1 ABSTRACT - NCSU

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.

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.

iii

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.

iv

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.

v

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

vi


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


4.3.3 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

vii

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

viii

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

ix

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

xi

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



) 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


), 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.


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

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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



) 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



) 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

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chosen to minimize boundary effect (i.e., further increase of model size does not change the

results).


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


(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:


=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.

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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)


*

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



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)


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)


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)


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.

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