Modelling in Geotechnics

Modelling in Geotechnics - Script

Part 1

Prof. Sarah Springman

In cooperation with Dr. Jan Laue & Dr. Jitendra Sharma

http://igtcal.ethz.ch/mig

ETH Zürich Modelling in GeotechnicsInstitute of Geotechnical Engineering

Page - 1

1 Design ...................................................................................................... 11.1 Modelling ............................................................................................................. 4

1.1.1 Numerical modelling .......................................................................................51.1.2 Physical modelling..........................................................................................61.1.3 Validation and calibration of models .............................................................12

1.2 Full scale (FS) 1g testing...................................................................................131.3 Geotechnical centrifuge modelling .................................................................... 15

1.3.1 History of geotechnical centrifuges...............................................................151.3.2 Types of centrifuge ....................................................................................... 151.3.3 Principles of modelling in the centrifuge....................................................... 161.3.4 Scaling effects ..............................................................................................171.3.5 Verification of models ...................................................................................18

1.4 References ........................................................................................................19

2 Principles of numerical modelling............................................................. 12.1 Why model numerically? .....................................................................................12.2 Validation of the finite element analysis (bench marking).................................... 3

2.2.1 El-Hamalawi (1997): mesh for a strip footing on clay .....................................42.2.2 Bransby (1995): mesh for lateral pressure on pile in clay ..............................5

2.3 Prediction............................................................................................................. 72.4 Styles of numerical analysis using a computer.................................................... 72.5 Idealisation for numerical modelling as before for physical modelling.................8

2.5.1 Geometry........................................................................................................82.5.2 Mesh design ...................................................................................................92.5.3 Structure ......................................................................................................... 92.5.4 Loading and construction effects.................................................................. 102.5.5 Ellis (1997): Piled full-height abutment: 3D problem as 2D.... ......................142.5.6 Soil ...............................................................................................................15

2.6 References ........................................................................................................17

3 Finite Element Method (FEM) in Geotechnical Engineering .................... 13.1 Introduction..........................................................................................................13.2 Numerical methods used in geotechnical engineering ........................................13.3 What is FEM? ......................................................................................................2

3.3.1 Historical Background.....................................................................................33.3.2 The fundamental steps of the FEM ................................................................33.3.3 Approximation of the Circumference of a Circle ............................................. 3

3.4 Basic formulation of the FEM ..............................................................................53.4.1 Interconnected elastic springs ........................................................................63.4.2 A plane truss element.....................................................................................83.4.3 A constant strain triangular finite element .................................................... 10

3.5 Approximations, accuracy and convergence in the FEM .................................. 133.6 Geotechnical finite element analysis .................................................................15

3.6.1 Plane strain and axisymmetric problems......................................................163.6.2 Different types of finite elements .................................................................. 17

3.7 Techniques for modelling non-linear stress-strain response .............................203.7.1 Tangential stiffness approach with carry over of unbalanced load ...............213.7.2 Modified Newton-Raphson method ..............................................................21

3.8 Techniques for modelling excavation and construction .....................................223.8.1 Excavation ....................................................................................................223.8.2 Construction .................................................................................................24

3.9 Advantages and drawbacks of the FEM............................................................253.9.1 Advantages...................................................................................................253.9.2 Drawbacks....................................................................................................25

3.10 Some popular commercial FEM programs ........................................................253.10.1ABAQUS ......................................................................................................25


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3.10.2SAGE CRISP ...............................................................................................263.10.3PLAXIS......................................................................................................... 273.10.4ZSOIL ........................................................................................................... 27

3.11 Guidelines for the use of FEM in geotechnical engineering ..............................283.12 Concluding remarks........................................................................................... 303.13 References ........................................................................................................30

4 Scaling laws and applications for centrifuge modelling ............................ 14.1 Introduction..........................................................................................................1

4.1.1 Scaling laws ...................................................................................................14.1.2 Scaling of time................................................................................................1

4.2 Scale effects ........................................................................................................24.2.1 Stress distribution in centrifuge model: Depth ................................................34.2.2 Stress distribution in a centrifuge model......................................................... 54.2.3 Particle size effects........................................................................................64.2.4 Coriolis acceleration .......................................................................................64.2.5 Boundary effects............................................................................................. 7

4.3 Scaling under earthquake conditions ................................................................ 10

5 Practical considerations: mechanical ....................................................... 15.1 Beam Centrifuges................................................................................................1

5.1.1 Capacity..........................................................................................................15.1.2 Swing platform, package and liner .................................................................2

5.2 Drum Centrifuges ................................................................................................45.2.1 Capacity..........................................................................................................4

5.3 Site investigation devices (penetrometers, vane)................................................85.3.1 Vane: ..............................................................................................................85.3.2 Penetrometer:.................................................................................................85.3.3 Cylindrical T-Bar: ............................................................................................8

5.4 Post-test investigation devices ............................................................................95.4.1 Photographic: camera & flash ........................................................................95.4.2 Digital Images and PIV analysis .....................................................................95.4.3 X-ray..............................................................................................................9

6 Practical considerations: geotechnical ..................................................... 16.1 Introduction..........................................................................................................16.2 Design of soil model: real or laboratory ...............................................................26.3 Kaolin as a model soil..........................................................................................76.4 Preparation of soil samples in the DRUM centrifuge......................................... 21

7 In-situ testing, instrumentation, data acquisition....................................... 17.1 Measurement of soil properties ........................................................................... 1

7.1.1 Vane shear testing to determine su at discrete locations ..............................1

7.1.2 Cone penetration testing (CPT): to determine a profile of su .......................... 37.1.3 T-Bar penetration testing: to determine su .................................................... 8

7.2 Measurement of displacement ..........................................................................107.2.1 Spotchasing.................................................................................................. 107.2.2 Digital Images and PIV analysis ...................................................................107.2.3 Displacement measurements ....................................................................... 117.2.4 Radiography ................................................................................................. 117.2.5 Excavation ....................................................................................................12

7.3 Electronic/electrical instrumentation .................................................................. 127.3.1 Other considerations: ...................................................................................17


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7.4 Summary ........................................................................................................... 197.5 References ........................................................................................................20

8 Finite Difference Analysis using FLAC ..................................................... 18.1 Basics .................................................................................................................. 1

8.1.1 Specific to Geotechnics via FLAC .................................................................. 18.2 Finite Difference .................................................................................................. 38.3 Details of FLAC Program................................................................................... 11

8.3.1 Null model group ..........................................................................................168.3.2 Elastic model group......................................................................................168.3.3 Plastic model group......................................................................................17

8.4 Example analyses .............................................................................................188.5 References ........................................................................................................26


Modelling in Geotechnics

Introduction to Modelling



Design Page 1 - 1

1 Design• The main focus of modelling is to achieve optimal design, which is cost effective, safe

and also aesthetic.

Figure 1.1: Design Process


• frequent modelling is necessary in geotechnics to 'engineer' solutions (design)• modelling implies idealisation of the real life 'prototype'• understanding 'system' behaviour in response to perturbations (various loads) is crucial:

→ e.g. fundamental understanding of soil behaviour is required and should bemodelled effectively,

→ i.e. directly in physical models or through constitutive modelling and numericalanalysis.

We need to decide the WORST CASE SCENARIO in terms of the worst possible combina-tions of loads, soil properties, geometry, local environmental and construction effects and todesign for them as indicated above by using Limit States to examine the potential structuraldamage as well as any unserviceable deformations.

So what are the relevant Limit States?

Ultimate Limit State Serviceability Limit State

ULS controlled by SLS controlled by

Figure 1.2: Ultimate Limit State (ULS) and Serviceability Limit State (SLS)

Structure

Codes

Ground

Load model Rock/soil layering/properties

Design

Dimensioning

Serviceability &Failure(ULS) deformations (SLS)

IdealisationPhysical model

Calculation model

Interaction


Design Page 1 - 2

Failure…or ULS

• avoiding 'failure' of engineering structures is central to the design process• includes social responsibility• together with a legal requirement• manifest in design codes via extremes of simple through to complex methods of analysis

(dependent on relative risk)• necessitating engineering judgement

e.g. Heathrow Airport: tunnel collapse

The parties found to have been responsible (after lengthy court proceedings) have beenfined recently, in the case of the main contractor, 1.2 million pounds or 3 million SFr.

• Human error is at the core of most engineering failures, due to

→ conceptual/modelling errors,

→ inadequate components,

→ poorly considered design/construction changes.

• often these combine to form a critical chain of events leading to failure.

• Failure October 1994• half face, 3 stage excavation• sprayed concrete lining• construction method not properly applied• 80m of tunnel and concourse collapsed (fortu-

nately without loss of life)• subsequently stabilised by structural and light-

weight foamed concrete• creating major delay and huge costs (tunnel

came into operation in 1998, nearly 3 yearslate)

Figure 1.3: Heathrow Airport: tunnel collapse

Figure 1.4: Failure of the Teton Dam, USA (1976)


Design Page 1 - 3

• e.g. failure of 90m high Teton Dam (Fig. 1.4 - 1.6): (1976) poor design of central core interms of material (silt), shape of core trench, and an irregular stepped longitudinalabutment cross section which promoted hydraulic fracture in the core.

But, we learn more from our failures than from our successes…

Factors of safety/reliability against failure may be defined as

→ available strength/required strength or

→ available resistance/required resistance

whereas at the moment of failure, we KNOW with certainty that

→ the available strength/resistance = the required strength/ resistance

and we can often establish the failure mechanism... which is extremely helpful for subse-quent back analysis.

Figure 1.5: Teton Dam during failure

Figure 1.6: Teton Dam after failure


Design Page 1 - 4

Development of engineering judgement is crucial and often underestimated and observingfailures of models provides a fast, comprehensive track to developing engineeringjudgement and experience,

... as long as the model is appropriate!

1.1 ModellingSo what models can we use?

e.g. simple small scale 1g physical model….

How do we model?

primarily through….

1. Numerical modelling

2. Physical modelling

Stochastic or statistical methods are also valid forms of modelling. These are, at present,less often employed in geotechnics, other than for hazard assessment of earthquakeengineering, and are not considered further in this course. Simple forms of these methodswill, however, be used in the future Eurocodes, in order to be able to reduce the relevantpartial factors.

Low-mediumrisk, quick andcheap

Medium-highrisk, more time

physical

• simple, theoretical or empirical• complex, iterative/computational

(relatively low cost)

• full scale (high cost)• small scale, 1 gravity• small scale, enhanced gravity

(medium cost)

numericallow

“acc

urac

y”

high

Figure 1.7: Types of models

• Domenico Fontana (in 1585)• 300m move: 330 ton, 30m Vatican obelisk• 1/50th model to demonstrate procedure• but geotechnical modelling - i.e. with soil - is more complex


Design Page 1 - 5

1.1.1 Numerical modelling

• specific fundamental (classical plasticity) or empirical models• finite element• finite difference• boundary element etc... (will be ignored here)

Type Sketch of

Model/Prototype

Key

assumptions

Advantages Disadvantages

specificfundamental

(classical plasticity)

Soil continuous

homogeneous, rigid perfectly

plastic

• exact solution from classical plasticity (when upper and lower bound agree)

• fundamental basis in (soil) mechanics

• no strain before yielding

• significant idealisation required

• uniform strength in failure zone

empirical model

Calculation method based on past tests - small/full scale measurement - and/or lab tests

and approximate constitutive

models

• quick and cheap (back of envelope)

• field validation of frequently used construction methods

• past data may not suit current design conditions

• basic assumptions may differ

• usually does not account for fundamental behaviour

finite element (FEM)

Soil continuawith partial differential

equations to describe physical

phenomenaand extensive

integration method with solution of

stiffness matrix

• general analytic tool• divide geometry into

elements• adaptive methods

can refine mesh and reduce errors

• spatial variation of material properties

• more descriptive constitutive models

• computing power up• ideal for

serviceabilty analysis

• approx. solution+ engineering judgement versus apparently complex analysis

• strain must vary according to type of element selected

• element concentration required for area of high strain variation

• numerical instability at large strain

finite difference

(FDM)

Soil continuaiterative finite

difference formulation i.e. similar to FEM with reduced integration

• competitive with FEM when highly non linear (large strain)

• range of constitutive models/applications inc. user-specified

• not ideal for linear problems

• strict limitations on mesh patterns unless at expense of calculation efficiency

• rel. stiffness can cause instability?

Tab. 1.1: Numerical modelling

y

x

in

out?


Design Page 1 - 6

1.1.2 Physical modelling

• 1g, full scale testing• (+ field monitoring)• 1g, small scale laboratory testing• ng, small scale centrifuge modelling

• 1g small scale testing is not useful for exacting work, because the stresses are notcorrect and since soil behaviour is nonlinear, the modelling is unsatisfactory.

• all other methods should have correct order of magnitude of stresses but need to takecare over stress paths.

Type Sketch of Model/prototype

σv at A

[kPa]


1g,

full scale

100to

140

• stress correct• can control soil

conditions

• time to construct and for diffusion processes

• boundary effects• cost

Field

monitoring

100to

140

• ‘the real thing’• stress correct• soil, geometry,

boundaries realistic

• time (for diffusion)• cost• failure not OK• boundary/soil

conditions often not clear

1g,

small scale

1 • time (very quick)• cost (very cheap)• good preliminary test to

check equipment and testing principles

• stresses incorrect• potential for suction

and dilatancy to affect results

• boundary effects

100g,

1/100th

scale in centrifuge

100to

140

• stress correct• idealise to reveal key

mechanism of behaviour

• select soil and soil parameters

• design stress history• control loading system• time• cost• allowed to fail: observer

witnesses deformation and failure mechanism

• radial ‘g’ field (in a beam centrifuge)

• ng varies with depth• coriolis effect• size of particles,

instrumentation, site investigation devices

• stress path may be different

• construction method different?

• boundary effects-> so take care over idealisation

Tab. 1.2: Physical modelling

A

7m

A7m

A7cm

A7cm

1 m

100g

ω


Design Page 1 - 7

• time, cost and of course technical factors (e.g. scaling laws, model preparation) areimportant.

• parametric studies can be extremely useful in exposing mechanisms of behaviourrelevant first at SLS and later at ULS.

What concerns do we have about modelling 'effectively'?

i.e. As designers:

• is it comparable or relevant to the design?• will it help the design process?• will it reveal the key mechanisms of behaviour?• will it reveal the secondary, more complex, mechanisms of behaviour (sometimes

important)?

i.e. As researchers (in addition):

• can we produce a robust design method which reproduces all the important character-istics?

• is this method potentially usable by engineers working in industry?

Idealisation: i.e. model an 'ideal' prototype - not necessarily the exact field condition.

• range of critical modes of behaviour identified• factors affecting these studied in detail• intelligent simplifications to replicate key features which control the prototype behaviour

pattern.

Will the idealisation reproduce the full scale 'prototype' behaviour?

e.g. for the Severn bridge approach embankments


Design Page 1 - 8

Idealise any combination of:

• geometry• soil• structure• loading• construction effects.

Figure 1.8 contains an idealisationexercise to emphasise these points interms of centrifuge modelling on a bridgeabutment and approach embankment(e.g. Severn Bridge).

Figure 1.8: Severn Bridge


Design Page 1 - 9

The prototype problem

• embankment on soft clay• clay moves laterally further than the piles• therefore “passive” lateral thrust on piles

→ ?magnitude and ?distribution

• increased pile bending moments (BM) andlateral deformation at deck (δu < 25 mm)

→ ?magnitude and ?distribution

1st idealisation

INPUT

• vertical load (pressurised air bag)• row of single free-headed piles• soft clay (su ≅ 15 kPa) over stiff sand

OUTPUT

• lateral “passive” pressure p as f(q)• BM & δup

3rd idealisation

INPUT

• pile group with cap • stiffer clay (su ≅ 40 kPa) over stiff sand

OUTPUT for both rows of piles

• lateral “passive” pressure p as f(q)• BM & δup

also

• drag under pile cap

4th idealisation

INPUT

• construct embankment in stages inflight(slow or fast build)

• soft clay (su ≅ 20 kPa) over stiff sand(with or without wick drains in clay)

Single free-headed pileSingle free-headed pile

Vertical load, q

Soft

Stiff

p?

δup

Pile groupPile group

Soft

Stiff

p?

δup

Bransby PhD, 1995

Vertical load, q

Inflight construction of embankmentInflight construction of embankment

Ellis PhD, 1997

Supply hopper

1

Piled full-height bridge abutmentPiled full-height bridge abutment

δu

Embankment

Soft

Stiff


Design Page 1 - 10

It is often advisable to omit some detail to focus on the key conditions which will affect thebehaviour.

Typically this will be a function of the:

geometry / soil / structure / loading / construction effects

Geometry

• simplify

→ from three dimensions to two dimensions

→ soil strata & structure

→ location for load application.

Soil

• constitutive model (numerical)• laboratory soil or real soil (physical)

→ create or design soil stress history

→ specify soil strength (for clays) or relative density (for silts and sands)

→ ensure homogeneity of the prepared soil model (when this is required).

4th idealisation

INPUT

• construct embankment in stages inflight• soft clay (su ≅ 20 kPa) over stiff sand

also NOTE

• drag under embankment• drag under pile cap

4th idealisation

OUTPUT load on abutment

• active lateral pressure on wall• arching of embankment loading onto wall

base• lateral “passive” pressure p as f(q)• BM & δups

(Springman, 2001)


Ellis PhD, 1997

p?

δuparching

Embankment loading on clayEmbankment loading on clay

Ellis PhD, 1997

Supply hopper


Design Page 1 - 11

Structure

• model key aspects which will affect soil-structure interaction

→ use hollow aluminium piles / retaining wall to have same flexural stiffness (E I)(Young's modulus in MPa x 2nd Moment of Area in m4) as equivalent concretepiles / wall

→ e.g. 1st idealisation - single row of free-headed piles

→ 2nd idealisation - pile group (2 vertical piles x 3) with pile cap raised aboveground surface

→ 3rd idealisation - as for 2nd, but pile cap rests on ground surface

→ 4th idealisation - as for 3rd, but with retaining wall (and embankment built in-flight).

Loadings

• point loading or line loading • normal, uniformly distributed or average pressure i.e. modelling embankment by a

surcharge load, ignoring tractions acting at surface of soft layer - idealisations 1-3 above(and these tractions really do have an effect).

Construction effects

it is quite difficult to build/excavate in-flight - soil is heavy under ng and equipment must besmall, light but strong and manoeuvrable!

• pile installation

→ lateral loading - 1g installation is OK: stress in upper layers of soil (whichcontrol the lateral response) is not too badly affected

→ axial loading - installation must be at correct stress levels (i.e. in-flight) sincepile response / load capacity is affected significantly by lateral stressesgenerated during this process

→ bored piles are not easily modelled in the centrifuge but better modellednumerically.

• tunnel construction

→ excavate clay at 1g and replace with airtight rubber membrane which can bepressurised later

→ pour sand around a polystyrene tunnel (with or without a liner) at 1g anddissolve the polystyrene in-flight

→ jack Tunnel Boring Machine from side of box! This is complex in a mechanicalsense, but can be and has been done.

• building an embankment

→ at 1g and embankment 'grows' as 'gravity' force increases (not same stresspath)

→ pour embankment in-flight (better stress path) but no allowance forcompacting the embankment material

→ NOTE: sand is dry & is not the same material as the 'real' embankment.


Design Page 1 - 12

1.1.3 Validation and calibration of models

A model is of no use if it fails to represent the prototype behaviour. The modelling lawsshould therefore be appropriate. Selection of relevant non-dimensional groups, whichcontrol model behaviour, is often necessary. These should remain constant if the basicmodelling premises are to be correct.

An example concerning stability of a slope of height H, soil density ρ and undrained shearstrength su and a key non-dimensional group Hρg/su is given in table 1.3

Similitude of models, when defined by Hρg/su, is guaranteed for Models 2-4 but not forModel 1, which is therefore not appropriate. Since it is very difficult to adjust either thedensity (model 3) or the shear strength (model 2) without affecting the response of themodel in other ways (e.g. modelling deformations), the recommended method is toincrease the gravity field (model 4).

Even if the agreement between non-dimensional groups is satisfied, it is often difficult tojudge whether the model is valid or not. Ideally it would be possible to compare resultsagainst a full scale prototype however it is extremely rare to obtain sufficiently good data tobe able to do this.

Normally the physical or numerical data is checked against a known numerical or funda-mental solution for validation. Provided this test is passed then there is confidence in themodelling method, which may in turn be used to calibrate either soil properties or additionalmodelling methods (e.g. new numerical algorithms).

For example, a strip footing on a homogeneous clay layer with uniform undrained shearstrength with depth, su, can be shown using plasticity theory to fail at a load of (2 + π)su. Aphysical test set up or a numerical calculation (or computer algorithm or run) may bechecked against this to ensure that the model system is valid before further models areused to calibrate/back-calculate for other prototypes (El-Hamalawi, 1997).

Prototype Model 1 Model 2 Model 3 Model 4

Geom. scale 1:1 1:10 1:10 1:10 1:10

Acceleration 1g 1g 1g 1g 10g

H [m] 5 0.5 0.5 0.5 0.5

su [kPa] 15 15 1.5 15 15

ρ [t/m3] 1.65 1.65 1.65 16.5 1.65

g [m/s2] 10 10 10 10 100

Hρg/su 5.5 0.55 5.5 5.5 5.5

Tab. 1.3: Instabile Hänge und andere risikorelevante natürliche Prozesse, Monte Verità (after Bucher, 1996)


Design Page 1 - 13

1.2 Full scale (FS) 1g testingFull scale 1g tests are not usually carried out on an 85m high earth dam or a 100m deepquarry which is about to be landfilled. It is much more likely that elements or aspects of afull scale test will be modelled. E.g. a trial embankment may be built quickly to failure in theappropriate clay to test the construction and compaction procedures to be used for a claycored dam, and to measure in situ density, permeability and strength. This can be backanalysed numerically and the results applied to model the behaviour of the completeprototype structure.

Equally Hertweck (1998) and Brinkmann (1999, 2001) have both used the IGT 5.5m x 4m x3m Big Box to model an aspect of the behaviour of a clay barrier which they have alsoanalysed numerically using finite element analysis with the same boundary conditions as inthe Big Box. Subsequently they have also modelled the full scale behaviour of the prototypewith the field boundary conditions.

Below are some figures from the doctoral work of Michael Hertweck (1998).

Figure 1.9: ETH Big Box for 1g models


Design Page 1 - 14

Investigation of stress-deformation behaviour for two-phase manufactured diaphragm wallsfor encapsulation of waste materials has been carried out by Andreas Brinkmann (2001).Critical external loads are related to internal stress and strains in the wall. Element tests,finite element calculations and full scale tests have been carried out.

