Influence of Excavation Rate and Tension Crack on the Stability of an Unsupported Vertical Cut in Unsaturated Soil
by
B.Tech. (Civil Engineering), Amity University Rajasthan, 2017
A Thesis Submitted in Partial Fulfillment of the requirements for the Degree of
Master of Science in Engineering
In the Graduate Academic Unit of Civil Engineering
Supervisor: Won Taek Oh, Ph.D., P.Eng., Department of Civil Engineering
Examining Board: Kripa Singh, Ph.D., P.Eng., Department of Civil Engineering Othman Nasir, Ph.D., P.Eng., Department of Civil Engineering Karl Butler, Ph.D., P.Geo., P.Eng., Department of Earth Sciences
This Thesis is accepted by the Dean of Graduate Studies
THE UNIVERSITY OF NEW BRUNSWICK
April 2020
many projects ranging from mining to infrastructure developments are initiated from
unsupported excavation. Unsupported excavation should be carried out with the utmost
caution since the failure of unsupported cuts can result in not only property losses but also
fatalities. This research focuses on the stability of unsupported vertical cuts in unsaturated
soils. For this, a series of numerical analyses is conducted in an unsaturated glacial till
considering three factors: i) excavation rate, ii) depth and location of tension crack, and iii)
rainfall infiltration into a tension crack. The results showed that the influence of excavation
rate is not significant if an unsupported vertical cut is made to a safe height (i.e. critical
height divided by factor of safety (1.2 in this study)). Tension crack is one of the major
factors that can lead to the failure in unsupported vertical cuts; however, the factor of safety
did not drop below unity if the location and depth of tension crack is limited within 20%
and 30% of the safe height from the cut wall and the ground surface, respectively. Rainfall
infiltration into a tension crack decreased the factor of safety with time and then eventually
led to the failure in unsupported vertical cuts for most cases. However, extremely long
duration of rainfall was required if the initial factor of safety with a tension crack is close
to 1.2. The proposed approaches are then applied to a deep unsupported vertical cut (9.75
m) made into a clay for its validation, which successfully estimated the critical location
and depth of a tension crack.
iii
ACKNOWLEDGEMENTS
Foremost, I would like to express my deep and sincere gratitude to my research supervisor
Dr. Won Taek Oh for giving me the opportunity by accepting me as his research student
and providing invaluable guidance throughout the study. He taught me methods to widen
my horizons on carrying out this research and present the works as clearly as possible. I
am extremely grateful for the freedom he offered me to pursue all possible avenues
diverging from academics. I would also like to thank him for his empathy, patience and all
the great tennis sessions shared with him.
I am extremely grateful to my parents for their love, prayers, and sacrifices for educating
and preparing me for my future. I am thankful to all my friends for making this journey a
lot more pleasant and becoming an extended part of my family in this home away from
home. I extend my special thanks to not just my colleagues but also great friends, Gregory
Brennan, and Mehdi Poormousavian. Greg, I couldn’t thank you more for the patience and
assistance you offered throughout my initial stage of study. It is safe to say that you’ve
successfully taught me the unconventionally effective principal of “fitting a square in a
circle”. Mehdi, I thoroughly enjoyed all the lectures we shared together and the
brainstorming sessions on soil mechanics which frequently ended up making us more
confused than we were.
Finally, I would like to thank the staff of UNB Civil Engineering for always being so
flexible to my needs. I could not have pursued the opportunities I did throughout my degree,
if it wasn’t always for the assistance provided by you.
iv
1. GENERAL INTRODUCTION .............................................................................. 1
1.1 Problem Statement ............................................................................................... 1
2. TENSION CRACKS ............................................................................................... 8
2.3 Estimating Depth of Tension Crack ................................................................... 15
2.4 Estimating Permeability Function of Tension Crack ......................................... 20
2.4.1 Zhou et al. (1998) ........................................................................................ 21
2.4.2 Hu et al. (2000) ........................................................................................... 22
2.4.3 Ping et al. (2005) ......................................................................................... 23
2.4.4 Zhang and Li (2012) ................................................................................... 24
2.4.5 Zhang et al. (2020) ...................................................................................... 25
2.4.6 Studies Questioning Modelling of Infiltration into Tension Crack ............ 26
3. EFFECTIVE AND TOTAL STRESS APPRAOCHES IN UNSATURATED SOILS ..................................................................................................................... 27
4. METHODOLOGY OF NUMERICAL MODELING ....................................... 31
4.1 Soil Properties .................................................................................................... 31
v
4.5 Simulation of Tension Crack and Rainfall Event............................................... 39
5. ANALYSIS RESULTS AND DISCUSSION ...................................................... 44
5.1 Influence of Excavation Rate on the Stability of UVC ...................................... 44
5.1.1 Estimation of Safe Height (FOS = 1.2) ....................................................... 44
5.1.2 Estimation of Critical Height (FOS = 1) ..................................................... 49
5.1.3 Summary and Conclusions ......................................................................... 55
5.2 Determining the Influence of Tension Crack on Stability of Unsupported Vertical
Cut…. ................................................................................................................. 56
5.2.1 Determining the Critical Height of UVC using Numerical Method ........... 56
5.2.2 Estimating a Depth of Tension Crack using Pufahl et al. (1983) approach 62
5.2.3 Determination of Critical Tension Crack .................................................... 66
5.2.4 Summary and Conclusions ......................................................................... 70
5.3 Influence of Rainfall Infiltration into Tension Crack on the Stability of UVC . 71
5.3.1 Methodology ............................................................................................... 71
5.3.2 Stand-Up time of UVC with Tension Crack under Different Rainfall
Intensities .................................................................................................... 76
6. CASE STUDY ........................................................................................................ 79
6.2 Soil Properties .................................................................................................... 82
7. GENERAL CONCLUSIONS ............................................................................... 94
REFERENCES ................................................................................................................ 97
CURRICULUM VITAE
LIST OF TABLES
Table 4.1. Basic soil properties of Indian Head Till (Vanapalli 1996) ............................. 32
Table 5.1. Variation of factor of safety for different combinations of DPtc and DStc with
GWT at 1, 3 and 5 m. ........................................................................................................ 61
Table 5.2. Stand-up time of UVC with a tension crack under different rainfall events
(Initial FOS of UVC is close to 1.2). ................................................................................ 77
Table 6.1. Material properties used in the numerical analysis (adopted from Kwan 1971
and Banerjee et al. 1988). ................................................................................................. 83
vii
LIST OF FIGURES
Figure 1.1. Variation of deformation, pore-water pressure, and FOS with time for (a) 10,
(b) 250 (c) 500, and (d) 750 seconds after 1.3m excavation stage in sand with initial
ground water table at 0.7 m (Richard 2018). ...................................................................... 3
Figure 2.1. Semi-infinite cohesive mass with horizontal surface: (a) stresses at boundaries
of prismatic element; (b) graphic representation of state of stress at failure; (c) shear
pattern for active state; (d) shear pattern for passive state; (e) stresses on vertical section
through the mass (Terzaghi 1943). ..................................................................................... 8
Figure 2.2. Monthly rainfall and FOS during the period concerned (Gofar et al. 2006). . 10
Figure 2.3. Schematic of test specimen and loading used to determine critical state energy
release rate (Lee et al. 1988). ............................................................................................ 12
Figure 2.4. Crack patters for UVC with different depth of cut (H) and Poisson’s ratio (ν):
(a) H = 6 m, ν = 0.48, (a) H = 12 m, ν = 0.41, (a) H = 18 m, ν = 0.41 (Lee et al. 1988) .... .
........................................................................................................................................... 14
Figure 2.5. Earth pressure distribution diagram used to calculate a factor of safety of the
temporary vertical cut against general failure using the field measurement data in
Whenham et al. (2007) (after Richard et al. 2020). .......................................................... 20
Figure 2.6. Simulating cracked upper layer as an equivalent weaker layer (Hu et al.
