foundation
Deep toe circle
through foundation
Deep toe circle
through foundation
Deep toe circle
tangential to sandstone
Deep toe circle
tangential to sandstone
Deep toe circle
through foundation
Deep toe circle
tangential to
foundation
When the foundation was not included in SLOPE/W, the software has assumed a stiff stratum,
causing a shallow toe failure arc and relatively high factor of safety. This was also evident
where the clay and alluvium foundations had failure surfaces tangential to the base of the
assumed two metre thick foundation (since the foundations were weaker than the earth fill).
This may not be representative of a real failure surface since the properties and depth of the
foundation are unknown. If the foundation properties at more locations beneath the
embankment and soil properties below the foundation were known then a more realistic
failure surface could be determined. If foundation properties are not known then an
assumption should be made for analysis purposes, if using SLOPE/W, to avoid unrealistic
factors of safety being calculated.
The weathered sandstone had the densest bulk unit weight of all materials in the case study
embankment, it acted as firm stratum below the embankment and so the critical failure surface
did not penetrate through it for the cases of 1 m clay and 1 m sandstone, and 2 m of
83
sandstone. When the foundation was only sandstone, the critical failure surfaces varied
between using apparent cohesion and using zero cohesion. When the earth fill apparent
cohesion was included the critical failure surface was a deep circle through the sandstone
layer which is not practically admissible since the sandstone layer is denser than the
embankment material. The factors of safety for all foundation assumptions were greater than
1.5, indicating that the embankment is stable using the given soil properties.
7.2
Upstream slope
The upstream slope was analysed assuming the pond level was at its current level, full and
empty. A clay foundation was assumed for these analyses, with a comparison to a sandstone
foundation for the case where the pond level was full.
Elevation (m)
1.303
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure 7.4: Critical failure surface of upstream slope with no apparent cohesion and at current tailings pond level
Table 7.3: Case study upstream slope factors of safety for 5 kPa and zero cohesion
Pond level
Factor of safety
c′ = 5 kPa
c′ = 0 kPa
Current level
1.467
1.303
Empty
1.475
1.308
Full
1.475
1.310
Full (sandstone
foundation)
1.475
1.310
The factors of safety calculated in SLOPE/W show the upstream slope was stable, since they
were greater than one. The critical failure surfaces for the upstream slope were all shallow toe
circles, propagating at the embankment crest. The shape of the critical failure surfaces were
close to resembling a translational or wedge failure since the failure occurs through the rock
armour and at the interface of the rock armour and earth fill. Since the rock armour is
predominantly cobbles and gravel, and is more granular than the earth fill, it is more likely to
84
fail by a planar translational slip surface rather than a circular failure surface (Smith, 2014).
This is more practically admissible than a deep toe circle with tangential sections. ANCOLD
(2012) states that “shallow surfaces are often identified by computer analyses as giving the
minimum factor of safety but do not lead to the critical breaching of the dam”. So SLOPE/W
identified the critical failure surface however a failure of this size is not likely to result in
complete collapse of the embankment.
7.3
Discussion
Assuming there is no apparent cohesion (zero effective cohesion) may result in
underestimating the stability of the embankment which could lead to unnecessarily high
construction costs. However, assuming 5 kPa apparent cohesion may not be accurate and
could result in an overestimation of the stability, which potentially could lead to embankment
failure. According to Table 6.1 and Table 6.2, cohesion values less than 20 kPa represent very
soft-soft soils and the accuracy of test methods at values less than 20 kPa may mean the
cohesion is not accurate, so values as low as measured in the case study should not be relied
upon in the design or back-analysis for tailings dam stability. It is safer to assume zero
cohesion in the design of an embankment so that it is not designed based on the assumption of
apparent strength.
ANCOLD (2012) recommends a factor of safety of 1.5 for a tailings dam embankment to be
considered stable (refer to Section 3.2). The analyses of the downstream slope showed that the
embankment was stable since all factors of safety were greater than 1.5. The upstream slope
had factors of safety lower than 1.5, but still greater than one which suggests the dam is stable
but the stability is within a lower tolerance than what ANCOLD recommends. However, it
was found that the failure surface of the upstream embankment would be unlikely to lead to
complete failure of the embankment.
There are a number of factors not included in the slope stability analysis that determine
whether the factors of safety are acceptable. ANCOLD (2012) lists infrastructure, population
at risk, business importance, public health, social dislocation, impact area, impact duration
and impact on natural environment as the factors that influence the severity of the dam failure
(as discussed in Section 2.3) . If the tailings are toxic and have the potential of significant
environment effects or would be difficult to remediate then the factor of safety should be
high, likewise if the tailings contain no chemicals hazardous to the environment then a lower
factor of safety may be acceptable. The surroundings of the dam also determine the required
factor of safety. If the dam is upstream from populated areas, drinking water supplies, national
85
parks, etc., then the required factor of safety should be high to ensure the consequence of
failure is acceptably low. Further investigation into the consequences of failure for this
tailings dam would be required to determine whether the factors of safety are sufficiently high
enough.
The analyses assumed that in SLOPE/W (2012) the entry and exit ranges were at the ground
surface (i.e. above the rock armour); it could be possible that the failure surface is within the
earth fill layer and passes through the toe of earth fill embankment before passing through the
rock armour. SLOPE/W (2012) does not allow the entry and exit ranges to be defined within
the embankment and it is not practical to investigate the stability without the rock armour
layer, since the weight of the rock armour at the toe increases the stabilising moment, thus
improving the factor of safety. Further analysis, for example, a finite element analysis, would
be suitable in this situation to analyse the earth fill embankment while assuming the rock
armour has no strength.
The depth and location of the bore holes suggest that the properties of the embankment and
foundation have been analysed after the dam has been constructed. The pocket of natural
alluvium at the trial pit beneath the rock armour, and the fact that there is limited foundation
data, suggests that little or no work was performed to increase the bearing capacity of the
foundation before the embankment was constructed. The slope stability analyses performed
using SLOPE/W have assumed that the tailings dam will not fail by bearing failure as a result
of a weak foundation. Therefore, the analysis may have shown that the slopes are stable but
bearing failure of the foundation could potentially lead to failure of the embankment.
Translational sliding between the embankment and foundation is unlikely due to the length
and subsequent weight of the embankment.
The analyses assumed the tailings material has the same bulk unit weight as water (9.81
kN/m3) and the same physical properties. Further investigation would be required to
determine whether the bulk unit weight was slightly different due to the chemical properties
of the tailings and what affect this would have on the stability.
86
8
PROBABILISTIC METHODS OF ANALYSIS
This thesis has until now focussed on analysing slope stability based on traditional methods of
analysis which output a single deterministic estimate as to whether, or not, an embankment is
stable. Limit equilibrium methods cannot neither quantify the level of risk, nor probability of
failure, associated with the slope: nor can they determine the criticality of the asset in the
event of a slope being seen, as a tailings dam would, in such terms. Probabilistic methods of
analysis produce a distribution of possible factors of safety rather than a single fixed value.
The parametric study in Section 6 has shown that the factor of safety changes when a single
variable changes for the same embankment. The study does not show how representative the
value was in terms of reliability or take into account the simultaneous variation of the input
variables. A reliability analysis allows the uncertainties to be identified, and quantified, to
determine how reliable the result is. It is therefore worthwhile to analyse the stability of a
tailings dam embankment using a probabilistic approach to account for the inherent variability
in a soil.
Probabilistic approaches were first applied to slope engineering in the 1970s (Alonso, 1976;
Ang and Tang, 1976). The methods have since developed and are well established and
documented in literature; however, probabilistic methods are not as commonly used by
practicing engineers. El-Ramly et al. (2002) suggest that this reluctance is attributed to four
factors:
1. Engineers’ training in statistics and probability theory is often limited to basic
information taught during the early years of their education.
2. A common misconception that probabilistic analyses require significantly more data,
time and effort than deterministic methods.
3. There are limited studies on the implementation and benefits of performing a
probabilistic analysis.
4. Acceptable probabilities of unsatisfactory performance are ill-defined, making it
difficult to understand the results of a probabilistic analysis.
A fifth reason why practicing engineers are reluctant to use deterministic methods could be to
do with insurance and indemnity issues. If an undesirable probability of failure were
calculated from back-analysis of an existing structure then this could be problematic with
regard to future insurance premium increases, or even a refusal to insure, and increases the
costs of maintaining the dam or result in its closure. More frequent maintenance inspections
87
may also be required to identify any signs of failure which will also increases costs and
potentially leads to closure if the failure probability is too high.
8.1
Uncertainties in soil properties
“Uncertainties in soil properties, environmental conditions and theoretical models are reasons
for a lack of confidence in deterministic analysis” (Alonso, 1976). El-Ramly et al., (2002)
describe factors contributing to uncertainty as geological anomalies, inherent spatial
variability of soil properties, scarcity of representative data, changing environmental
conditions, unexpected failure mechanisms, simplifications and approximations adopted in
geotechnical models, and human mistakes in design and construction”. Therefore it is
important to understand what uncertainties exist and the effect the uncertainties have on
analysing the stability of an embankment.
Geotechnical uncertainties can be divided into two categories; aleatory uncertainties and
epistemic uncertainties (Hacking, 1975). Many authors including Ang and Tang (2007) and
Lacasse and Nadim (1996) provide the following definitions:
•
Aleatory uncertainties are associated with the randomness and spatial variability of a
soil property. Aleatory uncertainties cannot be reduced or eliminated as they are
inherent in the soil.
•
Epistemic uncertainties are knowledge-based errors that arise from underlying
imperfect models of reality. Epistemic uncertainties can be reduced or even eliminated
calculation methods.