Figure 1.10: Dissertation Hertweck (1998)

Figure 1.11: Dissertation Brinkmann (2001)


Design Page 1 - 15

1.3 Geotechnical centrifuge modelling

1.3.1 History of geotechnical centrifuges

Centrifuge models have been recognised for more than 100 years as being capable ofcreating homologous points of stress and strain in both model and prototype. Bucky (1931)first used a centrifuge for mine roof stability investigations in Columbia USA, but the majorthrust of development came from the Soviet Union. This was largely stimulated by thescaling advantages applicable to the weapons industry, and large scale explosions inparticular. Pokrovsky made a major presentation concerning the use of a geotechnicalcentrifuge at the first International Conference on Soil Mechanics and FoundationEngineering at Harvard in 1936.

Since then, Professor Andrew Schofield FRS has been in the forefront of centrifuge devel-opments both at the University of Manchester Institute of Science and Technology and laterat Cambridge University (Schofield, 1980).

Further details are available in Chapter 1: Geotechnical Centrifuge Technology, Taylor(1995).

1.3.2 Types of centrifuge

BEAM

A beam is mounted on a central spindle and can rotate about this to allow models made inpackages located at each ends of the beam to be subjected to increased gravity. Usuallythese packages are fixed to a swinging platform so that they are hanging in the verticalplane initially and can swing up to lie in the horizontal plane as the centrifuge acceleration isincreased.

DRUM

A different style of centrifuge, in which a drum of diameter between 800mm and >2m (e.g.ETH Zürich) may be rotated respectively, between 90 and 650 r.p.m., to present a bed ofsoil between 0.1 up to 0.5 km wide by 1-3 km long in a gravity field of up to 500 g. Thisallows examination of soil behaviour for specific problems, which relate to shallow construc-tions (shallow foundations onshore and offshore, transport processes for environmentalgeotechnics, tunnelling, ice forces on structures, slope stability, etc.).

Centrifuge modelling: BEAM Centrifuge modelling: DRUM

g = 9.81 m/s2 g = 9.81 m/s2

ω

r

ngω

r

ng

Figure 1.12: Types of centrifuge

ng = ω2r - r..

ng = ω2r - r..


Design Page 1 - 16

1.3.3 Principles of modelling in the centrifuge

• Centrifuge testing accelerates the model to achieve full scale stress conditions

For non-linear materials in either small or full scale prototypes, correct stress-strain fieldsmust be replicated for subsequent meaningful interpretation.

PHYSICS

If a body is rotated about a spindle in a horizontal plane, then it is subject to a combinationof radial and tangential accelerations due to r and ω and the derivatives of these: dr/dt,d2r/dt2, dω/dt or

For static cases, there tends to be:

no radial acceleration dω/dt so rdω/dt disappears (tangential)

no change in radius dr/dt, d2r/dt2 so 2ωdr/dt and d2r/dt2 disappear (tangential andradial)

Example:

If the effective radius of the centrifuge is 4 m, and the gravity level is 100 g, then angularvelocity ω:

100 g = r ω2

ω = (100 g / 4)0.5

ω = 15.7 rad/sec = 15.7 * 60/(2*π) = 150 r.p.m.

A change in radius occurs during the construction of an embankment in flight: sand israined onto the fill, and the dr/dt term becomes important. This is described as the Corioliseffect and the particle trajectory follows a parabolic path. Deflector plates may be usedunderneath the sand pouring device to counter this effect.

Figure 1.13: Accelerations of a rotated body

ω

r ω2⋅ r··–

r

2 ω r·⋅ ⋅ r ω·⋅+

ng

r· r·· ω·, ,


Design Page 1 - 17

Optional explanation ... if a mass is rotated at a specific radius and angular velocity thenNewton's law of motion shows that the mass is pulled out of a straight line around the radialcurve causing a radial acceleration towards the centre. An associated inertial force mustapply, acting radially inwards.

Looking externally, the package must be accelerating inwards, but actually the massappears to be at rest in a radial sense, so relative to the package's frame of reference, theacceleration and body force act in the opposite direction - radially outwards. This meansthat the mass must be held by mechanical means strong enough to resist this outwardradial body force.

Newton first explained the concept of gravity - masses accelerated towards the centre ofthe earth in terms of a gravity force on each terrestrial mass. Relativity implies that thegravity force is identical to the inertial force - the small scale model will weigh more underangular velocity at ω than when the centrifuge is at rest.

1g = 9.81 m/s2. In the centrifuge, radial acceleration will be a factor of g, i.e. ng. So that ifthe model dimension is scaled down by a factor of n, i.e. 1/nth scale model, then thestresses will be equivalent.

Effectively, the gravity acts on the nuclei - the centre of mass of each atom - and the netpressure builds up with depth. This means that there is no gravity or inertial force at thesurface of a model, but this increases with depth.

1.3.4 Scaling effects

Parameter Unit Scale

(model/prototype)Acceleration m/s2 n

Linear dimension m 1/n

Stress kPa 1

Strain - 1

Density kg/m3 1

Mass or Volume kg or m3 1/n3

Unit weight N/m3 n

Force N 1/n2

Bending moment Nm 1/n3

Bending moment / unit width Nm/m 1/n2

Flexural stiffness/ unit width (EI/m) Nm2/m 1/n3

Time: diffusion s n2

Time: dynamic s n

Frequency 1/s n

Tab. 1.4:


Design Page 1 - 18

Similarity of stress (and strain) will be achieved in both modelm and prototypep, for asample in soil of the same density ρ and stiffness constructed at a scale of 1/n, located atan appropriate radius and rotated at an angular velocity to give a multiple of earth's gravity,ng at that radius so that the vertical stress at depth z equivalent to the radius is:

σv = ρm (ng)m (z/n)m = ρp gp zp

Scaling of time

As in fluid mechanics, it is not always possible to achieve correct scaling in all dimen-sionless groups, and so choices must be made.

In dynamics, where acceleration in m/s2 scales as n in the model, and the linear dimensionis modelled at 1/n prototype, then time is modelled n times faster in the centrifuge.

But the scaling factor for modelling time in terms of diffusion may be demonstrated to be:n2 faster in the centrifuge.

The non-dimensional time factor, Tv = f(time/depth2) = cvt/d2, becomes independent of

gravity level for a depth of sample reduced to 1/n of the original, if the model time is alsoreduced by 1/n2.

(1D Diffusion equation - saturated soil)

du/dt = cvd2u/dz2

where u is excess pore pressure and time t scales with length z2 provided cv m = cv p .

This offers a significant advantage because 27 years of prototype diffusion may bemodelled in 1 day using a centrifuge at 100 g, and is especially useful for environmentalproblems or heat loss by conduction where diffusion is the main transport mechanism.

However, in offshore foundations or earthquake problems, the pore pressures are createddynamically, with time scaling as: n times faster in the centrifuge and yet they decay in adiffusive process where time is modelled as: n2 faster in the centrifuge.

Solution: use pore fluid in the model with a viscosity of n times that of the prototype (andsame density) or reduce the value of permeability of the soil (Attention: this will cause achange in the properties).

1.3.5 Verification of models

If the prototype is at full scale under earth's gravity, then the model behaviour is scaledaccording to the value of n. If a 1/100th scale model at 100g predicts the same prototypebehaviour as a 1/200th scale model at 200g, then verification of models (often called‘modelling of models’) has been achieved and we can then use that predicted prototypebehaviour to check numerical analyses.


Design Page 1 - 19

1.4 References

1. El-Hamalawi, A., Adaptive refinement of finite element meshes for geotechnicalanalysis. PhD thesis, University of Cambridge, 1997.

2. Brinkmann, A. and Amann, P., Small and large scale tests for the determination of themechanical behaviour of a clay-cement stabilised slurry wall. Bauingenieur, Band 74,Heft 9, Sept. 1999,

3. Brinkmann, A., Untersuchungen zum mechanischen Verhalten von ton-zement-gebun-denem Dichtwandmaterial für Zweiphasen-Verfahren. PhD Thesis, Swiss FederalInstitute of Technology, ETH Zürich, 2001.

4. Hertweck, M., Untersuchung des Tragverhaltens von Steilwandbarrieren in Deponiebaumit grossmassstäblichen Modellversuchen. PhD Thesis, Swiss Federal Institute ofTechnology, ETH Zürich, 1998.

5. Springman, S. M., Soil structure interaction: idealisation, validation and calibration ofmodels. 1st Albert Caquot Conference, Paris, 2001.

6. Taylor, R. N., Geotechnical centrifuge technology. Geotechnical Engineering ResearchCentre, City University, London, 1995.

7. Schofield, A. N., Cambridge geotechnical centrifuge operations. 20th Rankine lecture,Géotechnique 30 , No.3, p. 227-268, 1980.

8. Bucky, P. B., Use of models for the study of mining problems. American Institution ofMining and Metallurgical Engineers, Tech. Pub. 425, p. 3-28, 1931.



Numerical Modelling



Principles of numerical modelling 2 - 1

2 Principles of numerical modelling

2.1 Why model numerically?This is inevitably part of any calculation and Code-based design methods, in which respon-sibility for safe design must be assured. This must be validated either by known experienceor another proven calculation or physical models.

A classic example of local experience applied in a foreign environment was the - at the time- surprising series of embankment failures which occurred in South East Asia whenextremely experienced and well known geotechnical engineers from Scandinavia designedthese embankments based on the undrained shear strength obtained from vane sheartests. What experience had NOT shown beforehand was that the plasticity index of the soilaffected the values of su obtained (LHA p.79, figure 6.17). The vane strength values of therelatively low plasticity Scandinavian clays required no correction factor but the strengths ofthe high plasticity S.E. Asian clays should have been reduced by 2/3rds.

The first lesson to learn about numerical modelling is that the results are only valid whenboth the input data and the calculation method (algorithm) are appropriate…..GIGO orGarbage In...Garbage Out.

The simplest forms of numerical modelling would be the 'back of the envelope' calculationsthat are carried out for a preliminary judgement on a particular engineering problem.

E.g. a relatively homogeneous clay deposit has an undrained shear strength su ~ 20 kPaand vertical load of 200 kN/m will be applied onto a strip footing / strip pile cap. Given thatthe width of the footing is limited for reasons of lack of space to 3m, will it be necessary touse piled foundations?

Figure 2.1: Understanding computer technology



Considering the Ultimate Limit State (ULS) at first and knowing that the maximum verticalload on a strip footing qmax is approximately 5su, qmax ~ 100 kPa. The load applied to thefooting = 200/3 = 67 kPa, so a global factor of safety would be 100/67 = 1.5 which iscertainly insufficient. We would also remember that if the load was inclined then this valuewould be reduced, and that even if the ULS conditions had been fulfilled that the Servicea-bility Limit State (SLS) should also be checked.

If the soil deposit is extremely variable (with uneven layering and fairly soft or sensitivecontents), and the structure to be built on it is extremely expensive (or indeed potentiallydangerous if failure occurred e.g. nuclear power station), then a far more extensivenumerical modelling process will be necessary. This may well entail more complexanalyses of continua using a computer to solve a series of equations based on a mesh withappropriate boundary conditions, a range of loading scenarios and a suitable constitutivemodel (e.g. elastic, elasto-plastic, critical state).

These are often called finite element or finite difference analyses and they differ only in themethod of solving the equations of equilibrium, compatibility and constitutive model (seetable on numerical modelling in chapter 1, table 1.1, page 5).

Even for these 'finite' models, there are ranges of complexity…..e.g.

• simple or complex meshes (e.g. with 2 elements for a 1/4 space (fig. 2.2 left) or anadaptive mesh for the whole sample and 229 or even 1791 elements depending on thelevel of strain in the soil(fig. 2.2 right)!)

• special purpose: calibration of soil parameters (back analysis of a specific event, e.g.Fig. 2.3) or prediction of behaviour (part of a design) or fundamental generic (investiga-tions into a specific class of problem)

• Class A prediction or validation of physical (centrifuge model) tests

→ e.g. primary focus on specific match to exact centrifuge model test, does thebehaviour agree (Fig. 2.3)?

→ parametric analyses are possible to match prototype more closely & to revealfurther trends

• numerical modelling is also quicker & cheaper than many forms of modelling, provided itis appropriate and the modellers are competent!

Figure 2.2: Range of mesh complexity for a triaxial sample

Undrained triaxial test on over-consolidated clay 229 elements 1791 elements



2.2 Validation of the finite element analysis (bench marking)Validation has also been mentioned in several formats already, in this and the first chapter.For any user to be able to accept the results of a computer analysis, some form ofvalidation is necessary. Sometimes this is called „bench marking“ carried out against aknown theoretical solution.

Access to an exact solution is one of the most approved modes of checking the validity of aparticular mesh set up and analysis. Both Bransby (1995) and El-Hamalawi (1997) haveshown that the mesh design influences the end result.

Mesh refinement entails either increasing the number of elements, or changing the densityof the elements according to the areas where either the most shear strain or the greatestpore pressure build up arises (e.g. adaptive meshing - see also on right pictures infigure 2.2). This means that the elements are smaller and hence the assumptions which arevalid for each finite element represent a smaller space and therefore

Figure 2.3: Calibration of soil parameters and match deformation response

Figure 2.4: Modelling “construction” processes

Centrifuge model test of stage-constructed embankment on soft clay after failure (Almeida 1984)

Finite element mesh for analysis of model embankment on soft clay

Computed and measured development of settlement at clay surface with time (Almeida 1984)

100mm

1g 2g

200mm

2m

20g h increases as time increases

h

Model “grows” as ng increases in centrifuge Layers added in prototype



→ greater variation is possible,

→ the model becomes closer to reality and

→ the solution becomes more exact.

2.2.1 El-Hamalawi (1997): mesh for a strip footing on clay

Comparing the case for a strip footing, loaded vertically in plane strain space q underundrained conditions on homogeneous isotropic rigid perfectly plastic soil with uniformshear strength su, the exact solution is (2+π)su (Prandtl, 1921, developed this based onclassical plasticity theory for metals). El-Hamalawi modelled this using finite elements andrepresented the soil by an elastic perfectly plastic constitutive model under drained condi-tions; uy is the settlement and b is the footing width.

answer

increasing mesh refinement

exact solution

Figure 2.5: Accuracy of result with mesh refinement

Initial mesh At start of yielding At failure

Figure 2.6: Foundation on clay

Mechanism at failure

Width of footing b



2.2.2 Bransby (1995): mesh for lateral pressure on pile in clay

The ultimate solution for lateral pressure pu caused by homogeneous isotropic rigidperfectly plastic soil with uniform shear strength su flowing around a pile in plane strainunder undrained loading conditions….. (Randolph and Houlsby, 1984).

smooth pile rough pile

Bransby's work can be used to check an undrained analysis in which soil under similarconditions is moved past a fully rough stationary pile. Following manual mesh refinementand development, the final agreement between the exact (11.94 su) solution and thecomputed result is within 2%, which is certainly close enough for most engineeringanalyses.

Figure 2.8: Mesh and boundary conditions

0 1 2 3 4 5 6

remeshing (5.147)

q/su

initial (5.444)

exact = (π + 2)

uy/b

Figure 2.7: Effect of mesh status on load - settlement curve

6 π+( ) su⋅ pu 4 2 2 π⋅+⋅( ) su⋅≤ ≤

Finite element modelling of a single pile in 2-d.



Figure 2.9: Load - transfer curve for pile under lateral load

Load-transfer curve for a single pile in elastic-plastic soil.



2.3 PredictionPrediction has already been mentioned in several formats. Determination of styles ofprediction are based on an alpha-numeric code which is given in the table below.

Poulos summarised the modes of modelling as shown in Table 2.2.

2.4 Styles of numerical analysis using a computerIt is worth considering WHAT a civil engineer might use a computer-based analysis for.

1. Reviewing results of someone elses' analysis (e.g. as a 'proof engineer')

a) Check validity of calculation model

b) Check input parameters as stated in the assumptions

c) Check answers are in the right zone to validate the work

d) Examine all data critically (deformations, stresses, strains etc.) and use as neces-sary

E.g. Modelling in Geotechnics: Exercise 1: GeoCAL SSI (Done Runs).

2. Setting up and running simple analyses

a) Simple mesh and boundary conditions

b) Simple loading conditions

Class Stage of prediction StatusA Calculation before or during design process or before event Results unknown

B Calculation during event (e.g. construction process) Results unknown

B1 Calculation during event (e.g. construction process) Results known

C Calculation after event (e.g. construction completes) Results unknown

C1 Calculation after event (e.g. construction completes) (back analysis)

Results known

Tab. 2.1: Classes of prediction

Analy-sis class

Characteristics and typical example


C Simplified methods, using closed form solutions. Sim-ple soil models used.

Easily applied, and allow rapid parametric studies.

Requires substantial ideali-zation, and experience in assessing parameters.

B Methods using boundary elements, with simplified soil models.

Relatively easy data input. Familiar soil model parame-ters used. Relatively rapid to run and interpret.

Requires some idealization, and experience in assess-ing parameters. Difficult to examine complex prob-lems.

A Complex numerical meth-ods (finite element, finite difference).

Can consider detailed and complex problems. Soil models can be more realis-tic.

Requires experience in as-sessing soil parameters which may be unfamiliar. Considerable effort to pre-pare data and interpret out-put.

Tab. 2.2: Classes of soil-structure interaction analysis



c) Simple constitutive model (e.g. elasticity)

d) ….then steps as above as in steps 1a-d

E.g. Modelling in Geotechnics: Exercise 1: GeoCAL SSI (Paint or Supermesh).

3. Setting up and running more complex analyses

a) As in 2a-c above

b) Development of more complexity in terms of mesh, loading conditions and constitu-tive model

c) ….then steps as above as in steps 1a-d

E.g. Modelling in Geotechnics: Exercise 2 (or 3).

2.5 Idealisation for numerical modelling.....as before for physical modelling

• geometry• soil• structure• loading• construction effects.

To reiterate, most of these remarks relate to continuum analyses - mainly by

• finite element method (FEM) or • finite difference method (FDM)

.....based on principle of discretization (meshing) - see p. 2 - 4 to solve complexboundary-value problems PLUS

• compatibility - kinematic conditions => geometry, displacement, strains must becompatible

• equilibrium - static conditions => forces and stress must be in equilibrium • stress-strain relationship - physical conditions => material-dependent relationship

between stress and strain must be specified at element level

2.5.1 Geometry

• try to represent 3-dimensional effects as 2-dimensional effects (cheaper, quicker)

→ may reproduce as a plane strain or axisymmetric problem by use of symmetryor asymmetry

• consider and idealise boundaries: soil/structure• draw outline section and plan with material/boundaries

→ (too much vertical deflection at rollers indicates boundaries may not be farenough away?)

• create mesh → nodes and elements

→ avoid large jumps in element size to < 3x (FEM) or < 1.5x (FDM)

→ refine mesh in regions of high strain but beware infinite stress concentrations

→ limit number of nodes and elements according to complexity, typically...



* non-critical structure 150~200 elements* dam with deep foundation 300~400 elements (or more)

Some finite element types:

→ 2 dimensional triangular constant strain

→ 2 dimensional quadrilateral linear strain

→ 3 dimensional hexahedral cubic strain

→ interface elements: relative movement between elements

(progressive slip on piles)

→ bar elements: capacity for tension (soil reinforcement)

/compression (props)

→ beam elements: capacity for axial force and bending moments

(structural inclusions)

→ infinite elements: models unbounded area e.g. in dynamics

where fixed boundary would reflect waves

2.5.2 Mesh design

This has been shown in the past to influence the results obtained and the major guidelineswill be presented in more detail in chapter 3. Several examples have been shown on pages2 - 4.

ADAPTIVE MESH REFINEMENT (e.g. El-Hamalawi, 1997) can be used to enrich andsubdivide mesh as regions of high strain develop - so that mesh choice does not precon-dition outcome of the analysis.

2.5.3 Structure

Material

• use 'drained' properties (not much pore pressure in steel or concrete!)• linear elastic (although can use linear elastic-perfectly plastic if trying to 'fail' structure)• much stiffer than soil so beware of numerical instabilities (sometimes need double

precision in FEM or need more time steps/finer mesh in FDM)

Equivalence

• row of piles as a sheet pile wall - equivalent bending rigidity

(EI)wall = n (EI)individual piles + (EI)soil between piles

• similar equivalence when modelling cylindrical sand drains as a 2-dimensional sanddrain wall.



2.5.4 Loading and construction effects

• in-situ stresses defined initially• loads primarily - normal and shear forces (tractions) on elements• excavation and fill: construction sequences for embankment and retaining wall• superposition of layers of soil or concrete (for geometric purposes) for subsequent

removal• displacement and rotation fixities (x, y, z, θ) - either the soil or structure can be moved

relative to rest of mesh (Bransby analysis p. 5)• pore pressure fixities - can be use to set up excess pore pressures, drains or free water

surfaces.

Figure 2.10: Equivalence

=

(EI)soil between piles

+ n(EI)piles (EI)wall

1 m

10 kN/m25 kN/m5 kN/m

2 kN/m2 1 kN/m 1 kN/m

Figure 2.11: Normal surface loading

Reality Model

321

123

Excavation Fill

Figure 2.12: Excavation and fill



Use pile response to various loadings as examples:

• axial loading: shaft friction and end bearing (3-dimensional to axisymmetry) seepage 12

e.g. the pile behaviour is a function of…..f ( Epile / G , l / ro , ν )

where

Epile is the pile Young's modulus

G is soil shear modulus

l is the pile length

ro is the pile radius

ν is the Poisson's ratio

• lateral thrust/loading due to embankment surcharge (3-dimensional to plane strain)• piled abutment (3-dimensional to plane strain)

Axisymmetry

Driven pile installation? not so good numerically (unless dynamic analysis); better in thecentrifuge

• spherical / cylindrical cavity expansion• remoulds soil around pile with massive strains• changes stress history• wish-in-place pile is normally adopted• it is artificially possible to change soil properties adjacent to pile or use interface

elements

Bored pile installation? not so good for centrifuge modelling; better in numerical analysis

• remove soil elements and replace with bentonite (relax circumferential stresses)• tremie concrete (heavy liquid) to reload excavated cylindrical hole circumferentially • replace concrete as a heavy liquid by hardened concrete.



Figure 2.13: Modelling an axially loaded pile

What is the vertical deformation pattern in the soil and in the pile due to the axial load on the pile?

Pile

Plan

θ

r

Plan axisymmetry r and θ plane: z common

Axial load

End bearing

δzs?

δzp?

