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1 ABSTRACT
JADID, ROWSHON. Strain-Based Stability Analysis of Earthen Embankments Subjected to
Cyclic Hydraulic Loading Associated with Extreme Events (Under the direction of Dr.
Mohammed Gabr and Dr. Brina Montoya).
Repeated rapid drawdown (RDD) and rapid rise in water level during extreme events lead to
progressive development of plastic shear strain zones within the earth embankments with subtle,
rather than obvious, visible signs of distress. The traditional analysis approach within the
framework of limit equilibrium method does not account for the accumulated permanent
deformation with repeated hydraulic loading.
This study investigates the effect of repeated rise and fall of water levels (representing severe flood
or drawdown cycles) on the stability performance aspects of embankment levees and dams.
Analysis are performed using unsaturated coupled transient seepage method and non-liner
advanced elasto-plastic constitutive relation in finite element (FE) program PLAXIS. Results show
a progressive development of internal distress within the embankment as the number of hydraulic
cycles is increased. This internal distress level is quantified in terms of level of shear strain. A
simple linear relationship between the shear strain and monitorable deformation at the toe of the
embankment is developed as a function of the geometry of the slope. This relationship provides a
simple means to estimate the performance limit state that corresponds to the instability of
embankment slopes, and the critical shear strain at the embankment toe, using the stress-strain data
obtained from triaxial testing. Results from the parametric study using numerical analyses show a
good agreement with the proposed analytical criterion. The proposed criterion is also compared
with data from the field studies by others and reasonable good agreement is obtained.
This study also assesses three remedial methods representing three different mechanisms to reduce
instability risk from the progressive development of deformation. These remedial methods
improve stability by providing reinforcement on the upstream slope (soil nails), reducing slope
height to decrease the shear stress (bench), and lowering phreatic surface to decrease pore water
pressure (drainage blanket). They are analyzed and compared in terms of probability of exceeding
the predefined ultimate limit state, where the limit state is associated with horizontal deformation
at slip surface toe that can be readily monitored in the field through periodic surveying. Given the
set of conditions used in this study, excavating a bench appears to be the most effective measure
in terms of associated risk among the three analyzed remedial methods due to the anticipated lower
probability of exceedance and shallower potential slip surface, which deems to cause lower
consequence.
For comparative study, pore water pressure and stability factor of safety are also calculated using
partially coupled and uncoupled transient seepage analysis. The uncoupled seepage analysis is
implemented in PLAXIS, whereas the partially coupled seepage analysis and stability analysis are
performed using FE program SEEP/W and limit equilibrium software SLOPE/W, respectively.
Results are presented and discussed on how pore water pressure predictions from different models
significantly affect the magnitude of stability factor of safety, the maximum thickness of potential
slip surface, and the required time to establish steady-state conditions.
© Copyright 2020 by Rowshon Jadid
All Rights Reserved
Strain-Based Stability Analysis of Earthen Embankments Subjected to Cyclic Hydraulic Loading
Associated with Extreme Events
by
Rowshon Jadid
A dissertation submitted to the Graduate Faculty of
North Carolina State University
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Civil Engineering
Raleigh, North Carolina
2020
APPROVED BY:
_______________________________ _______________________________
Dr. Mohammed Gabr Dr. Brina Montoya
Committee Co-Chair Committee Co-Chair
_______________________________ _______________________________
Dr. Shamim Rahman Dr. Celso Castro-Bolinaga
ii
2 DEDICATION
To my dearest parents, Mahmuda and Shamsul, my dear wife, Ishika, my beloved daughter,
Aleena, and my wonderful sister, Sabrin.
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3 BIOGRAPHY
Rowshon received his Bachelor’s and Master’s degree in Civil Engineering from Bangladesh
University of Engineering and Technology (BUET). Later, he joined as a faculty member at BUET
and was involved in several civil engineering projects as a consultant. His project experience
includes site exploration, technical report preparation, construction observation & quality control,
technical specification preparation, engineering design & analyses, and material testing. He has
served as a reviewer in multiple ASCE journals and also served as the President of ASCE Geo-
Institute Graduate Student Organization at NCSU.
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4 ACKNOWLEDGMENTS
Words are certainly not enough to express my gratitude to my advisors, Dr. Mo Gabr and Dr. Brina
Montoya, for their continuous guidance and support throughout my Ph.D. journey. It has been a
great honor and privilege to have worked with such advisors having exceptional knowledge,
experience, and attitude.
My sincere thanks to my committee members, Dr. Rahman and Dr. Castro, for their valuable
advice and suggestions. I am also honored and lucky to have incredible teachers, supporting staff,
and fellow graduate students at NCSU.
I would like to extend my gratitude to the U.S. Department of Homeland Security for funding my
research project. Lastly, I greatly appreciate my parents, wife, daughter, and relatives for their
endless love and support.
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5 TABLE OF CONTENTS
LIST OF TABLES………………………………………………………………………...........viii
LIST OF FIGURES ....................................................................................................................... ix
CHAPTER 1. INTRODUCTION ................................................................................................ 1 1.1 Background ........................................................................................................................ 1
1.2 Objectives ........................................................................................................................... 3
1.3 Dissertation Organization .................................................................................................. 4
CHAPTER 2. EFFECT OF REPEATED RISE AND FALL OF WATER LEVEL
ON SEEPAGE-INDUCED DEFORMATION AND RELATED STABILITY
ANALYSIS OF PRINCEVILLE LEVEE .................................................................................. 5 2.1 Introduction ........................................................................................................................ 7
2.2 Princeville Levee .............................................................................................................. 11
2.3 Domain Discretization and Model Properties .................................................................. 13
2.4 Analyses Approach .......................................................................................................... 16
2.4.1 Stability analysis ..................................................................................................... 16
2.4.2 Loading and boundary conditions ........................................................................... 17
2.5 Results and Discussion ..................................................................................................... 18
2.5.1 Model verification ................................................................................................... 18
2.5.2 Effect of storm cycles on stability .......................................................................... 19
2.5.3 Effect of small hydraulic loading cycles on shear strain ........................................ 24
2.5.4 Exceedance assessment ........................................................................................... 24
2.5.5 Effect of hydraulic conductivity anisotropy on LS ................................................. 29
2.6 Conclusions ...................................................................................................................... 31
CHAPER 3. ANALYSIS OF EARTHEN EMBANKMENTS USING
STRAIN-BASED PERFORMANCE LIMIT STATE APPROACH ..................................... 51 3.1 Introduction ...................................................................................................................... 53
3.2 Background ...................................................................................................................... 54
3.2.1 Monitoring and limit state approach ....................................................................... 54
3.2.2 Transient seepage analysis ...................................................................................... 56
3.3 Numerical Model ............................................................................................................. 57
3.3.1 Domain discretization and properties ..................................................................... 57
3.3.2 Modeling steps ........................................................................................................ 58
3.3.3 Coupled transient seepage analysis ......................................................................... 58
3.3.4 Hardening soil (HS) model ..................................................................................... 59
3.3.5 Material properties .................................................................................................. 60
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3.3.6 Stability analysis ..................................................................................................... 60
3.4 Results and Discussion ..................................................................................................... 60
3.4.1 Verification of pore pressure prediction ................................................................. 60
3.4.2 Stability analysis for repeated drawdown cycle ...................................................... 61
3.5 Correlation between the shear strain and displacement ................................................... 62
3.5.1 Effect of change in soil properties with drawdown cycles ..................................... 64
3.5.2 Effect of hydraulic conductivity of soil on developed correlation ......................... 65
3.5.3 Defining critical shear strain ................................................................................... 66
3.5.4 Performance limit state ........................................................................................... 67
3.6 Validation of the Developed Correlation ......................................................................... 68
3.6.1 IJkDijk levee ........................................................................................................... 68
3.6.2 Boston levee ............................................................................................................ 69
3.6.3 Elkhorn levee .......................................................................................................... 70
3.6.4 Lower Mississippi valley ........................................................................................ 70
3.7 Conclusions ...................................................................................................................... 71
CHAPTER 4. EFFICACY OF THREE SLOPE REPAIR METHODS IN TERMS
OF EXCEEDANCE PROBABILITY OF ULTIMATE LIMIT USING
COUPLED TRANSIENT SEEPAGE ANALYSIS .................................................................. 90 4.1 Introduction ...................................................................................................................... 92
4.2 Study Model ..................................................................................................................... 95
4.3 Domain Discretization and Modeling Approaches .......................................................... 96
4.3.1 Loading and boundary conditions ........................................................................... 97
4.3.2 Stability analysis ..................................................................................................... 98
4.3.3 Ultimate Limit State (ULS) .................................................................................... 98
4.3.4 Probabilistic approach ........................................................................................... 100
4.4 Pore Pressure Estimation ............................................................................................... 101
4.4.1 Verification of pore pressure prediction ............................................................... 101
4.4.2 Effect of pore pressure estimation on FS .............................................................. 102
4.5 Remedial Methods ......................................................................................................... 102
4.5.1 Installation of soil nails ......................................................................................... 103
4.5.2 Excavation of bench .............................................................................................. 105
4.5.3 Drainage blanket at upstream slope ...................................................................... 106
4.6 Comparison of Three Remedial Measures ..................................................................... 107
4.7 Summary and Conclusions ............................................................................................. 108
CHAPTER 5. SUMMARY, CONCLUSIONS, CONTRIBUTIONS, AND FUTURE
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WORKS ..................................................................................................................................... 128 5.1 Summary and Conclusions ............................................................................................. 128
5.2 Contributions .................................................................................................................. 131
5.3 Suggested Future Works ................................................................................................ 131
REFERENCES .......................................................................................................................... 133
APPENDICES ........................................................................................................................... 146 Appendix A ........................................................................................................................... 147
Appendix B ........................................................................................................................... 148
Appendix B ........................................................................................................................... 150
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6 LIST OF TABLES
Table 2.1. Soil Properties. ............................................................................................................. 33
Table 2.2. Sensitivity analysis results showing the most influencing soil parameters on shear
strain. ........................................................................................................................... 34
Table 2.3. The shear strain corresponding to each major variable (after 4 storm cycles). ........... 35
Table 2.4. Calculating the probability of exceeding LSs (after 4 storm cycles) using joint
variability of major variables. ..................................................................................... 35
Table 3.1. Soil properties. ............................................................................................................. 74
Table 3.2. Effect of friction angle on the number of cycles of loading, accumulated shear
strain, and velocity response. ...................................................................................... 75
Table 3.3. Summary of the case studies used for the verification of developed correlation. ....... 76
Table 4.1. Different types of slope repair methods with applicable soils. .................................. 111
Table 4.2. Soil properties. ........................................................................................................... 112
Table 4.3. Horizontal displacement corresponding to each major variable for soil nailing
at 43 days. ................................................................................................................. 113
Table 4.4. Calculating the probability of exceeding limit state (POELS) at 43 days using the
joint probability of major variables........................................................................... 114
Table 4.5. Pore pressure predictions from different methods. .................................................... 114
Table 4.6. Effect of pore water pressure prediction on FS after drawdown (at 43 days). .......... 115
Table 4.7. Properties of soil nail and facing. .............................................................................. 115
Table 4.8. Nail parameters adopted for FE simulations in PLAXIS. ......................................... 116
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7 LIST OF FIGURES
Figure 2.1. Princeville levee section (station 32+00): geometry and discretized mesh. ............... 36
Figure 2.2. SWCCs for SC, SP and CL layers. ............................................................................. 36
Figure 2.3. Flood stage hydrograph from Tarboro gage for 0.01 annual
exceedance probability [32]. ..................................................................................... 37
Figure 2.4. Deformation and flow boundary conditions. .............................................................. 37
Figure 2.5. Potential slip surface in- (a) Limit equilibrium approach (SLOPE/W);
(b) strength reduction approach (PLAXIS)................................................................. 38
Figure 2.6. Shear strained zone corresponding to factor of safety 0.98. ....................................... 39
Figure 2.7. Shear strain increase at (a) element A and element B with storm cycles; (b)
element A during the first storm cycle (water elevation y-scale is on the right). ..... 40
Figure 2.8. Stress paths during the first storm cycle at element A (top curve) and at element
B (bottom curve). ...................................................................................................... 41
Figure 2.9. Expanding of shear strained zone with cycles of loading. (a) After 1 cycle,
(b) after 3 cycles, (c) after 6 cycles. .......................................................................... 42
Figure 2.10. Distribution of shear strain along the slip surface. ................................................... 43
Figure 2.11. Factor of safety of Princeville levee using limit equilibrium method with
cycles of loading (water elevation y-scale is on the right)....................................... 43
Figure 2.12. Effect of drawdown rate on the factor of safety. ...................................................... 44
Figure 2.13. Gradual dropping of the phreatic surface after instantaneous drawdown. ............... 44
Figure 2.14. (a) Effect of small hydraulic loading cycles on shear strain at blanket toe;
(b) Increase in shear strain after the application of a small hydraulic loading
cycle with a scale factor = 0.5. ................................................................................. 45
x
Figure 2.15. Increase in shear strain with scale factor. ................................................................. 46
Figure 2.16. Variation of probability of exceeding limit state and factor of safety with
number of storm cycle. ............................................................................................ 46
Figure 2.17. Effect of soil anisotropy with respect to hydraulic conductivity and storm
cycles on- (a) shear stain; (b) probability of exceeding limit states and
factor of safety (for 𝑘𝑥/𝑘𝑧=2); and (c) flow rate at blanket toe. ............................. 47
Figure 2.18. Probability of exceeding LS3 for 2 SD and 𝑘𝑥/𝑘𝑧=2 versus consequence
curve showing the effect of load history on risk evaluation associated
with slope failure...................................................................................................... 50
Figure 3.1. Model geometry and discretized mesh. ...................................................................... 77
Figure 3.2. Selected points along the potential slip surface for stability analysis. ....................... 77
Figure 3.3. Comparison of pore pressure predictions obtained from different methods
after rapid drawdown. ................................................................................................ 78
Figure 3.4. Decrease in factor of safety with drawdown cycle. .................................................... 78
Figure 3.5. Shear strain and horizontal displacement increase at toe with drawdown cycles. ..... 79
Figure 3.6. Stress path meeting the failure envelope at fifth drawdown cycle. ............................ 79
Figure 3.7. (a) Deformed shape of the slope at fifth drawdown cycle; (b) Simplified
diagram of a deformed element at toe. ...................................................................... 80
Figure 3.8. Determination of the magnitude of 𝐶 for 𝑘= 10 − 9cm/s. ......................................... 80
Figure 3.9. Shear strain and horizontal displacement increase at toe with drawdown cycles;
(a) with 0.5 and 500 kPa increment after each cycle for ′ and 𝐸50𝑟𝑒𝑓,
respectively, (b) with 0.5 decrement after each cycle for ′. .................................... 81
Figure 3.10. Determination of 𝐶 using the data subjected to change in strength and/or
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stiffness parameters. ................................................................................................. 82
Figure 3.11. Effect of hydraulic conductivity on 𝐶; (a) 𝑘 = 10 − 6 cm/s,
(b) 𝑘 = 10 − 5 cm/s, and (c) 𝑘 = 10 − 4 cm/s. ........................................................ 83
Figure 3.12. Shear strained zone after fifth drawdown cycle for ′27;
(a) with 𝑘= 10 − 4 cm/s, (b) with 𝑘= 10 − 6 cm/s. ................................................ 86
Figure 3.13. Accumulation of plastic points for 𝜑 = 27; (a) after four drawdown cycles,
(b) after fifth drawdown phase. ................................................................................ 87
Figure 3.14. (a) Simulation of isotropic consolidated undrained triaxial tests of soil;
(b) comparison between shear strain obtained from model and from
stress-strain curve..................................................................................................... 88
Figure 3.15. Rapid increase of surface displacement at fifth drawdown cycle for 𝜑 = 27
(time is set to zero at the beginning of fifth cycle). ................................................. 89
Figure 3.16. Determination of the magnitude of 𝐶 from four case studies................................... 89
Figure 4.1. Model geometry and discretized mesh in PLAXIS 2D. ........................................... 117
Figure 4.2. Simulation of isotropic consolidated undrained triaxial tests of soil ....................... 117
Figure 4.3. Shear strained zone indicating potential slip surface after drawdown. .................... 118
Figure 4.4. Comparison of pore water pressure predictions by different methods after
drawdown. ............................................................................................................... 118
Figure 4.5. Prediction of pore water pressure with time at point 1 using different models-
(a) until the establishment of steady-state condition;
(b) for the first 43 days only. ................................................................................... 119
Figure 4.6. Factor of safety calculation in SLOPE/W- (a) using the slip surface
corresponding to coupled analysis; (b) using the critical slip
xii
surface from partially coupled analysis. .................................................................. 120
Figure 4.7. Model with soil nails (length of nail = 10 m and orientation of nail = 20)............. 121
Figure 4.8. (a) Influence of nail length on FS at 43 days with 15 nail orientation
(b) influence of nail orientation and strength parameters on FS at
43 days with 10 m long nail. ................................................................................... 122
Figure 4.9. Model with excavating a bench at EL=205.2 m with the inclination angle of
(a) = 0, (b) = +10, and (c) = -10 ................................................................. 123
Figure 4.10. Effect of bench location and inclination () on FS. ............................................... 124
Figure 4.11. (a) Model with upstream drainage blanket; (b) potential slip surface with
6.4 m thick drainage blanket. ................................................................................. 125
Figure 4.12. (a) Influence of blanket thickness on FS at 43 days with 𝑘𝑏 = 10 − 2 cm/s;
(b) influence of hydraulic conductivity of blanket on FS at
43 days with 𝑡𝑏= 6.4 m. ......................................................................................... 126
Figure 4.13. Effect of remedial measures on horizontal deformation at slip surface toe. .......... 127
Figure 4.14. Probability of exceeding limit state for three remedial measures. ......................... 127
1
1 CHAPTER 1. INTRODUCTION
1.1 Background
In recent decades, climate change has increased extreme precipitation in both frequency and
magnitude, which in turn has elevated flood risk in the U.S. [1]. In some areas, the increasing
temperature due to climate change is expected to cause more intense and prolonged droughts [2].
Earthen levees and dams are designed and constructed to play an important role during such
extreme events. They are critical infrastructure related to flood protection and water supply
management. While the importance of levees and dams as disaster defense infrastructures are ever-
increasing to fight against future extreme events, the health conditions of these earth structures are
deteriorating with age. The average age of levees and dams in the U.S. is more than 50 years, a
period considered as the nominal design life for heavy structures [3]. As a qualitative assessment,
ASCE assigned grade ‘D’ for dams and levees, which indicates that the infrastructure’s condition
and capacity are of serious concern with a strong risk of failure [4].
Levees and dams experience relatively rapid increase and decrease in water elevation during flood
events due to extreme precipitations associated with hurricanes or wet seasons. In addition, rapid
decrease in water level occurs due to excessive use of water supply from reservoirs during drought.
Several dam failures have indicated that repeated occurrence of such events may lead to breaching
failure as strain softening of the earth materials occurs. San Luis Dam failure in U.S. [5], Canelles
dam failure in Spain [6], and Vernago dam failure in Italy [7] are a few examples of such incidents.
While the quantification of internal damage due to repeated loading is essential to assess the health
condition of the embankments and manage the need for rehabilitation, the conventional slope
stability approach (e.g., limit equilibrium method) provides no means to account for such effect
2
[8]. In addition to that, significant amounts of research have been done on stability analysis of
embankment slopes based on a design water elevation that represents an extreme scenario [9, 5,
10]. In the context of climate change, the possibility that the embankment might experience
hydraulic loading similar to design loads on several occasions within the service life has been
ignored. The effect of multiple cycles or hydraulic loading history on the stability aspects of
embankment has not been studied in detail and thereby, one of the poorly understood design cases
in slope stability.
Monitoring surface movements of levees and dams is readily doable on a large scale and the use
of such deformation has the potential to indicate the gradual degradation of the structure’s stability
with cycles of hydraulic loading. In practice, the measured surface displacement with time or
deformation rate (referred to as velocity) for a given slope is used to compare the empirically-
defined critical threshold value [11]. Such an empirical approach is phenomenologically-based
and emphasizes the overall performance of the structural system while disregarding the underlying
mechanisms of failure. Thus, defining the critical thresholds, or performance limits, involves
subjectivity [12]. This limitation has led to failed predictions including the prediction of a landslide
failure near Innertkirchen in the Swiss Alps in 2001 [13]. Thus, there is a need to develop
performance limits based on underlying mechanisms of slope failure.
The aging dam and levee systems are considered deficient in some aspects of their structural
integrity and require on the order of $80 billion for levees and $45 billion for dams to repair and
upgrade their performance for future extreme events. Yet, a limited budget has been allocated
nationwide [4]. In practice, the performance of a repair method is assessed by increased stability
3
factor of safety, which does not provide a rational basis for condition assessment of dams and
levees as they are progressively loaded over time with repeated rise and fall of water levels as well
as the efficacy of remedial actions [14]. Thus, there is a need to assess the health condition and
predict the performance of earth structures used in flood defense and prioritize rehabilitation
measures based on improving functionality level and limiting damage under future extreme events.
1.2 Objectives
Based on the aforementioned research gaps and needs, this study aims to focus on the following
three major objectives:
1. To investigate the effect of repeated rise and fall of water level on the stability performance
aspects of embankment levees and dams in terms of probability of exceeding limit states.
This will lead to better understanding of the underlying kinematics of emerging shear band
and progressive instability due to cycles of hydraulic loading.
2. To define ultimate limit state that corresponds to the instability of earth embankments
forming levees and dams. This is accomplished by developing a correlation between the
shear strain at slip surface and the deformation at slope surface which are progressively
accumulated under repeated rise and fall of water level.
3. To assess three remedial actions representing three different mechanisms to reduce
instability risk from the progressive development of deformation. This will assist in
selecting the most effective approach among the three analyzed remedial measures to meet
the required stability performance aspects for future extreme events.
4
1.3 Dissertation Organization
This dissertation consists of five chapters. In chapter 2, strain-based limit state analyses and
conventional slope stability factor of safety (FS) approach are used to assess the effect of rise and
fall of water levels, representing severe storm cycles, on the stability of embankment slopes. The
effect of storm cycles on the probability of exceeding a prescribed performance limit state versus
the FS computed using the limit equilibrium method and strength reduction method is presented
in this chapter.
In chapter 3, a general criterion for performance limit state is developed which is defined based on
the framework of emergence of shear strain magnitude representing the onset of accelerated
deformation rate. A correlation between the magnitude of shear strain and the corresponding
deformation at toe is developed.
In chapter 4, three repair methods, representing three different mechanisms of remedial efforts, are
investigated to stabilize the upstream slope failure of an embankment slope. They are analyzed
and compared in terms of probability of exceeding a predefined performance limit state for a given
factor of safety, where the limit state is associated with horizontal deformation at slip surface toe.