Uncertainty in a
soil property
Aleatory
Spatial
variability
Epistemic
Randomness
Statistical
uncertainty
Measurement
and/or model
uncertainty
Figure 8.1: Soil uncertainties flow chart (after Lacasse and Nadim, 1996)
88
Baecher and Christian (2005) state that: “… much of the confusion in geotechnical reliability
analysis arises from a failure to recognise that there are these two different sources of
uncertainty and that they have different consequences”. The influence on the probability of
failure as a result of epistemic or aleatory uncertainties is fundamentally different (De Groot,
1988). For example, if the probability of failure due to spatial variability (aleatory) is 10%,
this implies that 10% of the structure will fail. The same 10% probability of failure due to
systematic (epistemic) errors implies there is a 10% chance that the entire structure will fail
(Baecher, 1987b). De Groot (1988) states it is important to separate and quantify the
contribution of the four sources of uncertainty to allow the effect of each source to be
analysed. The combined effect of all four sources allows the overall reliability of a design to
be calculated. The net result of such an error analysis is a quantification of the overall degree
of confidence that can be placed in the design (Baecher, 1987). This is difficult to apply to the
baseline cases and case study used in this thesis since only deterministic data is available. If
raw test data was available for each case then it would be suitable to determine and apply the
aleatory and epistemic uncertainties separately.
In 1982, Einstein and Baecher stated:
“In thinking about sources of uncertainty in engineering geology, one is left with the fact that
the uncertainty is inevitable. One attempts to reduce it as much as possible, but it must
ultimately be faced. It is a well-recognised part of life for the engineer. The question is not
whether to deal with uncertainty, but how?”
8.2
Random variables and probability density functions
Traditional deterministic methods (i.e. Bishop’s simplified method) use a single value for
each variable to determine a unique factor of safety. This assumes that the soil parameters are
homogenous and that the value of each input is known exactly (or at least within an
experimental error bound). However due to the uncertainties in a soil, and model
uncertainties, an exact value for most variables cannot be precisely defined and so a
distribution better represents the possible values. Properties such as shear strength, cohesion,
angle of shearing resistance, permeability, and unit weight vary within an embankment and
therefore can be better represented as random variables.
89
Deterministic description
0
Statistical description
Bulk unit weight
0
Bulk unit weight
Figure 8.2: Deterministic and statistical descriptions of a soil property (after Lacasse and Nadim, 1996)
Random variables are represented by a probability density function, such as normal and
lognormal curves. The normal distribution is the most common type of probability
distribution with many random variables in geotechnical engineering conforming to this
distribution (Rocscience, n.d.) When the mean (𝜇) and standard deviation (𝜎) are known, the
probability density function for a normal distribution is defined by:
𝑓(𝑥) =
1 𝑥−𝜇 2
exp �− 2 � 𝜎 � �
8.1
𝜎√2𝜋
The lognormal distribution is closely related to the normal distribution. “If x is distributed
lognormally with mean (𝜇) and standard deviation (𝜎), then ln(x) is distributed normally with
𝜇 and 𝜎” (MathWorks, 2015). The lognormal distribution is applicable when the value must
have a positive solution, since ln(x) only exists when x is positive.
8.2
𝑓(𝑥)
𝑓(𝑥)
𝑓(𝑥) =
(ln 𝑥 − 𝜇)2
�
2𝜎 2
𝑥𝜎√2𝜋
exp �−
Normal distribution
Lognormal distribution
Figure 8.3: Normal and lognormal distribution curves (US ACE, 1997)
90
The type of probability distribution applied to each soil property depends on the number of
observations. Where there are adequate amounts of data a cumulative distribution function of
the measurements can be used directly in the simulation process (El-Ramly, Morgenstern and
Cruden, 2002) The cumulative distribution function is representative of a histogram and may
show spikes that do not reflect actual data, so alternatively an average of the cumulative
probability can be used to smooth unwanted spikes (El-Ramly, Morgenstern and Cruden
2002). Where the observations are scarce or absent, the probability distribution for each soil
property can be assumed from the literature. There are a number of studies that have
estimated the probability distributions of soil properties and soil testing methods, including
those by Lumb 1966, Chowdhury 1984, Harr 1987, Kulhawy et al. 1991, and Lacasse and
Lee et al. (1983) say that the “normal or lognormal distributions are adequate for the large
majority of geotechnical distributions”. However care should still be exercised to ensure that
the variables remain within physical, practical, limits, for example, shear strength parameters
should not have negative values so the selected probability distribution function must not
imply any negative values.
The random variables characteristics used in probability analysis include the mean (𝜇),
variance (𝑉𝑎𝑟[𝑥]), standard deviation (𝜎𝑥 ) (refer to Appendix G for explanations on these
values) and the coefficient of variation (C.O.V).
Figure 8.4: Normal probability distribution function (US ACE, 2006)
91
Coefficient of variation (C.O.V) provides a dimensionless expression of the uncertainty
inherent in a random variable. It is the ratio of the standard deviation to the expected value,
expressed as a percentage.
𝐶. 𝑂. 𝑉 =
𝜎𝑥
× 100%
𝐸[𝑥]
8.3
The coefficient of variation for a number of soil properties and in situ tests are available from
literature but can also be determined from test data. A smaller C.O.V represents a smaller
uncertainty and a larger value represents a greater uncertainty.
Table 8.1: Coefficients of Variation (from Duncan, 2000)
Soil property
C.O.V (%)
References
3 to 7
Harr (1984), Kulhawy (1992)
2 to 13
Harr (1984), Kulhawy (1992)
13 to 40
Harr (1984), Kulhawy (1992), Lacasse and
Bulk unit weight (γ)
Effective angle of shearing
resistance (𝜙ʹ)
Undrained cohesion (cu)
8.3
Probability of failure
The probability of failure 𝑃(𝑓), in this case having a factor of safety less than one, can be
calculated using:
𝑃(𝑓) = 𝜙(−𝛽)
8.4
where 𝜙(−𝛽) is the area below the curve in a normal probability density function with a
mean of zero and standard deviation of one. The area can be found using the table shown in
Appendix F (US ACE, 2006). It can be seen that higher values of β imply lower probabilities
of failure; however the probability of failure incorporates only those factors that are included
in the analysis and excludes many important external factors that would affect the stability of
an embankment. Therefore the probabilities of failure must be considered as lower bound
since not all contributing factors are included.
Deterministic analysis suggest that an embankment with a higher factor of safety is less likely
to fail compared to an embankment with a lower factor of safety. Probabilistic analysis shows
that this is not always the case and there is no direct relationship between the deterministic
factor of safety and the probability of failure (GEO-SLOPE International Ltd, 2008). A slope
with a high factor of safety may not be less likely to fail than a slope with lower factor of
safety. Figure 8.6 shows two idealised cases where the factor of safety is assumed to be
normally distributed about the mean. The case with the higher mean factor of safety (1.5) and
92
high standard deviation (0.5) also has the greatest probability of failure (shown by the area
under the curve less than 1). The case with the lower factor of safety has a lower probability
of failure. This is due to the standard deviation being much smaller for the lower factor of
safety which could demonstrate less variability in the embankment. This shows that it is
important to adopt a reliability index or probability of failure design standard rather than
achieving a certain factor of safety alone.
Figure 8.6: Comparison of two situations with different means and distributions of factor of safety (Christian, Ladd,
Baecher, 1994)
8.4
Acceptable probabilities of failure
One of the factors attributing to the reluctance in probabilistic methods of analysis is the lack
of published acceptable probabilities of failure. Whitman (2000) describes the use of
probabilistic methods and quantifying risks in decision making is limited by standards for
acceptable risks. Similarly to setting a standard factor of safety, there are many factors
affecting the stability which are not included in either a probabilistic or deterministic analysis
(for example, severe weather conditions, mining company financial issues, etc.) which make
it difficult to set standard design values. Davies (2002) states that there are “…no two sites
that have identical foundation, tectonic, hydrogeological, tailings characteristics, operating
criteria, etc.”. He refers to the following axiom: “…tailings impoundments are not
automobiles and cannot be mass produced…”. Each embankment is ultimately different, they
vary in location, construction techniques, soil properties, type of tailings, governing
93
legislations, etc., and so an acceptable probability of failure for one dam may not be
acceptable for another.
The US ACE (1997) provided a relationship between the probability of failure and the
expected performance level:
Table 8.2: Expected performance levels (US ACE, 1997)
Probability of failure
(%)
Expected performance
level
5.0
0.00003
High
4.0
0.003
Good
3.0
0.13
Above average
2.5
0.62
Below average
2.0
2.28
Poor
1.5
6.68
Unsatisfactory
1.0
15.87
Hazardous
Reliability index β
Table 8.2 is useful in estimating the performance level of a dam based on its reliability index
and probability of failure. The values are based on a normal distribution, so if the distribution
is skewed or a different curve then the table does not reflect the expected performance levels.
It is still up to the judgement of the geotechnical engineer to determine what an appropriate
expected performance level is as the table provided by US ACE (1997) does not ascertain a
‘safe’ embankment.
EN 1990:2002 recommends targets for the reliability index for ultimate states β = 3.8; related
to a design working life of 50 years and applicable to a structure that has moderate
consequence with regard to loss of human life and considerable consequences for socioeconomic, or environmental factors. In reality, the life of a tailings dam may be sufficiently
longer than 50 years as it may still be containing tailings well after the mine has ceased
operations. The target is more applicable to structures (such as residential and office
buildings) and not necessarily to geotechnical structures. This places doubt over the
credibility of using the recommended reliability index for tailings dams, since the variance in
a soil is vastly different to the variance of a material property in a structural member. The
consequences of failure relative to the reliability index may also be different for a tailings dam
and a building (e.g. failure of a building may have little to no environmental impact since the
rubble can be cleaned up; failure of a tailings dam could lead to significant environmental
consequences as toxic tailings water is dispersed). The target values can only be used as
94
recommendations in the design or analysis of a tailings dam embankment. No exact reliability
index can accurately and confidently confirm that a tailings dam embankment will always be
stable.
