1
Geotechnical Engineering at the Grain ScaleT. Matthew Evans
Department of Civil, Construction, and Environmental Engineering
North Carolina State University
Graduate Students: Bahador, Cunningham, Kress, Ning, Wang, Zhao
Collaborators: Chall (RENCI), Gabr (NCSU), Rhyne (RENCI), Tayebali (NCSU), Frost (GT), Valdes (SDSU), Yun (Yonsei), Andrade (Caltech)
Funding: NSF, NASA, FHWA, NCDOT
NCSU: Geotechnical Engineering
5 Full-time faculty (one starting January 2012)
Excellent breadth in research interests, from the very small (particle-level) scale to full field scale, from highly theoretical to very applied, experimental and numerical
13 full-time graduate students (11 supported) and a comparable number of part-time students
Wide variety of courses offered:– Advanced Soil Mechanics (2
courses)
– Unsaturated Soil Mechanics
– Geosynthetics
– Foundation Design
– Laboratory Methods
– Soil Dynamics
– Numerical Methods
– Groundwater Hydrology
– Rock Mechanics
– Special Topics Courses
Preliminaries
2
NCSU: Geotechnical Engineering
Simulations of multiple laboratory tests
Quantification and analysis of microstructure evolution
Why does material response vary with boundary conditions?
Are common laboratory parameters sufficient for design?
Why or why not?
Sponsor: NCSU Department of Civil, Construction, and Environmental Engineering and the NCSU College of Engineering
The Effects of Loading Conditions on Material Response: Discrete Numerical Studies
Geotechnical systems are often subjected to plane strain loading, but designs are based on axisymmetric testing.
0 0.02 0.04 0.06 0.08 0.10
375
750
1125
1500
PS-M450
CTC -M450
PS-M450
CTC -M450
Axial Strain []
Devia
toric
Str
ess
[kPa]
Preliminaries
NCSU: Geotechnical Engineering
The microstructure of granular materials governs the design-scale behavior
Laboratory investigation of microstructure is expensive, time-consuming, and requires highly specialized equipment
Numerical studies are faster and less expensive, yet it has not previously been possible to directly compare experimental and numerical results
Current research will allow for equivalent measurements to be made on physical and numerical specimens
Sponsors: NCSU Department of CCEE, NCSU COE, North Carolina General Assembly
Visualization of Three-Dimensional Discrete Numerical Data
The microstructure of particulate systems can be studied experimentally or numerically, but it is often not possible to compare results from the two methods.
Collaborators: Theresa-Marie Rhyne and Steve Chall, Renaissance Computing Institute (RENCI@NCSU)
0
30
60
90
120
150
180
210
240
270
300
330
0.025
0.02
0.015
0.01
0.005
0
r 0.028 r 8.971 deg
Preliminaries
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7
Local Void Ratio [ ]
Perc
ent
Solid A
rea [
%]
eim 0.544
est 0.541
est 0.309
gamma 0.541
gamma 0.289
3
NCSU: Geotechnical Engineering
Rejecting material at the job site (or quarry) results in unnecessary expense and delays
Prepare specimens at varying GSD’s
Assess GSD impacts on material response
Use numerical tools to investigate underlying mechanisms
Can “out-of-spec” material be safely used in some projects?
The Effects of Grain Size Distribution Aggregate Base Course Performance
The current NCDOT requirement for ABC material is an ad-hocspecification based solely on the grain size distribution.
0.01 0.1 1 10 1000
0.2
0.4
0.6
0.8
15% Below
Lower Bound
Baseline
Upper Bound
5% Above
Best Fit
Diameter [mm]
Fra
ctio
n P
ass
ing [
]
Sponsor: North Carolina Department of Transportation (NCDOT)
Preliminaries
NCSU: Geotechnical Engineering
Field-Scale Research, Including Reinforced Earth, Pile Bents, and Undercut in Construction
Preliminaries
4
NCSU: Geotechnical Engineering
Preliminaries
Field-Scale Research, Including Reinforced Earth, Pile Bents, and Undercut in Construction
NCSU: Geotechnical Engineering
Two earth dams and a large load frame on the NCSU Centennial Campus have been instrumented for monitoring
Pore water pressures, dam movement, and strains in the steel superstructure are monitored
Data are transmitted wirelessly and automatically databased and posted to the project website
Visit http://www.ce.ncsu.edu/ccli-sensors
Sponsors: U.S. National Science Foundation
Remote Monitoring of Geostructural Health
Post-construction monitoring of large geostructures is increasingly frequent. We study the best approaches to monitoring and teach students to use these technologies in their careers.
Preliminaries
5
Geotech Faculty at NCSU
Dr. Roy Borden
– Classical and applied geotechnics
– Shallow and deep foundations
– Reinforced soil and earth walls
– Soil-structure interaction
Preliminaries
Geotech Faculty at NCSU
Dr. Mo Gabr
– Geoenvironmental engineering
– Geosynthetics
– Deep foundations
– Transportation geotechnics
Preliminaries
6
Brina Mortensen
– Bio-mediated soil improvement
– Identification and behavior of naturally cemented and aged sands
– Sustainable building materials
Geotech Faculty at NCSU
Preliminaries
Geotech Faculty at NCSU
Dr. Shamim Rahman
– Modeling and computing in geomechanics
– Soil dynamics
– Seabed mechanics
– Stochastic and neuro-fuzzy modeling
Preliminaries
7
Geotech Faculty at NCSU
Dr. Matt Evans
– Granular mechanics and particulate behavior
– Energy geotechnics
– Extraterrestrial geomechanics
– Multiphysics processes
– Unsaturated soil mechanics
– Image analysis and microstructure quantification
Preliminaries
Motivation
Soils are a fundamentally discrete, rather than a continuum, material
To capture the inherent nonlinearity, heterogeneity, and anisotropy in soils, their granular nature must be considered
Many engineering-scale behaviors can be explained (or at least inferred) by considering response at the granular level
From Tatsuoka, 2002
From Nübel, 2002
Preliminaries
8
Motivation (cont.)
Interfaces, grain crushing, bonded and unbonded material, and strain softening cannot be readily captured in continuum models
We can use the discrete element method (DEM) simulations to quantify granular response across a range of spatial and temporal scales
Preliminaries
Fn Fs
Fn
Overview
Soil micromechanics
Discrete element method (DEM)
Integrated Numerical-Experimental Study
DEM Simulations: Effects of Geometry
Soil-Structure Interaction (SSI)
Thermal Conductivity
Summary and Conclusions
Preliminaries
9
Hertzian Contacts
d
r
R
N
d1
2 R
1
3
3 1 g
Gg
N
2
3
F d( )2
32 r
Gg
1 g d
3
2
0 0.02 0.04 0.06 0.08 0.1 0.120
250
500
750
1000
Displacement [mm]
Forc
e [
N]
(after Santamarina, et al., 2001)
Soil Micromechanics
Tangential Loading:Hertz-Mindlin
1 0.5 0 0.5 1
1
0.5
0.5
1Backbone Curve
Masing Unload Loop
Masing Reload Loop
dtan
dy
T
N
dy3
82 g
N
Gg R
33 1 g N
8 Gg R2
dtan 1 1T
N
2
3
dy
Displacement at yield:
Tangential displacement:
(after Santamarina, et al., 2001)
Soil Micromechanics
10
Fabric Failure
6060
60
N
T 1
3 26060
60
N
T 1
3 2
N
F2
T
N
F2
T
T
F2N
60
T
F2N
60
Force required to cause fabric failure: Tf N tan 30( )
Soil Micromechanics
Shear Strain to Failure
gf
r12 1r12
4 R
2 R cos 30deg( )From geometry:
N 4 R2
cForce as a function of confining stress:
Material properties for quartz grains:
Gg 29 GPa g 0.5g 0.25
gfabric 1.4 105
kPa
2
3
gslip 6.7 106
kPa
2
3
Threshold failure strains:
0 5 106
1 105
1.5 105
0
250
500
750
1000
Fabric
Slip
Threshold Strain [ ]
Str
ess
[kP
a]
(after Santamarina, et al., 2001)
Soil Micromechanics
11
Rotational Frustration
Dilation in dense granular systems
?Even very simple models can be used to indicate the propensity for rotational frustration in a dense granular system. Energy must be dissipated through contact dissolution and formation if particles are unable to rotate even at very small strains.