Finally, the main findings, contributions to the state of the art and suggested future works are
summarized in Chapter 5.
5
2 CHAPTER 2. EFFECT OF REPEATED RISE AND FALL OF WATER LEVEL ON
SEEPAGE-INDUCED DEFORMATION AND RELATED STABILITY ANALYSIS
OF PRINCEVILLE LEVEE
The contents of this paper have been published in the Engineering Geology Journal.
Citation:
Jadid, R., Montoya, B. M., Bennett, V., & Gabr, M. A. (2020). Effect of repeated rise and fall of
water level on seepage-induced deformation and related stability analysis of Princeville levee.
Engineering Geology, 266, 105458.
6
Abstract
The Princeville levee, and flooding associated with Hurricanes Floyd and Matthew, is used as a
case study in which the analyses are focused on the effect of repeated rise and fall of water levels
(representing severe storm cycles) on the stability of the levee and the risk of failure. The analyses
included strain-based and strength reduction approaches and are conducted using the finite element
program PLAXIS 2D. The limit equilibrium stability software “SLOPE/W” was also used for
comparative study. The strain-based limit state approach considers the uncertainty of soil
properties and is used to characterize the levee performance under repeated storm loading in terms
of damage levels (or limit states). The strain-based analyses results show a progressive
development of plastic shear strain zone within the levee as the number of storm cycles is
increased. The accumulation of such shear strain leads to increasing the probability of exceeding
a given performance limit state. As more flooding cycles are introduced, the shear strain values
increase by a factor of 3.5 from cycle 1 to 6, and therefore reflect the increasing level of failure
risk. In parallel, the deterministic stability factor of safety obtained from limit equilibrium and
strength reduction approached slightly changed with an increased number of rises and falls of the
water level. The consideration of “rapid” drawdown conventionally used in limit equilibrium
stability analyses (where no consideration for time is included), instead of more realistic rate based
on drawdown hydrograph leads to conservative estimate of factor of safety. The analyses results
demonstrate the increase in risk with repeated hydraulic loading.
7
2.1 Introduction
Earthen embankment structures, including dams and levees, play important roles not only in flood
defense but also in storing water supply for irrigation, power generation, and transportation and in
providing means of sediment retention. The United States has thousands of levee systems, and 43
percent of the U.S. population lives in counties with at least one levee [15]. The levee and dam
systems are, however, aging, and their structural health are deteriorating. For example, the Task
Committee of the Association of State Dam Safety Officials [16] reported that approximately one-
third of the high hazard earth dams are considered deficient in some aspects of their integrity and
many are aged more than 50 years. The storm surge produced by Hurricane Katrina caused levee
failures that occurred at water levels well below their design due to the combination of
misinterpretation of geologic conditions and unforeseen failure mechanisms [17]. According to
ASCE [18], the 5-year funds needed for rehabilitation and repair of these structures is on the order
of $12.5 billion with less than half of such funds available to address the issue. Thus, there is a
need to assess the health condition and predict the performance of earth structures used in flood
defense and prioritize rehabilitation measures based on improving functionality level and limiting
damage under future flood events.
The levees and dams may experience large and rapid change in water elevation during the flood
events associated with hurricanes. The repeated occurrence of such events may cause major
distress to these earth structures and may lead to breaching failure. For example, Stark et al. [5]
reported that the cyclic hydraulic loading from the reservoir water level resulted in the upstream
slide of the San Luis Dam (now known as B.F. Sisk Dam) in 1981. The slide was about 550 m
(1,800 ft) along the centerline of the dam crest. The cyclic hydraulic loading from the reservoir
8
resulted in accumulated shear deformations that was sufficient to mobilize shear strengths between
fully softened and residual values. Consequently, the shear strength of the dam material as
significantly reduced causing the slope failure [5]. A similar observation was noted for the earth
dam of the Vernago hydroelectric reservoir in northern Italy which experienced large plastic
deformation and strain due to the fluctuations of the reservoir water level. This deformation
produced structural damages in a service shaft located within the dam [7].
While the accumulation of plastic strain may progressively compromise the stability of earth levees
and dams, the conventional slope stability approach, in which slope stability is assessed in terms
of singular factor of safety (FS), provides no means to account for the extent of the damage level
[8, 19, 20]. Such measure of damage extent is important to quantify especially in the aftermath of
extreme flooding events when the flood defense structure may have sustained damage but is
functioning with no imminent threat. The deterministic nature of the FS does not directly convey
the level of the structure performance under the repeated severe storm events and provides no
means of defining functionality under future events. The FS can be obtained using either limit
equilibrium method (LEM) or, numerical methods (e.g., strength reduction method, SRM) [21].
Several researchers have compared the results between the LEM and SRM and generally
concluded that two methods will give similar FS for most cases [22]. The detailed discussion on
the relative advantages, disadvantages, and the applicability of each approach can be found on
Griffiths and Lane [22] and Cheng, et al. [23]. Although the concept of FS is well established in
geotechnical engineering, its determination does not provide a quantitative measure of the
uncertainties in loading and soil properties. Beyond the issue of the binary assessment of
“safe/fail”, the use of FS also falls short when there is a need for the performance risk assessment
9
or life cost analyses especially with respect to the informed selection best approach among
alternative remediation and retrofit measures.
Khalilzad and Gabr [24] and Khalilzad et al. [25, 14] proposed strain-based limit states (LS) for
embankment dams and incorporated these into simple probabilistic analysis following an approach
introduced by Duncan [13]. Their motivation was to develop a sensor-based monitoring system
and model-aided approach that could enable early identification and warning of vulnerable earth
levees and enabling prioritized rehabilitation measures. In this approach, the performance LSs are
defined in terms of shear strain at key locations indicating basal stability of the embankment. They
defined the damage level associated with each LS as follows- LS 1: minor deformations, no
discernible shear zones (max shear strain less than 1%), low hydraulic gradients (i.e., i < 1)
throughout the embankment dam and foundation; LS 2: medium (repairable) deformations, limited
piping problems (i.e., i > 0.67 within a shallow depth at the location of toe), dispersed plastic zones
with moderate strain values (maximum shear strain less than 3%), tolerable hydraulic gradients
less than critical; LS 3: major deformations, breaches and critical hydraulic gradients at key
locations (i.e., i>1, boiling and fine material washing at the location of toe), high strain plastic
zones (maximum shear strain > 5%). Thus, this technique provides a graded measure (versus the
binary classification of safe/unsafe) of the safety margin under a specified storm loading. The
stain-based approach can be integrated with real-time monitoring programs to assess probability
of exceeding a predefined LSs. For example, satellite images and in-ground GPS sensors were
used for displacement monitoring of a levee section on Sherman Island, California [26]. This site
is underlain by highly fibrous peat. The monitored displacement data was used to calibrate the
numerical model for estimating the probability of exceeding a performance LS [26, 27] due to peat
10
deformation with time. The model predicted the magnitude of accumulated plastic shear strain at
embankment toe during the lifetime of the levee as 3.2% with the associated probability of
exceeding LS2 =95% due to peat decomposition [28]. As per the definition of LS2, the likelihood
of damage level is medium or repairable in this case. The importance of monitoring program for
levees led to instrumentation of many levees, including the New Orleans levees and Ritchard Dam
[29].
In this paper, strain-based limit state analyses and conventional slope stability factor of safety
approach are used to assess the effect of rise and fall of water levels, representing severe storm
cycles, on the stability of the Princeville levee, located on the Tar River, North Carolina. The
analyses are conducted using PLAXIS 2D for the SRM and strain-based analysis; and SLOPE/W
program for the LEM. The storm cycle is simulated based on flood stage hydrograph using
unsaturated transient seepage analysis that involves coupled-flow deformation method and non-
linear advanced elastic-plastic constitutive relation. The levee performance under repeated storm
cycles is investigated and characterized using strain-based limit state approach in view of
performance limit states. In parallel, the LEM and SRM are used to define the FS as conventionally
performed in practice. To estimate the probability of exceedance for each limit state, an approach
similar to Duncan [30] “3-sigma approach” is employed. A detailed sensitivity analysis is
performed to select random variables for the probabilistic analysis. The effect of repeating storm
cycles, the degree of uncertainty, and hydraulic conductivity anisotropy on the probability of
exceedance of a given LS versus the FS computed using the LEM and SRM is discussed. The
results from the strain-based approach are used for risk assessment to demonstrate the effect of
including hydraulic loading history on risk assessment. The primary novel contributions of the
11
study include: i. Explanation of the underlying kinematics of emerging shear band and progressive
instability due to repeated hydraulic loading due to storm cycles; ii) Comparison of strain-based
approach with the existing techniques of stability analysis (LEM and SRM) within the context of
cycles of hydraulic loading; iii. Incorporation of hydraulic loading history in the stability analysis
in order to quantify increased risk for the future storm event; and iv. Study of the effect of degree
of uncertainty and hydraulic conductivity anisotropy on the stability performance aspects of
embankment levees in terms of probability of exceeding LSs.
2.2 Princeville Levee
The Princeville levee system is located on the western, southwestern, and northern sides of the
Town of Princeville, North Carolina. The Town was established in 1865 by freed slaves (and
originally named Freedom Hill) and located in a natural flood zone adjacent to the Tar River.
Characterized by its low-lying topography, this zone experienced seven major flood events
between 1800 and 1958. Due to the frequent flood events in this region, FEMA declared
Princeville as National Disaster in the past. Consequently, U.S. Army Corps of Engineers
(USACE) built a three-mile long levee system beginning in 1967, along the south bank of the Tar
River, as a flood defense structure for the town. Since the construction of the levee system, the
town was overwhelmed by several major floods that occurred in conjunction with Hurricanes. The
levee system was severely damaged during the flood event associated with Hurricane Floyd in
1999. Flood water flowed around the northern and southern ends of the levee causing millions of
dollars of property loss. Additionally, several locations along the levee were impaired by erosion
due to overtopping. After the flood event, the damaged areas were repaired to bring the levee
system to its pre-flood condition. Another major flood event in conjunction with Hurricane
Matthew occurred in 2016 at the levee. Although this time the levee was not overtopped, as was
12
the case following Hurricane Floyd, the town was flooded with 3.0 m of water primarily from the
ends of the levee system and from an un-gated culvert running underneath the embankment [31].
The Princeville levee was analyzed herein using the finite-element software, PLAXIS 2016, and
the limit equilibrium software, SLOPE/W 2016. The geometry and soil layers of the analyzed
Princeville levee section at station 32+00 are shown in Figure 2.1. In a feasibility study, the
USACE identified the levee section at Station 32+00 as “critical” since it was overtopped by
flooding associated with Hurricane Floyd in 1999 [32]. The soil profile of the levee section consists
of a four-layer system that includes clayey sand (SC) layer over a poorly graded sand (SP) layer
with higher hydraulic conductivity. A thin silty sand (SM) layer was encountered beneath the SP
layer on the landside of the levee. A thin layer of stiff lean clay (CL) was placed at the embankment
toe. The levee is approximately 2.75 m (9 feet) high, as measured from the landside, with a top
elevation of 14.9 m (49 feet) and has a 3 m (10 feet) wide crest. The side slopes are 2.5 H to 1 V
for the landside and 3.0 H to 1 V for the riverside. The 2D plane strain model was utilized to
analyze the levee section. The levee segment containing the station 32+00 is straight with nearly
uniform cross-section and sufficiently long (approximately 986.4 m) compared to the other
dimensions of the levee section (e.g., height= 2.75 m, crest= 3.0 m). Therefore, the strain and
displacement in the long direction are expected to be zero (i.e. plane strain condition). Also, the
placement of clay blanket at embankment toe of the levee reduced the potential for erosion piping
[32]. While “concentrated” piping can be as a 3D phenomenon [33, 34, 35, 36, 37], the use of 2D
analyses for the levee section considered herein is further justified by the fact that USACE [32]
estimated the hydraulic gradient as 0.29 at toe for design flood scenario (Riverside EL =14.6 m,
Landside EL= 11.0 m) and, no erosion activity, such as sand boiling, was reported.
13
2.3 Domain Discretization and Model Properties
The levee section was modeled using 15-nodes triangular plain strain elements (TRI15) with a
domain having 6,849 nodes and 822 elements as shown in Figure 2.1. The use of TRI15 element
incorporates higher-order shape function (quartic) given the higher number of nodes (total 15) per
element. Also, the analyses mesh is composed of triangular elements which are less susceptible to
distortion errors [38]. However, the use of the higher-order element, the TRI15 requires more
computation time.
The constitutive model of the various layers in the analysis domain was defined by the hardening
soil (HS) model [39]. The HS model can simulate both soft and stiff soils and approximates non-
linear stress-strain behavior with a hyperbola similar to the hyperbolic model by Kondner [40] and
Duncan and Chang [41]. However, HS model supersedes hyperbolic model by using the theory of
plasticity rather than theory of elasticity, and accounts for soil dilatancy by introducing a yield cap.
The yield surface of a HS model is not fixed in principal stress space; rather it expands due to
plastic straining. The stress-strain behavior of soil shows a decreasing stiffness and simultaneously
irreversible plastic strains when subjected to primary deviatoric loading [42, 43]. The hyperbolic
relationship between vertical strain, 1, and the deviatoric stress, q, was formulated as:
휀1 =𝑞𝑎2𝐸50
𝑞
𝑞𝑎 − 𝑞 𝑓𝑜𝑟 𝑞 < 𝑞𝑓 (2.1)
Where 𝑞𝑎 is the asymptotic value of the shear strength and is determined from ultimate deviatoric
stress, 𝑞𝑓, and failure ratio, 𝑅𝑓. The 𝑞𝑓 is defined as:
𝑞𝑓 =6 sin𝜑′
3 − sin𝜑′ (𝜎3
′ + 𝑐′ cot 𝜑′) (2.2)
Where 𝑐 ′= cohesion; 𝜑′= friction angle, and 𝜎3′= minor effective principal stress.
14
The failure ratio (𝑅𝑓) is defined:
𝑅𝑓 =𝑞𝑓
𝑞𝑎 (2.3)
The confining stress-dependent stiffness modulus (𝐸50) under primary loading is given by:
𝐸50 = 𝐸50𝑟𝑒𝑓(𝑐′ cos𝜑′ − 𝜎3
′ sin𝜑′
𝑐′ cos𝜑′ + 𝑝𝑟𝑒𝑓 sin𝜑′)
𝑚
(2.4)
Where 𝐸50𝑟𝑒𝑓
= reference stiffness modulus at reference stress (𝑝𝑟𝑒𝑓). A default value of 100 kPa is
used as 𝑝𝑟𝑒𝑓 in PLAXIS. The actual stiffness depends on the minor principal stress, 𝜎3′ , which is
the confining pressure in triaxial tests. The amount of stress dependency is given by the power
coefficient “𝑚.” The stress-dependent stiffness modulus for unloading and reloading stress paths
is determined as:
𝐸𝑢𝑟 = 𝐸𝑢𝑟𝑟𝑒𝑓(𝑐′ cos𝜑′ − 𝜎3
′ sin𝜑′
𝑐′ cos𝜑′ + 𝑝𝑟𝑒𝑓 sin𝜑′)
𝑚
(2.5)
Where 𝐸𝑢𝑟𝑟𝑒𝑓
is the reference stiffness modulus for unloading and reloading, corresponding to the
reference pressure 𝑝𝑟𝑒𝑓. The shear hardening yield function,𝑓𝑠, is defined by:
𝑓𝑠 = 𝑓̅ − 𝛾𝑝 (2.6)
𝑓̅ =𝑞𝑎𝐸50
{(𝜎1′ − 𝜎3
′)
𝑞𝑎 − (𝜎1′ − 𝜎3
′)} −
2(𝜎1′ − 𝜎2
′)
𝐸𝑢𝑟 (2.7)
Where 𝜎1′and 𝜎2
′ the major and intermediate effective principal stresses respectively, and 𝛾𝑝 is the
plastic shear strain, and is approximated as:
𝛾𝑝 ≈ 휀1𝑝 − 휀2
𝑝 − 휀3𝑝 = 2휀1
𝑝 − 휀𝑣𝑝 ≈ 2휀1
𝑝 (2.8)
Where 휀1𝑝, 휀2𝑝, and 휀3
𝑝 are plastic strains along the principal axes, and 휀𝑣
𝑝 is the volumetric plastic
strain. The reference oedometer modulus (𝐸𝑜𝑒𝑑𝑟𝑒𝑓
) is used to control the amount of the plastic strains
15
that derive from the yield cap. Like 𝐸50 and 𝐸𝑢𝑟, the oedometer modulus (𝐸𝑜𝑒𝑑) also obeys the
stress dependency law as:
𝐸𝑜𝑒𝑑 = 𝐸𝑜𝑒𝑑𝑟𝑒𝑓(𝑐′ cos𝜑′ − 𝜎3
′ sin𝜑′
𝑐′ cos𝜑′ + 𝑝𝑟𝑒𝑓 sin 𝜑′)
𝑚
(2.9)
The flow rule in the HS model is defined as:
휀�̇�𝑝 = sin
𝑚�̇�𝑝 (2.10)
Where 𝑚
is the mobilized dilatancy angle; 휀�̇�𝑝 and �̇�𝑝 are the volumetric and shear plastic strain
rates. The input soil parameters for the levee section and foundation layers are presented in Table
2.1. The unit weight (), strength parameters (𝑐′ and 𝜑′) and the hydraulic conductivity (𝑘) values
were originally reported by USACE [32]. The stiffness parameters (𝐸50𝑟𝑒𝑓
, 𝐸𝑜𝑒𝑑𝑟𝑒𝑓
and 𝐸𝑢𝑟𝑟𝑒𝑓
) were
selected according to data presented by Schanz and Vermeer [44], Bowles [45], Sture [46] and
Brinkgreve [47] The value of failure ratio (𝑅𝑓) for granular and cohesive soils varies within a
narrow range, usually between 0.75 and 1.0 with the average value of 0.9 [48]. Since the test results
to estimate 𝑅𝑓 were not reported in the USACE [32], a value of 0.9 was selected for soil forming
the embankment and foundation layers based on recommendation from the literature [39, 47, 48].
Non-associated plasticity flow rule was considered for shear-plastic flow in this study since
associated flow rules with cohesionless soil models could predict far greater dilation than is
observed in reality [22]. The tensile stresses might develop in slopes and could lead to cracking
that would substantially reduce the stability of the embankment slope [49, 50]. The development
of tension crack was considered in the analyses using the tension cut-off mechanism available in
the HS model. In this case, the negative principal stresses are limited to the tensile strength (𝜎𝑡).
The magnitude of 𝜎𝑡 was assumed as zero all soils forming embankment and foundation layers
since soil can sustain none or very small tensile stresses [43]. Therefore, locations within the
16
embankment with Mohr circles having negative principal stresses formed tensile cracks in the
model. The 2D model used herein simulates these tensile cracks as continuous cracks in the long
direction. This assumption provides a lower (conservative) estimate of stability compared to the
case of discontinuous tensile cracks (3D case).
Since all or some portion of the embankment and foundation layers (i.e., the SC, SP, and CL layers
in Figure 2.1) might be above the phreatic surface at the various phases of modeling (e.g.,
simulation of rise and fall of water level of Tar River), unsaturated hydraulic properties were used
for these layers above the phreatic surface. Figure 2.2 shows the soil water characteristic curves
(SWCCs) for SC, SP and CL layers which were used to estimate unsaturated hydraulic
conductivities of these layers. The pertinent van Genuchten [51] model parameters (𝜃𝑟, 𝜃𝑠, 𝑔𝑎 and
𝑔𝑛) for the SWCCs shown in Table 1 were selected based on reported values for sand, clay and
clayey sand with similar soil gradations as SP, CL, and SC layers, respectively [43]. The SM layer
was assumed to be saturated through all phases of modeling.
2.4 Analyses Approach
2.4.1 Stability analysis
The analyses are conducted using two-dimensional SLOPE/W program for the LEM, and two-
dimensional finite element program PLAXIS 2D for the SRM and strain-based analysis. The levee
section was first modeled using SLOPE/W to determine the factor of safety (FS) using Spencer’s
procedure [52]. In this procedure, both force and moment equilibriums are taken into consideration
[53]. The SLOPE/W program uses an iteration scheme to find the critical slip surface and the
corresponding minimum factor of safety. The levee was then modeled as a continuum system in
17
PLAXIS 2D and the FS was obtained using SRM. In SRM, the factor of safety is defined as the
factor by which strength parameters (𝑐′and tan𝜑′) are divided in order to reach slope failure [54].
Coupled flow-deformation option was used in PLAXIS 2D which allowed to account deformation-
induced pore pressure in stability analysis resulting from the change in boundary loading.
2.4.2 Loading and boundary conditions
The flood events associated with hurricanes may cause a rapid rise in water level, followed by the
fall of water level with time. Failure in a levee or dam may occur due to the accumulation of plastic
shear strain resulting from the repeated rise and fall of water level, decreasing the soil strength in
the plastic zone [5]. The flood conditions, similar to elevations occurred in conjunction with
Hurricane Floyd and Hurricane Matthew, were simulated in this study to represent the storm
scenarios. The flood stage hydrograph from Tarboro gage for 0.01 annual exceedance probability,
as shown in Figure 2.3 [32], provided the basis to define the flood or storm cycle in the analysis.
A storm cycle, consisting of three consecutive phases- rise, peak, and drawdown shown in Figure
2.3, was simulated using transient seepage analysis in PLAXIS and SLOPE/W. Modeling steps for
simulating a storm cycle included- first generating the geostatic stress state in the levee and
foundation layers. Then, a steady-state seepage analysis was performed to establish the initial
groundwater conditions for transient seepage analysis. The flow boundary conditions for steady-
state analysis included a no flow (Q) boundary at the bottom of the model domain and a free-
seepage boundary at the landside of the levee. Constant total head boundaries (ℎ𝑡) were set to at
12.2 m and 11.0 m at the lefthand side and righthand side of the foundation layers, respectively,
as shown in Figure 2.4. Transient seepage analysis to introduce a storm cycle was performed in
three phases using the time-dependent total head boundaries on the riverside embankment face. In
the first phase (rise), the water level was raised in 4 days at a rate of 0.6 m/day from the total head
18
of elevation (EL) 12.2 m to an elevation (EL) of 14.6 m. Then, the water level was kept at EL
14.6m for 0.5 day to represent the storm duration or peak phase. Thereafter, the water level was
lowered at a rate of 0.6 m/day to EL 12.2 m in 4 days representing the drawdown phase. The rates
of water levels rise and fall at riverside were established on the basis of flood stage hydrograph
(Figure 2.3). A drained plastic phase of 11.5 days was used between two storm cycles to represent
the time lag between two storms. There is no significance to the 11.5 days’ time period other than
ensuring that the excess pore water pressure has dissipated before applying the next storm cycle
(i.e., the duration of time between each storm cycle can be any time period that is more than 11.5
days). Cycles of water level rise and fall are applied to the levee up to 6 cycles in PLAXIS for the
SRM and strain-based analysis and 2 cycles in SLOPE/W for the LEM. For strain-based analyses,
additional deformation boundary conditions were applied in PLAXIS which included-restriction
of horizontal deformation (𝑢𝑥) on the left and right edges of the domain as well as restriction of
horizontal (𝑢𝑥) and vertical deformations (𝑢𝑦) at the bottom boundary as shown in Figure 2.4. The
top surface of the model was maintained unconstrained during the analysis. The dimensions of the
model have been carefully chosen to minimize boundary effect (i.e., further increase of model size
does not change the results).