ANCOLD provides risk analysis criteria for water retention dams, but none for tailings dams.
In 1994, ANCOLD, developed individual and societal risk guidelines. The individual risk
guidelines stated “Do not subject any person, being a member of the public, to a risk greater
than 10-5 [0.0001] per annum”. These guidelines were later abandoned as it had proved
unworkable for dams and had no real meaning as a measure of equity (McDonald, 2008). The
societal risk guidelines show the probability of failure based on number of fatalities due to
dam failure (Figure 8.7).
Figure 8.7: ANCOLD (1994) societal risk guideline (McDonald, 2008)
The probabilities of failure from the 1994 guidelines were ‘very low’ (McDonald, 2008) and
many dams were not able to achieve or demonstrate this (due to foundations not fully known
or not feasible to fully investigate existing dams). This lead to situations where the lowest
level of risk had to be accepted since ‘society wants dams’ (McDonald, 2008). A few reasons
why ANCOLD has guidelines for water retention dams and not for tailings dams could
include:
95
•
If the contents of a tailings dam leaked into the environment, the physical and
chemical properties of the tailings can lead to more severe environment consequences
compared to failure of a water retention dam. The contents of water retention dams are
consistent – water – but the contents of tailings dams are different for each dam.
•
Water retention dams are generally not earth embankment dams. The variability in
construction materials for a water retention dam may be much smaller than the
variability in construction materials for a tailings dam, especially since tailings dams
use coarse tailings in their construction.
•
Water retention dams store a profitable material (water) whereas tailings dams are
storing waste. Consequently, there is greater attention in the design, monitoring and
maintenance for water retention dams than tailings dams.
•
Water retention dams can be closer to human populations due to their service in
supplying water. Tailings dams are at mine sites which are generally not as close to
public centres. This can make the public more concerned about the stability of water
retention dams rather than tailings dams.
There may be instances where the recommended probability of failure cannot be achieved,
e.g. if a tailings dam was already constructed and it was later identified that an endangered
species of flora or fauna lived downstream of the dam, then the consequences of
environmental failure have considerably changed since when the dam was first constructed.
The probability of failure may now have to be much lower to ensure, within certain
confidence levels, that failure will not lead to the species becoming extinct. The low level of
risk may be difficult to achieve since the embankment is already constructed and this could
cause the mining company or dam owners to be forced to close the dam.
According to Davies (2002) study, there are approximately 3500 tailings dams worldwide
with 2 to 5 major tailings dams failure incidents per year over the last 30 years. This equates
to an annual probability of failure between 1 in 700 (0.14%) and 1 in 1750 (0.06%). These
probabilities are ‘above average’ in terms of expected performance level, using Table 8.2.
Ultimately, it is difficult for authorities to set acceptable probabilities of failure, and it is
difficult for designers to ensure the acceptable probability of failure is achieved. Each
embankment should be individually analysed to determine whether the probability of failure
is suitably low enough to avoid the consequences of failure.
96
8.5
Types of probabilistic analysis
There are three levels of probabilistic analysis that can be used to assess reliability as first
defined by the Joint Committee of Structural Safety in 1982. The following definitions come
from Schueremans, Smars and Van Gemert (2001):
•
Level I methods verify whether or not the reliability of the structure is sufficient
instead of computing the probability of failure explicitly. Methods include expert
panel, event trees and decision trees.
•
Level II methods compute the probability of failure by means of an idealisation of the
limit state function where the probability density of all random variables are
approximated by equivalent normal distribution functions. The methods include
reliability index method, First Order Second Moment, First Order Reliability Method,
and Point Estimate Method.
•
Level III compute the exact probability of failure of the whole structural system using
the exact probability density function of all random variables. These types of methods
include Monte Carlo simulations and are considered the most accurate.
8.6
Level I analysis
8.6.1
Event tree
Baecher and Christian (2005) describe event trees as a convenient way for decomposing a
system reliability problem into smaller pieces that are easier to analyse. The event begins with
an initiating event, then has subsequent events (usually with dichotomous outcomes) until a
certain consequence is reached. Each path in the event tree, therefore, represents a specific
sequence of (subsequent) events results in a particular consequence and has a probability of
occurrence associated with it (Ang and Tang, 1975). The probability of a specific path
occurring is the product of the probabilities of all the events on that path (Ang and Tang,
1975). This example is used here to show how an event tree can be used to analyse the
consequences of a tailings dam failure; this method has not been applied to the baseline cases
or case study.
97
Figure 8.8: A portion of an event tree created by D’Appolonia (n.d.) as part of a risk assessment on the Zelazny Most tailings
dam in Poland
8.6.2
Fault tree
A fault tree diagram essentially decomposes the main feature event into unions and
intersections of sub-events or combinations of sub-events; the process of decomposition
continues until the probabilities of the sub-events can be evaluated as single-mode failure
probabilities (Ang and Tang, 1975). A fault tree analysis differs from an event tree analysis as
the fault tree looks at the causes leading to failure, and the event tree looks at the
consequences of failure.
Figure 8.9: Fault tree analysis vs event tree analysis (Taguchi, 2014).
98
The probabilities assigned for each event in both fault and event trees are usually determined
by the judgement of a small group of individuals, often experts in their field. The probability
estimates are based on “…physical reasoning, simple ‘back-of-the-envelope’ calculations and
the experience of the individuals” (Van Zyl, et al., 1996). US ACE (n.d.) say that proper
guidance and assistance to solicit and train the experts properly to remove all bias and
dominance is necessary. For these reasons, a detailed fault tree (or event tree) analyses have
not been performed for the case study in this thesis.
Neither the event tree nor fault tree consider failure over a period of time, the analyses can
only show instantaneous failure systems. This could result in not all failures being considered
as some failures can only occur a period of time after a preceding sub-event has occurred.
Figure 8.10: Fault tree for slope failure of a uranium mine tailings dam (Van Zyl and Robertson, 1987).
8.7
Level II analysis
The reliability index method is a practical, Level II method used to determine how reliable a
result from a model is. The reliability analysis determines the probability of failure of a slope
using traditional deterministic methods with different representations of the input variables.
Reliability is the “probability that a system will perform its intended function for a specific
period of time under a given set of conditions” (US ACE, n.d.).
99
The reliability index method is a First Order Second Moment method of analysis which
means that only the first two moments (mean and variance) are used to represent the
probability density function (US ACE, 2006). The reliability analysis is used to determine the
reliability index beta (β). β is the “number of standard deviations by which the expected value
of the factor of safety is away from the unsatisfactory performance conditions, or the factor of
safety equalling one” (US ACE, 2006). The equation for β (using lognormal distribution) as
represented by the US ACE (2006) is:
𝛽=
ln �
𝐸[𝐹𝑆]
�
�1 + 𝑉𝑎𝑟[𝐹𝑆]2
�ln(1 +
8.5
𝑉𝑎𝑟[𝐹𝑆]2 )
where E[FS] is the expected value of the factor of safety and Var[FS] is the variance in factor
of safety. The US ACE (2006) uses a lognormal probability density function to represent the
factor of safety. Since a factor of safety of 1 is considered safe and ln(1) = 0, the area under
the transformed normal distribution curve to the left of zero is equal to the probability of
failure:
Figure 8.11: Lognormal probability density function of factor of safety (FS) (US ACE, 2006)
100
Figure 8.12: Transformed probability density function using the natural log of the factor of safety (US ACE, 2006)
The level II analysis could only be used for simple calculations in Excel. The Case Study
could not be analysed using this method since the critical failure surface contains more than
one layer of soil and therefore could not be entered into the spreadsheet.
8.7.1
Drained analysis
The reliability index method was applied to Baseline Case 1. The analysis in Section 5.2.3
found the embankment had a factor of safety of 1.477 based on constant values of bulk unit
weight and angle of shearing resistance.
1.477
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 8.13: Baseline case 1 in drained conditions
101
The width, height and angle of each slice which define the most critical failure surface were
entered as deterministic values so the same failure surface was analysed as Section 5.2. The
angle of shearing resistance and the bulk unit weight were treated as a random variables with
normal distribution. The bulk unit weight was variable to confirm the findings in Section 6.4
which stated an overall change in unit weight does not affect the factor of safety for drained
conditions.
The factors of safety were then calculated using Bishop’s simplified method while keeping
one variable constant and varying the other by plus and minus one standard deviation. The
variance between the factors of safety for each parameter was then calculated which identified
that the angle of shearing resistance had 100% control of the analysis since the variance of
changing the bulk unit weight was zero (Table 8.4). This confirms that the bulk unit weight
does not have an effect on the factor of safety.
As test data was not available for the baseline cases, the coefficient of variation was estimated
using the upper limit from Table 8.1 as this represent the most variability. The standard
deviations are then equal to the coefficient of variation multiplied by the expected value.
