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FLACTraining CourseBasic Concepts and Recommended Procedures
for
Geotechnical Numerical Analysis
related to
Nuclear Waste IsolationAugust 7-11, 2006
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Instructors:
Dr. Roger HartYanhui Han
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Training ScheduleAugust 7, 2006 (morning)
09:00-12:00 Overview on Numerical Modeling forNuclear Waste Isolation
- Introduction and overview by IAEA
- Problems related to repository design and engineering
- Participant perceptions (each participant providesher/his perspective on numerical modeling in the contextof their national program ~ 10-15 min. per participant)
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Training ScheduleAugust 7, 2006 (afternoon)
01:00-02:00 Overview of Itasca and Itasca Software
Introduction to FLAC
- Overview of capabilities in geo-engineering analysisand design
- Modeling features specific to waste isolation studies
02:00-03:00 Introduction to the FLAC Graphical Interface
- Menu-driven versus command-driven operation
03:00-03:15 Break
03:15-05:00 FLAC Theoretical Background
- Explicit finite-difference solution
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Overview of Itasca
Consulting Services and Software for the
Mining, Civil, Petroleum, and Waste Isolation Industries
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Itasca office locations
plus software agents in 13 countries
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Itasca codes
1. FLAC two dimensional continuum, with joints
2. FLAC3D three dimensional continuum, with joints
3. PFC3D three dimensional DEM* spheres + clumps
4. PFC2D two dimensional DEM disks + clumps
5. UDEC two dimensional DEM polygonal bodies
6. 3DEC three dimensional DEM polyhedral bodies
All codes use an explicit, dynamic solution scheme, even to simulate
quasi-static problems. All include coupled fluid and thermal modes,
and include many nonlinear constitutive models.
All codes treat interactions between separate objects as boundaryconditions; there is no concept of a joint element. Thus, even for the
continuum codes, the DEM scheme is used for interactions.
* DEM (distinct/discrete element method)
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All Itasca codes
contain a built-in programming language, called FISH, thatallows users to:
add new plots or printout options
control a simulation (and the conditions) automatically
access and modify most of the internal variables & properties
set up special in situconditions & boundary conditions
add coupling between codes, or between physical entities.
Also, all codes can accept user-written constitutive(stress/strain) models, written in C++ orFISH(FLAC only).Many users have written their own models. Several models are
available that have been written by others.
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User support1. Extensive manuals, with many examples and useful FISH
functions, are provided, both on CD and in hard-copy.
2. Hundreds of references to papers describing applications of
all codes are available on the Itasca web site
(www.itascacg.com).
3. Worked examples are provided and updated regularly on
the web site; a new site provides a repository for newconstitutive models.
4. Latest code updates may be downloaded from the web.
5. International code-user symposia are held regularly.
6. Rapid answers to users queries are provided, both by
telephone and email (many hundreds of such questions are
handled every year).
7. Consulting agreements may be set up for more extensive
help with setting up models and interpreting the results.
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FLAC is a general-purpose code that can simulate a full range ofnonlinear static & dynamic problems, with coupled fluid flow, heat
flow and structural interaction. Any geometry can be represented,
and the boundary conditions are quite general.
FLAC simulates the behavior of nonlinear continua by the
generalized finite difference method (arbitrary element shapes),also known as the finite volume method.
FLACsolves the full dynamic equations of motion even for quasi-
static problems. This has advantages for problems that involve
physical instability, such as collapse, as will be explained later. To
model the static response of a system, damping is used to absorbkinetic energy.
What isFLAC?
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Advanced, Two and Three DimensionalContinuum Modeling for Geotechnical Analysis
of Rock, Soil, and Structural Support
Basic Features
Nonlinear, large-strain simulation of
continua
Explicit solution scheme, giving
stable solutions to unstable
physical processes
Interfaces or slip-planes are
available to represent distinct
interfaces along which slip and/or
separation are allowed, therebysimulating the presence of faults,
joints or frictional boundaries
Displacements resulting fromconstruction of a shallow tunnel
FLAC & FLAC3D
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Advanced, Two and Three DimensionalContinuum Modeling for Geotechnical Analysis
of Rock, Soil, and Structural Support
Basic Features
Built-in material models:
"null" model,
three elasticity models (isotropic,
transversely isotropic and
orthotropic elasticity),
eight plasticity models (Drucker-
Prager, Mohr-Coulomb, strain-
hardening/softening, ubiquitous-
joint, bilinear strain-
hardening/softening ubiquitous-
joint, double-yield, modified Cam-
clay, and Hoek-Brown)
User-defined models written in
FISH (FLAConly)
Continuous gradient or statisticaldistribution of any property may be
specified
Braced excavation
FLAC & FLAC3D
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Advanced, Two and Three DimensionalContinuum Modeling for Geotechnical Analysis
of Rock, Soil, and Structural Support
Basic Features
Built-in programming language
(FISH) to add user-defined
features
FLACand FLAC3Dcan becoupled to other codes via TCP/IP
links
Convenient specification of
boundary conditions and initial
conditions
Model grid for service tunnel connecting
two main tunnels
FLAC & FLAC3D
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Advanced, Two and Three DimensionalContinuum Modeling for Geotechnical Analysis
of Rock, Soil, and Structural Support
Basic Features
Automatic 3D grid generator
(FLAC3D) using pre-defined shapesthat permit the creation of intersecting
internal regions (e.g., intersecting
tunnels)
Full graphical user interface in FLAC;partial gui in FLAC3D(for plotting andfile handling)
Extensive plotting features
contours, vectors, tensors, flow, etc.)
Graphical output in industry-standard
formats includes PostScript, BMP,
JPG, PCX, DXF (AutoCAD), EMF, and
a clipboard option for cut-and-paste
procedures Sequential excavation and support for ashallow tunnel
FLAC & FLAC3D
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Advanced, Two and Three DimensionalContinuum Modeling for Geotechnical Analysis
of Rock, Soil, and Structural Support
Optional Features
Optional modules include:
thermal, thermal-mechanical, and thermal-poro-
mechanical analysis including conduction and
advection;
visco-elastic and visco-plastic (creep) material
models;
dynamic analysis capability with quiet and free-
field boundaries, and
user-defined constitutive models written in C++
two-phase fluid flow (FLAConly)
Liquefaction failure of apile-supported wharf
FLAC & FLAC3D
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FLAC Version 5 &FLAC3D Version 3
Features
1. Hysteretic dampingmore realistic and more efficient than
Rayleigh damping for dynamic analysis
2. Built-in Hoek-Brown constitutive model
3. Thermal advection (convection) logic for thermal / fluid-flow
analysis
4. Network key license version
5. More efficient calculation of fluid-flow / mechanical analysis(FLAC)
6. New structural element types: liner elements, rockbolt elements,
strip elements (FLAC)
7. Increased calculation speed (10-20% faster) due to optimization to
calculation cycle and updated compiler (FLAC3D)
8. New MOVIE facility in AVI or DCX format (FLAC3D)
9. Optional hexahedral-meshing preprocessor (3DShop) to facilitate
creation of complex meshes (FLAC3D)
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New Features in FLACVersion 5.1
1. Speedup of double-precision version by converting to Intel Fortran
compiler.
2. Automatic re-meshing logic.
3. Parallel processing on multiprocessor computers
(e.g., dual processors or dual core processor)
Pre-release available August 2006Official release in early 2007
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New Features in FLAC3D Version 3.1
1. Parallel processing on multiprocessor computers
(e.g., dual processors or dual core processor)
2. New structural element type Embedded Liner provides shear/slip and
normal interaction with the grid on both sides of the liner (e.g., to simulate
buried sheet pile walls)
3. New Mixed Discretization scheme for tetrahedral elements Nodal Mixed
Discretization provides more accurate solution of plasticity problems using
tetrahedral grids.
4. 64 bit version of FLAC3D*
5. Help File containing Command Reference, FISH Reference and Example
Applications.*
6. Tunnel extrusion grid generator tool.*
*not yet available
Pre-release available nowOfficial release in November 2006
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MODELLING-STAGE TABS
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Finite Difference FormulationofFLAC
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BASIS OFFLAC
FLACsolves the full dynamic equations of motion even for
quasi-static problems. This has advantages for problems that
involve physical instability, such as collapse, as will be
explained later.
