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FLAC Training Course Beijing, China October 17, 2005

Training Schedule October 17, 2005 (morning) 08:00-09:45

Introduction to FLAC - Overview of potential applications and capabilites in geo-engineering analysis and design - New features in FLAC 5.0 and FLAC3D 3.0 Introduction to the FLAC Graphical Interface - Menu-driven versus command-driven operation - Simple tutorial

09:45-10:00

Break

10:00-12:00

FLAC Theoretical Background - General-purpose versus limited-purpose analysis - Explicit finite-difference solution Practical Exercise - Slope stability analysis

FLAC & FLAC3D

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, thereby simulating the presence of faults, joints or frictional boundaries

Displacements resulting from construction of a shallow tunnel

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Built-in material models: •"null" model, •three elasticity models (isotropic, transversely isotropic and orthotropic elasticity), •eight plasticity models (DruckerPrager, Mohr-Coulomb, strainhardening/softening, ubiquitousjoint, bilinear strainhardening/softening ubiquitousjoint, double-yield, modified Camclay, and Hoek-Brown) User-defined models written in FISH (FLAC)

Braced excavation

Continuous gradient or statistical distribution of any property may be specified

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Built-in programming language (FISH) to add user-defined features FLAC and FLAC3D can be coupled to other codes via TCP/IP links Convenient specification of boundary conditions and initial conditions

Model grid for service tunnel connecting two main tunnels

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Water table may be defined for effective stress calculations Groundwater flow, with full coupling to mechanical calculation (including negative pore pressure, unsaturated flow, and phreatic surface conditions) Structural elements,such as tunnel liners, piles, sheet piles, cables, rock bolts or geotextiles, that interact with the surrounding rock or soil, may be modeled

Excavation supported by shotcrete wall, tiebacks and soilnails

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Automatic 3D grid generator (FLAC3D) using pre-defined shapes that permit the creation of intersecting internal regions (e.g., intersecting tunnels) Full graphical user interface in FLAC; partial gui in FLAC3D (for plotting and file 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 a shallow tunnel

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Optional Features

Optional modules include: • thermal, thermal-mechanical, and thermal-poromechanical analysis including conduction and advection; • visco-elastic and visco-plastic (creep) material models; • dynamic analysis capability with quiet and freefield boundaries, and • user-defined constitutive models written in C++

Liquefaction failure of a pile-supported wharf

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC Version 5 & FLAC3D Version 3 New Features 1.

Hysteretic damping – more 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)

MODELLING-STAGE TABS

FLAC Background 1. General-purpose vs Limited–purpose analysis 2. Explicit finite-difference solution

Geotechnical Software

General-purpose versus Limited-purpose methods

“Limited-purpose” programs “Limited-purpose” programs are commonly used in geoengineering practice because they provide rapid solutions and are generally very easy to operate. These programs are based upon simplifying assumptions. One example of a limited-purpose solution method is the limit-equilibrium method. This type of program executes very rapidly, and uses an approximate scheme – mostly the method of slices – in which a number of assumptions are made (for example, the location & angle of inter-slice forces). Several assumed failure surfaces are tested, and the one giving the lowest factor of safety is chosen. Equilibrium is only satisfied on an idealized set of surfaces.

Examples of Limited-purpose Programs Limiting condition Forces only (limit equilibrium) Linear properties (equivalent linear method)

Subgrade reaction (Winkler springs)

Example program SLOPE/W XSTABL SHAKE

LPILE WALLAP

“General-purpose” programs A “general-purpose” program provides a “full” solution of the coupled stress/displacement, equilibrium and constitutive equations. Given a set of properties, the change in both the deformation and stress state are calculated --- e.g., the system is either found to be stable or unstable, and the resulting deformation is determined.

The general-purpose approach is much slower than comparable limited-purpose methods, but much more general. Only in the past few years has it become a practical alternative to the limited-purpose methods (as computers have become faster).