Table 8.3: Random variable expected values and standard deviation
Expected
value
Coefficient of
variation (%)
Standard
deviation
Bulk unit weight (γ kN/m3)
18
7
1.26
Effective angle of shearing resistance (ϕ′ °)
35
13
4.55
Random variable
Table 8.4: Reliability analysis results for Case 1 in terms of effective stresses
Random variable kept
constant
γ
ϕ′
Random variable varied
ϕ′
γ
Varied by
Factor of safety
+σ
-σ
+σ
-σ
1.742
1.240
1.477
1.477
0.063
0.000
Percentage control of
analysis (%)
100
0
Standard deviation
0.25
0.00
Coefficient of variation
0.18
0.00
Reliability index β
1.58
Probability of failure (%)
5.71
Variance
102
The probability of failure (5.71%.) is approximately equal to a 1 in 17 chance that the
embankment will fail. The time dimension in this analysis is assumed to be the design life of
the dam so there is a 1 in 17 chance that the embankment will fail at any location during the
design life of the dam. Alternatively, the probability of failure shows that if 17 dams were
constructed with the exact same properties then one would fail, however no two tailings dams
are likely to ever have the exact some properties, so this interpretation is considered futile in
geomechanics.
The expected performance level (according to Table 8.2) is ‘unsatisfactory’. This is
predominantly due to the low reliability index which shows the factor of safety is within 1.58
standard deviations from unsatisfactory conditions. It would not be wise to construct a tailings
dam with uncertainty to this degree. The analysis was repeated using the coefficients of
variation at the lower end of the range specified in Table 8.1 (γ 3%, ϕ′ 2%) which gave a
reliability index of 10.75 and zero probability of failure. This shows that the probability of
failure is highly dependent on the coefficient of variation assigned to each random variable. A
coefficient of variation of 2% for angle of shearing resistance indicates the soil is relatively
uniform which is difficult to achieve in tailings dam construction. The embankment is
expected to perform ‘above average’ when the coefficient of variation for the angle of
shearing resistance is less than 7%. This could be used as criteria in the design of a new
embankment to ensure the variability in material results in a low coefficient of variation.
The reliability index method is useful in determining the probability of failure based on
assigned probability distributions for random variables. The extent or cause of the failure
cannot be identified using a reliability analysis. The analysis only shows the probability that
the embankment has variations of soil properties that lead to unsatisfactory conditions.
8.7.2
Undrained analysis
The deterministic analysis of baseline case 1 in terms of undrained conditions (from Section
5.2.2) showed that the embankment will fail since the factor of safety is less than one. In fact,
it is very unlikely that this embankment could even be constructed, due to the very low
undrained cohesion. The reliability analysis used the same deterministic values but now with
undrained cohesion and bulk unit weight as random variables.
103
0.735
Elevation (m)
15
10
5
0
0
5
10
15
20
25
30
35
40
Distance (m)
Figure 8.14: Baseline case 1 in undrained conditions
The standard deviations were calculated using the upper boundary coefficients of variation
from Table 8.1. The coefficient of variation for undrained cohesion is greater than that of the
angle of shearing resistance due to the wider range of possible values.
Table 8.5: Random variable expected values and standard deviation
Expected
value
Coefficient of
variation (%)
Standard
deviation
Bulk unit weight (γ kN/m3)
18
7
1.26
Undrained cohesion (cu kPa)
20
40
8.00
Random variable
Table 8.6: Reliability analysis results for Case 1 in terms of total stresses
Random variable kept
constant
γ
cu
Random variable varied
cu
γ
Varied by
Factor of safety
+σ
-σ
+σ
-σ
1.021
0.438
0.682
0.784
Variance
0.085
0.003
Percentage control of
analysis (%)
97.0
3.0
Standard deviation
0.29
0.05
Coefficient of variation
0.40
0.07
Reliability index β
-1.02
-4.56
Probability of failure (%)
84.61
0.0003
104
The reliability analysis shows that there is 84.61% probability of failure, showing that there is
a high chance of failure, however there is a 15.39% probability that the embankment will not
fail. This can be attributed to the variability in the undrained cohesion as the higher values of
cohesion increase the stability. The variability in bulk unit weight also calculated a reliability
index; however it has a very low probability of failure so failure is unlikely to occur as a
result.
Varying the undrained cohesion had 97.0% control of the analysis and the bulk unit weight
had 3.0%. This is consistent with the parametric study which showed that a change in bulk
unit weight does have an effect on the factor of safety in total stress conditions. It can be seen
in Table 8.6 that the factor of safety decreases with an increase in the bulk unit weight which
is consistent with the findings from the sensitivity analysis.
What has not been included in these reliability analyses is any correlation between undrained
cohesion and the bulk unit weight. The values have been varied individually to demonstrate a
large range of possible combinations of undrained cohesion and bulk unit weight. The
analysis could be performed again using a correlation factor (published in literature) between
random variables.
A limitation to using the reliability analysis applied to the Bishop’s simplified method
spreadsheet is that the same failure surface is analysed for each variation since the failure slice
dimensions are deterministic. It is not viable to analyse the slices as random variables since
there are 20 slices which must form an arc to be used in Bishop’s simplified method, and
using random values is unlikely to calculate the critical failure surface. If the baseline case
was modelled again using SLOPE/W for each variation then a slightly different failure surface
would appear, producing a different factor of safety and ultimately different probability of
failure to what is calculated here. This analysis could be performed again using SLOPE/W to
model each variation, i.e. keep the total unit weight constant while varying the undrained
cohesion value over the range 20 ± 8 kPa, etc. It is suitable, however, to keep the failure
surface constant so that the factors of safety from the reliability analysis can be compared to
the deterministic value.
105
8.8
Level III analysis
8.8.1
Monte Carlo simulation
Monte Carlo simulations are a stochastic problem-solving process commonly used to solve
complex problems with many random variables. Palisade’s @RISK software utilises Bishop’s
simplified method previously modelled in Microsoft Excel™ for the baseline cases in 5. The
input parameters are defined as either known or unknown variables. The unknown variables,
are described statistically by probability distributions, mean and standard deviation. The
Monte Carlo simulation is then performed which draws random values for each input variable
from within the assigned probability distribution. These values are used to solve the
spreadsheet model and calculate the corresponding factor of safety (El-Ramly et al., 2002).
This process is repeated sufficient times to estimate the statistical distribution of the factor of
safety within a certain confidence level. This allows the mean and variance of the factor of
safety and the probability of failure to be estimated. @RISK can be applied to any existing
spreadsheet making it a suitable, useful probability estimating tool, however with this also
comes the risk of misinterpreting the input requirements and therefore the results.
Monte Carlo simulations were run using @RISK for Microsoft Excel™ and using
GeoStudio™ SLOPE/W to determine the probability of failure for the baseline cases and the
slightly more complex case study. The simulation was not a true Level III analysis since only
changes in soil property are considered. This process can also be applied to other situations of
tailings dam design and construction such as construction productivity.
8.8.2
Number of simulations
Theoretically, the more simulations used in the analysis, the more accurate the solution will
be since more random cases are analysed. A large number of trials (in the order of millions)
begins to take time to analyse and may depend on available computer memory. The question
then arises on how many trials are suitably required to calculate the probability of failure to a
certain level of confidence.
Figure 8.15 shows a relationship between probability of failure (%) and the number of
simulations (using baseline case 1 in SLOPE/W). It can be seen that the data varies
significantly at low numbers of simulation and slowly converges as the number of trials is
increased. The number of trials analysed was from 50 to 10,000,000 at varying intervals.
Since the intervals varied, it is likely that the amplitude peaks were not recorded and the curve
appears to be converging faster than it would be if smaller intervals were used, nonetheless it
106
gives a good representation of how an increase in number of trials improves the level of
confidence.
Probabililty of failure (P(f) %) versus number of trials
2.5
P(f) %
2
1.5
1
0.5
0
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
Number of trials
Figure 8.15: Probability of failure versus number of trials
The GEO-SLOPE Stability Modelling document (2008) developed an equation to calculate
the number of trials required. The method is dependent on the desired level of confidence and
the number of random variables.
𝑚
𝑑2
𝑁=�
�
4(1 − 𝜀)2
where:
N
= number of trials
𝜀
= the desired level of confidence (0 to 100%) expressed as a decimal
m
= the number of variables
d
= the normal standard deviation corresponding to the level of confidence
8.6
Table 8.7: Normal standard deviate values for levels of confidence (Abramson, 2002)
𝜀%
80
d
1.282
90
1.645
95
1.960
99
2.576
107
Table 8.8 shows the number of trials required for different confidence levels, using two
random variables.
Table 8.8: Number of trials required for levels of confidence
𝜀%
80
N
106
90
4,577
95
147,579
99
275,209,520
Ideally, the highest level of confidence would be used to ensure the highest accuracy in the
data. SLOPE/W is unable to perform more than 10 million simulations so using a 99% level
of confidence is not possible. The level of confidence is therefore chosen as 95%. This means
that if the simulation runs for 147,579 trials, there is 95% confidence that the calculated
values will not differ by more than 5% from the value if the confidence was 100%. This is a
suitable level of confidence, considering there are many other factors that have not been
included in this analysis.
8.8.3
Drained analysis
The baseline case 1 in drained conditions was used to compare the @RISK software and
SLOPE/W. The @RISK software was applied to the spreadsheets developed in Section 5, and
the SLOPE/W Monte Carlo analysis was applied to the existing SLOPE/W model. The soil
parameters were changed accordingly for a probabilistic analysis, rather than deterministic. It
was ensured that the same range of values was entered into SLOPE/W and @RISK.
Table 8.9: Soil properties for Baseline Case 1 Monte Carlo simulation
Distribution
Expected
value
Standard
deviation
Bulk unit weight (γ kN/m3)
Normal
18
1.26
Effective angle of shearing resistance (ϕ′ °)
Normal
35
4.55
Random variable
The simulations were performed with 147,579 trials for a 95% level of confidence. Both
programs used randomly generated values for the bulk unit weight and effective angle of
shearing resistance into Bishop’s simplified method and produced 147,579 factors of safety.