2.5 Results and Discussion
2.5.1 Model verification
The design flood condition (riverside water EL=14.6 m and landside water EL= 11.0 m) as
reported in USACE [32] was simulated in both SLOPE/W and PLAXIS program. The limit
equilibrium model in SLOPE/W yielded a FS = 1.87 for design flood condition, and the
corresponding slip surface is shown in Figure 2.5(a). This model was used to verify the stability
19
analyses in the finite element program PLAXIS, which resulted in FS= 1.86 using the SRM for
design flood condition and yielded similar slip surface as the LEM using the SLOPE/W program
(Figure 2.5b).
The shear strength of soils forming the levee embankments changes over time due to the
accumulation of plastic shear strain resulting from the fluctuations of water level, swelling, and
creep, etc. [24, 49]. Once the reduced shear strength is no longer capable of resisting the hydraulic
force, the embankment experiences major deformation or failure. According to Khalilzad, et al.,
[25, 14], this damage level with the repeated rise and fall of water level can be defined as the
ultimate limit state (or LS3), and the maximum shear strain along the slip surface may be as high
as 0.05 at the location of the toe where axial extension stress path is induced. A scenario
representing LS3 was modeled in this study by reducing the strength parameters in Table 2.1 by a
factor of 1.87. A failure surface emerged in FE analysis as shown in Figure 2.6 and the associated
maximum shear strain was 0.051 at the toe. The corresponding FS was 0.98 from SRM method
and the levee experienced major deformations or instability. Such results are consistent with the
findings of Khalilzad, et al., [25, 14] (i.e., the maximum shear strain is greater than 0.05). It is
important to note that the failure potential slip surface shown in Figure 2.5(b) is obtained from the
SRM method, whereas Figure 2.6 shows the shear band which actually forms when the reduced
strength parameters are used as inputs.
2.5.2 Effect of storm cycles on stability
Two locations within the levee profile were considered to determine the influence of the loading
conditions on the levee performance. These key locations facilitate the assessment of basal stability
and are at the clay blanket toe (element A) and embankment toe (element B) as marked on Figure
20
2.1. The advantage of using these two locations is that they can be readily surveyed on regular
basis and are accessible to satellite imaging. They are located at a zone where the major principal
stress at failure is usually horizontal and the shear surface is inclined with respect to horizontal
plane [49]. Figure 2.8 shows the stress path during a storm cycle at element ‘A’ and element ‘B’.
The fluctuations of water level due to a storm cycle causes changes in riverside embankment face
boundary loading. For example, the rise of water level introduces higher external loading on the
upstream boundary and also increases the head driving seepage through the embankment [10]. As
a result, the levee experiences lateral deformation during the flood loading. Results indicate that
the stiff clay blanket is displaced by 4 mm horizontally during the rising phase and causing
deviatoric stress near element ‘A’ to increase due to lateral compression (Figure 2.8). Similarly,
the drawdown phase decreases the total stress since the head driving seepage flow is reduced. As
a result, the deviatoric stress decreased at element ‘A’ during drawdown phase. On the other hand,
the increase in pore pressure due to the seepage flow during the rise of water level reduces the
effective vertical stress at element ‘B’. Thus, the deviatoric stress decreased at element ‘B’ due to
axial extension during the flood loading (Figure 2.8). The cyclic loading from the river water
causes softening of the soil strength at these elements, and deviatoric stress due to lateral
compression or axial extension resulted in shear deformations. Therefore, these are potential
monitoring locations for the potential initiation of cascading failure.
Figure 2.7(a) shows the variation of the accumulated plastic shear strain with the number of storm
cycles at these two locations. The blanket toe (element A) yields greater shear strain compared to
element B. Thus, the element ‘A’ was considered as a key location for stability analysis. Results
indicated that the shear strain values increased at element ‘A’ by a factor of 3.5 as loading
21
progressed from cycle 1 to 6 (Figure 2.7a). These results, however, emphasize the importance of
considering the cumulative effect of successive storm given the corresponding excessive
accumulation of plastic strain. Figure 2.7(b) presents the accumulation of shear strain during the
first storm cycle only. The flooding condition (rise phase) mainly contributed to the generation of
shear strain at element A as the shear stress increases on the downstream side during flooding.
Figure 2.7(b) was also utilized to check the adequacy of the selected mesh size. As mentioned
earlier, fine mesh with 6849 nodes was selected for this model. The results using a very fine mesh
with 7697 nodes are also presented in Figure 2.7 (b). Changing the mesh from fine to very fine
caused indiscernible difference in the magnitude of the shear strain at point A after the first storm
cycle. Therefore, the fined mesh is used.
Figure 2.9 (a,b,c) shows the development of shear strain zone after 1, 3, and 6 cycles of loading,
respectively, with the onset of shear band emanating from blanket toe appearing at the sixth loading
cycle (Figure 2.9c). For a better understanding of this phenomenon, ten local points were selected
along the slip surface as shown in Figure 2.5(b) and the corresponding shear strain was obtained
with storm cycles and presented in Figure 2.10. As the storm cycles are introduced, shear strain is
gradually expanding from the toe to the crest with shear band progressively forming. The slope, in
this case, is experiencing cascading instability with the increasing number of storm cycles. This is
one of the key advantages of strain-based analysis since it allows for assessing the possibility of
progressive failure under impending storms. The LEM is in principle inappropriate for dealing
with such a phenomenon since it does not provide any information regarding deformations or
strains. The nonuniform distribution of strains along the slip surface as shown in Figure 2.10 could
22
complicate the stability analysis using LEM for brittle materials (soils with strain-softening
behavior); like overconsolidated clays and shales. In these materials, it is not possible to mobilize
the peak strength simultaneously at all points along the failure surface due to the strength reduction
with increasing strain [55]. Literature revealed that the average shearing resistance at the time of
failure is less than the peak shearing resistance and greater than the residual shearing resistance
[56]. Since the progressive failure is a strain-dependent process, the strain-based approach can be
used reliably to analyze the failure process with storm cycles for brittle materials.
The results from the conventional limit equilibrium method are shown in Figure 2.11. The critical
slip surface for minimum factor of safety develops as the pore pressure and stresses in the levee
respond to the changes in loading on the riverside embankment face. For instance, the lowering of
water level reduces the total stress on the riverside face. The shear stress increases in the upstream
boundary, which might lead to forming a slip surface emanating from the upstream face. On the
other hand, the rise of water level causes the shear stress to increase in downstream face and the
failure surface occurs at landside or downstream face [10]. However, the FS presented in Figure
2.11 was determined for the same slip surface at landside for both phases of water level rise and
fall to obtain consisted values reflecting the effect of the cycles of loading. In this case, the FS
does not change with cycles of loading as there is no provision for changing the soil strength and
hydraulic properties for the successive loading cycles associated with strain level. It is important
to note that SLOPE/W program perform uncoupled transient seepage analysis along with LEM to
determine the factor of safety of slopes due to change in hydraulic boundary conditions. In
uncoupled transient seepage analysis, the change in pore pressure is induced due to the change in
hydraulic boundary conditions only. However, the LEM ignores the coupling of pore pressures to
changes in total stress which has dramatic effects on the calculated pore pressure response [9, 57].
23
Although U.S. Bureau of Reclamation Embankment Dam Design Standards [58] allows using
uncoupled transient seepage analysis for stability analysis following rapid drawdown, the
assumption of uncoupled behavior predicts pore pressure that is not accurate [10]. Thus, the FS
presented in Figure 2.11 for transient condition might questionable.
The FS is however affected by rate of rise/drawdown of the water level as shown in Figure 2.12.
The use of the common assumption of instantaneous drawdown leads to the minimum FS (1.57)
as presented in Figure 2.12. The dissipation of excess pore water pressure due to storm cycle
depends on the rate of water level rise/drawdown. Therefore, the consideration of instantaneous
drawdown, instead of more realistic rate based on storm hydrograph yields 15.3% lower minimum
FS in this case (minimum FS drops from 1.85 to 1.57 in Figure 2.12). Similar observations were
also noticed by Sun et al. [59]. The USACE presents the method described in EM 1110-2-1913 for
stability analyses under transient seepage conditions [32]. In this method, the transient seepage
condition is considered by assuming instantaneous drawdown in the upstream face which leads to
conservative results. While the estimation of lower FS might be viewed as a safe design approach,
such an approach might also lead to excessively conservative design.
It is important to note that the FS values reported in Figure 2.12 did not change with time for both
cases (instantaneous and slow drawdown rates) after 4 days indicating the complete dissipation of
generated excess pore water pressure after drawdown phase. Figure 2.13 shows the gradual
lowering of the phreatic surface versus instantaneous drawdown. A steady-state condition was
established after 4 days as the phreatic surface within the levee did not change after 4 days as
shown in Figure 2.13 (phreatic surface for 4 days and 11.5 days are essentially same). Thus, the
assumed time gap (drained plastic phase) of 11.5 days between two storm cycles is long enough
24
to ensure the establishment of a steady-state condition before introducing the next storm cycle.
2.5.3 Effect of small hydraulic loading cycles on shear strain
The levee may experience many smaller hydraulic loading cycles due to tidal or seasonal variations
compared to the storm cycles considered in this analysis. Figure 2.14(a) shows that five smaller
hydraulic loading cycles, scaled-down by a factor of 0.5 from the storm cycle shown in Figure
2.7b, are simulated after the first storm cycle to investigate the effect of smaller cycles on the
accumulated shear strain. The shear strain is increased by 4.76 % due to the application of single
smaller hydraulic loading cycle (Figure 2.14b). This value, along with other values corresponding
to different scale factors, has been plotted in Figure 2.15. The accumulated plastic shear strain
increases with the higher loading cycle (i.e., higher scale factor in Figure 2.15) as the head driving
seepage through the embankment increases due to higher head differential associated with a higher
loading cycle. Loading cycles with a scale factor smaller than 0.3 has no or slight effect on shear
strain. The presence of repeated mini-cycles due to tidal variations could be a concern from the
erosion perspective [60]. However, Figure 2.14a indicates that the increase in shear strain
gradually decreases with the number of small hydraulic loading cycles (scale factor of 0.5), and it
becomes nearly 0% after the fifth cycle. Compared to the storm cycles, the effect of repeated mini-
cycles is slight to the accumulation of shear strain that causes progressive instability.
2.5.4 Exceedance assessment
Since the uncertainty involves on the determination of soil properties, the stability analysis based
on deterministic approach does not always ensure the safety or cost-effectiveness [61]. As such,
an approach similar to Duncan [30] was utilized here for reliability analysis against slope
instability. The unit weight and the strength parameters of soil are usually assumed as random
25
variables in probabilistic analysis using factor of safety approach [30]. Since the deformation of
levee due to cyclic loading also depends on the constitutive relation and the permeability of soil
[62], the stiffness parameters and hydraulic conductivities of soil could be considered as random
variables as well using strain-based approach. As such, sensitivity analyses were performed using
the PLAXIS program to identify the parameters most influencing the shear strain. The results are
presented in Table 2.2.
The “three-sigma rule” was used to estimate the standard deviation (SD) of normally distributed
random variables if they were not explicitly reported in USACE [32] as follow [30]:
𝑆𝐷 =𝐻𝐶𝑉 − 𝐿𝐶𝑉
6
(2.11)
Where HCV = highest conceivable value of the parameter and LCV = lowest conceivable value of
the parameter. For a lognormally distributed random variable, 𝑋, the SD was calculated as:
𝑆𝐷 = (𝑒2𝜇𝑙𝑛+𝜎𝑙𝑛2)(𝑒𝜎𝑙𝑛
2− 1) (2.12)
Where 𝜇𝑙𝑛 and 𝜎𝑙𝑛 is the mean and SD of ln (𝑋). Since the ln (𝑋) is normally distributed, the 𝜇𝑙𝑛
and 𝜎𝑙𝑛 was estimated as follow:
𝜇𝑙𝑛 = ln(𝐻𝐶𝑉) + ln(𝐿𝐶𝑉)
2
(2.13)
𝜎𝑙𝑛 = ln(𝐻𝐶𝑉) − ln(𝐿𝐶𝑉)
6
(2.14)
While calculating SD using the three-sigma rule, the range of values between HCV and LCV was
assumed as wide as possible to include both aleatory (or natural) and epistemic (or parameter and
model) uncertainty [30, 63]. A probability distribution was assumed for each parameter from the
literature and summarized in Table 2.2. Most of the soil parameters were assumed to follow normal
distribution, however, with exception to shear strength of clay [64, 65] and hydraulic conductivity
26
of soil which varies log-normally [66]. The mean values () of each parameter, as shown in Table
2.2 were taken from Table 2.1.
In this study, a sensitivity analysis was performed to determine input parameters that significantly
affect the output results (shear strain at blanket toe after first storm cycles in this case). The
sensitivity analysis was conducted in four steps. First, the sensitivity ratio (𝑆𝑅
), defined as the
percentage change in output divided by the percentage change in input for a specific input variable,
was estimated as follow [43, 67]:
𝑆𝑅= [𝑓(𝑋𝐿,𝐻) − 𝑓(𝑋)
𝑓(𝑋)] ∗ 100%
[𝑋𝐿,𝐻 − 𝑋𝑋 ] ∗ 100%
(2.15)
Where 𝑓(𝑋) is the reference value of the output or shear strain using the reference value of the
input variables (𝑋); 𝑓(𝑋𝐿,𝐻) is the value of the shear strain after changing the value of one input
variable from 𝑋 to 𝑋𝐿,𝐻. A total of 22 input variables, as shown in Table 2, including the soil
strength parameters (𝑐, ), stiffnesses (𝐸50, 𝐸𝑜𝑒𝑑, 𝐸𝑢𝑟), hydraulic conductivities (𝑘𝑥, 𝑘𝑧) and unit
weight () of SP, SC, and CL layer were considered in this study. Each input parameter was varied
across the range between +SD and -SD, whereas other parameters were deterministic. This
required a total 44 FE analyses for 22 input variables to obtain the values of 𝑓(𝑋𝐿,𝐻). In addition,
one FE analysis was conducted using the mean value () of all input variables in order to estimate
the 𝑓(𝑋). In the second step, the sensitivity score (𝑆𝑆
) which is the sensitivity ratio (𝑆𝑅
) weighted
by a normalized measure of the variability in an input variable was estimated as follow [43]:
𝑆𝑆=
𝑆𝑅∗(max𝑋 −minX)
𝑋 (2.16)
This normalization provided a unit independent estimation of sensitivity for each input variable.
27
The total sensitivity score (∑𝑆𝑆,𝑖
) of a given input variable was obtained by summing up the
sensitivity scores corresponding to +SD and -SD of that variable. In the third step, the relative
sensitivity () for each input variable was estimated using the following expression given by
Peschl [68]:
𝛼(𝑋𝑖) =∑
𝑆𝑆,𝑖
∑ ∑𝑆𝑆,𝑖
𝑁𝑖=1
∗ 100% (2.17)
Where 𝑁 is the total number of the input variable, which was 22 in this study. The relative
sensitivity indicates the percent contribution of a given input parameter towards the accumulation
of shear strain at the blanket toe. Thus, the higher the sensitivity score of a parameter, the greater
its influence on the shear strain. The sum of relative sensitivities of all input variables is 100%. In
the fourth step, a threshold value of 5% was used to select the ‘major’ variables from all input
variables. The parameters sensitivity analyses of the levee response, as presented in Table 2, show
that the unit weight (), stiffnesses (𝐸50, 𝐸𝑢𝑟) and hydraulic conductivities (𝑘𝑥, 𝑘𝑧) of SP and the
unit weight () of CL layer contribute significantly to the shear strain (i.e., their sensitivity scores
are higher than 5%). These properties were therefore considered as random variables during the
reliability analysis.
Sample calculations of shear strains (𝛾𝑠) from PLAXIS FE analysis and probabilities of exceeding
a limit state (LS), defined in terms of shear strain, are shown in Table 2.3 and Table 2.4,
respectively. The model was analyzed initially 4 times for each major variable up to 6 storm cycles
with the mean value () ± SD and ± 2SD of the variable. The mean value of shear strain (μs in
Table 2.4) was obtained from the finite-element analysis of the model using mean values of all
major variables ( in Table 2.3). The standard deviation (𝑆𝐷𝑠) and coefficient of variation (𝑉𝑠) of
28
shear strain (Table 2.4) were estimated using the formulas suggested by Duncan [30]. Then, the
reliability index (𝛽𝑙𝑛) was calculated which was used to estimate the probability of exceeding a
given LS. The original equation for reliability index in Duncan [30] yields the probability that the
factor of safety is smaller than 1.0. Since the focus herein is on computing the probability of
exceeding a given LS (1%, 3% and 5% shear strain for LS1, LS2, and LS3 respectively), the
reliability index was modified from Duncan [30] as follow:
𝛽𝑙𝑛 =
ln(𝐿𝑆) − 𝑙𝑛
(
𝑆𝐷𝑠
√1 + 𝑉𝑠2
)
√ln (1 + 𝑉𝑠2)
(2.18)
Where 𝛽𝑙𝑛 = lognormal reliability index; LS = performance-based limit states proposed by
Khalilzad and Gabr (2011), 𝑆𝐷𝑠= standard deviation of shear strain, and 𝑉𝑠 = coefficient of
variation of shear strain. The reliability value (𝑅) was based on normal distribution of the reliability
index. The reliability (𝑅) and the probability of exceeding each limit state, P (E.L.) were calculated
as follow [30]:
𝑅 = 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇(𝛽𝑙𝑛) (2.19)
𝑃(𝐸. 𝐿. ) = 1 − 𝑅 (2.20)
The probability of exceeding a given limit state was determined considering 1 and 2 standard
deviation (SD) and is presented in Figure 2.16 as a function of the number of loading cycle.
Approximately 68.2% of a given property values are within plus/minus one SD of the mean and
95.4% are within plus/minus two SD of the mean value. The FS obtained from the LEM and SRM
for the design flood scenario after each storm cycle is shown in Figure 2.16. Results indicated that
the probability of exceeding a given limit state is increased by 2 to 4 orders of magnitude,
depending on the degree of uncertainty (1 SD or 2 SD), as the number of storm cycles is increased
29
from 1 to 6. This increase is paralleled by the accumulation of shear strain after each storm cycle.
For example, considering 2 SD variability in material properties, the probability of exceeding LS3
is approximately increased from 10-8 after 1 storm cycle to 10-5 after 6 storm cycles (i.e.,
probability increased by 3 orders). The increase in the degree of uncertainty (1 SD to 2 SD) related
to material properties also leads to an increase in probability of exceeding a given limit state by 1
to 6 orders of magnitude, depending on the number of storm cycles. For example, the probability
of exceeding LS3 is approximately 10-10 after 6 storm cycles considering 1 SD variability in
material properties. This value is increased to 10-5 considering 2 SD variability in material
properties (i.e., probability increased by 5 orders). The FS remains as 1.87 from the LEM in this
case regardless of the number of storm cycles. The FS obtained from SRM also remains
approximately same (within 1.85 and 1.86) with storm cycles showing the inability to account load
history in this case.
2.5.5 Effect of hydraulic conductivity anisotropy on LS
Most of the natural soils are anisotropic with respect to hydraulic conductivity. The type of soil
and the nature of deposition controls the degree of anisotropy. The hydraulic anisotropy is
expressed as the ratio of hydraulic conductivity in the horizontal direction (𝑘𝑥) to that of vertical
direction (𝑘𝑧), i.e., 𝑘𝑥/𝑘𝑧. This ratio is similar for cohesive and cohesionless soil and is usually
less than 4 [69]. The analyses were conducted so far assuming the levee and foundation layers are
hydraulically isotropic or 𝑘𝑥/𝑘𝑧 = 1. To study the effect of anisotropy, the 𝑘𝑥 value is assumed
twice the 𝑘𝑧 for all the soil layers based on the suggested values in the USACE [32] from laboratory
permeability tests. The results are presented in Figure 2.17. The consideration of anisotropic
condition leads the shear strain to increase by 2 factors after 6 storm cycles (Figure 2.17a).
Consequently, the probability of exceeding a given LS is also increased by 2 to 5 orders of
magnitude compared to the isotropic condition, depending on the number of storm cycles and the
30
degree of uncertainty in material properties (Figure 2.16 and Figure 2.17b). For example, the
probability of exceeding LS3 is approximately 10-5 after 6 storm cycles for 2 SD variability in
material properties and for 𝑘𝑥/𝑘𝑧=1 (Figure 2.16). This value is increased to 10-2 when 𝑘𝑥/𝑘𝑧= 2
as shown in Figure 2.17(b) (i.e., probability increased by 3 orders). The flow rate at blanket toe
has also increased due to anisotropic condition Figure 2.17(c), causing more deformation to occur
at blanket toe. On the other hand, the FS drops from 1.87 to 1.82 for LEM (2.7% reduction) as the
degree of anisotropy is increased from 𝑘𝑥/𝑘𝑧=1 to 𝑘𝑥/𝑘𝑧= 2 and remains constant with storm
cycles. The FS obtained from SRM varies between 1.80 and 1.81 with storm cycle for 𝑘𝑥/𝑘𝑧= 2;
which was within 1.85 and 1.86 for 𝑘𝑥/𝑘𝑧= 1 (i.e., approximate 2.7% reduction).
The flood event related to Hurricane Floyd caused more than $6 million in property damage. The
Federal Emergency Management Agency (FEMA) allocated $26 million to the town to rebuild
after Floyd's floodwaters receded. Figure 2.18 shows the probability of exceeding LS3 for the
Princeville levee, plotted against risk criteria for traditional civil facilities as was presented by
Baecher and Christian [70]. A value for 𝑘𝑥/𝑘𝑧=2 and 2 SD variation in the soil properties was
assumed. The probability of exceedance was plotted against the property damage value ($6
million) as a consequence as was the case following Hurricane Floyd. Figure 2.18 shows the
probability of exceeding LS3 transitioned from ‘acceptable’ region after 1 storm cycle to the
‘unacceptable’ region after 6 cycles. Thus, using the strain-based approach the characterization of
the damage level and the associated probability of occurrence allow for forecasting the
consequences of future damage and therefore assist in informing decisions regarding rehabilitation
and retrofitting expenditures for mitigating future risk.
