Plaxis Mum Plaxisbai Course 2012

PLLAXISS STA

MU18-21 S

ANDA

UMBAI, ISEPTEM

ARD C

NDIA BER 201

COUR

2

RSE

CONTENTS Lectures & Exercises on 2D and 3D Modelling

CG1 Geotechnical Finite Element Modelling and Plaxis 2D 5

CG2 Introduction to Mohr-Coulomb Model 17

CG3 Exercise 1:Simple Foundation on Mohr-Coulomb Soil 38

CG4 Non-linear Computation in Plaxis 69

CG5 Hardening Soil Model 83

CG6 Exercise 2: Triaxial & Oedometer Test 143

CG7 Geometry, Meshing and Element Types in Plaxis 188

CG8 Structural Elements in Plaxis 205

CG9 Exercise 3: Anchored Excavation 218

CG10 Undrained and Drained Analysis in Plaxis 246

CG11 Modelling of Groundwater in Plaxis 262

CG12 Exercise 4: Excavation and Dewatering 288

CG13 Initial Geostatic Stresses 300

CG14 Safety Analysis using Phi-C Reduction Technique 306

CG15 Exercise 5: Stability Analysis of Slope Stabilised by Soil Nails 316

CG16 Overview of Soil Models 329

CG17 Consolidation Analysis in Plaxis 347

CG18 Exercise 6: Geotextile Reinforced Embankment with Consolidation 363

CG19 Introduction to Plaxis 3D 382

CG20 Modelling of Deep Foundations in Plaxis 3D 424

CG21 Exercise 7: 3D Piled Raft Foundation Analysis 460

CG22 Modelling of Tunnels and Tunnelling in Plaxis 3D 477

CG23 Modelling of Deep Excavations in Plaxis 3D 563

CG24 Exercise 8: 3D Excavation Modelling 607

DAY1 TUESDAY18.9.12THEME GEOTECHNICALFINITEELEMENTMODELLINGTime Module Description Lecturer

09:00 10:00 CG1 GeotechnicalFiniteElementModellingandPlaxis2D Dr.Juneja/DrWilliam10:00 10:15 Break

10:15 11:15 CG2 IntroductiontoMohrCoulombModel Dr.Juneja/DrWilliam11:15 12:45 CG3 Exercise1:SimpleFoundationonMohrCoulombSoil Mr.Siva

12:45 2:00 Lunch

2:00 3:00 CG4 NonlinearComputationinPlaxis Dr.Cheang

3:00 3:15 Break

3:15 4:15 CG5 HardeningSoilModel Dr.Cheang

4:15 5:30 CG6 Exercise2:Triaxial&OedometerTest Dr.Cheang

DAY2 WEDNESDAY19.9.12THEME Time Module Description Lecturer

09:00 10:00 CG7 Geometry,MeshingandElementTypesinPlaxis* Dr.Cheang

10:00 10:15 Break

10:15 11:15 CG8 StructuralElementsinPlaxis Dr.Cheang

11:15 12:45 CG9 Exercise3:AnchoredExcavation Mr.Siva

12:45 2:00 Lunch

2:00 3:00 CG10 UndrainedandDrainedAnalysisinPlaxis Dr.Cheang

3:00 3:15 Break

3:15 4:15 CG11 ModellingofGroundwaterinPlaxis* Dr.Cheang

4:15 5:30 CG12 Exercise4:ExcavationandDewatering Mr.Siva

3

DAY3 THURSDAY20.9.12THEME Time Module Description Lecturer

09:00 10:00 CG13 InitialGeostaticStressesinPlaxis Dr.Cheang

10:00 10:15 Break

10:15 11:15 CG14 SafetyAnalysisusingPhiCReductionTechnique Dr.Cheang

11:15 12:45 CG15 Exercise5:StabilityAnalysisofaSlopeStabilisedbySoilNails Mr.Siva

12:45 2:00 Lunch

2:00 3:00 CG16 OverviewofSoilModelsinPlaxis Dr.Cheang

3:00 3:15 Break

3:15 4:15 CG17 ConsolidationAnalysisinPlaxis Dr.Juneja/DrWilliam4:15 5:30 CG18 Exercise6:Geotextilereinforcedembankmentwithconsolidation Mr.Siva

DAY4THEME Time Module Description Lecturer

09:00 10:00 CG19 IntroductiontoPlaxis3D DrCheang

10:00 10:15 Break

10:15 11:15 CG20 ModellingofDeepFoundationsinPlaxis3D DrCheang

11:15 12:45 CG21 Exercise7:3DPiledRaftFoundationAnalysis(FleidenCase) DrCheang

12:45 2:00 Lunch

2:00 3:00 CG22 ModellingofTunnelsandTunnellinginPlaxis3D DrCheang

3:00 3:15 Break

3:15 4:15 CG23 ModellingofDeepExcavationsinPlaxis3D DrCheang

4:15 5:30 CG24 Exercise8:3DExcavationAnalysis DrCheang

4

1Basic concepts of FEMCiTG, Geo-engineering, http://geo.citg.tudelft.nl

Finite element modelling in geotechnical engineering

Ronald Brinkgreve, Plaxis bv / Delft University of Technology


Objectives:

To explain the basics of the finite element method

To show different types of elements and integration

To specify the components of the stiffness matrix

To formulate how the system of equations is formed

To explain how displacements and strains are calculated

Basic concepts of the Finite element method (deformations)

5


The Finite Element Method (FEM) is a numerical technique to find an approximate solution for a (set of) partial differential equation(s).

The Finite Element Method for deformations is based on the following principles:

Equilibrium (between external forces and internal stresses) Kinematics (displacements and strains) Constitutive relation (material behaviour)




load

displacement

strain

stress

equilibrium stiffness matrix

kinematicsconstitutive relation

6


Geometry is divided into finite elements (2D triangles or quadrilaterals; 3D tetrahedrals, bricks, other)

Elements consist of nodes which contain discrete values of primary quantities (displacement components)

Primary quantities are interpolated over the element using polynomials, and are continuous over element boundaries

In addition to nodes, elements contain (Gaussian) integration points (or stress points) for numerical integration

Integration points contain discrete values of secondary quantities (stress and strain components)



x

y

AA

0 1

23 4 5

6 7


7


Isoparametric elements with nodes for two-dimensional analysis:

Triangular elements

Quadrilateral elements

element

node



2D isoparametric elements with possible stress points:

Triangular elements

Quadrilateral elements

stress point


8


Isoparametric elements for three-dimensional analysis:

Tetrahedral elements

Brick elements



3D isoparametric elements with possible Gauss points:

Tetrahedral elements

Brick elements


9


Interpolation functions for linear 3-node triangular element:

x

y

1 2

3

v1x

v1y

0 1 2( , )xu x y a a x a y 0 1 2( , )yu x y b b x b y

1 1 2 2 3 3( , )x x x xu x y N v N v N v 1 1 2 2 3 3( , )y y y yu x y N v N v N v

1

2

3

1N x yN xN y

0 1

1 2 1

2 3 1

x

x x

x x

a va v va v v

0 1

1 2 1

2 3 1

y

y y

y y

b vb v vb v v

N : Shape functions



)1(44

)1(4)12()12(

)221)(1(

6

5

4

3

2

1

yxyNxyN

yxxNyyNxxN

yxyxN

Interpolation functions for quadratic 6-node triangular element:

x

y

1 2

3

4

56 v5x

v5y

254

23210),( yaxyaxayaxaayxux

254

23210),( ybxybxbybxbbyxuy

1 1 2 2 6 6( , ) ...x x x xu x y N v N v N v 1 1 2 2 6 6( , ) ...y y y yu x y N v N v N v

0 1

1 1 2 4

2 1 3 6

3 1 2 4

4 1 4 5 6

5 1 3 6

3 43 4

2 2 44 4 4 42 2 4

a va v v va v v va v v va v v v va v v v


10


Interpolation functions for quadratic 6-node triangular element:

1 1 2 2 6 6( , ) ...x x x xu x y N v N v N v 1 1 2 2 6 6( , ) ...y y y yu x y N v N v N v evNu

),(),(yxuyxu

uy

x

y

x

y

x

y

x

e

vv

vvvv

v

6

6

2

2

1

1

...