2000). ................................................................................................................................ 22
Figure 2.7. Permeability function of a tension crack used by Hu et al. (2000). ............... 23
Figure 2.8 Permeability functions for a fine-grained soil and a crack used by Zhang and
Li (2012). .......................................................................................................................... 25
Figure 3.1. Variation of load, stress, pore pressure, strength, and factor of safety at point
A due to excavation in saturated clay (modified after Bishop and Bjerrum 1960). ......... 28
Figure 3.2. Drops in pore-water pressure due to excavation measured from piezometer
sensors located near excavation and those estimated using FLAC (Galera et al. 2009). . 29
Figure 4.1. Grain size distribution curve of Indian Head Till (Oh and Vanapalli 2010) .. 32
Figure 4.2. SWCC of Indian Head Till (Oh and Vanapalli 2018). ................................... 33
Figure 4.3. Permeability function of Indian Head Till ...................................................... 33
viii
Figure 4.4. Meshes and boundary conditions established using ‘Insitu’ analysis type in
SIGMA/W. ........................................................................................................................ 35
Figure 4.5. Simulating staged excavation and assigning water total head hydraulic
boundary condition in SIGMA/W. ................................................................................... 36
Figure 4.6. Vertical stress contours computed with SIGMA/W (SIGMA/W manual) ..... 37
Figure 4.7. Slope stability analysis in SLOPE/W using ‘Entry and Exit’ surface option. 38
Figure 4.8. Example of analysis tree. ................................................................................ 39
Figure 4.9. Simulating tension crack using tension crack line feature in SLOPE/W. ...... 40
Figure 4.10. Example of stability analysis result using tension crack line feature in
SLOPE/W. ........................................................................................................................ 40
Figure 4.11. Slope stability analysis considering a tension crack. Tension crack was
simulated as a void. ........................................................................................................... 41
Figure 4.12. Simulating rainfall infiltration in SIGMA/W. .............................................. 42
Figure 4.13. Total head versus time relationships used to simulate the infiltration of
rainfall into a tension crack under different rainfall intensities. ....................................... 43
Figure 5.1. Slope stability analyses results with GWT at 1 m for different excavation rate:
(a) 1 second; (b) 4 hours; (c) 12 hours; and (d) 24 hours. ................................................ 45
Figure 5.2. Stability analyses results with GWT at 3 m for different excavation rate: (a) 1
second; (b) 4 hours; (c) 12 hours; and (d) 24 hours. ......................................................... 46
Figure 5.3. Sability analyses results with GWT at 5 m for different excavation rate: (a) 1
second; (b) 4 hours; (c) 12 hours; and (d) 24 hours. ......................................................... 47
Figure 5.4. Variation of safe height with respect to excavation rate for different levels of
ground water table (i.e. 1, 2, 3, 4, and 5 m). ..................................................................... 48
Figure 5.5. FOS of UVC (a) prior and (b) post equilibrium condition with respect to pore-
water pressure (excavation rate = 1 second, ground water table at 1 m). ......................... 50
Figure 5.6. Stability analyses with 1 second excavation rate for different levels of GWT
(1, 3, and 5 m): (a), (b), (c) coupled - SIGMA/W stress method; (d), (e), (f) Bishop’s
simplified method. ............................................................................................................ 52
Figure 5.7 Comparison of critical height estimated using coupled - SIGMA/W stress and
Bishop’s simplified methods at 1 second excavation rate for different levels of GWT. .. 53
ix
Figure 5.8. Variation of the critical height of UVC in sand (Unimin 7030) with respect to
the level of GWT from coupled – SIGMA/W stress (excavation rates = 10 and 10000 s)
and Morgenstern-Price method (limit equilibrium method) (modified after Richard 2018).
........................................................................................................................................... 54
Figure 5.9. Simulating tension crack for the safe height of 2.55 m with distance ratio
(DStc) fixed at 0.1 and six different depth ratios (DPtc): (a) 0; (b) 0.1; (c) 0.2; (d) 0.3; (e)
0.4; and (f) 0.5. .................................................................................................................. 57
Figure 5.10. Simulating tension crack for the safe height of 2.55 m with depth ratio (DPtc)
fixed at 0.5 and five different distance ratios (DStc): (a) 0.1; (b) 0.2; (c) 0.3; (d) 0.4; and
(e) 0.5. ............................................................................................................................... 58
Figure 5.11. Analysis tree for various depth ratio (DPtc) in SIGMA/W to determine
critical depth of tension crack. .......................................................................................... 59
Figure 5.12. Stability analysis considering tension crack with different combinations of
DStc and DPtc: (a) DStc = 0.4, DPtc = 0.4 (GWT at 1 m); (b) DStc = 0.5, DPtc = 0.5 (GWT
at 3 m). .............................................................................................................................. 60
Figure 5.13. Determination of depth of tension crack based on net active earth pressure
distribution in vadose zone (Eq.(2.15)) extending the approach by Pufahl et al. (1983). 62
Figure 5.14. Positive, negative, and net active earth pressure distribution (GWT = 1 m).63
Figure 5.15 Positive, negative, and net active earth pressure distribution (GWT = 3 m).64
Figure 5.16 Positive, negative, and net active earth pressure distribution (GWT = 5 m). 65
Figure 5.17. Contours of FOS for different combinations of DStc and DPtc with GWT at 1
m ....................................................................................................................................... 67
Figure 5.18. Contours of FOS for different combinations of DStc and DPtc with GWT at 3
m. ...................................................................................................................................... 68
Figure 5.19. Contours of FOS for different combinations of DStc and DPtc with GWT at
5 m. ................................................................................................................................... 69
Figure 5.20. Water pressure and water flux vector distribution in a tension crack
associated with rainfall infiltration. .................................................................................. 72
Figure 5.21. Variation of pore-water distribution with time around a tension crack under
25 mm/hr rainfall intensity (DStc = 0.2, DPtc = 0.3, GWT at 5 m). .................................. 73
x
Figure 5.22. Pore-water pressure distribution and factor of safety for different time step
under 25 mm/hr rainfall intensity with a tension crack (DStc = 0.4, DPtc = 0.1, GWT at 5
m). ..................................................................................................................................... 75
Figure 5.23. Variation of factor of safety with time based on the results in Figure 5.22
and the definition of stand-up time. .................................................................................. 76
Figure 6.1. Soil profile and variation of water content, shear strength parameters and
coefficient of permeability with depth at test site (after Kwan 1971). ............................. 80
Figure 6.2. Timeline of excavation (after Kwan 1970). ................................................... 81
Figure 6.3 Excavation profile and location of piezometers (after Kwan 1970) ................ 81
Figure 6.4. Soil-Water Characteristic Curve for materials used in numerical analysis. ... 84
Figure 6.5. Permeability function for materials used in numerical analysis. .................... 84
Figure 6.6. Mesh and boundary conditions used in the numerical analysis (11853 nodes,
4523 elements). ................................................................................................................. 85
Figure 6.7. Comparisons of measured and estimated (coupled analysis) pore-water
pressure contours (0 (i.e. phreatic line), 60, and 120 kPa) 12 days after removal of top
sediment layer. .................................................................................................................. 86
Figure 6.8. Comparison of measured and estimated pore-water pressure contours (0
(phreatic line) and 60 kPa) prior to failure. ....................................................................... 87
Figure 6.9. Pore-water pressure distribution and FOS prior to failure without tension
crack (coupled - SIGMA/W stress method). ..................................................................... 88
Figure 6.10. Pore-water pressure distribution and FOS prior to failure without tension
crack (Bishop’s simplified method). ................................................................................. 89
Figure 6.11. Identified potential slip surface with the first tension crack (after Kwan
1971). ................................................................................................................................ 90
Figure 6.12. Actual slip surfaces with the second tension crack that led to the failure of
UVC (after Kwan 1971). ................................................................................................... 90
Figure 6.13. Stability analysis with different locations of tension crack from the vertical
cut face: (a) 4.5 m; (b) 3.45 m; and (c) 2.75 m and (d) 2.43 m (coupled – SIGMA/W
stress method). .................................................................................................................. 92
C = total cohesion (kPa)
DPtc = depth ratio for tension crack
DStc = distance ratio for tension crack
E = elastic modulus (kPa)
xii
Hsafe = safe height of unsupported vertical cut (FOS = 1.2)
Hw = depth of water outside slope, measured above toe
hw = distance from ground surface to water table
Ip = plasticity index
K1c = fracture toughness
kc = coefficient of permeability of tension crack
ksat = coefficient of permeability for saturated condition
m, n, a = fitting parameters for Fredlund & Xing’s SWCC model (1994)
mvG, nnG, αvG = fitting parameters for van Genuchten’ s SWCC model (1980)
Ncf = stability number for φ > 0
No = stability number for φ = 0
Pa = atmospheric pressure (i.e. 101.3 kPa)
Pc = capillary pressure at rock joint
Pd = driving force term
ru = pore-pressure coefficient
t = thickness of specimen
uc = load point displacement
ua - uw = matric suction
v = Poisson’s ratio
γsat = saturated unit weight of soil
γw = unit weight of water
αt = ratio of effective cohesion to tensile strength
θ = volumetric water content
θs = volumetric water content for saturated condition
θc = contact angle
λ = specimen compliance
σn = total normal stress
σa = active earth pressure
xiv
σT = interfacial tension
τxy = shear strength in XY direction
φ’ = effective internal friction angle
φ = internal friction angle
b = angle describing rate of change between suction and shear strength
ψ = suction
μ = kinematic viscosity for water (10-6 m2/s in normal temperature)
μq = correction factor for surcharge
μt = correction factor for tension crack
μw = correction factor for water pressure
ξ, ζ = fitting parameters for undrained strength of unsaturated soil
1
most projects ranging from mining to infrastructure developments are initiated from
unsupported excavation. Unsupported excavation should be carried out with the utmost
caution because the failure of unsupported cuts can lead directly to work-related injuries
and deaths (Thompson and Tanenbaum 1977, Suruda et al. 1988, White 2008, Bureau of
Labor Statistics 2010). Between 2000 and 2009, an average of 39 fatalities were reported
annually in the U.S. in association with the failure of unsupported cuts (BLS 2010). Due
to this reason, authorities enforce their own regulations for excavation to safeguard workers
from injuries or deaths. For example, Canadian provinces specify the safe height,
maximum slope angles, benching angles, and minimum distance from an unsupported
vertical cut to stockpiling of excavated or backfill materials. Occupational Safety and
Health Administration (OHSA 2019) also revised excavation manual to make excavation
regulations and standards easier to understand.