To model the static response of a system, a
relaxation scheme is used in which damping absorbs kinetic
energy. This approach can model collapse problems in a more
realistic and efficient manner than other schemes, e.g.,
matrix-solution methods.
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A SIMPLE MECHANICAL ANALOG
m
F(t)
Newtons Law of Motion
dt
udmamF
For a continuous body, this can be generalized as
i
j
iji gxdt
ud
where = mass density,xi = coordinate vector (x,y)
ij = components of the stress tensor, andgi = gravitation
u,u,u
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STRESS-STRAIN EQUATIONS
In addition to the law of motion, a continuous
material must obey a constitutive relation -that is, a relation between stresses and strains.
For an elastic material this is:
In general, the form is as follows:
where
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A GENERAL FINITE-DIFFERENCE FORMULA
In the finite difference method, each derivative in the previous equations
(motion & stress-strain) is replaced by an algebraic expression relatingvariables at specific locations in the grid.
The algebraic expressions are fully explicit; all quantities on the right-hand
side of the expressions are known. Consequently each element (zone or
gridpoint) in a FLACgrid appears to be physically isolated from its neighbors
during one calculational timestep.
This is the basis of the calculation cycle:
(The time-step is sufficiently small that information
cannot propagate between adjacent elementsduring one step)
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Basic Explicit Calculation Cycle
Equilibrium Equation
(Equation of Motion)
Stress - Strain Relation(Constitutive Equation)
For all gridpoints (nodes)
For all zones (elements)
LnF jiji
new stresses
nodal forces
Gauss theorem
strain rates
velocities
i
j
iji g
xdt
ud
e.g., elastic
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FLACs grid is internally composed of triangles. These are
combined into quadrilaterals. The scheme for deriving
difference equations for a polygon is described as follows:
Overlaid Triangular element Nodal force vector
Elements with velocity vectors
FLAC:
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FLAC:For all gridpoints...
Once all stresses have been calculated, gridpoint forces
are derived from the resulting tractions acting on thesides of each triangle. For example,
Then a classical central finite-difference formula is used
to obtain new velocities and displacements:
( in large strain mode)
FLAC:F ll l
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CFor all elements...
Gauss theorem,
S Ai
i dAx
ffdSn
is used to derived a finite difference formula for elements of arbitrary shape.
)b(
iu nodal velocityb
a)a(
iu nodal velocity
S
For a polygon the formula becomes
S
i
i
SnfA
1
x
f
This formula is applied to calculating the strain increments, eij
, for a zone:
tx
u
x
u
2
1e
SnuuA2
1
x
u
i
j
j
iij
S
j
)b(
i
)a(
i
j
i
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Overlay & Mixed-Discretization Formulation of FLAC:
+ /2 =
Each is constant-stress/constant-strain:
Volume strain averaged over . Deviatoric strain evaluated for
and separately
(Mixed discretization procedure)
Solution is Updated Lagrangian (grid moves with the material), and
explicit (local changes do not affect neighbors in one timestep )
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Methods of solution in time domain
displacement
u
forceF
x
F
stress
u
numerical grid
EXPLICIT
All elements:
,ufF(nonlinear law)
All nodes:
tm
Fu
Repeat for
n time-steps
No iterations
within steps
Information cannot physically
propagate between elements during
one time step
Assume (u)
are fixed
Assume (F)
are fixed
Correct if
p
min
C
x
t
p-wave speed
IMPLICIT
uKF element
FuKum global
Solve complete set of equations
for each time step
Iterate within time step if
nonlinearity present
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Methods compared
Explicit, time-marching Implicit, static
1. Can follow nonlinear laws without
internal iteration, since
displacements are frozen within
constitutive calculation.
2. Solution time increases as N3/2 for
similar problems.
3. Physical instability does not cause
numerical instability.
4. Large problems can be modeled
with small memory, since matrix isnot stored.
5. Large strains, displacements and
rotations are modeled without extra
computer time.
1. Iteration of the entire process is
necessary to follow nonlinear laws
2. Solution time increases with N2 or
even N3.
3. Physical instability is difficult to
model.
4. Large memory requirements, or disk
usage.
5. Significantly more time needed for
large strain models.
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Strengths & Limitations
The explicit solution scheme used in FLAC enables the following
problems to be solved most efficiently:
1. Strongly nonlinear systems, with extensive yield and large
strain.
2. Systems in which localization occurs.
3. Systems that embody complex interactions, or which need
special user-defined conditions or material models.
Disadvantages are:
1. Slow execution (compared to say finite elements) for
linear (or well-behaved) systems.
2. Slow execution if there are great contrasts in material
stiffnesses or element sizes.
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DYNAMIC RELAXATION
In dynamic relaxation gridpoints are moved according to
Newtons law of motion. The acceleration of a gridpoint is
proportional to the out-of-balance force. This solution scheme
determines the set of displacements that will bring the system
to equilibrium, or indicate the failure mode.
There are two important considerations with dynamic relaxation:
1) Choice of timestep
2) Effect of damping
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TIMESTEP
In order to satisfy numerical stability the timestep must satisfy the
condition:
where Cp is proportional to 1 /mgp. For static analysis, gridpoint
masses are scaled so that local critical timesteps are equal ( )which provides the optimum speed of convergence. Nodal inertial
masses are then adjusted to fulfill the stability condition:
Note that gravitational masses are not affected.
1t
pCxt min
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DAMPING
Velocity-proportional damping introduces body forces that can
affect the solution.
Local damping is used in FLAC --- The damping force at a
gridpoint is proportional to the magnitude of the unbalanced
force with the sign set to ensure that vibrational modes are
damped:
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LOCAL DAMPING
The damping force, Fdis:
m
tuFFu
iiii
)(sgn||
Damping forces are introduced to the equations of motion:
where Fiis the unbalanced force
In FLACthe unbalanced force ratio (ratio of unbalanced force,Fi, to the
applied force magnitude, Fm
) is monitored to determine the static state.
By default, when Fi/ Fm < 0.001, then the model is considered to be in an
equilibrium state.
)sgn( iid uFF
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STATIC ANALYSIS
FLACis a dynamic solution method that provides a static
solution (with the effect of inertial forces minimized) provided
the unbalanced force ratio reaches a small value (~ 0.001 orless).
This is comparable to the level of residual error or convergence
criterion defined for matrix solution methods used in many finite
element programs. In FLAC, the level of error is quantified by the
unbalanced force ratio. In both FLACand FE solutions, the static
solution process terminates when the error is below a desired value.
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The collapse load can be determined from either :
1. A load-controlled test, i.e., apply a constant force and calculate the
solution. (stable or unstable?) Iterate until the difference between the stableand unstable load is smaller than a selected tolerance.
2. A velocity-controlled test, i.e., apply a small constant velocity until an
unstable state is reached.
load
settlement
T i i S h d l
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Training ScheduleAugust 8, 2006 (morning)
09:00-10:00 Numerical Analysis of Continuum and DiscontinuumMechanics
- DEM versus continuum analysis numerical methods
Introduction to Material Models to Simulate GeologicalMaterials
- Characteristics of soil and rock
- Constitutive models to represent continuum and
discontinuum behavior
- Selecting appropriate material models and properties
10:00-10:15 Break
10:15-12:00 Introduction to Material Models to Simulate GeologicalMaterials (continued)
T i i S h d l
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Training ScheduleAugust 8, 2006 (afternoon)
01:00-03:00 Model Building Grid Generation
- Grid building/altering/shaping tools; adding interfaces
Model Building Basic Material Models
- Assigning materials and properties in a FLAC model
03:00-03:15 Break03:15-05:00 Model Building Boundary Conditions / Initial Conditions
- Applying boundary and initial conditions
Model Building Solution
- Solving for equilibirum and monitoring model response
Model Building Result Interpretation
- Plotting unbalanced force, gridpoint velocities,
plasticity indicators
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DISCONTINUUM ANALYSIS TECHNIQUES
NUMERICAL SCHEMES TO MODEL CONTACTS
OR INTERFACES BETWEEN DISCRETE BODIES
Discrete Element Methods
(DEM)
Various DEM schemes exist.