Pile

Shaft friction component

Normal, uniformly distributed load

End bearing component

SoilPile

rz

Soil

zr

Circular ‘footing’ load

Mesh

Section

Soil

Soil

Shaft friction

Axial loading on a long flexible cylindrical pile

r z



Plane strain

1


δu

Embankment

Soft

Stiff

1


Ellis PhD, 1997

p?

δuparching

Figure 2.14: Piled full-height bridge abutment

δu

Plan

x

y

When pile is displaced laterally relative to the soil, what is the relative soil-pile movement?

Select half space: PLANE STRAIN x and y plane: z common

SoilPile

Section

x

z

Lateral thrust

CL

Figure 2.15: Idealisation for lateral thrust on a single row of piles from embankment loading



2.5.5 Ellis (1997): Piled full-height abutment: 3D problem as 2D....

• behaviour of soil under embankment of most interest• relative soil-pile displacement less critical than this• additional component of lateral thrust caused by arching is critical• model row of piles as a wall of equivalent bending rigidity• overlay soil and 'pile' wall with interaction law with relative soil-pile movement• soil may be displaced past 'pile' wall so lateral thrust on piles added to equilibrium

equation.

Figure 2.17: The finite element mesh with vertical drains (Ellis, 1997)

sand embankment

Figure 2.16: Finite element analysis: contours of horizontal stress (Ellis, 1997)

piled abutment wall

kPa

low sress

sand embankment high stress

soft clay

low sress

soft clay

SAND

SAND

CLAY



2.5.6 Soil

Why can we not use simple back of the envelope calculations probably based on elasticanalyses?

• simple elastic models do not reproduce key aspects of soil behaviour• we can select more appropriate soil models for design, to account for

→ pre-yield stiffness

→ yield and failure criteria

• OK for more complex analysis because computing power in design offices is growing

Must select?

• type of analysis• model of soil behaviour - or - constitutive model.

Type of analysis

• steady state (time-independent)

→ steady state seepage

→ static load-deformation problems.

• transient (time-dependent)

→ consolidation

→ dynamic loading (earthquakes, wave action)

→ contaminant transport processes

→ creep.

Drained analysis

• no excess pore pressure - highly permeable soils• all the loads will be transferred to the soil skeleton: effective stress• long-term condition - mostly interested in displacements.

Undrained analysis - low permeability soils

• loads will be carried by both soil skeleton and pore pressure • no volume change - very large bulk modulus K compared to shear modulus G: K>>G• short-term stability - mostly interested in (total) stresses - undrained failure of clays?• avoid using equal size elements if the solution is oscillating or use higher order elements,

or• set νu= 0.49 with a short time step within a consolidation analysis.



Consolidation analysis (Biot's equations) - more time consuming

• transition from undrained condition to drained condition• check the movement of the system with time

Which do you want to choose for your analysis?

Figure 2.18: Influence of the Poissons ratio on the settlement of a strip footing (Potts & Zdraykovic, 1999)

time

settlement

undrained

Figure 2.19: Settlement of a footing in time



2.6 References1. M.S.S. Almeida, Stage constructed embankment on soft clay. PhD thesis, University of

Cambridge. 1984.

2. M.F. Bransby, Piled foundations adjacent to surcharge loads. PhD thesis, University ofCambridge. 1995.

3. A. El-Hamalawi, Adaptive refinement of finite element meshes for geotechnicalanalysis. PhD thesis, University of Cambridge. 1997.

4. E.A. Ellis, Soil-Structure interaction for full-height piled bridge abutments constructed onsoft clay. PhD thesis, University of Cambridge. 1997.

5. D.M. Potts, L. Zdravkovic, Finite Element Analysis in Geotechnical Engineering. Vols.1 & 2. Thomas Telford, London.1999.

6. H.G. Poulos, Experiences with soil-structure interaction in the Far East. 2nd Int.Conference on Soil Structure Interaction in Urban Civil Engineering. Zürich, 2002.

7. L. Prandtl, Über die Eindringungsfestigkeit (Härte) plastischer Baustoffe und dieFestigkeit von Schneiden, Zeitschrift für angewandte Mathematik und Mechanik, 1921,1(1), 15-20.

8. M.F. Randolph and G.T. Houlsby, The limiting pressure on a circular pile loadedlaterally in cohesive soil. Géotechnique, 1984, 34, No. 4, pp. 613-623.



Numerical Modelling Finite Element Method

Dr. Jitendra Sharma


Finite Element Method (FEM) in Geotechnical Engineering Page 3 - 1

3 Finite Element Method (FEM) in Geotechnical Engineering

3.1 IntroductionThe importance of a carefully planned and executed experimental modelling can not beoverstated. However, experimental modelling can be expensive and time-consuming and isnormally used only for high-cost and high-risk projects. For “normal” projects, site investi-gation is undertaken in combination with laboratory testing to obtain soil parameters asaccurately as possible. These parameters are then used as input to either limit equilibriumbased programs (e.g. slope stability, bearing capacity, etc.) to predict failure loads (ultimatelimit state) or a numerical analysis program (e.g. finite element method, finite differencemethod, etc.) to predict the deformation under working load conditions (serviceability limitstate). In this chapter, we will focus on one of the most popular numerical analysistechnique used in geotechnical engineering – the finite element method or FEM. The aim ofthis chapter is to learn how to apply the FEM in solving a geotechnical engineering problem.The emphasis is on the application and not on the formulation of the FEM. A curious readermay well consult one of the numerous books that deal with the mathematics and thenumerical techniques used in the FEM, e.g. Zienkiewicz and Taylor (1989).

3.2 Numerical methods used in geotechnical engineering

Figure 3.1: Various ways of solving a geotechnical engineering problem

As stated in the beginning of this course, there are several different ways of findingsolutions to a geotechnical engineering problem. These are summarized in Figure 3.1. Inthis section, we will focus on the numerical methods. One of the characteristic features ofthe numerical methods is that they usually involve solving a set of simultaneous partialdifferential equations (PDEs). Since soil is essentially a non-linear elasto-viscoplastic,three-phase material, direct solution of the set of PDEs is often impossible. Therefore, aniterative numerical approach is used. There are five major types of numerical methods usedin geotechnical engineering – the finite element, the finite difference, the boundary element,the discrete element and the combined boundary/finite element. The way the PDEs areformulated and solved differs for each of these methods.

Solution of Geotechnical Problems

Solution of Geotechnical Problems

Empirical, Based on Experience“Exact” or

Closed Form Numerical

Finite Element

Boundary Element

FiniteDifference

Limit Equilibrium

Finite/Boundary Element

DiscreteElement



3.3 What is FEM?

Figure 3.2: Discrete vs. continuous problem

Before introducing the concept of the FEM, let us first explore the difference between adiscrete and a continuous system. For a discrete system, an adequate solution can beobtained using a finite number of well-defined components. Such problems can be readilysolved even with rather large number of components, e.g. the analysis of a building frameconsisting of beams, columns and slabs (Figure 3.2). For a continuous system, such as asoil layer, the sub-division is continued infinitely so that the problem can only be definedusing the mathematical fiction of infinitesimal. Depending on the level of complexityinvolved, there are two ways of solving such a problem. Simple, linear problems can besolved easily by mathematical manipulation. Solution of complex, non-linear problemsinvolves discretization of the problem into components of finite dimensions (Figure 3.2) andthen using a numerical method such as the FEM.

The most distinctive feature of the FEM that separates it from other numerical methods isthe division of a given domain into a set of simple subdomains, called finite elements. Anygeometric shape that allows computation of the solution or its approximation, or providesnecessary relation among the values of the solution at selected points, called nodes, of thesubdomain, qualifies as a finite element. Such a subdivision of a whole into parts has twoadvantages:

1. It allows accurate representation of complex geometries and inclusion of dissimilar materials.

2. It enables accurate representation of the solution within each element, to bring out local effects (e.g. large gradients of the solution).

Discrete Problem

Semi-infinite Continuum

A finite element

Discretization

Continuous Problem



3.3.1 Historical Background

The idea of representing a given domain as a collection of discrete parts is not unique to theFEM. It was recorded that ancient Greek mathematicians estimated the value of π by notingthat the perimeter of a polygon inscribed in a circle approximates the circumference of thecircle. They predicted the value of π to accuracies of almost 40 significant digits by repre-senting the circle as a polygon of finitely large number of sides. Searching for approximatesolution or comprehension of the whole, by studying the constituent parts of the whole isvital to almost all investigations in science, humanities, and engineering. The FEM is anoutgrowth of the familiar procedures such as the frame analysis and the lattice analogy for2- and 3-dimensional bodies. Its application is not exclusive to engineering. It has beenused in other fields such as mathematics & physics. One of the earliest examples of its usewas in mathematics by R. Courant who used it for the solution of equilibrium and vibrationproblems (Courant, 1943). However, Courant did not call his method the FEM. It was R.W.Clough who first coined the term finite element in 1960 when he applied the FEM to planestress analysis (Clough, 1960).

During the early days of the digital revolution, due to the excessive cost of using the bulky,not-so-easy-to-use mainframe computers, the FEM remained in the hands for those “elite”people of science who had access to this rather expensive computing power. Only after theadvent of the personal computer and the smaller, more manageable and efficient minicom-puters, did it manage to break the barriers. Now, with tremendous amount of rather cheapcomputing power at their disposal, FEM is the first choice for many engineers and scientistsembarking on the analysis of a wide variety of engineering problems – from designing anew ergonomic shoe sole to designing a supersonic fighter aircraft. Its use in the field ofbioengineering, for example, the modelling of knee prosthesis or stress analysis of brainoedema, is also fast becoming popular.

3.3.2 The fundamental steps of the FEM

The three fundamental steps of the FEM are:

1. Divide the whole into parts (both to represent the geometry as well as the solution of the problem).

2. Over each part, seek an approximation to the solution as a linear combination of nodal values and approximation functions.

3. Derive the algebraic relations among the nodal values of the solution over each part, and assemble the parts to obtain the solution of the whole.

We will consider the example of the approximation of the circumference of the circle inorder to understand each of these three steps. Although this is a trivial example, it illus-trates several (but not all) ideas and the steps involved in the finite element analysis of aproblem.

3.3.3 Approximation of the Circumference of a Circle

Consider the problem of determining the perimeter of a circle of radius R (Figure 3.3).Ancient mathematicians estimated the value of the circumference by approximating it byline segments, whose lengths they were able to measure. The approximate value of thecircumference is obtained by summing the lengths of all the line segments that were used.Let us now outline the steps involved in computing an approximate value of the circum-ference of the circle. In doing so, we will also learn about certain terms that are used in thefinite element analysis of any problem.



1. Finite element discretization: First, the domain (i.e. the circumference of the circle) is represented as a collection of a finite number of n subdomains, namely, line segments. This is called discretization of the domain. Each subdomain (i.e. the line segment) is called an element. The collection of elements is called the finite element mesh. The elements are connected to each other at points called nodes. In the present case, we discretize the circumference into a mesh of five (n = 5) line segments. The line segments can be of different lengths. When all elements are of same length, the mesh is said to be uniform; otherwise, it is called a non-uniform mesh (see Figure 3.3b).

2. Element equations: A typical element is isolated and its required properties, i.e. its length, are computed by some appropriate means. Let he be the length of the element

Ωe in the mesh. For a typical element Ωe, he is given by (see Figure 3.3c):

(3.1)

where R is the radius of the circle and θe < π is the angle subtended by the line segment atthe centre of the circle. The above equations are called element equations. Ancientmathematicians most likely made measurements, rather than using (3.1) to find he.

Figure 3.3: Approximation of the circumference of a circle by line elements

Assembly of element equations and solution: The approximate value of the circumference(or perimeter) of the circle is obtained by putting together the element properties in ameaningful way; this process is called the assembly of the element equations. It is based,in the present case, on the simple idea that the total perimeter of the polygon (assembledelements) is equal to the sum of the lengths of individual elements.

(a) (b)

(c)

Approximation of the circumferenceof a circle by line elements: (a) Circle of radius R; (b) Uniform and non-uniform meshes used to representthe circumference of the circle; (c) a typical element.

Element

Node

R

θ e

h e



(3.2)

Then, Pn represents an approximation to the actual perimeter, p, of the circle. If the mesh isuniform, i.e. he is the same for each element in the mesh, θe = 2π/n, and we have

(3.3)

3. Convergence and error estimate: For this simple problem, we know the exact solution:

(3.4)

We can estimate the error in the approximation and show that the approximate solution Pnconverges to the exact solution p in the limit as n → ∞.

In the summary, it is shown that the circumference of a circle can be approximated asclosely as we wish by a finite number of piecewise-linear functions. As the number ofelements is increased, the approximation improves, i.e. the error in the approximationdecreases.

3.4 Basic formulation of the FEM

In this section, the basic formulation of the FEM will be introduced using three simpleexamples: (1) a system of interconnected elastic springs; (2) a one-dimensional plane trusselement; and (3) a constant strain triangular finite element.

3.4.1 Interconnected elastic springs

a

d

b

c

d1

d2

d3

d4

1

2

3

4

2Ta

Tb TdW2

Equilibrium at Node 2



Figure 3.4: A system of interconnected springs

1. In this system, linear elastic springs are the finite elements.

2. From a structural mechanics point-of-view, the structure is statically indeterminate.

3. Let the stiffnesses of individual springs be ka, kb, kc and kd. Therefore, the tensions in these springs are given by:

(3.5)

where ea, eb, ec and ed are extensions of springs a, b, c and d, respectively.

4. Let us now invoke three fundamental principles of structural mechanics: compatibility, material behaviour and equilibrium for the calculation of the displacement of each spring. These three principles are applied in the order of compatibility – material behaviour – equilibrium.

5. The compatibility equations are:

(3.6)

where d1, d2, d3 and d4 are displacements of nodes 1, 2, 3 and 4, respectively. Here, weare making sure that the system does not fall apart, i.e. springs remain connected with eachother.

6. Material behaviour can be expressed using spring stiffnesses as:

(3.7)

7. Equilibrium (at node 2, see Figure 3.4):

or

(3.8)

which on rearrangement, results in:

(3.9)

8. Similar equations can be written for other nodes, giving four linear simultaneous equations in d1, d2, d3 and d4 that can be expressed in matrix form as:

(3.10)



The matrix on the left-hand-side is called the global stiffness matrix. Equation (3.10) canbe written in matrix notation as:

Kd = W

These simultaneous equations can be solved by elimination and values of displacementscan be obtained. From the values of displacements, the force in each spring can be calcu-lated.

9. The global stiffness matrix K consists of the sum of matrices of the following form (where ke is the stiffness of one particular spring):

10.

(3.11)

This matrix is called the element stiffness matrix. It relates the nodal displacements to theforces exerted on each spring at nodal points. One of these matrices is added into theglobal stiffness matrix for each spring in the system.

(3.12)

3.4.2 A plane truss element

Figure 3.5: A plane truss element

In this section, we will apply the same principles of compatibility, material behaviour andequilibrium to a one-dimensional plane truss element (Figure 3.5). The formulation is nowmore complex than that for a simple system of linear elastic springs. You may have noticedthat in the case of linear elastic springs, each node was allowed to move in only y-direction,i.e. up or down. Here, each of the two nodes of the plane truss elements has two degreesof freedom, i.e. it can move in both x- and y-direction. However, as we shall see, thegeneral solution procedure remains the same regardless of the increased complexity.

y

x

y’

x’

dy2

dx2

dy1

dx1

α

Length = L

1

2



In calculating the strains in this element, we are only interested in the displacements alongthe direction of the element. It is, therefore, logical to define a system of axes x’-y’ that islocal to the element, with x’-axis coincident with the direction of the element.

1. Let us first apply the condition of compatibility, i.e. the element should not break in the middle. Mathematically, it can be expressed in terms of the equation for displacement at a distance x’ along the element

: (3.13)

2. To obtain the element stiffness matrix, we need to write this expression in terms of the degrees of freedom dx1, dy1, dx2 and dy2. This is achieved by noting that

(3.14)

from simple geometric consideration.

3. Making this substitution, we obtain:

(3.15)

4. The strains inside the element can now be related to nodal displacements using a matrix that is obtained by differentiating equation (3.15) with respect to x’. This matrix is

called the B matrix in the FEM formulation and is given by:

In the matrix notation, the strain matrix is now written as:

(3.16)

where ae is the vector of nodal displacements – right-hand-side matrix in equation (3.15).

5. Assuming the plane truss element to be linear elastic, the stress inside the element can now be expressed in terms of nodal displacements as:

(3.17)



where D is the matrix of material behaviour or constitutive matrix for the element. In thiscase, it simply reduces to the Young’s modulus of the plane truss element, E.

6. The principle of virtual work can now be used to find the nodal forces Fe that are in equilibrium with this state of internal stress. A set of virtual nodal displacements applied to the element accompanies a set of virtual strains within the element according to the relation:

(3.18)

The principle of virtual work gives:

(3.19)

7. Substituting for σ and , we obtain:

(3.20)

From the above equation, can be cancelled out to give:

(3.21)

where K is the element stiffness matrix. For our plane truss element, it can be shown to begiven by:

(3.22)

where A is the cross-sectional area of the plane truss element, C = cosα and S = sinα.

8. For a typical plane truss problem, the forces acting on the nodes are known. Hence, equation (3.21) can be solved by first inverting the K matrix and then solving the resulting simultaneous equations for nodal displacements.

3.4.3 A constant strain triangular finite element

After having successfully formulated the FEM for the solution of two one-dimensionalproblems, we move to the formulation of a two-dimensional constant strain triangular finite

ε

Tea



element. Figure 3.6 shows the simplest triangular finite element used for two-dimensionalcontinuum analysis.

Figure 3.6: A constant strain triangular finite element

1. Each of its three nodes has two degrees of freedoms and the terms dx1, dy1, dx2, dy2, dx3, and dy3 denote the nodal displacements. In this case, the unknown variation of the displacement within the element adds to the complexity of the problem. Here, we are going to assume that this variation is linear, i.e.

and

(3.23)

Since the strain is the first derivative of the displacement, it will be constant within theelement. Hence, the element is called a finite element.

2. The coefficients c0, c1, etc. in equation (3.23) are obtained by substituting the coordi-nates of the three nodal points into these expressions. In this case, too, we assume a local coordinate system with origin at node 3 and x-axis along side 3-1 and y-axis along side 3-2. Solving the resulting sets of simultaneous equations, we obtain:

and

(3.24)

dy2

dx2

dy3

dx3

dy1

dx1

h

h

3 1

2

x

y



3. Equation (3.24) can be written in matrix notation as:

(3.25)

where N is the matrix of shape functions for the finite element and is given by:

(3.26)

4. Now, we can formulate the B matrix by partially differentiating the N matrix with respect to x and y as:

(3.27)

Here, the first row denotes strain in x-direction, second row denotes strain in y-direction andthe third row denotes the shear strain in the x-y plane.

5. Assuming plane strain conditions, the element stiffness matrix D can be easily obtained from Hooke’s law as:

(3.28)

where E is the Young’s modulus and ν is the Poisson’s ratio for the material.

6. Formulating the element stiffness matrix K is now a simple task of calculating the matrix product BTDB times the area of the element (h2/2) since the terms of all these matrices are constant. K is given by:

(3.29)

where a = 1 – ν; b = 0.5 – ν and c = 1.5 – 2ν.



Although the above three examples illustrate the basic idea of the FEM, there are severalother features that are either not present or not apparent from the discussion of theseexamples. These are summarized below:

1. Depending on its shape, a domain can be discretized into a mesh that contains more than one type of element. For example, in the discretization of an irregular two-dimen-sional domain, one can use a combination of triangular and quadrilateral finite elements. However, if more than one type of element is used, one of each kind should

be isolated and its equations developed. All the commercial FEM software take this into account and therefore, it is not a problem to mix element types during an analysis.

2. The governing (simultaneous) equations are generally more complex than those considered in these three examples. They are usually partial differential equations. In most cases, these equations cannot be solved over an element for two reasons. First, they do not permit exact solution. Second, the discrete equations obtained cannot be solved independent of the remaining elements because the assemblage of the elements is subjected to certain continuity, boundary and/or initial conditions.

3. The number and location of nodes in an element depend on (a) the geometry of the element, (b) the degree of polynomial approximation, and (c) the integral form of the equations. This point is elaborated further in the section dealing with types of finite elements.

4. There are three sources of errors in a solution obtained by the FEM: (a) those due to the approximation of the domain; (b) those due to the approximation of the solution; and (c) those due to numerical computations. The estimation of these errors is not a simple matter. The accuracy and convergence of a FEM solution depends on the differential equation, the integral form and the element used. Accuracy refers to the difference between the exact solution and the solution obtained by the FEM whereas conver-gence refers to the accuracy as the number of elements in the mesh is increased. This point is discussed in detail later in the chapter.

3.5 Approximations, accuracy and convergence in the FEM1. Engineers sometimes regard the finite elements in a mesh as being connected only at

the nodal points in the mesh. This is not a good conceptual picture of how the elements behave. Straining of finite elements results in a deformation pattern similar to that shown in Figure 3.7a rather than that shown in Figure 3.7b (i.e. there are no gaps that open up at the element boundaries). This is because the polynomials or shape functions that approximate the distribution of displacement are chosen in such a way that there is a continuity of displacements within the elements as well as between the adjoining elements.



Figure 3.7: Continuity of displacements in adjoining finite elements

2. Although strains will be continuous within a finite element, there will usually be a discon-tinuity of strains between adjacent elements. Some approximation (e.g. a smoothing zone as shown in Figure 3.8) is necessary so that the terms being integrated become continuous.

(a) (b)



Figure 3.8: The use of smoothing zone at element boundaries

3. The stress field within an element will be continuous but may not satisfy the equations of equilibrium. Except for very simple problems, stresses on either side of element boundaries will not be equal. Equilibrium is satisfied, however, in an average sense through the equilibrium equations at nodal points where the resultant forces equivalent to internal stress field balance the resultant forces due to external traction and body forces. The extent to which the local stresses appear not to be in equilibrium with the external forces gives some indication of the accuracy of the solution.

4. Before applying the FEM to solve real problems, it is advisable to test its accuracy by solving certain benchmark or validation problems for which an exact or closed-form solution exists. An error-free, robust FEM program should be able to reproduce the exact solution accurately. One of the most popular benchmark problem in geotechnical engineering is the calculation of undrained collapse load (qu) of a circular foundation on soft clay of uniform undrained shear strength (su) – qu = (π+2) su.

Dis

plac

emen

t uS

trai

n du

/dx

Rat

e of

str

ain

d2u/

dx2

∆

‘Smoothing’ zone

-∞



5. In addition to testing the accuracy of the FEM, a convergence test should be carried out for a given problem for which we do not have an exact solution. It involves conducting three or more FEM analyses with progressively finer mesh. Convergence is achieved when further refinement of mesh does not result in a significant increase in the accuracy of the solution (Figure 3.9).