Safe height of an unsupported vertical cut (hereafter referred to as UVC) is considered a
governing factor for ensuring safe excavation practices. Canadian provinces limit the
maximum allowable height of UVC to 1.2 m – 1.5 m. On the other hand, OHSA (2019)
allows UVC only in stable rock. However, previous studies showed that UVC with more
than several meters can remain stable in case where excavations are made into unsaturated
soils. (Tsidzi 1997, Whenham et al. 2007, De Vita et al. 2008, Stanier and Tarantino 2013,
Richard 2018).
2
Safe height of UVC in an unsaturated soil can be estimated using a critical height
(maximum excavation depth without failure, Factor of Safety = 1) divided by a certain
safety margin (i.e. factor of safety greater than unity). Critical height can be estimated
based on the net active earth pressure distribution with depth considering the influence of
matric suction on either the cohesion (Richard 2018, Ileme 2019) or the coefficient of
active earth pressure (Vahedifard et al. 2015). Various approaches are available in the
literature to estimate the active earth pressure in unsaturated soils (Pufahl et al. 1983,
Leshchinsky and Zhu 2010, Stanier and Tarantino 2013, Zhang et al. 2010, Vahedifard et
al. 2015).
Excavation causes a temporary drop in the phreatic line, which eventually rebounds with
time after the completion of excavation. In other words, stability of an unsupported cut
continuously varies throughout the excavation process. Richard (2018) conducted
numerical analysis to investigate the critical height of UVC in cohesionless soil. The results
showed that the factor of safety (hereafter referred to as FOS) decreases with time due to
the rebound of phreatic line until the pore-water pressure reaches an equilibrium condition.
The magnitude of drop in phreatic line and its rebound time are governed by excavation
rate and permeability function of a soil, respectively (Figure 1.1).
3
Figure 1.1. Variation of deformation, pore-water pressure, and FOS with time for (a) 10, (b) 250 (c) 500, and (d) 750 seconds after 1.3m excavation stage in sand with initial ground water table at 0.7 m (Richard 2018).
10
-6
2
-2
2
6
14
)
-3.0 -2.8 -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
4
8
12
16
-4
)
-3.0 -2.8 -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
-6
2
6
10
14
18
-2
)
-3.0 -2.8 -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
4
8
12
16
1.30
)
-3.0 -2.7 -2.4 -2.1 -1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0
Factor of Safety
≤ 1.00 - 1.10 1.10 - 1.20 1.20 - 1.30 1.30 - 1.40 1.40 - 1.50 1.50 - 1.60 1.60 - 1.70 1.70 - 1.80 1.80 - 1.90 ≥ 1.90
4
8
12
16
20
1.12
D ep
th (m
) -3.0 -2.7 -2.4 -2.1 -1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0
Factor of Safety
≤ 1.00 - 1.10 1.10 - 1.20 1.20 - 1.30 1.30 - 1.40 1.40 - 1.50 1.50 - 1.60 1.60 - 1.70 1.70 - 1.80 1.80 - 1.90 ≥ 1.90
4
8
12
16
1.32
)
-3.0 -2.7 -2.4 -2.1 -1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0
Factor of Safety
≤ 1.00 - 1.10 1.10 - 1.20 1.20 - 1.30 1.30 - 1.40 1.40 - 1.50 1.50 - 1.60 1.60 - 1.70 1.70 - 1.80 1.80 - 1.90 ≥ 1.90
(c) 500s
(b) 250s
(a) 10s
750s
4
Tension cracks are often found at the crest of slopes and cuts in case the tensile stress
exceeds the tensile strength of a soil (Baker 1981, Bagge 1985). It is well known that
tension cracks have adverse impact on the safe height of UVC due to following reasons: i)
Tension crack usually forms a part of slip surface. This shortens the length of slip surface
and subsequently reduces resistance to slope failure, ii) Additional driving force can be
generated if a tension crack is filled with water, and iii) Tension crack can act as a pathway
for rainfall to seep through the soil and further reduces its shear strength. To reliably
estimate the influence of tension crack on the stability of UVC, three factors should be
considered; i) depth of tension crack, ii) location of tension crack, and iii) penetration of
water into tension crack associated with rainfall events. Several closed-form equations or
approaches are available to estimate the depth and/or location of tension crack (Taylor
1948, Spencer 1968, Bagge 1985, Lee et al. 1988, Kutschke and Vallejo 2011, Baker and
Leshchinsky 2003, Michalowski 2013, Li et al. 2018). However, limited studies have been
undertaken to estimate the safe height and location and depth of tension cracks in UVC in
unsaturated soil. Research on the stability of slopes under rainfall events clearly showed
that rainfall infiltration into tension cracks significantly decreases the factor of safety (Ping
et al. 2005, Wang et al. 2012, Gofar et al. 2006, Sasekaran 2011, Sun et al. 2019, Zhang et
al. 2020). Although these studies successfully addressed the importance of considering
tension cracks under a rainfall event, it is still challenging to simulate the seepage into a
tension crack in numerical analysis.
5
1.2 Objectives of the Thesis
The main objective of this study is to investigate the stability of UVC in unsaturated soil
considering various practical scenarios. More details are as follows:
- Estimate the safe height of unsupported vertical cuts in an unsaturated cohesive soil
(Indian Head till) considering excavation rate.
- Estimate the influence of tension cracks on the stability of unsupported vertical cuts
considering various depths and locations of tension cracks
- Estimate the influence of rainfall infiltration into tension cracks on the stability of
unsupported vertical cuts.
- Validate the adopted methodologies/approaches through a case study (i.e. deep
unsupported cut in a clay, Kwan 1971).
1.3 Scope of the Thesis
This research is carried out for UVC excavated into an unsaturated glacial till (i.e. Indian
Head till). Multiple excavation scenarios were simulated in numerical analysis considering
four factors: i) level of the groundwater table, ii) excavation rate, iii) depth and distance
(i.e. distance from cut wall) of tension crack, and iv) infiltration of rainfall into tension
crack under several different rainfall intensities. It was assumed that UVC was initially
excavated to a safe height with factor of safety = 1.2. Slope stability analyses were
conducted based on the stress from finite element analysis. Geotechnical modelling
software, SIGMA/W and SLOPE/W (GEO-SLOPE International Ltd.) were used to
simulate excavation/rainfall events and to perform slope stability analysis, respectively.
6
This thesis consists of seven chapters including ‘General Introduction’ (Chapter 1) and
‘General Conclusions’ (Chapter 7).
Chapter 2 presents literature reviews on the formation and propagation of tension crack,
and existing methodologies/approaches to estimate the depth of tension cracks. Various
techniques to estimate the permeability function of tension cracks in unsaturated soils are
introduced.
Chapter 3 describes the mechanism of drop in phreatic line and the variation of factor of
safety of unsupported vertical cut (UVC) with time due to excavation. Two different
stability analysis methods; effective and total stress approaches are explained taking
account of the change in drainage condition of pore-water pressure during excavation
process.
Chapter 4 presents the numerical modelling technique adopted in this research along with
the soil properties. Details include the methodologies to i) simulate excavation and
subsequent change in phreatic line, tension crack and rainfall infiltration into tension crack,
and to ii) analyze slope stability. Descriptions of the numerical modeling software,
SIGMA/W and SLOPE/W used in this study are presented.
In Chapter 5, the influence of excavation rate on the stability of UVC is investigated. In
addition, the critical combinations of tension crack depth and location are determined for
different levels of ground water table. Lastly, the variation of factor of safety of UVC under
different rainfall intensities are studied.
7
Chapter 6 revisits an instrumented large-scale UVC failure case that took place in Welland,
Ontario (Kwan 1971). This case study is used to validate the methodologies and approaches
adopted in this research.
2.1 Introduction
Tension cracks are found near the crest of slopes and unsupported cuts when tensile stress
exceeds tensile strength (Bagge 1985). Figure 2.1 illustrates the state of stresses in a
cohesive soil. The diameter of circle Ct in Figure 2.1(b) represents the tensile strength of
soil at the surface. Desiccation, differential settlement and temperature changes can also
be the main reasons for the formation of tension cracks (Li and Zhang, 2018).
Figure 2.1. Semi-infinite cohesive mass with horizontal surface: (a) stresses at boundaries of prismatic element; (b) graphic representation of state of stress at failure; (c) shear pattern for active state; (d) shear pattern for passive state; (e) stresses on vertical section through the mass (Terzaghi 1943).
9
Gofar et al. (2006) performed numerical analysis on an open coal mine landslide caused
by rainfall infiltration through surface fissures in Airlaya, Indonesia using GeoStudio (i.e.
VADOSE/W and SLOPE/W). The slope consisted of three layers: Layer 1 was the dumped
mine material which exhibited high swelling and shrinkage characteristics under wet and
dry conditions, respectively; Layer 2 was a thin high organic material between fill material
and natural soil, which was attributed to lack of clearance before dumping; Layer 3 was
the natural clay stone material. A crack of 40 m deep existed, passing thorough Layer 2.
