Lecture 1 Rak-50 3149 a. l1- Introduction to Numerical Modelling

Numerical Methods of GeotechnicsNumerical Methods of Geotechnics

Prof. Minna KarstunenUniversity of Strathclyde

Tentative schedule: Jan 16-20, 2012

Introduction to numerical modelling & finite elements

Linear elasticity

Basic concepts of plasticity and Mohr Coulomb model

Non linear finite elements and solution techniques

Applied theory: Introduction to PLAXIS

T01 Tutorial: Soil testing tool, Mohr-Coulomb

Applied theory: Shallow foundations

Applied theory: Structural elements and interfaces

T02 Tutorial: Shallow foundation

Drained/undrained analysis

Consolidation analysis

Applied theory: Soil parameters for drained and undrained analysis

Applied theory: Slope stability and phi-c reduction

T03 Tutorial: Consolidation and phi-c reduction

Tentative schedule: Jan 16-20, 2012

Critical state models

Applied theory: Soil parameter for critical state models

T04 Tutorial: Analysis of an embankment (inc. stability)

Applied theory: Analysis of an embankment

T05 Tutorial: Boston embankment (I) Hardening Soil Model and Small Strain Stiffness

Applied theory: Soil parameters for Hardening Soil model

Applied theory: Excavations

Tentative schedule May 28-May 21, 2012

T06 Tutorial: Excavation in Limburg

Anisotropy, bonding and creep

Applied theory: Numerical modelling of ground improvement

T07 Tutorial: Boston embankment (II)

Assessment:

Coursework (100%) : Independent numerical analysis Part 1A: Identify a research paper with suitable numerical analysis

Date of submission April 2, 2012

Part 1B: Numerical analysis and report Part 1B: Numerical analysis and report

Date of submission June 15, 2012

Muir Wood, D. Geotechnical Modelling. Spon Press, 2004.

Potts, D. & Zdravkovic L. Finite element analysis in geotechnical engineering-Theory. Thomas Telford,1999.

Potts, D. & Zdravkovic L. Finite element analysis in geotechnical engineering-Application. Thomas Telford,1999.

Potts, D., Axelsson K., Grande, L. Schweiger, H. & Long, M. Guidelines for the use

Recommended reading:

Potts, D., Axelsson K., Grande, L. Schweiger, H. & Long, M. Guidelines for the use of advanced numerical analysis. Thomas Telford, 2002.

Azizi, F. Applied analysis in geotechnics. E & F. Spon, 2000.

Muir Wood, D. Soil behaviour and critical state soil mechanics. Cambridge University Press,1990.

Zienkiewicz & Taylor. The Finite Element Method available through various publishers

PLUS selected research papers available

Main aims of the module Give a comprehensive understanding of the role of soil

modelling and numerical analysis in practical geotechnical context.

The focus is on: The selection of appropriate soil model considering a particular

application and information available, application and information available, Interpretation of values for soil parameters for numerical analysis, Idealisation and modelling of geotechnical problems with 2D finite

element code PLAXIS Appreciation on the limitations of finite element modelling.

At the end of the course you will be competent (but not expert) on finite element modelling, and its opportunities and limitations, in geotechnical context.

Introduction to numerical modelling and finite elementsmodelling and finite elements

Real problem Idealized problem (conceptual model)

Mathematical

Introduction

Linear Momentum Balance

Strain displacement equation

Terzaghis principle

Continuity equation

ij j ib 0, + =

( )ij j i i j1 u u2 , , = += +

pTv

=

Real problem Idealized problem (conceptual model)

Mathematical

280

320

360

400

440m a.s.l.

411+100 412+000 413+000

North portal(Lleida)

Rele

vant

phenom

ena

Mathematical Model (PDE)Results

Solution

Analytical Numerical

Continuity equation

Mechanical constitutive law

Darcys law

vt

=

= D

v k H= T

f

H r=

g

280

320

360

400

440m a.s.l.

411+100 412+000 413+000

North portal(Lleida)

Quaternary Middle Eocene Early EoceneColluvion Limestone Claystone & Siltstone

Marl Anhydritic-Gypsiferous Claystone

Mathematical Model (PDE)

Solution

Analytical Numerical

ResultsQuaternary Middle Eocene Early Eocene

Colluvion Limestone Claystone & Siltstone

Marl Anhydritic-Gypsiferous Claystone

Introduction

A rigorous solution must satisfy the following Equilibrium Compatibility Compatibility Stress-strain relationship Boundary conditions

Empirical methods

Analytical methods

Numerical methods:

Introduction

The idealized problem can be solved by different methods:

Outside the scope of this course

Best alternative, but it is generally quite difficult for actual problems to obtain an analytical solution

Finite Difference Method (FDM) Finite Element Method (FEM) Boundary Element Method (BEM) Discrete Element Method (DEM)

Equations in integral form

Equations in differential form

Methods of analysis (1)

Closed form Constitutive behaviour linear elastic

Simple Limit equilibrium (e.g. compatibility not satisfied, soil rigid

with a failure criterion) Stress field (e.g. compatibility not satisfied, soil rigid with a

failure criterion) Limit analyses (e.g. lower bound does not satisfy

compatibility, upper bound not equilibrium, both assume soil ideally plastic with associated flow rule)

Settlement calculations (immediate, consolidation, creep) stresses from elastic theory, principle of superposition

Settlement calculations

Both assume elastic stresses & principle of superposition

The bearing capacity theory was first introduced for plane straincondition by Terzaghi and then developed by Meyerhof (1953), Vesic(1979) and Brinch-Hansen (1968) for more general conditions. The ultimate bearing capacity of a shallow foundation is expressedas:

1q B N D N c N' = + +

Bearing capacity

f f d cq B N D N c N2' = + +

where N, Nd and Nc are the so-called bearing capacity factors.

Formula is for general bearing capacity failure, so it does notconsider local failure or punching failure

Assumes principle of superposition, so in some cases error 25%(but on the safe side) Soil assumed isotropic and homogeneous

Errors increase when foundations subjected to moment loading.

Bearing capacity

Meyerhof (1953) and Vesic (1979) account for that by considering it tobe caused by a vertical load with a particular eccentricity e, and theeffective footing area.


Numerical Beam-spring approaches

Investigation of soil-structure interaction Soil modelled by springs or elastic interaction factors, so Soil modelled by springs or elastic interaction factors, so

need to make gross assumptions Only a single structure can be analysed Solutions include forces and movement of the structure,

but do not provide information on global stability or movements of the adjacent soil

Often neglect shear stresses


Numerical FEM- Finite Element Method (PLAXIS, SIGMA-W,

SAFE, ABAQUS, DIANA etc.) Simulation of the BVP attempting to satisfy all

theoretical considerations Accuracy depends on the ability of constitutive model to

represent real soil behaviour and the correctness of the idealization

User defines geometry, construction procedure, soil parameters and boundary conditions

Effect of time on pore pressure development can be modelled, including coupled consolidation


Numerical FDM- Finite Difference Method (FLAC)

Same as above, but equations in differential form Solution explicit, so results heavily dependent on step-

size No matrices are formed, so e.g. not possible to check

for convergence (you get a results but have no way of assessing its accuracy)

Finite Difference MethodFLAC Program

u&

v&

Typical FLAC Model

Slope at Failure


Numerical BEM- Boundary Element Method (mainly research

codes) Only surfaces are meshed, so less unknowns than in Only surfaces are meshed, so less unknowns than in

FEM/FDM Mathematically more complex than FEM, but

computationally more efficient Great for linear elastic materials Not possible to use advanced soil models


Numerical DEM- Discrete (or distinct) Element Method

(PFC2D, PFC3D, UDEC, 3DEC) Modelling of blocks or granular assemblies by Modelling of blocks or granular assemblies by

considering individual particles/clusters Particle contact properties need to be calibrated, as they

cannot be directly measured

Discrete Methods

Notation and FEM fundamentalsNotation and FEM fundamentals

Problem

PDE1f

2f

solutionexact PDE3f

Introduction

Solution

Function

Approximation

Function (approximation) Cordao-Neto

Approximation

Numerical methods are approximationsHow good are numerical models or approximations?

Introduction

It is difficult to distinguish difference between the circle and the approximation if N = 16.

N=5 N=8 N=16

Cordao-Neto

Vectors, Matrices & Notation (see also separate handout)

Stress and strain vectors

y

x

'

'

'

Tzxyzxyzyx

zx

yz

xy

z ]'''[''

=

=

Tzxyzxyzyx ddddddd ][ =

Vectors, Matrices & Notation

Stress-strain relationship

dDd ]['=

whereT

zxyzxyzyx ddddddd }'''{' =

and [D] is the elastic or elasto-plastic matrix (6 x 6 matrix in 3D).