Figure 3.9: Testing the convergence by progressive mesh refinement

3.6 Geotechnical finite element analysis

Most of the commercially available FEM programs are written with structural/mechanicalapplications in mind. These programs cater for materials that can be produced undercontrolled conditions and therefore, have well-defined physical or mechanical properties,e.g. metals, plastics, polymers, concrete, etc. The most important material in a geotechnicalanalysis is the soil. A soil’s physical or mechanical properties have to be measured insteadof being specified or specially fabricated. These properties vary enormously from site tosite and can be profoundly affected by factors such as sampling techniques, specimenhandling and preparation, characteristics of the measurement and data acquisitiontechniques. Therefore, the constitutive modelling takes the centre stage in a geotechnicalFEM program. The three phase (soil-water-air) nature of soil makes realistic constitutivemodelling of soil a formidable task. Since the shear strength of a soil at a given pointdepends on the effective stress at that point, the stress-strain response of a soil is highlynon-linear. For a geotechnical finite element analysis, the FEM program should have thefollowing features:

1. Material models that are capable of modelling non-linear stress-strain behaviour and that include options for undrained analysis (short-term behaviour), drained analysis (long-term behaviour), most importantly, coupled consolidation analysis.

qu

No. of Elements24 48 96

‘Exact’ solutionMesh A - 24 Elements

Mesh B - 48 Elements

Mesh C - 96 Elements



2. The ability to specify non-zero in-situ stresses.

3. The ability to add or remove elements during the analysis (for modelling the construction or excavation, respectively).

3.6.1 Plane strain and axisymmetric problems

While a three-dimensional finite element analysis is frequently used in structural ormechanical applications, it is rarely used in geotechnical engineering. Most of the geotech-nical problems can be assumed to be either plane strain or axisymmetric without signif-icant loss of the accuracy of the solution.

1. Plane strain problems: The characteristic feature of a plane strain problem (Figure 3.10) is that one dimension – in this case the dimension along the z-axis – is considerably greater than the other two dimensions. As a result, the strains in the direction of z-axis can be assumed to be zero. Therefore, we only have to solve for strains in the x-y plane and the problem reduces to a plane strain problem. For plane strain problems, the numerical integration is performed for a unit section (1 unit length) along the z-axis. Typical examples of plane strain geotechnical problems are embankments, retaining walls, tunnels (at sections sufficiently away from the head of the tunnel).

Figure 3.10: A plane strain problem

Axisymmetric problems: For an axisymmetric problem, both the structure and the loadingexhibit radial symmetry about the central vertical axis (Figure 3.11). Consequently, thecircumferential strains can be ignored in the solution and the problem reduces to a two-dimensional problem in a vertical radial plane. Keep in mind that the problem can only bereduced to an axisymmetric problem when both the structure and the loading are symmetricabout the central vertical axis. If one of the two does not exhibit radial symmetry, either theproblem has to be treated as a three-dimensional problem or techniques involving FastFourier Transforms (FFTs) have to be used. The numerical integration for an axisymmetricproblem is performed from zero to 2p, i.e. for the entire horizontal circular cross-section.Typical examples of axisymmetic geotechnical problems are pile foundation subject tovertical concentric loads, excavation of vertical shafts of circular cross-section, consoli-dation around a vertical drain.

y

xz



Figure 3.11: An axisymmetric problem

3.6.2 Different types of finite elements

There are many different types of finite elements available for use with a geotechnical FEMprogram. These elements can be classified based on either the dimensions of the problemor the order of the element. They can also be classified on the basis of whether the coupledconsolidation formulation is adopted or not.

1. 1-D, 2-D and 3-D elements (Figure 3.12): 1-D and 2-D elements are used mainly for the plane strain and axisymmetric problems. 3-D elements are used only for the truly three-dimensional problems.

• Typical 1-D elements include: (a) bar elements for the modelling of struts, geotextilereinforcement, ground anchors and any other structural element that is not capable ofresisting flexure, and (b) beam elements for the modelling of retaining walls, tunnellinings and any other structural element requiring flexural rigidity.

• Typical 2-D elements include (a) triangles and quadrilaterals for the modelling of soil andstructural components of significant dimensions, and (b) slip elements for modelling ofsoil-structure interface behaviour.

• Typical 3-D elements are hexahedrons and tetrahedrons for the modelling of soil andstructural components. Some FEM programs also have 3-D slip elements for modellingof soil-structure interface behaviour.

CL

r

y



Figure 3.12: 1-, 2- and 3-D elements

2. First-, second- and fourth-order elements (Figure 3.13): The order of the element is determined by the order of the polynomial used as the shape function.

• For a first-order element, a first-order polynomial, i.e. a straight line, is used as shapefunction. The constant strain triangle in the example above is a first-order element. Amesh containing only first-order elements requires a large number of elements for a suffi-ciently accurate solution.

• For a second-order element, a quadratic or second-order polynomial is used as shapefunction. As a result, the strain within the element is distributed linearly. Hence, theseelements are also called linear strain elements. Such elements usually have one or moremid-side nodes in addition to the vertex nodes. One does not need to use a largenumber of second-order elements in order to achieve sufficient accuracy.

• For a fourth-order element, a quartic or a fourth-order polynomial is used as shapefunction. The strains, therefore, have a cubic variation within the element and theelement is often called a cubic-strain element. Such elements have several mid-sidenodes as well as nodes inside the element in addition to the vertex nodes. It is notcommon to use such elements for a routine geotechnical analysis. Their use is limited tospecial situations such as testing a new constitutive model, unit cell radial consolidationproblems.

dy2

dx2

dy1

dx1

(a) Two-noded bar element

dx2dy1

dx1

dy2

θ1

θ2

(b) Two-noded beam element

(c) 2-D element

dy1

dx1

dy2

dx2

dy3

dx3

(e) 3-D elements

t

LSoil

Structure

(d) 2-D slip element



Figure 3.13: First-, second- and fourth-order finite elements

3. Consolidation elements (Figure 3.14): These elements are required when the FEM program adopts a coupled consolidation formulation. In a coupled-consolidation formu-lation, the excess pore pressures are treated as unknowns. Any variation in the magnitude of excess pore pressure at a given point is reflected simultaneously in the magnitude of effective stress at that point. In addition to the standard displacement nodes, consolidation elements have pore pressure nodes where the value of excess pore pressure is calculated. For second-order elements, pore pressure nodes are normally superimposed on vertex displacement nodes of the element. For higher-order elements, pore pressure nodes also exist inside the elements.

Figure 3.14: Consolidation element

Displacement

x

Displacement

x

Displacement

x

(a)

(b)

(c)

(a) First-order element

(b) Second-order element

(c) Fourth-order element

+ =

DisplacementElement

Pore PressureElement

ConsolidationElement



3.7 Techniques for modelling non-linear stress-strain response

The basic formulation of the FEM described in Section 3.4 is applicable only for materialsthat obey linear stress-strain laws (Figure 3.15a). However, as mentioned above, thestress-strain behaviour of a soil is highly non-linear (Figure 3.15b) and therefore, forsolution of geotechnical engineering problems the fundamental equation of the FEM(equation 3.21) cannot be used in its present form. First, the non-linear stress-strain curveshould be approximated by a set of interconnected straight lines (i.e. it is made piecewiselinear) and then an incremental form of equation 3.21 is used. This approach is illustrated inFigure 3.16. Depending on the degree of non-linearity, the imposed loading (ordisplacement) is divided into sufficient number of increments and equation 3.21 is solvedfor each increment in succession. This is the simplest way of modelling a non-linearmaterial. The trick here is to make sure that the piecewise linear approximation does notdrift from the true stress-strain curve by a certain tolerable amount. However, the appli-cation of this method is limited to material models that have a well-defined yield function,e.g. models based on critical state soil mechanics theory. This method is not suitable forelastic-perfectly plastic models such as the Mohr-Coulomb model. The reason for this isthat the yield function and the failure criterion are one and the same for such models andthere is no other way of detecting the yielding of the material than to cross (and go out of)the failure envelope (Figure 3.17a). Such a stress state is not admissible and will result ininternal forces that are not in equilibrium with external forces. Therefore, the stress statemust be corrected back to the failure criterion. This can be achieved in several differentways. The following two methods are commonly used in a geotechnical FEM software:

1. Tangential stiffness approach with carry over of unbalanced load

2. Modified Newton-Raphson method

Figure 3.15: Linear and non-linear materialbehaviour

Figure 3.16: Piecewise linear approximation ofnon-linear material behaviour

σ

σ

ε

ε

(a) Linear stress-strain response

(b) Non-linear stress-strain response

σ

ε

σ

εIncrement No. 1 2 3 4 5 6

(a) Non-linear stress-strain response

(b) Piecewise linear approximation

E4

Fe K ae⋅= δFe Ki δae⋅=



3.7.1 Tangential stiffness approach with carry over of unbalanced load

This approach is illustrated in Figure 3.17b. In this approach, the global stiffness matrix iscomputed based on the tangential stiffness at the beginning of an increment, say from 0 toa displacement d1 as shown in figure 3.17b. In other words, the stress-strain response isnow considered linear for this increment and is represented by the tangent drawn at thestarting point of the increment. The internal load at the end of this increment (∆P1) is nolonger in equilibrium with external load and this out-of-balance load (∆PC1) is re-applied tothe finite element mesh at the beginning of the next increment (from displacement d1 to d2).It is obvious that the accuracy of the solution will suffer considerably if the magnitude of theout-of-balance load is rather large. The accuracy of the solution can be assessed byexamining the global equilibrium error (percent difference between the sum of externalloads and sum of internal forces) at the end of each increment. For elastic-perfectly plasticmodels, this error should never be allowed to go beyond 15 to 20%. To achieve this goal, asufficiently large number of increments should be used. Another alternative is to divideeach increment into 5 or 10 sub-increments (Figure 3.17c). This will ensure that themagnitude of out-of-balance load for each sub-increment is small.

Figure 3.17: Methods of modelling non-linear material behaviour

3.7.2 Modified Newton-Raphson method

It is also known as the quasi Newton-Raphson method. In this method, similar to thetangential stiffness approach, the stiffness matrix is computed based on the tangentialstiffness at the beginning of an increment. However, the out-of-balance load is not carriedover to the next increment. Instead, an iterative procedure shown in Figure 3.17d isfollowed. The out-of-balance load (∆PC1) is re-applied to the mesh and the resulting incre-

∆τc11

2

yieldsurface

τ

σ ( )∫ ∆=∆ vol11 dP cc τTB

(a) Stress state correction

∆Pc1∆P1

∆P2

d1 d2

P

d

(b) Tangential stiffness approach

Sub-increment 1 2 3

P

d

(c) Use of sub-increments to applyout-of-balance load

∆P1

d

P

d1

(d) Modified Newton-Raphson method



mental displacements are added to the current displacements. If further yielding takesplace during the application of ∆PC1 then a second set of out-of-balance load (∆PC2) arecalculated and the above procedure is repeated until convergence is reached, i.e. theresulting incremental displacements or the out-of-balance load is less than a presettolerance. The main advantage of this procedure is that the stiffness matrix is computedonly at the beginning of an increment. However, rather large number of iterations requiredto achieve convergence compensates the savings on computation time thus achieved.Also, the method may fail to converge for some highly non-linear problems.

3.8 Techniques for modelling excavation and construction

3.8.1 Excavation

Geotechnical activities that involve excavation can be broadly classified into three maincategories: trenches, shafts and tunnels. Trenches can be rather small, e.g. for laying of adrainage pipe (Figure 3.18a), or big and deep, e.g. for the construction of basement carpark (Figure 3.18b). The effect of excavating a small trench on surrounding soil and struc-tures is not so great and, therefore, such a problem is rarely analyzed using the FEM.However, a deep excavation can result in significant ground movements capable ofdamaging the surrounding structures. It is, therefore, not surprising that its design almostinvariably involves conducting a few FEM analyses. The length and the width for a typicaldeep excavation are comparable and hence, it is a 3-D problem. However, a 3-D FEManalysis is rarely used and often, the problem is assumed to be a plane strain problem.Excavation of a shaft is modelled similar to a trench or a deep excavation; the onlydifference is that axisymmetric conditions are assumed.

Figure 3.18: Trenches and deep excavations

1 m

2 m

5~10 m 4~8

m6~

12 mStruts

Diaphragm Wall

(a)

CL

(b)



Figure 3.19: Excavation of a tunnel and its 2-D FEM approximations

The construction of tunnels is an urban necessity. It involves excavation of soil using atunnel boring machine (TBM) and installation of permanent lining for the excavated section.Excavation of a tunnel causes ground loss as well as stress relief at the face of the tunnel,resulting in significant surface settlements. These surface settlements can result in signif-icant damage to nearby structures. Tunnel excavation is also a 3-D problem (Figure 3.19).However, it is quite common to assume plane strain conditions (representing a verticaltransverse section sufficiently away from the face of the tunnel) and to use a volume lossparameter that takes into account the 3-D effect in an approximate manner. To study theground movements ahead of the tunnel face, a vertical longitudinal section is consideredand plane strain conditions are assumed.

For both the deep excavation and the tunnel, the modelling of excavation is achieved in thesame way – by removing the elements from the mesh. Here, it is worth noting that the bodyforces within the element are composed of both soil (effective stress) and water (porepressure) as shown in Figure 3.20. When an element is removed, both the soil and waterbody forces are removed. For the deep excavation, it represents an excavation that is dry,i.e. not filled with water. For an excavation in a clayey soil, this means that there arenegative pore pressures (pore suction) on the inner boundaries of the excavation. Unlesssome support in the form of a retaining wall is provided, the soil will eventually lose itssuction and the excavation will collapse. However, such removal of body forces is notrealistic for certain situations, e.g. installation of a diaphragm wall. The trench for adiaphragm wall is filled with either water or bentonite slurry. In this situation, one must eitherre-apply the water body forces or apply body forces corresponding to the bentonite slurryon the inner boundaries of the excavation.

Tunnel Face

GroundSurface

(a) Vertical Transverse Section

(b) Vertical Longitudinal Section

Unsupported Heading



Figure 3.20: Stress changes during modelling of an excavation

3.8.2 Construction

The word “construction” in geotechnical engineering usually means placing one or morelayers of soil over existing or made-up ground, e.g. construction of a highway embankmenton soft clay. The placing of a layer is modelled either by adding elements to the existingmesh or by applying pressure at the boundaries (Figure 3.21). The latter approach givessatisfactory solution provided the newly placed layers are not expected to undergo anyshear deformation. If this is not the case, the technique of adding elements to the meshshould be used. An element that is added is assumed to be unstressed and the self-weightof the element is the only contributor to the body forces of that element. For this reason, theadded elements must either have elastic properties or have a small non-zero value ofapparent cohesion c’ if elastic-perfectly plastic model is used. A constitutive model thatrequires specification of a stress history, e.g. Cam-clay or other critical state models, isunsuitable for modelling of added elements.

Figure 3.21: Techniques for modelling layered construction of an embankment

(a) Excavation (b) Stresses acting before excavation

(c) Stresses after excavation (zero) (d) Net effect of excavation

Effective stress + pore pressure

Second LayerFirst Layer

Soft Clay

EmbankmentCL CL

(a) by adding elements (b) by pressure loading



Compaction is an integral part of a geotechnical construction activity and its effects shouldideally be included in the modelling. However, the effects of compaction are difficult toquantify in terms of stresses. In addition, the soil that is being compacted is usually partiallysaturated. These factors make the modelling of compaction activity quite complicated andtherefore, most commercial geotechnical FEM software simply ignore it.

3.9 Advantages and drawbacks of the FEM

3.9.1 Advantages

1. It is relatively easy to use and, therefore, it is one of the most popular methods for advanced geotechnical modelling.

2. There are many commercial FEM programs available that are capable of geotechnical modelling (discussed later).

3. Since each element’s properties are modelled and evaluated separately, it is quite easy to incorporate non-homogenous ground conditions such as layers of different soils.

4. Any shape of domain can be modelled with the possibility of including holes, gaps, etc.

5. Boundary conditions can be applied easily.

6. It is possible to couple different physical phenomena such as diffusion and thermal conduction within the same formulation. This is possible because all of these phenomena can be described by the Laplacian equation.

7. Construction and excavation of soil layers in geotechnical engineering can be done easily by adding or removing elements from the mesh (discussed later).

3.9.2 Drawbacks

1. While an FEM program is relatively easy to use, interpretation of its output can be a formidable task and usually requires considerable expertise and experience.

2. It is not suitable for highly non-linear problems or problems that involve large strains, e.g. cone penetration test, consolidation of a hydraulic fill or a clay slurry. For such problems, a finite difference formulation incorporating fast Lagrangian analysis procedure is more suitable.

3. It is also not suitable for the modelling of brittle materials that exhibit discontinuities in the form of cracks, faults and fissures, e.g. rock. For such materials, a discrete element formulation is more suitable.

3.10 Some popular commercial FEM programs

3.10.1 ABAQUS

ABAQUS is a general-purpose FEM program that contains many useful features:

• Static stress-displacement, transient dynamic stress-displacement, heat transfer, masstransport and steady-state transport analyses.

• Coupled formulations that include: Biot’s consolidation theory, thermo-mechanicalcoupling, thermo-electrical coupling, fluid flow-mechanical coupling, stress-massdiffusion coupling, piezoelectric and acoustic-mechanical coupling. The most importantof these from a geotechnical point-of-view is Biot’s consolidation theory.



• Dynamic stress-displacement analysis, determination of natural modes and frequencies,transient response via modal superposition, steady-state response resulting fromharmonic loading, response spectrum analysis, and dynamic response resulting fromrandom loading. These features are mainly for earthquake or other dynamic applications.

• It has a huge library of all finite elements developed in the literature such as 1-D, 2-D and3-D continuum elements, shell, membrane, pipe, beam and elbow elements, springs,dashpots, joint, interface and infinite elements. User-defined elements can also be used.

• Similarly, it has an impressive collection of constitutive models including general elastic(linear and non-linear), elasto-plastic, elasto-viscoplastic, hyper and hypo-elasticmodels. Constitutive models that are useful for geotechnical analysis are von Mises,Mohr-Coulomb, Drucker-Prager, Extended Drucker-Prager (non-associated flow), CamClay, Modified Cam Clay, Capped Drucker-Prager (Cam Clay with Extended Drucker-Prager for use in tunnel excavation), and strain-rate dependent plastic laws.

• User-defined constitutive models can also be incorporated with the help of a subroutineinterface.

• It is possible to simulate excavation and construction.• It can deal with large strain and large deformations.• Presently, it is the only commercial program except ZSOIL (described below) that can

deal with partially saturated soils.• It can model seepage problems with phreatic surfaces and capillary effects.• It even allows cracks and rock joints to be modelled and it can model creep, too.• It can perform adaptive mesh refinement for undrained and drained problems only.• Operating System: Windows NT, UNIX, Sun Solaris and a host of other systems running

mainly on multiprocessor or parallel computers.• It is very expensive but a cheaper, educational version with limited capabilities is

available for teaching and research use.• More information can be obtained from http://www.abaqus.com/

3.10.2 SAGE CRISP

SAGE CRISP has evolved from CRISP – CRItical State Program – developed by theCambridge University Soil Mechanics Group in the 1970s and 80s. CRISP was one of thefirst FEM programs dedicated to geotechnical analysis. In the early 1990s, SAGEEngineering Ltd., UK developed the pre- and post-processors for this program and beganmarketing the program by the name SAGE CRISP. The main features of SAGE CRISP areas follows:

• It can perform static stress-displacement and coupled consolidation analyses in one-,two- and three-dimensions. At present, there is no facility to do dynamic analysis but thedevelopers of SAGE CRISP are in process of incorporating this facility.

• Its element library includes 1-D, 2D and 3D continuum, bar, beam and interfaceelements.

• Almost all of its constitutive models cater for geotechnical applications. These includegeneral elastic (linear and non-linear), anisotropic elastic, elastic-perfectly plastic withvon Mises, Tresca, Drucker-Prager, Mohr-Coulomb failure criteria, Cam Clay, ModifiedCam Clay, Schofield, 3-Surface Kinematic Hardening (for small-strain modelling) andhyperbolic (Duncan and Chang type) models.

• It can model excavation and construction.



• It does not include a large-strain formulation but it can deal with large strain problems inan approximate manner by using large number of increments and updating of geometryat the end of each increment.

• Operating System: Windows (older versions of CRISP run under MS DOS).• More information can be obtained from http://www.crispconsortium.com/

3.10.3 PLAXIS

PLAXIS is a geotechnical FEM program developed by PLAXIS BV of the Netherlands. Itsname is a combination of PLane strain and AXISymmetric. As the name suggests, it canonly do 1-D and 2-D analyses although a 3-D version is being developed. Its featuresinclude:

• 1-D and 2-D static stress-displacement and coupled consolidation analyses.• The element library consists of 1-D and 2-D continuum, beam, spring and interface

elements.• Its library of constitutive models includes general elastic (linear and non-linear) aniso-

tropic elastic, Mohr-Coulomb (associated as well as non-associated flow), Soft soil (CamClay), Soft soil creep and Hardening soil (hyperbolic) models.

• It can model excavation and construction. In addition, it can do analysis of tunnelexcavation that incorporates a volume loss parameter that represents the contractionaround tunnel lining due to overcut by the tunnel boring machine and the loss of pressureat the face of the tunnel.

• It can deal with large strain and large deformation situations and can also model creep.• It is able to select the optimum number of increments needed for efficient convergence of

non-linear problems.• It is able to model seepage problems involving phreatic surfaces and capillary effects.• It allows for incorporation of safety factors into an analysis of, for example, foundations

or slopes.• Operating System: Windows.• More information can be obtained from http://www.plaxis.nl/

3.10.4 ZSOIL

ZSOIL is a geotechnical FEM program developed by Zace Services AG, Switzerland. Itsfeatures include:

• 1-D and 2-D static stress-displacement and coupled consolidation analyses.• Elements include 1-D and 2-D continuum, beam, spring, shell, cable and interface

elements.• Constitutive models include general elastic (linear and non-linear), anisotropic elastic,

elastic-perfectly plastic with Mohr-Coulomb and capped Drucker-Prager failure criteria,and Hoek-Brown models.

• It can model excavation and construction.• It is able to deal with large strain and large deformation problems.• It can model seepage problems involving phreatic surfaces and capillary effects.• It can model partially saturated flow problems and problems involving creep.• Operating System: Windows.• More information can be obtained from http://www.zace.com/



3.11 Guidelines for the use of FEM in geotechnical engineeringThere are no shortcuts for learning to use the FEM effectively. One becomes an FEMexpert by experience and a lot of hard work. However, the following guidelines will makesure that one has a good start to the learning endeavour.