The analyses results showed that if the tension crack was neglected, only limited infiltration
took place through the layers with no remarkable rise in the ground water table (hereafter
referred to as GWT). Figure 2.2 shows the monthly rainfall data and the variation of factor
of safety (hereafter referred to as FOS) from the slope stability analyses. It was noticed that
FOS became lowest on two occasions, June 21, 2001 and November 27, 2002. However,
the failure only took place on the latter, which was attributed to the development of tension
crack shortly after the end of dry season. This tension crack has allowed the water to seep
into Layer 2, and it started flowing horizontally within the layer while generating a weak
plane, which led to failure eventually. The numerical analyses performed with a tension
crack successfully captured this failure condition.
10
Figure 2.2. Monthly rainfall and FOS during the period concerned (Gofar et al. 2006).
Sasekaran et al. (2011) conducted slope stability analysis on a homogenous silty clay slope,
taking account of tension crack location and depth and rainfall intensity. The results
showed that, with increase in the depth of tension crack, pore-water pressure around its
vicinity increases as water infiltrates into a deeper section of the soil, further raising the
GWT. This led to a decrease in matric suction and drop in FOS. However, the tension crack
did not have any significant impact on stability of slope when it was located at more than
30 m from the crest of the slope. When rainfall was simulated it was noted that, for the
same duration, the higher intensity rainfall has the greater influence towards slope failure.
Similar research was also carried out by Zhang and Li (2012) considering the same factors
as Sasekaran et al. (2011). The lowest FOS was observed with a tension crack located on
the crest rather than on the slope since the slip surface initiated from the tension crack when
it is on the crest. The FOS decreased with increasing the depth of tension crack. It was also
observed that light rainfall for long duration was more detrimental than a storm like rainfall
lowest FOS
11
for short duration. This is because rainfall with longer duration will have extended seepage
period, which allows water to further infiltrate into the soil.
These examples clearly show that tension crack is one of the governing factors that affect
the stability of slopes. Tension cracks become detrimental during rainfall as they act as
pathway for rainfall to seep through the soil. As rainfall infiltrated through the tension
cracks, the pore-water pressure around the cracks increase, which leads to decrease in
matric suction and shear strength of the soil (Hu 2000). Typically, pre-existing cracks are
highly influential towards the deep failures (Hu 2000, Ping et al. 2005, Wang et al. 2010).
In practice, however, this aspect is often overlooked in the design of unsupported cuts.
This chapter discusses the formation mechanism and methodologies to estimate the depth
of a tension crack. In addition, techniques to estimate the permeability function of tension
cracks in numerical seepage analysis are revisited.
2.2 Formation and Propagation of Tension Crack
Lee et al. (1988) investigated the initiation and propagation of tension cracks in an over-
consolidated marine clay specimen. For this, the concept of critical state energy release
rate (Gc) was adopted, which was determined using an experimental setup as shown in
Figure 2.3.
12
Figure 2.3. Schematic of test specimen and loading used to determine critical state energy release rate (Lee et al. 1988).
A slot of 0.5 mm was cut in an over-consolidated clay specimen, and a force Fc was applied
by drawing the pins apart. Movement of the pins were then read using a dial gauge and
exerted tension on the specimen was measured. Crack propagation from tip of the crack
was then determined at various stages of applied force using a microscope positioned
directly above the specimen.
Gc is defined as the difference between rate of work done by applied loading and rate of
increase in strain energy in a specimen (Eq.(2.1))
( )21 2c c
λ= (2.1)
where, Fc is applied force, t is thickness of specimen, dλ is change in specimen compliance,
and dcL is change in crack length
cF cF
Lc cu
The specimen compliance, λ can be calculated using Eq. (2.2)
c
c
where, uc is load point displacement
The plot of Gc against crack length, cL remained fairly constant, which justifies that Gc can
be adopted as a material constant. The crack was then assumed to propagate from the tip
of an existing crack in direction of normal to maximum circumferential tensile stress,
(σθθ)max (Eq. (2.3)).
max 1( ) 2 cr Kθθσ π = (2.3)
where, r is radial length of crack, and K1c is fracture toughness (Erdogan and Sih 1963)
Fracture toughness, K1c can be corelated to Gc by following Irwin (1958)’s expression for
plane stress condition (Eq. (2.4)).
2 1 2(1 )c
c KG v E
= − (2.4)
where, E is elastic modulus, and v is Poisson’s ratio
K1c in Eq. (2.4) can be estimated based on Gc, E and ν determined through laboratory test.
Eq. (2.3) then can be used to estimate the direction of crack propagation.
Lee et al. (1988) extended this approach to numerical modelling to study the influence of
Poisson’s ratio and depth of cut on crack patterns in UVC. It turned out that lateral extent
14
of the crack zone is a function of ν (i.e. the higher ν the greater extent of crack zone),
while an increase in the depth of cut increased both depth and lateral extent of cracking
(Figure 2.4). However, no attempt was made to determine depth and location of crack that
are critical to the failure of UVC.
Figure 2.4. Crack patters for UVC with different depth of cut (H) and Poisson’s ratio
(ν): (a) H = 6 m, ν = 0.48, (a) H = 12 m, ν = 0.41, (a) H = 18 m, ν = 0.41 (Lee et al.
1988).
15
Kutschke and Vallejo (2011) performed finite element analysis to study stability of UVC
in over-consolidated clays considering tension cracks. Analyses were performed on a
homogenous domain with elastic-plastic model. Excavation was simulated by removing
elements from the in-situ condition and the slope was allowed to achieve the equilibrium
condition in terms of stress. During excavation, horizontal movement was observed along
the slope face due to stress relief, which led to the formation of tension cracks. It was
inferred that tension cracks were formed when lateral stress induced due to stress relief
exceeds tensile strength of the soil. This indicates that the formation and propagation of
tension cracks are highly dependent on in-situ earth pressure coefficient at-rest (i.e. K0).
To study the influence of tension crack on failure plane, a crack with zero tensile strength
(or stiffness) was simulated on the crest of UVC. The analysis results with various in-situ
K0 values showed that a failure in an UVC takes place when the depth of tension crack
ranges between 0.45 and 0.57 times the height of UVC.
2.3 Estimating Depth of Tension Crack
Maximum expected depth of tension crack in UVC can be estimated extending either
effective stress approach or total stress approach. Most traditional equations to estimate the
depth of a tension crack, zt were proposed by Taylor (1948) for drained (Eq.(2.5)) and
undrained (Eq.(2.6)) conditions.
2 tan 45
= (2.6)
16
where, zt is depth of tension crack, c’ is effective cohesion, φ’ is effective internal friction
angle, cu is undrained shear strength, and γ is unit weight of soil
Taylor (1948)’s equations are applicable only to UVC and do not consider the influence of
pore-water pressure. Janbu (1968) developed charts that can be used to estimate the critical
height of UVC. The FOS can be calculated using Eq. (2.7) and Eq. (2.8) extending total
and effective stress approaches, respectively taking account of surcharge, submergence,
seepage, and depth of tension crack. According to Janbu (1968), critical slip surface passes
through the toe for both drained and undrained conditions.
0 ( 0)u
φ′ = > (2.8)
where No and Ncf are stability numbers , and Pd is driving force term (Eq. (2.9))
w w d
q w t
H q HP γ γ μ μ μ + −= (2.9)
where, γw is unit weight of water, H is height of UVC, μq, μw, μt are correction factors for
surcharge, water pressure, and tension crack, respectively, and Hw is depth of water outside
slope, measured above toe.
Spencer (1973) used Eq. (2.10) to estimate the depth of tension crack in embankment
extending the effective stress approach. The stability analysis was carried out based on zero
17
lateral effective stress considering geometry of slope and influence of pore-water pressure
(i.e. pore-pressure coefficient, ru).
u
′ ′ = + − (2.10)
The approach proposed by Baker (1981) can be used to determine location and depth of
tension crack in UVC for both drained and undrained conditions. According to Baker
(1981), maximum depth of tension crack takes place in UVC and it never exceeds one
quarter of the slope height. Bagge (1985) suggested that maximum depth of tension crack
in UVC can be estimated considering the change in pore-water pressure due to stress relief
during vertical cutting. Eqs (2.11) and (2.12) can be used to estimate the depth of tension
crack for undrained and drained conditions, respectively. If a water table is at the ground
( )
w
K A K γ α
γ γ ′− + − − =
γ γ
(2.12)
where A is Skempton pore-pressure parameter, αt is ratio of effective cohesion to tensile
strength, and hw is distance from ground surface to water table
These conventional analytical solutions can be more effectively used for saturated
homogeneous soils. However, in reality, shear strength of soil is governed by the location
of water table and varies with respect to matric suction. Pufahl et al. (1983) investigated
18
lateral earth pressure in UVC extending the unsaturated soil mechanics considering the
influence of matric suction on the shear strength of soil. Based on the conventional Rankin
earth pressure theory, the active earth pressure, σa, for unsaturated condition can be written
as Eq. (2.13) assuming air-pressure is atmospheric pressure.
( )2 tan b a a a w azK c u u Kσ γ φ′ = − + − (2.13)
where z is depth from the ground surface, (ua – uw) is matric suction, ua is pore-air pressure,
uw is pore-water pressure, and φb is angle indicating the rate of increase in shear strength
with respect to a change in matric suction
Eq. (2.13) can be rewritten as Eq. (2.14) considering the nonlinear variation of unit weight
and shear strength of unsaturated soil with depth (Vanapalli et al. 1996).