Soil Micromechanics
Lateral Earth Stressat Rest (K0 = Px/Py)
Px
Py
From a force balance:
From continuum mechanics ( = 0.3):
K0 tan 30( ) 0.577
K0
1 0.429
Jaky, 1948 (f’ = 30°): K0 1 sin 30( ) 0.5
Note that for f’ = 25° and = 0.366, the geometric, continuum, and Jaky (1948)
solutions are equivalent (i.e., K0 = 0.577).
Jaky, 1944 (f’ = 30°):
K0 1 sin f( )( )
12
3sin f( )
1 sin f( ) 0.444
Soil Micromechanics
12
Filter Criteria: GrainSize Distribution
0.01 0.1 10.1
1
10
100
Normalized Diameter [ ]
Perc
ent
Pass
ing
[%]
0.01 0.1 10
20
40
60
80
100
Normalized Diameter [ ]
Perc
ent
Pass
ing
[%]
Pp 100d
dmax
Talbot relationship:
Soil Micromechanics
Unsaturated Soils
(after Santamarina, et al., 2001; Lu and Likos, 2004)
Soil Micromechanics
The force acting between the two particles is:
F u r22
Ts 2 r2
We can get u from the Young-Laplace equation:
u Ts1
r1
1
r2
A simple geometric argument tells us that:
R r1 2 R2
r1 r2 2
Solving for r1, we can express Young-Laplace in terms of r2 and R only: r11
2
r22
R r2 u Ts
1
1
2
r22
R r2
1
r2
Note that r1 and r2 must have different signs in the Young-Laplace
equation because they contribute to opposite pressures. Simplifying
the expression for u gives:
u Ts
2 R 3 r2
r22
13
Unsaturated Soils
(after Santamarina, et al., 2001; Lu and Likos, 2004)
Soil Micromechanics
Thus, the expression for the force between the two
particles due to capil larity may be expressed as:
F Ts
2 R 3 r2
r22
r22
Ts 2 r2
Or, simplifying: F 2 R r2 Ts
Normalizing by an area of (2R)2 provides an expression
for the equivalent effective stress due to cap il larity:
'F
2 R( )2
'Ts
4 R2
2 R r2
From the above argument it is possible to calculate effective stress without measuring matric suction. This is significant because matric suction is such an elusive quantity to measure.
Using a similar approach, effective stress between to face-to-face platy particles may be calculated as:
'
4Ts
Sa gw
w g
where Sa is specific surface area.
Unsaturated Soils
(after Santamarina, et al., 2001; Lu and Likos, 2004)
Soil Micromechanics
Now let's calculate the water content of our system.The
half-height of the meniscus can be expressed as:
R2
r22
R h( )2
h R R2
r22
If we assume that the meniscus is a cylinder of height
2h and radius r2, we can calculate its volume:
Vw r22
2 h( ) 21
3 h
2 3 R h( )
Noting that the coordination number (cn) for a simple
cubic (SC) packing is 6, each particle wil l be
associated with six half-menisci. Water content is
then calculated as:
w
6Vw
2 w
4
3 R
3 s
w9
4
Vw
R3
Gs
After a bi t of algebraic acrobatics, we get the following:
w9
8
4
Gs
where:
r2
R
This calculation of water content (an easily measurable quantity) is significant because it allows us to remove r2 (which isn’t easy to measure) from our subsequent equations.
14
Unsaturated Soils
Soil Micromechanics
0 0.02 0.04 0.06 0.080.1
1
10
100
Simple Cubi cTetrahedral
Gravimetric Wa ter Content [g/g]
Su
ctio
n [
kPa
]
R 0.1mm
0.01 0.1 1 101 10
4
1 103
0.01
0.1
1
10
100
w = 0.1%w = 1%w = 6.3%Particle We ight
Particle Radius [mm]
Effe
ctiv
e St
ress
[kP
a]
Overview
Soil micromechanics
Discrete element method (DEM)
Integrated Numerical-Experimental Study
DEM Simulations: Effects of Geometry
Soil-Structure Interaction (SSI)
Thermal Conductivity
Summary and Conclusions
Preliminaries
15
DEM: Introduction
Increasingly popular in research
– Used to gain insight into particulate behavior
– Calibrated to simulate macroscale laboratory results
– Microstructure output consists of quantities not readily measurable in real soils
– Microstructure can also be quantified using many of the same approaches used for physical experiments
Not yet widely used in practice
– Inertia / skepticism / unfamiliarity
Discrete Element Method
Model Fundamentals
uab
nb
rb
rana
Solution of Newton’s EOM for each particle:
– Calculate contact normals
– Determine overlap and contact locations
– Calculate relative positions and velocities
– Use constitutive relations
– Calculate new forces/moments
Discrete Element Method
16
Model Assemblies:Particle Shape
Discrete Element Method
Model Assemblies:Specimen Geometry
Discrete Element Method
17
Microstructural Parameters
Normal Contact ForcesDisplacement Vectors Particle Rotations
Discrete Element Method
Bridging Scales: the Stress-Force-Fabric Relationship
Discrete Element Method
fs
fn
x2
x1
θ
r
At any contact in the assembly, the mechanics can be defined by the magnitude and orientation of the contact normal and shear force vectors.
By assembling this information for every contact in the assembly, we can know something about the stresses at the specimen scale.
18
Bridging Scales: the Stress-Force-Fabric Relationship
E 1
2 1 ac cos 2 a N N0 1 an cos 2 f T N0 at sin 2 f
Contact Orientation Normal Force Tangential Force
Note the fitting parameters to the Fourier expansions above: ac, an, and at.
(Rothenburg and Bathurst, 1992)
Discrete Element Method
Stress-Force-Fabric Relationship
sin11 22
11 22
1
2ac an at
The theoretical stress-force-fabric relationship derived from fabric tensors and microscale stress quantities:
(Rothenburg and Bathurst, 1992)
Discrete Element Method
19
Simulation of RealSystems and Processes
Direct Shear Test (Liu, 2006)
Particle Crushing (Lobo-Guerrero, 2006)
Cone Penetration(Jiang, et al., 2006)
Layered Systems (e.g., pavements)
Discrete Element Method
Overview
Soil micromechanics
Discrete element method (DEM)
Integrated Numerical-Experimental Study
DEM Simulations: Effects of Geometry
Soil-Structure Interaction (SSI)
Thermal Conductivity
Summary and Conclusions
Preliminaries
20
Plane Strain Testing
LVDT Load
Cell
1
2
3 6
7
5 4 547
6
2
3
1
2 Frictionless
Slider
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
21
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
22
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
23
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
24
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
25
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
26
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
27
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
28
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
0 2 4 6 8 100
1 105
2 105
3 105
4 105
5 105
Top Platen
Base PlatenSide Wall
System Friction
Axial Stress vs. Axial Strain
Strain [%]
Str
ess
[Pa]
0 2 4 6 8 1038
40
42
44
46
48
50
Upper Thickness
Lower Thickness
Sample Thickness vs. Axial Strain
Axial Strain [%]
Sam
ple
Thic
knes
s [m
m]
0 2 4 6 8 100
5
10
15Sled Movement
Axial Strain [%]
Sle
d M
ov
emen
t [m
m]
7.5
2.5
5
44
43
42
41
40
39
Integrated N-E Study
29
Biaxial Testing Results
0 1 2 3 4 5 6 7 8 9 10 11 122.5
2
1.5
1
0.5
0
0.5
1
1.5
2
BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)BT-2030-16 (0.56, 10 psi)
Axial Strain [%]
Volum
etric
Strain
[%]
0
100
200
300
400
500
600
700
800
900
1000
Axi
al S
tress
[kPa
]
1.5
1.0
0.5
-2.5
0
-0.5
-1.0
-1.5
-2.0
900
800
700
100
600
500
400
300
200
1 2 3 4 5 6 87 109 1211
Axial Strain [%]
Volu
metr
ic S
train
[%
]Axia
l Str
ess
[kPa]
0 1 2 3 4 5 6 7 8 9 10 11 122.5
2
1.5
1
0.5
0
0.5
1
1.5
2
BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)BT-2030-16 (0.56, 10 psi)
Axial Strain [%]
Volu
metr
ic S
train
[%
]
Integrated N-E Study
Biaxial Testing Results
1.5
1.0
0.5
-2.5
0
-0.5
-1.0
-1.5
-2.0
900
800
700
100
600
500
400
300
200
1 2 3 4 5 6 87 109 1211
Axial Strain [%]
Volu
metr
ic S
train
[%
]Axia
l Str
ess
[kPa]
0
100
200
300
400
500
600
700
800
900
1000
Axi
al S
tress
[kPa
]
0 1 2 3 4 5 6 7 8 9 10 11 122.5
2
1.5
1
0.5
0
0.5
1
1.5
2
BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)
BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)
BT-2030-16 (0.56, 10 psi)
Axial Strain [%]
Volum
etric
Strain
[%]
0 1 2 3 4 5 6 7 8 9 10 11 122.5
2
1.5
1
0.5
0
0.5
1
1.5
2
BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)BT-2030-16 (0.56, 10 psi)
Axial Strain [%]
Volu
metr
ic S
train
[%
]
0 1 2 3 4 5 6 7 8 9 10 11 122.5
2
1.5
1
0.5
0
0.5
1
1.5
2
BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)
BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)
BT-2030-16 (0.56, 10 psi)
Axial Strain [%]
Vo
lum
etr
ic S
train
[%
]
0 1 2 3 4 5 6 7 8 9 10 11 122.5
2
1.5
1
0.5
0
0.5
1
1.5
2
BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)
BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)
BT-2030-16 (0.56, 10 psi)
Axial Strain [%]
Volu
metr
ic S
train
[%
]
Integrated N-E Study
30
Shear Band Inclination
Test Designation fp yp fcs fs ycs e C R A
Slightly dilatant 38.8° 8.8° 27.7° 27.7° 1.0° 59° 64° 49° 57°
Highly dilatant 41.8° 11.5° 27.9° 27.9° 0.2° 61° 66° 51° 58°
59°
61°
C
4
fp
2
R
4
yp
2
A
4
1
4fp yp
Slightly Dilatant Highly Dilatant
Integrated N-E Study
80 mm (typ.)