Comparison of Limited-Purpose and General-Purpose Solutions

Comparison of

“General-purpose” to “Limited-purpose” programs Limiting conditions can be prescribed for “general-purpose” programs to approximate the simplifying assumptions built into “limited-purpose” programs. In this way, the “generalpurpose” program can be validated. Further, when the limiting condition is removed from the “general-purpose” program, the influence of the simplifying assumption in the “limited-purpose” program can be assessed.

We suggest using both general-purpose and limitedpurpose methods in parallel, to get confidence in the general-purpose method. - if they give the same result, this provides reassurance - if they give different results, then the reasons can be explored; for example, is there a different mechanism?

The combined approach can be justified in terms of quality assurance.

Finite Difference Formulation of FLAC

BASIS OF FLAC

FLAC solves 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.

A SIMPLE MECHANICAL ANALOG

m

u, u , u F(t)

Newton´s Law of Motion

F ma m

du dt

For a continuous body, this can be generalized as

du i ij gi dt x j where = mass density, xi = coordinate vector (x,y) ij = components of the stress tensor, and gi = gravitation

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

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 relating variables 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 FLAC grid appears to be physically isolated from its neighbors during one calculational timestep.

(The time-step is sufficiently small that information cannot propagate between adjacent elements during one step)

This is the basis of the calculation cycle:

Basic Explicit Calculation Cycle For all gridpoints (nodes) velocities

Equilibrium Equation (Equation of Motion) du i ij gi dt x j

Gauss´ theorem

nodal forces

Fi ijn jL

For all zones (elements)

strain rates

e.g., elastic

Stress - Strain Relation (Constitutive Equation)

new stresses

FLAC’s 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

Elements

with velocity vectors

Nodal force vector

FLAC:

For all elements...

Gauss’ theorem,

f dA A x i

nifdS S

is used to derived a finite difference formula for elements of arbitrary shape.

u (i b ) nodal velocity b

S a

u (i a ) nodal velocity For a polygon the formula becomes

f 1 f n i S x i A S

This formula is applied to calculating the strain increments, eij, for a zone: u i 1 u (i a ) u (i b ) n jS x j 2A S eij

1 u i u j t 2 x j x i

FLAC:

For all gridpoints...

Once all stresses have been calculated, gridpoint forces are derived from the resulting tractions acting on the sides of each triangle. For example,

Then a “classical” central finite-difference formula is used

to obtain new velocities and displacements:

(… in large strain mode)

Overlay & Mixed-Discretization Formulation of FLAC:

+ Each

is constant-stress/constant-strain:

Volume strain averaged over and

/2 =

. Deviatoric strain evaluated for

separately (Mixed discretization procedure)

Solution is “Updated Lagrangian” (grid moves with the material), and explicit (local changes do not affect neighbours in one timestep )

Methods of solution in time domain numerical grid

F

displacement

u

stress

u

force

F

x

EXPLICIT All elements:

F f u,

IMPLICIT

Assume (u) are fixed

(nonlinear law)

All nodes:

u F t m

Repeat for n time-steps No iterations within steps

Assume (F) are fixed

element

F Ku global

mu Ku F

Correct if t

x min Cp

p-wave speed

Information cannot physically propagate between elements during one time step

Solve complete set of equations for each time step Iterate within time step if nonlinearity present

Methods compared Explicit, time-marching

Implicit, static

1. Can follow nonlinear laws without internal iteration, since displacements are “frozen” within constitutive calculation.

1. Iteration of the entire process is necessary to follow nonlinear laws

2. Solution time increases as N3/2 for similar problems.

2. Solution time increases with N2 or even N3.

3. Physical instability does not cause numerical instability.

3. Physical instability is difficult to model.

4. Large problems can be modeled with small memory, since matrix is not stored.

4. Large memory requirements, or disk usage.

5. Large strains, displacements and rotations are modeled without extra computer time.

5. Significantly more time needed for large strain models.

DYNAMIC RELAXATION

In dynamic relaxation gridpoints are moved according to Newton’s 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

TIMESTEP In order to satisfy numerical stability the timestep must satisfy the condition:

xmin t Cp

where Cp is proportional to 1 /mgp. For static analysis, gridpoint

masses are scaled so that local critical timesteps are equal ( t 1 ) 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.