... main differences are associated with:
Contacts
Solid
Materials
Solution
Rigid
Deformable
Rigid
Deformable
Static
Dynamic
Continuum Methods
For example:
Finite Elements with Joints
Finite Differences with interfaces
Limit Equilibrium Methods
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Definitions
The nameDiscrete
Element Method(DEM)
should be applied to a method only if it:
1. allows finite displacements and rotations of
discrete bodies; including complete detachment
2. recognizes new interactions (contact)
automatically as the calculation progresses
The name DistinctElement Methodis used for aDEM that uses an explicit dynamic solution to
Newtons laws of motion.
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A discrete element code will embody an efficient
algorithm for detecting and classifying contacts. Itwill maintain a data structure and memoryallocation scheme that can handle many hundredsor thousands of discontinuities or contacts.
Finite element codes for modeling discontinuaare often modified continuum programs, whichcannot handle general interaction geometry (e.g.
many intersecting joints). Their efficiency maydegenerate drastically when connections arebroken repeatedly.
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Overview of DEM & explicit, dynamic method
( / 2) ( / 2) ( )
( ) ( ) ( / 2)
/t t t t t
t t t t t
u u F t m
x x u t
( )
( ) ( )
If , 0
If , ( )
t
t t
n
x R F
x R F R x k
The formulation of is very simple. For example, for a ball impacting a wall,
R
x
F
mmass One time step, t
unknowns knowns
(all contacts, in general)
(all particles, in general)
Full dynamic equations(integration of Newtons 2nd law)
} Explicitsolution scheme
u
Three consequences of this formulation are as follows
(central difference 2ndorder accurate)
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1. Treating each body as discrete(DEM)allows discontinuousmaterial (such as a rock mass) to be modeled easily.
2.Full dynamic equations of motion allow the evolution of unstable
systems to be simulated realistically.
3.Explicit solution scheme makes the task of handlingnonlinearity trivial. Examples of nonlinearities are: (a) contact
making & breaking; (b) softening material behavior (rock-like); e.g.,
force
displacement
INPUT
OUTPUT
The explicit schemeuses a time step sosmall thatinformationcannot propagatebetween neighbors inone step.
Thus, each element isisolatedduring one step, enabling
mt
k
COMPUTATION CYCLE IN THE DEM
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ks
kn
F
n
F
s
u
n
us
ssns
ssss
nnnn
FsgnF,FminF
ukFF
ukFF
All the contacts
CONSTITU
TIVE
c
iF
+M
xi
I/M
m/Fu
FxeM
FF
ii
jiij
i
c
i
At the centroid
ALL THE BLOCKS
MOVEMENT
zone
c
iF
node
At the element
,...,C
tdx
ud
dx
ud
2
1
ijijij
i
j
j
iij
At the node
m/Fu
FFF
dsnF
ii
c
i
e
ii
z
jij
e
i
MOVEME
NT
ALL THE BLOCKS
tttGo to
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What is the applicability of each code?
In general, if there are few discontinuities in the application,
FLACorFLAC3Dmay be used.
If the application contains many discontinuities, UDECor3DEC should be used, because these codes allow easy
specification of multiple joint sets.
For granular materials or solids that may fracture, PFCisthe best choice.
Note that all Itasca codes may be coupled e.g., a FLACmodel may contain regions represented by PFC.
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Ch t i ti f il & k
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1. Behavior changes in character, according to stress state (e.g axial splitting
in unconfined test; shear bands when confined).
2. Memoryof previous stress or strain excursions, in both magnitude anddirection. (c.f. - moving yield surfaces, evolving anisotropic damage tensors,
Kaiser effect)
3. Dilatancythat depends on history, mean stress and initial state.
4. Continuously nonlinearstress-strain response, with ultimate yield, followedby softening or hardening.
5. Hysteresis at all levelsof cyclic loading/unloading.
6. Transition from brittle to ductileshear response as the mean stress isincreased.
Characteristics of soil & rock
continued
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7. Dependence of incremental stiffnesson mean stress and history.
8. Induced anisotropyof stiffness and strength with stress & strain path.
9. Nonlinear envelopeof strength.
10. Spontaneous appearance ofmicrocracksand localized macro-fractures in rock, and shear bandsin soil.
11. Spontaneous emission ofacoustic energy.
Characteristics of soil & rock continued
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It would take a complicated constitutive law to reproduce allofthese phenomena.
If such a model existed (which it doesnt), very manymaterialparameters or internal state variables would be needed.
(For example, some existing laws have 20 parameters, and/or
families of yield surfaces involving perhaps 100 state variables).
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What mechanisms should be included in a model?
Only include those things that actually impact the relevant behaviorof the system (i.e., things that are important to successful design).
The following examples illustrate the modeling approach for several
particular requirements:
Collapse or ultimate failure of the system:- use elastic/plastic law (no effect of moduli); try FLAC/Slope
Monotonic loading; displacements are important:- use simple hardening law (yield stress increases with strain)
Cyclic loading; damping is important:- use hysteretic damping option in FLAC/FLAC3D
General loading paths; several nonlinear effects important
- must consider complex constitutive model, OR
Cyclic loading; volume-change is important (e.g.,liquefaction):- use empirical void-collapse scheme in FLAC/FLAC3D
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use a micromechanical approach, in which complexity arises
automatically from the interaction of many simple objects
(emergent behavior*).
Note thatall11 of the characteristics of soil & rock (listed earlier)are reproduced by a micromechanical model consisting of anassembly of frictional and/or bonded particles.
(Calibration is needed to match the observed magnitude of each effect)
* Often, a collection of simple objects exhibits complex
behavior at the system level. This is an example of emergentbehavior(e.g., see Emergence by Steven Johnson, Scribner
2001).
In this case it is not necessary to invent complex constitutive
laws just create a system of the appropriate micro-elements,
and the complex behavior will emerge automatically.
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Constitutive Models for FLAC and FLAC3D
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Constitutive Models for FLACand FLAC3D
Built-in Models User-defined Models*
Elasticity models:Isotropic
Transversely isotropic
Orthotropic
Plasticity models:Drucker-Prager
Mohr-Coulomb
Ubiquitous-joint
Strain-hardening/softening
Bilinear strain-hardening/softening/ubiquitous-jointDouble-yield
Modified Cam-clay
Hoek-Brown
Dynamic Liquefaction models:Finn (Martin et al., 1975) model
Bryne, 1991 model
Creep models:Viscoelastic
Burgers substance viscoelastic
Two-component power law
Reference creep formulation (WIPP)
Burger-creep/Mohr-Coulomb viscoplastic
Two-component power law/Mohr-Coulomb viscoplastic
WIPP-creep/Drucker-Prager viscoplastic
Crushed-salt*partial list of models created by
or developed for code users
Elasticity models:
Hyperbolic elasticDuncan-Chang, 1980
Plasticity models:NorSand
Jardine et al., 1986
Manzari-Dafalias, 1997
Kleine et al., 2006
Concrete hydration
vonWolffersdorff hypo-plastic
Dynamic Liquefaction models:UBCSAND
UBCTOT
Wang, 1990
Roth et al.,2001
Andrianopoulos, 2005
Creep models:Minkley viscoplastic
Hein-crushed salt
Salzer creep
Lubby2 creep
FLACCONSTITUTIVE MODELS
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Model Representative material Example application
Null void holes, excavations, regions in whichmaterial will be added at later stage
Elastic homogeneous, isotropic continuum;
linear stress- strain behavior
manufactured materials (e.g. steel)
loaded below strength limit; factor ofsafety calculation
Anisotropic thinly laminated material exhibiting
elastic anisotropy
laminated materials loaded below
strength limit
Drucker-Prager limited application; soft clays with
low friction
common model for comparison to
implicit finite-element programs
Mohr-Coulomb loose and cemented granular materials
soils, rock, concrete
general soil or rock mechanics
(e.g., slope stability and undergroundexcavation)
Strain-hardening/softening
Mohr-Coulomb
granular materials that exhibit nonlinear
material hardening or softeningstudies in post-failure (e.g., progressive
collapse, yielding pillar, caving)
Ubiquitous-joint thinly laminated material exhibitingstrength anisotropy (e.g., slate)
excavation in closely bedded strata
Bilinear strain-hardening/
softening ubiquitous-joint
laminated materials that exhibit non-
linear material hardening or softening
studies in post-failure of laminated
materials
Double-yieldlightly cemented granular material inwhich pressure causes permanentvolume decrease
hydraulically placed backfill
Modified Cam-clay materials for which deformability and shear
strength are a function of volume change
geotechnical construction on soil
Hoek-Brown * isotropic rock material geotechnical construction in rock
*new in FLAC 5
CONSTITUTIVE MODELS
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FOR CONTINUUM ELEMENTS
NULL all stresses are zero: for use as a void - e.g., for excavated regions
ELASTIC isotropic, linear, plane strain or plane stressANISOTROPIC elastic,assumes that the element is transversely anisotropic:
b
b planes are planes of symmetry. The b axes may be at any angle to the x, y axes:
b
x
y
FLAC PLASTICITY MODELS
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Drucker-Prager
Mohr-Coulomb
Ubiquitous-Joint
Strain-Hardening-Softening
Double-YieldModified Cam-clay
Hoek-Brown
1. All models are characterized by yield functions, hardening/softening functions and flow rules.
2. Plastic flow formulation is based on plasticity theory that total strain is decomposed into elastic
and plastic components and only the elastic component contributes to stress increment via theelastic law. Also, elastic and plastic strain increments are coaxial wuth the principal stress axes.
3. Ducker-Prager, Mohr-Coulomb, Ubiquitous Joint and Strain-Softening models have a shear yieldfunction and non-associated flow rule.
4. Drucker-Prager, Mohr-Coulomb, Ubiquitous Joint and Strain-Softening models define the tensilestrength criterion separately from the shear strength and use an associated flow rule.
5. All models are formulated in terms of effective stresses.
6. Double-yield and modified Cam-clay models take into account the influence of volumetric changeon material deformability and volumetric deformation (collapse).
7. Hoek-Brown incorporates a nonlinear failure surface with a plasticity flow rule that varies with
confining stress.
CONSTITUTIVE MODELSDRUCKER-PRAGER
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Drucker-Prager elastic/plastic with non-associated flow
rule: shear yield stress is a function of
isotropic stress
C
tk/q
Bk ft=0
A
Drucker-Prager Failure Criterion in FLAC
CONSTITUTIVE MODELSMOHR-COULOMB
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Mohr-Coulomb elastic / plastic with non-associated flow rule: operates onmajor and minor principal stresses
C
B
A N
c2 t tanc
1
ft=0
3
Mohr-Coulomb Failure Criterion in FLAC
shear
stress
slope = G
(for constant n)
shear strain
CONSTITUTIVE MODELSUBIQUITOUS-JOINT MODEL
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Ubiquitous-Joint Model uniformly distributed slip planes embedded in a
Mohr-Coulomb material
element
Mohr-Coulomb
n
rigid-plastic, dilatant
tanc njmax
Note: rotates with the element in large-strain mode
t
j C
B
j
j
tan
c
22
ft=0cj
A
CONSTITUTIVE MODELSSTRAIN-SOFTENING / HARDENING
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Strain-softening / hardening identical to the Mohr-Coulomb model except that , C and are arbitrary functions of accumulated plastic strain (p)*
produces
p
p
pInput by user Output
v
21
2P
12
2dP
22
2dP
11p eee
C
CONSTITUTIVE MODELS
BILINEAR STRAIN HARDENING/SOFTENING MODEL
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BILINEAR STRAIN-HARDENING/SOFTENING MODEL
Bilinear model a generalization of the ubiquitous-joint model. The failure envelopes for
the matrix and joint are the composite of two Mohr-Coulomb criteria with
a tension cut-off. A non-associated flow rule is used for shear plastic flow
and an associated flow rule for tensile-plastic flow.
DCB
AN2
N11
1 t1
1
tan
c2
2
tan
c
1
3
FLAC bilinear matrix failure criterion
A
B
D
C
33
Cj1
Cj2
jtj1
j2
FLAC bilinear joint failure criterion
CONSTITUTIVE MODELS DOUBLE-YIELD MODEL
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CONSTITUTIVE MODELS DOUBLE YIELD MODEL
Double-yield model extension of the strain-softening model to simulate
irreversible compaction as well as shear yielding.
CONSTITUTIVE MODELS - MODIFIED CAM-CLAY MODEL
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Modified Cam-Clay model incremental hardening/softening elastic-plastic model,
including a particular form of non-linear elasticity and
a hardening/softening behavior governed by volumetric
plastic strain (density driven).
vl
vkA
vkB
ln p
v
N
A
Bk1
l1
ln p1
swelling lines
normal
consolidation line
Normal consolidation line and swelling line
for an isotropic compression test
plastic compaction
p
0 pe
plasticdilation
0 pe
q
2c
cr
pp
2c
cr
pMq
pc
Cam-Clay failure criterion in FLAC
CONSTITUTIVE MODELSHOEK-BROWN MODEL
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Hoek-Brown model empirical relation that is a nonlinear failure surface which
represents the strength limit for isotropic intact rock and
rock masses. The model also includes a plasticity flow
rule that varies as a function of confining stress.
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BUILT-IN MATERIAL MODELS
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FLAC Interface Model
FLAC (OR CONTINUUM CODE)
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Use for problems at either end of the joint-density spectrum
single or isolated discontinuities multiple, closely-packed blocks
interface ubiquitous jointing
problems
INTERFACES
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Interfaces represent planes on which sliding or separation can occur:
- joints, faults or bedding planes in a geologic medium
- interaction between soil and foundations
- contact plane between different materials
To join regions that have different zone sizes
Elastic-plastic Coulomb sliding:
- tensile separation of the interface, and
- axial stiffness to avoid inter-penetration
INTERFACE MECHANICS
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Each node on the surface of both bodies owns a length, L, of interface for the purpose of converting
from stress to force. L is calculated in the following way
Body 1
Body 2
A1 D1
E2
B1 C1
C2B2
A2 D2
LB2 LC2 LB1 LD2 LC1 LD1
LINEAR MODEL
n= -Knun
= -Ksus = max (max, ) sgn ()
max= ntan +c
Fn = nL
Fs = L
[Kn]=stress/disp
INTERFACE ELEMENTSPROCEDURE
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PROCEDURE
1. Form interface using grid generation commands
2. Null out region
3. Move grid halves together
4. Declare interface
int n aside from i1, j1 to i2, j2 bside from i3, j3 to i4, j4
5. Input the interface properties
int n ks =... kn = ... fric =... coh =...
(i3, j3)
(i1, j1)
(i4, j4)
(i2, j2)
bside
aside
INTERFACE PROPERTIES
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kn : normal stiffness [stress/displacement]
ks : shear stiffness [stress/displacement]
cohesion : cohesion [stress]
friction : friction angle [degrees]
dilation : dilation angle [degrees]
tbond : tensile strength [stress]
If the interface is used to attach two sub-grids,it is necessary to declare itglued.
Properties estimation
Sub-grids attached:
- declare glued
- set kn and ks = 10 *
Geologic joints
- shear tests; considering scale effect
- kn and ks for rock mass joints, can vary between 10-100 MPa/m
for joints with soft clay in-filling, to over 100 GPa/m for tight joints
in basalt or granite.
INTERFACE CONDITIONS
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INTERFACE CONDITIONS
1. Glued Interface --- No slip or separation is allowed, but elastic displacement, defined by
kn and ks, occurs.
2. Unbonded Interface --- Slip occurs as defined by Coulomb shear-strength criterion
(and including dilation at onset of slip). The interface has zero tensile bond strength.
3. Bonded Interface ---- It a tensile bond (tbond) strength is specified, the interface acts
as if glued while the normal stress is below the bond strength. If magnitude of normal
stress exceeds bond strength, the bond breaks (tbond is set to zero) and the interfacebehaves as an unbonded interface.
A shear bond strength is also specified when tbond is set, in which case the bond will break
if either the shear stress exceeds the shear bond strength (sbratio*tbond) or the normal
stress exceeds the normal bond strength (tbond). The interface then reverts to unbonded.