( )(1 ) 2 tan (1 )
G e zK c u u S K e
κθσ γ φ + + ′ ′ = − + − + (2.14)
where S is degree of saturation, κ is fitting parameter (function of plasticity index), Gs is
specific gravity, θ is volumetric water content, and e is void ratio
Hence, the depth of tension crack is obtained for σa = 0 as shown in Eq. (2.15)
( )2 tan
(1 ) (1 )
G e K e
κ φ θ γ
(2.15)
More recently, Baker and Leshchinsky (2003) studied the spatial distribution of safety
factors in a cohesive UVC by utilizing the safety map notion originally proposed by Baker
19
and Leshchinsky (2001). Michalowski (2013) proposed a method that can be used to
estimate the maximum depth of tension crack based on limit analysis. The location of
tension crack was determined as the one with the most adverse influence on the stability.
A closed-form solution proposed by Li et al. (2018) can also be used to determine depth of
tension crack, which was based on limit equilibrium method while taking account of
linearly increasing undrained strength. Numerical analyses were also carried out to study
the influence of tension crack on the stability of UVC (Lee et al. 1998; Kutschke and
Vallejo 2011). The main advantage of numerical analysis is that progressive failure
mechanism can be taken into account during analysis.
In the present study, Eq. (2.15) was used to estimate the maximum depth of tension crack
analytically. Richard et al. (2020) used Eq. (2.15) to estimate critical height of an
instrumented temporary large scale UVC (3 m deep, 6 m wide and 20 m long) in an
unsaturated soil (Whenham et al. 2007) taking account of rainfall infiltration (Figure 2.5).
The factor of safety at the moment of general failure matched the one obtained extending
the concept in Eq. (2.15). However, it should be noted that Eq. (2.15) can only be used to
determine theoretical maximum depth of a tension crack, not the location.
20
Figure 2.5. Earth pressure distribution diagram used to calculate a factor of safety of the temporary vertical cut against general failure using the field measurement data in Whenham et al. (2007) (after Richard et al. 2020).
2.4 Estimating Permeability Function of Tension Crack
As mentioned earlier, tension crack acts as a pathway for rainfall to seep through the soil,
which results in a significant decrease in shear strength of unsaturated soil. Hence, it is
important to estimate permeability of tension crack in case the stability of UVC is carried
out through numerical analysis considering rainfall events. Seepage of fluids such as water,
gasoline or oil through unsaturated soil is far more complicated than single-phase saturated
percolation. This is because the coefficient of permeability of an unsaturated soil is
governed not only by the properties of fluids and pore-size distribution of a soil, but also
by the degree of saturation (or matric suction). This indicates that permeability of a tension
crack in an unsaturated soil is also governed by both the width of tension crack and matric
suction. Following subsections summarize the methodologies or techniques used to
γunsat = 18.8 kN/m3
p1 p2 p5
(ua - uw) = 18 kPa at 3.0 m
(ua - uw) = 8.5 kPa at 2.5 m
p1 = γsat x 1.5 m x Ka p2 = γunsat x 1.5 m x Ka
p3 = 2c' Ka p4 = {[(ua - uw)Sκtanφ'] Ka} at 2.5 m P5 = {[(ua - uw)Sκtanφ'] Ka} at 3.0 m
P1
P2
P3
P4
21
2.4.1 Zhou et al. (1998)
Zhou et al. (1998) introduced an analytical model to determine the permeability function
of openings in rock joints. For this, van Genuchten’s (1980) Soil-Water Characteristic
Curve (SWCC) model was adopted to establish the relationship between capillary pressure
( )
(2.16)
where, Se is effective degree of saturation (Eq.(2.17)), Pc (Eq.(2.18)) is capillary pressure
at rock joint, and αvG, mvG, nvG are fitting parameters
1
(2.17)
where, S is degree of saturation and Sr is residual degree of saturation
2 cosT c c
σ θ= (2.18)
Where, σT is interfacial tension, θc is contact angle, and bs is critical opening at which joints
begin to excrete under pressure, Pc
Based on the fitting parameters obtained from Eq. (2.16), the permeability function of a
rock joint, k(Se) can then be estimated as shown in Eq. (2.19).
22
( ) 1/1/2 2[1 (1 ) ]vG vG e sat e e
m mk S k S S= − − (2.19)
where, ksat is coefficient of permeability of rock joint for saturation condition
2.4.2 Hu et al. (2000)
In the study by Hu et al. (2000), the upper cracked soil was modeled as a special type of
soil with relatively high permeability and low strength. The thickness of this equivalent
‘weaker’ layer was approximated as the mean crack depth (Figure 2.6).
Figure 2.6. Simulating cracked upper layer as an equivalent weaker layer (Hu et al. 2000).
Strength of the weaker soil was estimated by adopting strength reduction factor that is a
function of crack spacing. The permeability of crack was assumed to be two magnitudes
higher than uncracked soil (Figure 2.7). The SWCC of the crack was estimated extending
the approach by Zhou et al. (1998).
23
Figure 2.7. Permeability function of a tension crack used by Hu et al. (2000).
2.4.3 Ping et al. (2005)
Ping et al. (2005) conducted numerical analysis to simulate rainfall infiltration into a slope
considering the depth, width, and location of tension cracks. The coefficient of
permeability of a tension crack was estimated using Eq. (2.20) based on research on the
seepage into fractured rock by Wang and Su (2002).
2
12 c
c gwk
μ = (2.20)
where, kc is coefficient of permeability of tension crack, g is gravitational acceleration, wc
is width of tension crack, and μ is kinematic viscosity for water (10-6 m2/s in normal
temperature)
Analyses results suggested that, as the depth of tension crack increases, the profile of
infiltration becomes prominent near the lower boundary of the model. For the analysis with
24
a significantly narrow tension crack (0.29 mm), the influence of infiltration through the
tension crack on the increment of pore-water pressure was negligible. Whereas, the tension
crack widths of 0.63 mm and 1.35mm were wide enough for water to reach the bottom of
tension crack. The infiltration of rainfall into a tension crack increased with increasing
rainfall intensity.
2.4.4 Zhang and Li (2012)
Zhang and Li (2012) conducted numerical analyses using SLOPE/W and SEEP/W (product
of GeoStudio) to study the influence of tension crack on stability of slope taking account
of various crack characteristics and conditions. Wang (2011) proposed a methodology to
estimate SWCC and permeability function of a tension crack by analyzing random aperture
distribution of cracks. Zhang and Li (2012) used the methodology proposed by Wang (2011)
to estimate the SWCC and permeability function, assuming a tension crack as a material
with a distinct SWCC and permeability function rather than as a boundary condition.
Figure 2.8 shows the permeability function used by Zhang and Li (2012) as an example.
25
Figure 2.8. Permeability functions for a fine-grained soil and a crack used by Zhang and Li (2012).
2.4.5 Zhang et al. (2020)
Zhang et al. (2020) investigated the stability of shallow slope considering tension crack
extension state, which is dependent on soil type and water content. According to Valentin
et al. (2005), the water-tension causes a compaction effect on soil and the void ratio
decreases accordingly. Cracks are then developed as tension exceeds the soil bearing
capacity. Once the initial cracks are opened, depth and width of cracks continuously
increases due to the repetition of dry-wet cycles. The effect of soil cracking state on the
permeability of soil was considered by having different ksat/q ratios (where q is rainfall
intensity).
C oe
ffi ci
2.4.6 Studies Questioning Modelling of Infiltration into Tension Crack
Sun et al. (2019) assumed that, in unsaturated soil slope stability analysis, water infiltration
quickly fills cracks; however, no seepage takes place into the deeper soils. On the other
hand, Deng and Shen (2006) and Li et al. (2018) suggested that water infiltration into a
tension crack can be neglected since tension cracks formed in clays can close up upon
wetting due to expansion.
3. EFFECTIVE AND TOTAL STRESS APPRAOCHES IN UNSATURATED SOILS
During excavation, an overall mean total stress at a local point decreases, which leads to a
decrease in pore-water pressure. This stress relief also continuously increases the applied
shear stress and becomes maximum at the end of excavation. However, if undrained
condition is maintained during excavation process the shear strength remains constant until
the completion of excavation (i.e. undrained shear strength). Following the completion of
excavation, water flows towards the excavation and pore-water pressure increases until
equilibrium condition is achieved. Due to this reason, the shear strength starts decreasing
with time and the fine-grained soil swells. This phenomenon is well presented by Bishop
and Bjerrum (1960), showing the variation of load, applied shear stress, pore-water
pressure, shear strength, and factor of safety with time due to excavation (Figure 3.1). The
short-term analysis (i.e. φ = 0 analysis) is required to analyze the stability of excavated
slope until the completion of excavation, and the long-term stability analysis thereafter.
28
Figure 3.1. Variation of load, stress, pore pressure, strength, and factor of safety at point A due to excavation in saturated clay (modified after Bishop and Bjerrum 1960).