...

yyy

xxx

NNNNNN

N621

621

0......000......00

N : Shape functions



Strains for 6-node triangular element:

yaxaaxuyx xxx 431 2),( ybxbbyuyx yyy 542 2),(

xxxx

xx vdxdNv

dxdNv

dxdN

dxdu

66

22

11 ...

yyyy

yy vdydNv

dydNv

dydN

dydu

66

22

11 ...

yxyxyx

xy vdxdNv

dydNv

dxdNv

dydN

dxdu

dydu

66

22

11

11 ...x

y

1 2

3

4

56 v5x

v5y

ybaxbaabxuyuyx yxxy )2()2()(),( 453421


11


Strains for 6-node triangular element:

evB

xN

yN

xN

yN

xN

yN

yN

yN

yN

xN

xN

xN

B

662211

621

621

......

0......00

0......00

y

x

y

x

y

x

e

vv

vvvv

v

6

6

2

2

1

1

...

...

),(),(),(

yxyxyx

xy

yy

xx

B : Strain interpolation matrix



Nodal forces for 6-node triangular element:

y

x

y

x

y

x

e

ff

ffff

f

6

6

2

2

1

1

...

...x

y

1 2

3

4

56 f5x

f5y


12


Element stiffness matrix:

kk

TTe wBMBdVBMBK

B : Strain interpolation matrixM : Material stiffness matrixwk : Weight factor of integration point k

12

1 01 0

(1 2 )(1 )0 0

EM D

Hookes law:



yyxyyyxyyyxy

yxxxyxxxyxxx

yyxyyyxyyyxy

yxxxyxxxyxxx

yyxyyyxyyyxy

yxxxyxxxyxxx

e

KKKKKKKKKKKK

KKKKKKKKKKKKKKKKKKKKKKKK

K

666626261616

666626261616

626222221212

626222221212

616121211111

616121211111

......

......................................................

......

......

......

......

Element stiffness matrix (12x12 for 6-node triangular element):


13


Elements and nodes in a FE mesh (global node numbers are indicated):



Global stiffness matrix, displacement vector, force vector:

Global system of equations from which vs are to be solved:

Or, in non-linear computations:

elements

eKK nodes

nodesvv elements

eff

fvK

fvK vvv ii 1


14


2

222

21

212

211

122

121

112

111

ee

eeee

ee

KK

KKKK

KK

0

0

elements

eKK

Global stiffness matrix


Strains, stresses

Once vs are known:

Or, in non-linear computations:

MevB

Mii 1evB

15


Read input dataForm stiffness matrix

New stepForm new load vectorForm reaction vectorCalculate unbalanceReset displacement increment

New iterationSolve displacementsUpdate displacement incrementsCalculate strain incrementsCalculate trial stressesCalculate constitutive stressesForm reaction vectorCalculate unbalanceCalculate errorAccuracy check

Update displacementsWrite output data (results)If not finished > new step

Finish

ii

TTe wBMBdVBMBK elements

eKK

ex

i

ex

i

exfff 1

kk

ic

Tic

T

inwBdVBf 11

in

i

exinfff

0v1 jj

fKv 1vvv jj 1

vB vB eictr D1

gDd

f etrtrjic

)(,

kk

ic

Tic

T

inwBdVBf

in

i

exinfff

iexffe

iterationneweeif tolerated vvv ii 1

1 ii

Finite element method (deformations)

Soil model

16

1MC model and soil stiffnessCiTG, Geo-engineering, http://geo.citg.tudelft.nl

Mohr-Coulomb model and soil stiffness

Objectives:

To indicate features of soil behaviour To formulate Hookes law of isotropic linear elasticity To formulate the Mohr-Coulomb criterion in a plasticity framework To identify the parameters in the LEPP Mohr-Coulomb model To give suggestions on the selection of parameters To indicate the possibilities and limitations of the MC model


Typical results from soil lab tests

P

F

1

3 31

v

Triaxial test (axial loading)

1-3

-1

-1v

stiffness

strength

dilatancy

17


Typical results from soil lab testsOedometer test (one-dimensional compression)

1

1

1

1

Pre-consolidation stress

primary loading

unloading

reloading


Typical results from soil lab testsOedometer test (constant load; secondary compression)

1

1

time

1

creep

18


Typical results for soil stiffnessStiffness at different levels of strain

Modulus reduction curve after Benz (2007)


Features of soil behaviour

Elasticity (reversible deformation; limited) > stiffness Plasticity (irreversible deformation) > stiffness, strength Failure (ultimate limit state or critical state) > strength Presence and role of pore water Undrained behaviour and consolidation Stress dependency of stiffness Strain dependency stiffness Time dependent behaviour (creep, relaxation) Compaction en dilatancy Memory of pre-consolidation pressure Anisotropy (directional strength and/or stiffness)

19


Concepts of soil modelling

Relationship between stresses (stress rates) and strains (strain rates)

Elasticity (reversible deformations) d=f (d) Example: Hookes law

Plasticity (irreversible deformations) d=f (d,,h) Perfect plasticity, strain hardening, strain softening Yielding, yield function, plastic potential, hardening/softening rule Example: Mohr-Coulomb yielding

Time dependent behaviour (time dependent deformations) Biots (coupled) consolidation d=f (d,,t) Creep, stress relaxation Visco elasticity, visco plasticity

yy yz yx xy

xx xz zx zz

zy


Types of stress-strain behaviour

Linear-elastic Non-linear elastic Elastoplastic

Lin. elast. perfectly-plast. EP strain-hardening EP strain-softening

20


xx

yy

zz

xy

yz

zx

xx

yy

zz

xy

yz

zx

E

( )( )1 1 2

1 0 0 01 0 0 0

1 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

12

12

12

xx

yy

zz

xy

yz

zx

xx

yy

zz

xy

yz

zx

E

1

1 0 0 01 0 0 0

1 0 0 00 0 0 2 2 0 00 0 0 0 2 2 00 0 0 0 0 2 2

Inverse:

Hookes law


1 1

2 2

3 3

11

(1 )(1 2 )1

E

Hookes law

In principal stress / strain components:

00 3

v

s

p Kq G

In isotropic and deviatoric stress / strain components:

1 1 2 33p 2 2 2

1 2 2 3 3 11 ( ) ( ) ( )2

q

21


Two parameters:

- Youngs modulus E- Poissons ratio

Meaning (axial compr.):

E dd

1

1

dd

3

1

- d1d3

- 1

- 1

3

E1

1

Model parameters in Hookes law:d1


Shear modulus:

Gdd

Exyxy

2 1

dxy

dxy

E

dd

Eoed

1

1

11 1 2

Oedometer modulus:

Kdpd

Ev

3 1 2

Bulk modulus:

- d1- d1

dp

dv

Alternative parameters in Hookes law:

22


Stress definitions

In general, soil cannot sustain tension, only compression PLAXIS adopts the general mechanics definition of stress and strain:

Tension/extension is positive; Pressure/compression is negative

In general, soil deformation is based on stress changes in the grain skeleton (effective stresses)

According to Terzaghis principle: = - pw

yy

yyxx xx

yy

yyxx xx


The modeling of non-linear soil behaviour requires a relationshipbetween effective stress rates (d ) and strain rates (d)

Symbolic: 1' 'e ed D d d D d

12

12

12

1 ' ' ' 0 0 0'' 1 ' ' 0 0 0'' ' 1 ' 0 0 0' '

0 0 0 ' 0 0' (1 ')(1 2 ')0 0 0 0 ' 0'0 0 0 0 0 ''

xx xx

yy yy

zz zz

xy xy

yz yz

zx zx

d dd dd dEd dd dd d

Hookes law for effective stress rates

23


Basic principle of elasto-plasticity:

pij

eijij (total strains)

pij

eijij ddd (strain rates)

Elastic strain rates:

klijkleeij dDd '1

Plasticity


Basic principle of elasto-plasticity:

pij

eijij (total strains)

pij

eijij ddd (strain rates)

Plasticity

Plastic strain rates:

ij

pij

gdd'

d = scalar; magnitude of plastic strainsdg/d = vector; direction of plastic strainsg = plastic potential function

24


When do plastic strains occur?