varies
varies
Portion of
specimen
extracted
80 mm (typ.)
varies
varies
Portion of
specimen
extracted
Specimen Impregnation and Sectioning
Sealed ContainerGlue Reservoir
Atmosphere
Biaxial cell
Specimen
Sealed Container
Overflow
Low vacuum
Sealed ContainerGlue Reservoir
Atmosphere
Biaxial cell
Specimen
Sealed Container
Overflow
Low vacuum
Sealed ContainerGlue Reservoir
Atmosphere
Biaxial cell
Specimen
Sealed Container
Overflow
Low vacuum
Resin Submersion
Glue Impregnation
Locked Microstructure
Solidified and Sectioned x1
x3
x2
x1
x3
x2
Surfaces for imaging are on the negative x2 plane
Integrated N-E Study
31
Surface Preparationand Image Capture
Surface Planing
Original Subimages
Grinding/Polishing Image Capture
Stitched Image Mosaic Binary of Mosaic
Integrated N-E Study
Local VoidRatios
Area of solid Particles, AS
Area of Voids, AV
Local Void Ratio = AV/AS
Area of solid Particles, AS
Area of Voids, AV
Local Void Ratio = AV/AS
Area of solid Particles, AS
Area of Voids, AV
Local Void Ratio = AV/AS
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
Unsheared
Sheared
Local Void Ratio [ ]
Pe
rcen
t S
oli
d A
rea
[%]
eun 0.605 un 0.598 un 0.417
esh 0.670 sh 0.667 sh 0.446
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
Unsheared
Sheared
Local Void Ratio [ ]
Pe
rcen
t S
oli
d A
rea
[%]
eun 0.572 un 0.569 un 0.389
esh 0.638 sh 0.636 sh 0.439
Shearing tends to flatten the histogram and shift it to the right
Slightly Dilatant
Highly Dilatant
(after Park, 1999)
Integrated N-E Study
32
Void Ratio Distributions
0 1 2 3 40
2
4
6
8
Histogram
Gamma Distribution
Weibull Distribution
Lognormal Distribution
Local Void Ratio
Pe
rcen
t S
oli
d A
rea
[%]
0 1 2 3 40
2
4
6
8
Histogram
Gamma Distribution
Weibull Distribution
Lognormal Distribution
Local Void Ratio
Pe
rcen
t S
oli
d A
rea
[%]
Estimates of distribution parameters from curve fitting of CDF's to entire dataset:
0 0.5 1 1.5 2 2.5 30
0.5
1
Measured Data
Gamma CDF
Weibull CDF
Lognormal CDF
Cumulative Distribution Functions
Local Void Ratio [ ]
Cu
mu
lati
ve
Pe
rce
nt
So
lid
Are
a [
%]
0 1 2 3 40
2
4
6
8
Histogram
Gamma Distribution
Weibull Distribution
Lognormal Distribution
Local Void Ratio
Pe
rcen
t S
oli
d A
rea
[%]
Maximum Likelihood PDF Fitting CDF Fitting
Integrated N-E Study
MicroscaleExperimental Results
Subregional Analyses Strip Analyses
Integrated N-E Study
33
Unsheared
Slightly Dilatant
Sheared
Highly Dilatant
Integrated N-E Study
Void Ratio Contours
Strip Analyses
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Void Ratio Orthogonal to the Shear Band
Distance Along Transect [mm]
Vo
id R
atio
[ ]
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Incremental Void Ratio
Accumulated Void Ratio (L to R)
Accumulated Void Ratio (R to L)
Distance Along Transect [mm]
Vo
id R
atio
[ ]
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Incremental Void Ratio
Accumulated Value (LHS)
Accumulated Value (shear band)
Accumulated Value (RHS)
Distance Along Transect [mm]
Vo
id R
atio
[ ]
LHS RHS
Integrated N-E Study
34
Strip Analyses
Slightly Dilatant Highly Dilatant
Unsh
eare
dSheare
d
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Distance Along Transect [mm]
Vo
id R
atio LHS RHS
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Distance Along Transect [mm]
Vo
id R
atio LHS RHS
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Distance Along Transect [mm]
Vo
id R
atio LHS RHS
0 5 10 15 20 25 30 35 40 45 500.2
0.4
0.6
0.8
Distance Along Transect [mm]
Vo
id R
atio LHS RHS
Integrated N-E Study
Strip Analyses: LVRD
Slightly Dilatant Highly Dilatant
Unsh
eare
dSheare
d
0 0.5 1 1.5 2 2.5 3 3.5 40
5
10
Inside Shear Band
Outside Shear band
Local Void Ratio [ ]
Per
cent S
olid A
rea
[%]
est 0.557 in 0.558 out 0.556
est 0.363 in 0.360 out 0.366
0 0.5 1 1.5 2 2.5 3 3.5 40
5
10
Inside Shear Band
Outside Shear band
Local Void Ratio [ ]
Per
cent S
olid A
rea
[%]
est 0.641 in 0.721 out 0.584
est 0.441 in 0.484 out 0.397
0 0.5 1 1.5 2 2.5 3 3.5 40
5
10
Inside Shear Band
Outside Shear band
Local Void Ratio [ ]
Per
cent S
olid A
rea
[%]
est 0.666 in 0.756 out 0.615
est 0.441 in 0.490 out 0.401
0 0.5 1 1.5 2 2.5 3 3.5 40
5
10
Inside Shear Band
Outside Shear band
Local Void Ratio [ ]
Per
cent S
olid A
rea
[%]
est 0.610 in 0.604 out 0.614
est 0.407 in 0.401 out 0.410
Integrated N-E Study
35
LVRDStatistics
Mean
(est)
Standard Deviation
(est)
Slightly Dilatant Specimen
Unsheared 0.598 0.417
Sheared 0.667 (11.1%) 0.444
Inside Shear Band 0.747 (24.6%) 0.490
Outside Shear Band 0.622 (3.6%) 0.409
Highly Dilatant Specimen
Unsheared 0.569 0.389
Sheared 0.636 (13.6%) 0.439
Inside Shear Band 0.715 (27.6%) 0.482
Outside Shear Band 0.588 (4.9%) 0.403
0.55 0.6 0.65 0.7 0.750.3
0.35
0.4
0.45
0.5
0.55
SDU
HDU
SDS (outside)
HDS (outside)
SDS (intside)HDS (inside)
Unsheared + Sheared Outside
Inside
trace 9
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
0.55 0.6 0.65 0.7 0.750.7
0.72
0.74
0.76
0.78
0.8
SDU
HDU
SDS (outside)
HDS (outside)
SDS (inside)HDS (inside)
Unsheared + Sheared Outside
Inside
trace 9
Estimated Mean Void Ratio
Voi
d R
atio
Ent
ropy
0.55 0.6 0.65 0.7 0.750.3
0.35
0.4
0.45
0.5
0.55
SDU
SDS
HDU
HDS
Unsheared Best Fit Line
Sheared Best Fit Line
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion 0.634 0.029
R2
0.643
1.342 0.513
R2
0.875
0.502 0.106
R2
0.296
Average void ratios inside, outside, and total -- compare change, ASTM emax
0.55 0.6 0.65 0.7 0.750.3
0.35
0.4
0.45
0.5
0.55
SDU
HDU
SDS (outside)
HDS (outside)