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:

LOCAL DAMPING • Damping forces are introduced to the equations of motion: ui Fi | Fi | sgn (ui )

t m

where Fi is the unbalanced force • The damping force, Fd is:

Fd Fi sgn( ui ) • In FLAC the 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.

PRACTICAL EXERCISE SLOPE STABILITY ANALYSIS

Training Schedule October 17, 2005 (afternoon) 01:00-02:45

FLAC Operation – System requirements, installation structure, manual volumes, files, nomenclature, system of units – Grid Generation : [Build], [Alter] and [Interface] tools Material Models : [Material] tool Practical Exercise – Biaxial load tests

02:45-03:00

Break

03:00-05:00

Boundary Conditions / Initial Conditions : [ In Situ] tool Histories / Tables / Fish Library : [Utility] tool Global Settings : [Settings] tool Solution : [Run] tool

Result Interpretation : [Plot] tool Practical Exercise – Determination of failure

SYSTEM REQUIREMENTS FOR FLAC

Processor – Recommended minimum clockspeed of 1 GHz Hard Drive – Recommended minimum disk space of 100 MB

RAM – RAM required to load FLAC is 60 MB; 24 MB is provided by default for models and memory can be increased by the user if needed Display – Recommended screen resolution is 1024 x 768 pixels and 16-bit color palette Operating System – Any Intel-based computer running Windows 98 and upward is suitable Operation on PC Networks – A network-license version of FLAC 5.0 is available

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC Files Project File (*.prj) – ASCII file describing state of model and GIIC at the stage the file is saved; includes FLAC commands, link to save files, and plot views for the project

Save File (*.sav)

– Binary file containing values of all state variables and user-defined conditions at stage that file is saved

Data File (*.dat)

– ASCII file listing FLAC commands that represent the problem being analyzed

History File (*.his) – ASCII file record of input or output history values Material File (*.gmt) – ASCII file containing material properties (can be updated). Plot File

– Graphics plot file (in various standard formats)

Movie File (*.dcx)

– String of PCX images that can be viewed as a ―movie‖

FLAC Nomenclature

Zone Numbers

Gridpoint Numbers

System of Units

GRID GENERATION

Build Tools

Alter Tools

BASIC MATERIAL MODELS

FLAC CONSTITUTIVE MODELS Model

Representative material

Example application

Null

void

Elastic

homogeneous, isotropic continuum; linear stress- strain behavior

holes, excavations, regions in which material will be added at later stage manufactured materials (e.g. steel) loaded below strength limit; factor of safety calculation

Anisotropic

thinly laminated material exhibiting elastic anisotropy

laminated materials loaded below strength limit

Drucker-Prager

limited application; soft clays with low friction loose and cemented granular materials soils, rock, concrete

common model for comparison to implicit finite-element programs

Mohr-Coulomb Strain-hardening/softening Mohr-Coulomb

granular materials that exhibit nonlinear material hardening or softening

Ubiquitous-joint

thinly laminated material exhibiting strength anisotropy (e.g., slate) laminated materials that exhibit nonlinear material hardening or softening lightly cemented granular material in which pressure causes permanent volume decrease materials for which deformability and shear strength are a function of volume change

Bilinear strain-hardening/ softening ubiquitous-joint Double-yield Modified Cam-clay Hoek-Brown * *new in FLAC 5

isotropic rock material

general soil or rock mechanics (e.g., slope stability and underground excavation) studies in post-failure (e.g., progressive collapse, yielding pillar, caving) excavation in closely bedded strata studies in post-failure of laminated materials hydraulically placed backfill

geotechnical construction on soil

geotechnical construction in rock

CONSTITUTIVE MODELS 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 stress

•ANISOTROPIC elastic,assumes that the element is transversely anisotropic:

g

b

b planes are planes of symmetry. The b axes may be at any angle f to the x, y axes: y b

f

x

FLAC PLASTICITY MODELS Drucker-Prager Mohr-Coulomb Ubiquitous-Joint Strain-Hardening-Softening Double-Yield Modified 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 the elastic 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 yield function and non-associated flow rule. 4. Drucker-Prager, Mohr-Coulomb, Ubiquitous Joint and Strain-Softening models define the tensile strength criterion separately from the shear strength, and 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 change on 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 MODELS — DRUCKER-PRAGER •Drucker-Prager

elastic/plastic with non-associated flow rule: shear yield stress is a function of isotropic stress

A

kf

B

ft=0

C

t

kf /qf Drucker-Prager Failure Criterion in FLAC

CONSTITUTIVE MODELS — MOHR-COULOMB •Mohr-Coulomb

elastic / plastic with non-associated flow rule: operates on major and minor principal stresses 3

ft=0 B C 2c Nf

A

t

c tan f 1

Mohr-Coulomb Failure Criterion in FLAC

(for constant n)

shear stress slope = G shear strain

g

CONSTITUTIVE MODELS – UBIQUITOUS-JOINT MODEL •Ubiquitous-Joint Model

uniformly distributed slip planes embedded in a Mohr-Coulomb material

element

n

rigid-plastic, dilatant

max c j n tanf Mohr-Coulomb Note: rotates with the element in large-strain mode

A

cj

f t=0

B tj

22

C cj

tan f j

CONSTITUTIVE MODELS — STRAIN-SOFTENING / HARDENING •Strain-softening / hardening

identical to the Mohr-Coulomb model except that f, C and are arbitrary functions of accumulated plastic strain (gp )*

f gp f

g

produces ev

gp

g gp

Output

Input by user

2

1 2 2

P d e22Pd e12P g p e11 2

CONSTITUTIVE MODELS 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.

3 A B

B

A

1 Nf2

Nf1

1 C

D

t

c1 tanf1

Cj2

c2 tanf2

Cj1

C

1

D

jt

FLAC bilinear matrix failure criterion

fj2 fj1

3’3’

FLAC bilinear joint failure criterion

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 • 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).

v

q

N normal consolidation line

plastic dilation

vl A

vkA k

vkB

1

swelling lines

ln p1

qcr M

B

pc 2

e p 0

plastic compaction

e p 0

l 1

ln p

Normal consolidation line and swelling line for an isotropic compression test

pcr

pc 2

pc

Cam-Clay failure criterion in FLAC

p

CONSTITUTIVE MODELS – HOEK-BROWN MODEL • 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.

FLAC Interface Model

FLAC (OR CONTINUUM CODE) Use for problems at either end of the joint-density spectrum

single or isolated discontinuities “interface”

multiple, closely-packed blocks “ubiquitous jointing”

problems

INTERFACES • 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 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

A1

C1

B1

D1

B2 C2

Body 1 E2

A2

Body 2

D2

LB2 LC2

LB1

LD2

LC1

LD1

LINEAR MODEL n= -Knun

[Kn]=stress/disp

= -Ksus = max (max, ) sgn () max= ntan f+c Fn = nL Fs = L

INTERFACE ELEMENTS PROCEDURE 1. Form interface using grid generation commands

2. Null out region

bside (i3, j3) (i1, j1)

(i4, j4) (i2, j2) aside

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 =...

INTERFACE PROPERTIES Kn :

normal stiffness

Ks :

shear stiffness

coh :

cohesion of the joint

fric :

friction angle of the joint

ten :

tensile strength of the joint

If the interface is used to attach two sub-grids,it is necessary to declare it glued. Properties estimation • Sub-grids attach: - declare glued -

4 K 3 G K n K s 10.max l

• Geologic joints - shear tests; considering the “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.