If f


Origin: F

T

n

Coulomb: T A + F tan c - n tan

A

T

Fc

n

The Mohr-Coulomb failure criterion


In general:n

3

1The condition c - n tan must hold for arbitrary angle


26


c

c cos-s* sin

-n-s*

t*

-1-3

MC criterion:

t* c cos - s* sin



MC criterion: t*c cos - s* sint* = (3 - 1)s* = (3+1)

'sin'''cos''' 13211321 c

31 ''sin1'sin1

'sin1'cos'2'

c

Note: Compression is negative and 1 2 3


27


c

n

a-3

-1

1b 'sin1

'cos'2

ca

'sin1'sin1

b

Visualisation of the M-C failure criterion


'sin'''cos''' 23212321 c 'sin'''cos''' 32213221 c 'sin'''cos''' 13211321 c 'sin'''cos''' 31213121 c 'sin'''cos''' 12211221 c 'sin'''cos''' 21212121 c

1

32

Full Mohr-Coulomb criterion

28


'sin'''cos''' 13211321 c

Reformulation into yield functions

'cos''sin'''' 132113212 cf b


'cos''sin'''' 232123211 cf a 'cos''sin'''' 322132211 cf b 'cos''sin'''' 312131212 cf a 'cos''sin'''' 132113212 cf b 'cos''sin'''' 122112213 cf a 'cos''sin'''' 212121213 cf b

1

32

Reformulation into yield functions

Parameters: Effective cohesion (c) and effective friction angle ()

29


cos'sin'''' 232123211 cg a cos'sin'''' 322132211 cg b cos'sin'''' 312131212 cg a cos'sin'''' 132113212 cg b

cos'sin'''' 212121213 cg b Dilatancy angle instead of friction angle Motivation based on simple shear test

cos'sin'''' 122112213 cg a

Plastic potentials of the M-C model


0'1 klijkleeij dDd ij

pij

gdd'

sin*4

''' 2

1

tdgdd yyxx

xx

pxx

sin*4

''' 2

1

tdgdd xxyy

yy

pyy

*'

' tdgdd xy

xy

pxy

0xxd

0

d sin

d cos

tan pxy

pyy

xy

yy

dd

dd

Failure in a simple shear test: xx t*

yy

30


tan pxy

pyy

xy

yy

dd

dd xy

xyyy

xy

Failure in a simple shear test:

dilatancy


Linear-elastic perfectly-plastic stress-strain relationship

- Elasticity: Hookes law- Plasticity: Mohr-Coulomb failure criterion

For this model: Plasticity = Failure

This does NOT apply to all models!!!

The LEPP Mohr-Coulomb model

The LEPP model with Mohr-Coulomb failure contour is in PLAXIS called the Mohr-Coulomb model

31


Model parameters:

- Youngs modulus (stiffness) E- Poissons ratio - Cohesion c- Friction angle - Dilatancy angle

Model parameters must be determined such that real soil behaviour is approximated in the best possible way

The LEPP Mohr-Coulomb model


Parameter determination

Parameter determination from:

Laboratory tests (triaxial test (CD, CU), oedometer test or CRS, simple shear test, )

Field tests (SPT, CPT, pressure meter (Menard, CPM, SBP), dilatometer, )

Correlations with qc , PI , RD and other index parameters Rules-of-thumb, norms, charts, tables Engineering judgement

32


E 50

1-2

1-3

-1

-1

v

3 = confining pressureMC approximation of a CD triax. test

32 'cos ' 2 ' sin '1 sin '

c

sin1sin2


-1

-1

MC approximation of a compr. test

Eoed

(1 )(1 2 )(1 )oed

E E

33


refref ppE '350 500..150

Loose Dense

Order of magnitude for E50:

- Sand:

- Clay: or[%]

1500050

p

uuI

cE

Ip = plasticity index

505000

[%]u

p

cGI

Stiffness parameter suggestions


-1

-1

pref Eoed

refrefoed ppE '1500..150

Loose Dense

oedoed EE

EddE

)1()21)(1(

2111

1

1

This Evalue applies to primary compression

Order of magnitude Eoed (sand):

coed qE 3..1 (correlation)


34


-1

-1

pref Eoed

1500 'oed

p

EI

oedoed EE

EddE

)1()21)(1(

2111

1

1

This Evalue applies to primary compression

Order of magnitude Eoed (clay):

3..5oed cE q (correlation)

(correlation)



-1

-1

1(0)Eoed

Secant oedometer stiffness:

1(1)1

1oedE


35


cuG



Duncan & Buchignani


36


Possibilities and limitations of the LEPP Mohr-Coulomb model

Possibilities and advantages

Simple and clear model First order approach of soil behaviour in

general Suitable for many practical applications Limited number and clear parameters Good representation of failure behaviour

(drained) Dilatancy can be included

1

32


Possibilities and limitations of the LEPP Mohr-Coulomb model

Limitations and disadvantages

Isotropic and homogeneous behaviour Until failure linear elastic behaviour No stress/stress-path/strain-dependent stiffness No distinction between primary loading and

unloading or reloading Dilatancy continues for ever (no critical state) Be careful with undrained behaviour No time-dependency (creep)

1

32

37

Elastoplastic analysis of a footing

ELASTOPLASTIC ANALYSIS OF AFOOTING

Computational Geotechnics 1

38


INTRODUCTION

One of the simplest forms of a foundation is the shallow foundation. In this exercise we willmodel such a shallow foundation with a width of 2 meters and a length that is sufficiently longin order to assume the model to be a plane strain model. The foundation is put on top of a 4mthick clay layer. The clay layer has a saturated weight of 18 kN/m3 and an angle of internalfriction of 20.

Figure 1: Geometry of the shallow foundation.

The foundation carries a small building that is being modelled with a vertical point force.Additionally a horizontal point force is introduced in order to simulate any horizontal loadsacting on the building, for instance wind loads. Taking into account that in future additionalfloors may be added to the building the maximum vertical load (failure load) is assessed. Forthe determination of the failure load of a strip footing analytical solutions are available from forinstance Vesic, Brinch Hansen and Meyerhof:

QfB

= c Nc + 12B NNq = e

pi tan tan2(45 + 12)

Nc = (Nq 1) cot

N =

2(Nq + 1) tan

(V esic)

1.5(Nq 1) tan (BrinchHansen)(Nq 1) tan(1.4) (Meyerhof)

This leads to a failure load of 117 kN/m2 (Vesic), 98 kN/m2 (Brinch Hansen) or 97 kN/m2

(Meyerhof) respectively.

2 Computational Geotechnics

39


SCHEME OF OPERATIONS

This exercise illustrates the basic idea of a finite element deformation analysis. In order tokeep the problem as simple as possible, only elastic perfectly-plastic behaviour is considered.Besides the procedure to generate the finite element mesh, attention is paid to the input ofboundary conditions, material properties, the actual calculation and inspection of some outputresults.

Aims

Geometry input

Initial stresses and parameters

Calculation of vertical load representing the building weight

Calculation of vertical and horizontal load representing building weight and wind force

Calculation of vertical failure load.

A) Geometry input

General settings

Input of geometry lines

Input of boundary conditions

Input of material properties

Mesh generation

B) Calculations

Initial pore pressures and stresses

Construct footing

Apply vertical force

Apply horizontal force

Increase vertical force until failure occurs

C) Inspect output


40


GEOMETRY INPUT

Start PLAXIS by double-clicking the icon of the PLAXIS Input program. The Quick selectdialog box will appear in which you can select to start an new project or open an existingone. Choose Start a new project (see Figure 2). Now the Project properties window appears,consisting of the two tabsheets Project and Model (see Figure 3 and Figure 4).

Figure 2: Quick select dialog

Project properties

The first step in every analysis is to set the basic parameters of the finite element model.This is done in the Project properties window. These settings include the description of theproblem, the type of analysis, the basic type of elements, the basic units and the size of thedrawing area.

The Project tabsheet

Figure 3: Project tabsheet of the Project Properties window

In order to enter the proper settings for the footing project, follow these steps:


41


In the Project tabsheet, enter Exercise 1 in the Title box and type Elasto-plasticanalysis of drained footing or any other text in the Comments box.

In the General options box the type of the analysis (Model) and the basic element type(Elements) are specified. As this exercise concerns a strip footing, choose Plane strainfrom the Model combo box. Select 15-node from the Elements combo box.

The Acceleration box indicates a fixed gravity angle of -90, which is in the verticaldirection (downward). Independent acceleration components may be entered for pseudo-dynamic analyses. Leave these values zero and click on the Next button below thetabsheets or click on the Model tabsheet.

The Model tabsheet

Figure 4: Model tabsheet of the Project properties window

In the Model tabsheet, keep the default units in the Units box (Length = m; Force = kN;Time = day).

In the Geometry dimensions box the size of the considered geometry must be entered.The values entered here determine the size of the draw area in the Input window.PLAXIS will automatically add a small margin so that the geometry will fit well withinthe draw area. Enter Xmin=0.00, Xmax=14.00, Ymin=0.00 and Ymax=4.25.