SDS (intside)HDS (inside)
Unsheared + Sheared Outside
Inside
trace 9
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
0.55 0.6 0.65 0.7 0.750.7
0.72
0.74
0.76
0.78
0.8
SDU
HDU
SDS (outside)
HDS (outside)
SDS (inside)HDS (inside)
Unsheared + Sheared Outside
Inside
trace 9
Estimated Mean Void Ratio
Voi
d R
atio
Ent
ropy
corr out_un Hvout_un 20.517 corr ins Hvins 2
0.545
0.55 0.6 0.65 0.7 0.750.7
0.72
0.74
0.76
0.78
0.8
SDU
SDS
HDU
HDS
Unsheared Best Fit Line
Sheared Best Fit Line
Estimated Mean Void Ratio
Voi
d R
atio
Ent
ropy
Hv 0.700 0.128
R2
0.836
Hv 0.511 0.383
R2
0.624
Hv 0.559 0.326
R2
0.517
0.55 0.6 0.65 0.7 0.750.3
0.35
0.4
0.45
0.5
0.55
SDU
SDS
HDU
HDS
Unsheared Best Fit Line
Sheared Best Fit Line
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
0.55 0.6 0.65 0.7 0.750.3
0.35
0.4
0.45
0.5
0.55
SDU
HDU
SDS (outside)
HDS (outside)
SDS (intside)HDS (inside)
Unsheared + Sheared Outside
Inside
trace 9
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
0.55 0.6 0.65 0.7 0.750.7
0.72
0.74
0.76
0.78
0.8
SDU
HDU
SDS (outside)
HDS (outside)
SDS (inside)HDS (inside)
Unsheared + Sheared Outside
Inside
trace 9
Estimated Mean Void Ratio
Voi
d R
atio
Ent
ropy
corr out_un Hvout_un 20.517 corr ins Hvins 2
0.545
0.55 0.6 0.65 0.7 0.750.7
0.72
0.74
0.76
0.78
0.8
SDU
SDS
HDU
HDS
Unsheared Best Fit Line
Sheared Best Fit Line
Estimated Mean Void Ratio
Voi
d R
atio
Ent
ropy
Hv 0.700 0.128
R2
0.836
Hv 0.511 0.383
R2
0.624
Hv 0.559 0.326
R2
0.517
0.55 0.6 0.65 0.7 0.750.3
0.35
0.4
0.45
0.5
0.55
SDU
SDS
HDU
HDS
Unsheared Best Fit Line
Sheared Best Fit Line
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
Hv
1
n
i
Pi logn Pi
Distribution entropy:
Integrated N-E Study
GammaDistributions
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7
SD (total)
HD (total)
SD (inside shear band)
HD (inside shear band)
SD (outside shear band)
HD (outside shear band)
Local Void Ratio
Per
cent
Soli
d A
rea
[%]
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7
SD (total)
HD (total)
SD (inside shear band)
HD (inside shear band)
SD (outside shear band)
HD (outside shear band)
Local Void Ratio
Per
cent
Soli
d A
rea
[%]
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7
SD (unsheared)
SD (total)
SD (inside shear band)
SD (outside shear band)
Local Void Ratio
Per
cent
Soli
d A
rea
[%]
0.3940.622Outside
0.4770.747Inside
0.4270.667Sheared
0.4050.598Unsheared
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7
HD (unsheared)
HD (total)
HD (inside shear band)
HD (outside shear band)
Local Void Ratio
Per
cent
Soli
d A
rea
[%]
0.3880.588Outside
0.4760.715Inside
0.4250.636Sheared
0.3730.569Unsheared
Integrated N-E Study
36
Critical State
AA stackpns13c
kPa
p313
kPa
p313
kPa
BB stackpns16c
kPa
p316
kPa
p316
kPa
10 100 1 103
0.55
0.6
0.65
0.7
0.75
Slightly Dilatant
Highly Dilatant
CS Line for CTC (Santamarina and Cho, 2001)
Mean Effective Stress [kPa]
Void
Rat
io
Inside Shear
Band
Global Values
Outside Shear
Band
e – log(p’) Plot
Integrated N-E Study
Above and Belowthe Shear Band
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.74
0.76
0.78
0.8
0.82
0.84
SDS - inside
HDS - inside
SDS - above
HDS - above
SDS - belowHDS - below
Estimated Mean Void Ratio
Void
Rat
io E
ntr
opy
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.35
0.4
0.45
0.5
0.55
SDS - inside
HDS - inside
SDS - above
HDS - above
SDS - belowHDS - below
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
Inside Shear
Band
Dead Zone
No Dead Zone
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.74
0.76
0.78
0.8
0.82
0.84
SDS - inside
HDS - inside
SDS - above
HDS - above
SDS - belowHDS - below
Estimated Mean Void Ratio
Void
Rat
io E
ntr
opy
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.35
0.4
0.45
0.5
0.55
SDS - inside
HDS - inside
SDS - above
HDS - above
SDS - belowHDS - below
Estimated Mean Void Ratio
Sta
ndar
d D
evia
tion
Inside Shear
Band
Dead Zone
No Dead Zone
PlatenDeadZone?Side
ViewPlanView
Integrated N-E Study
37
Shear Band ThicknessSlightly Dilatant Highly Dilatant
(mm) (d50) (mm) (d50)
Void ratio strips
20 11.0 15 12.5 17
40 9.4 13 11.0 15
60 11.0 15 12.5 17
80 12.5 17 11.0 15
Virtual
surfaces5.9 8 7.8 11
Image measurement
20 8.7 12 8.4 11
40 8.5 11 9.7 13
60 9.4 13 9.2 12
80 7.6 10 9.7 13
External
measurement8.3 11 9.0 12
9.24 mm
Integrated N-E Study
The Transition Zone Concept
Primary
Block 1
Primary
Block 2
Transition
Zone
Primary
Block 1
Primary
Block 2
Shear
Block
Transition
Zones
Initial Condition
1 block
Just after peak
2 blocks
1 transition
zone
At large strain
3 blocks
2 transition
zones
Void Ratio
Strips
Virtual Surfaces
Integrated N-E Study
38
Numerical Simulations
ParameterNumerical
Simulations
Physical
ComparisonReference
Particle normal
stiffness108 N/m 4 × 106 N/m
(Santamarina et
al., 2001)
Particle shear
stiffness107 N/m n/a n/a
Particle friction
coefficient0.31 0.31
(Proctor and
Barton, 1974)
Particle specific
gravity2.65 2.65 (Yang, 2002)
Platen
stiffness108 N/m n/a n/a
Membrane
stiffness107 N/m 500 N/m (Frost, 1989)
Platen/membrane
friction coefficient0.31 n/a n/a
Membrane specific
gravity1.50 1.10
(MatWeb,
2005)
Integrated N-E Study
Velocity-Controlled Flexible Wall
Numerical-Experimental Comparison
Test Designation fp yp fcs fs ycs e C R A
SD (E) 38.8° 8.8° 27.7° 27.7° 1.0° 59° 64° 49° 57°
SD (N) 39.5° 16.7° 28.3° 28.2° 3.0° 57° 65° 53° 59°
HD (E) 41.8° 11.5° 27.9° 27.9° 0.2° 61° 66° 51° 58°
HD (N) 42.8° 9.5° 27.8° 27.8° 3.8° 58° 66° 50° 58°
Integrated N-E Study
0
