Flac Training

FLAC Training Course

Beijing, ChinaOctober 17, 2005

Training ScheduleOctober 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

Advanced, Two and Three Dimensional Continuum 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, thereby simulating the presence of faults, joints or frictional boundaries

Displacements resulting from construction of a shallow tunnel

FLAC & FLAC3D



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 (FLAC)

Continuous gradient or statistical distribution of any property may be specified

Braced excavation

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

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

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

FLAC & FLAC3D



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++

Liquefaction failure of a pile-supported wharf

FLAC & FLAC3D

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 are commonly used in geo-engineering 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.

“Limited-purpose” programs -

Examples of Limited-purpose Programs

Limiting condition Example program

Forces only (limit equilibrium)

SLOPE/W XSTABL

Linear properties(equivalent linear method)

SHAKE

Subgrade reaction(Winkler springs)

LPILE WALLAP

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).

“General-purpose” programs -

Comparison of Limited-Purpose and General-Purpose Solutions

Limiting conditions can be prescribed for “general-purpose” programs to approximate the simplifying assumptions built into “limited-purpose” programs. In this way, the “general-purpose” 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.

Comparison of

“General-purpose” to “Limited-purpose” programs -

We suggest using both general-purpose and limited-purpose 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

F(t)

Newton´s Law of Motion

dtud

mamF

For a continuous body, this can be generalized as

ij

iji gxdt

ud

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

u,u,u

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.

This is the basis of the calculation cycle:

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

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

ij

iji gxdt

ud

e.g., elastic

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 Nodal force vector

Elements with velocity vectors

FLAC:For all elements...

Gauss’ theorem,

S Ai

i dAxf

fdSn

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

Si

i

SnfA1

xf

This formula is applied to calculating the strain increments, eij, for a zone:

tx

u

xu

21

e

SnuuA21

xu

i

j

j

iij

Sj

)b(i

)a(i

j

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:

+ /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), andexplicit (local changes do not affect neighbours in one timestep )

Methods of solution in time domain displacement u

force F

x

F

stress

u

numerical grid

EXPLICIT

All elements:

,ufF(nonlinear law)

All nodes: t

mF

u

Repeat for n time-steps

No iterationswithin steps

Information cannot physicallypropagate between elements duringone time step

Assume (u)are fixed

Assume (F)are fixed

Correct if

p

min

Cx

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

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 is not 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.

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:

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

pC

xt min

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

• The damping force, Fd is:

m

tuFFu iiii

)(sgn||

• Damping forces are introduced to the equations of motion:

where Fi is the unbalanced force

• 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.

)sgn( iid uFF

PRACTICAL EXERCISESLOPE STABILITY ANALYSIS

Training ScheduleOctober 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 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 of safety calculation

Anisotropic thinly laminated material exhibiting elastic anisotropy

laminated materials loaded belowstrength limit

Drucker-Prager limited application; soft clays withlow friction

common model for comparison to implicit finite-element programs

Mohr-Coulomb loose and cemented granular materialssoils, rock, concrete

general soil or rock mechanics(e.g., slope stability and undergroundexcavation)

Strain-hardening/softeningMohr-Coulomb

granular materials that exhibit nonlinearmaterial hardening or softening

studies in post-failure (e.g., progressivecollapse, 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 laminatedmaterials

Double-yield lightly cemented granular material inwhich pressure causes permanentvolume decrease

hydraulically placed backfill

Modified Cam-clay materials for which deformability and shearstrength 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 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:

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

x

y

FLAC PLASTICITY MODELSDrucker-PragerMohr-CoulombUbiquitous-Joint

Strain-Hardening-SofteningDouble-Yield

Modified Cam-clayHoek-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

Ct

k/q

B k ft=0

f s=0A

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

C

B

A

fs=0

Nc2 t tan

c

1

+- ft=0

Mohr-Coulomb Failure Criterion in FLAC

shear stress

slope = G

(for constant n)

shear strain

CONSTITUTIVE MODELS – UBIQUITOUS-JOINT MODEL

•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

tj C

B

j

j

tanc

22

f t=0cj

A f s=0

CONSTITUTIVE MODELS — STRAIN-SOFTENING / HARDENING

•Strain-softening / hardening identical to the Mohr-Coulomb model except that , C and are arbitrary functions of accumulated plastic strain (p)*

produces

p

p

p

Input by user

Output

v

2

12P

12

2dP22

2dP11p eee

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.

DCB

Af2

s =0f1

s =0

f t =0

N

N11 t

1

1

tanc

2

2

tanc

3

FLAC bilinear matrix failure criterion

A

B

D

C

3’3’

f t =0

f2

s =0

f1 s =0Cj1

Cj2

jt

j1

j2

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

vA

vB

ln p

v

N

A

B1

1

ln p1

swelling lines

normal consolidation line

Normal consolidation line and swelling linefor an isotropic compression test

plastic compaction

p

critic

al sta

te lin

e

0 pe

plastic dilation

0 pe

q

2c

cr

pp

2c

cr

pMq

pc

Cam-Clay failure criterion in FLAC

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 multiple, closely-packed blocks

“interface” “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

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

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

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

-

• 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.

l

GKmaxKK sn

34

.10

Boundary and Initial Conditions

Histories, Tables, FISH Library

Global Settings

Result Interpretation - Plotting

Solution

Documents

Flac Training