The Grid box contains values to set the grid spacing. The grid provides a matrix of dotson the screen that can be used as reference points. It may also be used for snapping toregularly spaced points during the creation of the geometry. The distance of the dots isdetermined by the Spacing value. The spacing of snapping points can further be dividedinto smaller intervals by the Number of snap intervals value. Enter 1.0 for the spacingand 4 for the intervals.

Click on the Ok button to confirm the settings. Now the draw area appears in which thegeometry model can be drawn.


42


Hint: In the case of a mistake or for any other reason that the project propertiesshould be changed, you can access the Project properties window byselecting the Project properties option from the File menu.

Creating the geometry

Once setting the project properties have been completed, the draw area appears with anindication of the origin and direction of the system of axes.

The cursor is automatically switched in the Geometry line drawing mode. If not, the user canchange the drawing mode to Geometry line by clicking the geometry line button .

In order to construct the contour of the proposed geometry as shown in Figure 5, follow thesesteps. (Use Figure 5 for orientation, it represents the completed geometry).

Figure 5: Geometry model

Create sub-soil

Position the cursor (now appearing as a pen) at the origin (point 0) of the axes (0.0; 0.0).Click the left mouse button once to start the geometry contour.

Move along the x-axis to (14.0; 0.0). Click the left mouse button to generate the secondpoint (number 1). At the same time the first geometry line is created from point 0 to point1.

Move upward to point 2 (14.0; 4.0) and click again.

Move to the left to point 3 (0.0; 4.0) and click again.

Finally, move back to the origin (0.0; 0.0) and click the left mouse button again. Sincethe latter point already exists, no new point is created, but only an additional geometryline is created from point 3 to point 0. PLAXIS will also automatically detect a cluster(area that is fully enclosed by geometry lines) and will give it a light colour.


43


Click the right mouse button to stop drawing.

This action created the sub-soil cluster. The next step is to introduce the footing.

Create footing

Position the cursor at point 4, (6.0, 4.0) and click the left mouse button once.

Move vertical to point 5, (6.0; 4.25). Click the left mouse button to generate a verticalline.

Move horizontal to point 6, (8.0; 4.25). Click the left mouse button to generate a horizontalline.

Generate a second cluster by clicking the left mouse button on coordinate (8.0; 4.0).


This action created the footing.The proposed geometry does not include plates, hinges, geogrids, interfaces, anchors ortunnels. Hence, you can skip the corresponding buttons in the second toolbar.

Hints: Mispositioned points and lines can be modified or deleted by first choosing the

Selection button from the toolbar. To move a point of line, select the point orthe line and drag it to the desired position. To delete a point or a line, select thepoint or the line and press the Delete key on the keyboard.

> Undesired drawing operations can be restored by pressing the Undo button

from the toolbar or by selecting the Undo option from the Edit menu or bypressing on the keyboard.

Hint: The full geometry model has to be completed before a finite element mesh can begenerated. This means that boundary conditions and model parameters must beentered and applied to the geometry model first.

Hint: During the input of geometry lines by mouse, holding down the Shift key willassist the user to create perfect horizontal and vertical lines.

Input of boundary conditions

Boundary conditions can be found in the second block of the toolbar and in the Loads menu.For deformation problems two types of boundary conditions exist: Prescribed displacementsand prescribed forces (loads). In principle, all boundaries must have one boundary conditionin each direction. That is to say, when no explicit boundary condition is given to a certainboundary (a free boundary), the so-called natural condition applies, which is a prescribedforce equal to zero and a free displacement. In order to avoid the situation where the displacementsof the geometry are undetermined, some points of the geometry must have prescribeddisplacements. The simplest form of a prescribed displacement is a fixity (zero displacement),but non-zero prescribed displacements may also be given.To create the boundary conditions for this exercise, follow the steps below.


44


Prescribed displacements

Click on the Standard fixities button on the toolbar or choose the Standard fixities optionfrom the Loads menu to set the standard boundary conditions. As a result PLAXIS willautomatically generate a full fixity at the base of the geometry and roller conditions at thevertical sides (ux=0; uy=free). A fixity in a certain direction is presented as two parallel linesperpendicular to the fixed direction. Hence, the rollers appear as two vertical parallel lines andthe full fixity appears as cross-hatched lines.

Hint: The Standard fixities option is suitable for most geotechnical applications. It isa fast and convenient way to input standard boundary conditions.

Vertical load

Click on the Point load - load system A button on the toolbar or choose the Point load- static load system A option from the Loads menu to enter another point force. Click on thecoordinate (7.0, 4.25) to enter a point force. As a result PLAXIS will automatically generate avertical point force on the indicated point with a unity force (f = 1).

Horizontal load (see also next step "Changing direction .....")

Click on the Point load - load system B button on the toolbar or choose the Point load -static load system B option from the Loads menu to enter a point force. Click on the coordinate(7.0, 4.25) to enter a point force. As a result PLAXIS will automatically generate a vertical pointforce on the indicated point. As a horizontal force is needed, the direction of load B needs tobe changed.

Changing direction and magnitude of loads

Choose the Selection button from the toolbar. Double click on the geometry point 8 withcoordinate (7.0, 4.25) which will display a box as indicated in Figure 6. Select Point Load -load system B, click OK and enter 1.0 as x-value and 0.0 as y-value. These values are theinput load of point force B. Click OK to close the window.

Input of material properties

In order to simulate the behaviour of the soil, a proper soil model and corresponding parametersmust be applied to the geometry. In PLAXIS, soil properties are collected in material data setsand the various data sets are stored in a material database. From the database, a data setcan be assigned to one or more clusters. For structures (like walls, plates, anchors, geogrids,etc.) the system is similar, but obviously different types of structures have different parametersand thus different types of data sets. PLAXIS distinguishes between material data sets for Soil


45


Figure 6: Select window and Point load window

& Interfaces, Plates, Anchors and Geogrids. The creation of material data sets is generallydone after the input of boundary conditions. Before the mesh is generated, all material datasets should have been defined and all clusters and structures must have their appropriate dataset.

Table 1: Material properties of the clay layer and the concrete footing.Parameter Symbol Clay Concrete Unit

Material model Model Mohr-Coulomb Linear elastic Type of behaviour Type Drained Non-porous

Weight above phreatic level unsat 16.0 24.0 kN/m3

Weight below phreatic level sat 18.0 kN/m3

Youngs modulus Eref 5.0103 2.0107 kN/m2Poissons ratio 0.35 0.15

Cohesion c 5.0 kN/m2

Friction angle 20 Dilatancy angle 0

The input of material data sets can be selected by means of the Material Sets button onthe toolbar or from the options available in the Materials menu.

Create material data sets

To create a material set for the clay layer, follow these steps:

Select the Material Sets button on the toolbar.

Click on the button at the lower side of the Material Sets window. A new dialogbox will appear with five tabsheets: General, Parameters, Flow parameters, Interfacesand Initial (see figure 7).

In the Material Set box of the General tabsheet, write Clay in the Identification box.

Select Mohr-Coulomb from the Material model combo box and Drained from the Materialtype combo box.


46


Enter the proper values for the weights in the General properties box according to thematerial properties listed in table 1

See also figure 8 and figure 9. In these figures the Advanced parameters part has beencollapsed.

Figure 7: General tabsheet of the soil and interface data set window for Clay

Click on the Next button or click on the Parameters tabsheet to proceed with the input ofmodel parameters. The parameters appearing on the Parameters tabsheet depend onthe selected material model (in this case the Mohr-Coulomb model).

Enter the model parameters of table 1 in the corresponding edit boxes of the Parameterstabsheet. The parameters in the Alternatives and Velocities group are automaticallycalculated from the parameters entered earlier.

Since the geometry model does not include groundwater flow or interfaces, the third andfourth tabsheet can be skipped. Click on the OK button to confirm the input of the currentmaterial data set.

Now the created data set will appear in the tree view of the Material Sets window.

For the concrete of the footing repeat the former procedure, but choose a Linear Elasticmaterial behaviour and enter the properties for concrete as shown in table 1 (see alsofigures 9 and 10).


47


Figure 8: Parameters tabsheet of the soil and interface data set window for Clay

Figure 9: General tabsheet of the soil and interface data set window for Concrete

Figure 10: Parameters tabsheet of the soil and interface data set window for Concrete


48


Assigning material data sets to soil clusters

Drag the data set Clay from the Material Sets window (select it and keep the left mousebutton down while moving) to the soil cluster in the draw area and drop it there (releasethe left mouse button). Notice that the cursor changes shape to indicate whether or notit is possible to drop the data set. When a data set is properly assigned to a cluster, thecluster gets the corresponding colour. Drag the concrete material set to the footing anddrop it there.