200
400
600
800
Str
ess
[kP
a]
0 1 2 3 4 5 6 7 8 9 103
2
1
0
Experimental Results
Numerical Results
Axial Strain [%]
Vo
lum
etri
c S
trai
n [
%]
0
100
200
300
400
Str
ess
[kP
a]
0 1 2 3 4 5 6 7 8 9 104
2
0
Experimental Results
Numerical Results
Axial Strain [%]
Vo
lum
etri
c S
trai
n [
%]
Slightly Dilatant Highly Dilatant
Experimental
Numerical
39
Shear Band Formation
eax = 0% eax = 2% eax = 4% eax = 6% eax = 8% eax = 10%
Slig
htly
Dila
tant
Hig
hly
D
ilata
nt
Integrated N-E Study
Subregional VoidRatio Analyses
eax = 0% eax = 2% eax = 4% eax = 6% eax = 8% eax = 10%
Slig
htly
Dila
tant
Hig
hly
D
ilata
nt
Integrated N-E Study
40
Evolution of LVRD
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
2
4
6
8
10
0% Strain
2% Strain
4% Strain
6% Strain8% Strain
10% Strain
Local Void Ratio
Per
cen
t S
oli
d A
rea
[%]
eax
0
2
4
6
8
10
%
0.159
0.162
0.184
0.190
0.192
0.195
0.049
0.049
0.065
0.069
0.070
0.073
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
2
4
6
8
10
0% Strain
2% Strain
4% Strain
6% Strain8% Strain
10% Strain
Local Void Ratio
Per
cent
Soli
d A
rea
[%]
eax
0
2
4
6
8
10
%
0.162
0.171
0.196
0.208
0.215
0.219
0.049
0.052
0.067
0.078
0.083
0.086
Slightly Dilatant Highly Dilatant
Integrated N-E Study
Numerical Results:Strip Analyses
0 2 4 6 8 100.14
0.16
0.18
0.2
0.22
0.24
Image Void Ratio
Mean of Local Void Ratios
Mean of Void Ratios Inside the Shear Band
Mean of Void Ratios Outside the Shear Band
Axial Strain [%]
Vo
id R
atio
0 2 4 6 8 100.14
0.16
0.18
0.2
0.22
0.24
0.26
Image Void Ratio
Mean of Local Void Ratios
Mean of Void Ratios Inside the Shear Band
Mean of Void Ratios Outside the Shear Band
Axial Strain [%]
Vo
id R
atio
Slightly Dilatant Highly Dilatant
Integrated N-E Study
41
Shear BandParameters
0.14 0.16 0.18 0.2 0.22 0.24 0.260.45
0.5
0.55
0.6
0.65
0.7
0.75
SD - total
HD - total
SD - inside
HD - inside
SD - outside
HD - outside
trace 7
Estimated Mean Void Ratio
Void
Rat
io E
ntr
opy Hv 2.4 0.1
R2
0.982
0 2 4 6 8 100.45
0.5
0.55
0.6
0.65
0.7
SD - total
HD - total
SD - inside
HD - inside
SD - outside
HD - outside
Global Axial Strain [%]
Vo
id R
atio
En
tro
py
Void Ratio
Strips
Direct
Measurement
(L) (d50) (L) (d50)
Slightly Dilatant Specimen
eax = 4% 0.97 15 0.91 15
eax = 6% 0.97 15 0.97 15
eax = 8% 1.1 18 1.1 18
eax = 10% 1.1 18 1.3 21
Highly Dilatant Specimen
eax = 4% 0.65 10 0.99 16
eax = 6% 0.65 10 0.99 16
eax = 8% 0.81 12 1.1 18
eax = 10% 0.97 15 1.2 19
Integrated N-E Study
Overview
Soil micromechanics
Discrete element method (DEM)
Integrated Numerical-Experimental Study
DEM Simulations: Effects of Geometry
Soil-Structure Interaction (SSI)
Thermal Conductivity
Summary and Conclusions
Preliminaries
42
Simulation ofLaboratory Tests
Assemblies consisted of ~15,000 2-ball particles in the sample volume
– 2H:1W for axisymmetric
– 7H:2W:4D for plane strain
Specimens assembled in “loose” (e0 = 0.67), “medium” (e0 = 0.54), and “dense” (e0 = 0.47) states
Consolidated under “low” (ζ3' = 75 kPa) and “high” (ζ3' = 450 kPa) confining stresses
Stacked wall entities used for servo control of confining stresses
Model and material parameters constant across simulations
Zhao, X. and T.M. Evans. (2009). “Discrete Simulations of Laboratory Loading Conditions,” International Journal of Geomechanics, 9(4), pp. 169-178.
Effects of Geometry
Macroscale Simulation Results
Plane Strain Simulations
• Note effects of e0 and ζ3'
• Convergence to ultimate/terminal state (=f(e0))
Axisymmetric Simulations
• Effects of e0 and ζ3' similar to PS
• No ultimate/terminal state at ε1=10%
500
1000
1500
2000PS-L75
PS-M75
PS-D75
PS-L450
PS-M450
PS-D450
Devia
toric
Str
ess
[kPa]
0 0.02 0.04 0.06 0.080.08
0.06
0.04
0.02
0
0.02
0.04
Axial Strain [ ]
Volu
metr
ic S
train
[ ]
500
1000
1500
2000CTC -L75
CTC -M75
CTC -D75
CTC -L450
CTC -M450
CTC -D450
Devia
toric
Str
ess
[kPa]
0 0.02 0.04 0.06 0.080.08
0.06
0.04
0.02
0
0.02
0.04
Axial Strain [ ]
Volu
metr
ic S
train
[ ]
Effects of Geometry
43
Plane Strain Simulations
• Region of high localized strain (shear band)
• Post-peak stress-strain response controlled by shear band
Axisymmetric Simulations
• Diffuse (bulging) failure, single shear band not easily identified
• Entire specimen still deforming post-peak
Macroscale Deformation
ε1 = 0%
Plane Strain Axisymmetric Compression
ε1 = 10%ε1 = 10% ε1 = 0%
Effects of Geometry
Small-Strain Response
Young’s Modulus: Calculated from CTC Results versus Measured in PS
Poisson’s Ratio: Calculated from CTC Results versus Measured in PS
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
Simulation
1:1 Line
= 0.294 (C TC )
Measured [ ]
Cal
cula
ted [
]
0 50 100 1500
50
100
150
Simulation
1:1 Line
Best-F it Line
= 0.294 (CTC )
Measured [MPa]
Calc
ula
ted [
MPa]
Ec 9.08 MPa 1.03 Em
R2
0.999
21
CTCPS
CTC
EE
1
CTCPS
CTC
Effects of Geometry
44
Shear Strength
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Peak
1:1 Line
Peak
1:1 Line
Measured [ ]
Calc
ula
ted [
]
tan ' tan ' cos 'ds ps cvf f f
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Peak
1:1 Line
Peak
1:1 Line
Measured [ ]
Calc
ula
ted [
]
1
sec' tan (1.2 tan ' )ps dsf f
Rowe (1962) Bolton (1986)
Effects of Geometry
Shear Strength
0 0.5 1 1.50
0.5
1
1.5
Peak
1:1 Line
Peak
1:1 Line
Meausred [ ]
Calc
ula
ted [
]
2
2
( 1) 12 3tan ' cos '
4 3ps cv
KD D D
KD KD Df f
1
1d
Dd
e 2 '
tan 452
cvKf
Hanna (2001)
Effects of Geometry
Ramamurthy and Tokhi (1981)
f
1
sin ctc
1 1.5 2 2.5 31
1.5
2
2.5
3
Peak
A t 10% Global Strain
1:1 Line
b = (σ2'-σ3')/(σ1'-σ3')