Differential matrix for 3D continuum:

00

00

00

y

x

continuum:

=

0

0

0

00

xz

xz

yz

z


Nabla operator is a vector defined as:

=

xIt is an operator because it assumes a meaning onlywhen applied to a matrix or vector, such as:

=

z

ywhen applied to a matrix or vector, such as:

z

w

yv

x

u

w

v

u

zyxvT

+

+

=

=


Gradient:

The gradient of a scalar function f(x,y,z) is a vector whose components are the partial derivatives of f with

= fx

f

fpartial derivatives of f with respect to x, y, and z

=

z

fyff

Soils are multiphase materials whichare commonly assumed to have twophases:

General Formulation (3D)

pores1. Solid particles

The resolution of equilibrium equations involves the static equilibrium of soilparticles, subjected to self weight, external loads and buoyancy due to thepresence of wetting fluid.

The simultaneous fulfilment of static and hydraulic equations is commonlyreferred to as fully coupled problem.

grains1. Solid particles 2. Water (saturated soils) or air (dry

soils)

soil

H Balance Equations (global)

Constitutive Equations (related to the behaviour of the materials)


x

y

zdx

dy

dz

the materials)


Equation of motion (which reduces to Newtons second law if the quasi-static problem is approached (no time derivatives) are considered) :

Linear Momentum Balance

0=+ b where b is a vector of body forces

zyyzzxxzyxxy

zzzyzx

yyzyyx

xxzxyx

bzyx

bzyx

bzyx

===

=+

+

+

=+

+

+

=+

+

+

;;

0

0

0

General Formulation (3D) Small deformation formulation (1)

Assumed that all elongations, distortions and the rates of change of displacements are small. All higher order terms are ignored and it is assumed that strains are linear function of displacements and their derivatives.

+

=

+

=

+

=

=

=

=

+

=

z

u

x

w

yw

z

v

x

v

yu

z

w

yv

x

u

x

u

x

u

zxyzxy

zyx

i

j

j

iij

21

;21

;21

;;

21

General Formulation (3D) Small deformation formulation (2)

We normally use the Cartesian notation:

wvu

z

u

x

w

yw

z

v

x

v

yu

z

w

yv

x

u

zxyzxyxy

zyx

+

=

+

=

+

==

=

=

=

;;2

;;

General Formulation (3D) Terzaghis principle

= + where is the vector (1,1,1,0,0,0)T and the pore water pressure.

Continuity equation

When a fluid is flowing through the soil mass, the mass conservation law has to be fulfilled. Such a conservation equation (the continuity equation) reads

pTvt

=

within the hypothesis of incompressible fluid and without sources or sinks. v is the flow velocity

is the volumetric strain, given by the trace of the strain tensor (=x+y+z). p


Mechanical problem

A further set of equations are the constitutive equations (or constitutivelaws) for the mechanical and hydraulic problem.

The mechanical constituve law can be simply considered as a law which correlates strain and effective stress tensor.

= D Hydraulic problem

The link between velocity of fluid flowing through the soil mass (v) and hydraulic head (H) is established via the Darcys equation, as a function of soil permeability k:

v k H= Note that Bernoullis equation defines the hydraulic head (neglecting the dynamic term).


Equation Name Mathematical form NEquationsN

Unknowns

Linear Momentum Balance 3 6 ()

Strain displacement equation 6 3 (u)( )ij j i i j1 u u2 , , = +

0=+ b

Terzaghis principle 6 6 ()

Continuity equation 1 1 ()

Mechanical constitutive law 6 6 ()

Darcys law 3 + 3 (v)

Total 25 25

= + pTvt

=

= D v k H=

What we do?

System the Partial Differential

equations (PDEs)+

Boundary Conditions

System of Equations in Integral form

System of algebraic equations

Finite Element Method

We solve this systemDomain + Boundary

Different techniques can be used to obtain the system of algebraic equations:

Weighted Residual Method

Variational Method

Virtual Work Principle (very common in structural problems)

systemDomain + Boundary

Typical 2D meshElements connected together in nodes

Primary variables (displacements) are calculated in nodes

Footing width = B

Node

Gauss point

Typical 2D mesh

Footing width = B

Node

Displacements interpolated within element to calculate strains at Gauss points

Constitutive law is used to relate strains Node

Gauss point

used to relate strains to stresses

Forces acting at nodes are calculated (inc. body forces and surface tractions)

b: body forcesDifferential Equation of Equilibrium

=+

+

=+

+

0

0

yyxy

x

xyx

byx

byx

x 0

2D FEM - Mechanical Problem

x

y

xy

=

0

0

x y

y x

=

TLLLL

T + b = 0LLLL

=

+

00

0

0

y

x

xy

y

x

bb

x

y

y

x

=+

+

=+

+

0

0

yyxy

x

xyx

byx

byx

( ), ux yv

= =

u u

u

V

x

y


Displacements

u

v

= =

u

3

i i 1 1 2 2 3 3i 1

u u N u N u N u N=

= = + +3

= = + +e

3

i i 1 1 2 2 3 3i 1

v v N v N v N v N=

= = + +

1

1

3 21 2

3 21 2

3

3

e e e

u

v

N uN NuN vN Nv

u

v

= = = =

u N a

xux

=

yx

y

=x

v

yu

xy

+

= u

u

V

x

y


Deformations (1)