(By default, sbratio = 100.)
Ifbslip=on is specified, slip (defined by the Coulomb criterion) can occur even though
the interface is still bonded. Dilation is suppressed in this case.
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INTERFACE MODEL
Create interfaceand assign properties
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Key Features ofFLACfor Grid Generation
1. FLACis command-driven.
2. GIICBuild tools provide
mouse-driven facilities forgrid generation from
templates.
3. FISHtools in the FISHLibrary are used to create
complicated grid shapes.
Geometry grid setup
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Geometry grid setup
1. Always start with a coarse grid*; work out the bugs,
and increase the grid density only as much as
necessary (are results converging?).
2. Avoid badly-shaped zones, and sudden jumps in zone
widths.
3. Avoid high aspect ratios in regions of high straingradients.
4. Make sure the boundaries are far enough away to
avoid influencing the results.
5. Try to avoid triangular zones at free surfaces,
especially if performing large-strain plasticity analysis.
* For dynamic analysis, the zone size should be small
enough to model wave propagation accurately.
Boundary conditions
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Boundary conditions
There are two main classes of boundary conditions: velocity
conditions and stress conditions (although there are additional
conditions in dynamic simulations).
Both can be activated with the APPLY command: e.g.,
APPLY XVEL=1.0 I=1 J=1,5 ; FLAC velocity
APPLY SXX=-1e5 J=21 ; FLAC stress
APPLY SXX=-1e5 RANGE Z=19.9 20.1 ; FLAC3D
For historical reasons, the velocity conditions can also be
set with a FIX command and an INI command: e.g.,
FIX X I=1 J=1,5
INI XVEL=1.0 I=1 J=1,5
The latter 2 commands achieve the same effect as the
first APPLY command above.
Boundary locations
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y
Extreme gridstunnelsizes are the same
stress
displacement
ATTACH - accuracy
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ATTACH accuracyLoad applied here
Note smooth displacement contours
Grid Generation
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Create Mesh
Alter Mesh to Fit Shape
M i l M d l d P i
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Material Models and Properties
Boundary and Initial Conditions
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Boundary and Initial Conditions
Histories, Tables, FISH Library
Global Settings
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Result Interpretation - Plotting
Solution
Training Schedule
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August 9, 2006 (morning)
09:00-10:00 Introduction to FISH
- FISH variables, arithmetic, systax ad data types
- Writing FISH functions
- Simple exercises using the FISH Editor & Library
10:00-10:15 Break
10:00-12:00 Factor of Safety Calculation
- Implementation of the strength reduction method in FLAC
- Application of FLAC for factor-of-safety calculations
Training Schedule
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August 9, 2006 (afternoon)
01:00-03:00 Soil/Rock Structure Interaction
- Beams, liners, cables and rockbolts
- 2D/3D equivalence
03:00-03:15 Break
03:00-05:00 Simulating Support for Underground Excavations andEmplacement Drifts
- Using interface elements for tunnel liner and rockinteraction
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FISH- The programming language
ofFLAC
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FISH (1)
FISHis a compiler. Functions are entered via a data file and
are translated into a list of instructions stored in the memory
space of the code.
Variable names and values are available for monitoring and
changing at any time.
FISH (2)
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Common ways to use FISH:
1. Special-purpose operations; e.g., grid generation, profile ofmaterial properties, automation of input commands, plot or
print user-defined variables.
2. Use as a HISTORY variable.
3. Automatic execution during stepping; e.g., use as a servo-control for
numerical test (with WHILE_STEPPING command).
4. Drive a data file; e.g., change parameters while calculation
progresses (using COMMAND statements).
5. Use as a constitutive model function; e.g., apply a user-written
constitutive model.
FISHVariables, Functions and Operations
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Some remarks
FISHis case-insensitive. All characters after a semi-colon (;) are ignored.
If parameters are to be passed to functions, then they must be set beforehand
by using the SET command.
If a number is expected in FLAC, it can be substituted by FISHsymbols.
As soon as a variable is mentioned in a validFISHprogram line, it is
globally recognized both in FLACcommands and FISHcode.
User-defined variables or function names.
Pre -defined scalar variables.
Grid variables (e.g., stresses, properties).
Intrinsic functions.
Tables, general memory access.
FISH handles definitions of:
FISHControl Statements (1)
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DEF
...END
To define a FISHfunction
CASE_OFexpr IFexpr 1 test expr 2
CASEn ELSE END_CASE END_IF
LOOPvar(expr1, expr2) LOOP WHILEexpr1 test expr 2 END_LOOP END_LOOP
Conditional statements
Looping statements
FISHControl Statements (2)
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SECTION END_SECTION
COMMAND END_COMMAND
EXITEXIT SECTION
Sectioning statements
FISHSpecification Statements
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WHILESTEPPING (execution of the function at every FLAC step)
WHILE_STEPPING
CONSTITUTIVEMODEL (the function is taken to be a new constitutive model)
CONSTITUTIVE_MODEL
INT (change the type of the associated variable)
FLOAT
STRING
ARRAYvar(n1, n2) (definition of an array)
FISHFunctions
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Mathematical functions
atan atan2 cosexp tan ln
log sin sqrt
abs max min
sgn
Type conversion
float int string
type
Message functions
in out
Random generator
grand urand
Logical operators
and not or
Others
fc_arg get_mem lose_mem
Tables
xtable ytable table
Memory Access
imem fmem
FISH Editor
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The FISH Editor allows you to create and edit text files that contain FISH functions.
FISH functions defined in this way can be executed using the UTILITY/FishLib toolif they are stored within the /flac/gui/fishlibdirectory. You can also run FISHfunctions directly using the Run/Execute menu item from the FISH Editor.To automate the execution ofFISH functions, special comment lines are included inthe file. There are four types of input field:
1. Name: This is the name of the primary FISH function to run.(A file can have more than one FISH function.)
2. Diagram: This is the name of an optional file name of an image (GIF/JPG)that shows what the FISH function does.
3. Input: This contains the input values for the function.
4. Note: This contains notes and comments that describe the FISH function.
FISH Input Parameter Data
http://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htm7/30/2019 Iaea Flac Course
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The Input/Define parameters menu item brings up a dialog that allows input
parameters to be defined. These will have values requested when the FISHfunction is run either through the FISH Library (using the UTILITY/FishLib tool),or executed here.
The input parameters forFISH functions are entered as a comment string of theform:
;Input: name/type/value/description
in which
1. Name - FISH variable name.
2. Type - int/float/string corresponding to data type:integer, floating-point or string.
3. Value - Default value for parameter.
4. Description - Helpful string describing what the parameter is.
FISH Input Parameter Data
http://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishtoolinput.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishtoolinput.htm7/30/2019 Iaea Flac Course
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The Input/Define notes menu item brings up a text area where comments
can be saved.
Here you can describe the FISH function and these comments will be shown whenyou try to execute the function from either the FISH Editor or the FISH Library(using the UTILITY/FishLib tool).
These lines are added to a FISH file as comments prefixed by [Note:]
FISH Notes
FISH Library
http://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishnote.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishnote.htm7/30/2019 Iaea Flac Course
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The directory flac/gui/fishlib/contains files with FISH functions accessed from theUTILITY/FishLibtool.
These FISH functions have special comment lines included to allow the GIIC to identifyinput parameters, notes and diagrams.
The directory structure inside flac/gui/fishlib/is mirrored in this tool as a tree structure.