108
Probability distribution curves
9
@RISK
8
SLOPE/W
7
Frequency (%)
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
Factor of safety
Table 8.10: Output data from @RISK and SLOPE/W
Figure 8.16: Probability distribution curves for drained conditions
Outputs
@RISK
SLOPE/W
Minimum factor of safety
0.538
0.457
Maximum factor of safety
3.192
3.288
Mean factor of safety
1.486
1.486
Standard deviation
0.253
0.255
Reliability index β
-
1.486
1.8*
1.734
Probability of failure (%)
*value only given to two significant figures
The mean factors of safety calculated using @RISK and SLOPE/W were the same (to three
decimal points) with very similar values for minimum and maximum factor of safety and
standard deviations between the two methods. @RISK does not give a reliability index so this
value cannot be compared. The probabilities of failure are quite similar however @RISK only
provided the value to two significant figures so determining the accuracy is constrained by
this. The difference in values between @RISK and SLOPE/W can be attributed to rounding
errors and slightly different upper and lower end parameters.
109
Probabilities of failure vary quite significantly to what was analysed using the reliability
index method (failure probability 5.71%). This is because the distribution of the factor of
safety was assumed normal for the reliability index method, whereas the results of the Monte
Carlo simulation show the factor of safety is skewed normally distributed. The reliability
index method calculated the factor of safety between one standard deviation from the mean
which could also contribute to the higher factor of safety.
The Monte Carlo simulation has shown that the probability of failure is actually lower than
what was calculated using the reliability index method. If a tailings dam were to be
constructed with this variability in soils it is worth doing a Monte Carlo simulation to
determine a more accurate probability of failure. The failure still shows the dam would have
below average to poor performance (based on 1.8% probability of failure).
8.8.4
Undrained analysis
The baseline case was then analysed in undrained conditions. Note, in real life the undrained
conditions would be analysed first; the drained conditions were analysed first in this chapter
since in the undrained conditions the embankment would fail. Lognormal distribution was
chosen for undrained cohesion as recommended by Lacasse and Nadim (1996).
Table 8.11: Soil properties for Baseline Case 1 Monte Carlo simulation undrained
Distribution
Expected
value
Standard
deviation
Bulk unit weight (γ kN/m3)
Normal
18
1.26
Undrained cohesion (cu kPa)
Lognormal
20
8.0
Random variable
The simulations were performed with 147,579 trials for a 95% level of confidence.
110
Probability distribution curves
9
8
@RISK
Frequency (%)
7
SLOPE/W
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Factor of safety
Figure 8.17: Probability distribution curves for undrained conditions
Table 8.12: Output data from @RISK and SLOPE/W
Outputs
@RISK
SLOPE/W
Minimum factor of safety
0.113
0.334
Maximum factor of safety
3.941
1.467
Mean factor of safety
0.746
0.829
Standard deviation
0.304
0.215
Reliability index β
-
-0.794
82.7*
77.3
Probability of failure (%)
*value only given to one decimal place
The probabilities of failure were found to vary by approximately 7% in this analysis. The
same range of variables was used in both programs to limit any potential epistemic errors.
Even with this, @RISK calculated a much greater maximum factor of safety. This difference
is assumed to be due to default functions in the programs on how they treat lognormal
distributions. The mean factors of safety were approximately 10% different with @RISK
having a lower mean factor of safety, and therefore a greater probability of failure.
The failure distribution curves are lognormal since the input variable (cohesion) was
lognormal. This does not affect the probabilities of failure; the distribution can be easily
turned into a normal distribution by taking the log of x-axis values. If lognormal distribution
111
was not used for the undrained cohesion, then negative factors of safety would be calculated
resulting in low probability of failure which would not accurately represent the embankment.
The high probabilities of failure confirm the findings throughout this thesis that the undrained
analysis of baseline case 1 is not stable.
8.9
Probabilistic analysis of the case study
The case study tailings dam introduced in Section 7 is used again in this analysis to determine
the probability of failure. The embankment is analysed with a 2 m sandstone foundation as
this produced the highest factor of safety in Section 7. The embankment height, phreatic
surface and soil layer widths are assumed constant. In reality these values are likely to be
variable due to natural variability and construction techniques. Similar to the random
variables already discussed in this thesis, the variability can be observed from in situ tests that
would show differences in embankment height, phreatic surface and soil layer widths at
various locations around the embankment. It would be difficult to accurately model the exact
conditions or calculate suitable ranges for each variable for the case study without having
Therefore the values at this cross section are assumed instantaneous and hence deterministic.
A more detailed analysis that includes using embankment height, phreatic surface and soil
layer widths are random variables could be undertaken to determine the effect these would
have on the factor of safety and probability of failure.
The bulk unit weight and angle of shearing resistance of each layer are considered random
variables. The rock armour layer is made up of cobbles, boulders and gravel with some silty
fines and sand. A coefficient of variation for the bulk unit weight is chosen as 7%, and for
angle of shearing resistance 13%, to represent the variability in the rock armour layer. These
values are from the upper end of the typical range specified in Table 8.1.
The earth fill layer is a highly variable mixture of gravelly clay with some pockets of clayey
sand and cobbles. Since this layer is highly variable, a higher coefficient of variation is
estimated. Due to the lack of data from the geotechnical observations, the coefficients of
variation are estimated based on the information provided. The PP (pocket penetrometer) test
results (Appendix D) show values varying from 200 to 600 kPa which highlight the variability
in strength. For these reasons, the coefficient of variation for the bulk unit weight is 8%, and
19% for the angle of shearing resistance. Rétháti (2012) found that for the angle of shearing
resistance of granular soils, the coefficient of variation increase with the decreasing grain size,
112
and for the compact state it is higher than those for the loose state. This confirms that the earth
fill layer should have a higher coefficient of variation than the rock armour layer.
The sandstone foundation is assumed to be 2 m thick and relatively uniform below the
embankment. It is ‘extremely weathered’ meaning that the rock substance has soil propertiesi.e. it can be remoulded and classified as a soil but the texture of the original rock is still
evident (Douglas Partners, 2010). It is given a lower coefficient of variation to represent this
assumption.
Table 8.13: Random variable properties
Random variable
Distribution
Expected
value
C.O.V (%)
Standard
deviation
Normal
20 kN/m3
7
1.40
Normal
40°
13
5.20
Normal
20 kN/m3
8
1.6
Normal
35°
19
6.65
Normal
21 kN/m3
4
0.84
Normal
36°
7
2.52
Rock armour
Bulk unit weight (γ)
Effective angle of shearing
resistance (ϕ′)
Earth fill
Bulk unit weight (γ)
Effective angle of shearing
resistance (ϕ′)
Sandstone foundation
Bulk unit weight (γ)
Effective angle of shearing
resistance (ϕ′)
The critical slip surface is initially found using the expected values only which gave a
deterministic factor of safety of 2.025.
Once the critical failure surface was identified, a Monte Carlo simulation was run using the
random variable properties in Table 8.13 over 147,579 trials. The critical failure surface was
found first to drastically reduce the computational time, i.e. to find the failure surface using
expected values only 4851 trials are required, using the probabilistic values there are over 48
million trials required. The Monte Carlo simulation produced the histogram of factors of
safety shown in Figure 8.18. The x-axis shows the distribution of factors of safety over 100
intervals.
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Factor of safety distribution
8
7
Frequency %
6
5
4
3
2
1
5.45
5.24
5.03
4.82
4.61
4.40
4.19
3.98
3.77
3.56
3.35
3.14
2.92
2.71
2.50
2.29
2.08
1.87
1.66
1.45
1.24
1.03
0.82
0.61
0.40
0
Factor of safety
Figure 8.18: Case study histogram of factors of safety from SLOPE/W
Table 8.14: Case study probabilistic results from SLOPE/W
Minimum factor of safety
0.40
Maximum factor of safety
5.66
Mean factor of safety
2.06
Standard deviation
0.397
Reliability index β
2.67
Probability of failure (%)
0.165
The mean probabilistic factor of safety (2.06) is above ANCOLD’s recommended minimum
of 1.5 so would be considered safe in this regard. The difference between the deterministic
factor of safety (2.025) and the mean probabilistic factor of safety (2.060) is minimal (1.7%),
which shows that the deterministic value can be used with confidence.
The reliability index of 2.67 places the expected performance level of the dam as slightly
better than ‘below average’ (using Table 8.2). To determine whether this probability of failure
is acceptable, a risk assessment would have to be performed to investigate the potential
damage if the tailings dam was to fail. Referring back to ANCOLD’s consequence categories
introduced in Section 2.3, if the dam were to have minor consequences then an ‘average’
expected performance level could be deemed acceptable, in this situation. In situations where
there is potential for major consequences, whether they be infrastructure damage, lives lost or
impact to the environment, then it may be better to set a reliability index in the design and
maintenance of a tailings dam, rather than achieving a minimum factor of safety.
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9
COMPARATIVE ANALYSIS
In the past, engineers have used deterministic approaches for the design and back-analysis of
tailings dams. The deterministic methods calculate a factor of safety. These methods are
popular and effective as the results can be easily interpreted: a factor of safety of one has
destabilising and resisting forces in instantaneous equilibrium; a factor of safety greater than
one has greater resisting, than destabilising, forces so is therefore ‘safer’. Deterministic
methods (such as stability charts and limit equilibrium methods) do not account for
uncertainties in the soil parameters as they use a unique set of values to generate a single
factor of safety. In recent years, geotechnical engineers have recognised the need to deal with
the inherent variability and uncertainty in soil properties. Not only have geotechnical
engineers recognised this need, so have dam owners: “Knowledge about the factors that
control the behaviour of tailings dams has improved greatly… also the consequences and
public perceptions of tailings dam failures has increased considerably, causing managers and
owners to become more aware of the risks involved” (ICOLD Bulletin 121). Bowles, et al.