0x

(Deformation Vector)

(DisplacementVector ) Note. The unknowns of theproblems are the nodal

displacements

e{0 ex

e e

y

xy

x

u u

v vy

y x

= = = =

B

N aL L LL L LL L LL L L

{3 x 2 2 x 6 6 x 1

3 x 6e

e e

B

N a123LLLL

uu

V

x

y


Deformations (2)

{

0

0e

x

e e

y

xy

x

u u

v vy

y x

= = =

B

N aL LL LL LL L

B is the matrix that relates nodal displacements to strains

ey x

31 2

31 2

1 1 2 2 3 3

00 0

0 0 0e e

NN Nxx x

NN Ny y y

N N N N N Ny x y x y x

= =

N BLLLL

e e B a

Constitutive laws expresses the relationship between stress () and the strain () vectors in a matrix equation of the form:

= D

where D is often referred to as the material stiffness matrix.For the case of linear plane strain isotropic elasticity, the matrix is:


Stressese e

B a Deformations Mechanical law Stresses

For the case of linear plane strain isotropic elasticity, the matrix is:

D = +

Ev v

v v

v vv( )( )1 2 1

1 01 0

0 0 1 22

where E is the Young's modulus and is the Poisson's ratio. Because the coefficients are constants (for the case where geometry changes are assumed to be small) the resulting FE equations are linear and can be solved in a one step.


Stresses and element matrix

= LLLL

= D ;= D LLLL e e e= N a

{e e e e e

e

= =

B D N a DB aLLLL

1

2

3

fy1

fx1

fx3

fx2

fy2

fy2

( )T T Ti i iN d N d N d = + b tL

e e e

e e

T e T Ti i

e e e

d N d N d = +

=

K f

B D B a b t

K a f

14424 43 14 4 4424 4 4 43

6 x 3 3 x 3 3 x 6 6 x 6

eT e=B D B K

Equation of equilibrium applied to the element as a whole


e e e=K a f

1

2

3

fy1

fx1

fx3

fx2

fy2

fy2

11

11

22

22

33

33

;

x

y

xe e

y

x

y

fufvfufvfufv

= =

a f

Local matrix (element)

Body forces and surface tractions applied to the element may be generalized into a set of forces acting at the nodes.

Footing width = B

Node

Gauss point

=

Assembly of the global matrix

2D FEM- Solution

The global stiffness matrix generally contains many terms that are zero. If node numbering is efficient these are clustered in a band along the diagonal. In this case it is necessary to store only element within the bandwidth saves considerably on amount of storage.

Ns s

0

0

N

s s

0

2D FEM- Solution

PLAXIS uses skyline procedure and some of the zero terms within the band are not stored.

Skyline profileBand profile

To determine the nodal displacements corresponding to the applied forces it is necessary to solve the global stiffness equation ( KU = P ) subject to the appropriate boundary conditions.

FEM- Solution

Forces are related to displacement via stiffness equations which are solved within the FE code to find within the FE code to find values of nodal displacements

Elements for 2D analysis

(a) Triangular elements

(b) Lagrange elements

A polynomial interpolation function is used to describe the displacements within each element (higher order polynomial yields more accurate results)

Elements for 3D analysis

(a) Tetrahedron (b) Hexahedron (c) Pentahedron

Some practical aspectsSome practical aspects

FE meshes for geotechnical analysis

Typically the aspect ratio of the elements should be smaller than 3

Typically for triangular elements in internal angles should be 15 in internal angles should be 15 < < 165

Fine mesh for area with high gradients

FE meshes for geotechnical analysis

Summary of FE method FEM is a computational procedure that may be used to obtain

approximate solutions to mathematical problems

Governing mathematical equations (generally continuous) are approximated by a series of algebraic equations involving approximated by a series of algebraic equations involving quantities that are evaluated at discrete points within the region of interest

Linear elasticityLinear elasticity

Constitutive equations

Relate generally stress and strain, i.e. not just in 1D or for a special stress path

A variety of constitutive equations available and important to adopt a model that is most appropriate for the particular analysis to be appropriate for the particular analysis to be carried out. Choice depends on:

material (clay, sand) information available (in situ & lab. tests) type of problem (embankment, excavation..) knowledge of the user

Constitutive equations (cont.)