31
2.6 Conclusions
Strain-based limit state analyses and conventional factor of safety approaches were used to
investigate the effect of rise and fall of water levels, representing severe storm cycles, on the
stability of the Princeville levee. The effect of storm cycles on the probability of exceeding a
prescribed performance LS versus the FS computed using the LEM and SRM was presented in
this paper. This comparison revealed the need for analyzing the progressive failure under
impending storm. The importance of using strain-based stability analyses to account hydraulic load
history was demonstrated and discussed. Based on the results of this study, the following
conclusions are drawn:
The strain-based analyses results show a progressive development of plastic shear strain
within the levee as the number of storm cycles is increased. The shear strain is gradually
expanding form the toe to the crest with shear band progressively forming and causing
cascading instability with increasing number of storm cycles. In this case, the strain-based
approach reflects the damage levels based on the loading history and facilitates the
estimation of the increased level of instability risk for the next storm cycle.
The deterministic FS obtained from LEM remains unchanged and slightly changed for
SRM with increased number of storm cycles. The progressive instability of slope was not
explicitly expressed due to the disregard of induced plastic deformation after each storm
cycle. Therefore, the conventional factor of safety approach does not reflect the potential
probability of failure in the case of repeated hydraulic loading for Princeville levee.
32
The FS is affected by rate of rise/drawdown of the water level. The consideration of
instantaneous drawdown, instead of a more realistic rate based on storm hydrograph, yields
a lower minimum FS. However, the consideration of instantaneous drawdown may result
in an excessively conservative design.
The increase in number of storm cycles, the degree of uncertainty, and anisotropy
associated with material properties all lead to an increase in probability of exceeding a
given LS. While the deterministic FS is unaffected by the number of storm cycles and the
degree of uncertainty in material properties, the probability of exceeding a given LS is
increased by several orders of magnitude considering these two factors.
For a given consequence associated with a flood event, the increase in probability of failure
due to increased number of storm cycles led to the transition from acceptable to an
unacceptable risk, based on comparison with a published criteria. Thus, the strain-based
approach is best suited for the performance of risk assessment study.
33
Tables:
Table 2.1. Soil Properties.
Soil Parameters SC SP CL SM
, lb/ft3 (kN/m3) 125 (19.6) 120 (18.9) 132 (20.7) 125 (19.6)
𝑐′, lb/ft2 (kPa) 75 (3.6) 0 3000 (144) 0
𝜑′, degrees 30 28 0 33
𝐸50𝑟𝑒𝑓
, lb/ft2 (kPa) 8.50 x 105
(4.07 x 104)
3.13 x 105
(1.50 x 104)
7.50 x 104
(3.60 x 103)
3.13 x 105
(1.50 x 104)
𝐸𝑜𝑒𝑑𝑟𝑒𝑓
, lb/ft2 (kPa) 6.80 x 105
(3.25 x 104)
2.50 x 105
(1.20 x 104)
6.00 x 104
(2.88 x 103)
2.50 x 105
(1.20 x 104)
𝐸𝑢𝑟𝑟𝑒𝑓
, lb/ft2 (kPa) 2.55 x 106
(1.22 x 105)
9.39 x 105
(4.50 x 104)
2.25 x 105
(1.08 x 104)
9.39 x 105
(4.50 x 104)
k, ft/day (m/day) 5 (1.5) 60 (18.3) 2x10-3(6.1x10-4) 10 (3.0)
𝜃𝑟a 0.065 0.045 0.068 N/A
𝜃𝑠a 0.41 0.43 0.38 N/A
𝑔𝑎a, 1/ft (1/m) 2.29 (7.5) 4.42 (14.5) 0.24 (0.8) N/A
𝑔𝑛a 1.89 2.68 1.09 N/A
avan Genuchten parameters.
34
Table 2.2. Sensitivity analysis results showing the most influencing soil parameters on shear strain.
Soil
Type
Material
Parameter LCV HCV
Probability
Distribution
Std. dev.
(SD) +SD -SD
(%)
Reference
SP (kN/m3) - - Normal 1.10 18.85 19.95 17.75 19 [32], [71], [72]
SP E50 (kN/m2) 12.50 x 103 30.02 x 103 Normal 2.92 x 103 14.99 x 103 17.91 x 103 12.50 x 103* 14 [43], [44], [46]
CL (kN/m3) - - Normal 1.10 20.74 21.84 19.64 9 [32], [71], [72]
SP Eur (kN/m2) 37.50 x 103 90.01 x 103 Normal 8.76 x 103 44.96 x 103 53.63 x 103 37.5 x 103* 8 [43], [44], [46]
SP kx (m/day) 2.13 45.72 Log-normal 6.22 18.29 24.51 12.07 7 [73], [74], [75], [76]
SP kz (m/day) 2.13 45.72 Log-normal 6.22 18.29 24.51 12.07 6 [73], [74], [75], [76]
SC kz (m/day) 4.87 x 10-4 8.63 Log-normal 89.61 x 10-2 1.52 2.42 62.78 x 10-2 4 [73], [77]
SC kx (m/day) 4.87 x 10-4 8.63 Log-normal 89.61 x 10-2 1.52 2.42 62.78 x 10-2 4 [73], [77]
SP Eoed (kN/m2) 10.01 x 103 23.99 x 103 Normal 2.33 x 103 11.97 x 103 14.32 x 103 10.01 x 103* 4 [43], [44], [46]
SP ' (deg.) 27.00 41.00 Normal 2.33 28.00 30.33 27.00* 4 [73], [78], [79]
SC Eoed (kN/m2) 43.28 x 103 21.83 x 103 Normal 3.58 x 103 32.56 x 103 36.15 x 103 28.97 x 103 4 [43], [45]
SC Eur (kN/m2) 16.23 x 103 81.87 x 103 Normal 13.45 x 103 122.09 x 103 135.50 x 103 108.69 x 103 4 [43], [45]
CL Eur (kN/m2) 4.93 x 103 23.41 x 103 Normal 3.08 x 103 10.77 x 103 13.84 x 103 7.71 x 103 3 [43], [47]
SC (kN/m3) - - Normal 1.10 19.64 20.74 18.54 2 [32], [71], [72]
SC c (kN/m2) - - Log-normal 0.48 3.59 4.07 3.11 2 [30], [32], [71], [80]
CL kx (m/day) 4.35 x 10-5 4.33 x 10-3 Log-normal 5.18 x 10-4 6.07 x 10-4 1.13 x 10-3 9.02 x 10-5 2 [73]
CL E50 (kN/m2) 1.65 x 103 7.80 x 103 Normal 1.03 x 103 3.59 x 103 4.62 x 103 2.56 x 103 1 [43], [47]
CL kz (m/day) 4.35 x 10-5 4.33 x 10-3 Log-normal 5.18 x 10-4 6.07 x 10-4 1.13 x 10-3 9.02 x 10-5 1 [73]
SC E50 (kN/m2) 54.10 x 103 27.29 x 103 Normal 4.48 x 103 40.70 x 103 45.20 x 103 36.25 x 103 1 [45]
CL Su (kN/m2) - - Log-normal 19.15 143.64 162.79 124.49 1 [30], [32]
SC ’ (deg.) 30.00 40.00 Normal 1.67 30.00 31.67 30.00* 0 [73]
CL Eoed (kN/m2) 1.32 x 103 6.27 x 103 Normal 8.24 x 102 2.87 x 103 3.70 x 103 2.05 x 103 0 [47]
*LCV was used when MLV-SD is less than LCV Relative sensitivity, =100
35
Table 2.3. The shear strain corresponding to each major variable (after 4 storm cycles).
Soil Type Soil Parameter SD +/-SD γs Δγs
SP (kN/m3) 18.9 1.1
20 8.92 x 10-4
-1.05 x 10-3
17.8 1.94 x 10-3
SP E50 (kPa) 1.5 x 104 2.9 x 103
1.8 x 104 6.98 x 10-4
-9.85 x 10-4
1.3 x 104 1.68 x 10-3
SP Eur (kPa) 4.5 x 104 8.8 x 103
5.4 x 104 9.61 x 10-4
-5.40 x 10-4
3.8 x 104 1.50 x 10-3
SP k (m/day) 18.3 6.2
24.5 1.24 x 10-3
7.00 x 10-5
12.1 1.17 x 10-3
CL sat (kN/m3) 20.7 1.1
21.8 9.55 x 10-4
-5.29 x 10-4
19.6 1.48 x 10-3
Table 2.4. Calculating the probability of exceeding LSs (after 4 storm cycles) using joint variability
of major variables.
Parameter LS 1 LS 2 LS 3
Standard deviation (𝑆𝐷𝑠) 8.14 x 10-4 8.14 x 10-4 8.14 x 10-4
Mean (𝜇𝑠) 1.17 x 10-3 1.17 x 10-3 1.17 x 10-3
Coefficient of variation (𝑉𝑠) 0.70 0.70 0.70
Reliability index (𝛽𝑙𝑛) 3.73 5.48 6.29
Reliability, R= (𝛽𝑙𝑛) 0.9999042 0.9999999 0.9999999
Probability of exceeding a LS, P (E.L.) 9.58 x 10-5 2.13 x 10-8 1.56 x 10-10
36
Figures:
Figure 2.1. Princeville levee section (station 32+00): geometry and discretized mesh.
Figure 2.2. SWCCs for SC, SP and CL layers.
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Deg
ree
of
Sat
ura
tion (
Sr)
Suction (m)
SC
SP
CL
37
Figure 2.3. Flood stage hydrograph from Tarboro gage for 0.01 annual exceedance
probability [32].
Figure 2.4. Deformation and flow boundary conditions.
38
(a)
(b)
Figure 2.5. Potential slip surface in- (a) Limit equilibrium approach (SLOPE/W); (b) strength
reduction approach (PLAXIS).
39
Figure 2.6. Shear strained zone corresponding to factor of safety 0.98.
40
(a)
(b)
Figure 2.7. Shear strain increase at (a) element A and element B with storm cycles; (b) element A
during the first storm cycle (water elevation y-scale is on the right).
0.0000
0.0004
0.0008
0.0012
0.0016
0.0020
0 1 2 3 4 5 6 7
Sh
ear
stra
in a
t em
ban
km
ent
toe
Number of cycle
Element A
Element B
0
2
4
6
8
10
12
14
16
18
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 1 2 3 4 5 6 7 8
Wat
er e
levat
ion (
m)
Shea
r st
rain
at
elem
ent
A
Time (days)
Shear strain (7697 nodes)
Shear starin (6849 nodes)
Water elevation
Rise phase Drawdown phase
Peak phase
41
Figure 2.8. Stress paths during the first storm cycle at element A (top curve) and at element B
(bottom curve).
1.5
2.0
2.5
3.0
3.5
2.0 2.5 3.0 3.5
Dev
iato
ric
stre
ss, q
(kP
a)
Mean effective stress, p' (kPa)
Rise phase (element A)Peak phase (element A)Drawdown phase (element A)Rise phase (element B)Peak phase (element B)Drawdown phase (element B)
42
(a)
(b)
(c)
Figure 2.9. Expanding of shear strained zone with cycles of loading. (a) After 1 cycle, (b) after 3
cycles, (c) after 6 cycles.
43
Figure 2.10. Distribution of shear strain along the slip surface.
Figure 2.11. Factor of safety of Princeville levee using limit equilibrium method with cycles of
loading (water elevation y-scale is on the right).
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
1 2 3 4 5 6 7 8 9 10
Sh
ear
stra
in
Local points
1 cycle
2 cycles
3 cycles
4 cycles
5 cycles
6 cycles
0
2
4
6
8
10
12
14
16
18
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 10 20 30 40
Wat
er e
levat
ion (
m)
Fac
tor
of
safe
ty
Time (days)
FS
Water elevation
Rise phaseDrawdown phase
Plastic phase
Peak phase
44
Figure 2.12. Effect of drawdown rate on the factor of safety.
Figure 2.13. Gradual dropping of the phreatic surface after instantaneous drawdown.
1.0
1.5
2.0
2.5
3.0
0 2 4 6 8
Fac
tor
of
safe
ty
Time (days)
Instantaneous drawdown
Slow drawdown (0.6 m/day)
EL=14.6
m EL=12.2 m
0 day
4 days, 11.5 days
45
(a)
(b)
Figure 2.14. (a) Effect of small hydraulic loading cycles on shear strain at blanket toe; (b)
Increase in shear strain after the application of a small hydraulic loading cycle with a scale
factor = 0.5.
0
2
4
6
8
10
12
14
16
18
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
0 20 40 60 80 100 120
Wat
er e
levat
ion
(m
)
Dev
iato
ric
shea
r st
rain
at
bla
nket
to
e
Time (days)
Shear strain
Water level
0.0E+00
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
0 10 20 30
Dev
iato
ric
shea
r st
rain
at
bla
nket
toe
Time (days)
4.76%
Storm cycle
(scale factor=1.0) Small loading cycle
(scale factor=0.5)
46
Figure 2.15. Increase in shear strain with scale factor.
Figure 2.16. Variation of probability of exceeding limit state and factor of safety with number of
storm cycle.
0
20
40
60
80
100
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Incr
ease
in
sh
ear
stra
in (
%)
Scale factor
1.00
1.20
1.40
1.60
1.80
2.00
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
0 1 2 3 4 5 6 7 8
Fac
tor
of
safe
ty
Pro
bab
ilit
y o
f ex
ceed
ing l
imit
sta
te
Number of cycle
LS 1 (1 SD)
LS 1 (2 SD)
LS 2 (1 SD)
LS 2 (2 SD)
LS 3 (1 SD)
LS 3 (2 SD)
FS (LEM)
FS (SRM)
47
Figure 2.17. Effect of soil anisotropy with respect to hydraulic conductivity and storm cycles
on- (a) shear stain; (b) probability of exceeding limit states and factor of safety (for 𝑘𝑥/𝑘𝑧=2);
and (c) flow rate at blanket toe.
48
(a)
(b)
0.0000
0.0004
0.0008
0.0012
0.0016
0.0020
0.0024
0.0028
0.0032
0.0036
0 1 2 3 4 5 6 7
Sh
ear
stra
in a
t b
lan
ket
to
e
Number of cycle
Isotropic (kx=kz)
Anisotropic (kx=2kz)
1.00
1.20
1.40
1.60
1.80
2.00
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
0 1 2 3 4 5 6 7 8
Fac
tor
of
safe
ty
Pro
bab
lity
of
exce
edin
g l
imit
sta
te
Number of storm cycle
LS1 (1 SD)
LS1 (2 SD)
LS2 (1 SD)
LS2 (2 SD)
LS3 (1 SD)
LS3 (2 SD)
FS (LEM)
FS (SRM)
49
(c)
0
2
4
6
8
10
12
14
16
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
Wat
er e
levat
ion
(m
)
Flo
w r
ate
at b
lan
ket
to
e (m
/day
)
Time (days)
Isotropic (kx=kz)
Anisotropic (kx=2kz)
Rise phase
Peak phase
Drawdown phase
50
Figure 2.18. Probability of exceeding LS3 for 2 SD and 𝑘𝑥/𝑘𝑧=2 versus consequence curve
showing the effect of load history on risk evaluation associated with slope failure.
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
1.0E+4 1.0E+5 1.0E+6 1.0E+7 1.0E+8 1.0E+9 1.0E+10
Pro
bab
ilit
y o
f ex
ceed
ing L
S3
Consequence($ Lost)
Probability of exceeding LS3
Acceptable (Baecher and Christian 2003)
51
3 CHAPER 3. ANALYSIS OF EARTHEN EMBANKMENTS USING STRAIN-BASED
PERFORMANCE LIMIT STATE APPROACH
Some of the contents of this chapter have been published in the proceedings of Dam Safety
National Conference.
Citation:
Jadid, R., Montoya, B. M., & Gabr, M. A. (2020). Strain-based approach for stability analysis of
earthen embankments. Proceedings from Dam Safety 2020, Association of State Dam Safety
Officials (ASDSO).
52
Abstract
Repeated rapid drawdown (RDD) and rapid rise in water level during extreme events lead to
progressive development of plastic shear strain zones within the earth embankments with subtle,
rather than obvious, visible signs of distress. The traditional analysis approach within the
framework of limit equilibrium method does not account for the accumulated permanent
deformation with repeated hydraulic loading. The research presented herein is focused on the
quantification of internal distress level in terms of level of shear strain. A simple linear relationship
between the shear strain and monitorable deformation at the toe of the embankment is developed
as a function of the geometry of the slope. This relationship provides a simple means to estimate
the performance limit state that corresponds to the instability of embankment slopes, and the
critical shear strain at the embankment toe, using the stress-strain data obtained from triaxial
testing. Results from the parametric study using numerical analyses show a good agreement with
the proposed analytical criterion. The proposed criterion is also compared with data from the field
studies by others and reasonable good agreement is obtained.
53
3.1 Introduction
Earth embankment dams and levees are critical infrastructure related to flood protection and water
supply management. These earth structures experience relatively rapid increase and decrease in
water elevation during flood events due to extreme precipitations associated with hurricanes or
wet seasons. In addition, rapid decrease in water level occurs due to excessive use of water supply
from reservoirs during drought. Repeated occurrence of such events may lead to breaching failure
as strain softening of the earth materials occurs. For example, San Luis Dam (now known as B.F.
Sisk Dam) experienced an upstream slide in 1981, 14 years after construction, during the eighth
cycle of drawdown of the reservoir [81]. The cyclic hydraulic loading from the reservoir water
level resulted in shear deformation level that was sufficient to mobilize shear strength between
fully softened and residual values. Consequently, the shear strength of the earth dam material was
significantly reduced leading to the slope failure [5]. A similar observation was noted for the
Canelles dam in Spain which experienced several cycles of drawdown before the embankment
slope failure in 2006. The significant decrease of annual rainfall in the period 2005-2006 combined
with high irrigation demand in the spring and summer resulted in a considerable reduction of water
level at a high drawdown velocity (1.2 m/day) [6]. The reservoir hydrograph indicated that the
dam experienced a similar scale of drawdown (larger and faster) in the period of 1990-1991 as
well, but no records of landslide activity are reported in the literature.
The cycles of rising and falling water levels may cause the progressive development of plastic
shear strain zones within the earth embankments with subtle, rather than obvious, visible signs of
distress on the surface [8]. While the quantification of such an internal distress level is essential to
assess the health condition of the embankments and manage the need for rehabilitation , the
54
conventional slope stability approach (e.g., limit equilibrium method) provides no means to
account for such effect [8]. On the other hand, the monitoring of the surface movements is readily
doable on a large scale and the use of such deformation has the potential to indicate the gradual
degradation of the structure’s stability with cycles of hydraulic loading. There is however a need
for the interpretation of such measured quantities in the context of induced shear strain and
comparing the estimated shear strain to established performance limit states.
The primary objective of this study is to define a strain-based performance limit state that
corresponds to the instability of embankment slopes and to develop an approach to quantify it in
terms of deformation level that can be readily monitored in the field through periodic surveying.
The focus of the approach is on mechanisms causing the progressive instability due to the effect
of repeated rise and fall of water level. The stability and deformational response of an embankment
slope is first investigated using numerical analyses and a performance limit state is defined. A
correlation between the shear strain and the corresponding deformation at toe of the embankment
is developed. The robustness of the proposed approach is then assessed through comparison with
data from literature. The use of the proposed correlation to estimate the performance limit state is
presented and discussed.
3.2 Background
3.2.1 Monitoring and limit state approach
The monitoring of slope movements has become common practice in important and critical
earthwork projects [12]. Such monitoring provides data for assessing the health, predicting the
failure and implementing countermeasures. In practice, the measured surface displacement with
55
time or deformation rate (referred to as velocity) for a given slope is used to compare the
empirically-defined critical threshold value. This empirical approach based on surface monitoring
is discussed by Saito [82], Salt [83], Voight [84], Fukuzono [85], Bhandari [86], Federico et al.
[87], Crosta and Agliardi [88], and Hungr et al. [12], among others. Such an empirical approach
is phenomenologically-based and emphasizes the overall performance of the structural system
while disregarding the underlying mechanisms of failure. Thus, defining the critical thresholds, or
performance limits, involves subjectivity [12]. This limitation has led to failed predictions
including the prediction of a landslide failure near Innertkirchen in the Swiss Alps in 2001 [13].
In parallel, numerical approaches provide mechanistic means to study the deformational instability
of a slope in terms of emerging failure modes and the associated deformation of a slope. Therefore,
numerical approaches may provide a rational basis to define performance limits that can be used
to compare with field measurements. To establish a correlation between the field observation and
model-aided approach, Khalizad and Gabr [24] and Khalilzad et al. [89, 25] proposed a
deformation-based limit state (LS) approach for embankment dams and incorporated these into a
probabilistic analysis, following an approach introduced by Duncan [30]. In this approach, the
performance LSs are defined in terms of deformation at the toe where axial extension stress
condition is common and the initiation of basal instability of the embankment occurs. They
defined the damage level associated with each LS as follows- LS1: minor deformations, no
discernible shear zones, low hydraulic gradients (i.e., i < 1) throughout the embankment dam and
foundation; LS2: medium (repairable) deformations, limited piping problems (i.e., i > 0.67 within
a shallow depth at the location of toe), dispersed plastic zones with moderate strain values,
tolerable hydraulic gradients less than critical; LS3: major deformations, breaches and critical
56
hydraulic gradients at key locations (i.e., i > 1, boiling and fine material washing at the location of
toe), high strain plastic zones. However, this approach provides qualitative definitions of limit
states and requires sophisticated numerical analysis to determine LSs for a specific case.
3.2.2 Transient seepage analysis
The fluctuation of water level due to rapid drawdown or flood loading changes the boundary
stresses and hydraulic boundary conditions with time. As a result, both stress-induced pore
pressure due to the change in boundary loads and flow-induced pore pressure due to transient flow
develop simultaneously within an embankment slope. There are three methods available in the
literature to predict the pore pressure response due to water level changes [10]. The first procedure
is Bishop’s [90] method in which the change in pore water pressure is assumed equal to the change
in total vertical stress resulting from the change in water elevation above the point in consideration.
However, Barrett and Moore [91] using results from numerical analyses showed that the change
in pore pressure is 0.7 to 0.9 times the change in vertical stress caused by lowered water level.
Both Bishop’s and Barrett & Moore’s approaches usually overestimate (conservative) the pore
pressure after rapid drawdown [10]. The second procedure is an “uncoupled” analysis in which
pore pressure response is assumed uncoupled from the change in boundary total stresses. Stated
differently, the stress-induced pore pressure component is neglected in the analysis, otherwise the
constitutive (stress-strain) relation of soil would be required. The “coupled” analysis (third
procedure) best represents the in-situ condition in the form of joint consideration of transient
seepage flow and stress deformation analysis. However, the coupled analysis is relatively
complicated because of the need to solve the governing equations of transient flow and
deformation simultaneously and requires extensive input parameters for advanced constitutive
relation. The difference between coupled and uncoupled analysis can be explained by examining
57
the governing equation (Eq. 3.1) for two-dimensional transient flow through an isotropic porous
medium. In an uncoupled analysis, Eq. 3.1 is solved by assuming the void ratio of soil is constant
which drops the second term of the right-hand side of the equation.