• Use smaller elements in regions where the rate of change of stress with distance isgreater. This happens, for example, near the edges of a loaded area, near a re-entrantcorner in the mesh or where adjacent parts of a mesh have significant differences instiffness (e.g. soil reinforcement, retaining wall, pile foundation) as shown in Figure 3.22.Note that some of these situations result in stress concentrations where the stressestend to infinity. The smaller we make the elements near the concentration, the higher arethe stresses. Sometimes, a stress concentration will “spoil” the solution locally, leading tooscillation of stresses. Here, it is worth remembering that infinite stress concentrationsare mathematical fiction that may be unimportant in describing real behaviour andtherefore, it is often advisable to ignore them.

• When increasing the element size from area of interest to the far boundaries, avoidincreasing the element size by more than a factor of 2 between adjacent elements.

• Wherever possible, make use of symmetry of the problem (if any) - it will save both yoursand the computer’s time.

• Keep the triangular elements as equilateral as possible and the quadrilateral elements assquare as possible.

Figure 3.22: Areas of FEM domains that require finer elements

• Avoid using curved elements as interior edges between elements in a mesh - only usethem at external boundaries or internal boundaries (e.g. inside of a tunnel) if absolutelynecessary.

• Where you place the boundary of a mesh can make a big difference to the outcome ofthe analysis. If you are unsure of the boundary effect, try two different meshes – one witha close and the other with a far boundary.

Re-entrantCorner

Edge of theLoaded Area

Interface betweenpile and soil

Stress Concentration

Zone of Interest



• When modelling an axisymmetric undrained problem where collapse is expected (e.g.cylindrical cavity expansion in a pressuremeter test), use only fourth-order (cubic strain)elements.

• Always check that the in-situ stresses specified at the start of the analysis are inequilibrium. If the in-situ stresses are not in equilibrium, either you have not inputconsistent values of soil unit weight or have not applied correct fixity conditions to one ormore of mesh boundaries.

• When using elastic-perfectly plastic constitutive models with either a Mohr-Coulomb or aDrucker-Prager failure criterion, specify a small value for c’ (0.1 or 1 kPa) even if thematerial has c’ = 0 kPa. This will ensure that the initial state of stress for the material isnot on the failure surface.

• In order to model the incompressibility of a saturated soil under undrained conditions, aPoisson’s ratio (ν) of 0.5 should ideally be used. However, if ν = 0.5 is input into an FEManalysis, it will result in serious ill-conditioning of the equations. The reason for this isthat the bulk modulus (K) of the soil approaches infinity as ν → 0.5. In such situations, ν= 0.49 usually gives satisfactory results.

• Treat pore pressure boundary conditions with respect. They are the most likely source ofdisaster in a geotechnical FEM analysis. Before applying these boundary conditions,make sure that you fully understand the ground water conditions for your problem. MostFEM programs treat any mesh boundary as impermeable by default. Setting the excesspore pressure to zero on a boundary means that the boundary is now able to drain.However, the task is not complete by just “switching on” the pore pressure boundary. Itseffect must be felt by the adjacent elements in the next time step. Otherwise, oscillationof pore pressures can occur. The minimum time step required for this purpose can becomputed based on the parabolic isochrone solution to the consolidation equation asshown in Figure 3.23.

Figure 3.23: Minimum time-step for dissipation of excess pore pressures

• When modelling excavation or construction by removing or adding elements to themesh, respectively, use several layers of elements and remove/add these elements layerby layer, applying each layer over several increments. This will ensure that the stiffnessof the soil being removed or added is correctly modelled.

y

uumax

L

y

uumax

Oscillation of pore pressuredue to insufficient time-steptcL v12= or

vc

Lt

12

2

min =



3.12 Concluding remarksIn the beginning of the chapter, we stated that site investigation and laboratory testing areused to obtain the input soil parameters for an analysis using FEM. As a geotechnicalengineer, one must never forget that soil is an extremely difficult material to characterize.Sampling disturbances, poorly controlled laboratory experiments, failure to interpret theresults from laboratory tests in a scientific manner are some of the factors that introduceerrors and uncertainty in the values of soil parameters. Therefore, the results of a FEManalysis must always be critically examined by comparing them with the results of anotherFEM analysis of a successfully completed project in similar ground conditions. Otherwise,one is likely to fall victim to the simplest equation of them all: Garbage In = Garbage Out !!

3.13 References1. Clough, R.W. (1960). The finite element method in plane stress analysis. Proc. Second

Conference on Electronic Computation, ASCE, Pittsburgh.

2. Courant, R. (1943). Variational methods for the solution of problems of equilibrium andvibrations. Bulletin of American Mathematics Society, Vol.49.

3. Zienkiewicz, O.C. and Taylor, R.L. (1989). The Finite Element Method, Vol. 1, BasicFormulation and Linear Problems, McGraw-Hill, London.



Centrifuge Modelling 1



4 Scaling laws and applications for centrifuge modelling

4.1 Introduction

4.1.1 Scaling laws

It is necessary to take the scaling laws into account before conducting any test series. Thescale effects need to be worked out as a basis for the planned modelling. The followingtable shows the different scaling factors. A brief discussion of these has been given at theend of chapter one. Some limitations to the modelling process and ways of solving some ofthese problems arising from the associated errors are mentioned briefly in this chapter.

4.1.2 Scaling of time

As in fluid mechanics, it is not always possible to achieve correct scaling in all dimen-sionless groups, and so choices must be made.

In dynamics, where acceleration in m/s2 scales as n in the model, and the linear dimensionis modelled at 1/n prototype, then time is modelled n times faster in the centrifuge.

But the scaling factor for modelling time in terms of diffusion may be demonstrated to be:n2 faster in the centrifuge.

The non-dimensional time factor, Tv = f(time/depth2) = cvt/d2 , becomes independent ofgravity level for a depth of sample reduced to 1/n of the original, if the model time is alsoreduced by 1/n2.

(1D Diffusion equation - saturated soil)

where u is excess pore pressure and time t scales with length z2 provided cv m = cv p .


(model/prototype)Acceleration m/s2 n

Linear dimension m 1/n

Stress kPa 1

Strain - 1

Density kg/m3 1

Mass or Volume kg or m3 1/n3

Unit weight N/m3 n

Force N 1/n2

Bending moment Nm 1/n3

Bending moment / unit width Nm/m 1/n2

Flexural stiffness/ unit width (EI/m) Nm2/m 1/n3

Tab. 4.1: Scaling laws

∂u ∂t⁄ cv∂2u ∂z2⁄=

Scaling laws and applications for centrifuge modelling 4 - 1


This offers a significant advantage because 27 years of prototype diffusion may bemodelled in 1 day using a centrifuge at 100 g, and is especially useful for environmentalproblems or heat loss by conduction where diffusion is the main transport mechanism.

However, in offshore foundations or earthquake problems, the pore pressures are createddynamically, with time scaling as: n times faster in the centrifuge and yet they decay in adiffusive process where time is modelled as: n2 faster in the centrifuge.

Solution: use pore fluid in the model with a viscosity of n times that of the prototype (andsame density) or reduce the value of permeability of the soil (Attention: this will cause achange in the properties).

4.2 Scale effects

The range and magnitude of possible shortcomings exposed by the scaling laws may bedescribed as scale effects. These must be reviewed to ensure that they will not affect theoutcome of the experiments and so the verification or modelling of models techniquemay well be a useful way of checking this.

Typical scale effects are:

• non-uniform acceleration field (with depth (beam & drum) - & width of model (beam notdrum))

• particle size effects• Coriolis acceleration• boundary effects (due to being at small scale)


(model/prototype) Time: diffusion s n2

Time: inertia s n

Time: viscous s 1

Frequency 1/s n

Tab. 4.2: Scaling of time



4.2.1 Stress distribution in centrifuge model: Depth

Since the inertial radial acceleration in the centrifuge is not linear with depth, but propor-tional to centrifuge radius, the depth to radius ratio of the model is of major importance.

Equation 1 ; ρ const with z.

density of soil = ρ radius to top of model =

model gravity level = n g depth below surface = z = r - r*

In the centrifuge: since radial acceleration varies with radius the vertical stress at z = r - :

r >

So distribution of vertical stress with depth in a centrifuge forms a parabola / quadratic.

Vertical stresses in the prototype are (nominally) linear.

We must select the best way of minimising the error... so we should set thecentrifuge (parabolic) and prototype (linear) stresses to be equal at some depth toachieve this over the important section (depths) of the model.

R

2/3rdsmodeldepth

over-stress

under-stress

δr

aR

r*

z

zz

σv σv

error

Model Vertical stress in Centrifuge Vertical stress in Prototype

r

Figure 4.1: Vertical stress distribution

σvm ρω2

2------ r2 r∗2

–( )= σvp ρg nz( )=

r∗

r∗

σv ρrω2 rdr∗

r

∫=ρr2ω2

2---------------

r

r∗ρω2

2---------- r2 r∗2 )–(== r∗



where (Equation 1) => σvm = σvp at z = aR

Effective radius of model = R occurs at a depth below surface of model = a R

where total depth of model = 3aR/2

so stresses are equal at r = R where = R(1-a) and z = aR

σvp = σvm

Check error (which will be close to the maximum understress) at z = aR / 2 :where r = R - aR/2 r*=R(1- a)

r∗

ρngaR ρω2

2------ R2 R2 1 a–( )2

–( )=

ρω2

2------R2 1 1– 2a a2

–+( )=

gn ω2R2

----------- 2 a–( )=

ω2 2ng 2 a–( )R ⁄=

ρ n g R a– a2 4⁄ 2 a⋅ a2–+ + ⋅ ⋅ ⋅ ⋅

2 a–( )------------------------------------------------------------------------------------------------=

ρ n g R a 1 3 a⋅ 4⁄– ⋅ ⋅ ⋅ ⋅ ⋅2 a–( )

--------------------------------------------------------------------------=

ρ n g R a 4 3 a⋅–( )⋅ ⋅ ⋅ ⋅ ⋅4 2 a–( )⋅

-----------------------------------------------------------------=

z a R⋅2

------------⎝ ⎠⎛ ⎞=Error at

ρ n g R a 4 3 a⋅–( )⋅ ⋅ ⋅ ⋅ ⋅4 2 a–( )⋅

----------------------------------------------------------------- ρ n g R a⋅ ⋅ ⋅ ⋅ 2⁄–⎝ ⎠⎛ ⎞

ρ n g R a 4 3 a⋅–( )⋅ ⋅ ⋅ ⋅ ⋅4 2 a–( )⋅

-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

4 3 a⋅–( )2 a–

------------------------- 2–⎝ ⎠⎛ ⎞

4 3 a⋅–2 a–

------------------------------------------------------------=

4 3 a⋅ 4 2 a⋅+––( )4 3 a⋅–

----------------------------------------------------= a 4 3 a⋅–( )⁄–=

σvm σvp–( ) σvm⁄=

understress

σvm ρ ω⋅ 2 R2 1 a 2⁄–( )2⋅ R2 1 a–( )2⋅– 2⁄=



If you use the overall height of the soil for your model, taking the effective radius of thecentrifuge at z=2/3 rds is a good choice to minimize the error. Looking at the problemsrelating to near surface (e.g. foundations or wave propitiation) other choices may be better.

Example: From the error which will occur at a depth of z = aR/2 in the model (see above) ifthe effective radius R is taken at 2/3rds model depth, where z = aR, calculate the error fora = 1/6, a = 1/8, a = 1/10, a = 1/12, a = 1/15. For R = 1m, calculate the total model depth.

Now calculate the overstress at z = 3aR/2 for the same range of values of a.

4.2.2 Stress distribution in a centrifuge model

States of equistress in the soil sample exist at common radii.

For a drum centrifuge, the surface of the model is at constant radius from the centrifugespindle, so this is not relevant.

However, soil models in the beam centrifuge (unless the ground surface is curved) willhave higher stresses at the package boundary. Likewise the ground water surface will alsobe curved and higher at the boundary (and this will affect the total stress (but not theeffective stress) at depth) ⇒ e.g. + 20 kPa here. The larger the radius of the beam, thelower is this effect. When investigating the response of a sheet pile wall, the structureshould be placed centrally in the mould.

Answers a = 1 / 6 a = 1 / 8 = 1 / 10 a = 1 / 12 a = 1 / 15

z = aR / 2 -1 / 21 -1 / 29 -1 / 37 -1 / 45 -1 / 57

z = 3 a R / 2 1 / 23 1 / 31 1 / 39 1 / 47 1 / 59

Model depth mm 250 182.5 150 125 100

Tab. 4.3: Example

1m

Width (Exaggerated scale)

Figure 4.2: Stress distribution as a function of width in a centrifuge



4.2.3 Particle size effects

How can the centrifuge be used to model soil if the particles are not reduced in size by afactor n?

• Model: clay particles of mean size d50 = 1µm. Prototype: sand at 100g with100 x 0.001 mm = 0.1 mm particle size?

→ not sensible: clay and sand have different particle shapes etc. and hence theywill have different stress-strain-volume change characteristics.

• Model: sand particles of 0.1mm. Prototype: sand (with 1mm diameter) at 100gwith 100 x 0.1 = 10 mm particle size?

• Model: sand particles of 0.1mm. Prototype: gravel at 100g with 100 x 0.1 = 10mmparticle size?

→ correct modelling process, but check particle shapes (rounded/angular) andparticle hardness (crushing) which will affect dilatancy + also beware of anyboundary effects.

Conclusion

• model appropriate stress-strain-volume change characteristics• particle size dimensions to be at least 1/15th but preferably >1/30th of the relevant

„model“ dimension

4.2.4 Coriolis acceleration

This may become relevant when particles are changing radius with some speed:

For example - building an embankment in-flight, but here the problem is really only relevantto the positioning of the embankment in the hopper.

d50

D• not equivalent• OK provided soil grain size is not significant compared to model

dimensions and to the boundary effects• D > 15 d50 (preferably > 30 d50)

Sand embankment

too far!

Clay

Wooden spacer block

Hopper

??

2 ωr·

rω2

Figure 4.3: Positioning of an embankment in the hopper



Consider error as a function of the nominal centrifugal acceleration:

so for r = 4m, @ 100 g, then ω ~ 15 rad/sec and for error < 10 %: radial velocity mustbe < 3.27 m/s

cf. angular velocity = r ω = 4 x 15 = 60 m/s (typically for example above), the range of theCoriolis error may typically be ~5% in terms of radial velocity over angular velocity.

So, for an earthquake model which is subject to changes in velocity, additional terms willbecome important.

horizontal shaking due to E/Q:

vertical shaking due to E/Q:

On the other hand, high velocity particles in blast loading are also less affected by errorbecause they will tend to move in straight (almost) lines, when the radius of curvature oftheir particle trajectory rc ~ r (and r > 3m).

4.2.5 Boundary effects

NB the pile or penetrometer must be reduced in scale (where the diameter D will be approx.10 mm and should be greater than 30 particle diameters) with a comparable size of the"stress bulb" within which the value of q is increased. The load cell used to measure q will'report' the hard layer 5-10 D below

• side friction- consolidation - soil movements from loading/unloading ⇒ DRAG

• stiffness of container (plane strain means ε2 = 0)

• base effects• refraction and reflection of waves

2r·ωng

---------- 10%<

r·

x· x··, r·ω( term )

y·· r··( term )

τσh’

SAND

CLAY

q

z

qq

“stress bulb”

same

boundaryeffects

D

>5D

Figure 4.4: ‚Plane strain‘ sand embankment on clay and pile installation or penetrometer penetration



Side friction

Lubricants are applied to the strongbox walls (which have also been coated with a lowfriction paint) to minimise side friction on the soil model. The principal stresses may thenbecome almost vertical / horizontal at the model boundary.

Similarly, latex sheets may be marked with a grid, greased and placed between a sandembankment and strongbox walls or perspex face to reduce soil-wall adhesion to ~5° atstress levels up to 300 kPa.

τ = σh' tan δ ∴ Allow for reduced "weight" in embankment.

Base effects

In-flight penetration of piles or penetrometers close to side boundaries or near the base ofthe box or a stiffer layer will affect the data because the rigid surfaces will influence thestrain field. It is recommended that there should be 5 penetrometer or pile diameters to aside wall. Likewise, for stiff soils, there should be 5 - 10 diameters below the pile tip to therigid layer.

Refraction and reflection of waves

Typically waves may be generated by blast loading or earthquakes, and these are reflectedor refracted at the boundaries. These boundaries can be rigid with vibration suppressingmaterials or stacked ring (flexible) systems. Both have advantages and disadvantages.However researchers are mainly interested in the first passage of the shock wave.

Example:

a) For a drum centrifuge with a radius of 1.1 m to the base wall (i.e. r = R + aR/2), cal-culate the maximum depth of model which would limit the over or under stress to+/- 5%.

b) Which other errors might be relevant and why?

c) If the centrifuge is able to achieve from 100 to 400 g, what range of rotation speeds is required and what range of soil depths are possible?

d) How many years of diffusion may occur in 1 day (24 hours) in the drum centrifuge during this same range of gravities?

Interface Lubricant OCR adhesion, δ°kaolin/kaolin Nil 1 18.8

“ “ 8 28.3

kaolin/perspex Nil 1 11.9

“ “ 8 18.8

“ Adsil spray* 1 6.3

“ “ 8 14.0

“ Silicone grease** 1 2.3

“ “ 8 5.1

*) transparent**)most effective; not transparent

Tab. 4.4: Kaolin/perspex residual friction characteristics from shear box (Waggett, 1989)



a)

errors +/- 5% ∴ a = 1/6 is o.k.⇒ ex.: Radius to drum wall = 1.1 m

R(1+a/2) = 1.1 m ∴ R(1+1/12) = 1.1 ⇒ R = 13.2/13 = 1.015 m

Depth of model = 3aR/2 = (3/12)*1.015 m = 254 mm

b)

Other relevant errors in a drum:

• particle size• possibly Coriolis acceleration from moving particles• boundary effects

NOT: stress distribution due to width of model because top surface is curved

c)

Rω2 = 100 g = 1.015 ω2 ∴ω2 = (100*9.81/1.015)*(30/π)2

ω = 296.8 r.p.m.

Rω2 = 400 g = 1.015 ω2 ω = 593.7 r.p.m.

Depth: 100 g for 254 mm model depth ⇒ 25.4 m prototype depth

400 g for 254 mm model depth ⇒ 101.6 m prototype depth (but reducing the scale by a factor of 400 can be very diffi-cult to achieve physically in terms of size of structures etc.)

d)

Diffusion is modelled as 1/n2

prototype 1 day ⇒ 1 x 1002 days @ 100 g

⇒ 27.4 years

prototype 1 day ⇒ 1 x 4002 days @ 400 g

⇒ 438.4 years

(a) 254 mm: (c) 296.8 r.p.m., 25.4 m@ 100g: up to 600 r.p.m., 101.6 m @ 400g (d) 27.4 yrs:438.4 yrs!

z

σv

<+5%

254 mmcentrifuge

prototype

<+5%



4.3 Scaling under earthquake conditions

Plane stress conditions, with gravity and base shaking, so resolving orthogonally:

→ δσx / δx + δτxy / δy + X =0

↑ δσy / δy + δτxy / δx + Y =0

For sinusoidal base motions, vertically propagating horizontal shear waves give:

prototype: model:

In the prototype, the body forces are inertial where:

So, for correct inertial modelling at 1/n scale:

• the model amplitude is 1/n times the embankment amplitude

• the max model velocity = max embankment velocity

cos (2πfmt) = cos (2πfpt) = 1

• the model acceleration is n times the embankment acceleration

y··– ng=

x··

E/Q

acceleration

X

Ybody forces

base shaking

x

y

prototype

freq.: fpamp.: ap

σx

σy

σx

σy

τxy

τxy

model

freq.: fmamp.: am

1/fp

-ap ap

t

xp

-ap /n = -amxm

t

1/fm = 1/ (n fp)

Figure 4.5: Scaling under earthquake conditions

2/fp

xp ap– 2πfpt( )sin⋅= xm am 2πfmt( )sin–=

x·p a– p2πfp 2πfpt( )cos= x·m a– m2πfm 2πfmt( )cos=

x··p ap 2πfp( )2 2πfpt( )sin= x··m am 2πfm( )2 2πfmt( )sin=

y··p g–= y··m ng–=

X ρap 2πfp( )2 2πfpt( )sin=

Y ρg–=

am∴ ap n⁄=

amfm∴ apfp=

fm∴ nfp=

y··m∴ ny··p=



so that:

→ δσx / δ(x/n) + δτxy / δ(y/n) + n X = 0

↑ δσy / δ(y/n) + δτxy / δ(x/n) + n Y = 0 cancels

• same equilibrium equations in model ⇒ same stresses at homologous points in theembankment

Referring to additional components of acceleration: y terms refer to vertical shakingand x terms refer to lateral shaking

Example

A prototype has 10 cycles of a 1Hz earthquake (duration 10 secs) with amplitude 0.1 m.Table 4.5 shows the implications of modelling this in the centrifuge with very small ampli-tudes and extremely short durations due to the high frequency.

Example

For a 50 litre model subjected to an earthquake as described in table 4.5 at 100g, calculatethe equivalent prototype size and the peak accelerations in model and prototype. At whatangle should the model be tilted to achieve the same % lateral acceleration? Now calculatethe model details for a scaling factor of 50 on the prototype quoted above.

model: prototype:

fm = 100 Hz fp = (100/100) = 1 Hz

am = 1 mm ap = 1 x 100 mm = 0.1 m

Volm = 50 litres Volp = 50 x (100)3 litres

n = 100 = 50,000 m3

Model gravity [g] Frequency [Hz] Duration [s] Amplitude [mm]100 100 0.1 1

200 200 0.05 0.5

Tab. 4.5: Modelling in a centrifuge

n∴

n = 100g

40.2 g θ

100g

0.402 g

Vol.:50,000 m3

1g

Figure 4.6: Model (left & centre) and equivalent prototype (right)



θ = tan-1 (40.2 g/100 g)

= 21.9 °

∴ For a model at 1/50 scale of this prototype:

Volume = 50,000 m3 / 503 = 400 litres ⇒ 8 x larger!