FISH Library
http://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htmhttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Itasca_tetFLACguihelpengfishlib.htm7/30/2019 Iaea Flac Course
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Application of the shear strengthreduction method in design:
using numerical solutions for factor of
safety
Factor of Safety (FS) in
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Factor of Safety (FS) in
Geotechnical Engineering
loadacting
capacityloadFS
structural mechanics approach
load = force, moment, pressure
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Footings
Q
B
Q
u
b cq
qFS
B
Q
BNqNcNq
qcu2
1
bearing capacity theory
FoS calculation independent of
load capacity and acting load calculation
q
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Retaining Walls
P
earth pressure theory
bearing capacity theory
o
r
oM
MFS
s
r
s
F
FFS
maxq
qFS u
b c
po
wr
rPM
rWM
PF
AWF
s
r
tan
BNqNcNq
qcu2
1
S
M
b
Wq max
aKHP2
2
1
W
B
FoS calculation independent of
load capacity and acting load calculation
Slopes
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Slopes
q
qFS ubc ?FS
***
2
1
BNqNcNqqcu
q
load ?
unit weightbearing capacity theory
FS calculation independent ofload capacity and acting load calculation
fFS
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Structural Mechanics Approach
o
r
oM
MFS
s
r
sF
FFS
specified failure modestatically determinate cases
global equilibrium
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Strength Reduction Approach
dddddc
c
c
cFS
tan
tan
tan
tan
cd
d
c
d
dcc
tantan
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Alternative Strength Reduction
constc
cFS
d
c
c
constcFSd
tan
tan
cd
c
d
tanand cvarying nonlinearly
Method of Slices
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(Limit Equilibrium)
ddc
cFS
tan
tan
specified failure mode
slices
global equilibrium
statically undeterminate cases
strength reduction
A full solution of the coupled stress/displacement,
Numerical-modeling approach -
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equilibrium and constitutive equations is made with
codes like FLAC.
Given a set of properties, the system is either found
to be stable or unstable.
By performing a series of simulations, with various
properties, the factor of safety can be found that
corresponds to the point of stability.
This approach is much slower, but much more general,
than the limit-equilibrium solution. Only in the past few
years has it become a practical alternative to the limit
equilibrium method (as computers have become faster).
What is a full numerical solution and how does it
differ from the limit equilibrium method?
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Numerical solution Limit equilibrium
Equilibrium Satisfied everywhereSatisfied only for specific
objects (slices)
StressesComputed everywhere using
field equations
Computed approximately on
certain surfaces
Deformation Part of the solution Not considered
Failure
Yield condition satisfied
everywhere; failure surfaces
develop automatically as
conditions dictate
Failure allowed only on
certain pre-defined surfaces;
no check on yield condition
elsewhere
Kinematics
The mechanisms that
develop satisfy kinematic
constraints
Kinematics are not
consideredmechanisms
may not be feasible
A single numerical simulation with given properties
will show eitherfailure orstability (like a single
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The strength properties (e.g., cohesion and friction)
are reduced by trial values of the factor of safety, as
follows -
y ( g
physical model).
How do we get a factor of safety?
Several simulations are performed, with different
properties.
CF
trial
1trial
F
tantrialC =trial tan= { }
-1
How can the exact value of be found quickly,trialF
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q y
with the smallest number of simulations?
Dawson et al(Gotechnique 49, 1999) give the basisfor the bracketing methodof finding numerically thefactor of safety. In essence, the interval between
values of giving failure and stability is
repeatedly halved. The process quickly converges,and is stopped when the interval becomes small
(e.g., < 0.005).
In more detail, the scheme implemented in the codeFLACis as follows
trialF
Steps in the strength-reduction solution scheme forFLAC
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1. Determine the characteristic response time of the systemin terms of steps needed for equilibrium call it Nc.
2. Set F=1.0, and keep halving it until lower bound (first
stable case) is found call it Fs.
3. Keep doubling F until upper bound (first unstable case) is
found call it Fu.4. Set F = (Fu+Fs)/2, and determine ifstable (then set Fs=F)
orunstable (then set Fu=F).
5. If Fu-Fs < 0.005, then stop, else go to 4.
How is instability (failure) determined?
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The condition of stability or instability is determined witha program-specific method.
For example, with implicit, matrix-solution finite element
codes, the condition of instability is often based on thenon-convergence of the system of equations (see Griffiths
and Lane, 1999).
In FLAC, instability is determined by monitoring the kinetic
energy in the model. The change in kinetic energy ismeasured by the unbalanced force ratio.
How is instability (failure) determined?
Definition of stability/instability in FLAC
S f bili /i bili
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1. Do up to Nc steps. Record unbalanced force ratio, Ru.2. If Ru falls below 0.001 during stepping, exit as stable.
3. If (RuRu(old)) / Ru < 0.1, exit as unstable.
4. If total iterations (steps 1-3) > 6, exit as unstable.
5. Go to 1.
Steps to test for stability/instability:
During the whole process, the following information is displayed
the number of calculation steps completed in 1 as a % ofNc,
the number of completed solution cycles (steps 1-3),
the current values of Fu and Fs (brackets).
How good is the scheme? We can compare it with exact solutions.The following example solved analytically by Chen (1975) has a
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The following example, solved analytically by Chen (1975), has a
factor of safety of1.0.
This example was set up with FLAC, using two differentgrids. The results are
Non-associated
Associatedflow rule
C id (20 20) 0 99 1 03
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FLAC (Version 4.00)
LEGEND
9-Oct-01 18:09step 18546
-1.167E+00
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Design by URS Corporation
10-mile long corridor
33-foot deep, 51-feet wide
freight rail connection
from ports of Long Beach
and Los Angeles to rail hub
in downtown Los Angeles
Practical application
of the strengthreduction method
LOS ANGELES ALAMEDA CORRIDOR
T h ll 3 f t di t t i l i f d t il
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URS Corporation
Trench walls are 3-foot diameter cast-in-place reinforced concrete piles,
4 feet on center with shotcrete on inside, and supported by pre-cast concrete
struts at top.
Stage 3 of construction is critical because potential for kick-out failure
governs required pile length.
LOS ANGELES ALAMEDA CORRIDOR
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URS Corporation
FLAC analysis for
Factor of Safety includessoil-structure interaction
producing factor of 1.3.
Limit-equilibrium methodpredicts failure, which
would result in over-
design of pile length by
up to 8 feet.
FLAC analysis resulted
in cost savings of several
million dollars.
What are the advantages of using a numerical FoS solution?
1. Any failure mode develops naturally no need to specify a
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1. Any failure mode develops naturally no need to specify a
range of trial surfaces in advance.
2. There are no restrictions on geometry all situations (slopes,
footings, tunnels, etc) are modeled in the same way.
3. No artificial parameters (e.g., functions for inter-slice force
angles) need to be given as input.
4. Multiple failure surfaces (orcomplex internal yielding) evolve
naturally, if the conditions give rise to them.5. Structural interaction is modeled realistically as fully-coupled
deforming elements, not simply as equivalent forces.
6. Kinematics is respected!
Stable with FLAC
Unstable by LE solution
weakplanes
What are the disadvantages?
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- only one: Speed. A limit equilibrium program typically executes ina fraction of a second. A numerical solution for FoS using a coarse
grid often takes less than a minute. A medium-grid solution may take
several minutes, and is usually quite accurate. A fine-grid solution
may take an hour or two.
Thus, there is no real drawback, given that most problems can
be solved in a few minutes.
One further perceived problem
Programs that perform full numerical solutions are oftendifficult to use FLAC is no exception!
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To make numerical solution for factor of safety easily accessible, a
new programFLAC/Slopehas been produced. This is as easy(or perhaps, easier) to use than limit equilibrium programs such as
XSTABL or SLOPE/W.
FLAC/Slopehas a simple graphical interface that is oriented tosetting up slope stability cases. It uses FLACas the computational
engine but the user is completely insulated from it. Point-and-clickoperations are all that are needed for example:
Select slope type; thenenter dimensions
FLAC/Slope allows -
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p
Several types of slope: e.g., benched slope, dam, embankment Several layers, and fairly general layer geometry
Library of material properties built-in and user-defined
Water table specified as an arbitrary surface
Structural reinforcement: e.g., geo-grids, soil nails, rockbolts Weak plane, modeled as an interface
Surface loads
Regions can be excluded from the FoS calculation
Instant comparison of runs using different parameters, and evencomparison of results from different projects
Hard-copy reports and plotting in several formats
Various parameters may be included or excluded from variation
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Various parameters may be included orexcluded from variation
during the FoS solution. The following factors are included by
default: material friction and cohesion.The following items can also be included:
1. material tension tensile strength of materials
2. interface strength cohesion and friction of interfaces
3. reinforcement grout strength cohesion and friction ofreinforcement grout (soil/reinforcement interface) *
4. group regions of space included in, or excluded from,
the scope of the parameter-variations *
* new in FLAC/Slope 5
Note that, at present, onlyMohr-Coulombmaterial can be assigned
in FLAC/Slope. (Mohr-Coulombandubiquitous-joint materials
can be assigned in FLAC.)