∇[𝜌𝑤(𝐾∇ℎ)] = 𝜌𝑤𝑛𝛿𝑆
𝛿𝑡+ 𝜌𝑤𝑆
𝛿𝑛
𝛿𝑡 (3.1)
Where, ∇= gradient operator; 𝜌𝑤= unit weight of water; 𝐾 = hydraulic conductivity; h= total
hydraulic head; n = porosity; t = time; and S = saturation.
3.3 Numerical Model
3.3.1 Domain discretization and properties
A simple earth embankment slope with 3H: 1V inclination and 3m height is modeled using a two-
dimensional finite element software PLAXIS 2D 2018. The analysis is performed using plane
strain 15-nodes triangular elements. The geometry and the discretized mesh of the model are
shown in Figure 3.1. The fine mesh is observed to be optimum size from the mesh sensitivity
analysis and is used herein with a domain having 9,657 nodes and 1,174 elements. The dimensions
of the model have been carefully chosen to minimize boundary effect (i.e., further increase of
model size does not change the primary results output). The deformation boundary conditions
included restriction of horizontal deformation on the left and right edges of the model as well as
restriction of horizontal and vertical deformations at the bottom boundary. The crest and the slope
surface of the model are maintained unconstrained during the analysis. The flow boundary
conditions included impervious boundaries at the bottom and left side of the model. A constant
head boundary that is equal to the total hydraulic head is applied at the right side of the model.
Time-dependent total head boundaries (transient seepage analysis) are applied on the slope surface
to simulate repeated rise and fall of water level.
58
3.3.2 Modeling steps
Modeling steps included first generating the geostatic stress state in the domain with the initial
water level assumed at an elevation (EL) of 0.0 m (at toe level), as shown in Figure 3.1. Then, the
boundary water pressure is gradually applied to the upstream slope by raising the water level in
several stages, allowing a steady-state condition to occur, until water level reaches the crest level
(EL = 3.0 m). The slope is subjected to repeated drawdown and water level rise cycles to
investigate the effect of hydraulic loading on the deformational response of the slope. To simulate
a drawdown cycle, first, the water is lowered at a rate of 0.5 m/day to the elevation 0.0 m. Then,
the water level is assumed to be at that elevation for four days, based on the reservoir hydrograph
from literature [6], to represent the lower water level condition. After that, the water level is raised
at a rate of 0.5 m/day to the elevation of 3.0 m and followed by a plastic phase of 3000 days. There
is no significance to the 3000 days’ time period other than ensuring that a steady-state condition
has been established before applying the next drawdown cycle (i.e., the duration of time between
each storm cycle can be any time period that ensures the end of consolidation). Cycles of water
level fall and rise are applied until the failure occurs, as indicated by excessive shear strain.
3.3.3 Coupled transient seepage analysis
The coupled transient seepage analysis is performed in PLAXIS to account for both flow-induced
and stress-induced pore pressure response of soil with the change in water level. With time, the
stress-induced pore pressure also dissipates with volume change occuring. PLAXIS utilizes Biot’s
theory of consolidation coupled with constitutive relation to determine dissipated pore pressure
and associated deformation at any stage of drawdown cycle. The accuracy of stress-induced pore
pressure prediction depends on how well the constitutive (stress-strain) relation of soil accounts
59
the pore pressure response. The advanced non-linear elasto-plastic hardening soil (HS) model was
used in this study to simulate the stress-induced pore pressure as accurately as possible. The HS
model can simulate both soft and stiff soils and approximates non-linear stress-strain behavior with
a hyperbola similar to the hyperbolic model by Kondner [40] and Duncan and Chang [41].
However, HS model supersedes hyperbolic model by using the theory of plasticity rather than
theory of elasticity, and accounts for soil dilatancy by introducing a yield cap. The yield surface
of a HS model is not fixed in principal stress space; rather it expands due to plastic straining. The
stress-strain behavior of soil shows a decreasing stiffness and simultaneously irreversible plastic
strains when subjected to primary deviatoric loading [42, 43].
3.3.4 Hardening soil (HS) model
In HS model, three stiffness input parameters (𝐸50𝑟𝑒𝑓
, 𝐸𝑜𝑒𝑑𝑟𝑒𝑓
and 𝐸𝑢𝑟𝑟𝑒𝑓
) are used to model the soil
behavior along with the strength parameters, angle of internal friction (𝜑) and the cohesion
intercept (𝑐). The shear behavior of soil is controlled by the reference stiffness modulus, 𝐸50𝑟𝑒𝑓
;
whereas the volumetric behavior is controlled by the reference oedometer modulus, 𝐸𝑜𝑒𝑑𝑟𝑒𝑓
; and the
unloading-reloading characteristics is modelled by using the reference loading-unloading stiffness
modulus, 𝐸𝑢𝑟𝑟𝑒𝑓
[9]. The amount of stress dependency is given by the power coefficient ‘m’ as
follow:
𝐸 = 𝐸𝑟𝑒𝑓 (𝑐 cos𝜑 − 𝜎𝑖 sin𝜑
𝑐 cos𝜑 + 𝑝𝑟𝑒𝑓 sin𝜑)𝑚
(3.2)
Where, 𝐸 = stress dependent moduli (𝐸50, 𝐸𝑜𝑒𝑑 and 𝐸𝑢𝑟) corresponding to the reference stiffness
𝐸𝑟𝑒𝑓(𝐸50𝑟𝑒𝑓
, 𝐸𝑜𝑑𝑒𝑟𝑒𝑓
and 𝐸𝑢𝑟𝑟𝑒𝑓
), 𝑝𝑟𝑒𝑓= reference stress of the stiffness, 𝜎𝑖= minor effective principal
stress for 𝐸50 and 𝐸𝑢𝑟, and major principal stress for 𝐸𝑜𝑒𝑑.
60
3.3.5 Material properties
The soil parameters for the HS model are presented in Table 3.1. These parameters represent soft
Bangkok clay as Surarak et al. [42] showed that the HS model can well approximate the stress-
strain relationship and the pore pressure response of soft Bangkok clay under shearing. The soil-
water characteristic curves (SWCCs) are assigned to the model domain above the initial water
level based on the soil gradation with the pertinent van Genuchten parameters presented in Table
3.1.
3.3.6 Stability analysis
The stability analysis is numerically performed using the strength reduction method (SRM) in
PLAXIS. In SRM, the factor of safety (FS) is defined as the factor by which strength parameters
(𝑐′and tan𝜑′) are reduced in order to reach slope failure as shown in Eq. 3.3 [92, 93].
FS =tan𝜑𝑖𝑛𝑝𝑢𝑡
′
tan𝜑𝑟𝑒𝑑𝑢𝑐𝑒𝑑′ =
𝑐𝑖𝑛𝑝𝑢𝑡,
𝑐𝑟𝑒𝑑𝑢𝑐𝑒𝑑, (3.3)
To verify the FS obtained from the SRM, the FS is also calculated using the computed principal
stresses from finite element method (FEM). The major and minor principal stresses at any given
point along the potential slip surface are utilized for the calculation of maximum available shear
strength and mobilized shear stress. The FS is then calculated by dividing the total maximum
available shearing resistance by the total amount of mobilized shear stress along the slip surface.
3.4 Results and Discussion
3.4.1 Verification of pore pressure prediction
The accuracy of the prediction of stability factor of safety and the associated deformation under
transient seepage condition largely depends on the accuracy of the predicted pore water pressure
61
in the model. VandenBerge et al. [10] recommended that the Bishop’s or Barrett and Moore’s
method provide an approximate upper bound of pore water pressures after rapid drawdown; hence
should be used to verify the more complicated numerical analysis. For this purpose, the predicted
pore water pressure from coupled transient seepage analysis at several locations along the slip
surface (points: 2,4,6 and 8 in Figure 3.2) immediately after rapid drawdown (EL 3.0 m to 0.0 m
in Figure 3.1) is presented in Figure 3.3 along with the predicted pore pressure from Bishop’s and
Barrett and Moore’s method. The detailed sample calculations are presented in Appendix A.
Results indicated that the coupled transient seepage analysis predicted approximately equal pore
pressure at point 2 and smaller pore pressures at points 4, 6, and 8 compared to the upper bound
set by Bishop’s method, and they were smaller at all points compared to the and Barrett and
Moore’s method. For comparison, the predicted pore pressure from the uncoupled analysis is also
presented in Figure 3.3. The phreatic surface within the slope essentially remained at the initial
level (near crest) after rapid drawdown which is attributable to the very low hydraulic conductivity
of soil. Therefore, the predicted pore pressure from the uncoupled analysis is same as the initial
pore pressure for this case since the uncoupled analysis does not model the pore pressure response
to the change in confining stress.
3.4.2 Stability analysis for repeated drawdown cycle
The slope of an embankment might experience several cycles of rise and fall of water level (WL)
over its service life. To investigate the effect of hydraulic loading history, the slope shown in
Figure 3.1 is subjected to several cycles of drawdown. As mentioned earlier, a drawdown cycle is
composed of a drawdown phase (lowering WL from EL 3.0 m to 0.0 m at a rate of 0.5 m/day), a
plastic phase (WL at EL 0.0 m for 4.0 days), and a rise phase (raising WL from EL 0.0 m to 3.0 m
at a rate of 0.5 m/day). The stability factor of safety is calculated after each drawdown phase using
62
the SRM with the results presented in Figure 3.4. For verification, the factor of safety was also
obtained from the principal stresses as shown in Figure 3.4 (sample calculation is presented in
Appendix B). The two methods predict similar results and the maximum difference is
approximately 2% for the second drawdown phase (Figure 3.4). Data in Figure 3.4 indicate that as
the number of cycles is increased, the factor of safety gradually reduced from 1.09 after first
drawdown phase to approximately 0.99 after fifth drawdown phase. Once the factor of safety
reached unity at fifth cycle, the shear strain and the movement of the slope at toe increased rapidly
as shown in Figure 3.5 and led to instability failure of the slope. Each cycle results in accumulated
plastic shear strain and displacement at a point (dist.=-0.42 m and EL=-0.35 m in Figure 3.1) near
toe. For example, the accumulated plastic shear strain has increased from 0.10 at toe after the first
cycle to approximately 0.18 between fourth and fifth cycle, and after fifth cycle, strain increases
rapidly (Figure 3.5). This magnitude of accumulated shear strain and the associated displacement
causes the effective confining stress to decrease with drawdown cycles and is sufficient to mobilize
marginal stability shearing resistance at fifth cycle (Figure 3.6).
3.5 Correlation between the shear strain and displacement
Figure 3.7 shows that the slope has experienced some rotational movements when it is subjected
to drawdown cycles. In continuum mechanics, the rotation of a rigid body is quantified by the
engineering shear strain, which is the ratio of transverse displacement and the perpendicular
distance. The deformation pattern of the soil at the toe over a relatively short distance is
approximated as “rigid.” Based on this analogy, the deformed shape of a triangular element at toe
is simplified as shown in Figure 3.7(b) and the magnitude of the displacement (𝑢) may be
correlated with the mobilized shear strain (𝛾𝑠) at toe as follow:
63
𝑢
𝑟= 𝜃 ∝ 𝛾𝑠 (3.4)
Where 𝑟 = slope length of the element; 𝜃 = the magnitude of the slope rotation. The slope length
(𝑟) in Figure 3.7(b) can be determined as:
𝑟 =ℎ
𝑠𝑖𝑛𝛽 (3.5)
Where ℎ = height of the element, =inclination of the slope. Substitution of Eq. (3.5) into
Eq. (3.4) and rearranging:
𝑢𝑠𝑖𝑛𝛽ℎ⁄
𝛾𝑠= 𝐶′ (3.6)
Where 𝐶′ is a proportional constant.
If the height of the potential slip surface (𝐻𝑠), as shown in Figure 3.2, is 𝑚 times higher than ℎ,
then 𝐻𝑠 can be expressed as:
𝐻𝑠 = 𝑚ℎ (3.7)
Substitution of Eq. (3.7) into Eq. (3.6) gives:
𝑢𝑠𝑖𝑛𝛽𝐻𝑠⁄
𝛾𝑠= 𝐶 (3.8)
Where 𝐶 = 𝐶′
𝑚⁄
To determine the magnitude of 𝐶, a parametric study is numerically performed by varying the
effective friction angle (𝜑′) of soil as 25, 27, 30and 33 (where 𝜑 = 27 is assumed as the base
case). The safety factor after first drawdown phase and the required number of drawdown cycles
that initiates rapid slope movement for each friction angle are presented in Table 3.2. For a given
value of 𝜑′, the accumulated shear strain and the associated horizontal displacement after each
64
drawdown cycle at toe are obtained from the model and plotted in Figure 3.8. For instance, total
number of data points corresponding to 𝜑′ = 27 is five since the required number of drawdown
cycles to induce instability failure is five. Figure 3.8 shows a good linear relationship between the
shear strain and the displacement at toe for different effective friction angle with a R-squared value
of 0.962. The magnitude of 𝐶 is found to be 0.155.
3.5.1 Effect of change in soil properties with drawdown cycles
In the previous analyses, only the change in slope geometry due to the application of a drawdown
cycle is considered as an initial condition for the next drawdown cycle. However, the soil
parameters might also change along with the geometry due to the accumulation of shear strain. In
addition to that, many soils exhibit time-dependent strength and stiffness properties due to
weathering, creep, or consolidation phenomenon. The void ratio along the slip surface does not
change here with cycles of loading due to undrained shearing. At undrained condition, the change
in volumetric strain (𝜖𝑣) is zero. Thus, the change in soil properties due to the change in void
ratio is not expected in this case. Time-dependent strength and stiffness change are considered here
to check the validity of the developed correlation. The angle of internal friction (′) of Bangkok
clay increases from 27 to approximately 29 due to weathering [42, 94]. The reference stiffness
parameter (𝐸50𝑟𝑒𝑓
) also increases from around 1000 kPa for normally consolidated clay to around
3000 kPa for overconsolidated clay [47]. Based on this data, ′ and 𝐸50𝑟𝑒𝑓
are assumed to increase
0.5 and 500 kPa, respectively after each drawdown cycle; so that after the fifth cycle, ′ is
increased from 27 to 28.5 and 𝐸50𝑟𝑒𝑓
is increased from 800 kPa to 2800 kPa. As shown earlier,
instability occurred after the fifth cycle if the soil parameters are not changed with drawdown
cycles (i.e., ′ and 𝐸50𝑟𝑒𝑓
were constant throughout simulation). Figure 3.9(a) shows that instability
65
does not occur after fifth cycle due to the increase of ′. To investigate the effect of reduction of
soil strength on the deformational response, ′ is also decreased by 0.5 after each cycle, which
induces instability after third cycles (Figure 3.9b). The stiffness parameter is not changed in the
latter case to avoid numerical convergence issues due to excessive deformation. The accumulated
shear strain and the associated horizontal displacement after each drawdown cycle at toe are
obtained from the model and plotted in Figure 3.10. Figure 3.10 shows a good linear relationship
between the shear strain and the displacement at toe where strength and/or stiffness parameters
change with drawdown cycles. The magnitude of 𝐶 is found to be 0.148 with a R-squared value
of 0.94. Figure 3.5 and Figure 3.9 indicate that horizontal deformation and shear strain changes in
equal proportion in response to the change in soil properties. Thus, their normalized parameter, 𝐶
does not get affected by the change in their magnitudes as long as the underlying assumption for
developing the correlation, such as the rotational movement of the slope, holds true.
3.5.2 Effect of hydraulic conductivity of soil on developed correlation
The previous analyses were conducted assuming the magnitude of hydraulic conductivity of soil
(𝑘) as 10−9cm/s, which represents an approximately lower limit value of 𝑘 for Bangkok clay [95].
To investigate the effect of 𝑘 on the developed correlation, analyses are performed assuming 𝑘 as
10−6 cm/s, 10−5 cm/s, and 10−4 cm/s and results are presented in Figure 3.11. According to
Casagrande [96], a 𝑘 of 10−9cm/s is approximately lower limit of permeability of soils, 10−6 cm/s
is the approximate boundary between poor drained and practically impervious soil; whereas 10−4
cm/s is the approximate boundary between pervious and poorly drained soils [97]. Figure 3.11(a)
and (b) indicates a good linear correlation between the shear strain and the corresponding
horizontal displacement with R2 = 0.970 and 0.883 for 𝑘 = 10−6 cm/s and 𝑘 = 10−5 cm/s,
66
respectively. On the other hand, a poor correlation is observed for 𝑘 = 10−4 cm/s with R2 = 0.335
(Figure 3.11c) as the shear band associated with 𝑘 = 10−4 cm/s is not explicitly circular (Figure
3.12a). A local shear strained zone is observed near the crest which triggers relatively complicated
movement pattern compared to the rotational movement associated with classical circular slip
surface (Figure 3.12a). Conversely, the shear band associated with 𝑘 = 10−6 cm/s indicates a
circular shear band (Figure 3.12b), which concurs to the underlying assumption (i.e., slope
experiences rotational movement) for the developed correlation. Therefore, 𝐶 value of 0.150 for
𝑘 = 10−6 cm/s is very close to 0.155 for 𝑘 = 10−9cm/s (Figure 3.8 and Figure 3.11a). 𝐶 value of
0.210 for 𝑘 = 10−5 cm/s is also comparable with the 𝐶 values for 10−6 cm/s and 10−9 cm/s.
3.5.3 Defining critical shear strain
Eq. 3.8 represents a simple approach to predict the surface displacement corresponding to a given
shear strain at toe. If the mobilized shear strain at toe at the initiation of rapid movement is defined,
then the accumulated shear displacement at the toe, which can readily be monitored through
periodic surveying, indicating the initiation of instability can be predicted. The magnitude of the
mobilized shear strain at any given point depends on its location along the slip surface. Figure 3.13
(a, b) shows the development of plastic points after four and five drawdown cycles, respectively,
for 𝜑′ = 27, with the formation of potential slip surface after fifth drawdown cycle. Only the
vicinity of toe experienced plastic deformation after four drawdown cycles as the toe is subjected
to an axial extension stress path where the shear strength is expected to be relatively low [8].
Previous studies also showed that the toe is the potential location for the initiation of cascading
failure, e.g. [8].
The average initial mean confining stress at toe is approximately 6 kPa, as obtained from the
67
model, and is used to simulate the response of isotropic consolidated undrained (ICU) triaxial tests
for the various magnitudes of friction angle utilized herein. Figure 3.14(a) shows the simulated
stress-strain curves which indicate that the peak deviatoric stress is achieved when the mobilized
shear strain value is within the range of 0.13 to 0.15 for the friction angles used herein. The ICU
triaxial tests by Surarak et al. [42] for 𝜑 = 27 indicates that the peak deviatoric stress value is
achieved when the magnitude of axial strains are 0.15, 0.14 and 0.14 corresponding to the
confining stress of 138 kPa, 276 kPa and 414 kPa respectively. It is important to note that the shear
strain is related to the directly measurable axial strain for an undrained triaxial test (see
Appendix C). The accumulated shear strain at toe before the initiation of rapid movement (Table
3.2) is greater compared to the shear strain corresponding to the peak deviatoric stress obtained
from the stress-strain curve (Figure 3.14b). This implies the requirement of greater shearing at toe
to mobilize sufficient shear strain at other points along the potential slip surface to form a slip
surface. Thus, the use of shear strain corresponding to peak deviatoric stress in Eq. 8 will probably
yield smaller horizontal displacement at toe compared to the accumulated displacement before the
initiation of failure.
3.5.4 Performance limit state
Figure 3.15 represents data from the fifth drawdown cycle for 𝜑 = 27, and shows the slope
experiences an immediate increase in displacement rate, or velocity (𝑣1 and 𝑣2), from 0.008 m/day
to 0.102 m/day at toe, which is approximately 13 times increase, as presented in Table 3.2. The
velocity increases further, causing the instability of the slope. In cases where the slope movement
can be monitored with time, the accumulated deformation accompanied by rapid acceleration
could form a rational basis to define a performance corresponding to the ultimate limit state (ULS).
Stated differently, if the magnitude of monitored deformation approaches the value of ULS, the
68
stability of the slope is a concern and remedial measures should be implemented. Figure 3.15
shows that the magnitude of ULS is approximately 0.25 m for the slope analyzed in this study.
The shear strain corresponding to peak deviatoric stress is 0.14 for 𝜑 = 27 (Figure 3.14a) and is
used in Eq. 3.8 to predict ULS as follow:
ULS =𝐶𝛾𝑠𝐻𝑠𝑠𝑖𝑛𝛽
=0.155 ∗ 0.14 ∗ 3.0
sin (18.43)= 0.21 𝑚 (3.9)
Thus, the predicted ULS from Eq. 3.8 is 0.21 m, which is 1.2 times smaller (hence conservative)
compared to the value obtained from the model. For 𝜑 = 25, 30, and 33, the slope experiences
accelerated deformation at the horizontal displacement values of 0. 25 m, 0.24 m, and 0.22 m,
respectively, which are smaller than the predicted ULS. Accordingly, Eq. 3.8 provides a simple
approach to predict ULS for earth embankment slopes.
3.6 Validation of the Developed Correlation
Analyses are performed for four case studies from literature including IJkDijk levee, Boston levee,
Elkhorn levee, and a lower Mississippi valley levee to verify the developed correlation between
shear strain and displacement. A brief description of the analysis for each case study is presented
here. The readers are referred to Melnikova et al. [98, 99], and Khalilzad et al. [25, 89] for detailed
description of the case studies and their associated analyses results. The soil properties (𝜑′and 𝑘),
embankment geometry (𝐻𝑠 and 𝛽), and deformational response (𝛾𝑠 and 𝑢) at toe for these
embankments are summarized in Table 3.3.
3.6.1 IJkDijk levee
Melnikova et al. [98] simulated the experimental slope failure of a full-scale earthen levee, known
as IJkDijk levee breach experiment, at Bad Nieuweschans, the Netherlands in September 2012.
69
The IJkDijk levee was 4 m high and 50 m long and was constructed using sand with a 50 cm clay
cover flanking the side. The side slope was 1V: 1.5 H (𝛽 = 33.6°). The failure of the slope as
indicated by excessive shear strain was initiated by excavating a 2-meter deep trench along the
right slope. Melnikova et al. [98] used a finite element module “Virtual Dike” for stability analysis
with a 2D Drucker-Prager linear elastic perfectly plastic constitutive model for all levee layers.
Their analyses were based on reference monitored data from the sensors (piezometers,
inclinometers, strain and temperature meters, and settlement gauges) to predict the displacements
as realistically as possible. Results showed that a shear band was formed emanating from the levee
toe and propagated towards the crest at the collapse stage. Their predicted mode of failure agreed
well with the experimental study. The horizontal deformation (𝑢) and shear strain (𝛾𝑠) at toe, at
the collapse stage, were obtained from their model as shown in Table 3.3.