Boundary and Initial Conditions

Histories, Tables, FISH Library

Global Settings

Solution

Result Interpretation - Plotting

View more...
Training Schedule October 17, 2005 (morning) 08:00-09:45

Introduction to FLAC - Overview of potential applications and capabilites in geo-engineering analysis and design - New features in FLAC 5.0 and FLAC3D 3.0 Introduction to the FLAC Graphical Interface - Menu-driven versus command-driven operation - Simple tutorial

09:45-10:00

Break

10:00-12:00

FLAC Theoretical Background - General-purpose versus limited-purpose analysis - Explicit finite-difference solution Practical Exercise - Slope stability analysis

FLAC & FLAC3D

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, thereby simulating the presence of faults, joints or frictional boundaries

Displacements resulting from construction of a shallow tunnel

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Built-in material models: •"null" model, •three elasticity models (isotropic, transversely isotropic and orthotropic elasticity), •eight plasticity models (DruckerPrager, Mohr-Coulomb, strainhardening/softening, ubiquitousjoint, bilinear strainhardening/softening ubiquitousjoint, double-yield, modified Camclay, and Hoek-Brown) User-defined models written in FISH (FLAC)

Braced excavation

Continuous gradient or statistical distribution of any property may be specified

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Built-in programming language (FISH) to add user-defined features FLAC and FLAC3D can be coupled to other codes via TCP/IP links Convenient specification of boundary conditions and initial conditions

Model grid for service tunnel connecting two main tunnels

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Water table may be defined for effective stress calculations Groundwater flow, with full coupling to mechanical calculation (including negative pore pressure, unsaturated flow, and phreatic surface conditions) Structural elements,such as tunnel liners, piles, sheet piles, cables, rock bolts or geotextiles, that interact with the surrounding rock or soil, may be modeled

Excavation supported by shotcrete wall, tiebacks and soilnails

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Basic Features

Automatic 3D grid generator (FLAC3D) using pre-defined shapes that permit the creation of intersecting internal regions (e.g., intersecting tunnels) Full graphical user interface in FLAC; partial gui in FLAC3D (for plotting and file 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 a shallow tunnel

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC & FLAC3D

Optional Features

Optional modules include: • thermal, thermal-mechanical, and thermal-poromechanical analysis including conduction and advection; • visco-elastic and visco-plastic (creep) material models; • dynamic analysis capability with quiet and freefield boundaries, and • user-defined constitutive models written in C++

Liquefaction failure of a pile-supported wharf

Advanced, Two and Three Dimensional Continuum Modeling for Geotechnical Analysis of Rock, Soil, and Structural Support

FLAC Version 5 & FLAC3D Version 3 New Features 1.

Hysteretic damping – more 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)

MODELLING-STAGE TABS

FLAC Background 1. General-purpose vs Limited–purpose analysis 2. Explicit finite-difference solution

Geotechnical Software

General-purpose versus Limited-purpose methods

“Limited-purpose” programs “Limited-purpose” programs are commonly used in geoengineering practice because they provide rapid solutions and are generally very easy to operate. These programs are based upon simplifying assumptions. One example of a limited-purpose solution method is the limit-equilibrium method. This type of program executes very rapidly, and uses an approximate scheme – mostly the method of slices – in which a number of assumptions are made (for example, the location & angle of inter-slice forces). Several assumed failure surfaces are tested, and the one giving the lowest factor of safety is chosen. Equilibrium is only satisfied on an idealized set of surfaces.

Examples of Limited-purpose Programs Limiting condition Forces only (limit equilibrium) Linear properties (equivalent linear method)

Subgrade reaction (Winkler springs)

Example program SLOPE/W XSTABL SHAKE

LPILE WALLAP

“General-purpose” programs A “general-purpose” program provides a “full” solution of the coupled stress/displacement, equilibrium and constitutive equations. Given a set of properties, the change in both the deformation and stress state are calculated --- e.g., the system is either found to be stable or unstable, and the resulting deformation is determined.

The general-purpose approach is much slower than comparable limited-purpose methods, but much more general. Only in the past few years has it become a practical alternative to the limited-purpose methods (as computers have become faster).