(1996) demonstrate the application of risk assessment in dam engineering (not specifically to
tailings dams) and that the factor of safety by itself is no longer a sufficient measure for the of
analysis of stability. Phoon, et al. (2006) state that “…collective experience (from practice
and research) suggests that is may be time for a shift to an uncertainty-based [probabilistic]
perspective which may be, on the whole more convenient in terms of safety, performance and
economy”.
Probabilistic methods (such as First Order Second Moment and Monte Carlo simulations)
take into account variability and uncertainties in soil properties by assigning a probability
distribution function to each random variable. This gives a range of values that are
representative of each soil property. Such methods are run over a number of trials with each
trial using different values of the input random variables. The result gives a mean factor of
safety, a reliability index, and a probability of failure. Probabilistic methods have often been
hindered in geotechnical analysis by their more complex approach; however, with the everincreasing capabilities of computers and software, some of the barriers encountered in the past
have been removed. This was evident in this thesis as a Monte Carlo simulation of nearly
150,000 trials took only a few minutes to complete, whereas if 150,000 trials of Bishop’s
simplified method were to be performed, it would take an engineer an economically
unfeasible time. The geotechnical engineer performing any probability analysis needs to have
some understanding of comprehension of statistics to be able to interpret the results (Phoon, et
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al., 1996; and Duncan 2000). This may be where some reluctance in using probabilistic
methods lies as some engineers are limited to basic statistical theories. The decisions, and
judgements, in assessing the stability of an embankment should not be completely replaced by
computers; an engineer should still analyse and interpret both inputs and outputs (and perform
a hand calculation) for verification and to ensure practical admissibility. Whitman (2000) says
that “Traditional engineers rightly worry that too much emphasis upon analysis might drive
out engineering judgement and lead to unsatisfactory designs”. Lacasse and Nadim (1996) say
that it is necessary to find a balance between appropriate technology and adequate
complexity, given the budget and consequences of the failure of the project.
There is also the misconception that probabilistic methods require more data, time, and effort
than are economically available (Duncan, 2000). Nowadays there is no extra time or effort
required in performing a probabilistic analysis and the number of observations depends on the
budget. More tests and observations from different locations throughout a dam will give a
more accurate analysis for both probabilistic, and deterministic, analyses; one does not require
more tests than the other. In practice, a deterministic test would be likely to require more tests
to determine conservative, representative values.
Phoon et al. “Addressing uncertainty does not increase the level of safety, but allows a more
rational design as the engineer can consciously calibrate his decisions on a desired
performance level of a structure”. Designing the tailings dam to a desired performance level
(rather than a conservative factor of safety) allows undesired conservatism (which would be
likely to lead to increased costs) to be reduced, without affecting the stability of the dam.
Lacasse and Nadim (1996) demonstrate schematically the capacity obtained from a traditional
design approach and a reliability based design approach (see Figure 9.1). Lacasse and Nadim
(1996) describe how the traditional approach gives a calculated, and a design, capacity and a
traditional factor of safety. The reliability-based approach (referred to as the ‘refined’
approach in their paper) also gives a calculated, and design, capacity, a refined safety margin,
and a calibrated safety factor, as well as an indication of the unnecessary safety margin
implied by the traditional approach. The aim is to reduce the margins of physical capacity and
safety which then makes it “…feasible to document why the factor of safety should be
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Deterministic design approach
Physical capacity
Probabilistic design approach
Physical capacity
Refined safety margin
Refined calculated capacity
Calculated capacity
Calibrated factor of safety
Factor of safety
Refined design capacity
Unnecessary safety margin
Design capacity
Figure 9.1: Capacities of deterministic and probabilistic design approaches (after Lacasse and Nadim, 1996)
Phoon, et al. (2006) state that: “In principle, no category of approaches is preferable over
another. The quality of geotechnical characterisation or design is not conceptually bound to
the level of explicitness of soil variability modelling”. Whitman (2000) discusses how good
communication with the client is essential in determining the type of analysis required. If the
client (or regulator) is not interested in quantifying risk as part of their decision-making then
traditional deterministic methods will continue to be used. When the risks are large and the
cost of absolute safety is large, then clients may be interested in discussing risks and a
probabilistic analysis may then be required. If there are no guidelines or regulated standards
on acceptable probabilities of failure then deterministic approaches will continue to be used as
there are accepted, published, factors of safety (e.g. ANCOLD recommends a minimum factor
of safety of 1.5). The published values are generally recommended minima; setting exact
factors of safety or probabilities of failure for a design is unsuitable in geotechnical
engineering as each structure is inherently different.
Table 9.1 shows the results of case 1 (undrained and drained conditions) and the case study
results from the deterministic and probabilistic analyses (where F is factor of safety, P(f) is
the probability of failure and β is the reliability index).
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Table 9.1: Comparison of results from deterministic and probabilistic analyses
Deterministic methods
Stability
charts
Limit
equilibrium
methods
Reliability
Index method
Monte Carlo
simulations
0.656
0.735
0.729
0.746
P(f)
n/a
n/a
84.6%
82.7%
β
n/a
n/a
-1.02
-0.79
F
1.502
1.477
1.357
1.486
P(f)
n/a
n/a
5.7%
1.8%
β
n/a
n/a
1.58
1.49
F
n/a
2.025
n/a
2.060
Case
Undrained case 1 F
Drained case 1
Case study
Probabilistic methods
The table shows that the factors of safety from Monte Carlo simulations were higher (by less
than 2%) than the factors of safety calculated using limit equilibrium methods. For design
purposes it would be conservative to use the lower factor of safety so that the stability is not
falsely assumed to be safer, however this is likely to increase the costs of materials,
construction, etc. as a higher factor of safety is assigned. The case study gave similar factors
of safety using the limit equilibrium methods and Monte Carlo simulations. If rounding to two
decimal places, there is a difference of 0.03 between the values which is insignificant in the
design of an embankment. Even using three significant figure when interpreting the factor of
safety could be deemed inappropriate: for example, an embankment with a factor of safety of
1.03 should not be considered safer than one at F = 1.00 since it is only fractionally greater
than 1; likewise, an embankment with factor of safety of 0.995 should not be rounded-up to
1.0, deeming the embankment safe. This is one of the reasons why setting national minimum
factors of safety could be dangerous, as it could limit the depth of thought with which an
engineer analyses their results, for example, if the results give the factor of safety as 1.03 and
the minimum recommended is 1 then the designer has technically achieved this criterion
however relying on an embankment to be stable with a factor of safety as low as this may not
be wise.
Having a national required minimum factor of safety (from deterministic analysis) could also
lead to tampering with the data and falsifying analyses to reach the minimum values. The tests
could knowingly ignore weaker portions of the embankment to make it appear stronger so that
the minimum standard is achieved. Again, technically the designer has reached the minimum
requirement and if the case came before a court it would be difficult to prove that the soil
118
properties used in the analysis were, or were not, a representative and conservative sample.
This is especially true since soil properties vary both horizontally and vertically within an
embankment and vary over time.
On the other hand, embankments constructed in the past to a deterministic approach are still
stable; however it is not known to what factor of safety each dam was designed, and if
globally, whether, or not those dams that have failed were designed to a lower factor of safety.
There is no way of quantifying a risk attached to a factor of safety through a deterministic
approach so it would be difficult to prove that an embankment has a higher chance of failing
compared to one with a higher factor of safety. Studies by Whitman (2000) and Duncan
(2000) have demonstrated that an embankment is not necessarily safer with a higher factor of
safety.
Therefore, setting minimum requirements for factor of safety alone is an unreliable approach
by itself. Setting a probability of failure appropriate to the performance of a dam and the
consequences of its failure, would not only account for uncertainties but would also provide a
more reliable standard with which to design and analyse tailings dams. This should especially
be applied to tailings dams since the consequences of failure can be much greater than in the
failure of a water-retaining dam (due to the chemical properties of the tailings stored). Setting
a probability of failure relevant to the expected performance level and consequence of failure
allows each tailings dam to be individually analysed without setting a ‘blanket’ acceptable
probability of failure for all dams. This has been attempted by ANCOLD (1994), B.C. Hydro
(Salmon and Hartford, 1995) and USBR (1986) (from Von Thun, 1996) for water retention
dams, but have faced difficulties in quantifying the consequences of risk. Further
investigations in this area are required to produce probabilities of failure associated with
consequence categories and performance levels.
Mining companies would not want to spend as much money on ensuring the stability of a
tailings dams (compared to Government authorities constructing a reservoir for public water
supply) so bringing in standards of probability of failure will force mining companies to put
more effort into ensuring stability of their tailings dams, or considering other options for the
storing of tailings (and hopefully this is a more sustainable option). Mining companies must
also be liable for the cost of environmental and social consequences; otherwise there is no
incentive to spend more money on making the tailings dams safer in the first place. They must
be liable for any issues for the life of the dam and the life of the tailings that were stored in the
dam, as some of environmental effects (such as seepage) have cumulative environmental
119
damage over time and mitigation works will be more expensive the longer it is unnoticed and
untreated. This would be difficult to monitor and enforce.