Amplitude = ap/n = 100/50 mm = 2 mm

Frequency = fp×n = 1×50 Hz = 50 Hz@ 0.2 secs duration

= 395 m/s2=40.2 g = 3.95 m/s2 = 0.402 g

xm am 2πfmt( )sin–=

x··m( )max∴ 10 3– 2π( )⋅2

1002 1 m

s2-----⋅ ⋅=

x··m +am 2πfm( )2 2πfmt( )sin=

xp ap 2πfpt( )sin–=

x·p ap2πfp 2πfpt( )cos–=

x··p( )max∴ 0.1 2π( )⋅ 2 12 1 m

s2-----⋅ ⋅=

x··p +ap 2πfp( )2 2πfpt( )sin=

x·m am2πfm 2πfmt( )cos–=



Some typical examples of applications for centrifuge modelling(mainly at Cambridge and Bochum)

Italics show projects completed in a small / large drum centrifuge: others are in a beamcentrifuge.

Dams and slope stability problems formed early topics of investigation in the centrifuge.Padfield (1978) exposed mechanisms responsible for ongoing and unpredictable slides inthe Mississippi levees. Endicott (1971), Beasley (1973), Goodings (1979) and Horner(1979) used 'real' soils in their slopes and they failed them vigorously. Ma (1994) studiedthe stabilisation of a kaolin dam by means of vertical concrete ribs. Schofield conductedextensive investigations into the Teton Dam failure with assistance from Avgherinos (1970).

Subsequently Davies (1981), Almeida (1984) and Barker (1998) constructed sand embank-ments on clay to failure. Jewell (1980) revealed mechanisms of reinforced soil behaviourthat have led recently to publication as a UK Design Code. Sharma (1994) reinforcedembankments on soft clay with various ground improvement measures to enhance shortterm 'undrained' stability.

Basic footing problems have also provided excellent benchmarks for comparison betweenfundamental methods (clay and sand - undrained or drained) Lau (1988), Shi (1988), Tan(1990). Bakir (1994) investigated the influence of foundations near to slopes, which led to anew french design code. Cyclically loaded footings have been studied e.g. by Laue (1996)or Bay Gress (2000). Nater (2004) studied the effects of layering of the ground on thebearing bahaviour. Deep foundations have also offered an excellent source of problemsowing to the ability to model long piles at small scale. Ah Teck (1982) and Barton (1982)investigated lateral pile group behaviour in sand. Grundhoff et al. (1998) used thecentrifuge to evaluate the moment distribution down into a pile due to the dynamic impactlateral of loads. (e.g. car colliding with a bridge pier). Gui (1995) reviewed pile andpenetrometer installation in sand, and there have been many examples of piling carried outfor offshore applications in clays. Gurung (1985) carried out pull-out tests on transmissionline foundations.

Combining foundation problems with soil structure interaction for bridge abutments on clayswas first investigated by Springman (1989) and Sun (1989) with subsequent work fromStewart (1993 - UWA), Bransby (1995) and Ellis (1997). Norrish (1996) created integralbridge abutments in sand and then applied a range of lateral cyclic deck movements tomodel temperature effects on backfill.

Powrie (1986) followed by Stewart (1989) conducted seminal work on the stability ofretaining walls in clay and the latter author extended this to nailing of clay excavations. Mak(1984) loaded a strip behind a retaining wall in sand. Kusakabe (1982) and Phillips (1987)investigated the stability of excavations, trench headings and shafts in clay. Reinforcedwalls in sand were recently investigated by Balachandran (1996).

Thin-walled buried pipes and culverts (Britto, 1979) and uplift mechanisms associated withpipelines (Ng & Springman, 1994; White, 2001) on and offshore have been modelled in thecentrifuge.

The offshore and coastal engineering industries have benefited more than most fromcentrifuge model testing in the development of current design methods (Craig, 1988). Theyare operating at such large scales, these can only be replicated at huge cost at full scale,and so the centrifuge provides a thoroughly economic alternative.



Mechanisms observed, in clay, for gravity platforms (Rowe, 1975), anchors, rupture due tocollapse of abyssal plains (Stone, 1988), the installation and response to a variety of axialand lateral forces as well as moments on a 3-legged jack up rig mounted on spud cans(Tan, 1990, Tsukamoto, 1994, Wong et al., 1993) other spud can investigations (Craig andChua, 1991, Springman 1991), monotonic and cyclic loading in an axial sense on tensionpiles (Nunez, 1989) and laterally on a variety of large tubular open ended piles (Hamilton etal., 1991), suction piles and anchors (Fuglsang and Steensen-Bach, 1991; Renzi et al.,1991), seabed mechanics and many other applications have been investigated usingcentrifuges around the world.

An offshore jack up platform has a complex interaction between the rig, leg and spudgeometry, depth of embedment in, and characteristics of, the founding stratum, and thedynamic wave and wind loading. To identify the basic controlling mechanisms, a single legwith a central column of an equivalent, scaled, bending rigidity, instrumented to allow anevaluation of the shear force and bending moment distribution is used. To mimic the lateralload transfer from the latticework to the surrounding soil, plates with an equivalent crosssectional area are mounted to fixings attached to the central shaft. For the general case ofa jack up platform, the soil-structure interaction is controlled by the combination of strengthand stiffness of the soft soil. By replicating a typical strength profile in the centrifuge model,the mechanisms which lead to failure in the prototype will be reproduced

Tunnels have also been investigated very successfully in the centrifuge. Mair (1979, 1984)inserted semi-cylindrical rubber membranes into clay and inflated them before he increasedgravity in a centrifuge. Subsequently he reduced the pressure and observed the defor-mation above the tunnels and the failure mechanisms which formed for different cover-depth-tunnel diameter ratios, for varying lengths of unsupported heading, for both 2 and 3dimensional effects. Design stability charts were produced to aid estimates of ultimate limitstate and serviceability limit state analysis. Taylor (1984), and Taylor, Stallebrass and Grant(1996) have continued this work in clay, following on from preliminary tunnel investigationsin sand by Potts (1976) and Atkinson (1977). More recently Bolton and Sharma (1996)succeeded in dissolving polystyrene tunnels (with various ‘tunnel’ linings) in sand to createand record surface deformations. König (1998) has built a tunnel drilling machine to followthe influence of the excavation process inflight. Compensation grouting and hydro-fracturein relation to limiting settlements above tunnels has also been achieved by Chin (1996) &Lu (1997).

Environmental effects are ideally suited to modelling in a centrifuge. Time for diffusionprocesses is increased by a factor of n2 and so 27 years of pollution migration may bemodelled in one centrifuge day at 100 g and nearly 700 years in one day at 500 g! Clearlyboundary effects become very important but there is potential to model transport processesHensley (1989), flow in multi-layered soils Boyce (1994), hydraulic, density and electricaleffects on transport processes Hellawell (1994), Potter (1996), contaminant migrationthrough intact or damaged liners Evans (1994), isokinetic clean up Penn (1997). Gronow(1984), Edwards (1986) and Price (1996) have also used the two drum centrifuges atCambridge to model transport processes.

Hot thermal influences, e.g. sub-seabed disposal of high level radioactive waste, Savvidou(1984) are also well modelled in the centrifuge. A heat source buried in a deep clay layergenerated pore pressure and caused cracking due to the differential thermal coefficients ofexpansion of both the soil and fluid phases. Her analysis considered the coupling of heatand fluid flow.



Poorooshasb (1988) fired projectiles of differing shapes (nominally containing such a radio-active waste) into a soft clay layer and then heated the environment around which theprojectile came to rest. The primary question was whether a gap would be left behind theprojectile, creating a preferential leakage path or whether the soil would deform in a suffi-ciently plastic fashion to close that gap. The effect of heat on this region was found toincrease the likelihood of leakage along this softened zone due to thermal cracking.

Cold regions engineering will have applications in Switzerland. Previous work at Cambridgefocused mainly on the influence of ice forces on offshore structures (Vinson, 1982; Lovell,unpublished, Jeyatharan 1991) or iceberg scour above pipelines buried below the seabed(Lach, 1992).

Experiments in which the ground has been frozen have been concentrated on mainlyonshore applications. Smith (1992) investigated the thaw-induced settlement of pipelinesand Vinson (1983) discussed the effect of hot fluids flowing through pipelines in frozenground.

There are few studies on the response to earthquake excitation for structures built on clay.The difficulties of reconciling scaling laws for diffusion time and inertial time necessitate theuse of a more viscous pore fluid, which is complicated in fine grained clayey soils. However,work has been carried out on basic dynamic behaviour (Morris, 1979), embankments(Kutter, 1982; Lee, 1985), slopes and dams (Habibian, 1987; Pilgrim, 1993), retaining walls(Steedman, 1984; Xeng 1990), piles (Maheetharan, 1990), towers (Madabhushi, 1991).Most of these have been in sand with a replacement pore fluid with a viscosity 'n' greaterthan water. This ensures that pore pressure dissipation rates are modelled at n times fasterthan prototype time, which is the same time scale followed for the generation of porepressures in a dynamic sense.

The basis of the use of the centrifuge for dynamic problems such as e.g. wave propagationhave been studied by Siemer (1996).

The centrifuge is not suited to examining chemical effects or long term timedependent processes such as creep.







Practical considerations: mechanical Page 5 - 1

5 Practical considerations: mechanical

5.1 Beam Centrifuges

5.1.1 Capacity

Of major importance in categorising centrifuges is:

• the acceleration level possible (ng), and• the maximum payload (tonnes)

This is frequently cited as the g-ton capacity of a centrifuge.

Cambridge (UK): (1972) 0.9 t x 125 g = 112.5 gton at 4 m radius

ISMES (Italy): (1983) 0.4 t x 600 g = 240 gton at 2 m radius

Bochum Z1 (Germany): Krupp (1985) 2 t x 250 g = 500 gton at 4 m radius

Nantes (France): 2 t x 100 g = 200 gton at 5.5 m radius

UWA, Perth (Australia): (1989)City University, London (GB)Bochum Z2 (Germany): (1992)

0.2 t x 200 g or0.4 t x 100 g

= 40 gton at 1.8 m radius

Tab. 5.1: Geotechnical centrifuges (international)

0.4

0.2

100 200acceleration g

payl

oad

t

Figure 5.1: Acutronic 661 (UWA, City and Bochum Z2)



5.1.2 Swing platform, package and liner

The soil model must be prepared in a container (or liner) and may need to be transferred toa 'strongbox' which can carry the loads applied due to enhanced gravity. There are somecentrifuges that have their model integral with the beam system, but allowing model makingin a separate liner. However, most centrifuges have a swinging platform which providessupport to the base of a strongly reinforced box. This platform is usually in the form of aswing and must be mounted on the centrifuge arm. This is ideal because the model hangsalways in the resultant direction of acceleration (e.g. vertical at 1g and horizontal at ng).

Figure 5.2: Arrangement of centrifuge arm during test (Cambridge)

Figure 5.3: Plan view of centrifuge arm with camera mounting to shoot vertically downwards



Beam balance

It is important for continued good health of the centrifuge to protect the bearings byensuring that the beam remains in balance. The weights applied to the other end of thebeam are designed to achieve this. Therefore, careful calculations of the mass of all itemson the package times their centroidal heights are required to find the total mass and meancentroidal height. For example, the Cambridge beam is limited to difference in mass xradius of +/-5 kgm between each end. More modern machines generally have an automaticbalancing system, based usually on a vibration measuring system.

Users must take into account movements occurring during the test - e.g. passing largevolumes of fluid from a storage tank to the model, pouring sand from a hopper, installing/penetrating a structural device. If necessary, two calculations should be carried out toestablish the range of centroidal heights to check the mass x radius is acceptable, so thatthe counterweight may be set between the two limiting cases.

Plumbing: water, air, other fluids

It is possible to pass water, air and other fluids (e.g. a 'model' pollutant) through the sliprings on the central spindle to reach the centrifuge model. This is useful in setting up basicgroundwater conditions, steady state flows within a model, providing water or air into arubber bag to provide a normal load, providing a supply of 'pollutant' to a landfill for a shortperiod of time, pressurising one side of a hydraulic jack to 'actuate' a device (installing apile, simple monotonic load, opening the hopper etc.).

Actuators

All actuators designed for 'fitness for purpose':

• can apply or measure suitable range of loads with sufficient accuracy,• where and how are they mounted on the package,• must be as light as possible, but robust & manoeuvrable,• think about the strength of soil to be tested to design

→ size of 'motor' or pressure ranges to apply load etc.,

→ size of load cell to measure strength accurately (i.e. range of linear responsederived from calibration).

• NOTE: must remember that fixings should allow for Newton's 3rd Law.......for everyaction is an equal and opposite reaction!

Load (static, monotonic, cyclic, construction)

Are the actuators load controlled or displacement controlled???

static: pressure bags - made from latex rubber to suit required size & shape - LOAD

monotonic: single thrust actuators (maybe hydraulic rams - lattice leg) - LOAD

cyclic: electrically (lateral loading on piles) or mechanically (abutment movements atintegral bridges) induced - DISPLACEMENT

hydraulic: LOAD and DISPLACEMENT but be aware of change of stiffness of the wholesystem

construction: use of self weight and hydraulic actuation for installation (piles, lattice leg) -LOAD, or opening of ports at the base of the hopper (e.g. electrical solenoid) also LOAD



5.2 Drum Centrifuges

5.2.1 Capacity

Date of commiss-ioning

Drum dimensions

mDepth x Perimeter x Width

Payload

t

Max g level

g

Capacity

gton

Location

1971 0.025 x (0.25 x π) x 0.12 ~ 0.003m3 0.006° 1000* 6.1° UMIST, UK(Schofield,1976)

1979 0.13 x (1.2 x π) x 0.23 ~ 0.11m3 0.2 650* 145 Davis, USA(Fragaszy & Cheney, 1981)

1986 0.1 x (0.8 x π) x 0.3 ~ 0.075m3 0.195° 150* 43° Utsonomiya, Japan (Kusakabe et al., 1988)

1988 0.15 x (2 x π) x 1 ~ 0.95m3 ~1.7 400 ~ 675 Cambridge, UK(Dean et al., 1990)

1995 0.115 x (0.74 x π) x .185 ~ 0.05m3 0.13° 416 ~ 50° Hiroshima, Japanformerly Cambridge minidrum Mk I (Kusakabe & Gurung, 1997)

1995 0.12 x (0.74 x π) x 0.18 ~ 0.05m3 0.13° 400 ~ 50° Cambridge minidrum Mk II(Barker, 1998)

1996 0.17 x (1 x π) x 0.25 ~ 0.13m3 0.2 450 90 COOPE, Brazil† (Gurung et al., 1998)

1997 0.15 x (1.2 x π) x 0.3 ~ 0.23m3 0.6° 484 290° UWA, Perth‡ (Stewart et al., 1998)

1998 0.2 x (2.2 x π) x 0.8 ~ 1.1m3 ~3.7 440 ~ 1600 Toyo, Japan (Miyake & Yanagihara, 1999)

1999 0.3 x (2.2 x π) x 0.7 ~ 1.45m3 2.0 440 880 ETHZ, CH(Springman et al., 2001)

Tab. 5.2: Drum centrifuge capacity* horizontal axis

° estimated for better comparison with a fill of density of 2600 kg/m3 as no detailed data wasavailable† a similar machine is working at the MIT‡ an identical machine is in operation by Kiso Jiban Consultants in Tokyo and another with thesame size drum at the TIT Tokyo



The ETH drum allows the model to be made in the drum, which is mounted on the outer oftwo shafts and rotated by a 37 kW motor. An inner shaft is connected to a separate toolplatform on which actuators can be mounted (e.g. excavate soil, load foundations etc.).These shafts can be linked to rotate together or to be operated independently of each other,in which case the inner shaft is powered by a separate motor. If the drum is still rotating butthe tool platform is stationary, a safety shield may be lowered and locked in place, whileadjustments are made to the tools mounted on the platform.

Figure 5.4: ETHZ Drum Centrifuge, Ø 2.2m, ETH Hönggerberg



Full field or small package configuration

The options exist in all drum centrifuges to fill the entire drum ring to create a large deposit(e.g. equivalent to 40 m deep, 1 km long, 80 m wide in a minidrum or 100 m deep, 2.5 kmlong, 250 m wide in the ETHZ drum), or to place a smaller model within a 'box' (e.g. equiv-alent to 40 m deep, 60 m long, 60m wide or 10 m deep, 15 m long, 15 m wide). If the latterwas provided with a perspex face, then it would be possible to take lateral photographs ofthe cross section with depth.

Figure 5.5: Cambridge Geotechnical Drum Centrifuge



Balance

Balance is also a concern on the drum to prevent it from stressing bearings unequally andto cause the drum shell to go out of shape. The maximum out of balance of the ETHZ drumthat is allowed will be 10 kgm at 440 g or 600 r.p.m. and proportionally greater as the gravitylevel reduces. Accelerometer transducers have been installed to measure vibrations and tohelp ensure that the drum remains within the balance limits. There is a similar limit to themaximum out of balance, which is allowed on the tool table (1 kgm at 440 g).

Plumbing: water, air, other

This is very similar to requirements for the beam centrifuge, but it is also necessary tosupply water/air to both:

• the central console or tool plate• the drum ring

There are two porous toroidal pipes inside the drum, which distribute the water suppliedfrom above the centrifuge via an open channel on top of the drum. A series of switches linkto a standpipe and a set of valves below the drum to control the water flow or to maintainthe water at a specific level within the drum.

Actuators

These are similar in complexity to beam actuators but there is much more space to mountthem and for them to act since they are generally mounted on the tool plate (Fig. 5.6). It isintended that:

• they should be able to rotate with the drum ring and hence carry out operations on themodel,

• disengage and become stationary so that activity could take place within a shieldedregion around the tool plate,

• the spindle could move at differing speeds to the drum ring

→ either to locate to a new position relative to the drum or

→ to plough a furrow for example....or tow a drag anchor

• there are fewer drums, so more opportunity for development!

Figure 5.6: Tool table with horizontal actuators (a) and piecocone placed in the actuator (d =11.3mm) (b)

(a) (b)



5.3 Site investigation devices (penetrometers, vane)

The main questions are:

• what mode of actuation is required?• what speed of testing - given the centrifuge scaling laws?• what are the geotechnical aspects of what we are measuring....

5.3.1 Vane:

• this device is for clay only• it is necessary to drive the vane down to the required depth and then rotate it• speed of rotation should be about 72°/min• we are finding su,vane (torque is measured & su,vane deduced)

• separate the shaft friction, peak friction and residual values: a 15° slip coupling is used

5.3.2 Penetrometer:

• continuous penetration• displacement rates between 0.6 - 12 mm/min• consider whether this is into soft clay, stiff clay, dense sand?• motor/actuation must be able to achieve this axial push• for sand, need a stronger, stiffer load cell than for clay

• diameter likely to be 10 mm rather than area of 10 cm2 at full scale• measuring tip resistance qc and sometimes the pore pressure u

5.3.3 Cylindrical T-Bar:

• continuous penetration• displacement rates between 0.6 - 100 mm/min• only for soft soils (clay)• measuring the total force to deduce su

• interpretation based on plasticity calculation of soil around a cylinder (and shearing pastthe end faces)

Figure 5.7: Vane Figure 5.8: Penetrometer



5.4 Post-test investigation devices

5.4.1 Photographic: camera & flash

Both still photographs and video films are useful in revealing soil deformation mechanisms.These require a solid base on which to mount the cameras and sufficient light. Tiny videocameras (15 mm diameter, 50 mm long) are now available with adequate definition torecord marker positions to the necessary degree of accuracy. They may also be able topass the signals by radio waves. When combined with novel methods of 'grabbing' videoframes or digitising these directly, it is possible to use these in preference to still cameras.

5.4.2 Digital Images and PIV analysis

Image capture using an inexpensive 2-megapixel digital camera provides a significantincrease in resolution and image stability compared with video. Particle Image Velocimetry(PIV) is a velocity-measuring technique in which patches of texture are tracked through animage sequence. Image processing algorithms are available to apply the PIV principle toimages of soil. The software has a precision of 1/15th of a pixel when tracking themovement of natural sand or textured clay. The system allows displacements to bemeasured to a precision greater than with still photographs or video films without installingintrusive target markers in the soil.

5.4.3 X-ray

It is common practice to insert mixtures of lead, water and soluble oil into a clay stratum orlead balls into a sand stratum, at the model making stage. Before testing at higher gravities,exposure of a suitable film to X-rays reveals the initial location of these lead markers.Subsequently, the soil deformations caused by the loading sequences will allow post-testexamination of these internal movements by identical radiographic techniques. Clearly thisrequires a radiographic device with sufficient power to penetrate soil of certain thicknesses,the aluminium liner and sometimes the strongbox. It also requires sufficient shielding tocomply with local safety requirements.






Practical considerations: geotechnical Page 6 - 1

6 Practical considerations: geotechnical

6.1 Introduction

What happens to soil as gravity increases in a centrifuge model?

Following model making, there are oftensuctions present in a clay (maybe up to -30to -40 kPa). This causes extra effectivestress in the soil. Once stresses have beenincreased to those caused by 100 gravities,the total stress increases together with thepore pressure. Subsequent dissipation to asteady state, long term, stress condition is afunction of the non-dimensional groupdesignated by Tv:

Tv = cv t /d2

where cv is coefficient of consolidation, t isthe time and d is the drainage path. Dissi-pation will often take between 2 - 20 hoursin the centrifuge at ng depending on thethickness of the clay layer and whether theclay can drain in one or two directions (d or2d) and the permeability of the soil.

The model must be in equilibrium before further perturbation commences.

100 mm 10 m

short long long

z

σv , σv’

u u

Model: 1g Model: 100g

σv = 1.5 σv’ = 11.5 u = -10 kPa σv = 150 σv’ = 50 u = 100 kPa LONG TERM

σv = 150 σv’ = 11.5 u = 138.5 kPa SHORT TERM

Figure 6.1: Stress distribution with depth

Figure 6.2: Time since start of initial accelerationof the centrifuge

Time (sec)



6.2 Design of soil model: real or laboratory

We aim:

• to describe a variable matrix of rock and soil in terms of zones and layers, and if possible• appropriate properties are designed to suit the "prototype" soil conditions and to allow

measurement and analysis of performance under some loading regime.

Selection of soil for a centrifuge model:

• 'real' or ‘natural’ prototype soil with less classification data and knowledge of parametersand their likely ranges

• specific laboratory soil - known parameters, classification details.

For clay deposits:

• undisturbed samples of ‘real’ or ‘natural’ prototype soils, local structure is maintained,and naturally occurring preferential seepage paths remain (unsmeared),

• remoulded natural soil,• most appropriate remoulded 'laboratory' soil (i.e. kaolin) with an extensive database of

properties.

For cohesionless deposits:

• disturbed, re-poured and compacted real soil, or• known laboratory soil:

→ poured and compacted to a specific relative density

→ with particle size selected as function of some controlling dimension

→ this could be single size, uniform or well or gap graded.