Comparison of Limited-Purpose and General-Purpose Solutions

Comparison of

“General-purpose” to “Limited-purpose” programs Limiting conditions can be prescribed for “general-purpose” programs to approximate the simplifying assumptions built into “limited-purpose” programs. In this way, the “generalpurpose” program can be validated. Further, when the limiting condition is removed from the “general-purpose” program, the influence of the simplifying assumption in the “limited-purpose” program can be assessed.

We suggest using both general-purpose and limitedpurpose methods in parallel, to get confidence in the general-purpose method. - if they give the same result, this provides reassurance - if they give different results, then the reasons can be explored; for example, is there a different mechanism?

The combined approach can be justified in terms of quality assurance.

Finite Difference Formulation of FLAC

BASIS OF FLAC

FLAC solves 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.

A SIMPLE MECHANICAL ANALOG

m

u, u , u F(t)

Newton´s Law of Motion

F ma m

du dt

For a continuous body, this can be generalized as

du i ij gi dt x j where = mass density, xi = coordinate vector (x,y) ij = components of the stress tensor, and gi = gravitation

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

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 relating variables 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 FLAC grid appears to be physically isolated from its neighbors during one calculational timestep.

(The time-step is sufficiently small that information cannot propagate between adjacent elements during one step)

This is the basis of the calculation cycle:

Basic Explicit Calculation Cycle For all gridpoints (nodes) velocities

Equilibrium Equation (Equation of Motion) du i ij gi dt x j

Gauss´ theorem

nodal forces

Fi ijn jL

For all zones (elements)

strain rates

e.g., elastic

Stress - Strain Relation (Constitutive Equation)

new stresses

FLAC’s 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

Elements

with velocity vectors

Nodal force vector

FLAC:

For all elements...

Gauss’ theorem,

f dA A x i

nifdS S

is used to derived a finite difference formula for elements of arbitrary shape.

u (i b ) nodal velocity b

S a

u (i a ) nodal velocity For a polygon the formula becomes

f 1 f n i S x i A S

This formula is applied to calculating the strain increments, eij, for a zone: u i 1 u (i a ) u (i b ) n jS x j 2A S eij

1 u i u j t 2 x j x i

FLAC:

For all gridpoints...

Once all stresses have been calculated, gridpoint forces are derived from the resulting tractions acting on the sides of each triangle. For example,

Then a “classical” central finite-difference formula is used

to obtain new velocities and displacements:

(… in large strain mode)

Overlay & Mixed-Discretization Formulation of FLAC:

+ Each

is constant-stress/constant-strain:

Volume strain averaged over and

/2 =

. Deviatoric strain evaluated for

separately (Mixed discretization procedure)

Solution is “Updated Lagrangian” (grid moves with the material), and explicit (local changes do not affect neighbours in one timestep )

Methods of solution in time domain numerical grid

F

displacement

u

stress

u

force

F

x

EXPLICIT All elements:

F f u,

IMPLICIT

Assume (u) are fixed

(nonlinear law)

All nodes:

u F t m

Repeat for n time-steps No iterations within steps

Assume (F) are fixed

element

F Ku global

mu Ku F

Correct if t

x min Cp

p-wave speed

Information cannot physically propagate between elements during one time step

Solve complete set of equations for each time step Iterate within time step if nonlinearity present

Methods compared Explicit, time-marching

Implicit, static

1. Can follow nonlinear laws without internal iteration, since displacements are “frozen” within constitutive calculation.

1. Iteration of the entire process is necessary to follow nonlinear laws

2. Solution time increases as N3/2 for similar problems.

2. Solution time increases with N2 or even N3.

3. Physical instability does not cause numerical instability.

3. Physical instability is difficult to model.

4. Large problems can be modeled with small memory, since matrix is not stored.

4. Large memory requirements, or disk usage.

5. Large strains, displacements and rotations are modeled without extra computer time.

5. Significantly more time needed for large strain models.

DYNAMIC RELAXATION

In dynamic relaxation gridpoints are moved according to Newton’s 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

TIMESTEP In order to satisfy numerical stability the timestep must satisfy the condition:

xmin t Cp

where Cp is proportional to 1 /mgp. For static analysis, gridpoint

masses are scaled so that local critical timesteps are equal ( t 1 ) 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.