• Peak strength is consistent with the theoretical relationship of R&T (1981) and Hanna (2001)
• Results at ultimate strain are inconsistent, with the exception of specimens which do not dilate
45
Friction and Dilation
0 10 20 30 400
10
20
30
40
Predicted Dilation A ngle [deg]
Measu
red D
ilation A
ngle
[deg]
1 max
1
0.048
vp
d
d
ey
e
1.25p p csy
1 max
10 vp
d
d
ey
e
Bolton’s (1986) Dilatancy Relationships (note that black
points are TXC)
• Dilation angle measurements are not strictly consistent with the semi-empirical relationships of Bolton
• In general, boundary measurements will tend to overpredict the dilation angle
Effects of Geometry
Overview
Soil micromechanics
Discrete element method (DEM)
Integrated Numerical-Experimental Study
DEM Simulations: Effects of Geometry
Soil-Structure Interaction (SSI)
Thermal Conductivity
Summary and Conclusions
Preliminaries
46
Definition ofthe Granular Material
Material Properties:
ParameterNumerical
Simulation
Physical
ComparisonReference
Grain Normal
Stiffness108 N/m 4x106 N/m
(Santamarina
et al., 2001)
Grain Shear
Stiffness107 N/m n/a n/a
Grain Friction 0.31 0.31(Proctor and
Barton, 1974)
Grain Specific
Gravity2.65 2.65 (Yang, 2002)
Wall Stiffness 108 N/m n/a n/a
L
W
Aspect Ratio
Soil-Structure Interaction
Shear Strengthof the Granular Material
Confining
Stress
(kPa)
Peak Friction
Angle (deg)
Critical State
Friction
Angle (deg)
25 34.8 28.8
50 32.5 26.3
75 31.6 26.6
100 31.0 24.9
Mean 32.5 26.6
Friction Value for Confining Stress Cases
Soil-Structure Interaction
Biaxial Compression Deviator Stress
0 0.02 0.04 0.06 0.08 0.10
50
100
150
200
250
σ’3 = 100 kPaσ’3 = 75 kPaσ’3 = 50 kPaσ’3 = 25 kPa
Axial Strain [ ]
Dev
iato
r St
ress
[kP
a]
47
Shear Strengthof the Granular Material
Mohr-Coulomb Method at Peak Mohr-Coulomb Method at Critical State
0 40 80 120 160 200 240 280 3200
30
60
90
120
150
Normal Effective Stress [kPa]
Shea
r St
ress
[kP
a]
0 50 100 150 200 2500
25
50
75
100
Normal Effective Stress [kPa]
σ’3 = 100 kPaσ’3 = 75 kPaσ’3 = 50 kPaσ’3 = 25 kPa
σ’3 = 100 kPaσ’3 = 75 kPaσ’3 = 50 kPaσ’3 = 25 kPa
Soil-Structure Interaction
Elastic Propertiesof the Granular Material
Initial Elastic Region of the Deviator Strain Curve
E
0 0.02 0.04 0.06 0.08 0.10
50
100
150
200
250
Axial Strain [ ]
Dev
iato
r St
ress
[kP
a]
0.001 0.002 0.003 0.0040
50
100
Axial Strain [ ]
Small Strain Region of the Deviator Strain Curve
EBest
EWorst
σ’3 = 100 kPaσ’3 = 75 kPaσ’3 = 50 kPaσ’3 = 25 kPa
σ’3 = 100 kPaσ’3 = 75 kPaσ’3 = 50 kPaσ’3 = 25 kPa
Soil-Structure Interaction
48
1 x 10-2
0.014
8 x 10-3
2 x 10-3
4 x 10-3
Elastic Propertiesof the Granular Material
3
1 1
1 ve e
e e
Biaxial Compression Volumetric Strain
0 0.016 0.024 0.032 0.040.020
Axial Strain [ ]
Vo
lum
etri
c St
rain
[ ]
0.008
σ’3 = 100 kPaζ’3 = 75 kPaζ’3 = 50 kPaζ’3 = 25 kPa
Soil-Structure Interaction
Elastic Propertiesof the Granular Material
Biaxial Compression Volumetric Strain
Axial Strain [ ]
Vo
lum
etri
c St
rain
[ ]
σ’3 = 100 kPaζ’3 = 75 kPaζ’3 = 50 kPaζ’3 = 25 kPa
3 x 10-3
2.25 x 10-3
1.5 x 10-3
7.5 x 10-4
00 2.5 x 10-3 5 x 10-3 7.5 x 10-3 1.0 x 10-2
1
ve
e
Soil-Structure Interaction
49
Interface Strengthof the Granular Material
Soil-Structure Interaction
DEM Interface Shear Results for Friction Angle
0 0.1 0.2 0.30
10
20
30
0.00 x D50
0.05 x D50
0.25 x D50
0.50 x D50
0.75 x D50
1.00 x D50
2.00 x D50
3.00 x D50
Biaxial Test
Sawtooth Size [m]Fr
icti
on
An
gle
[deg
]
Coefficient of Friction Experimental Results (after Uesugi & Kishida 1986b)
Shallow Foundation Models
Soil-Structure Interaction
50
Shallow Foundation Models
12 m
30 m
1 m
2 m
Shallow Foundation Baseline Assembly
60 m
30 m
24 m 12 m
Shallow Foundation Extended Width Assembly
Shallow Foundation Extended Height Assembly
Soil-Structure Interaction
Shallow Foundation Results
0 20 40 60 80 100
0
0.15
0.1
0.05
Baseline
Extend WidthExtend Height
Resistance (kN)
No
rmal
ized
Dis
pla
cem
ent
[ ]
Shallow Foundation Model Dimensions Total Resistance
Shallow Foundation Friction Total Resistance
0 20 40 60 80 100
0
0.15
0.1
0.05
Foundation Wall Friction 0.155
Foundation Wall Friction 0.31 Foundation Wall Friction 0.46
Foundation Wall Friction 0.62
Resistance (kN)
No
rmal
ized
Dis
pla
cem
ent
[ ]
Soil-Structure Interaction
51
Shallow Foundation Results
Shallow Foundation Grain Rotations at 0.00B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.00B
Soil-Structure Interaction
Shallow Foundation Grain Rotations at 0.05B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.05B
Soil-Structure Interaction
Shallow Foundation Results
52
Shallow Foundation Grain Rotations at 0.10B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.10B
Soil-Structure Interaction
Shallow Foundation Results
Shallow Foundation Grain Rotations at 0.15B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.15B
Soil-Structure Interaction
Shallow Foundation Results
53
Shallow Foundation Grain Rotations at 0.20B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.20B
Soil-Structure Interaction
Shallow Foundation Results
Shallow Foundation Grain Rotations at 0.25B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.25B
Soil-Structure Interaction
Shallow Foundation Results
54
Shallow Foundation Force Chains at Baseline, Extended Height, and Extended Width Model
Size (CW Respectively). Vertical Displacements are 0.20B for all cases shown.
Soil-Structure Interaction
Shallow Foundation Results
Deep Foundation Models
25m
26 m
15 m
2 m
Parametri
c Case
No.