Another current example of mining indemnity issues is the St Barbara Gold Ridge tailings
dam on the Solomon Islands (introduced in Section 2.2.4). The level of the tailings dam is
critical and the tailings dam is nearing collapse (Mining Australia, 2015). After Cyclone Ita,
the consequences of failure were compounded when it was discovered that a number of
minors had illegally moved to the mine site. Amid escalating safety and security concerns, St
Barbara withdrew all workers from the site, ceased mining operations and brought in a United
Nations team of specialists to assess the stability of the tailings dam and consequences of
failure (ABC, 2014). The results of this analysis were not available, but, in early 2015 St
Barbara began discussions to hand over the operation to the Solomon Islands Government.
This example shows that an increase in the risk of failure can lead to abandonment of a
tailings dam, and in this case the mining company would rather pass the risk on to someone
else. If the mine is handed back to the Solomon Islands Government and it subsequently fails,
then the responsibility lies with the Government. The Solomon Islands Government have
stated that the tailings dam issue is “…only St Barbara’s problem to solve” (Mining Australia,
2015).
For the cases analysed in this thesis, there was little difference in using limit equilibrium
methods and Monte Carlo simulations (less than 2%). The uncertainty in probabilistic
analysis was described by estimates of coefficient of variation and distribution type. In reality
these parameters would be determined by observations so that more accurate descriptions of
the uncertainty could be used, also helping to reduce the level of conservatism and therefore
unnecessary costs.
Performing a probabilistic analysis on an existing dam that was designed using deterministic
approaches could lead to issues becoming apparent. If the deterministic analysis showed that
the embankment was safe, then the probabilistic analysis showed that the embankment had a
high probability of failure (or in the case of a back-analysis the embankment had already
failed) then either the values used in the deterministic analysis (for both short- and long-term)
were not conservative enough to account for variability, or something unforeseen had
occurred that was not addressed in the traditional approach. Once the client/dam owner is
aware of the high probability of failure they then must act on improving the safety of their
dam; criminal negligence – “saw a risk and went ahead with the action anyway” (Hartford,
2013) – meant that not only are they responsible for improving the stability, their insurance is
likely to be increased which could have negative effects on the operation of the tailings dam.
120
Environmental concerns or public safety may be forefront issues which the dam owner must
now address to ensure that the consequences of dam failure are as low as possible. Attempts
to mitigate any impending failure are likely to become expensive and force it into a cessation
of operations (and maintenance and monitoring) of the dam if these costs become too great.
The probability of failure would significantly increase without the monitoring and
maintenance and failure could have significant impacts for which the dam owner will not
want to take responsibility (since the initial design of the embankment showed it would be
stable). An event such as this shows that performing probabilistic analyses (as well as their
deterministic counterparts) is beneficial to the quantification of risk over the life of the dam.
In saying this, neither deterministic nor probabilistic methods are time-dependent, and since
the soil properties change over time as a result of consolidation, continued assessments (using
new observations) would be required to assess stability over the life of the dam.
Limitations to this thesis include the fact that cases where a load is applied at the top of the
embankment were not analysed. These loads could be from machinery, infrastructure or
construction of subsequent dykes and would considerably change the factor of safety of the
existing embankment. As is common with tailings dam construction, the embankment height
is raised as the level of impoundment is increased so analysing the change in factor of safety
of the existing dyke when a subsequent dyke is constructed would be beneficial.
Another limitation in the comparison of methods is that only circular slip surfaces were
analysed. In reality the failure surface could be a combination of rotational and translational
failures or any of the other methods aforementioned in Chapter 3. The main limit equilibrium
method used in this thesis was Bishop’s simplified method. If non-circular failure surfaces
were to be analysed then a different method (such as Morgenstern and Price’s (1965)), and
therefore the spreadsheets developed for this thesis, could not be used as Bishop’s simplified
method can only be used for circular failure surfaces.
This thesis only analysed failure of the embankment, there was no attempt to include the
failure of other infrastructure such as pipework, monitoring devices, spillways, etc. that could
lead to a release of tailings. To gain a fuller perspective of the likelihood and consequences of
failure then these issues should also be included in future research and indeed current stability
analyses.
121
10 CONCLUSION
Four baseline cases were analysed using deterministic methods such as: slope stability charts
(Taylor, 1937; Bishop & Morgenstern, 1960; Spencer, 1967; and Janbu, 1968), and limit
equilibrium methods (Fellenius, 1936; Bishop’s simplified, 1955; Morgenstern and Price,
1965; and Janbu’s simplified, 1968) in terms of undrained and drained conditions. It was
found that the stability charts gave a good indication of the stability of an embankment since
they calculated factors of safety similar to the factors of safety calculated using limit
equilibrium methods.
The deterministic analyses assumed that the soil was homogeneous and isotropic giving the
embankment constant soil properties. In reality, this would not be true due to variations in the
materials. For this reason, a parametric study was performed to investigate the effects, on the
factor of safety, of varying the embankment geometry (slope angle β), shear strength
parameters for undrained conditions (cohesion cu) and drained conditions (apparent cohesion
c´ and angle of shearing resistance φ´), bulk unit weight (γ) and groundwater conditions
(location of phreatic surface zw). The parametric study showed that the factor of safety
improved linearly with increases in cu and tan φ´ and improved when the phreatic surface was
lowered and when the embankment slope angle was reduced.
The parametric study was performed in an attempt to consider the inherent variability in a
given soil profile and assess the effect that the variability had on the factor of safety. The
study involved performing a large number of trials in each of which, the input properties were
varied separately. The method was somewhat analogous to that performed during a
probabilistic analysis, however probabilistic methods, such as a Monte Carlo simulation,
account for input variability by assigning probability distribution functions to each random
variable and performing a large number of trials simultaneously to gain a distribution of
output factors of safety. Probabilistic methods (namely the US ACE reliability index method
and Monte Carlo simulations) were then used in this thesis to investigate whether
probabilistic analyses calculate a factor of safety that better represents the stability of an
embankment by accounting for variability in the soil properties, rather than using a single
deterministic value for each soil property as used in the stability charts and limit equilibrium
methods.
Monte Carlo simulations were performed on the baseline case (in terms of undrained and
drained conditions) and the case study which gave probabilities of failure and mean factors of
122
safety. The factors of safety were compared to those calculated using limit equilibrium
methods which showed that there was little difference (less than 2%). This difference was
quite insignificant when considering that the error bound in either analysis could be greater
than the difference between methods, or that a number of factors (such as consequences of
failure) have not been included in the analysis.
The probabilistic methods were used to calculate a refined range of factors of safety by
considering the variability in soil properties; this increased the confidence which the engineer
can place in the factor of safety calculated using probabilistic methods as opposed to limit
equilibrium methods where the soil properties were single idealised representations of the
given soil profile. Nowadays, due to the increased availability of more powerful computerbased routines, it is not difficult to perform a Monte Carlo simulation; however, assessing the
results requires a more detailed understanding of statistics than that generally taught at a
tertiary level. There are also no recommended guidelines on what is an acceptable probability
of failure, often making the stability of an embankment analysed using probabilistic methods
difficult to assess. Based on the literature review and analyses in this thesis, setting regulatory
guidelines on probabilities of failure and utilising probabilistic analysis methods in the design
and to assess stability of existing tailings dams is recommended. The regulatory guidelines
would take into account the expected performance level of the tailings dam and the
consequences of failure so that each dam would be analysed individually, as opposed to
setting a standard minimum probability of failure to which all tailings dams adhere.
In this thesis it was shown that analysing the stability of a tailings dam embankment is not a
simple task. The stability is dependent on a number of different factors including (but not
limited to): failure mode, variation in soil properties, foundation conditions, phreatic surface
location, pore water pressures (short- or long-term), analysis method, and how to assess
stability based on the outcomes of the analyses (i.e. acceptable factors of safety and
probabilities of failure). Further investigation into acceptable factors of safety and
probabilities of failure, based on the expected performance level of a tailings dam and the
consequences of its failure, is recommended. Studies in this area would help to set regulatory
standards for the stability of tailings dams and subsequently allow probabilistic methods to be
used more effectively in practice.
123
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12 BIBLIOGRAPHY
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Compression Test on Cohesive Soils, ASTM D2850, ASTM International
ASTM International 2011, Standard Test Method for Consolidated Undrained Triaxial
Compression Test for Cohesive Soils, ASTM D4767, ASTM International
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130
APPENDIX A
Embankment construction
There are three methods of construction based on sequentially raising the embankment as the
pond level rises, namely; upstream, downstream, and centreline construction. Each method
begins with a starter dam constructed at the downstream toe. The tailings waste is piped into
the dam from the embankment crest to form a beach area composed of the coarse components
thereof. The beach section becomes the foundation for subsequent embankment layers in
upstream construction. Upstream construction has many advantages, including being
economical since less fill material is required, a long beach allows for a low phreatic surface
and design adjustments to the downstream slope are possible during staged construction.
Despite the advantages, upstream construction also carries the highest risk of failure as the
process relies on the strength and stiffness of the beached tailings which will change over time
due to consolidation (ANCOLD, 2012). Chambers (2012) state that “… the performance of
tailings dams constructed by the centreline or downstream methods has been markedly better
than dams constructed by the historically common upstream method”. This was evident in
Section 2 where the number of incidents has been separated by construction methods from
ICOLD Bulletin 121. Yet still “the upstream construction technique has been used extensively
for many tailings embankments worldwide, largely because of the relatively low construction
costs” (Chambers, 2012). A risk analysis could be undertaken to ascertain the most
sustainable construction method which satisfies economic, environmental and social
responsibilities to provide a stable embankment.
Tailings water
Coarse tailings
Starter dam
Toe
Figure A.1: Upstream construction method
The downstream method was developed because of the risk of failure of dams built by the
upstream method (ICOLD Bulletin 121). The subsequent dykes in downstream construction
are built on top of the starter dyke and are increasing in size, moving the centreline of the dam
downstream as work progresses. This construction method requires more material than the
upstream method, and often uses borrow material, so becomes more expensive. The
downstream method is less susceptible to failure compared to the upstream method, since the
131
structure is not dependent upon the strength of the coarse tailings as a foundation (US EPA,
1994).