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:

LOCAL DAMPING • Damping forces are introduced to the equations of motion: ui Fi | Fi | sgn (ui )

t m

where Fi is the unbalanced force • The damping force, Fd is:

Fd Fi sgn( ui ) • In FLAC the 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.

PRACTICAL EXERCISE SLOPE STABILITY ANALYSIS

Training Schedule October 17, 2005 (afternoon) 01:00-02:45

FLAC Operation – System requirements, installation structure, manual volumes, files, nomenclature, system of units – Grid Generation : [Build], [Alter] and [Interface] tools Material Models : [Material] tool Practical Exercise – Biaxial load tests

02:45-03:00

Break

03:00-05:00

Boundary Conditions / Initial Conditions : [ In Situ] tool Histories / Tables / Fish Library : [Utility] tool Global Settings : [Settings] tool Solution : [Run] tool

Result Interpretation : [Plot] tool Practical Exercise – Determination of failure

SYSTEM REQUIREMENTS FOR FLAC

Processor – Recommended minimum clockspeed of 1 GHz Hard Drive – Recommended minimum disk space of 100 MB

RAM – RAM required to load FLAC is 60 MB; 24 MB is provided by default for models and memory can be increased by the user if needed Display – Recommended screen resolution is 1024 x 768 pixels and 16-bit color palette Operating System – Any Intel-based computer running Windows 98 and upward is suitable Operation on PC Networks – A network-license version of FLAC 5.0 is available

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC 5.0 MANUAL

FLAC Files Project File (*.prj) – ASCII file describing state of model and GIIC at the stage the file is saved; includes FLAC commands, link to save files, and plot views for the project

Save File (*.sav)

– Binary file containing values of all state variables and user-defined conditions at stage that file is saved

Data File (*.dat)

– ASCII file listing FLAC commands that represent the problem being analyzed

History File (*.his) – ASCII file record of input or output history values Material File (*.gmt) – ASCII file containing material properties (can be updated). Plot File

– Graphics plot file (in various standard formats)

Movie File (*.dcx)

– String of PCX images that can be viewed as a ―movie‖

FLAC Nomenclature

Zone Numbers

Gridpoint Numbers

System of Units

GRID GENERATION

Build Tools

Alter Tools

BASIC MATERIAL MODELS

FLAC CONSTITUTIVE MODELS Model

Representative material

Example application

Null

void

Elastic

homogeneous, isotropic continuum; linear stress- strain behavior

holes, excavations, regions in which material will be added at later stage manufactured materials (e.g. steel) loaded below strength limit; factor of safety calculation

Anisotropic

thinly laminated material exhibiting elastic anisotropy

laminated materials loaded below strength limit

Drucker-Prager

limited application; soft clays with low friction loose and cemented granular materials soils, rock, concrete

common model for comparison to implicit finite-element programs

Mohr-Coulomb Strain-hardening/softening Mohr-Coulomb

granular materials that exhibit nonlinear material hardening or softening

Ubiquitous-joint

thinly laminated material exhibiting strength anisotropy (e.g., slate) laminated materials that exhibit nonlinear material hardening or softening lightly cemented granular material in which pressure causes permanent volume decrease materials for which deformability and shear strength are a function of volume change