Galera et al. (2009) investigated a drop in pore-water pressure due to an open pit excavation
in Cobre Las Cruces mine. The mine consisted of 150 m overlying marls which exhibited
properties of over-consolidated clay on the copper ore. The water table was 30 m below
the ground level. The coefficient of permeability of the marls was in the range of 10-9 to
10-10 m/s, making them almost impermeable in nature. Multiple piezometers were installed
at mine pit excavation to read the pore-pressure profile throughout the excavation. A
significant pore-water pressure drops up to -864 kPa was observed on the piezometers
. A
Excavation
Excavation
Time
installed near the excavation (Figure 3.2). Whereas, for the piezometers installed farther
from the excavation, the range of pore-water pressure drop was between -6.2 kPa and -258
kPa. This example clearly shows that a drop in pore-water pressure continuously takes
place throughout the excavation process. Good agreement was observed between the
measured pore-water pressures and those estimated through coupled hydro-mechanical
analysis using FLAC.
Figure 3.2. Drops in pore-water pressure due to excavation measured from piezometer sensors located near excavation and those estimated using FLAC (Galera et al. 2009).
As explained using Figure 3.1, total stress approach is required to analyze the stability of
UVC before the redistribution of pore-water pressure initiates. The duration of undrained
condition during excavation process varies depending on soil type (National Bureau of
Standards 1988, Irvine and Smith 1983, Leroueil et al. 1990). However, Banerjee et al.
(1988) concluded that dissipation of the excess negative pore-water pressure starts
30
immediately after excavation, which justifies the use of effective stress approach (Lambe
and Turner 1970, Kwan 1971, DiBagio and Roti 1972, Dysli and Fontana 1982) in
analyzing the stability of UVC.
In unsaturated soil, the variation of shear strength with respect to matric suction can be
estimated using either Eq. (3.1) (Vanapalli et al. 1996) or Eq. (3.2) (Oh and Vanapalli 2018)
extending the effective stress or total stress approach, respectively. GeoStudio (2019 R2)
adopts Eq. (3.1) and allows users to conduct drained stability analysis (i.e. effective stress
approach). However, in case of undrained stability analysis, the undrained shear strength
values need to be manually assigned to the elements based on the redistribution of pore-
water pressure due to excavation (Oh and Vanapalli 2018). In this study, stability analyses
were performed assuming drained condition using Eq. (3.1).
( ) ( )tan tanr unsat n a a w
s r
c u u u θ θτ σ φ φ θ θ
−′ ′ ′= + − + − − (3.1)
ζ
ξ
(3.2)
where τunsat is shear strength of unsaturated soil under drained condition, (σn – ua) is net
normal stress, σn is normal stress, θ is volumetric water content (subscript s is saturated
condition and r is residual condition), cu(unsat) is shear strength of unsaturated soil under
undrained condition, cu(sat) is shear strength of saturated soil under undrained condition, ζ
and ξ is fitting parameters, and Pa is atmospheric pressure (i.e. 101.3 kPa)
31
4. METHODOLOGY OF NUMERICAL MODELING
SIGMA/W and SLOPE/W (product of GeoStudio 2019 R2) were jointly used to simulate
excavation, tension crack, rainfall and to conduct stability analysis of UVC, respectively.
4.1 Soil Properties
In this thesis, it was assumed that the UVC was excavated into a well compacted glacial
till obtained from Indian Head, Saskatchewan, Canada (i.e. Indian Head till, IHT). Basic
soil properties of the soil are summarized in Table 4.1. The grain size distribution curve
and the SWCC are shown in Figure 4.1 and Figure 4.2, respectively. The Fredlund and
Xing's (1994) model used to achieve the best-fit curve of the SWCC and the fitting
parameters are included in Figure 4.2 as insets. Figure 4.3. shows the permeability function
of IHT estimated using the Fredlund and Xing’s (1994) model using the coefficient of
permeability for saturated condition and the SWCC.
32
Table 4.1. Basic soil properties of Indian Head Till (Vanapalli 1996).
Properties Value
Saturated water content, θs (%) 47
Void ratio, e 0.55
Specific gravity, Gs 2.72
Effective internal friction angle, φ’ () 23.1
Saturated coefficient of permeability, ksat (m/s) 1 × 10-7
Elastic modulus, E (kPa) 10,000
Poisson’s ratio, ν 0.33
Figure 4.1. Grain size distribution curve of Indian Head Till (Oh and Vanapalli 2010).
Grain size (mm) 0.001 0.01 0.1 1 10
Pe rc
ng
0
20
40
60
80
100
33
Figure 4.2. SWCC of Indian Head Till (Oh and Vanapalli 2018).
Figure 4.3. Permeability function of Indian Head Till.
Suction (kPa) 0.1 1 10 100 1000
D eg
re e
of s
at ur
at io
( ) 1
ln
C oe
ffi ce
nt o
4.2 Initial In-Situ Stresses
The initial stress condition was established using ‘Insitu’ analysis type by applying body
load to the elements. The size of domain was 10 m × 10 m. Figure 4.4 shows the defined
mesh and boundary conditions used for the domain. Initial pore-water pressures were
specified by drawing an initial water table, which distributes hydrostatic positive and
negative pore-water pressures below and above the water table, respectively. Stress/strain
boundary conditions were assumed to be restrained in horizontal (X) direction at the
vertical ends (i.e. fixed-X displacement boundaries; hollow red triangles) and restrained in
both horizontal (X) and vertical (Y) directions at the bottom (i.e. fixed-XY boundaries
along the base of the domain). Fine meshes were used in the vicinity of the excavation to
obtain accurate and reliable results, which were generated using quadrilateral and
triangular mesh pattern. Four and three-point integration order was used for quadrilateral
elements and triangular elements, respectively, with a linear interpolation model for
calculating stresses.
Figure 4.4. Meshes and boundary conditions established using ‘Insitu’ analysis type in SIGMA/W.
4.3 Simulation of Excavation
Staged excavation was performed by deactivating regions in 0.15 m increments for various
excavation rates in SIGMA/W (Figure 4.5) using ‘Coupled Stress/PWP’ analysis type to
consider the variation of GWT due to excavation. In this study, excavation rate defines the
intervals between excavations; for example, 5 min excavation rate indicates that every 5
min, 0.15 m thickness of soils are removed. During excavation, water total head hydraulic
boundary condition which was equal to the elevation of initial water table, was assigned
along the lateral extents of the soil region on the right side (i.e. solid circles). This boundary
condition was used to maintain a constant hydraulic total head along the right edge of the
domain regardless of the fluctuation of water table due to excavation.
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Figure 4.5. Simulating staged excavation and assigning water total head hydraulic boundary condition in SIGMA/W.
4.4 Slope Stability Analysis
After each excavation, slope stability analyses were conducted using SLOPE/W based on
finite element-computed stresses, which are then imported into a conventional limit
equilibrium analysis (i.e. ‘SIGMA/W stress’ analysis type). In other words, the stress
conditions and redistribution of pore-water pressure from finite element analysis in
SIGMA/W were used as the parent analysis in the stability analysis. Figure 4.6 illustrates
the vertical stress contours computed with SIGMA/W. As can be seen, the 50 kPa vertical
stress contour is not a constant distance from the slope surface, but instead, closer to the
slope surface in the vicinity of the toe. This is because the vertical stress is affected by both
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
37
overburden weight and shear stress. In ‘SIGMA/W stress’ analysis type, σx, σy, and τxy at
the mid-point of base for each slice is first computed. This information is then used to
determine the mobilized shear stress and available shear strength along the base of each
slice. By integrating the mobilized shear stress and shear strength over the length of slip
surface, FOS can be estimated using Eq. (4.1)
r
m
F =
(4.1)
where, Fr is total resisting shear force and Fm is total mobilized shear force
Figure 4.6. Vertical stress contours computed with SIGMA/W (SIGMA/W manual).
‘Entry and Exit’ surface option was used to define the range of entry and the exit point of
potential slip surfaces in SLOPE/W (Figure 4.7). It was assumed that the slip surfaces pass
through the toe of UVC (i.e. exit point) based on the existing studies (Janbu 1968, Dunlop
and Duncan 1970, Kutschke and Vallejo 2011). This procedure was repeated until the depth
of which FOS = 1.2 is achieved. This depth was denoted as safe height.
38
Figure 4.7. Slope stability analysis in SLOPE/W using ‘Entry and Exit’ surface option.
For soil with negative pore-water pressure (i.e. matric suction), total cohesion is computed
with Eq. (4.2) using effective cohesion and the SWCC. The residual volumetric water
content was taken as 5% of the volumetric water content at saturation for calculating total
cohesion (C) in SLOPE/W. The Mohr-Coulomb failure criterion was used as the material
model.
(4.2)
Figure 4.8 shows an example of analysis tress, including establishing in-situ condition,
simulating excavation, and slope stability analysis. The slope stability analysis method
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
39
used in the study is denoted as ‘coupled - SIGMA/W stress’ to distinguish to from the
conventional limit equilibrium method.
4.5 Simulation of Tension Crack and Rainfall Event
After a safe height (FOS =1.2) is achieved, the influence of tension crack on the stability
of UVC is studied. In SLOPE/W, tension cracks can be defined using ‘tension crack line’
feature, which specifies constant crack depth along the surface of a soil (Figure 4.9). Each
potential slip surface extends up vertically as the slip surface meets tension crack line. The
location of tension crack that has the most adverse impact on the stability of UVC is then
Coupled Stress/PWP analysis
40
determined based on the minimum FOS. Figure 4.10 shows an example of stability analysis
using tension crack line feature.
Figure 4.9. Simulating tension crack using tension crack line feature in SLOPE/W.