Tailings water
Coarse tailings
Starter dam
Figure A.2: Downstream construction method
The centreline construction method is similar to both the upstream and downstream methods,
however sequential raising takes place in a vertical direction and the centreline does not move
beyond local, short-term, deviations during non-symmetrical placing of material. The
Australian Government LPSDP (2007) states that the centreline method is not commonly used
in Australia as it is unsuitable for long-term storage.
Tailings water
dyke
dyke
dyke
Coarse tailings
Starter dam
Toe
Figure A.3: Centreline construction method
Stability analyses of tailings dams differ from those used for the analysis of conventional
dams as a result of the staged construction methods used for tailings storage facilities
(ANCOLD, 2012). Conventional water retention dams are built to their full design capacity
prior to the dam’s operation and use essentially one construction method – downstream
(Chambers, 2012). Whereas tailings dams are designed and constructed in stages coinciding
with the operation of the mine. Therefore it is important to understand the implications of
consolidation and rate of construction for upstream, downstream, and centreline construction
methods during their stability analysis. Tailings dams also differ to water retention dams in
that the design life for tailings dams is much longer (perpetuity) compared to the finite life of
water supply dams (Chambers, 2012). The construction sequence is also important in
conditions are either short-term, assuming the embankment has just been constructed, or longterm after consolidation has occurred (see Section 3.1).
132
APPENDIX B
Factor of safety versus number of slices
The number of slices and resultant factor of safety were investigated for the drained
embankment in Case 1.
Factor of safety vs number of slices
1.62
Bishop
1.6
Fellenius
Factor of safety
1.58
Janbu Simplified
1.56
Morgenstern & Price
1.54
1.52
1.5
1.48
1.46
1.44
1.42
0
5
10
15
20
Number of slices
Figure B.1: Factor of safety vs number of slices
An analysis with one slice was trialled and the resultant factor of safety was approximately
9% higher compared to the factor of safety found when using 20 slices. An analysis with one
slice is impractical because:
•
The area of the slice used to determine the weight would be overestimated since the
weight of a slice is equal to the width (b) multiplied by its mid-height (h) of slice and
the bulk unit weight (γ) of the soil.
•
The angle α varies over the base of each slice. Only one angle is used in the
calculations for a single slice which then treats the failure plane as translational or a
wedge.
•
There are no interslice or interslice shear forces.
•
The method of slices does not accurately calculate a circular slip surface.
The difference between the factor of safety for five and 20 slices was 0.14% for Case 1
(drained analysis). This suggested that, by increasing the number of slices, the plot of number
of slices versus factor of safety becomes asymptotical around a converged factor of safety.
133
The factor of safety was also tested for 100 slices and was found to give the same values as a
20-slice analysis, to four significant figures.
The asymptotic behaviour suggested that a greater number of slices will give a more accurate
factor of safety; however, restrictions apply to the maximum number of slices possible. The
maximum number of slices is limited to a number such that the soil particle size exceeded the
slice width. If the slice width was less than the particle size (more specifically its maximum
particle size D100), the analysis suggested that the shear stresses would be great enough to
fracture the soil particles, which in reality is not practically admissible.
The difference between factors of safety calculated with five or 20 slices is minimal and 20
slices gave the same values as 100 slices, thus they give an acceptable upper and lower bound
to the number of slices required for analysis.
134
APPENDIX C
Bishop’s simplified method and Fellenius method- proof they are the same in undrained
conditions.
Bishop’s simplified method (BSM):
Fellenius’ method:
𝑭𝑩𝑺𝑴 =
∑�
𝐹𝐹𝑀 =
Substituting FM into BSM:
Rearranging:
𝐹𝐵𝑆𝑀
𝐹𝐵𝑆𝑀
𝐹𝐵𝑆𝑀 =
∑�
𝒄𝑳 𝒄𝒎𝒔 𝜶 + 𝑾 𝒕𝒂𝒏 𝝓
𝒔𝒊𝒏 𝜶 𝒕𝒂𝒏 𝝓 �
𝒄𝒎𝒔 𝜶 +
𝑭
∑ 𝑾 𝒔𝒊𝒏 𝜶
∑ (𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙)
∑ 𝑊 sin 𝛼
⎡
⎤
⎢
⎥
𝑐𝐿 cos 𝛼 + 𝑊 tan 𝜙
⎥
∑⎢
sin 𝛼 tan 𝜙
⎢cos 𝛼 +
⎥
∑ (𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙) ⎥
⎢
⎣
∑ 𝑊 sin 𝛼
⎦
=
∑ 𝑊 sin 𝛼
𝑐𝐿 cos 𝛼 + 𝑊 tan 𝜙
∑�
�
∑ 𝑊 sin 𝛼 sin 𝛼 tan 𝜙
cos 𝛼 +
∑ (𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙)
=
∑ 𝑊 sin 𝛼
𝑐𝐿 cos 𝛼 + 𝑊 tan 𝜙
�
∑(𝑐𝐿cos 𝛼 + 𝑊 cos 𝛼 cos 𝛼 tan 𝜙) + ∑ 𝑊 sin 𝛼 sin 𝛼 tan 𝜙
∑ (𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙)
∑ 𝑊 sin 𝛼
Using trigonometry identity sin2 𝛼 + cos 2 𝛼 = 1:
𝐹𝐵𝑆𝑀
∑(𝑐𝐿 cos 𝛼 + 𝑊 tan 𝜙)
∑ (𝑐𝐿 cos 𝛼 + 𝑊 tan 𝜙)
�
�
∑ (𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙)
=
∑ 𝑊 sin 𝛼
𝐹𝐵𝑆𝑀 =
∑(𝑐𝐿 + 𝑊 cos 𝛼 tan 𝜙)
∑ 𝑊 sin 𝛼
∴ 𝐹𝐵𝑆𝑀 = 𝐹𝐹𝑀
135
APPENDIX D
Case study data:
136
APPENDIX E
Elevation (m)
2.279
18
16
14
12
10
8
6
4
2
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
75
80
75
80
75
80
Distance (m)
Figure E.1: Critical failure surface for case study with no foundation and 5 kPa apparent cohesion
Elevation (m)
2.082
18
16
14
12
10
8
6
4
2
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Distance (m)
Figure E.2: Critical failure surface for case study with no foundation and no apparent cohesion
Elevation (m)
1.914
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Distance (m)
Figure E.3: Critical failure surface with clay foundation and earth fill apparent cohesion 5 kPa
Elevation (m)
1.846
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Distance (m)
Figure E.4: Critical failure surface with clay foundation and earth fill zero effective cohesion
137
Elevation (m)
1.826
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure E.5: Critical failure surface with alluvium foundation and earth fill apparent cohesion 5 kPa
Elevation (m)
1.759
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure E.6: Critical failure surface with alluvium foundation and zero effective cohesion
Elevation (m)
2.002
16
14
12
10
8
6
4
2
0
-2
0
10
20
30
40
50
60
70
80
Distance (m)
Figure E.7: Critical failure surface with clay and sandstone foundation and earth fill apparent cohesion 5 kPa
Elevation (m)
1.901
16
14
12
10
8
6
4
2
0
-2
0
10
20
30
40
50
60
70
80
Distance (m)
Figure E.8: Critical failure surface with clay and sandstone foundation and zero effective cohesion
138
Elevation (m)
2.179
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure E.9: Critical failure surface with sandstone foundation and earth fill apparent cohesion 5 kPa
Elevation (m)
2.070
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
75
80
Distance (m)
Figure E.10 Critical failure surface with sandstone foundation and zero effective cohesion
Elevation (m)
1.467
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Distance (m)
Figure E.11: Critical failure surface of upstream slope with 5 kPa apparent cohesion, at current tailings pond level
Elevation (m)
1.303
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
FigurE.12: Critical failure surface of upstream slope with no apparent cohesion and at current tailings pond level
139
Elevation (m)
1.475
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure E.13: Critical failure surface of upstream slope with 5 kPa apparent cohesion and no tailings water
Elevation (m)
1.308
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Distance (m)
Figure E.14: Critical failure surface of upstream slope with no apparent cohesion and no tailings water
Elevation (m)
1.475
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
75
80
Distance (m)
Figure E.15: Critical failure surface of upstream slope with 5 kPa apparent cohesion and full pond
Elevation (m)
1.310
16
14
12
10
8
6
4
2
0
-2
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Distance (m)
Figure E.16: Critical failure surface of upstream slope with no apparent cohesion and full pond
140
APPENDIX F
Figure F.1: Area under the standard normal distribution curve (US ACE, 2006)
141
APPENDIX G
Characteristics of random variables
The mean (𝜇) is generally known as the average value of the data set. For use in reliability
index it is also called the expected value (E(x)) (US ACE, 2006). The mean or expected value
of x is used to represent the first moment in reliability analysis. It is represented by the
equation:
𝜇𝑥 = �
𝑥
𝑁
where 𝑥 is a data point and N is the number of data points.
The variance (𝑉𝑎𝑟[𝑥]) is a measurement of the spread between numbers in a data set. It is the
average squared difference of each number from its mean and is represented by:
𝑉𝑎𝑟[𝑥] = �
[(𝑥 − 𝜇𝑥 )2 ]
𝑁−1
The standard deviation (𝜎𝑥 ) is equal to the square root of the variance and it is a measure of
how much the data points differ from the mean.
𝜎𝑥 = �𝑉𝑎𝑟[𝑥]
142
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