St t l l t i FLAC
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Structural elements in FLAC
General Application : Design and analysis of structural support to stabilize a rock or soilmass.
Types of elements available: Beam
Liner Cable
Pile
Rockbolt
Strip
Support Member
Geometry:
Linear element with 2 end nodes
Structural elements in FLAC
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There are seven types of structural element available in FLAC:
1. Beam elements. These allow bending, and are connected to thegrid (soil) either at nodes (rigid connection) or via interfaces, which
allow separation & slip. Plastic yield occurs as a function of axial
thrust, plastic moments can be specified.
2. Liner elements.* Similar to beams, and also include bendingstresses in yield criterion.
3. Cable elements. No bending resistance. Cable nodes are slaved togrid motion in the normal direction, and via shear springs & slip
elements in the shear direction. Yield may occur axially. Cable
nodes may also be connected rigidly to gridpoints.
4. Pile elements. Bending resistance is included. Connection to thegrid is via yielding springs in both the normal and shear directions.
* new in FLAC 5.0
Structural elements in FLAC
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There are seven types of structural element available in FLAC:
5. Rockbolt elements.* Similar to piles, and also can account forchange in confining stress, strain-softening of grout, and tensile
rupture of element.
6. Strip elements.* Similar to rockbolts, but cannot sustain bending.
Shear behavior at strip/soil interface is defined by nonlinear shearfailure envelope that varies as a function of confining stress.
7. Support members. Simple 1D nonlinear spring elements that linkpoints on free surfaces (used in mining to represent for example
wooden props).
* new in FLAC 5.0
Structural elements in FLAC
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Structural elements in FLAC
Formulation: Each element type is characterized by a combination of
a) structural behavior
b) medium/structure interaction.
The structural element logic is implemented in the framework of
FLAC two-dimensional Lagrangian, Explicit Finite-Difference
scheme, which uses a Dynamic Relaxation Method to solve static
problems.
Beam ElementsStructural behavior:
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Structural behavior:
3 degrees of freedom per node (2 translations
+ 1 rotation)
Constant axial force, F; constant shear force,
T; linear moment, M
Linear axial displacement, cubic deflection.
Axial peak and residual strengths
Can be joined together and/or the grid
Nodal behavior may also include plastic
hinges.
Applications:
Modeling of structural support in which
bending resistance is important, includingsheet piles, support struts in an open-cut
excavation.
Beam Properties
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Beam Properties
Structural: Cross-sectional area [or height and width, or radius]
Elastic modulus
Moment of inertia
Axial peak and residual yield strengths
Optional:
Plastic moment
Density
Thermal expansion coefficient
Liner Elements
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Structural behavior:
similar to beam elements
bending stresses are included in the yieldcriterion
Applications:
Modeling of structural support in which
bending resistance, limited bending
moments and yield strengths are important,such as concrete or shotcrete tunnel linings
Typical moment-thrust diagram for liner elements
Liner Properties
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Liner Properties
Structural: Cross-sectional area [or height and width, or radius]
Elastic modulus
Moment of inertia
Cross-sectional shape factor
Thickness
Axial peak and residual yield strengths
Optional:
Density Thermal expansion coefficient
Cable Elements
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Structural behavior:
One degree of freedom per node (axialtranslation).
Can also fail in tension and compression,
no flexural resistance.
Medium/structure interaction:
Can be point-anchored or grouted so that
the cable element develops forces along itslength resisting relative motion between
cable and grid.
May be pre-tensioned, if desired.
Applications: supports for which tensile
capacity is important, including
rock bolts, cable bolts and
tie-backs.
Grout behavior accounted for in Cables
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Conceptual Model
Constitutive Model
Cable Properties
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Cable Properties
Structural: Elastic modulus
Tensile yield
Compressive yield
Grout: Stiffness
Cohesive strength
Frictional resistance
Optional:
Density
Thermal expansion coefficient
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Rockbolt Elements
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Structural behavior:
Similar to pile elements
Can also account for:
- effect of changes in confining stress
- strain-softening of grout
- tensile rupture of element.
Applications: rock reinforcement in whichnonlinear effects of confinement,
grout bonding or tensile rupture
are important.
Rockbolt PropertiesStructural:
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Cross-sectional area [or radius]
Elastic modulus
Moment of inertia [automatic calculation for radius]
Yield strength
Tensile failure strain
Optional: Plastic moment
Density
Tables relating cohesion and friction to shear displ.
Thermal expansion coefficient
Medium/structure interaction, Shear and Normal:
Stiffness
Cohesive strength
Frictional resistance
Exposed perimeter
Structural Boundary Conditions
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Structural Boundary Conditions
Free/fixed velocities (translation and rotation)
Applied forces and moments
Pin connection
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2D / 3D Equivalence in FLACTwo dimensional modeling of structural features:
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g
FLACstructural elements can be used to model structures which are either long or
short but regularly spaced in the out of plane direction.
1. The element structural behavior is formulated in plane stress. To model longfeatures, the stiffness property should be divided by (1- 2) to account for plane-
strain conditions.
2. Reducing 3D problems with regularly spaced beams, liners, cables, piles,rockbolts or support involves averaging the effect of 3D over the structurespacing, S. FLACuses a linear scaling of properties method proposed byDonovan et al. (1984)* to distribute effects of elements over a discrete spacing.
3. The three dimensional effects associated with the flow of soil through a row ofpiles can also be accounted for, in an approximate manner, by calibration ofproperties associated with the normal component of the pile medium/structureinteraction.
n
*
Property Scaling2D/3D Equivalence
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Property Scaling2D/3D Equivalence
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Property Scaling2D/3D Equivalence
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Property Scaling 2D/3D Equivalence
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Property Scaling 2D/3D Equivalence
When the spacing keyword is specified in FLACVersion 5, structural
element properties are scaled automatically to account for regular
spacing. Gravity loads and pre-tensioning values are also scaled.
Actual structural element forces and moments will automatically be
printed and plotted, accounting for spacing.Note, any loads or pre-tensioning that are applied to structural elements
(e.g., pre-loaded struts) using the STRUCT node n load command
should be scaled by dividing by S.
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Training ScheduleAugust 10, 2006 (morning)
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09:00-10:00 Coupled Modeling -Introduction to Effective Stress and
Groundwater Analysis
- Effective stress calculation
- Governing equations for transient fluid flow and coupledanalysis
- Recommended approaches for coupled calculations
- Two-phase flow analysis
10:00-10:15 Break
10:15-12:00 Coupled Modeling Introduction to Effective Stress andGroundwater Analysis (continued)
Training ScheduleAugust 10, 2006 (afternoon)
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01:00-03:00 Coupled Modeling Introduction to Thermal and
Thermal-mechanical Analysis
- Governing equations for thermal, thermal-mechanical andporo-thermal-mechanical analysis
- Procedures for performing thermal and thermal-mechanicalcalculations
- Constitutive models in coupled analyses
- Thermal loading and boundary conditions
03:00-03:15 Break
03:15-05:00 Coupled Modeling Introduction to Thermal andThermal-mechanical Analysis(continued)
Groundwater flow and
consolidation
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consolidation
FLACmodels the flow of groundwater through a permeable solid,
such as soil.
The modeling of flow may be done:
- by itself, independent of the usual mechanical calculation ofFLAC
- in parallel with the mechanical modeling, so as to capture the effects
of fluid / solid interaction.
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Mathematical Formulation
Transport Law Compatibility Equation
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Transport Law Compatibility Equation
Balance Laws
Constitutive laws
iji w k k j
q P g xxk
iv
i
t x
1
2
jiij
j i
uu
x x
1
wP
t n
K
n
t t
s
t t t
, ,ij ij ij ijd
P Hdt
k
mobility coefficient
Biotcoefficient
wet density
fluid bulk modulus
porosity
ij iis s
j
dug
x dt
Total versus Effective Stress Formulations
In FLAC equilibrium is expressed using total stress:
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In FLAC, equilibrium is expressed using total stress:
By definition ofeffective stress:
Substitution of the last 2 equations in the first, gives:
0ij
s i
jgx
s d wn s
ij ij ijp
d
w
n
s
: material dry density: porosity
: saturation
: fluid density
1 0
ij
d i wj i i
pg n n
x x x
l l
w
w w
x gp
g g
g
BuoyancyDrag
(seepage force)
Solid weight
Groundwater Modeling Approaches (1)
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Choose simplest model consistent with mechanisms.