Figure 4.10. Example of stability analysis result using tension crack line feature in SLOPE/W.
0.82
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-30 - -20 kPa -20 - -10 kPa -10 - 0 kPa 0 - 10 kPa 10 - 20 kPa 20 - 30 kPa 30 - 40 kPa 40 - 50 kPa 50 - 60 kPa 60 - 70 kPa
Tension crack line
41
However, ‘tension crack line’ feature is not useful when a user wants to estimate FOS of
UVC considering a specific location and depth of a tension crack. Moreover, simulating
seepage or infiltration through crack is not possible using this feature. To overcome this
disadvantage, in this research, tension crack was simulated as void (0.1 m opening) in
SIGMA/W as shown in Figure 4.11. In case where a tension crack is included in the
analysis, the bottom of tension crack and the toe of UVC were specified as entry range and
exit point, respectively. In other words, it was assumed that failure is initiated from the
bottom of a tension crack. Various locations and depths of tension cracks were considered
in the analyses to define the most critical combination of tension crack location and depth.
Figure 4.11. Slope stability analysis considering a tension crack. Tension crack was simulated as a void.
Tension crack
Entry range
Exit point
42
Seepage analyses were carried out to investigate the influence of rainfall infiltration into a
tension crack on the stability of UVC. For this, unit flux boundary (hollow triangles in
Figure 4.12) conditions were applied to the ground surface and the bottom of the excavation.
To simulate seepage through tension crack, water total head (hollow circles) boundary
condition was assigned to the crack’s geometry. It was assumed that tension cracks are
gradually filled with water at the same rate as rainfall intensity and the water level in the
tension crack is maintained at the ground level once it is filled with water to consider worst
case scenario. Example of total head increments with time for different rainfall intensities
are shown in Figure 4.13.
Figure 4.12. Simulating rainfall infiltration in SIGMA/W.
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Unit flux
43
Figure 4.13. Total head versus time relationships used to simulate the infiltration of rainfall into a tension crack under different rainfall intensities.
Time (hours) 0 20 40 60 80 100 120 140 160
To ta
5.1 Influence of Excavation Rate on the Stability of UVC
5.1.1 Estimation of Safe Height (FOS = 1.2)
A series of analyses was carried out to investigate the influence of excavation rate (i.e. 1
sec, 1 min, 5 min, 15 min, 30 min, 1 hour, 4 hours, 12 hours, 18 hours and 1 day) on the
FOS for various levels of GWT (1, 2, 3, 4, and 5 m). For this, UVC was excavated at a
constant rate up to a certain depth until FOS = 1.2 is achieved (i.e. safe height). In some
cases, FOSs were slightly higher than 1.2 since excavations were conducted at 0.15 m
increments. Figure 5.1, Figure 5.2, and Figure 5.3 show the stability analysis results with
GWT at 1, 3, and 5 m, respectively for different excavation rates (i.e. 1 sec, 4 hrs, 12 hrs
and 24 hrs). As expected, drop in pore-water pressure in the vicinity of excavated area is
more predominant with the quick excavation rate. This is simply because there was not
enough time to reach equilibrium condition in terms of pore-water pressure when the
duration between the staged excavations is short.
45
(a) (b)
(c) (d)
Figure 5.1. Slope stability analyses results with GWT at 1 m for different excavation rate: (a) 1 second; (b) 4 hours; (c) 12 hours; and (d) 24 hours.
1.24
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa 80 - 100 kPa
1.22
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-10 - 10 kPa 10 - 30 kPa 30 - 50 kPa 50 - 70 kPa 70 - 90 kPa
1.23
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa 50 - 70 kPa 70 - 90 kPa
1.23
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
E le
va tio
n (m
Water Pressure
-10 - 10 kPa 10 - 30 kPa 30 - 50 kPa 50 - 70 kPa 70 - 90 kPa
46
(a) (b)
(c) (d)
Figure 5.2. Stability analyses results with GWT at 3 m for different excavation rate: (a) 1 second; (b) 4 hours; (c) 12 hours; and (d) 24 hours.
1.24
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-40 - -20 kPa -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa
1.24
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -30 - -10 kPa -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa 50 - 70 kPa
1.24
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -30 - -10 kPa -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa 50 - 70 kPa
1.24
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-30 - -10 kPa -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa 50 - 70 kPa
47
(a) (b)
(c) (d)
Figure 5.3. Stability analyses results with GWT at 5 m for different excavation rate: (a) 1 second; (b) 4 hours; (c) 12 hours; and (d) 24 hours.
1.20
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-60 - -40 kPa -40 - -20 kPa -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa
1.20
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -50 - -30 kPa -30 - -10 kPa -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa
1.20
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-50 - -30 kPa -30 - -10 kPa -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa
1.20
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-50 - -30 kPa -30 - -10 kPa -10 - 10 kPa 10 - 30 kPa 30 - 50 kPa
48
The result showed that the depth of UVC to achieve FOS = 1.2 (i.e. safe height) was not
affected by excavation rate regardless of the level of GWT, shown in Figure 5.4.. This
indicates that the redistribution of matric suction (or pore-water pressure) between the
ground surface and the toe of UVC does not affect the overall stability of UVC in case
where UVC is excavated with certain safety margin (i.e. 1.2 in this research).
Figure 5.4. Variation of safe height with respect to excavation rate for different levels of ground water table (i.e. 1, 2, 3, 4, and 5 m).
Excavation rate (sec) 1 10 100 1000 10000 100000Ex
ca va
to in
d ep
th w
ith F
O S
1.0
1.5
2.0
2.5
3.0
3.5
4.0
GWT at 1 m GWT at 2 m GWT at 3 m GWT at 4 m GWT at 5 m
49
5.1.2 Estimation of Critical Height (FOS = 1)
Figure 5.5 shows the variation of FOS of UVC prior and post equilibrium state with respect
to pore-water pressure excavated to its critical height (i.e. maximum excavation depth
without failure, FOS = 1) at 1 second excavation rate (GWT at 1 m). Unlike the cases
excavated with FOS = 1.2, the UVC failed as GWT rebounds. Therefore, field works
should pay more attention in case UVC is excavated in a fine-grained soil to its critical
height at fast excavation rate.
50
Figure 5.5. FOS of UVC (a) prior and (b) post equilibrium condition with respect to pore-water pressure (excavation rate = 1 second, ground water table at 1 m).
1.02
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa 80 - 100 kPa
0.97
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa 80 - 100 kPa
(a)
(b)
51
For the purpose of comparison, additional stability analyses were carried out using
Bishop’s simplified method (Limit Equilibrium Method) for GWTs at 1, 2, 3, 4, and 5 m.
It was assumed that the GWTs remained unchanged, which represents minimum influence
of matric suction on the stability. Figure 5.6 shows stability analysis results using both
coupled - SIGMA/W stress and Bishop’s simplified methods with GWT at 1 m, 3 m, and
5 m. Figure 5.7 shows the comparison of critical height estimated from both methods with
increasing level of GWT at 1 second excavation rate. The maximum difference in the
critical height was estimated to be 1 m for the GWT at 5 m. This result demonstrates that,
for a deep GWT, the critical height of UVC can be significantly overestimated in case
where UVC is excavated at fast rate with FOS close to unity.
52
Figure 5.6. Stability analyses with 1 second excavation rate for different levels of GWT (1, 3, and 5 m): (a), (b), (c) coupled - SIGMA/W stress method; (d), (e), (f) Bishop’s simplified method.
1.01
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa 80 - 100 kPa
1.01
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa 80 - 100 kPa
0.99
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-40 - -20 kPa -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa
1.04
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-40 - -20 kPa -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa 60 - 80 kPa
1.01
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-60 - -40 kPa -40 - -20 kPa -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa
1.01
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-60 - -40 kPa -40 - -20 kPa -20 - 0 kPa 0 - 20 kPa 20 - 40 kPa 40 - 60 kPa
(a) (d)
(b) (e)
(c) (f)
53
Figure 5.7. Comparison of critical height estimated using coupled - SIGMA/W stress and Bishop’s simplified methods at 1 second excavation rate for different levels of GWT.
Richard (2018) studied the critical height of UVC in a sand (Unimin 7030) for various
levels of GWT. The critical heights were estimated using both coupled – SIGMA/W stress
at two different excavation rates (i.e. 10 and 1000 seconds) and Morgenstern-Price method
(limit equilibrium method). Comparison of the critical heights estimated using both
methods are shown in Figure 5.8.
GWT (m) 1 2 3 4 5 6
C rit
ic al
h ei
gh t (
54
Figure 5.8. Variation of the critical height of UVC in sand (Unimin 7030) with respect to the level of GWT from coupled – SIGMA/W stress (excavation rates = 10 and 10000 s) and Morgenstern-Price method (limit equilibrium method) (modified after Richard 2018).