3.6.2 Boston levee
Melnikova et al. [99] also simulated the instability of Boston levee, England, due to tidal
fluctuations of the river Haven. The levee is mainly composed of soft brown clay which is overlain
by a fine sand layer. The foundation of the levee is formed by dark brown sand. The strength
parameters of the soils are obtained from Cone Penetration Tests. The levee has been equipped
with sensors registering pore pressures and media temperatures of the levee. The hydraulic
conductivity of the levee analysis was calibrated using the pore pressure values registered by the
sensors. The levee was modeled using finite element software package COMSOL. The stability
analysis was carried out by Melnikova et al. [99] also for two river water level (RL) conditions,
namely high tide, where RL was +4m above the mean sea level, and low tide, where RL was -
1.1m. The results indicated the instability of the levee and agreed well with the field observations.
The shear band is similar for both cases and was entirely located in the clay layer. The shear strain
70
and the corresponding displacement at toe were obtained herein from the reported distributions of
shear strain (defined as effective plastic strain by the authors) and displacements for the high tide
and low tide conditions (Table 3.3).
3.6.3 Elkhorn levee
Khalilzad et al. [25] simulated the Elkhorn levee on the Sacramento River, California. They
analyzed the stability of the levee under sustained flood loading. The levee is constructed from
silty sand over a thin layer of sandy clay. A berm with a side slope of 1V:3.3H, a width of 3.4 m
and a height of 2.3 m is placed on the downstream side of the levee. The numerical model of the
levee by Khalilzad et al. [25] was built in several stages for stability analysis using the finite-
element program, PLAXIS, and the limit equilibrium program, SLOPE/W. The input soil
parameters for the model were obtained from the laboratory and field tests, as well as from the
literature. The flood condition was simulated by raising the river water level up to 0.1 m below the
crest. High strain zone gradually developed from the vicinity of berm toe and reached at the berm
top due to several days of the sustained high-water level. Table 3.3 includes their reported shear
strains and associated horizontal deformations at three stages of flooding that corresponded to the
minor (LS1), medium (LS2), and major (LS3) levee damages respectively.
3.6.4 Lower Mississippi valley
Khalilzad et al. [89] also investigated the deformation response of a Lower Mississippi Valley
levee due to flood loading. The soil profile consisted of a three-layer soil system: a shale layer at
the bottom; Alluvium soil layer in the middle, the foundation layer and body of the embankment
dam at the top. The side slopes were 4H:1V on both the downstream and upstream sides of the
embankment. The model embankment in the numerical analyses was constructed in several stages
71
and the flood condition was simulated by raising the river water level at 3.1 m below the crest. The
water level kept at this level until the instability of the slope is observed. The failure surface was
initiated near the toe zone and expanded up to the crest. They varied the embankment size from
4.4 m to 44.0 m to study the effect of the change in height on the magnitude shear strain and
corresponding horizontal deformation at toe. Similar to the case of Elkhorn levee, they reported
the shear strain and deformation for each case at three stages of flood loading associated with
damage levels (LS1, LS2 and LS3 in Table 3.3).
For the four case studies, the ranges of soil property and embankment geometry are: 15.9 to 30
for effective friction angle (𝜑′); 0 to 2 kPa for 𝑐′, 2 to 30 MPa for stiffness (𝐸), 1.16E-05 cm/s to
1.20E-02 cm/s for hydraulic conductivity (𝑘); 2.4 m to 44 m for height of slip surface (𝐻𝑠); and
14 to 33.7 for side slope (𝛽). The shear strain (𝛾𝑠) and displacement (𝑢) at embankment toe from
Table 3.3 are used to plot Figure 3.16. Data in Figure 3.16 show a good linear relationship between
the shear strain and toe vertical displacement obtained from the literature with a R-squared value
of 0.948. The magnitude of 𝐶 is found to be 0.171 which is within the range of 0.148 and 0.210
obtained herein.
3.7 Conclusions
Work in this study develops a general criterion for performance limit state that is defined based on
the framework of emergence of shear strain magnitude representing the onset of accelerated
deformation rate. A correlation between the magnitude of shear strain and the corresponding
deformation at toe is developed. Based on the results obtained from this study, the following
conclusions can be drawn:
72
The stability analysis results show a gradual decrease in FS of the upstream slope as the
number of drawdown cycles is increased. As more cycles are introduced, the FS after
drawdown is reduced from 1.09 to 0.99 (from cycle 1 to 5) for the base case, and therefore
reflects the instability risk due to repeated hydraulic loading.
The onset of the instability of the slope is preceded by a gradual accumulation of surface
displacement with its rate accelerating with the continuity of hydraulic load cycles. The
performance limit corresponding to the ultimate state (ULS) is quantified by the
accumulated displacement at toe before the emergence of rapid movement due to
instability.
A simple linear relationship between the shear strain and deformation at toe is developed
as a function of the geometry of the slope. The results from the parametric studies show a
good agreement with the correlation when slope experiences rotational movement. Also,
the criterion from the correlation shows good agreement with data from field studies by
others. This relationship provides a simple means to estimate the performance limit state
using the stress-strain data obtained from triaxial testing.
The predicted pore pressure after rapid drawdown from the coupled analysis is observed to
be smaller compared to the uncoupled analysis since the latter case does not account for
the decrease in total stress due to the lowering of the water level. The coupled analysis
yields lower pore pressure compared to the calculated upper bound set by Barrett and
Moore’s method.
73
The performance limit state approach proposed in this study can be used in conjunction with the
surface monitoring and surveying techniques to: a) assess the real-time health condition of earth
slopes; b) predict the performance of earth structures under future flood events; c) prioritize
rehabilitation measures based on improving functionality level and limiting damage under future
flood events.
74
Tables:
Table 3.1. Soil properties.
Soil Parameters
Symbol
(unit)
Value
Unit weight 𝛾 (kN/m3) 18
Effective cohesion 𝑐′(kPa) 1
Effective angle of internal friction 𝜑′(degrees) 27
Reference secant stiffness in standard drained triaxial test 𝐸50𝑟𝑒𝑓 (kPa) 800
Reference tangent stiffness for oedometer loading 𝐸𝑜𝑑𝑒𝑟𝑒𝑓
(kPa) 850
Reference unloading/ reloading stiffness 𝐸𝑢𝑟𝑟𝑒𝑓
(kPa) 800
Hydraulic conductivity 𝑘 (m/day) 8.64 x 10-7
Unsaturated properties
(van Genuchten parameters)
𝜃𝑟 0.068
𝜃𝑠 0.38
𝑔𝑎 (1/m) 0.80
𝑔𝑛 1.09
75
Table 3.2. Effect of friction angle on the number of cycles of loading, accumulated shear strain,
and velocity response.
’()
FS after first
drawdown phase
No. of
drawdown
cycles before
failure
Accu. shear
strain at toe
before failure
Velocity increase,
(𝑣2/𝑣1)
25 1.01 1 0.15 16.26
27 1.09 5 0.18 12.75
30 1.22 12 0.16 10.01
33 1.36 19 0.14 12.99
76
Table 3.3. Summary of the case studies used for the verification of developed correlation.
Case Studies 𝜑′(°) 𝑐′(𝑘𝑃𝑎) 𝐸(𝑀𝑃𝑎) 𝑘 (c𝑚/𝑠) 𝐻𝑠(𝑚) 𝛽(°) 𝛾𝑠 𝑢(𝑚) Comments References
IJkDijk
levee,
Netherlands
30.0 0.0 30.0 - 4.0 33.7 0.063 0.071 - Melnikova et al. [98]
Boston
levee,
England
25.0 2.0 2.0 1.16E-05 4.0 28.0a 0.022 0.032 High tide Melnikova et al. [99]
25.0 2.0 2.0 1.16E-05 4.0 28.0 0.025 0.038 Low tide
Elkhorn
levee, U.S.
15.9 1.7 3.5 7.60E-05 2.4 16.9 0.018 0.026 LS1
Khalilzad et al. [25] 15.9 1.7 3.5 7.60E-05 2.4 16.9 0.032 0.041 LS2
15.9 1.7 3.5 7.60E-05 2.4 16.9 0.047 0.058 LS3
Lower
Mississippi
valley, U.S.
17.5 0.0 6.7 4.98E-04 44.0 14.0 0.018 0.740 LS1
Khalilzad et al. [89]
17.5 0.0 6.7 4.98E-04 44.0 14.0 0.035 1.170 LS2
17.5 0.0 6.7 4.98E-04 44.0 14.0 0.052 1.700 LS3
17.5 0.0 6.7 4.98E-04 22.0 14.0 0.014 0.230 LS1
17.5 0.0 6.7 4.98E-04 22.0 14.0 0.028 0.460 LS2
17.5 0.0 6.7 4.98E-04 22.0 14.0 0.042 0.760 LS3
17.5 0.0 6.7 4.98E-04 8.8 14.0 0.009 0.060 LS1
17.5 0.0 6.7 4.98E-04 8.8 14.0 0.018 0.120 LS2
17.5 0.0 6.7 4.98E-04 8.8 14.0 0.028 0.180 LS3
17.5 0.0 6.7 4.98E-04 4.4 14.0 0.008 0.030 LS1
17.5 0.0 6.7 4.98E-04 4.4 14.0 0.015 0.050 LS2
17.5 0.0 6.7 4.98E-04 4.4 14.0 0.022 0.070 LS3 aAverage value was used
77
Figures:
Figure 3.1. Model geometry and discretized mesh.
Figure 3.2. Selected points along the potential slip surface for stability analysis.
78
Figure 3.3. Comparison of pore pressure predictions obtained from different methods after rapid
drawdown.
Figure 3.4. Decrease in factor of safety with drawdown cycle.
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0 2 4 6 8 10
Po
re w
ater
pre
ssu
re (
kP
a)
Points along the potential slip surface
Bishop (1954)
Barrett and Moore (1975)
Calculated (Coupled)
Calculated (Uncoupled)
Initial pore pressure
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00
Fac
or
of
safe
ty (F
S)
Time (years)
FS from SRM
FS from principal stresses
Cycle-1
Cycle-2
Cycle-3
Cycle-4
Cycle-5
79
Figure 3.5. Shear strain and horizontal displacement increase at toe with drawdown cycles.
Figure 3.6. Stress path meeting the failure envelope at fifth drawdown cycle.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00
Ho
rizo
nta
l d
isp
lace
men
t (m
) an
d S
hea
r
stra
in
Time (years)
Displacement
Shear strain
Cycle-1
Cycle-5
80
Figure 3.7. (a) Deformed shape of the slope at fifth drawdown cycle; (b) Simplified diagram of a
deformed element at toe.
Figure 3.8. Determination of the magnitude of 𝐶 for 𝑘= 10−9cm/s.
0.00
0.01
0.02
0.03
0.04
0.00 0.05 0.10 0.15 0.20 0.25
u s
in(
)/H
s
Shear strain, s
C = 0.155
R2 = 0.962
81
(a)
(b)
Figure 3.9. Shear strain and horizontal displacement increase at toe with drawdown cycles; (a)
with 0.5 and 500 kPa increment after each cycle for ′ and 𝐸50𝑟𝑒𝑓
, respectively, (b) with 0.5
decrement after each cycle for ′.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00
Ho
rizo
nta
l d
isp
lace
men
t (m
)/S
hea
r st
rain
Time (years)
Deformation
Shear strain
Cycle-1
(′27 and 𝐸50𝑟𝑒𝑓
= 800 𝑘𝑃𝑎
Cycle-5
(′29 and 𝐸50𝑟𝑒𝑓
= 2800 𝑘𝑃𝑎
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.00 5.00 10.00 15.00 20.00
Hori
zonta
l dis
pla
cem
ent
(m)/
Shea
r st
rain
Time (years)
Deformation
Shear strain
Cycle-1
(′27)
Cycle-3
(′26)
82
Figure 3.10. Determination of 𝐶 using the data subjected to change in strength and/or stiffness
parameters.
0.00
0.01
0.02
0.03
0.04
0.00 0.05 0.10 0.15 0.20 0.25
usi
n(
)/H
s
Shear strain, s
+0.5°/cycle & +500 kPa
-0.5°/cycle
C = 0.148
R2 = 0.94
83
Figure 3.11. Effect of hydraulic conductivity on 𝐶; (a) 𝑘 = 10−6 cm/s, (b) 𝑘 = 10−5 cm/s, and
(c) 𝑘 = 10−4 cm/s.
84
(a)
(b)
0.00
0.01
0.02
0.03
0.04
0.00 0.05 0.10 0.15 0.20 0.25 0.30
uxsi
n(
)/H
Shear strain, s
C = 0.150
R2 = 0.970
0.00
0.01
0.02
0.00 0.02 0.04 0.06 0.08 0.10
uxsi
n(
)/H
Shear strain, s
C = 0.210
R2 = 0.883
85
(c)
0.000
0.001
0.002
0.003
0.004
0.00 0.01 0.01 0.02
uxsi
n(
)/H
Shear strain, s
C = 0.220
R2 = 0.335
86
(a)
(b)
Figure 3.12. Shear strained zone after fifth drawdown cycle for ′27; (a) with 𝑘= 10−4 cm/s,
(b) with 𝑘= 10−6 cm/s.
87
(a)
(b)
Figure 3.13. Accumulation of plastic points for 𝜑 = 27; (a) after four drawdown cycles, (b)
after fifth drawdown phase.
88
(a)
(b)
Figure 3.14. (a) Simulation of isotropic consolidated undrained triaxial tests of soil; (b)
comparison between shear strain obtained from model and from stress-strain curve.
0.00
1.00
2.00
3.00
4.00
5.00
0.00 0.05 0.10 0.15 0.20
Dev
iato
ric
stre
ss, q
(kP
a)
Shear strain, s
0.00
0.05
0.10
0.15
0.20
0.00 0.05 0.10 0.15 0.20
Shea
r st
rain
fro
m s
tres
s-st
rain
cu
rve
Shear strain from embankment model
89
Figure 3.15. Rapid increase of surface displacement at fifth drawdown cycle for 𝜑 = 27 (time is
set to zero at the beginning of fifth cycle).
Figure 3.16. Determination of the magnitude of 𝐶 from four case studies.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 2.0 4.0 6.0
Ho
rizo
nta
l d
isp
lace
men
t (m
)
Time (days)
v1=0.008 m/dayv2= 0.102 m/day
ULS= 0.25 m
v2/v1= 12.75
ULS from correlation
=0.21 m
0.000
0.004
0.008
0.012
0.016
0 0.02 0.04 0.06 0.08
u s
in(
)/H
s
Shear strain, s
IJkDijk levee
Boston levee
Elkhorn levee
Lower Mississippi dam
C = 0.171
R2 = 0.948
90
4 CHAPTER 4. EFFICACY OF THREE SLOPE REPAIR METHODS IN TERMS OF
EXCEEDANCE PROBABILITY OF ULTIMATE LIMIT USING COUPLED
TRANSIENT SEEPAGE ANALYSIS
91
Abstract
Three repair methods, representing three different mechanisms of remedial efforts, are investigated
here to stabilize the upstream slope failure of Pilarcitos dam due to rapid drawdown. These
methods improve stability by providing reinforcement on the upstream slope (soil nails), reducing
slope height to decrease the shear stress (bench), and lowering phreatic surface to decrease pore
water pressure (drainage blanket). They are analyzed and compared in terms of probability of
exceeding a predefined ultimate limit state, where the limit state is associated with horizontal
deformation at slip surface toe that can be readily monitored in the field through periodic
surveying. All the analyses are performed using unsaturated coupled transient seepage method and
non-liner advanced elasto-plastic constitutive relation in finite element (FE) program PLAXIS.
Given the set of conditions used in this study, excavating a bench appears to be the most effective
measure in terms of associated risk among the three analyzed remedial methods due to the
anticipated lower probability of exceedance and shallower potential slip surface, which deems to
cause lower consequence. For comparative study, pore water pressure and stability factor of safety
are also calculated using partially coupled and uncoupled transient seepage analysis. The
uncoupled seepage analysis is also implemented in PLAXIS, whereas the partially coupled seepage
analysis and stability analysis are performed using FE program SEEP/W and limit equilibrium
software SLOPE/W, respectively. Results are presented and discussed on how pore water pressure
predictions from different models affect the magnitude of stability factor of safety (FS), location
of potential slip surface, and the required time to establish steady-state conditions significantly.
92
4.1 Introduction
In recent decades, climate change has increased extreme precipitation in both frequency and
magnitude, which in turn has elevated flood risk in the US [1]. In some areas, the increasing
temperature due to climate change is expected to cause more intense and prolonged droughts [2].
Earthen levees and dams are designed and constructed to play an important role during extreme
flood and drought events. The average age of levees and dams in the US is more than 50 years, a
period considered as the nominal design life for heavy structures [3]. As a qualitative assessment,
ASCE assigned grade ‘D’ for dams and levees, which indicates that the infrastructure’s condition
and capacity are of serious concern with a strong risk of failure [4]. These aged structures are
considered deficient in some aspects of their structural integrity and require on the order of $80
billion for levees and $45 billion for dams to rehabilitate and upgrade their performance for future
extreme events, yet, limited budget has been allocated nationwide [4]. Therefore, there is a clear
need to assess the existing health condition to prioritize repair measures.
Dams and levees experience large and rapid increase in water elevation during extreme flood
events associated with hurricanes and rapid decrease in water level due to excess supply of water
during droughts. Studies show that repeated occurrence of such extreme events (hurricanes or
droughts) causes major displacement to these earth structures and may lead to breaching
failure [8]. The conventional slope stability approach (e.g., limit equilibrium method) provides no
means to account for the effect of such displacement on the structural integrity aspects of levees
and dams. On the other hand, this accumulated displacement may be monitored by instrumentation
and data management systems and can be compared with the established performance limit sate
for assessment of structure’s vulnerability. To this end, strain-based ultimate limit state approach,
93
proposed by Jadid et al. [100], is associated with horizontal deformation at slip surface toe that can
be readily monitored in the field through periodic surveying to assess the real-time health condition
of earth slopes and to prioritize rehabilitation measures based on improving functionality level and
limiting damage under future extreme events.
Many methods have been implemented to repair slope failure in the past. Table 4.1 summarizes
different types of slope repair methods from the literature [49, 101, 102, 103, 104]. Each method
has both advantages and disadvantages and is found to be suitable for a particular set of conditions.
Several remedial methods can be technically feasible to stabilize a slope under given
circumstances. For example, the reservoir drawdown caused upstream slide of the San Luis Dam
in California (now known as B.F. Sisk Dam) in 1981. After the slide, a rockfill berm was
constructed as a repair action at the toe to minimize the slope movement [5]. A similar set of
conditions was also observed in Canelles dam in Spain, which experienced an upstream slide due
to rapid lowering of the Canelles reservoir’s water level in 2006. As a remedial measure, weight
transfer from near the crest to the near toe was proposed to stabilize the Canelles dam [6]. The
performance of a repair method is usually assessed by increased stability factor of safety, which
does not provide rational basis for condition assessment of dams and levees as they are
progressively loaded over time with repeated rise and fall of water levels as well as efficacy of
remedial actions [14].
In general, earth slopes experience changes in external water levels during flood or drawdown
events, which lead to modification of internal pore water pressure within the levees and dams [57].
This modification of pore water pressure has three components- (i) seepage-induced pore water
94
pressure component due to transient flow, (ii) stress-induced pore water pressure component due
to changes in boundary loads applied by the weight of water on the slope, and (iii) consolidation-
related dissipation of pore water pressure with time [9]. Several transient seepage analyses
approaches are available in the literature to predict the pore water pressure response due to water
level changes. The first procedure is “coupled” transient seepage analysis, which considers all
three pore water pressure components and perhaps best representing the in-situ condition.
However, the coupled analysis is relatively complicated because of the need to solve the governing
equations of transient flow and deformation simultaneously and requires extensive input
parameters for advanced constitutive relations as well as longer computational time [10]. The
second procedure is partially coupled transient seepage analysis in which pore water pressure
response is assumed uncoupled from the change in boundary loads. Stated differently, the stress-
induced pore water pressure component is neglected in the analysis. On the other hand, the
uncoupled analysis (third procedure) ignores both consolidation and stress-induced pore water
pressure components.
A simplified theoretical procedure proposed by Bishop [90] is commonly used to estimate pore
water response after rapid drawdown. In this method, the change in pore water pressure is assumed
equal to the change in total vertical stress resulting from the change in water elevation above the
point in consideration. Bishop assumed the pore pressure parameter A as 1.0, which results in
coefficient �̅� as 1.0 for saturated soil. On the other hand, Barrett and Moore [91] using results from
numerical analyses showed that the change in pore pressure is 0.7 to 0.9 times the change in
vertical stress caused by lowered water level. Both Bishop’s and Barrett and Moore’s approaches
usually overestimate (conservative) the pore pressure after rapid drawdown [10].
95
The primary objective of this study is to investigate the efficacy of three repair methods
representing three different categories to stabilize an earth embankment which geometry and soil
profile are representative of Pilarcitos Dam. These methods improve stability by providing
reinforcement on the upstream slope (soil nails), reducing slope height to decrease the shear stress
(bench), and lowering phreatic surface to decrease pore water pressure (drainage blanket). They
are analyzed and compared in terms of probability of exceeding the ultimate limit state associated
with horizontal deformation at slip surface toe. All the analyses are performed using unsaturated
coupled transient seepage method and non-liner advanced elasto-plastic constitutive relation in
finite element (FE) program PLAXIS. For comparative study, pore water pressure and stability
factor of safety are also calculated using partially coupled and uncoupled transient seepage
analysis. The uncoupled seepage analysis is also implemented in PLAXIS, whereas the partially
coupled seepage analysis and stability analysis are performed using FE program SEEP/W and limit
equilibrium software SLOPE/W, respectively.
4.2 Study Model
An embankment dam is modeled using a two-dimensional finite element software PLAXIS 2D
2018 for SRM and deformational analysis. For comparative study, finite element software
SEEP/W 2016 and limit equilibrium software SLOPE/W 2016 are also used for seepage analysis
and stability analysis, respectively. The geometry and soil layer of the analyzed dam section is
obtained from VandenBerge [105], and is shown in Figure 4.1. The model represents an earth dam
section, which geometry and soil profile are representative of the Pilarcitos Dam. The dam is
approximately 23.8 m (78 ft) high and was built from homogeneous compacted sandy clay. The
upstream slope is 2.5H to 1.0V from the embankment toe having an elevation (EL) of 189.0 m up
to the EL of 206.7 m. From this EL, the slope is inclined at 3.0H to 1.0V up to the crest (EL= 212.8
96
m). The water level in the Pilarcitos reservoir was lowered from EL of 211.0 m to EL of 200.3 m
between October 7 and November 19, 1969, for inspection and repair purposes, which resulted in
a rapid drawdown failure of the upstream slope [106, 10]. The exposed portion of the failure
showed a circular slip with an approximate maximum depth of 3.7 m. While the top of the slip
surface emanated from the EL of 209 m, the toe of the failure plane was submerged and could not
be located. The drawdown rate was nearly 0.52 m/day for 14 days prior to the failure [107].