Bilinear strain-hardening/ softening ubiquitous-joint Double-yield Modified Cam-clay Hoek-Brown * *new in FLAC 5

isotropic rock material

general soil or rock mechanics (e.g., slope stability and underground excavation) studies in post-failure (e.g., progressive collapse, yielding pillar, caving) excavation in closely bedded strata studies in post-failure of laminated materials hydraulically placed backfill

geotechnical construction on soil

geotechnical construction in rock

CONSTITUTIVE MODELS 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 stress

•ANISOTROPIC elastic,assumes that the element is transversely anisotropic:

g

b

b planes are planes of symmetry. The b axes may be at any angle f to the x, y axes: y b

f

x

FLAC PLASTICITY MODELS Drucker-Prager Mohr-Coulomb Ubiquitous-Joint Strain-Hardening-Softening Double-Yield Modified 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 the elastic 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 yield function and non-associated flow rule. 4. Drucker-Prager, Mohr-Coulomb, Ubiquitous Joint and Strain-Softening models define the tensile strength criterion separately from the shear strength, and 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 change on 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 MODELS — DRUCKER-PRAGER •Drucker-Prager

elastic/plastic with non-associated flow rule: shear yield stress is a function of isotropic stress

A

kf

B

ft=0

C

t

kf /qf Drucker-Prager Failure Criterion in FLAC

CONSTITUTIVE MODELS — MOHR-COULOMB •Mohr-Coulomb

elastic / plastic with non-associated flow rule: operates on major and minor principal stresses 3

ft=0 B C 2c Nf

A

t

c tan f 1

Mohr-Coulomb Failure Criterion in FLAC

(for constant n)

shear stress slope = G shear strain

g

CONSTITUTIVE MODELS – UBIQUITOUS-JOINT MODEL •Ubiquitous-Joint Model

uniformly distributed slip planes embedded in a Mohr-Coulomb material

element

n

rigid-plastic, dilatant

max c j n tanf Mohr-Coulomb Note: rotates with the element in large-strain mode

A

cj

f t=0

B tj

22

C cj

tan f j

CONSTITUTIVE MODELS — STRAIN-SOFTENING / HARDENING •Strain-softening / hardening

identical to the Mohr-Coulomb model except that f, C and are arbitrary functions of accumulated plastic strain (gp )*

f gp f

g

produces ev

gp

g gp

Output

Input by user

2

1 2 2

P d e22Pd e12P g p e11 2

CONSTITUTIVE MODELS 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.

3 A B

B

A

1 Nf2

Nf1

1 C

D

t

c1 tanf1

Cj2

c2 tanf2

Cj1

C

1

D

jt

FLAC bilinear matrix failure criterion

fj2 fj1

3’3’

FLAC bilinear joint failure criterion

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 • 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).

v

q

N normal consolidation line

plastic dilation

vl A

vkA k

vkB

1

swelling lines

ln p1

qcr M

B

pc 2

e p 0

plastic compaction

e p 0

l 1

ln p

Normal consolidation line and swelling line for an isotropic compression test

pcr

pc 2

pc

Cam-Clay failure criterion in FLAC

p

CONSTITUTIVE MODELS – HOEK-BROWN MODEL • 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.

FLAC Interface Model

FLAC (OR CONTINUUM CODE) Use for problems at either end of the joint-density spectrum

single or isolated discontinuities “interface”

multiple, closely-packed blocks “ubiquitous jointing”

problems

INTERFACES • 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 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

A1

C1

B1

D1

B2 C2

Body 1 E2

A2

Body 2

D2

LB2 LC2

LB1

LD2

LC1

LD1

LINEAR MODEL n= -Knun

[Kn]=stress/disp

= -Ksus = max (max, ) sgn () max= ntan f+c Fn = nL Fs = L

INTERFACE ELEMENTS PROCEDURE 1. Form interface using grid generation commands

2. Null out region

bside (i3, j3) (i1, j1)

(i4, j4) (i2, j2) aside

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 =...

INTERFACE PROPERTIES Kn :

normal stiffness

Ks :

shear stiffness

coh :

cohesion of the joint

fric :

friction angle of the joint

ten :

tensile strength of the joint

If the interface is used to attach two sub-grids,it is necessary to declare it glued. Properties estimation • Sub-grids attach: - declare glued -

4 K 3 G K n K s 10.max l

• Geologic joints - shear tests; considering the “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.

Boundary and Initial Conditions

Histories, Tables, FISH Library

Global Settings

Solution

Result Interpretation - Plotting

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