Unlike the results in Figure 5.5 and Figure 5.7, the estimated critical heights using coupled
– SIGMA/W stress method for the two excavation rates (i.e. 10 and 1000 seconds) were
approximately the same regardless of the levels of GWT. This can be attributed to the
relatively high permeability of sand, which brought the equilibrium condition within short
period of time. However, the critical heights estimated using Morgenstern-Price method
were significantly low for the level of ground water table less than 0.9 m. This phenomenon
GWT (m) 0.0 0.5 1.0 1.5 2.0
C rit
ic al
h ei
gh t (
m )
0.0
0.5
1.0
1.5 coupled - SIGMA/W stress (excavation rate = 10 s) coupled - SIMGA/W stress (excavation rate = 10000 s) Morgenstern-Price method (Limit equilibrium method)
Richard (2018)
55
is relevant to concentration of shear stress at the toe of UVC in coupled – SIGMA/W stress
analysis, not the excavation rate.
5.1.3 Summary and Conclusions
The influence of excavation rate on the safe and critical heights of UVC was investigated.
For this, stability analyses were carried out for various excavation rates (from 1 second to
24 hrs) and levels of GWT (1, 2, 3, 4, and 5 m). The coupled - SIGMA/W stress analysis
results showed that the excavation rate does not affect the overall stability of UVC in case
where excavation is performed considering a certain safety margin (i.e. FOS = 1.2 in this
research). However, the UVC eventually failed if UVC is excavated up to a critical height
(i.e. FOS = 1) due to the rebound of the GWT. The critical heights estimated using the
coupled - SIGMA/W stress method were significantly overestimated when compared with
those from the Bishop’s simplified method (Limit Equilibrium Method). However,
research by others has shown that the influence of excavation rate on the critical heights of
UVC in sand is negligible due to its high permeability. This indicates that permeability
function of a soil is a key parameter in estimating the critical height of UVC when
excavation rate is considered.
56
5.2 Determining the Influence of Tension Crack on Stability of Unsupported Vertical Cut
5.2.1 Determining the Critical Height of UVC using Numerical Method
As mentioned earlier, tension crack was considered and simulated as a void in SIGMA/W.
UVC was first excavated to a depth where FOS = 1.2 (safe height) under the equilibrium
condition with respect to matric suction. The stability analyses were performed for various
depths (up to 70% of safe height) and distances (up to 50% of safe height from the
excavation wall) of tension cracks with three levels of GWT (i.e. 1, 3, and 5 m). Each
analysis is denoted by its depth ratio and distance ratio as shown in Eq.(5.1) and Eq. (5.2),
respectively.
( ) ( 1.2)tc
= = (5.1)
( ) ( 1 .2 )tc
= =
Analyses were performed keeping one of the two above-mentioned parameters constant
while varying the other until the factor of safety fell below unity; namely, if distance ratio
(DStc) was kept constant then depth ratio (DPtc) is varied till FOS < 1 (Figure 5.9) and vice
versa (Figure 5.10). This analysis pattern efficiently determines the depths and locations of
tension cracks that cause failure in UVC. Figure 5.11 shows an example of analysis tree
with various DPtc of tension crack.
57
Figure 5.9. Simulating tension crack for the safe height of 2.55 m with distance ratio (DStc) fixed at 0.1 and six different depth ratios (DPtc): (a) 0; (b) 0.1; (c) 0.2; (d) 0.3; (e) 0.4; and (f) 0.5.
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
10
58
Figure 5.10. Simulating tension crack for the safe height of 2.55 m with depth ratio (DPtc) fixed at 0.5 and five different distance ratios (DStc): (a) 0.1; (b) 0.2; (c) 0.3; (d) 0.4; and (e) 0.5.
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
10
(e)
59
Figure 5.11. Analysis tree for various depth ratio (DPtc) in SIGMA/W to determine critical depth of tension crack.
Examples of stability analysis results with different combinations of DPtc and DStc are
shown in Figure 5.12. It was assumed that a slip surface begins at bottom of the tension
crack. Analyses were carried out for multiple scenarios to determine critical combinations
of DPtc and DStc. Table 5.1 summarizes FOS for various DPtc and DStc combinations with
GWT at 1, 3, and 5 m.
Safe height (FOS = 1.2)
DPtc = 0.1 (0.255 m)
DPtc = 0.2 (0.510 m)
DPtc = 0.3 (0.765 m)
DPtc = 0.4 (1.02 m)
DPtc = 0.5 (1.275 m)
60
(a)
(b)
Figure 5.12. Stability analysis considering tension crack with different combinations of DStc and DPtc: (a) DStc = 0.4, DPtc = 0.4 (GWT at 1 m); (b) DStc = 0.5, DPtc = 0.5 (GWT at 3 m).
1.00
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
0
1
2
3
4
5
6
7
8
9
10
Water Pressure -10 - 0 kPa 0 - 10 kPa 10 - 20 kPa 20 - 30 kPa 30 - 40 kPa 40 - 50 kPa 50 - 60 kPa 60 - 70 kPa 70 - 80 kPa 80 - 90 kPa
0.98
Distance (m) 0 1 2 3 4 5 6 7 8 9 10
El ev
at io
n (m
Water Pressure
-30 - -20 kPa -20 - -10 kPa -10 - 0 kPa 0 - 10 kPa 10 - 20 kPa 20 - 30 kPa 30 - 40 kPa 40 - 50 kPa 50 - 60 kPa 60 - 70 kPa
61
Table 5.1. Variation of factor of safety for different combinations of DPtc and DStc with GWT at 1, 3 and 5 m.
GWT = 1 m (Hsafe = 1.5 m)*
DPtc DStc
0.1 0.2 0.3 0.4 0.5 0.1 1.86 1.4 1.21 1.14 1.12 0.2 1.62 1.24 1.1 1.04 1.03 0.3 1.41 1.11 1.02 1 1 0.4 1.3 1.1 - - - 0.5 1.27 1.08 - - - 0.6 1.27 1.08 - - - 0.7 1.23 1.05 - - -
GWT = 3 m (Hsafe = 2.55 m)*
DPtc DStc
0.1 0.2 0.3 0.4 0.5 0.1 2.45 1.56 1.29 1.2 1.16 0.2 2.11 1.38 1.19 1.1 1.13 0.3 1.82 1.23 1.07 1.05 1.11 0.4 1.59 1.09 0.98 1.07 1.21 0.5 1.4 0.98 0.92 0.92 0.98 0.6 1.39 - - - - 0.7 1.25 - - - -
GWT = 5 m (Hsafe = 3.45 m)*
DPtc DStc
0.1 0.2 0.3 0.4 0.5 0.1 2.32 1.55 1.24 1.18 1.13 0.2 2.01 1.38 1.13 1.1 1.11 0.3 1.73 1.22 1.04 1.04 1.1 0.4 1.5 1.1 0.92 0.92 0.95 0.5 1.32 0.99 - - - 0.6 1.05 - - - - 0.7 0.98 - - - -
Hsafe = Safe height estimated using couple – SIGMA/W stress method
62
5.2.2 Estimating a Depth of Tension Crack using Pufahl et al. (1983) approach
For the purpose of estimating the depth of tension cracks, the approach proposed by Pufahl
et al. (1983; Eq. (2.15)) was adopted. In this method, the maximum expected depth of
tension crack in vadose zone can be estimated by locating the depth with zero net active
earth pressure as shown in Figure 5.13.
Figure 5.13. Determination of depth of tension crack based on net active earth pressure distribution in vadose zone (Eq. (2.15)) extending the approach by Pufahl et al. (1983).
The positive, negative, and net active earth pressure diagrams with GWT at 1, 3, and 5 m
are shown in Figure 5.14, Figure 5.15, and Figure 5.16, respectively.
Area I
+ =
γsat
z
+H
γunsat
e
63
Figure 5.14. Positive, negative, and net active earth pressure distribution (GWT = 1 m).
Pressure (kPa) -12 -8 -4 0 4 8 12 16
D ep
th (m
) 0.0
0.5
1.0
1.5
2.0
Posive earth pressure (σvKa) Negative earth pressure (2C Ka) Net earth pressure (σvKa - 2c Ka)
zt
64
Figure 5.15. Positive, negative, and net active earth pressure distribution (GWT = 3 m).
Pressure (kPa) -20 -10 0 10 20 30
D ep
th (m
) 0.0
0.5
1.0
1.5
2.0
2.5
3.0
zt
Positive earth pressure (σvKa) Negative earth pressure (2C Ka) Net earth pressure (σvKa - 2C Ka)
65
Figure 5.16. Positive, negative, and net active earth pressure distribution (GWT = 5 m).
Pressure (kPa) -20 -10 0 10 20 30 40 50
D ep
th (m
0
1
2
3
4
5
Positive earth pressure (σvKa) Negative earth pressure (2C Ka) Net earth pressure (σvKa - 2C Ka)
zt
66
5.2.3 Determination of Critical Tension Crack
Based on the obtained FOS in Table 5.1, contours of FOS for different combinations of
DPtc and DStc are plotted in Figure 5.17, Figure 5.18, and Figure 5.19 with GWT at 1, 3,
and 5 m, respectively. The depths of tension crack estimated using Eq. (2.15) were 0.83,
1.34, and 1.56 m with GWT at 1 m, 3 m, and 5 m, respectively, which leads to an average
depth ratio of 0.59 (i.e. 0.55, 0.64, and 0.57 with GWT at 1, 3, and 5 m, respectively). This
depth ratio is similar to the range proposed by Kutschke and Vallejo (2011) based on the
finite element analysis of UVC in stiff clay (i.e. 0.45 – 0.57). As expected, FOS decreases
with increasing the depth ratio. In cas