4.3 Domain Discretization and Modeling Approaches
The dam section is modeled using plane strain 15-nodes triangular elements, as shown in Figure
4.1. The fine mesh is observed to be optimum mesh size from the mesh sensitivity analysis and is
used herein to develop the model with a domain having 59057 nodes and 7272 elements.
The constitutive model of the soil layer in the analysis domain is defined by the hardening soil
(HS) model [39]. Three stiffness input parameters (𝐸50𝑟𝑒𝑓
, 𝐸𝑜𝑒𝑑𝑟𝑒𝑓
and 𝐸𝑢𝑟𝑟𝑒𝑓
) are used in HS model to
simulate the soil behavior along with the strength parameters, the cohesion intercept (𝑐) and the
angle of internal friction (𝜑). The reference stiffness modulus (𝐸50𝑟𝑒𝑓
) controls the shear behavior
of soil; whereas the reference oedometer modulus (𝐸𝑜𝑒𝑑𝑟𝑒𝑓
) simulates the volumetric behavior; and
the reference loading-unloading stiffness modulus (𝐸𝑢𝑟𝑟𝑒𝑓
) models unloading-reloading
characteristics of soil [9]. The input soil parameters for the Pilarcitos dam section are presented in
Table 4.2. The magnitude of unit weight (γ), strength parameters (c′ and φ′), and the hydraulic
conductivity (𝑘) are reported in Wahler and Associates [108] and Wong et al. [107]. The reference
stiffness parameters (𝐸50𝑟𝑒𝑓
, 𝐸𝑜𝑒𝑑𝑟𝑒𝑓
and 𝐸𝑢𝑟𝑟𝑒𝑓
) are selected based on data presented by VandenBerge
[109] and Obrzud and Truty [48]. The unsaturated hydraulic properties of soil above the phreatic
97
surface is simulated using the van Genuchten model [51]. Table 4.2 presents the van Genuchten
model parameters (𝜃𝑟 , 𝜃𝑠, 𝑔𝑎 and 𝑔𝑛) which are selected based on the reported value for soil with
similar gradation as material comprising Pilarcitos dam [43].
4.3.1 Loading and boundary conditions
As mentioned earlier, the Pilarcitos dam experienced an upstream slide in 1969 when the reservoir
water level was lowered by approximately 10.7 m in 43 days. The drawdown rate was nearly
constant at about 0.52 m/day for the last 14 days causing 7.28 m drawdown [106]. Hence, the water
level was dropped by 3.42 m in the first 29 days, with an average rate of 0.12 m/day. The drawdown
condition similar to water elevations and rate occurred in conjunction with Pilarcitos dam failure
is modeled in this study. Modeling steps for simulating the drawdown included- first generating
the geostatic stress state in the dam section. Then, the reservoir water level is raised in several
steps from toe to the crest, and the initial condition is established under a steady-state condition at
an elevation (EL) of 211.0 m, as shown in Figure 4.1. Thereafter, transient seepage analysis is
performed by lowering the water level to the elevation of 200.3 m at a rate of 0.12 m/day for the
first 29 days, and 0.52 m/day for the last 14 days of drawdown (Figure 4.1).
The flow boundary conditions for the seepage analysis included an impervious boundary at the
bottom of the model and a free-seepage boundary at the upstream slope of the dam. The steady-
state seepage condition for the initial condition is modeled as a constant pore pressure boundary,
whereas the transient condition due to drawdown is modeled as a time-dependent pore pressure
boundary. Deformation boundary conditions included- restriction of horizontal deformation on the
upstream and downstream slope edges of the domain as well as restriction of horizontal and
vertical deformations at the bottom boundary. The dimensions of the model have been carefully
98
chosen to minimize boundary effect (i.e., further increase of model size does not change the
results).
4.3.2 Stability analysis
The stability analysis is performed using the two-dimensional finite element program PLAXIS 2D
for the strength reduction method (SRM). In SRM, the factor of safety (FS) is defined as the factor
by which strength parameters (𝑐′and tan𝜑′) are reduced in order to reach slope failure. For
comparison, the two-dimensional program SLOPE/W is also used for the limit equilibrium method
(LEM). In LEM, the factor of safety (FS) is determined using Spencer's procedure [52]. Both force
and moment equilibriums are taken into consideration in Spencer's method [53]. The SLOPE/W
program utilizes an iteration scheme to determine the critical slip surface and the corresponding
minimum factor of safety.
4.3.3 Ultimate Limit State (ULS)
Jadid et al. [100] defined a strain-based ultimate limit state that corresponded to the instability of
embankment slopes and developed an approach to quantify it in terms of monitorable deformation
level that can be readily monitored in the field through periodic surveying. In this approach, the
performance limit corresponding to the ultimate state (ULS) is quantified by the accumulated
horizontal displacement at the toe of the potential slip surface before the emergence of rapid
movement due to instability. In this location, there is a tendency for failure to begin as the stress
path follows the form of an axial extension loading [14, 49]. Jadid et al. [100] proposed a simple
approach, as shown in Eq. 4.1, to predict the ULS using the shear strain (𝛾𝑠) that corresponds to
peak deviatoric stress of stress-strain diagram obtained from triaxial testing.
99
ULS =𝐶𝛾𝑠𝐻𝑠𝑠𝑖𝑛𝛽
(4.1)
Where 𝐻𝑠= height of the potential slip surface, 𝛽=inclination of the slope within the potential slip
surface, and 𝐶 = proportional constant = 0.155. To determine 𝛾𝑠 for the Pilarcitos dam case,
isotropic consolidated undrained (ICU) triaxial tests are simulated in PLAXIS 2D using the
material properties shown in Table 4.2. Figure 4.2 presents the simulated stress-strain curves,
which indicate that the peak deviatoric stress is achieved when the mobilized shear strain
magnitude is approximately 0.12 for all three assumed initial cell pressures of 10 kPa, 100 kPa,
and 500 kPa. Figure 4.3 shows a strained shear zone indicating potential slip surface after
drawdown, and the factor of safety (FS) is found as 1.03 using the SRM method, indicating
marginal stability condition. The height of the potential slip surface (𝐻𝑠) and the average slope
inclination within the slip surface (𝛽) are approximated as 11.0 m and 20 respectively from Figure
4.3. Using these values, the ULS is predicted for the Pilarcitos dam failure condition as:
ULS =𝐶𝛾𝑠𝐻𝑠𝑠𝑖𝑛𝛽
=0.155 ∗ 0.12 ∗ 11.0
sin (20)= 0.60 𝑚 (4.2)
This magnitude of ULS is used for probabilistic analysis, which is discussed in the following
section.
It is important to note that the horizontal deformation (𝑢) and shear strain (𝛾𝑠) at the slip surface
toe (Point A in Figure 4.3), at marginal stability condition, are obtained as 0.65 m and 0.14,
respectively. These values are then utilized to check the 𝐶 value proposed by Jadid et al. [100] as
follow:
𝐶 =
𝑢 𝑠𝑖𝑛𝛽𝐻𝑠⁄
𝛾𝑠=0.65 sin(20) /11
0.14= 0.145 (4.3)
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The magnitude of the 𝐶 parameter is found to be 0.145 based on data from the Pilarcitos dam. This
value is comparable to the value of 0.155 proposed by Jadid et al. [100]. Also, the predicted ULS
from Eq. 4.2 is 0.60 m is smaller (hence conservative) compared to the value of 0.65 m obtained
from the model.
4.3.4 Probabilistic approach
The probability of exceeding limit state (POELS) is estimated, based on the horizontal
displacement at the slip surface toe (Point A in Figure 4.3), as the water level drops in the reservoir.
The lowering of reservoir water level decreases total stress on the upstream slope and reduces the
head driving seepage through the dam [10]. Consequently, shear stress increases in the upstream
face that contributes to the increase of shear strain and horizontal displacement. Table 4.3 and
Table 4.4 show sample calculations of estimating the POELS based on the results from numerical
simulations for soil nailing and using an approach similar to Duncan [30]. The parameters
sensitivity analyses based on the deformational response at the upstream slope show that the unit
weight, stiffness, angle of internal friction, and the permeability parameters of the soil contribute
significantly to the horizontal deformation. These properties are, therefore, considered as random
variables during the reliability analysis. The model is analyzed two times for each random variable
with the mean value () plus/minus a standard deviation () of the variable. The mean value of
horizontal displacement (j in Table 4.4) is calculated from the finite-element analysis of the model
using mean values of all input parameters ( in Table 4.3). Then, the reliability index (βln) is
calculated, which is used to estimate the probability of exceeding limit state (POELS).
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4.4 Pore Pressure Estimation
4.4.1 Verification of pore pressure prediction
VandenBerge et al. [10] suggested that Barrett and Moore’s [91] method presents an approximate
upper bound of pore water pressure prediction after rapid drawdown; thus should be used to verify
complex numerical simulation. Figure 4.3 shows that several locations along the slip surface
(points: 1, 2, and 3) have been selected, and the calculated pore water pressures from different
models after drawdown at these locations are presented in Figure 4.4. The detailed calculations are
shown in Table 4.5. Compared to the Barrett and Moore’s, the coupled transient seepage analysis
in PLAXIS predicts smaller pore water pressures in all three locations; partially coupled analysis
in SEEP/W calculates higher pore water pressures at points 1 and 2; whereas uncoupled analysis
using PLAXIS overestimates at all three locations. Therefore, coupled analysis used in this study
calculates pore water pressures more realistically compared to the partially coupled and uncoupled
analysis.
The pore water pressure prediction not only affects the magnitude of stability factor of safety (FS)
and location of potential slip surface; but also controls the required time to establish steady-state
conditions. Figure 4.5 shows the dissipation of pore water pressures from the onset of lowering of
reservoir water level at point 1. The required times for complete dissipation of excess pore water
pressures are estimated as 45 days, 180 days, and 2050 days from coupled, partially coupled and
uncoupled analysis, respectively, as presented in Figure 4.5(a). The rate of change in pore water
pressure changes after 29 days as the reservoir drawdown rate increases from 0.12 m/day to 0.52
m/day at this stage (Figure 4.5b).
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4.4.2 Effect of pore pressure estimation on FS
Table 4.6 presents the calculated FS after drawdown using different models. While the coupled
analysis predicts the FS as 1.03 indicating a marginal stability condition for Pilarcitos dam failure
case, the uncoupled analysis underestimates FS as 0.82. Drawdown event causes a reduction of
total stress at the upstream slope resulting in a decrease in pore water pressure. The partially
coupled method does not account for the pore water pressure change due to the change in total
stress. Therefore, it predicts higher pore pressure compared to the coupled analysis and leads to
predict lower FS. The FS using uncoupled analysis could not be reported as the calculated pore
water pressures exceed the total overburden stress along part of the slip surface. It is important to
note that the slip surface corresponding to partially coupled analysis (Figure 4.6a) is kept same as
coupled analysis (Figure 4.3) in order to entirely focus on the effect of pore pressure on FS. The
critical slip surface corresponding to the minimum factor of safety of 0.23 is shown in Figure
4.6(b) for partially coupled analysis. Therefore, the pore pressure prediction also influences the
location of the potential slip surface, which in turn affects the design of remedial actions for slope
stabilization.
4.5 Remedial Methods
Among the remedial methods that are technically feasible to stabilize the upstream slope of
Pilarcitos dam, three methods are selected from three different categories in this study: (i) installing
soil nails from mechanical category; (ii) excavating a bench from earthwork category; and (iii)
constructing a drainage blanket from drainage category. These three methods represent three
different mechanisms of remedial efforts. For example, stability is improved by reinforcing slopes
for soil nailing technique, reducing slope height for bench excavation, and lowering phreatic
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surface to decrease pore water pressure for drainage blanket. USACE [32] recommends a
minimum required FS of 1.20 for drawdown from the maximum storage pool level, likely to persist
for a long period to establish steady-state condition. Therefore, parametric studies are performed
for each method before adopting the final design that ensures a FS of 1.20 for the upstream slope.
Later, the performances of each remedial methods are investigated and compared in terms of
probability of exceeding the limit state.
4.5.1 Installation of soil nails
Soil nails are installed in stabilizing the upstream slope of Pilarcitos dam, as shown in Figure 4.7,
and the effect of nailing on the exceedance probabilities is investigated here. If the upstream failure
wedge in Figure 4.3 starts to move, tension force will rapidly develop in the nails to prevent further
movement. Where the potential slip surface passes the soil nails perpendicularly, the movement of
failure wedge also induces shear and bending resistance in nails.
Table 4.7 shows the nail parameters used in this study, which are selected according to the
recommendation from Babu and Singh [110], Fan and Luo [111], Rawat and Gupta [112], and
FHWA [113]. Since soil nails are discrete circular structures, they are modeled as ‘equivalent
plate’ in the two-dimensional plane strain analysis in PLAXIS. The discrete nail element is
replaced by the plate element of one unit width [111]. The flexural rigidity (𝐸𝐼), axial stiffness
(𝐸𝐴) and equivalent plate thickness (𝑑𝑒𝑞) are the important input parameters for plate element.
Therefore, the nail parameters in Table 4.7 are used to estimate 𝐸𝐼, 𝐸𝐴 and 𝑑𝑒𝑞 using an approach
described in Babu and Singh [110], and presented in Table 4.8. Soil nails are modeled as an elastic
material in PLAXIS. The field pullout tests showed that the coefficient of soil-reinforcement
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interaction is significantly more than unity [110]. Therefore, the use of interface elements between
nail and soil is eliminated conservatively by assuming the magnitude of interface strength (𝑅𝑖𝑛𝑡𝑒𝑟)
as unity.
In addition to the nail properties and spacings as presented in Table 4.7, the nail length and
orientation with respect to the horizontal also affect the factor of safety (FS) of the reinforced
slope. Parametric studies are performed here by varying the nail length from 6 m to 14 m first and
then the orientation from 10 to 30 to achieve the required FS of 1.20. Figure 4.8(a) shows that
the FS is improved by 23% when the nail length is increased from 6 m to 14 m with 15 nail
orientation. Since 10 m long nails provide a FS of 1.18, closer to the required FS of 1.20, it is
chosen for the second parametric study to investigate the effect of nail orientation on FS. Figure
4.8(b) shows that FS is increased by 4.3% for the base case when the nail orientation changes from
10 to 30. For probabilistic analysis, a layout of soil nails with 10 m length and 20 orientation is
chosen in this study (Figure 4.7), as it provides the desired FS of 1.20.
Wahler and Associates [108] performed a number of isotopically consolidated-undrained triaxial
tests in the stress range of 34.5 kPa to 690 kPa. The soil strength parameters (𝜑′=45 and 𝑐′= 0)
reported in Table 4.2 are considered as the base case here. They were obtained from the strength
envelope fitted within the low-stress range (0-69 kPa), which is applicable for shallow slip surface
like in the Pilarcitos dam failure case. However, strength envelope fitted within the high-stress
range (0-690 kPa) resulted in strength parameters of 𝜑′=32 & 𝑐′= 8 kPa. With zero cohesion
intercept, this strength parameters become 𝜑′=34 and 𝑐′= 0 kPa. Since the soil nail pushes the
slip surface deeper, the effective overburden pressure along the slip surface also increases.
105
Therefore, two sets of strength parameters corresponding to high-stress range are used here to
investigate their effect on FS. They are denoted as case A for with cohesion intercept and Case B
for without cohesion intercept. The FS is not available for nail orientation less than 20 for case B
due to numerical converge problems. While case A yields a 3 % increase in FS compared to the
base case corresponding to 20 nail orientation, Case B results in a 17.7% decrease of FS. Thus,
the consideration of cohesion intercept in strength parameters contributes to the FS significantly.
This example demonstrates the importance of selecting appropriate stress range and curve fitting
techniques for determining strength parameters.
4.5.2 Excavation of bench
The upstream slope can be made more stable by excavating a bench to reduce its height, as shown
in Figure 4.9(a). The reduction of slope height decreases the driving shear stress along the potential
slip surface and increases the stability factor of safety. While flattening the upstream slope would
facilitate similar benefits, it was not chosen for this case study because it requires complete
dewatering the reservoir to ensure the site is accessible to construction equipment. It is important
to remember that the excavation of bench requires sacrificing useful areas at the crest of the top.
The effect of bench location (EL) and the bench inclination () on the factor of safety at 43 days
has been investigated, and results are presented in Figure 4.9. The reduction of crest width (B) is
assumed as 4.0 m for each analysis, and the slope above the bench is maintained as 3H to 1V
(Figure 4.9a). Three inclinations of bench with respect to horizontal () are assumed: 0
(horizontal), +10(clockwise) and -10(anti-clockwise) for analysis, while horizontal bench (=0)
is considered as the base case. Figure 4.10 shows that FS decreases with the increase of elevation
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(EL) for all differently inclined benches due to the increased height of the slope below the bench.
However, the bench inclined with -10 causes more increase in FS from the base case compared
to the decrease caused by +10 inclined bench, since greater amount of earth is removed for the
former case resulting in greater reduction in shear stress along the potential slip surface. Moreover,
the potential slip surface is deeper and larger for -10 inclined bench compared to other
configurations (Figure 4.9). The required design FS of 1.20 can be achieved if the horizontal bench
is excavated at an EL=205.2 m (Figure 4.10), which has been selected for the probabilistic analysis.
4.5.3 Drainage blanket at upstream slope
The potential failed soil mass from the upstream slope can be entirely removed and replaced with
a blanket consisting of porous drainage material, as shown in Figure 4.11(a). The strength and
stiffness properties of the upstream blanket are assumed same as the embankment soil in order to
entirely focus on the hydraulic impacts of the blanket. Blanket improves upstream slope stability
by lowering phreatic surface (Figure 4.11 a & b). Thus, it decreases pore water pressure within the
upstream slope, hence increases effective stress and shear strength. The effect of excavation
thickness (𝑡𝑏) and the hydraulic conductivity of blanket (𝑘𝑏) are investigated, and results are
presented in Figure 4.12. As expected, a thicker blanket lowers the phreatic surface more from the
upstream slope resulting in greater reduction of pore water pressures. Thus, the FS increases with
the increase of blanket thickness (Figure 4.12a). Similarly, the greater hydraulic conductivity
ensures faster dissipation, thereby increases FS corresponding to 43 days (Figure 4.12b). However,
the hydraulic conductivity higher than 10−3cm/s ensures nearly maximum lowering of the phreatic
at 43 days; thus, it does not provide additional improvement of the stability. Blanket thickness of
6.4 m with 𝑘𝑏 = 10−2 cm/s are chosen for probabilistic analysis as it yields a FS of 1.20. The
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selected hydraulic conductivity can be achieved by using sandy gravel fill material. Continuous
maintenance is required for the blanket since it may be susceptible to surface erosion due to
fluctuation of reservoir water level, and the drains may be clogged due to siltation.
4.6 Comparison of Three Remedial Measures
Figure 4.13 and Figure 4.14 show a comparison of three remedial measures discussed here in terms
of horizontal displacement and probability of exceeding a LS (POELS) with time at point ‘A’
(Figure 4.3). The rate of displacement or velocity changes after 30 days as the drawdown rate
increases from 0.12 m/day to 0.52 m/day at this stage, resulting in faster removal of stabilizing
hydraulic boundary loads (Figure 4.13). Although all of these repair actions yield approximately a
FS of 1.20, they differ in terms of horizontal movement and, thereby, the probability of exceeding
a LS (POELS) with time. The POELS corresponding to without any remedial measure could not
be calculated, since the upstream slope is in marginal stability condition and causes numerical
convergence issues when the variables are changed plus/minus one standard deviation for
calculating POELS. However, at marginal stability condition, the POELS can be expected to
approach unity.
The excavation of bench leads to lowest displacement and POELS compared to other methods
since it does not only reduce shear stress along the potential slip surface but also causes smaller
and shallower potential slip surface in this case (Figure 4.9a). Installation of soil nails causes
highest displacement and POELS compared to other methods as the nails are not prestressed, and
the upstream slope must experience movements to develop resistance against sliding. Compared
to the bench, soil nails cause 44.4 % higher displacement and twice more POELS at 43 days
(Figure 4.13 and Figure 4.14). The potential slip surface corresponding to the soil nail is also
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deeper and larger (Figure 4.7). On the other hand, drainage blanket causes intermediate
deformation and hence POELS with relatively smaller potential slip surface compared to soil nail
(Figure 4.11b).
The excavation of bench seems to be the most effective approach among the three analyzed
methods as it offers lowest risk associated with slope failure (risk=probability of exceedance ×
consequences). The consequences associated with the shallower slip surface is smaller compared
to the deeper sliding [49]. Thus, excavating a bench not only causes a lower probability of
exceedance, but it may also reduce anticipated consequences owing to potential shallower slip
surface. However, the crest width of the dam will be reduced by 4.0 m if the bench option is
adopted and implemented in the field (Figure 4.9a).
4.7 Summary and Conclusions
Three methods representing three different categories are investigated here to stabilize the
upstream slope failure of Pilarcitos dam. These methods improve stability by providing
reinforcement on the upstream slope (soil nails), reducing slope height to decrease the shear stress
(bench), and lowering phreatic surface to decrease pore water pressure (drainage blanket). They
are analyzed and compared in terms of probability of exceeding the ultimate limit state associated
with horizontal deformation at slip surface toe. For comparison, pore water pressure and stability
factor of safety are also calculated using partially coupled and uncoupled transient analysis. The
following major conclusions can be drawn based on the results presented here:
Coupled transient seepage analysis model predicts lower pore water pressures after
drawdown compared to the partially coupled and uncoupled analyses. Only coupled
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analysis yields lower pore water pressure at the selected three points along the embankment
slope compared to the values computed using upper bound equation by Barrett and Moore’s
method.
The use of a given pore water pressure prediction model significantly affects the magnitude
of stability factor of safety (FS), and location and size of potential slip surface. Only the
coupled analysis yielded representative FS and maximum thickness of potential slip surface
for Pilarcitos dam failure compared to the partially coupled and uncoupled analysis.
Soil nails tend to cause highest horizontal movement and probability of exceeding the limit
state compared to other two methods as they do not generate resisting force until there is
sufficient movement within the soil mass.
On the other hand, excavating a bench leads to lowest horizontal deformation and
exceedance probability since lowering the height of slope by constructing a bench does not
only reduce shear stress along the potential slip surface but also causes smaller and
shallower potential slip surface.
Given the set of conditions used in this study, excavating a bench appears to be the most
effective measure in terms of risk among the three analyzed remedial methods due to the
anticipated lower probability of exceedance and shallower potential slip surface, which
deems to cause lower consequence.
