iaea flac course

7/30/2019 Iaea Flac Course

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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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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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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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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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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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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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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)


76/203

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


87/203

ATTACH accuracyLoad applied here

Note smooth displacement contours

Grid Generation


88/203

Create Mesh

Alter Mesh to Fit Shape

M i l M d l d P i


89/203

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


95/203

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


101/203

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


102/203

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.htm


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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.htm


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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.htm


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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.htm


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Application of the shear strengthreduction method in design:

using numerical solutions for factor of

safety

Factor of Safety (FS) in


107/203

Factor of Safety (FS) in

Geotechnical Engineering

loadacting

capacityloadFS

structural mechanics approach

load = force, moment, pressure


108/203

Footings

Q

B

Q

u

b cq

qFS

B

Qq

Q

BNqNcNq

qcu2

1

bearing capacity theory

FoS calculation independent of

load capacity and acting load calculation

q


109/203

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


110/203

Slopes

q

qFS ubc ?FS

***

2

1

BNqNcNqqcu

q

load ?

unit weightbearing capacity theory

FS calculation independent ofload capacity and acting load calculation

fFS


111/203

Structural Mechanics Approach

o

r

oM

MFS

s

r

sF

FFS

specified failure modestatically determinate cases

global equilibrium


112/203

Strength Reduction Approach

dddddc

c

c

cFS

tan

tan

tan

tan

cd

d

c

d

dcc

tantan


113/203

Alternative Strength Reduction

constc

cFS

d

c

c

constcFSd

tan

tan

cd

c

d

tanand cvarying nonlinearly

Method of Slices


114/203

(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 -


115/203

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?


116/203

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


117/203

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


118/203

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


119/203

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?


120/203

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


121/203

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


122/203

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


123/203

FLAC (Version 4.00)

LEGEND

9-Oct-01 18:09step 18546

-1.167E+00


124/203

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


125/203

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


126/203

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


127/203

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?


128/203

- 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!


129/203

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 -


130/203

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


131/203

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



133/203

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



134/203

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



135/203


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:


136/203

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


137/203

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


138/203


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


139/203

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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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



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Rockbolt Elements


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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.


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 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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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)



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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

qq

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



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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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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.exe


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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.exe


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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

iaea flac course

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