Click on the OK button in the Material Sets window to close the database.

Hint: PLAXIS distinguishes between a project database and a global database ofmaterial sets. Data sets may be exchanged from one project to another usingthe global database. In order to copy such an existing data set, click on theShow global button of the Material Sets window. Drag the appropriate data set(in this case Clay) from the tree view of the global database to the projectdatabase and drop it there. Now the global data set is available for the currentproject. Similarly, data sets created in the project database may be draggedand dropped in the global database.

Hints: Existing data sets may be changed by opening the material sets window,selecting the data set to be changed from the tree view and clicking on the Editbutton. As an alternative, the material sets window can be opened by doubleclicking a cluster and clicking on the Change button behind the Material set boxin the properties window. A data set can now be assigned to the correspondingcluster by selecting it from the project database tree view and clicking on theOK button.

> The program performs a consistency check on the material parameters and willgive a warning message in the case of a detected inconsistency in the data

Mesh generation

When the geometry model is complete, the finite element model (mesh) can be generated.PLAXIS includes a fully automatic mesh generation procedure, in which the geometry isautomatically divided into elements of the basic element type and compatible structural elements,if applicable. The mesh generation takes full account of the position of points and lines in thegeometry model, so that the exact position of layers, loads and structures is reflected bythe finite element mesh. The generation process is based on a robust triangulation principlethat searches for optimised triangles, which results in an unstructured mesh. This may lookdisorderly, but the numerical performance of such a mesh is usually better than for regular(structured) meshes. In addition to the mesh generation itself, a transformation of input data(properties, boundary conditions, material sets, etc.) from the geometry model (points, linesand clusters) to the finite element mesh (elements, nodes and stress points) is made.In order to generate the mesh, follow these steps:

Click on the Generate mesh button in the toolbar or select the Generate option fromthe Mesh menu. After the generation of the mesh a new window is opened (PLAXISOutput window) in which the generated mesh is presented (see Figure 11).


49


Click on the Close button to return to the geometry input mode.

Figure 11: Generated finite element mesh of the geometry around the footing

If necessary, the mesh can be optimised by performing global or local refinements. Meshrefinements are considered in some of the other exercises. Here it is suggested to accept thecurrent finite element mesh.

Hints: By default, the Global coarseness of the mesh is set to Medium, which isadequate as a first approach in most cases. The Global coarseness settingcan be changed in the Mesh menu. In addition, there are options available torefine the mesh globally or locally.

> At this stage of input it is still possible to modify parts of the geometry or to addgeometry objects. In that case, obviously, the finite element mesh has to beregenerated.

Press the close button to close the output program and return to PLAXIS input.Creating the input for this project now finished. Press the green Calculation button on thetoolbar to continue with the definition of the calculation phases.


50


CALCULATION

After the finite element model has been created, the calculation phases need to be defined.This analysis consists of four phases. In the initial phase the initial pore pressures andstresses are generated, in the first phase the footing is constructed, during the second phasethe vertical load is applied and in the third phase the horizontal load is applied.When starting the PLAXIS Calculation program the Calculation mode window appears. Inthis window the user can choose how he wants PLAXIS to handle pore pressures during thecalculation. This is important when calculating with undrained behaviour and/or groundwaterflow. In this first exercise this is not important and so the default setting of Classical mode ischosen. Press to close the Calculation mode window. PLAXIS now shows the Generaltabsheet of the initial phase (see Figure 12).

Figure 12: General tabsheet of the initial calculation phase

Initial phase (generation of initial conditions)

Before starting the construction of the footing the initial conditions must be generated. Ingeneral, the initial conditions comprise the initial groundwater conditions, the initial geometryconfiguration and the initial effective stress state. The clay layer in the current footing project isfully saturated with water, so groundwater conditions must be specified. On the other hand, thesituation requires the generation of initial effective stresses. As we want to include the footingconstruction in the simulation process, the footing should not be present in the initial situation(prior to construction). In PLAXIS it is possible to switch off clusters in order to calculatecorrect initial effective stresses. The initial stresses in this example case are generated usingthe K0-procedure. The initial conditions are entered in separate modes of the Input program.In order to generate the initial conditions properly, follow these steps:

In the phase list select the initial phase


51


Make sure the Calculation type is set to K0-procedure on the General tabsheet. This isthe default setting.

Go to the Parameters tabsheet by clicking the Parameters button or by directly selectingthe tabsheet.

On the Parameters tabsheet press the Define button located in the Loading input box.This will start a window presenting the problem in Staged construction mode. In Stagedconstruction mode it is possible to switch on and off various parts of the geometry,change loads, apply strains etc.

In the initial condition of this exercise, that is the situation before we start constructingour project, the footing is not present. Therefore the footing has to be deactivated. Inorder to do so, click on the area that represents the footing so that it will change colorfrom the material set color to white. The footing is now disabled.

Click on Water conditions in the button bar in order to move to the Water conditionsmode of the program.

Select the Phreatic level button .

Position the cursor (appearing as a pen) at coordinate (0.0, 4.0) and click the left mousebutton to start the phreatic level.

Move along the x-axis to position (14.0, 4.0). Click the left mouse button to enter thesecond point of the phreatic level.


Press the Water pressures button to view the pore pressures.

The pore pressures are generated from the specified phreatic level and the water weight.Directly after the generation, a PLAXIS Output window is opened, showing the pore pressureas presented in Figure 13. The colors indicate the magnitude of pore pressure. The porepressures vary hydrostatically, ranging from 0 kN/m2 at the top to -40 kN/m2 at the bottom.

Close the output program in order to return to the input program.

Click on Update in order to save the changes made and return to the PLAXIS Calculationsprogram. This completes the definition of the initial conditions.

Hints: For the generation of initial stresses based on the K0procedure it is necessaryto specify the coefficient of lateral earth pressure, K0. This K0value is definedper material set and therefore has to be set when entering material set data. Ifthe K0value is not explicitly set PLAXIS uses a value according to Jakysformula (K0 = 1-sin()).

> The K0 procedure may only be used for horizontally layered geometries with ahorizontal ground surface and, if applicable, a horizontal phreatic level.


52


Figure 13: Initial pore pressures

First calculation phase (construction of footing)

Click on the Next button . This will introduce a new calculation phase andpresent the corresponding tabsheets for the first calculation stage. Enter a suitable namein the Number/ ID box (e.g. Construction of footing).

Select the second tabsheet called Parameters. On this sheet Staged construction isselected by default in the Loading input combo box. Click the Define button. This willopen the window presenting the problem in Staged construction mode.

Click on the cluster that represents the strip footing, in order to switch on the footing(original colour should reappear).

Click on Update to conclude the definition of the first calculation phase. Updating willautomatically present the calculation window.

Second calculation phase (apply vertical load)

Click on the Next button . This will introduce a new calculation phase and present thecorresponding tabsheets for the second calculation stage. Enter a suitable name in theNumber/ ID box (e.g. apply vertical load).

Select the Parameters tabsheet. On this tabsheet accept the selection Staged constructionin the Loading input combo box. Click on the Define button. This will open the windowpresenting the problem in Staged construction mode.

Click on the point forces in the middle of the footing, a Select items window comes up.Select the Point load - Load System A to activate point load A and press the Changebutton to change the load value. Change the y-value to -50 kN/m and press the Okbutton.


53


Figure 14: Parameters tabsheet of the first calculation phase

The point load A is now active (blue) and has a load value of 50 kN/m.

Press Update.

Figure 15: Select items window

Third calculation phase (add horizontal load)

Click on the Next button to add another phase. This will present the tabsheets forthe third calculation stage. Enter a suitable name in the Number/ID box (e.g. applyhorizontal load).

Select the second tabsheet called Parameters. On this sheet accept the selectionStaged construction in the Loading input combo box. Click on the Define button.

Click on the point forces in the middle of the footing, select the Point load - load system Bto activate point load B and press the Change button to change the load value. Changethe load x-value to 20 kN/m2 and press the Ok button.

Press the Ok button to closed the Select items window.

Press Update.


54


Fourth calculation phase (vertical load to failure)

Click on the Next button . This will present the tabsheets for the fourth calculation stage.Enter a suitable name in the Number/ID box (e.g. vertical load failure).

Directly below the Number/ID box select from the Start from phase dropdown list thesecond calculation phase. By selecting this the 4th phase will be a continuation of the2nd phase, hence we will continue to apply the vertical load without having the additionalhorizontal load that was applied in phase 3.