Description
Model
Dimensions
(Height x
Width)
Foundatio
n Wall
Friction
1 Low Friction 25 m x 26 m μ = 0.155
2 Baseline 25 m x 26 m μ = 0.31
3 High Friction 25 m x 26 m μ = 0.46
4Very High
Friction25 m x 26 m μ = 0.62
5Extended
Width36 m x 25 m μ = 0.31
6Extended
Height26 m x 35 m μ = 0.31
Deep Foundation Parametric Analysis
Deep Foundation:
Soil-Structure Interaction
55
35 m
25 m
36 m
26 m
Deep Foundation Extended Width Assembly
Deep Foundation Extended Height Assembly
Soil-Structure Interaction
Deep Foundation Models
0 60 120 180 240 300
0.15
0.1
0.05
0
Baseline Extend WidthExtend Height
Resistance (kN)
No
rmal
ized
Dis
pla
cem
ent
[ ]
0 60 120 180 240 300
0
0.15
0.1
0.05
Foundation Wall Friction 0.155Foundation Wall Friction 0.31
Foundation Wall Friction 0.46Foundation Wall Friction 0.62
Resistance (kN)
No
rmal
ized
Dis
pla
cem
ent
[ ]
Deep Foundation Model Dimensions Total Resistance
Deep Foundation Friction Total Resistance
Soil-Structure Interaction
Deep Foundation Results
56
Deep Foundation Results
Soil-Structure Interaction
Deep Foundation Toe versus Side Resistance
0 50 100 150 200 250
0.15
0.1
0.05
0
Foundation Base Friction 0.155
Foundation Base Friction 0.31 Foundation Base Friction 0.46Foundation Base Friction 0.62
Foundation Base Extended WidthFoundation Base Extended Height
Foundation Side Friction 0.155
Foundation Side Friction 0.31Foundation Side Friction 0.42Foundation Side Friction 0.62
Foundation Side Extended WidthFoundation Side Extended Height
Resistance (kN)
Nor
mal
ized
Dis
plac
emen
t [ ]
Deep Foundation Results
Deep Foundation Grain Rotations at 0.00B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.00B
Soil-Structure Interaction
57
Deep Foundation Grain Rotations at 0.05B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.05B
Soil-Structure Interaction
Deep Foundation Results
Deep Foundation Grain Rotations at 0.10B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.10B
Soil-Structure Interaction
Deep Foundation Results
58
Deep Foundation Grain Rotations at 0.15B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.15B
Soil-Structure Interaction
Deep Foundation Results
Deep Foundation Grain Rotations at 0.20B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.20B
Soil-Structure Interaction
Deep Foundation Results
59
Deep Foundation Grain Rotations at 0.25B, Vertical Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.25B
Soil-Structure Interaction
Deep Foundation Results
Laterally Loaded Pile Model
Laterally Loaded Pile:
Pile Center of Mass
0.01 rad/s rotation
0.1B total rotation
1.8° total rotation
Shallow Foundation Model Dimensions Total Resistance
25 m
26 m
15 m
Soil-Structure Interaction
60
Laterally LoadedPile Results
Soil-Structure Interaction
Deep Foundation Lateral Loading Lateral Stresses on Foundation Wall
Deep Foundation Lateral Loading Vertical Stresses on Foundation Wall
1 1.50
100
200
300
400
Bottom Pile Wall
Left Pile Wall
Right Pile Wall
Rotation [deg]
Str
ess
[kP
a]
1 1.50
50
100
150
Bottom Pile Wall
Left Pile Wall
Right Pile Wall
Rotation [deg]
Str
ess
[kP
a]
Deep Foundation Lateral Loading Grain Rotations at 0.00B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.00B
Soil-Structure Interaction
Laterally LoadedPile Results
61
Deep Foundation Lateral Loading Grain Rotations at 0.05B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.05B
Soil-Structure Interaction
Laterally LoadedPile Results
Deep Foundation Lateral Loading Grain Rotations at 0.10B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.10B
Soil-Structure Interaction
Laterally LoadedPile Results
62
Deep Foundation Lateral Loading Grain Rotations at 0.15B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.15B
Soil-Structure Interaction
Laterally LoadedPile Results
Deep Foundation Lateral Loading Grain Rotations at 0.20B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.20B
Soil-Structure Interaction
Laterally LoadedPile Results
63
Deep Foundation Lateral Loading Grain Rotations at 0.25B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.25B
Soil-Structure Interaction
Laterally LoadedPile Results
Deep Foundation Lateral Loading Grain Rotations at 0.30B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.30B
Soil-Structure Interaction
Laterally LoadedPile Results
64
Deep Foundation Lateral Loading Grain Rotations at 0.35B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.35B
Soil-Structure Interaction
Laterally LoadedPile Results
Deep Foundation Lateral Loading Grain Rotations at 0.40B Pile Rotation
< -35
-35 ~ -8
> +35
+8 ~ +35
+2 ~ +8
- 2 ~ +2
-8 ~ -2
δ = 0.40B
Soil-Structure Interaction
Laterally LoadedPile Results
65
Deep Foundation Lateral Loading Force Chains at 0.1B Pile Rotation
Soil-Structure Interaction
Laterally LoadedPile Results
Deep Foundation Lateral Loading Vertical Stress at 0.1B Pile Rotation (Units: Pa)
Deep Foundation Lateral Loading Coordination Number at 0.1B Pile Rotation
Deep Foundation Lateral Loading Shear Stress at 0.1B Pile Rotation
(Units: Pa)
Deep Foundation Lateral Loading Lateral Stress at 0.1B Pile Rotation
(Units: Pa)
Soil-Structure Interaction
Laterally LoadedPile Results
66
Rigid Retaining Walls
Soil-Structure Interaction
Rigid Retaining Wall Load-Displacement Active Case
Rigid Retaining Wall Load-Displacement Passive Case
0 0.05 0.1 0.15 0.20
200
400
600
Left WallRight Wall
Displacement [m]
.
0 0.03 0.06 0.09 0.12 0.150
50
100
150
Left WallRight Wall
Displacement [m]
Late
ral T
hru
st [
kN]
Rigid Retaining Wall:
30 m x 12 m (W x H)
Models sizes same as for shallow foundations
Lateral displacement of rigid wall
Rigid Retaining Wall Active Grain Rotations at 0.00B, Vertical
Displacement
Rotations Color Bar for Rotations Figures (deg)
δ = 0.00B
Rigid Retaining Walls
Soil-Structure Interaction
67
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Active Grain Rotations at 0.05B, Vertical
Displacement
δ = 0.05B
Rigid Retaining Walls
Soil-Structure Interaction
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Active Grain Rotations at 0.10B, Vertical
Displacement
δ = 0.10B
Rigid Retaining Walls
Soil-Structure Interaction
68
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Active Grain Rotations at 0.15B, Vertical
Displacement
δ = 0.15B
Rigid Retaining Walls
Soil-Structure Interaction
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Active Grain Rotations at 0.20B, Vertical
Displacement
δ = 0.20B
Rigid Retaining Walls
Soil-Structure Interaction
69
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Passive Grain Rotations at 0.05B, Vertical
Displacement
δ = 0.05B
Rigid Retaining Walls
Soil-Structure Interaction
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Passive Grain Rotations at 0.10B, Vertical
Displacement
δ = 0.10B
Rigid Retaining Walls
Soil-Structure Interaction
70
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Passive Grain Rotations at 0.15B, Vertical
Displacement
δ = 0.15B
Rigid Retaining Walls
Soil-Structure Interaction
Rotations Color Bar for Rotations Figures (deg)
Rigid Retaining Wall Passive Grain Rotations at 0.20B, Vertical
Displacement
δ = 0.20B
Rigid Retaining Walls
Soil-Structure Interaction
71
Rigid retaining wall active force chains (top) and passive force chains (bottom). Retaining wall lateral displacement is 0.10 m.
Rigid Retaining Walls
Soil-Structure Interaction
Akbas & Kulhawy (2009a) Hyperbolic Method:
Shallow FoundationLoad-Displacement
Shallow Foundation Normalized DEM & Hyperbolic Fit
Definition of QL2 (After Akbas and Kulhawy 2009a)
2L
Q B
Q a B b
Hyperbolic fit equation. Parameters a and b are 0.70 and 1.77 respectively for residual soils and 0.69 and 1.68 respectively for cemented soils
0 0.38 0.75 1.5
0.075
0.05
0.025
0
DEM QL2 PeakDEM QL2 CriticalAkbas & Kulhawy (2009a)
Normalized Resistance [ ]
Norm
aliz
ed S
ett
lem
ent
[ ]
1.13
Soil-Structure Interaction
72
Shallow FoundationBearing Capacity
Shallow Foundation:
1 0 5ult f q q .q D ( N )d BN dg gg g
Bearing Capacity (effective stress analysis):
2
4 2
tan
qN e tan f f
6 520 477
.N . eg
fult
a fF.S.
qq Dg
Allowable Load:Ueno et al. (1998):
Where:
Soil-Structure Interaction
Shallow FoundationBearing Capacity
Soil-Structure Interaction
0
0 50 100
0.3
0.2
0.1
DEM Baseline Load-DisplacementImmediate Settlement (Gazetas et al. 1985)Total Settlement (Mayne & Poulos 2001)Allowable Load (Peak. FS = 3.0)
Resistance [kN]
Dis
plac
emen
t [m
]
Shallow Foundation Bearing Capacity:
At Ф’p– Ultimate Load = 246.6 kN
– Allow FS 1.5 = 168.9 kN
– Allow FS 3.0 = 86.8 kN
73
Shallow FoundationFailure Surfaces
Soil-Structure Interaction
Grain Rotations Shallow Foundation Baseline at 0.20B Vertical Displacement (computed at Ф’peak )
0tan( )
r( ) r e f 4 2 f
Pile Lateral Load-Deflection
Analytical Solution (Matlock and Reese, 1962)
mh is related to the subgrade modulus for the soil response along the pile
M = 0 ; Px is reaction at ground surface from compressive forces acting on the pile in the subgrade
Lateral deflection at ground surface to be approximately 0.065 m and rotation of the pile walls to be approximately 0.318°
2 3
2 43 1 624 4
x c cL
h h
P l lM. .