In the field:

• if field samples are taken from an overconsolidated deposit, they should be sampledfrom the weakest stratum (lowest value of su ; point X for o.c. clays, point Y for n.c. clays)

• subsequent stress fields imposed in the centrifuge may lead to the best approximation ofthe key field strength conditions, but in consequence, the strains will not be entirelycomparable.

o.c.

Depth

su

n.c.

YX

Figure 6.3: Variation of shear strength with depth and influence of stress history



Comparison between the use of real or laboratory clayey soils in (beam) centrifuge model tests

Activity Field prototype soil Laboratory alternative soil

Aim To reproduce the exact soil behaviour from the field in the centrifuge model.

To model the relevant prototype soil behaviour using a known laboratory soil.

Sampling and

disturbance

Inserting a cutter to extract a block sample, preparation for transport and subsequent

trimming to fit a centrifuge strongbox, is likely to cause softening and loss of peak strength

(extent depending on OCR of the soil).Disturbance will be significant.

Mixed from slurry under vacuum, consolidated in liner/consolidometer before transfer of liner

containing the soil to the centrifuge strongbox. Sample is assumed to be uniform and

homogeneous in stress state and properties. Disturbance is not likely to be a problem.

Local structure

If 'fabric' exists, it is difficult to quantify & is likely to confuse back analysis of model test.

No local cementation exists; critical states may be adopted to explain soil behaviour.

Aging Effect of local cementation, particle shape variations, load cycling: soils will be disturbed during sampling, and are difficult to model or

interpret.

The repetition of 1g - ng loading cycles appears to cause a slight increase in strength. Further

research is underway at the moment.

Particle size

For soil deposits which contain random large particles, it should be remembered that these

will be enlarged by a factor of n with respect to the model dimensions. Therefore, shell

fragments in remoulded calcareous deposits assume a reinforcing role out of proportion to

the rest of the soil layer, and should be removed. Root systems in naturally occurring

clays may become major tunnel networks.

The size of any structure/probe acting on granular soil particles of mean diameter, d should

be > 15 d. The same field and model particle size/void ratio/pore fluid will not affect

permeability. Time taken for dissipation of excess

pore pressures (diffusion) will be n2 faster. Sometimes pore fluid with n times higher

viscosity may be used so that both diffusion / inertial velocity will be factored by n.

Permea-bility

Some natural clays (e.g. montmorillonite) take lengthy consolidation time at n g.

Relatively permeable clays such as kaolin may be used to minimise consolidation time.

Homo-geneity

Unless the sample is remoulded, it is difficult to assess this until after the test.

This is completely controlled in the centrifuge sample.

Hetero-geneity

The field sample is more likely to be anisotropic.

1d consolidation: no radial strain at sample boundaries; stress field assumed uniform.

Properties Exhaustive sets of laboratory tests may be necessary to establish soil parameters for design of test/subsequent back analysis.

Extensive databases exist for. Classification and extra laboratory tests using appropriate stress

paths may be carried out.

Stress history

Undisturbed samples will retain the stress history and sampling effects at the nominal

depth of sampling +/- 200 mm. The centrifuge model will reproduce stress history for n times the depth of real samples, from ground surface

downwards. The prototype may have been subjected to differing stress histories and

paths.

This may be designed to suit the requirement of strength and stiffness with depth, remembering

that the sample is consolidated at 1g under uniform total stress. The effective stress profile

may be manipulated by using upward / downward hydraulic gradients to create a linear

pore pressure profile.

Global structure

Samples taken from one/several number depths do not represent the global variation of

OCR, strength, stiffness with depth.

Realistic variation of some properties (e.g. OCR) with depth, but less realism in others (e.g.

anisotropy).

Results Distortion of global effects due to samples not representing fully the in-situ soils.

Global effects modelled better except soil properties may different from field soil.

Conclusion Direct scaling from model to prototype possible in rare instances only. Usually, model data needs to be understood in an analytical framework, and then applied to the field situation.

Tab. 6.1: Comparison between ‘real/natural/field’ prototype soil and a laboratory alternative



Design of stress history

Similar arguments may be employed when considering whether to use natural orremoulded soils in the drum centrifuge (Züst, 2000; Fauchère, 2000).

• future soil behaviour is a function of past stress history, recent & anticipated stress path,• we can create a centrifuge model in the laboratory to control both stress history/stress

path due to 1D consolidation prior to model making and reconsolidation in the centrifuge,• it is possible to design soil deposits to exhibit a chosen range of strength and stiffness,

although there are some anisotropic details which cannot be modelled effectively.

How do we design strength profile?

• decide strength profile required => su

• consider link between su & σv'

• design stress history to achieve required profile of σv'

Figure 6.4: Undrained shear strength versus depth, shown as f(g-level); for stress history seeFigure 6.10

undrained shear strength su [kPa]



What strength?

If:

• current effective vertical stress = σv'

• past maximum effective vertical stress in one dimensional normal compression= σv,max’ = σvc'

• so that overconsolidation ratio OCR = σv,max' / σv'

su / σv' = a OCR b (1)

where a and b are constants: see Table 6.2 for data for speswhite kaolin. (For OCR = 1,su = a σv' )

Does vane shear strength vary when compared against test methods or classificationparameters?

Strength ratio su /σv' or su /σv,max' (in Fig. 6.5) is dependent on plasticity index Ip for vane:correct via factor µ to calculate su for design:

where: su design = µ su vane (2)

su design = µ σv' a OCR b (3)

so µ = 1, Ip = 20 % & µ = 0.9, Ip = 32 % (kaolin).

Vane data for the jack up 'lattice leg' sample arecompared with the expected values and in-flighttests at 100 g (uncorr. by Bjerrum's factor µ). Areasonable approximation has been achieved forsu,vane peak over the top 200 mm of kaolin,assuming that average su,vane is represented bythe mid-depth su,vane (see Fig. 6.6).

Researcher Date a b Comment

Nunez 1989 0.22 0.62 based on vane test

Phillips 1987 0.19 0.67 “

Springman 1989 0.22 0.71 “

Tab. 6.2: Speswhite kaolin

Figure 6.5: Correction factor, µ(Bjerrum, 1973)

Figure 6.6: Expected and deduced values ofsu



Also: vane prediction may not relate to the actual sumobilised in triaxial compression (TXC)/extension(TXE), simple shear, plane strain compression/extension etc. (see Fig. 6.7).

How does su from major laboratory tests vary withvane shear test data ?

Equivalent su in triaxial compression (txc) / extension (txe) for normally consolidated clays,together with vane data corrected for strain rate (Bjerrum, 1973) shows su,txc ~ 50% greaterthan su,vane,corr, su,txe ~ 33% smaller then su,vane,corr (for Ip = 32%).

Results in simple shear, on normally consolidated speswhite kaolin (Ip = 32%) at constantvolume gave φ' = 21.8° (Airey, 1984) and su/σv,max' = 0.18, whereas Al Tabbaa (1984)found φtxc' = 23° when OCR = 2.

Figure 6.7: Strength ratio su/σvc' =

su/σv,max' versus plasticity

index Ip for vane tests

Laboratory and field tests (Bjerrum, 1973)



6.3 Kaolin as a model soil

Notes: φtxc', φtxe', φpsp', φpsa', are values of φcrit' for triaxial (tx) compression (c),extension (e) and plane strain (ps) passive (p), active (a) respectively.

Since 1978, speswhite kaolin has been supplied for laboratory testing in place of spestonekaolin, and Mair (1979) commented that both clays exhibited similar soil properties.

So given we can design a soil strength profile.... how do we fix the stress history?

• using preconsolidation at 1 g in the laboratory (e.g. as below for σvc’=100 kPa), there areat least three possible methods:

→ soil in a liner under a hydraulic press ⇒ almost constant σv' with depth,

→ can apply a hydraulic gradient to allow σv' to vary with depth,

→ allow partial consolidation to create a stiffer crust (& base) as a second phase1 g consolidation,

• allowing consolidation in the centrifuge

→ either after preconsolidation at 1 g, so some of sample will usually be o.c.,

→ from slurry, so sample will be entirely n.c.,

→ changing g-level for short periods of time,

→ adding and removing a surcharge,

→ applying a hydraulic gradient.

Clay Method ofconsolidation

Ip

%

φtxc‘ (°) or

φpsa‘

φtxe' (°) or

φpsp’

Source

Kaolin Isotropic, tx 20 24 24 Yong & McKyes (1971)

Kaolin Isotropic, tx 25 29.2 36 Broms & Casbarian (1965)

Spestone kaolin Isotropic, tx 32 22.6 20.5 Nadarajah (1973)

Spestone kaolin K0 , tx 32 20.8 28.0 Nadarajah (1973)

Spestone kaolin K0 , ps 32 20.9 21.7 Sketchley (1973)

Tab. 6.3: Properties of kaolin clays: frictional strength (Airey, 1984)

Clay Method ofconsolidation

Ip

%

(su/σv,max‘)

txc or psa

(su/σv,max‘)

txe or psp

Source

Kaolin Isotropic, tx 25 0.43 0.34 Broms & Casbarian (1965)

Spestone kaolin Isotropic, tx 32 0.215 0.205 Nadarajah (1973)

Spestone kaolin K0 , tx 32 0.205 0.175 Nadarajah (1973)

Spestone kaolin K0 , ps 32 0.20 0.16 Sketchley (1973)

Spestone kaolin K0 , ps 32 0.30 0.18 Ladd et al. (1977)

Tab. 6.4: Properties of kaolin clays: undrained strength (Airey, 1984)



Consolidation at 1g, Centrifuge test at 100g

Depth Depth

Overconsolidated

Normally consolidated

1 2 10100

at 1g

σv,max’

200 mm20 m

Vertical effective stress [kPa] Overconsolidation ratio Undrained shear strength [kPa]

0 10 20

200 mm20 m

Figure 6.8: Consolidated @ 1g, test @ 100g

Figure 6.9: (a) plane strain 1g consolidometer, (b) axisymmetric 1g consolidometer/downward hydraulic gdt.



What is consolidation with downward hydraulic gradient?

Figure 6.10: Development of stress history at 1g

su

Figure 6.11: Variation of su,vane and OCR with depth at 100g



Example: kaolin layer, 350 mm deep:

• initial σv,max' = 110 kPa after drainage to an excess pore pressure of zero,

• if we consolidated at 100 g in soil with γ' = 6 kN/m3 => σv' = 6 x 0.35 x 100 = 210 kPa atthe base of the sample: 0.35 x 100 = 35 m depth in the prototype, but before this,

• a 2nd stage consolidation may follow: σv is increased to 225 kPa, and

→ a seal in the circumference of top disk press allows additional pore waterpressure to be created, so downward hydraulic gradient with u = 165 kPa atthe top of sample, decreasing to 0 at the base,

• σv = u + σv' : σv' profile is assumed to be linear: 225 - 165 = 60 kPa at the surface and225 kPa at the base,

• OCR profile is then given for equilibrium at 100g, together with the ideal profile ofsu,vane ,

• σv' exceeds previous σv,max' below model depths of 170 mm (17 m prototype; point X inFig. 6.11),

• below this, OCR = 1, predicted profile of su,vane is linear with depth.

Example: Using the stress history from the example above and equation (1), verifythe undrained strength profile (su,vane) shown in Figure 6.4. What values of a and bappear to have been used from the Table 6.2?



How can we create a stiffer crust?

• carry out first phase 1g consolidation at 100 kPa,• apply second phase 1g consolidation at 200 kPa for a short period of time, • parabolic isochrones indicate extent of excess pore pressure dissipation and show

increasing depth penetration from drainage boundaries with time, = (12 cv t )0.5 (shownbelow in Fig. 6.12 for drainage in both directions).

• so total value of σv,max' increases with time at shallow depths,

• so OCR much higher above 100mmm / 10mp,

• and strength increases with time allowed for 2nd phase consolidation,• more extreme values of su,vane in the crust are achieved with much greater temporary

values of σv,max'.

2nd phase consolidation => stiffer crust

Figure 6.12: Two phase consolidation (1st phase 100 kPa @ 1g, 2nd phase 200 kPa @2g)

What happens if slurry is consolidated directly in the centrifuge?

Normal consolidation in the centrifuge

Depth

Depth

Overconsolidated

Normally consolidated

1 2 40100

2nd at 2g

σv,max’

200 mm20 m

Vertical effective stress [kPa] Overconsolidation ratio Undrained shear strength [kPa]

0 10 20

200 mm20 m

Stiffer crust1st at 1g

200

at 100g

Depth Depth

all depths normally consolidated

σv,max’

20 m

Vertical effective stress [kPa] Undrained shear strength [kPa]

0 10 20

20 m

at 100g

Figure 6.13: Consolidation from slurry in the centrifuge



The g-level could also be increased for a period of time although this would tend to increasethe maximum effective stress at the base of the sample first (theory of parabolicisochrones). If full consolidation (dissipation of excess pore pressure) was allowed then thegradient of the lines describing or dsu/dz will double but remain triangular inshape. Drainage of these excess pore pressures will develop first at the base.

An alternative in the ETHZ drum centrifuge will be to place temporary surcharge of∆σv = γdhSand (e.g. sand) over the consolidating clay layer (Fig. 6.14 & Fig. 6.15a). Eitherpartial or full dissipation of excess pore pressures could be permitted prior to removal of thissurface layer (with a scraper tool) depending upon desired undrained shear strength.

dσ'v max, dz⁄

Depth of clay Depth

σv,max’

20 m

Vertical effective stress [kPa] Undrained shear strength [kPa]

0 10 20

20 m

at 100g

Figure 6.14: Consolidation from slurry in the centrifuge

1 OCR

∆σv = γdhSand

n.c.

Overconsolidation ratio Undrained shear strength [kPa]

assume ∆σv

const. over

clay layer



A hydraulic gradient could also be applied in the drum centrifuge by ponding water on top ofthe clay layer and ensuring there was no pore pressure in the sand layer at the base of theclay (Fig. 6.15b & Fig. 6.16).

These techiques are very much under development currently. Successful attempts havebeen made in the Cambridge mini drum centrifuge to date.

On of the major questions remains the installation of pore pressure transducers and howthese are

- protected to ensure the ceramic filters remain saturated and- that they are located at the expected depths.

How do we prepare our centrifuge models in the laboratory?

z zw

waterclay layer

base sand layer

drain open:u = 0 kPa

zhSand

clay layer

base sand layer

surcharge γd

(a) (b)

Figure 6.15: Drum centrifuge test setup(a) Sand surcharge(b) Downward hydraulic gradient

Depth of clayz

Total vertical stress [kPa]

at 100g

Figure 6.16: Application of a downward hydraulic gradient

γwzw

Pore pressure [kPa] Effective vertical stress [kPa]

γwzw

at 100g

σ'v,max= +

γwzw



clay mixing

• using reconstituted laboratory soils, clay powder is mixed into a slurry with w > wL (e.g.the Cambridge University Soils Group used to mix kaolin at w = 120%, University ofBochum mix kaolin at w = 70-80% but for a longer period) under a vacuum for a period ofnot less than 2 hours (so that the mixture is homogeneous/fully saturated),

• the slurry is then placed inside the liner/consolidometer or strongbox, trying to ensure noair bubbles are included,

• porous strips and filter papers, saturated with de-ionised water are located above andbelow the sample.

N.B. ETHZ are currently developing their own methods for application to clay placement inthe drum centrifuge.

NOTE: add rock flour to kaolin to change stress ratios at failure, φ'crit , permeability,stiffness etc.

consolidation process in strongbox

• consolidation pressure will be applied to a rigid plate in contact with drainage layers / soildeposit,

• pressure is regulated in stages => σv' on the consolidating deposit,

• cavitation effects due to suctions (negative values of u) from unloading from σvc' may beminimised:

→ by using stress decrements ∆σv < 100 kPa with all drains open,

→ allowing for equilibration of the pore pressures at each stage,

→ possible air entry at the edges of the model is limited by permitting controlledswelling,

• allow for any subsequent consolidation settlement or swelling in planning installationdepth of pore pressure transducers (PPTs), which are inserted between last 2 loadingphases via temporary unload,

• holes obtained following PPT insertion are backfilled with slurry injected through asyringe to ensure that any air will be excluded from the soil matrix (procedure specifiedby Phillips and Gui, 1992),

• transducer wires are taken out through special ports in the side of the 850 mm diametertubular strongboxes or pressed into the rear of the plane strain soil sample, with thewires passing through a small slot in the piston,

• any possible reinforcing effect of the transducers and their cables must be considered,• consolidation process in drum centrifuge.

How to calculate how much clay is required at model-making stage?

e.g. For fully saturated speswhite kaolin, with specific gravity γs = 2.61 × γw (Airey, 1984),the following relationship may be used to calculate void ratio e following consolidationunder σ’v,max :

e = 2.767 - 0.26 ln (0.8 σ'v,max ) (Critical State Soil Mechanics) (4)

and this can be checked by taking samples for moisture content w determination:

e = (γs / γw)× w = 2.61 w (5)



The height of the sample is directly proportional to specific volume v = 1+e, the initialvolume (and height) of clay slurry may be determined.

(6)

Saturated unit weight may also be calculated:

γsat = ((γs / γw) + Sr e) γw / (1 + e) = (2.61 + e) 10 / (1 + e) (7)

Example: Calculate the compression expected in a fully saturated speswhite kaolin slurry(model offshore soil in the jack up 'lattice leg' test) placed at a nominal w = 120% & consol-idated to equilibrium σ’v,max = 110 kPa, with samples expected to show w ~ 60%. We needa post-1g-consolidation model height of 300 mm.

e = 2.767 - 0.26 ln (0.8 x 110) = 1.603

w = 1.603 / 2.61 = 61.4%

ew=120% = 2.61 x 1.2 = 3.132

hw=120%= 4.132 x 300 / 2.603 = 476 mm

γsat = (2.61 + 1.603) 10 / 2.603 = 16.2 kN/m3

γ' = 6.2 kN/m3

Aim for at least 500 mm to allow for trimming/error in calculation, so a reduction in height to~ 60% of original slurry was allowed to ensure sufficient depth of clay following consoli-dation at 1g. (N.B. In a drum centrifuge, maximum depth of clay would be about 200 mm, soclay will need to be placed slowly over a period of time to allow some consolidation to takeplace because depth of the drum is only 300 mm).

Example: What depth of speswhite kaolin slurry (γs = 2.61×γw) to be placed at w = 120% inthe drum centrifuge so that after consolidating under a surcharge of ∆σv = 100 kPa at 100 gthere is a post-1g-consolidation model height of 60 mm. Calculate the respective weights ofwater & kaolin needed to mix the slurry if the drum has dimensions 2.2 m (diameter) x700 mm (width) x 300 mm (depth)? After consolidation under ∆σv = 100 kPa what is w atmid-depth of the 60 mm layer and hence what is the saturated unit weight (assume initialvalue)?

At depth z = 30mm · 100 = 3m

hσ'v max,hw=120% 1 eσ'v max,

+( ) 1 ew=120%+( )⁄=

50 mm

h

γ = 20 kN/m3, ∆σv = 100 kPa

Clay

Depth of clayz

100 kPa

at 100g

30 mm x 100 = 3 m

γ · zassume γ = 16.4 kN/m3

Figure 6.17: Example (@ 100g)



e = 2.767 - 0.26 ln (0.8 x (16.4 x 3 + 100)) = 1.524

w = e / (γs / γw) = e / 2.61 = 58.4%

ew=120% = 2.61 x 1.2 = 3.132

hw=120%= 4.132 x 60 / (1+1.524) = 98.2 mm

γsat = (2.61 + 1.524) 10 /(1+1.524) = 16.4 kN/m3

also calculate values at 100 mm (1 m prototype) and 50 mm (5 m prototype):

Answer: 100 mm clay will be necessary for placement at w = 120% although some“wastage” should be allowed for. Assume minimum of 120 mm clay depth.

Volume:

Tot: = π/4 · (2.22 - 1.962) · 0.7 = 0.549 m3

Water : Clay = 1.2 : 1

γsat,w=120% = 13.9 kN/m3

Clay: 0.25 m3 347 kg (i.e. 14 x 25 kg bags)

Water: 0.3 m3 30 l

Consolidation: time

Non-dimensional consolidation time factor Tv = cv t / h2 where cv is the vertical coefficientof consolidation, t is time and h is length of the drainage path (for 2 way drainage, modeldepth = 2h).

1m 5m

e = 2.767 - 0.26 ln (0.8 x (16.4 x z + 100)) 1.588 1.472

w = e/2.61 60.8% 56.4%

hw=120%= 4.132 x 60 / (1+e) 95.8mm 100.3mm

γsat = (2.61 + e) 10 /(1+e) 16.2kN/m3 16.5kN/m3

Clay Range of σv‘

kPa+= current σvc‘

e cv

mm2/s

kv

10-6

mm/s

kv pred

10-6

mm/sEqn (8)

Source

Speswhite kaolin 256+- 450+

450+- 120

450+ -60

-

1.21

-

0.3

0.57

0.58

0.72

0.34

0.35

-

0.9

-

Bransby (1993)

Speswhite kaolin 100+- 200+ 1.30 0.18 0.95 1.17 Ellis (1993)

Speswhite kaolin 54+-91+ 1.54 0.25 2.87 2.03 Sharma (1993)

Speswhite kaolin 43+- 86 1.54 0.27 2.06 2.03 Springman (1989)

Tab. 6.5: Consolidation data derived from centrifuge models in a large 1 g consolidometer

∴ →

∴ →



Al Tabbaa (1987) quotes values of kv and horizontal permeability kh (from independentfalling head and consolidation tests) for speswhite kaolin with respect to void ratio e:

mm/s (8)

mm/s (9)

for normally and overconsolidated states with 0.98 < e < 2.2.

Data given above shows that values of kv,pred by equation (8) are:

• up to 3 times larger than those measured in the 1 g consolidometer for unloading incre-ments, and

• in quite good agreement for virgin consolidation.

Example: If cv = 10-7 m2/s for unloading - reloading conditions, for a clay depth of 60 mm,with top and bottom drainage, 90% consolidation.

Tv 90 = 0.848 and t90 = 0.848 x (0.06/2)2/10-7 = 2.1 hours;

Example: For a clay depth of 350 mm, draining to top and bottom of sample, σv' > 86 kPain the lower half of the model where the sample would be normally consolidated, calculatetime for 90% consolidation to occur. Adopt a conservative value of cv. How would youreduce this time for consolidation?

Between 26-40 hrs: depends on cv selected (2.7 to 1.8*10-7 m2/s): place a thin sand layeracross most of the clay at mid-depth to aid dissipation of pore pressures. Link into thedrainage system at the top/bottom of clay to facilitate drainage: reduces consolidation timeby a factor of 4 (to ~ 7-10 hours).