Select the second tabsheet called Parameters. On this sheet choose the selection Totalmultipliers in the Loading input group box. Select the third tabsheet called Multipliers byeither clicking on the Define button or directly selecting the tabsheet.

Enter a MloadA of 10. In this way the working force is increased to a maximum load of10 x 50 = 500 kN/m.

In PLAXIS two methods exist to increase an active load. The magnitude of theactivated load is the input load multiplied by the total load multiplier. Hence, inthis excersise MloadA x (input load of point load A) = Active load AThe value of the input load A can be changed using Staged construction asLoading input while using Total multipliers as Loading input may be used tochange the load multiplier.

Define load displacement points

After the calculation it is possible to create load-displacement curves. These can be usedto inspect the behaviour in a node during the calculation steps. In order to create load-displacement curves it is first necessary to indicate for which node(s) the displacementsshould be traced.

Click on the Select points for curves button in the toolbar. This will result in a plot ofthe mesh, showing all generated nodes. Click on the node, located in the centre directlyunderneath the footing. For a correct selection of this node it may be necessary to usethe zoom option . After selection of the node it will be indicated as point A. Press

the Update button to proceed to calculations.

Start the calculation

After definition of the last calculation phase, the calculation process is started by clicking theCalculation button . This will start the calculation. During the calculation a calculationwindow appears showing the status and some parameters of the current calculation phase.


55


INSPECT OUTPUT

After each successful execution of a calculation phase PLAXIS will indicate the phase witha green check mark ( ). This indicates a successful calculation phase. If during executioneither failure or an error occurs, PLAXIS marks the stage with a red cross ( ).

Figure 16: Calculation window with all phases calculated

While phase 3 is highlighted, press the View calculation results button that will startthe output program, showing the deformed mesh for the situation with both horizontaland vertical load applied, as presented in figure 17.

Figure 17: Deformed mesh at the end of phase 3


56


Check the various types of output, such as the deformed mesh, displacement contours,effective (principal) stresses etc. These can be found from the Deformations andStresses menus.

Still in the Output program, select from the dropdown list at the right of the toolbar theoutput step belonging to phase 4.

From the Displacements menu in the Output program now select Incrementaldisplacements and then the option |u|. Display the incremental displacements ascontours or shadings. The plot clearly shows a failure mechanism (see Figure 18).

Figure 18: Shadings of displacement increments after phase 4

Load displacement curves

In the Output program, select the Curves manager from the Tools menu. The Curvesmanager has 2 tabsheets, one for the curves defined in this project (currently none) andone for the points selected to make load-displacement curves (currently 1 node that waspre-selected, that is before the calculation).

In the Curves manager select the button New to define a new curve. Now the Curvegeneration window opens.

On the x-axis we want to plot the settlement of our chosen point in the middle of thefooting. In the x-axis box choose point A from the dropdown list and then below inDeformations and then Total displacements choose |u|.

On the y-axis we want to plot the force applied on the footing, which is a global valuenot connected to a specific node or stress point. In y-axis box choose Project from thedropdown list to indicate we want to plot a global value, and then in Multipliers chooseMLoadA.


57


Figure 19 shows the Curve generation window after applying the steps mentioned.

Press OK to show the resulting curve. See also figure 20.

Figure 19: Curves generation window

Figure 20: Load displacement curve for the footing

The input value of point load A is 50 kN/m and the load multiplier MloadA reaches approximately4.6. Therefore the failure load is equal to 50 kN/m x 4.6 = 230 kN/m. You can inspect the loadmultiplier by moving the mouse cursor over the plotted line. A tooltip box will show up with thedata of the current location.


58


RESULTS DRAINED BEHAVIOUR

In addition to the mesh used in this exercise calculations were performed using a very coarsemesh with a local refinement at the bottom of the footing and a very fine mesh. Fine mesheswill normally give more accurate results than coarse meshes. In stead of refining the wholemesh, it is generally better to refine the most important parts of the mesh, in order to reducecomputing time. Here we see that the differences are small (when considering 15-nodedelements), which means that we are close to the exact solution. The accuracy of the 15-noded element is superior to the 6-noded element, especially for the calculation of failureloads.

Hint: In plane strain calculations, but even more significant in axi-symmetriccalculations, for failure loads, the use of 15-noded elements is recommended.The 6-noded elements are known to overestimate the failure load, but are okfor deformations at serviceability states.

The results of fine/coarse and 6-noded/15-noded analyses are given below.

Table 2: Results for the maximum load reached on a strip footing on the drained sub-soil fordifferent 2D and 3D meshes

Mesh size Elementtype

Nr. ofelements

Max.load

Failureload

[kN/m] [kN/m2]very coarse mesh with local refinementsunder footing

6-noded 79 281 146

coarse mesh 6-noded 121 270 141very fine mesh 6-noded 1090 229 121very coarse mesh with local refinementsunder footing

15-noded 79 236 124

coarse mesh 15-noded 121 248 130very fine mesh 15-noded 1090 220 116Analytical solutions of:- Vesic- Brinch Hansen- Meyerhof

1179897

In this table the failure load has been calculated as:

QuB

= MaximumforceB

+ concrete d = Maximumforce2 + 6

From the above results it is clear that fine FE meshes give more accurate results. On the otherhand the performance of the 15-noded elements is superior over the performance of the lowerorder 6-noded elements. Needless to say that computation times are also influenced by thenumber and type of elements.


59


ADDITIONAL EXERCISE:

UNDRAINED FOOTING


60



61


INTRODUCTION

When saturated soils are loaded rapidly, the soil body will behave in an undrained manner, i.e.excess pore pressures are being generated. In this exercise the special PLAXIS feature forthe treatment of undrained soils is demonstrated.

SCHEME OF OPERATIONS

In PLAXIS, one generally enters effective soil properties and this is retained in an undrainedanalysis. In order to make the behaviour undrained one has to select undrained as the Typeof drainage. Please note that this is a special PLAXIS option as most other FE-codes requirethe input of undrained parameters e.g. Eu and u.

Aims

The understanding and application of undrained soil behaviour

How to deal with excess pore pressures.

A) Geometry input

Use previous input file

Save as new data file

Change material properties, undrained behaviour for clay

Mesh generation, global mesh refinement

B) Calculations

Re-run existing calculation phases

Construct footing

Apply vertical force

Apply horizontal force

C) Inspect output

Inspect excess pore pressures


62


GEOMETRY INPUT

Use previous input file

Start PLAXIS by clicking on the icon of the Input program.

Select the existing project file from the last exercise (drained footing).

From the File menu select Save As and save the existing project under a new file name(e.g. exercise 1b)

Change material properties

Change material properties by selecting the item Soils & Interfaces from the Materials menuor click on the Material sets button . Select the clay from the Material sets tree view and clickon the Edit button. On the first tab sheet, General, change the Drainage type to "UndrainedA" and close the data set.

Figure 21: Set drainage type to "Undrained A"

Mesh generation

The mesh generator in PLAXIS allows for several degrees of refinement. In this examplewe use the Refine global option from the Mesh menu, which will re-generate the mesh,resulting in an increased number of finite elements to be distributed along the geometry lines.Notice the message that appears about staged being reconstructed: the program will take intoaccount the newly generated mesh for the previously generated initial conditions and stagedconstruction phases. From the output window, in which the mesh is shown, press the continue

button to return to the Input program.Hint: After generation of a finer mesh, the geometry may be refined until a

satisfactory result appears. Besides the option Refine global several othermethods of refinement can be used.


63


Hint: After re-generation of the finite element mesh new nodes and stress pointsexists. Therefore PLAXIS has to regenerate pore water pressures and initialstresses. This is done automatically in the background when regenerating themesh. Also, the new mesh is taken into account for any change to calculationphases with the exception of ground water flow analysis.

After generating the mesh one can now continue to the calculation program. Click on theCaculations button to proceed to the calculations program. Click yes to save the data.

CALCULATIONS

Re-run existing calculation list

The calculation list from example 1 appears, as indicated below. All phases are indicatedby (blue arrows). After mesh (re)generation, staged construction settings remain and phaseinformation has been rewritten automatically for the newly generated mesh. However, this isnot the case for points for load displacement curves due to the new numbering of the meshnodes.

Click on the Select points for curves button in the toolbar. Reselect the node locatedin the centre directly underneath

Click on the Calculate button to recalculate the analysis. Due to undrained behaviourof the soil there will be failure in the 3rd and 4th calculation phase.