In order of complexity we can have:
1. No fluid-mechanical interaction (pore pressure distribution is
needed to compute correct effective stresses).
2. Flow calculation is used to obtain pore pressure distribution
(medium can be saturated or partially saturated with phreaticsurface).
Groundwater Modeling Approaches (2)
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3. Pore pressure generated by mechanical deformation.There is no flow, the analysis can be static or dynamic
(e.g., undrained pore pressure buildup or liquefaction).
4. Coupled mechanical deformation and fluid flow.
a) time scale not important
b) time scale is important
Groundwater Modeling in FLAC(1)
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1. Effective Stress Calculation- pore pressure fixed
- no groundwater flow
- specify pore pressures with the INITIAL pp command
orWATER table command (in non CONFIG gw mode)
- in non CONFIG gw mode, wet and dry densities of materialare supplied by the user
- in CONFIG gw mode:
- wet and dry densities are calculated by FLAC
- SET flow off
- set WATER bulk = 0- if pore pressures changed instantaneously (e.g., dewatering),
use CONFIG ats to automatically adjust existing total stresses
Groundwater Modeling in FLAC(2)
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2. Flow-Only Calculation
- specify CONFIG gw and SET mech off
- pore-pressure distribution and phreatic surface location will be calculated
- specify correct permeability, but low fluid bulk modulus if only steady-state
condition is required
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Groundwater Modeling in FLAC(4)
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4. Coupled-Flow and Mechanical Calculation
- specify CONFIG gw and SET flow on
- specify realistic fluid-bulk modulus and permeability
- for basic-flow logic:
- use SET nmech SET ngw (default: nmech=1 ngw=1)
- SET force SET sratio (default: force=0 sratio=10-3)
- SET step SET clock (default: step=100000 clock=1440 min.)
- SOLVE auto on age
When to Use Fast-Flow Schemes
*
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* Consider flow incompressible if Kw >>> K + 4G/3
Common Fluid-Flow Boundaries
Impermeable Boundary
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Impermeable Boundary
- default conditions- pore pressure free to vary
- saturation free to vary
Free Surface
- pore pressure fixed to zero (FIX pp)
- saturation free to vary if pore pressure fixed at zero
Applied Pore-Pressure Boundary
- pore pressure fixed (FIX pp)
Permeability of Porous Medium (1)
Darcys Law expressed in terms of pressure is
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y p p
dxdPkq
where q is the specific discharge (in units of velocity - e.g., ft/s or m/s)
dP/dx is the pressure gradient (e.g., in psf/ft or Pa/m)
k is the mobility coefficient (e.g., in ft4/lb-sec or m2/Pa-sec )
dx
dhKq H
where his the head (e.g., in ft or m)
KH is the hydraulic conductivity (e.g., in ft/s or m/s).
The more usual expression of Darcys Law is
Since P=gwh
( h i th it ti l l ti d i th d it f t )
Permeability of Porous Medium (2)
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(where g is the gravitational acceleration and w is the mass density of water),
w
H
g
Kk
Another constant that is sometimes used is intrinsic permeability,
k, which is related to kand Kby
kg
K
w
H m
mk
where m is the dynamic viscosity (e.g., units of lb-s/ft2 or Pa-s).
The units ofkare [length]2 (e.g., ft2 or m2).
Bulk Modulus of Water
P
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VV
PKw
/
Kw = 4.18x107 psf (or 2.0 GPa) for pure water
Steady-State Flow
(a) fully saturated - solution-time independent(b) partially saturated - solution time reduced by lowering Kw
(IfKwis too low, results are erratic. Set )
Transient Flow
(a) flow-field solution (high modulus)
(b) phreatic surface migration (low modulus)
(c) use SET funsat algorithm to alternate solutions automatically
gzK ww 3.0
Groundwater - tipsA fully coupled simulation with FLAC (e g a consolidation process)
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A fully coupled simulation with FLAC(e.g. a consolidation process)using the basic fluid-flow scheme can be very time-consuming. The
FLACmanual provides detailed suggestions about variousapproximations that can be made to reduce the solution time. The
important factors to consider are:
1. The ratio between the required simulated time and the
characteristic time of the diffusion process in the system.
2. The nature of the imposed perturbation (fluid or mechanical).
3. The ratio of fluid to solid stiffness.
Increasing time step
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g p
When the fluid modulus is much larger than the bulk modulus of thesolid material, the timestep is small, and the simulation time long
for the basic fluid-flow scheme. It is possible to reduce (artificially)
the fluid modulus, without affecting the results; the allowed
reduction factor (for given error) depends on the problem
constraints, but in almost all cases the following upper limit of fluidmodulus gives minimal error:
43
20 ( )wK n K G
It can often be reduced further. Rapid, partial simulations can be
made to assess the error introduced by various reduction factors.
Caution!
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If the timestep is small, and there are many steps, it may benecessary to use the double precision version ofFLAC4.0. Theregular version uses single precision, which corresponds to an
accuracy of 1 part in 106. If for example a million timesteps are
executed, then accumulated quantities (such as pore pressure
increments) may be lost.
Note that the regular version ofFLAC5.0 uses double precision.
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Applications
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Slope stability
Groundwater contamination control
Design of hydraulic structures
References
Richards (1931), Philip et al. (1989)
van Genuchten (1982), Fredlund (1987), Forsyth (1995)
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Capillary pressure
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Pcdepends on
saturation
geometry of the void space
nature of solid and liquid
Micro-observation Macro-observation
Saturation
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Definitions:
Fluid phase saturation
Residual saturation
Effective saturation
1w a
S S w
w
VS
nV
aa
VS
nV
1
w rwe
rw
S SSS
rwS
a=0.336 (clay)
a=0.6 (sand)Pc/P0 Cc/(P0tan )
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Yield criterion for partially saturated soil:
saturation saturation
van Genuchten relation Capillary-induced cohesion
Steady unsaturated flow around a
drift
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drift
movie
Conclusions
http://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Aug10_01_Groundwater/twophase/drift/drift_movie/movie.exehttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Aug10_01_Groundwater/twophase/drift/drift_movie/movie.exe7/30/2019 Iaea Flac Course
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1. Water is deflected from the drift roof and driplobes are formed in which saturation
and flow velocity are increased (compared to initial steady state).
2. A dryshadow is formed, sheltered by the drift cavity.
Rainfall on a Slope
Stable slope with initial
water table
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water table
(soil saturation above the
water table is ~ 0.5)
Steady rainfall of 9 inches over
4 days results in slope failure
movie
Conclusions
http://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Aug10_01_Groundwater/twophase/slope/rain_movie/movie.exehttp://localhost/var/www/apps/conversion/releases/20121107221618/Local%20Settings/Temp/Rar$DI00.953/Aug10_01_Groundwater/twophase/slope/rain_movie/movie.exe7/30/2019 Iaea Flac Course
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1. A coupled analysis evaluates the transient response for the given infiltration rate.
2. The saturation is seen to increase toward a steady value consistent with the higher
magnitude of the rainfall event.
3. The increase in saturation near the slope surface causes a reduction in soil cohesionand failure of the slope.
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The thermal option includes both conduction and advection
Thermal Option
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p
models.
ConductionTransient transfer of heat based on Fouriers
law of heat conduction.
AdvectionTransient transfer of heat by convection inporous media, by:
forced convectionheat carried by fluid motion,
and
free convectionfluid motion caused by fluiddensity difference due to temperature variation.
Mathematical Formulation for Conduction
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Energy-Balance Equation:
where heat-flux vector
volumetric heat source intensity
stored heat per unit volume
Thermal constitutive law relates temperature changes to the heat storage ,
so the energy-balance equation can be rewritten as:
(1)
tqTT
v
T
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