The simplest: linear elastic model (Hookes law)a

xx b yy

xx

xx2b

In the case of a uniaxial tensile stress applied to an elastic bar, the strains produced in two perpendicular directions are related by the expression:

where is an elastic constant known as Poisson's ratio.

2a

vx

y=

Constitutive equations (cont.)In 3D the stress-strain equations take the form:

yy

)(1 zyxx vvE =

xx

zz

)(1)(1

yxzz

zxyy

vvE

vvE

E

=

=


If a shear strain in applied to an elastic material, a shear strain is produced.

xy =

yx

xy

Gxy

xy

=

yx

xy xy

where G is the shear modulus.


Four elastic parameters are commonly used : Young's modulus, E Shear modulus, G, Poisson's ratio Bulk modulus, K

An elastic material is fully specified, however, whenvalues of two of these parameters are given.

)21(3)1(2 vEK

v

EG

=

+=

2D elastic analysis

x

z

zz = 0

P

To carry out FE analysis of 2D problems, it is necessary to specify the condition in the third dimension. The plane straincondition is most commonly used in soil mechanics.

yP

y

x

2D elastic analysis (cont.)

In plane strain condition the out-of plane strain is set to zero and Hookes law gives:

x

y

( )( )

xyxy

xyy

yxx

G

vvvv

E

vvvv

E

=

++

=

++

=

)1()1)(21(

)1()1)(21(

Drained and Undrained Analysis

u= =0

Dissipation of u with time

u=0=

1. Undrained 2. Consolidation

3. Drained

Drained and Undrained Analysis (cont.)(a) The shear modulus is identical for drained and undrained loading.

(b) The drained and undrained Young's moduli are related by the expression:

G G Gu = =

Note that for most soils the value of generally lies in the range 0.3 to 0.35 for sands and 0.2-0.3 for clays. Youngs modulus values, however, may vary substantially between different materials and stress levels.

uEvE )1(32

+=

y x

y

xyyx

yz

zy

Isotropic elasticity in 3D

)'''''(1)'''''(

'

1

)'''''('

1

vv

vvE

vvE

zxyy

zyxx

=

=

=

x

yz

x

z

xz zx

zy

'

'

'

)'''''('

G

G

G

vvE

zxzx

yzyz

xyxy

yxzz

=

=

=

=

y x

y

xyyx

yz

zy

Cross-isotropic elasticity (around y-axis)

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

zhhvh

z

v

hh

v

yx

v

hhy

z

v

vhy

v

hh

h

xx

vv

Ev

EEv

Ev

Ev

E

+=

+=

=

Sampling direction

vh

x

yz

x

z

xz zx

zy

'

'

'

'

'

'

'

'

hh

zxzx

vh

yzyz

vh

xyxy

h

zy

v

hhx

h

vhz

G

G

G

EEE

=

=

=

+=

General 3D elasticity would require the specification of 21 elastic constants!

Needs 5 elastic

constants!

Oedometer test

Soil

Load

yVertical

strainyy

E'

oed'/1 Emv =

1

yy

Primarycompression

E'oed,ur

1

E'oed

Unload/reloadpath

Lateral strains are zero, therefore measure modulus is not Youngs modulus. Based on Hookes law:

=

+ E v E

v voed

( )( )( )

11 2 1

Triaxial test (see also Muir Wood 1990)

E'ur

Deviatorstress

q qf

E'501 1

Axial strain, a

qf / 2

50

Shearing with constant cell pressure

Secant or Tangent E?

Es

'1

Et1

'1

Youngs modulus E

1

Es1

1

Poissons ratio 5.0'1

Non-Linear Elasticity

Non linear elasticity

')1(')1('' pededp

eddpK +=+==

1

ln(p')eBulk modulus K

)1('

dee

dK

v

=+== 1

')'1(2)'21(3

' KG

+

=

Shear modulus G

5.0'1

Elasticity vs. Plasticity

In elasticity, there is a one-to-one relationship between stress and strain. Such a relationship may be linear or non-linear. An essential feature is that the application and removal of a stress leaves the material in its original condition

Elasticity vs. Plasticity

for elastic materials, the mechanism of deformation depends on the stress increment

for plastic materials which are yielding, the mechanism of (plastic) deformation depends on the stress

reversible = elastic irreversible = plasticreversible = elastic irreversible = plastic

Next lecture will look at the Basic Concepts of Plasticity and Mohr Coulomb model

Lecture 1 Rak-50 3149 a. l1- Introduction to Numerical Modelling

Documents

finite element analysis

independent numerical

soil parameters

geotechnical modelling

role of soil modelling

mohrcoulomb applied

soil behaviour

finite elementsmodelling