INSPECT OUTPUT

As mentioned in the introduction of this example, the compressibility of water is taken intoaccount by assigning undrained behaviour to the clay layer. This results normally, afterloading, in excess pore pressures. The excess pore pressures may be viewed in the outputwindow by selecting:

Select in the calculation program the phase for which you would like to see output results.

Start the output program from the calculation program by clicking the View output button .

Select from the Stresses menu the option Pore pressures and then pexcess, this results inFigure 22 .

The excess pore pressures may be viewed as stress crosses ( ), contour lines ( ),

shadings ( ) or as tabulated output ( ). If, in general, stresses are tensile stressesthe principal directions are drawn with arrow points. It can be seen that after phase 3 on theleft side of the footing there are excess pore tensions due to the horizontal movement of thefooting. The total pore pressures are visualised using the option of active pore pressures.These are the sum of the steady state pore pressures as generated from the phreatic leveland the excess pore pressures as generated from undrained loading.


64


Figure 22: Excess pore pressures at the end of the 3rd phase

Select from the Stresses menu the option Pore pressures and then pactive. The resultsare given in Figure 23.

From the load displacement curve it can be seen that the failure load in the last phase isconsiderably lower for this undrained case compared to the drained situation, as expected.For the undrained case the failure load is approx. 70 kPa.


65


Figure 23: Active pore pressures at the end of phase 3


66



67


APPENDIX A: BEARING CAPACITY CALCULATION

Given the formula for bearing capacity of a strip footing:

QfB

= c Nc + 12B NNq = e

pi tan tan2(45 + 12)

Nc = (Nq 1) cot

N =

2(Nq + 1) tan

(V esic)

1.5(Nq 1) tan (BrinchHansen)(Nq 1) tan(1.4) (Meyerhof)

Filling in given soil data:

Nq = epi tan(20) tan2(55) = 6.4

Nc = (6.4 1) cot(20) = 14.84

N =

2(6.4 + 1) tan(20) = 5.39 (V esic)

1.5(6.4 1) tan(20) = 2.95 (BrinchHansen)(6.4 1) tan(28) = 2.97 (Meyerhof)

The effective weight of the soil:

= w 10 kN/m3 = 18 10 = 8 kN/m3For a strip foundation this gives:

QfB

= c Nc + 12B N =

5 14.83 + 1

2 8 2 5.39 117kN/m2 (V esic)

5 14.83 + 12 8 2 2.95 98kN/m2 (BrinchHansen)

5 14.83 + 12 8 2 2.87 97kN/m2 (Meyerhof)

L =

Qf

B

I

II

III


68

1 / 27Non-linear calculations

Non-linear calculationsin PLAXIS


Content Learning objectives Introduction Multipliers Iteration process Plastic points Recommendations

69


Learningobjectives To recognize the items in the calculation progress window

To be able to evaluate the progress of a calculation

To use the calculation control parameters appropriately

To understand and explain the calculation procedure


Introduction

Load multipliers

70


Introduction

Load multipliers

Miscellaneous parms


Introduction

Load multipliers

Miscellaneous parms

Load-displ. curve

71


Introduction

Load multipliers

Miscellaneous parms

Load-displ. curve

Iteration process


Introduction

Load multipliers

Miscellaneous parms

Load-displ. curve

Iteration process

Plastic points

72


LoadmultipliersApplied load = Load multiplier x Input load

Defaults: Load multiplier = 1 Input load = 1 unit

Loading input: Staged construction: Change Input load Total multipliers: Change Load multiplier (M) Incremental multipliers: Change Load multiplier (M)

Total multiplier (phase) = Sum of incremental multipliers (step)


LoadmultipliersMdispX : Tot. mult. prescribed x-displacementsMdispY : Tot. mult. prescribed y-displacementsMloadA : Tot. mult. loads system AMloadB : Tot. mult. loads system BMweight : Tot. mult. soil & structural weightsMaccel : Tot. mult. pseudo-static accelerationMsf : Tot. mult. Phi-c reduction processMstage : Tot. mult. staged-construction process

73


Loadmultipliers Incrementalmultipliersinput


Loadmultipliers Totalmultipliersinput

74


MiscellaneousparametersPMax : Maximum (excess) pore pressure in the modelMarea : Relative part of the mesh area currently activeForce-X : Reaction force due to horizontal prescr. displ.Force-Y : Reaction force due to vertical prescribed displ.Stiffness : Current (relative) Stiffness ParameterTime : Elapsed model time (usually in days)Dynamic time : Elapsed model time for dynamics (s)


Loaddisplacementcurve Evaluation of calculation progress:

Multipliers Stiffness (CSP) Pmax Load-displacement curve Iterations Global error Plastic points

75


IterationprocessCalculation phase

Load steps (q) Equilibrium iterations

constitutive model q displacement strain stress reaction

Equilibrium?


IterationprocessLoad q

Settlement of Node A

Elastic stiffness (K)

Non-linearbehaviour

Load step q qin

qex

Unbalance

iterations

76


IterationprocessCurrent step Max. step Additional stepsIteration Max. iterations Maximum iterat.Unbalance Global error Tolerance Tolerated error

Control parameters


Iterationprocess Controlparameters

77


Iterationprocess Overrelaxation

u uo

q0+q

q0

A BOverrelaxationA

A

B

Standard setting: 1.2Absolute maximum: 2.0Low s (


Iterationprocess Arclengthcontrol

uuo

q0 + q

q0

P0 + P

P0K

1 P = IPe Pc I

uuo

q0 + q

q0

P0 + P

P0

K

1 P = IPe PcI = const.

Arc length control


Iterationprocess Desiredminimum/maximum

Standard setting: Des. min = 6Des. max = 15

Purpose: Automatic loadadvancement

79


Iterationprocess Desiredminimum/maximum

u

qScaling up

Scaling down

Scaling up

Converged within desired minimum number of iterations: Scaling up load step by a

factor 2 Not converged within desired

maximum number of iterations: Scaling down load step by a

factor 2


Plasticpoints

-3

Cap (HS, SS and SS-Creep model)-1

Mohr-Coulomb pointf


PlasticpointsLocal error criterion:

eq

c

||||||||

c

eqcErrorLocal

Constitutive stress c:Stress that follows from the constitute model (Mohr- Coulomb)

Equilibrium stress eq:Stress that is in equilibrium with the external load

Inaccurate point: Local error > Tolerated error

Convergence requirement:Inaccurate stress points 3 + (plastic soil points) /10Inaccurate interface points 3 + (plastic interface points) /10


Recommendations Use mostly defaults Monitor and evaluate calculation progress In case of bad convergence or numerical failure, check input Use output facilities to trace input errors In case input is right, consider control parameters Dont change control parameters without understanding

consequences! Dont increase tolerated error to speed up convergence!

81


82

Hardening Soil ModelWilliam Cheang

Notes by:

Professor Helmut Schweiger ( TU Graz)

Professor Pieter Vermeer

A/Professor Tan Siew Ann (NUS)

83

INTRODUCTION

INTRODUCTION

84

LINES OF EQUAL SHEAR STRAINS (Tatsuoka & Ishihara, 1974)

90

Do you need plasticity when unloading (back into the yield locus)?Yes..if the accumulation plastic volumetric strains are important in cyclically loaded soils..dynamic liquefaction related boundary value problems

91

Surface Heave in Initial Exc./Cantilever Wall

20kPa

3 m deep excavation with cantilever wall

FSP III sheetpile

3m

7m

5m

Dry sandy material

3 analyses with Mohr Coulomb, Hardening Soil & Hardening Soil-Small models using equivalent soil input parameters

Compare ground movements, wall displacements & wall stability

Soil Input Parameters for 3 Analyses

Analyses Material Model

c' ' (or ur)

Rinter (kN/m3) (kPa) (Deg) [-] [-] 1 MC 20 5 35 0.3 0.426 0.67 2 HS 20 5 35 0.2 0.426 0.67 3 HSsmall 20 5 35 0.2 0.426 0.67

Parameters for soil strength & initial stress state

Parameters for soil stiffness prior to failure Analyses Material

Model Eref

(or E50ref or Eoedref) Eurref pref m G0 0.7

(MPa) (MPa) (kPa) [-] (MPa) [-] 1 MC 30 - - - - - 2 HS 30 90 100 0.5 - - 3 HSsmall 30 90 100 0.5 150 210-5

For derivation of soil stiffness parameters,

- HS model from standard drained triaxial compression tests

- HSsmall model from small-strain triaxial tests or field tests (e.g. downhole / crosshole seismic survey)