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Selection of stress range and curve-fitting techniques in determining strength parameters
influence the FS significantly. Strength parameters corresponding to the high stress-range
with zero cohesion intercept results in 17.7% decrease of FS compared to strength
parameters corresponding to lower stress-range.
In this study, each method is analyzed independently in order to focus on their individual
performance on improving the embankment slope. However, a combination of several methods
might be the most suitable approach for the analysis configurations used herein.
111
Tables:
Table 4.1. Different types of slope repair methods with applicable soils.
Categories Methods Applicable soil
Drainage
methods
Surface drainage Most types of soils*
Horizontal drains Fine soils**
Drain wells and stone columns Fine soils
Wellpoints and deep wells Fine soils
Trench drains Fine soils
Drainage galleries Fine soils
Finger drains Fine soils
Earthwork
methods
Excavate bench Clay and weathered rock
Flatten slope Most types of soils
Rebuilding and compaction Most types of soils
Soil substitution Most types of soils
Buttress fills Most types of soils
Mechanical
methods
Prestressed anchors and anchored
walls Most types of soils
Gravity walls, MSE walls, and soil
nailed walls Most types of soils
Reinforcing piles and drilled shafts Most types of soils
Tire bales High plasticity clay
Geosynthetics Most types of soils
Recycled plastic pins Most types of soils
Gabions Silt & clay
Soldier piles and laggings Most types of soils
Sheet piles Most types of soils except cobbles and
boulders
Additives
Cement Most types of soils
Lime Clay, clayey silt and dry clayey sand
Fly ash Silt and clay with high plasticity
Injection
methods
Lime piles and lime slurry piles Most types of soils
Cement grout Most types of soils
Microbially Induced Calcium
Carbonate Precipitation (MICP) Silt & coarse-grained soils
112
Table 4.1 (continued).
Biotechnical
stabilization
Vegetation Most types of soils (not suitable for dry or
acidic soil)
Brush layering
Log terracing
Live stacking
Live fascine
Branch packing
Live crib wall
Other
methods
Thermal treatment Clay
Bridging Most types of soils
* Gravel, sand, silt, and clay
** Silt and clay
Table 4.2. Soil properties.
Soil Parameters Symbol (unit) Value
Unit weight 𝛾 (kN/m3) 21.2
Effective cohesion 𝑐′(kPa) 0
Effective angle of internal friction 𝜑′(degrees) 45
Reference secant stiffness in standard drained triaxial test 𝐸50𝑟𝑒𝑓 (MPa) 10.8
Reference tangent stiffness for oedometer loading 𝐸𝑜𝑑𝑒𝑟𝑒𝑓
(MPa) 10.8
Reference unloading/ reloading stiffness 𝐸𝑢𝑟𝑟𝑒𝑓
(MPa) 43.2
Hydraulic conductivity 𝑘 (cm/s) 4.0 x 10-8
Unsaturated properties
(van Genuchten parameters)
𝜃𝑟 0.068
𝜃𝑠 0.38
𝑔𝑎 (1/m) 0.80
𝑔𝑛 1.09
113
Table 4.3. Horizontal displacement corresponding to each major variable for soil nailing at 43
days.
Soil Parameter
(unit)
μ σ μ-/+σ 𝑢𝑥 Δ𝑢𝑥
kN/m3 21.20 1.85
19.35 0.496
0.361
23.05 0.135
Eoed(MPa) 10.8 1.00
9.80 0.266
0.026
11.80 0.240
45.00 3.15
41.85 0.262
0.076
48.15 0.186
kx (cm/s) 4.00E-08 2.48E-08
1.52E-08 0.383
0.156
6.48E-08 0.227
kv (cm/s) 4.00E-08 2.48E-08
1.52E-08 0.185
-0.089
6.48E-08 0.274
114
Table 4.4. Calculating the probability of exceeding limit state (POELS) at 43 days using the joint
probability of major variables.
Standard deviation (𝜎𝑗) 0.20555
Mean (j) 0.26000
Coefficient of variation (Vj) 0.79058
Reliability index (βln) 1.54854
Reliability, R= (βln) 0.93925
Probability of exceeding limit state (POELS) 0.06075
Table 4.5. Pore pressure predictions from different methods.
Coordinates
(m)
Barrett and Moore (1975)
*
Partially
coupled
(SEEP/W)
Coupled
(PLAXIS)
Uncoupled
(PLAXIS)
Poi-
nts
Dist. EL
1
(kPa)
u
(kPa)
u(kPa) u(kPa) u(kPa) u(kPa)
1 29.08 199.34 101.0 90.9 22.8 29.7 17.4 73.9
2 48.03 204.18 26.3 23.7 39.7 41.8 31.6 59.5
3 56.10 209.28 0.3 0.2 13.0 6.1 2.7 9.6
* Pore pressure factor was assumed as 0.9 for Barrett and Moore’s method
115
Table 4.6. Effect of pore water pressure prediction on FS after drawdown (at 43 days).
Methods Programs Pore water pressure components
FS
Coupled PLAXIS
Seepage-induced, stress-induced,
and consolidation
1.03
Partially coupled SLOPE/W
Seepage-induced and
consolidation
0.82
Uncoupled PLAXIS Seepage only -
Table 4.7. Properties of soil nail and facing.
Parameter Unit Type/Value
Nailing type - grouted
Elastic modulus of nail (𝐸𝑛) GPa 200
Elastic modulus of grout (𝐸𝑔) GPa 22
Diameter of reinforcement (d) mm 25
Drill hole diameter (𝐷𝐷𝐻) mm 100
Spacing (𝑆ℎ𝑋 𝑆𝑉) m x m 1.0 x1.0
Shotcrete facing thickness (t) mm 80
116
Table 4.8. Nail parameters adopted for FE simulations in PLAXIS.
Parameter Unit Type/Value
Nail element and material model - Plate and elastic
Axial stiffness (𝐸𝐴) kN/m 228.7 x 103
Flexural rigidity (𝐸𝐼) kN m2/m 142.9
Equivalent plate thickness (𝑑𝑒𝑞) mm 86.6
Poisson’s ratio () - 0.3
Interface strength (𝑅𝑖𝑛𝑡𝑒𝑟) 1.0
117
Figures:
Figure 4.1. Model geometry and discretized mesh in PLAXIS 2D.
Figure 4.2. Simulation of isotropic consolidated undrained triaxial tests of soil
0
100
200
300
400
500
600
700
0.00 0.05 0.10 0.15 0.20 0.25
Dev
iato
ric
stre
ss, q (
kP
a)
Axial strain
Initial cell pressure = 10 kPa
Initial cell pressure = 100 kPa
Initial cell pressure = 500 kPa
118
Figure 4.3. Shear strained zone indicating potential slip surface after drawdown.
Figure 4.4. Comparison of pore water pressure predictions by different methods after drawdown.
0.0
20.0
40.0
60.0
80.0
0 1 2 3 4
Pore
wat
er p
ress
ure
(kP
a)
Points along the potential slip surface
Uncoupled (Plaxis)
Partially coupled (Slope/w)
Coupled (Plaxis)
Barrett and Moore (1975)
Point A
119
(a)
(b)
Figure 4.5. Prediction of pore water pressure with time at point 1 using different models- (a) until
the establishment of steady-state condition; (b) for the first 43 days only.
0
20
40
60
80
100
120
0 500 1000 1500 2000
Po
re w
ater
pre
ssu
re (
kP
a)
Time (days)
Uncoupled (Plaxis)
Partially coupled (Slope/w)
Coupled (Plaxis)
0
20
40
60
80
100
120
0 10 20 30 40
Pore
wat
er p
ress
ure
(kP
a)
Time (days)
Uncoupled (Plaxis)
Partially coupled (Slope/w)
Coupled (Plaxis)
45 180
2050
120
(a)
(b)
Figure 4.6. Factor of safety calculation in SLOPE/W- (a) using the slip surface corresponding to
coupled analysis; (b) using the critical slip surface from partially coupled analysis.
121
Figure 4.7. Model with soil nails (length of nail = 10 m and orientation of nail = 20).
122
(a)
(b)
Figure 4.8. (a) Influence of nail length on FS at 43 days with 15 nail orientation (b) influence of
nail orientation and strength parameters on FS at 43 days with 10 m long nail.
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
5 7 9 11 13 15
Fac
tor
of
Saf
ety (
FS
)
Length of nail (m)
0.80
0.90
1.00
1.10
1.20
1.30
0 10 20 30 40
Fac
tor
of
Saf
ety (
FS
)
Orientation of nail (degree)
phi=45°, c= 0 kPa (base case)
phi=32°, c= 8 kPa (case A)
phi=34°, c= 0 kPa (case B)
123
(a)
(b)
(c)
Figure 4.9. Model with excavating a bench at EL=205.2 m with the inclination angle of (a) =
0, (b) = +10, and (c) = -10
= 0
= +10
= -10
B=4m
124
Figure 4.10. Effect of bench location and inclination () on FS.
1.00
1.10
1.20
1.30
1.40
1.50
1.60
200 202 204 206 208 210
Fac
tor
of
Saf
ety, F
S
Elevation, EL (m)
125
(a)
(b)
Figure 4.11. (a) Model with upstream drainage blanket; (b) potential slip surface with 6.4 m thick
drainage blanket.
𝑡𝑏=6.4 m
𝑡𝑏=6.4 m
126
(a)
Figure 4.12. (a) Influence of blanket thickness on FS at 43 days with 𝑘𝑏 = 10−2 cm/s; (b)
influence of hydraulic conductivity of blanket on FS at 43 days with 𝑡𝑏= 6.4 m.
1.00
1.05
1.10
1.15
1.20
1.25
5.5 5.7 5.9 6.1 6.3 6.5 6.7 6.9 7.1
Fac
tor
of
Saf
ety
Blanket thickness (m)
1.00
1.05
1.10
1.15
1.20
1.25
1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Fac
tor
of
safe
ty
Hydraulic conductivity of blanket, kb (cm/s)
127
Figure 4.13. Effect of remedial measures on horizontal deformation at slip surface toe.
Figure 4.14. Probability of exceeding limit state for three remedial measures.
0.00
0.20
0.40
0.60
0.80
0.00 10.00 20.00 30.00 40.00
Ho
rizo
nta
l d
efo
rmat
ion
(m
)
Time (days)
Nail
Bench
Blanket
w/o measure
1.0E-10
1.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 10 20 30 40 50 60
Pro
bab
ilit
y o
f ex
ceed
ing U
LS
Time (days)
Nail
Bench
Blanket
128
5 CHAPTER 5. SUMMARY, CONCLUSIONS, CONTRIBUTIONS, AND FUTURE
WORKS
5.1 Summary and Conclusions
In this work, the performance of earthen embankments subjected to cyclic hydraulic loading
associated with extreme events are evaluated using strain-based limit state approach. Analysis are
performed using unsaturated coupled transient seepage method and non-liner advanced elasto-
plastic constitutive relation in finite element (FE) program PLAXIS. For comparative study,
unsaturated transient seepage analysis and stability analysis are also conducted using FE program
SEEP/W and limit equilibrium software SLOPE/W.
In chapter 2, strain-based limit state (LS) analyses and conventional slope stability factor of safety
(FS) approach are used to assess the effect of rise and fall of water levels, representing severe
storm cycles, on the stability of the Princeville levee. The effect of repeating storm cycles, the
degree of uncertainty, and hydraulic conductivity anisotropy on the probability of exceedance of
a given LS versus the FS computed using the limit equilibrium method (LEM) and strength
reduction method (SRM) is discussed. The results from the strain-based approach are used for risk
assessment to demonstrate the effect of including hydraulic loading history on risk assessment.
Based on the results of this study, the following conclusions are drawn:
The strain-based analyses results show a progressive development of plastic shear strain
within the levee as the number of storm cycles is increased.
The shear strain is gradually expanding form the toe to the crest with shear band
progressively forming and causing cascading instability with increasing number of storm
cycles.
129
The deterministic FS obtained from LEM remains unchanged with increased number of
storm cycles.
The FS is affected by rate of rise/drawdown of the water level. The consideration of
instantaneous drawdown, instead of a more realistic rate based on storm hydrograph, yields
a lower minimum FS.
The increase in number of storm cycles, the degree of uncertainty, and anisotropy
associated with material properties all lead to an increase in probability of exceeding a
given LS.
For a given consequence associated with a flood event, the increase in probability of failure
due to increased number of storm cycles led to the transition from acceptable to an
unacceptable risk, based on comparison with a published criteria.
In chapter 3, strain-based performance limit state that corresponds to the instability of embankment
slopes is defined. A simple linear relationship between the shear strain and monitorable
deformation at the toe of the embankment is developed as a function of the geometry of the slope.
This relationship provides a simple means to estimate the performance limit state that corresponds
to the instability of embankment slopes, and the critical shear strain at the embankment toe, using
the stress-strain data obtained from triaxial testing. Based on the results obtained from this study,
the following conclusions can be drawn:
The stability analysis results show a gradual decrease in FS of the upstream slope using
SRM as the number of drawdown cycles is increased.
The onset of the instability of the slope is preceded by a gradual accumulation of surface
displacement with its rate accelerating with the continuity of hydraulic load cycles.
130
Parametric studies show a good agreement with the developed correlation between the
shear strain and deformation at toe for rotational slope movements. Also, the criterion from
the correlation shows good agreement with data from field studies by others.
The predicted pore pressure after rapid drawdown from the coupled analysis is observed to
be smaller compared to the uncoupled analysis.
In chapter 4, the efficacy of three repair methods representing three different categories to stabilize
an earth embankment under rapid drawdown is investigated. These methods improve stability by
providing reinforcement on the upstream slope (soil nails), reducing slope height to decrease the
shear stress (bench), and lowering phreatic surface to decrease pore water pressure (drainage
blanket). They are analyzed and compared in terms of probability of exceeding the ultimate limit
state associated with horizontal deformation at slip surface toe. For comparison, pore water
pressure and stability factor of safety are also calculated using partially coupled and uncoupled
transient analysis. The following major conclusions can be drawn based on the results presented
in chapter 4:
Soil nails tend to cause highest horizontal movement and probability of exceeding the limit
state compared to other two methods as they do not generate resisting force until there is
sufficient movement within the soil mass.
Excavating a bench leads to lowest horizontal deformation and exceedance probability
since lowering the height of slope by constructing a bench does not only reduce shear stress
along the potential slip surface but also causes smaller and shallower potential slip surface.
Given the set of conditions used in this study, excavating a bench appears to be the most
effective measure in terms of risk among the three analyzed remedial methods due to the
anticipated lower probability of exceedance and shallower potential slip surface, which
131
deems to cause lower consequence.
The selection of pore water pressure prediction model significantly affects the magnitude
of stability factor of safety (FS), maximum depth of potential slip surface, and the required
time to establish steady-state conditions.
5.2 Contributions
The primary contributions of this study are listed as follow:
Explanation of the underlying kinematics of emerging shear band and progressive
instability due to repeated hydraulic loading due to storm.
Incorporation of hydraulic loading history in the stability analysis in order to quantify
increased risk for the future storm event.
Definition of ultimate limit state in terms of accumulated deformation that correspond to
the instability of slopes.
Development of correlation between the shear strain and the corresponding surface
deformation.
Demonstration of selecting most effective approach from several feasible approaches
within the context of reducing exceedance probability of ultimate limit state.
5.3 Suggested Future Works
The work presented in this study attempted to define a strain-based ultimate limit state (ULS) that
corresponds to the instability of embankment slopes and to develop a correlation between shear
strain and corresponding surface deformation to quantify the ULS. Several topics may be worthy
to explore in future to fill the gaps of this this study, such as:
The developed correlation to predict the ultimate limit state has been verified herein using
132
a two-dimensional (2-D) plane strain finite element model. In 2-D analysis, either the
maximum cross-section (highest or maximum amount of soil involved in potential sliding)
or the cross-section that gives a minimum factor of safety is generally considered for
stability analysis. A three-dimensional model can be simulated in the future study to
validate the developed correlation for different sections and to identify the potential section
that gives maximum deformation.
Analyses are performed here using the hardening soil model, which cannot simulate the
strain-softening behavior of soil. Future studies using a constitutive relation that can model
strain-softening behavior (e.g., hypoplastic model) can ensure the applicability of the
developed correlation for soils with strain-softening characteristics (e.g., stiff clays).
The application of the developed correlation has been studied here in relation to the failure
associated with cyclic hydraulic loading. Similar correlations may be developed in future
for cases where different failure mechanism works (e.g., creep).
The relationship between shear strain and surface deformation has been developed by
assuming a rotational movement of failure wedge. Slopes may experience translational
movement as well during sliding, which may be investigated.
133
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7 APPENDICES
147
Appendix A
Pore pressure predictions from different methods.
Points
(Figure
3.2)
Coordinate Bishop (1954)
Barrett and
Moore (1975)
Calculated
(Coupled)
Calculated
(Uncoup-
led)
x y
uo
(kPa)
v
(kPa)
u
(kPa)
u
(kPa)
u*
(kPa)
u
(kPa)
u
(kPa)
u
(kPa)
2 -1.0 -0.2 31.7 26.2 26.2 5.5 21.0 10.7 5.5 31.6
4 -2.8 0.0 29.4 20.3 20.3 9.1 16.2 13.2 7.7 29.4
6 -6.1 0.6 23.5 9.4 9.4 14.1 7.6 16.0 9.7 23.6
8 -8.4 1.7 12.3 1.9 1.9 10.5 1.5 10.9 5.6 12.3
* Change in pore pressure is assumed as 0.8 times the change in vertical stress.
148
Appendix B
Sample calculation of FS after first drawdown phase using the principal stresses obtained from
FEM.
Points
(Figure
3.2)
x
(m)
y
(m)
𝜎1,𝑖′
(kPa)
𝜎3,𝑖′
(kPa)
𝜑
(deg)
𝑐
(kPa)
𝛼𝑓
(deg)
𝜎𝑛,𝑖′
(kPa)
𝜏𝑚𝑜𝑏,𝑖
(kPa)
𝜏𝑚𝑎𝑥,𝑖
(kPa)
𝑙
(m)
𝐷𝑖
(kN)
𝑅𝑖
(kN)
1 -0.1 0.0 3.3 0.1 27.0 1.0 58.5 0.9 1.4 1.5 0.5 0.8 0.8
2 -1.0 -0.2 9.2 2.6 27.0 1.0 58.5 4.4 3.0 3.2 0.9 2.7 2.9
3 -1.9 -0.2 11.8 3.7 27.0 1.0 58.5 5.9 3.6 4.0 0.9 3.3 3.7
4 -2.8 0.0 13.4 4.5 27.0 1.0 58.5 6.9 4.0 4.5 1.2 4.7 5.3
5 -4.2 0.1 17.6 6.0 27.0 1.0 58.5 9.2 5.2 5.7 1.7 8.8 9.6
6 -6.1 0.6 18.7 6.4 27.0 1.0 58.5 9.8 5.5 6.0 1.6 8.9 9.7
7 -7.3 1.1 16.3 5.5 27.0 1.0 58.5 8.4 4.8 5.3 1.3 6.2 6.8
8 -8.4 1.7 13.5 4.3 27.0 1.0 58.5 6.8 4.1 4.5 1.0 4.2 4.6
9 -9.1 2.2 11.2 3.4 27.0 1.0 58.5 5.5 3.5 3.8 0.8 2.9 3.2
10 -9.6 2.9 3.3 0.3 27.0 1.0 58.5 1.1 1.3 1.6 0.6 0.7 0.9
sum= 43.2 47.7
The major and minor principal stress (𝜎1,𝑖 and 𝜎3,𝑖) at any given point, 𝑖 along the potential slip
surface is utilized for the calculation of normal stress (𝜎𝑛,𝑖) and mobilized shear stress (𝜏𝑚𝑜𝑏,𝑖)
using the Eq. B1 and Eq. B2 respectively [97, 114].
𝜎𝑛,𝑖 =𝜎1,𝑖 + 𝜎3,𝑖
2+𝜎1,𝑖 − 𝜎3,𝑖
2cos(2𝛼𝑓) (B1)
𝜏𝑚𝑜𝑏,𝑖 =𝜎1,𝑖 − 𝜎3,𝑖
2sin(2𝛼𝑓) (B2)
Where, 𝛼𝑓 = angle of failure plane with respect to the minor principal stress that can be determined
149
using the following expression:
𝛼𝑓 = 45° +𝜑
2 (B3)
The maximum available shear strength (𝜏𝑚𝑎𝑥,𝑖) at any given point may be computed using the
Mohr-Coulomb failure criterion as follow:
𝜏𝑚𝑎𝑥,𝑖 = 𝜎𝑛,𝑖 tan𝜑 + 𝑐 (B4)
Now, the maximum available shearing resistance (𝑅𝑖) for a given segment (𝑙𝑖) along the slip
surface is determined from Eq. B5. Similarly, the mobilized shearing or driving forces (𝐷𝑖) can be
obtained from Eq. B6.
𝑅𝑖 = 𝜏𝑚𝑎𝑥,𝑖𝑙𝑖 (B5)
𝐷𝑖 = 𝜏𝑚𝑜𝑏,𝑖𝑙𝑖 (B6)
The FS is now calculated by dividing the total available maximum shearing resistance by the total
amount of mobilized shear stress along the slip surface:
𝐹𝑆 =∑ 𝑅𝑖𝑛𝑖=1
∑ 𝐷𝑖𝑛𝑖=1
(B7)
Where n= number of segments considered along the slip surface.
𝐹𝑆 =∑ 𝑅𝑖𝑛𝑖=1
∑ 𝐷𝑖𝑛𝑖=1
=47.7
43.2= 1.10
150
Appendix C
The shear strain (𝛾𝑠) is defined by:
𝛾𝑠 =√2
3√(𝜖1 − 𝜖2)2 + (𝜖2 − 𝜖3)2 + (𝜖3 − 𝜖1)2 (C1)
Where 𝜖1, 𝜖2, and 𝜖3 are the major, intermediate, and minor principal strains, respectively.
The volumetric strain (𝜖𝑣) can be calculated as:
𝜖𝑣 = 𝜖1 + 𝜖2 + 𝜖3 (C2)
The principal strains 𝜖2 and 𝜖3 are equal for triaxial test conditions. Thus, Eq. (C1) and (C2) is
reduced to:
𝛾𝑠 =2
3 (𝜖1 − 𝜖3) (C3)
𝜖𝑣 = 𝜖1 + 2𝜖3 (C4)
The 𝜖𝑣 can be assumed zero at undrained condition. Therefore, Eq. (C4) can be rearranged as:
𝜖1 = −1
2𝜖3 (C5)
Substitution of Eq. (C5) into Eq. (C3) gives:
𝛾𝑠 = 𝜖1 (C6)
Thus, the shear strain (𝛾𝑠) can be obtained from the axial strain value (𝜖1) of undrained triaxial
test.