m m
3 4
1 62 1 734 4
x c c
h h
P l lM. .
m m
1 5
4p p
c
h
E Il
m
Soil-Structure Interaction
74
Deep Foundation Laterally Loaded Pile Rotation Quantification
1.8°
Q
Laterally Loaded Pile:
DEM Results (AutoCAD measurements)
Pile Lateral Load-Deflection
Soil-Structure Interaction
Laterally Loaded Pile:
DEM Results (AutoCAD measurements)
Set to 0.25⁰, determine lateral
displacement along ground surface
Deep Foundation Laterally Loaded Pile Rotation at 0.25 (deg)
θ
ρ
Pile Lateral Load-Deflection
Soil-Structure Interaction
75
Rigid Wall Force Mobilization
Soil-Structure Interaction
Rigid Retaining Wall Normalized Lateral Force Active Case
Rigid Retaining Wall Normalized Lateral Force Passive Case
Deflections required for full mobilization of active or passive thrust (after Salgado 2008)
Soil TypeActive State
(δh/H0)Passive State
(δh/H0)
Dense Sand 0.001 0.020
Loose Sand 0.004 0.060
Stiff Clay 0.010 0.020
Soft Clay 0.020 0.040
Active Case Analytical Versus DEM Results (at Ф’cs)
Description
Formulation
(Theoretical)
Theoretical
Value (A)
DEM Simulation
Value (B)
Orientation [deg] 58.3 60 – 65 (rot figs)
Normalized Lateral Wall
Deflection [ ]0.001 0.001 - 0.004
Earth Pressure
Coefficient [ ]0.38 0.32
Lateral Thrust [kN] 48.4 37.9
1
1a
sin( )K
sin( )
f
f
0
0
H
a aP K zdzg
4 2a
f
0
a
H
d
Rigid Wall EarthStress Coefficients
Soil-Structure Interaction
76
Passive Case Analytical Versus DEM Results (at Ф’cs)
Description
Formulation
(Theoretical)
Theoretical
Value (A)
DEM Simulation
Value (B)
Orientation [deg] 31.7 30 – 36 (rot figs)
Normalized Lateral Wall
Deflection [ ]0.020 0.004 - 0.015
Earth Pressure
Coefficient [ ]2.62 3.16
Lateral Thrust [kN] 533.4 507.9
1
1a
sin( )K
sin( )
f
f
0
0
H
a aP K zdzg
4 2a
f
0
a
H
d
Rigid Wall EarthStress Coefficients
Soil-Structure Interaction
Overview
Soil micromechanics
Discrete element method (DEM)
Integrated Numerical-Experimental Study
DEM Simulations: Effects of Geometry
Soil-Structure Interaction (SSI)
Thermal Conductivity
Summary and Conclusions
Preliminaries
77
Heat in Soils
Historically ignored or considered insignificant in geotechnical engineering
98% of Earth’s volume has T>1000°C
Heat flux in oceans and climate change are observable manifestations of geothermal processes
Important in alternative energy and waste isolation applications
Coupled thermo-hydro-mechanical response incredibly difficult to model (fully-coupled multiphysics simulations)
Thermal Conduction
Thermal Properties of Soils
Specific Heat: heat required to raise the temperature of a unit mass of material by 1 K (bulk property, scalar, J·kg-1K-1)
Thermal Conductivity: heat flow per unit temperature gradient (transport property, scalar, W·m-1K-1)
Thermal Diffusivity: governs rate of spread of temperature disturbances (bulk property, scalar, m2s-1)
Coefficient of Thermal Expansion: measures fractional change in size per unit temperature change (bulk property, scalar, K-1)
Pringle, UA-F, GEOS 692, 2006
Thermal Conduction
Q m Tc
q T
Dc
L
dLL
dT
78
Thermal Properties of Soils
SpecificHeat
[J·kg-1K-1]
Volumetric Heat Capacity
[J·m-3K-1]
Thermal Conductivity[W·m-1K-1]
Thermal Diffusivity
[m2s-1]
Coefficient of Volume Thermal
Expansion[K-1]
Solid 750 2×106 8 1×10-6 15×10-6
Water 4184 4×106 0.56 9×10-8 207×10-6
Air 1000 1×103 0.024 4×10-9 1/T0
Additional Notes:1. Values given for solids are representative of soil minerals.2. Volumetric heat capacity is specific heat multiplied by mass density.3. Can you prove that αV=1/T0 for air?
Thermal Conduction
Thermal Conductivity of the Matrix
Consider a dry (two-phase) soil. We can approximate thermal conductivity of the matrix using semi-empirical or analytical approaches.
Semi-empirical:
Analytical: next slide
Equation Notes Reference
λ in W/m·KGs=2.7
Johansen, 1975, lower estimate
λ in W/m·KJohansen, 1975, upper
estimate
λ in W/m·K ρ in g/cm3 Gavriliev, 2004
(after Yun and Santamarina, 2008)
Thermal Conduction
79
Thermal Conductivity of the Matrix
Model Equation Reference
SeriesDeVera, and Strieder, 1977
ParallelDeVera, and Strieder, 1977
Geometric Mean Sass, et al., 1971
Hashin and Shtrikman Boundary
Hashin and Shtrikman,1962
Self-Consistent Hill, 1965
NOTE: For HSL, 1=solid, 2=pore; for HSU, 1=pore, 2=solid.
(after Yun and Santamarina, 2008)
Thermal Conduction
Heat Transport in Soils
Five primary mechanisms
– Conduction (dominates in solid phase)
– Convection (dominates in liquid phase)
– Radiation (no material medium required)
– Vaporization/condensation (partially saturated soils)
– Ion exchange (flow of water through clays)
Big players: conduction, convection
Thermal Conduction
80
Heat is transferred by molecular excitation of stationary materials
Governed by Fourier’s Law
– Differential form:
– Integral form:
Conduction
(after McCartney, 2009)
Thermal Conduction
q T
S
QT dA
t
Convection
Thermal Conduction
Transfer of heat between two thermodynamic systems moving relative to one another
Newton’s Law of Cooling
– For water: w w w wq c T T
81
The Heat Equation
Thermal Conduction
Assuming conduction only, we can combine Fourier’s Law with the continuity equation to get the Heat Equation
– Fourier’s Law:
– Continuity:
– Combining:
– Which gives us:
q T
Tc q
t
T
c Tt
2TT
t c
Heat Conduction in the Ground
One-dimensional heat conduction equation is:
During the year, the surface temperature is a sinusoidal function:
The analytical solution is:
2
2
, ,T x t T x t
t c x
0(0, ) sinT t T t T
0, exp sin2 2
c cT x t T x t x T
(after Rongère, 2009)
Thermal Conduction
82
Ground Temperature
Ground temperature remains constant around the year for a depth greater than 10 m.
Ground characteristics:– T0= 20oC
– ω = 2.10-7 s-1
– ρ = 2,300 kg·m-3
– C = 900 J·kg-1·K-1
– k = 1.5 W·m-1·K-1
Ground Temperature Profiles
0
5
10
15
20
25
-20 -10 0 10 20
Temperature (C)
De
pth
(m
) Spring
Summer
Fall
Winter
(after Rongère, 2009)
Thermal Conduction
Thermal Conductivity:Existing Models
Thermal Conduction
83
Thermal Conductivity:Discrete Models
Yun, T.S. and T.M. Evans. (2010). “Three-dimensional random network model for thermal conductivity in particulate materials.” Computers and Geotechnics, in press.
Thermal Conduction
Summary and Conclusions
Many (all?) macroscale behaviors are driven by microscale processes
Enhanced understanding of soil micromechanics can lead to better understanding of design-scale soil behavior
Discrete simulations can be used for micromechanical studies or applied to a variety of design problems
Well-calibrated discrete simulations can reasonably predict soil behavior across multiple scales
Top Related