112

Pre-failure Stress-strain Behaviour

1e-6 1e-5 1e-4 1e-3 1e-2 1e-1

1: Mohr Coulomb

2: Hardening Soil3:Hardening Soil + Small Strain Overlay

1: Linear elastic, perfectly plastic

2: Hyperbolic stress-strain curve (stiffness degradation for > 1E-4)3: Non-linear stiffness from very small strains (1E-6)

Predicted Surface Settlement Behind Wall

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.0060 5 10 15 20 25 30

Distance behind wall (m)

Set

tlem

ent (

m)

MCHSHSsmall

Heave

Settlement

MC predicts unrealistic surface heave 4 mm

HS & HSsmall predict max. surface settlement 9 mm

113

Predicted Heave at Exc. Level in Cofferdam

-0.005

0.000

0.005

0.010

0.015

0.020

0.025-5 -4 -3 -2 -1 0 1 2 3

Distance in front of wall (m)H

eave

(m)

MCHSHSsmall

MC predicts 20 mm heave at cofferdam centreline

HS & HSsmall predict 11 mm & 8 mm respectively

Wall

Predicted Wall Resultant Displacement

Ux=6mmUx=11mm Ux=10mm

HS HSsmallMC

Ux: wall horizontal displacement

114

Predicted Stability of Wall3

2.5

2

1.5

FOS=2.8

FOS=2.8

FOS=2.8

MC

HS

HSsmall

Rotation mechanism with FOS 2.8

Phi-c' reduction for predicting FOS

FSP III sheetpile properties:

EI=34440 kNm2/m; EA=3.92106kN/mMp=369 kNm/m; Np=3575 kN/m

3

2.5

2

1.5

3

2.5

2

1.5

Summary of Predictions

MC predicts incorrect surface heave behind wall

- related to soil stiffness (E) prior to failure different ways of modelling E in 3 constitutive models

Stability of wall has FOS = 2.8 for 3 analyses

- related to soil shear strength all 3 constitutive models use Mohr Coulomb failure criterion with c'=5 kPa & '=35

Analyses Surface settlement behind wall

Heave at excavation level

Wall horizontal displacement

FOS for wall stability

MC Heave 4 mm (not OK)

Heave 20 mm 6 mm 2.8

HS Settle 9 mm Heave 11 mm 11 mm 2.8 HSsmall Settle 9 mm Heave 8 mm 10 mm 2.8

115

Variation of Soil Stiffness in ExcavationA. Soil stiffness is not constant and varies with

1. stress-level. Higher stress, higher stiffness

2. strain-level. Higher strain (or displacement), lower stiffness

3. stress-path (recent soil stress history). Rotation of stress path, higher soil stiffness

4. anisotropy, destructuration

B. During excavation, soil elements at different locations experience different changes in

1. stress,

2. strain

3. stress-path direction

Soil Stress Paths Near ExcavationGCO No.1/90

A: unloading compression;

B: unloading extension

Rotation of stress paths at A & B

116

Soil Stress Paths Near Excavation

A: unloading compression

B: unloading extension-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50 60s' (kPa)

t (kP

a)

AB

A

B

3m

7m

20kPa

20kPa

20kPa

Exc.

Exc.K0

K0

Failure line

Failure line

5m

A

B

Rotation of stress path at A, A 90 w.r.t. K0 directionRotation of stress path at B, B 160 w.r.t. K0 direction

Stress Path Dependent Soil Stiffness

0.1

Shear modulus, 3G (MPa)

Shear strain (%)10.01

=0, no change in stress path direction=180, full reversal of stress path direction

t

s'K0

=0

=90=180

Stress path rotation,

Atkinson et al. (1990)

Triaxial tests on London Clay

-1 -0.1 -0.01

117

Stress Path Dependent CDG Stiffness

Wang & Ng (2005)

At s 0.01%, shear stiffness in extension 60% higher than in compression

Extension

Compression

Extension

Compress

=90

Stress-level Test series

Why MC Predicts Incorrect Surface Heave? MC models a constant soil stiffness prior to failure not realistic

In reality, stiffness of soil elements near excavation varies according to

1. stress-level

2. strain-level

3. direction of stress-path

Realistic prediction of wall deflections & ground settlements in all excavation stages requires a constitutive model that considersabove factors, e.g. HS & HSsmall models

HS & HSsmall consider the interplay between factors (1), (2) & (3) in determining the operational soil stiffness (E), i.e. E is changing during excavation

118

APPENDIX

73

Hardening Soil (HS)

Characteristics:

1. Stress-dependent stiffness behaviour according to a power law

2. Hyperbolic Stress-strain relationship 3. Deviatoric hardening4. Volumetric hardening5. Elastic unloading / reloading6. Failure behaviour according to the Mohr-Coulomb criterion7. Small-strain stiffness (HS-small model only)

119

Hyperbolic stress-strain relationship in (tri)axial loading:

E0

Eur

qult

q

1

Rf qult

E0 = initial stiffnessqult = asymptotic value of q (related to strength)Rf = failure ratio (standard value 0.9)

ultqEq

//1 101

ultqqEq/1

/ 01

(Duncan-Chang model)

1.Hardening Soil model

sin1sin'2cos2 3

cqR ultf

m

refref

pEE

300 '

m

refrefurur p

EE

3'

pref = 100 kPa (1 bar)

Unloading / reloading

Hyperbolic stress-strain relationship in (tri)axial loading:

2.Hardening Soil model

120

Elastoplastic formulation of hyperbolic q-1 relationship:p

fric ff *

'cos'sin)()( 13211321 cf f (MC failure)

ura Eq

qqq

Ef 2

1150

* m

refref

pccEE

'cot''cot 3

5050 m

refrefurur pc

cEE

'cot''cot 3

Yield function: (non-associated)

3.Shear hardening in the HS model

pvpp 12

pfric ff *

'cos'sin)()( 13211321 cf f

ura Eq

qqq

Ef 2

1150

* m

refref

pccEE

'sin'cos'sin''cos 3

5050 m

refrefurur pc

cEE

'sin'cos'sin''cos 3

Elastoplastic formulation of hyperbolic q-1 relationship:

(MC failure)

Yield function: (non-associated)


pvpp 12

121

p

q MC failure line

1p,fric2p,fric

3p,fric


Elastoplastic formulation of hyperbolic q-1 relationship:

m

q MC failure lineElastic

plastic

Flow rule:m

fricpfricpv dd sin,,

cvm

cvmm

sinsin1sinsinsin

with:

'cot2''''sin

31

31

cm

sin'sin1sin'sinsin

cv

p

q MC failure line

1p,fric2p,fric

3p,fric


m

122

sin'sin1sin'sinsin

cv

Flow rule:m

fricpfricpv dd sin,,

cvm

cvmm

sinsin1sinsinsin

with:

p

q MC failure line


cvm>0

m

9.Compaction and Shear hardening in the HS model

Cone

Cap

10.Compaction and Shear hardening in the HS model-Summary

Relevance of Compaction hardening:

Plastic compaction in primary loading Distinction between primary loading and unloading/reloading

Relevance of Shear Hardening:

Decreasing stiffness (increasing plastic shear strains) in deviatoric stress paths (principal stress differences, shearing)

124

Strain(path)-dependent elastic overlay model:

11.Small-strain stiffness in the HS model (HSsmall)

7.0

0

/385.01 G

Gs

GurG starts again at G0after full strain reversal

urt GGG 27.0/385.010


G0Gt

Gs

+c

G0

-c

G0

Cyclic loading leads to Hysteresis

Energy dissipation Damping

125

GsGt

0.7G0

Gur


Relevance of small-strain stiffness:

Very stiff behaviour at very small strains (vibrations) Reduction of stiffness with increasing strain; restart after load reversal Hysteresis in cyclic loading:

Energy dissipation Damping

Also relevant for applications like: Excavations (settlement trough behind retaining wall) Tunnels (settlement trough above tunnel)


126

Parameters:E50ref Secant stiffness from triaxial test at reference pressureEoedref Tangent stiffness from oedometer test at prefEurref Reference stiffness in unloading / reloadingG0ref Reference shear stiffness at small strains (HSsmall only)0.7 Shear strain at which G has reduced to 70% (HSsmall only)m Rate of stress dependency in stiffness behaviourpref Reference pressure (100 kPa)ur Poissons ratio in unloading / reloadingc Cohesion Friction angle Dilatancy angleRf Failure ratio qf /qa like in Duncan-Chang model (0.9)K0nc Stress ratio xx/yy in 1D primary compression

Plaxis Mum Plaxisbai Course 2012

Documents

Transcript of Plaxis Mum Plaxisbai Course 2012