Stage 1 of 2 stage consolidation process => mid-depth sand layer in a deep model in clay (Nunez, 1989)

• drainage pipe to middle sand layer fixed in tub, • bottom drain placed,• slurry poured and first clay layer consolidated to

σv' = 90 kPa.

kv 0.5 e3.25 10 6–⋅ ⋅=

kh 1.43 e2.09 10 6–⋅ ⋅=

Figure 6.18: Stage 1



Stage 2

• pressure released and swelling allowed,• middle sand layer poured with gaps to allow for pile /

penetrometer tests,• filter fitted to top of drainage pipe,• filter paper placed on sand.

Stage 3

• slurry poured and first and second layers consolidatedto σv' = 260 kPa,

• sand layer to settle onto filter and drainage pipe,• final clay surface to be cut flush with the top of the tub.

A similar method could be adopted in the drum centrifuge (although placing filter paperwould be a bit complicated and fiddly)!





Inflight downward hydraulic gradient with intermediate drain

Clay: final model preparation

Having consolidated the clay sample in a strongbox, some final preparation is required priorto loading on the beam:

• close drains, unload press, then the liner is removed from consolidometer,• clay excavated/cut to size,• deformation markers added (internal and external),• structures, instrumentation and site investigation equipment installed,• final assembly of strongbox and installation on swing,• installation of swing on centrifuge arm,• connection of all fluid/air supplies, electrical/electronic equipment.

Clearly this procedure is slightly different in the drum but the principles are the same.

How does this modelmaking procedure vary for granular materials?

Sand: placement

• dry sand will be placed to a uniform relative density D or (ID) by pouring from a hopper ata specific height with a uniform flow rate,

• pouring may be halted at any time to allow placement of deformation markers, spots orlines of coloured sand, for subsequent displacement or strain determination,

• known weight of sand poured into a specific volume allows determination of e, andhence D (or ID),

• the sand will be saturated either by upward flow (but with low differential head to avoidpiping) or by using a vacuum under a restraining sheet,

Figure 6.21: Inflight downward hydraulic gradient with intermediate drain



• then, as for a clay model: structures, instrumentation & site investigation equipmentinstalled........etc.

•

Properties: strength

The peak angle of friction φ'max and the critical state angle of friction φ'crit may be based onD (ID) as follows (Bolton, 1986):

ID = (emax - e) / (emax - emin) x 100% (D is f(n)) (10)

where:

emax = maximum void ratio

emin = minimum void ratio

φ'max = φ' crit + A Ir (11)

where:

A = 3 for triaxial and 5 for plane strain, and

Ir = ID (Q - ln p') - 1 (12)

where:

Q is f(particle crushing strength), (usually given as 10 for quartz,i.e. crushing strength > 20 MPa),

p' is mean effective stress in kPa.

Note: Most quartzitic sands have φ'crit = 32 - 33°

(Dilatancy is f(φ'max - φ'crit))

This shows more easily how these equations may be used to determine possible dilatancy.

Triaxial data @ failure

Figure 6.22: Dilatancy for quartz sands: φ'max - φ'crit v. p’ (Bolton, 1986)



Stiffness

Initial stiffness Gmax may be found from

• lab testing,• by empirical relationships, e.g.

Gmax = 50’000 (p‘)0.5 / (1 + e)3 (kPa)

Permeability

This can be found from simple lab permeability tests on soil samples prepared at equivalentD (ID).

6.4 Preparation of soil samples in the DRUM centrifuge

Many of the techniques described for beam models are used on the drum centrifuge,however generally modelmaking is done in the centrifuge under ng.

• sand may be sprayed from a nozzle near the central console or onto a spinning diskwhich moves vertically to place a uniform sample across the vertical height of the drum,

• saturation is possible by varying the groundwater levels, using an elevating standpipedriven by a small motor,

• clay samples may be placed as a slurry in the same manner (although there areconcerns about achieving full saturation), and sand may be placed on top to achieveoverconsolidation,

• downward hydraulic gradient consolidation is also possible by raising the water tableabove the clay and draining to zero pore pressure at the base of the sample, or

• block samples may be prepared separately and placed in the drum at a later stagefollowing preliminary placement of a sand base.

N.B.: More will be added to this chapter as experience with the Zurich drum centrifugedevelops ...



(1) A Sand delivery pipe can move up and down B Air blows sand out through nozzle onto cylindrical surface C Berm at base of drum wall assists stability of layer later

(2) D Water delivery pipe is at base of drum E Water table in sand layer after hydration

(3) F Water now discharged through pipe G Water table in sand layer after de-hydration

(4) H Cutting frame attached to central column I Drum stationary @ 1g. Central column rotates. Blade trims surface, moved

outwards as cutting process; debris collected from base

Figure 6.23: Placement of a sand sample in a drum centrifuge (Dean et al., 1990)



Figure 6.23: Placement of a clay sample in a drum centrifuge (Dean et al., 1990)

(1) A Upper buoyancy foam blocks fixed to drum B Lower buoyancy foam blocks holding drain pipeworks C Base drain for clay layer D Water drains from clay as self-weight consolidation occurs

(2) E Valve opened when sand layer reaches level of second drain

(3) F Water drains from sand; clay continues to consolidate under higher load of dry sand

(4) G Dry sand falls away when drum stopped H Solid clay layer has apparent cohesion and remains standing @ 1g I Rotating blade trims clay surface @ 1g






In-situ testing, instrumentation, data acquisition Page 7 - 1

7 In-situ testing, instrumentation, data acquisition

7.1 Measurement of soil properties

7.1.1 Vane shear testing to determine su at discrete locations

In the field su,vane for a homogeneous clay varies according to:

• the vane aspect ratio L / D (should be reduced in the centrifuge),• the rotation rate of the vane ω.

Peak and residual torque T, may be measured from a torsion load cell (which is calibratedunder torsional load) mounted on the vane shaft (above soil penetration level), and causesa nominal failure surface of a cylinder of diameter D, and height L (and also potentiallyround the shaft):

T = π su,vane ( D3 + 3D2 L) / 6 => peak and residual su,vane, (13)

This is derived from the shear surface x su x lever arm (integrating an annulus on top andbottom surfaces and direct calculation for the cylinder).

Modelling principles

• su,vane is averaged over 3 surfaces: 1 cylindrical (vertical) and 2 horizontal discs

• su,vane is also averaged over depth if su is not constant (OK if dsu / dz is linear),

• So if L/Dcentrifuge is 14/18 cf. L/Dfield = 130/65 (dimension mm) then:

→ more emphasis is placed on su in the horizontal plane cf. su (vertical) - (i.e.horizontal slip planes),

→ can conduct more vane tests per depth of clay model (> 1 vane depth nottested between each shearing event at a specific depth),

→ the effect of the shaft diameter: vane diameter ratio is reduced (relative shaft/vane resistance is smaller and disturbance from soil displacement due toinsertion will be relatively smaller),

slow fast

speed of rotation

su

su,vane

ω

dz

L

D

Torque Trotation speed ω

Measure T, ω & dz

L/Dcentrifuge ~0.77 (14/18)

L/Dfield ~2 (130/65)

Figure 7.1: Shear vane geometry and influence of speed of rotation



• the shaft is greased and is fitted with a 15o slip coupling to separate the component offriction from that due to the shear around the vane,

• at fast ω, viscous effects would cause the measured su,vane to increase,

• at very slow ω, significant consolidation occurs, causing higher values of su,vane,

• optimal rotation speed gives lowest value of su,vane.

Scaling effects:

• consider testing procedures with respectto difference in scaling with respect totime,

• time relative to the strain rate will beidentical in both cases,

• time taken for dissipation of u followinginsertion of the vane (by diffusion), scalesas n2 faster,

• the centrifuge vane data will tend toindicate higher strengths (greaterdrainage) for the same rotation speeds inprototype and model; therefore increasemodel vane rotation speed.

Operation:

• vane is driven vertically (6-12V electric motor) at between 2 - 6 mm/min,• a linear potentiometer reveals when depth for the next test has been reached,• a (5V) rotary motor is then engaged, and shearing is begun at n g after 1 minute, • in the field, this delay is usually 5 minutes:

→ for radial drainage at the vane circumference (cv t / (D/2)2 )m = (cv t / (D/2)2 )p,

→ for cv equivalent in both models, tm = 5 x 60 / (65/18)2 = 23 seconds,

• centrifuge model clay will have consolidated more during this pause of 1 minute, andhigher strengths would be anticipated,

• surface water should be prevented from entering the vane 'bore',• minimum su,vane for the 14 x 18 mm vane was achieved in kaolin at 72°/min at 100 g,

• these data should be reviewed again - as a function of new equipment, building new soildatabase etc.

Output:

• relationship between su vane , σv' & OCR (Eqn (1)) as used in design,

• must allow for Bjerrum's factor µ (Eqn (2)) when converting su,vane to design strengthfor analysis.

Figure 7.2: Vane shear test data



7.1.2 Cone penetration testing (CPT): to determine a profile of su

Modelling principles:

• the cone penetrometer presents different modelling difficulties & advantages (somerelative to vane):

→ the tests are quicker to perform and

→ a continuous profile of soil resistance is obtained, and

• since content of most centrifuge models is generally known,

→ can be used to check consistency of each test sample and to compare with in-flight vane test data,

• but extrapolation from empiricism at full scale to model scale prediction leads to someuncertainty........

Scaling effects: geometry

• modelling a prototype 10 cm2 diameter field cone at 100 g implies a centrifuge probe of0.36 mm diameter! - impossible to manufacture a working device, let alone strain gaugea load cell,

• and so we model (effectively) a pile installation,• it is general European practice to use a standard cone of diameter 10-11.6 mm, with a tip

load cell and usually total resistance from tip and shaft together,• Cambridge have a piezocone of 12.7 mm diameter, with a porous sintered stone at the

tip of a 60o cone, and a rosette load cell located at the top of the shaft, but isolated fromthe friction exerted on the shaft.

• the ETH cone has a diameter of 11.3 mm allowing separate measurement of the tipresistance and total resistance from tip and shaft together.

Scaling effects: time

• faster insertion rates are likely to be subject to viscous effects, • whereas slow penetrations will allow significant dissipation of pore pressures,• penetration rates of between 3 and 26 mm/s have been investigated in clay, and these

were found to fall in the intermediate range, with less than 13% difference in qc (tipresistance).

Output:

• for clay, soil strength:

→ f ( qc tip resistance, field cone factor Nc)

→ f (OCR, φcrit', u @ cone shoulder, location of u measurement, what is themost appropriate penetration rate?):

→ Nc = (qc - σv ) / su (14)

→ can plot su / σv ' to reveal OCR,

→ pore pressure acting on shoulder of a piezocone should be allowed for,



→ depth should be corrected (gravitational field depth effect error) to zpc from zm

→ zpc = n zm (1 + (zm / 2Rs )) (15)

where zm is depth of cone below soil surface, and n is gravity level atcentrifuge radius

Rs at the surface of the soil.

• for sand, tip resistance: (N.B. pc = crushing strength)

→ f (D, z/B (depth: cone dia), B/d (cone dia: particle dia), OCR, σv'/pc , φcrit')

→ Nq = (qc - σv ) / σv', (16)

→ using Equations (11) & (12),

→ p' = (σv' qc) 0.5, (17)

→ also allows determination of φ'max

→ normalised tip resistance, Q = Nq - 1 (18)

→ normalised depth, zm / B = Z (19)

Figure 7.3: Tip resistance profile in kaolin clay (Gui, 1995)



Mechanism Relative density effect (Note: Id = ID)

Figure 7.4: Mechanisms and influence of relative density (Gui, 1995)

ID effect



Operation:

Grain size effect

Figure 7.7: Grain size effect on LB sand (Gui, 1995)

• drive continuously between 3-26 mm/sin sand, or 3-12 mm/s in clay,

• dimensional limitations > 5 B (5 conediameters) from the location of anotherimportant section of the model or fromthe edge of the strongbox,

• the proximity of the base of thestrongbox or a stiffer granular foundingstratum will increase the measured loadfor between 5 - 10 B above thatboundary,

• these interactions may be lessnoticeable in soft clays.

Figure 7.5: Penetration rate and OCR effect(Gui, 1995)

Figure 7.6: Grain size effect on LB sand (Gui, 1995)



European cooperation => unified testing methods:

• recommendations regarding the operational and cone design standards have beenreviewed by a group of European Universities under an EC contract (Bolton et al., 1999),

• better agreement was achieved in sand samples (Fig. 7.18a) than for clay (Fig. 7.18b),

but there were some differences in sample preparation, strongbox shape and size, conediameters and radius of centrifuges.

Figure 7.9: Cone penetration data from various European centrifuge laboratories (Renzi et al., 1994)a) sandb) clay

Stress level effect Dilatancy & mobilised friction

Figure 7.8: Influence of dilatancy (Gui, 1995)

a) b)



Nonetheless, the agreement confirms that the CPT is the main method of site investigationin the centrifuge and that it can be comparable, within an Institution, and between manyInstitutions, provided testing methods are similar.

7.1.3 T-Bar penetration testing: to determine su

Modelling principles:

• a semi-rough cross bar, of length approximately 5 bar diameters db (db = 7 mm), withsmooth ends, mounted on a shaft instrumented to read axial load (Fig. 7.9), and

• a classical plasticity solution in plane strain (Randolph & Houlsby, 1984) => the forcerequired to pull or push the cylinder through a rigid-perfectly plastic medium of shearstrength su (graphically shown in Fig. 7.10),

• a fundamental relationship is advantageous in finding su from a miniature SI device.

Scaling effects:

• the bar diameter must be small so that dsu /dzprototype is insignificant, and can beassumed to be constant in the near field to the bar, and

• the end effects of pushing or pulling the bar vertically adjacent to the soil surface or aninterface with a stiffer layer must also be considered,

• time/rate effects should also be investigated as for penetrometer and vane tests.

Output:

• bar factor Nb obtained from measuring force / unit length P on the cylinder wheresurface roughness is equal to α su and adhesion factor α = 0 (smooth bar), and α = 1(rough bar mobilising friction up to su )

Nb = P/(sudb) (20)

• comparisons with su predicted from in-flight cone penetrometer, post-test vane shearand independent triaxial tests (Stewart and Randolph, 1991) were fair - the bar gave ansu profile:

Figure 7.10: Schematic diagram(Stewart and Randolph, 1991)

Figure 7.11: Variation of Nb with surface roughness

(Stewart and Randolph, 1991)



→ which was very similar to the triaxial prediction for normally & lightly overcon-solidated samples,

→ about 20% greater for OCR > 3,

→ vane shear results are expected to be < equivalent values at 100g (ingress ofwater during deceleration / prior to inserting the vane) - dependent on timeafter deceleration & clay cv .

Comparisons between su obtained from in-flight bar and cone penetrometer tests, post-testshear vane tests and predictions based on triaxial data are given in Figures 7.10 - 7.13(Stewart and Randolph, 1991).

Figure 7.12: 1 OCR < 3 Figure 7.13: OCR > 3

(Stewart and Randolph, 1991)

≤



7.2 Measurement of displacement

7.2.1 Spotchasing

7.2.2 Digital Images and PIV analysis

Image capture using an inexpensive 2-megapixel digital camera provides a signif-icant increase in resolution and imagestability compared with video. Particle ImageVelocimetry (PIV) is a velocity-measuringtechnique in which patches of texture aretracked through an image sequence. Imageprocessing algorithms are available to applythe PIV principle to images of soil. Thesoftware has a precision of 1/15th of a pixelwhen tracking the movement of natural sandor textured clay. The system allows displace-ments to be measured to a precision greaterthan with still photographs or video filmswithout installing intrusive target markers inthe soil.

• insertion of black marker 'bullets'=> evaluation of relativemovements at various stages inthe loading programme by backanalysis of high quality photo-graphs taken in-flight,

Figure 7.14: Vectors of outward horizontal movement postinstallation of spudcan (after Phillips, 1990)

• these differential movements =>appropriate values for input intoa strain evaluation computerprogram to estimate contours ofshear strain, directions of linesof no-extension etc.,

• in some cases it has beenpossible to derive sufficientinformation for use in theanalysis of geostructural mecha-nisms (Bolton and Powrie, 1988;Sun, 1989),

• e.g. for installation of an offshorejack up rig near to piles whichare intended to support anoperational platform (Phillips,1990), vectors of outwardhorizontal movement may beobtained.

Figure 7.15: Measurement of deformation in atriaxial test (White et al., 2001)



7.2.3 Displacement measurements

Sometimes it is helpful to measure post-test displacements - often this should be doneimmediately after the test while the model is still on/in the centrifuge so that it is notdisturbed. This might be particularly relevant for:

• the shape of an embankment or • backfill behind an abutment wall, or • the failed shape of a retaining wall or • the collapse of an anchored wall in sand after an earthquake etc......

7.2.4 Radiography

Information may be obtained from post-centrifuge model test radiographs:

• of the location of rupture zones (via detection of diagonal lead threads) or

a) Plane strain calibration chamber for pile testing b) Selected node paths during pile penetration

Figure 7.16: Application: pile penetration in a calibration chamber (White et al., 2001)

Figure 7.17: Forced subsidence of a model abyssal plain; dashes represent ruptures (Stone, 1988)



• of movement of soil between piles due to surcharge loading (Springman, 1989),

7.2.5 Excavation

Much useful information may be obtained from careful excavation. In particular, it isnecessary to determine the actual position of the pore pressure transducers (allowing forpost test swelling or desiccation) so that back analysis of the pore pressure test data makessense!

An alternative to using lead threads is to create a small hole and to insert a piece of noodleor spaghetti. This softens on contact with water. The noodle can be exposed and its positionmeasured (approximately) during excavation.

7.3 Electronic/electrical instrumentation• should be small and rugged,• instrumentation calibration ideally should be linear but always repeatable,• beware of creating reinforcement or preferential drainage paths with cables,• maybe bought off-the-shelf or custom-made.

Internal

Common types of instrumentation inserted in a soil matrix are:

• pore pressure transducers (clay),• total pressure/stress cells (Garnier, 1999),• for contamination problems, resistivity probes (Hensley, 1989; Hellawell, 1993),• for dynamic applications (e.g. earthquake studies), accelerometers (Kutter, 1982),• indicators of displacement for exposure to X-rays.• matrixes to measure deformation ⇒ stress distributions (e.g. Tekscan; Springman et

al.,2002, Laue et al. 2002)

Figure 7.18: Internal soil-pile interaction for a row offree-headed piles (Springman, 1989)

Figure 7.19: Internal soil-pile interaction for a pile group (Springman, 1989)



External

Other forms of measurement are mounted externally:

• thermocouples (TC), thermistors or digital thermometers (and CCTV),• strain gauge bridges (as bending moment transducers - BMT),• load cells (LC),• displacement measurement,

→ linear variable differential transformer (LVDT) say 30 mm range,

→ linear potentiometer (LP) say 300 mm range,

→ laser profilometer,

→ external measurement,

→ wave height gauges.

Air/pore pressure transducers (PT/PPT)

• in future, PPTs may be made in the form of minuscule chips, interrogated by a remotetrigger so that the difference in the measured and the true pore pressure (that wouldhave existed in the soil in the absence of the PPT) due to interference between the soil,the instrumentation cabling and transducer will be reduced significantly,

• in the meantime, care is necessary when designing and installing PPT layouts,• air pressure transducers (PT) can be subjected to several bar of pressure and are inter-

posed on air supply lines, mainly for actuator control.

Instrument Input Voltage Amplify/Filter

mV or Voutput

Comments

PPT 0 to -5V 10x mV actually better

LVDT -5V to 5V - / Filtered V Linear over 60-80% of range, typically +/- 15

mm

LP 0 to 10V - V 300 mm range

LC 0 to 2V (or 3V, 5V) ~100x mV

TC -12V to 12V - mV separate J-Box

ACC 10V chargeamplifiers

1000x

7 picoCoulombs/g

=> V

separate J-Box, convert pressure change on

piezoceramic crystal to charge

pH sensor

2 or 4 electrode Resistivity probe

1V square wave @ 100Hz approx

yes mV electrical conduction

Tab. 7.1: Instrumentation: Input, output, errors, signal processing



Resistivity probes

Fibre optic pH sensor

• used for electrokinetic remediation,• non-electrical method required,• light passed from LED, via fibre-optic cable, through liquid, filter, ball lens & a non-bleed

pH paper,• transmitted light collected by second fibre-optic cable and passed to a silicon photo-

diode,• a logarithmic amplifier enhances the output signal,• absorbance measured and according Beer's Law => concentration can be determined.

• Pore pressure transducers (PPTs)measure various magnitudes of porepressures (typically, a 350 kPa range).The company DRUCK sells thesmallest ones used as standard trans-ducers in centrifuge.

Figure 7.20: Diagram of a Pore Pressure Transducer

• cylindrical in shape, 6.4 mm in externaldiameter, with a main body 13 mmlong,

• a porous stone protects the load cellface so that pore pressure and not totalstress is measured,

• the porous stone should be de-aired,the PPT should be inserted prior to theaddition of the final loading incrementin the consolidometer, havingunloaded the clay, in stages of 100 kPaor less, to zero excess load at 1 g,

• a typical 4 electrode resistivity probeshown with calibration test data,

• electrical conduction occurs throughpore fluid,

• this is affected by soil porosity andchemical composition,

• but the only variation tends to be thechemistry,

• signals are generally multiplexed.

Figure 7.21: Diagram of a resistivity probe



Strain Gauge Bridges and Load Cells

Strain gauge bridges: using properties of changing electrical resistance due to strain of thinfoil elements, suitably attached to host material - connected together in electrical circuits tomeasure:

• shear• normal load• torsion• bending moment

Overview of system

Single channel

logyellow l.e.d.

sensor photodiode

CUED Environmental Geotechnics

Ball lens coupler

non-bleed pH paper

fibre - light infibre - light out

Fibre-optic pH sensorBall lens coupler

non-bleed pH paper

fibre - light infibre - light out

Fibre-optic pH sensor

Schematic design of the sensor

Figure 7.22: Schematic of system and sensor (Lynch, 1998)

Experimental arrangement for plume detection

Light in Light out

porous basesensor sand

pollutant

optical fibres

Light in Light out

porous basesensor sand

pollutant

optical fibres

Sodium hydroxide pollution plume in sand



Constant voltage energisation:

in ABC, current i = E / (R1 + R2)

in ADC, current i= E / (R3 + R4)

V(B) - V(D) = i(AB) R1 - i(AD) R4 = V

For R1 = R2 = R3 = R4 = R, V = 0

For R1 = R + dR1, R2 = R + dR2 etc., V may be calculated

Gauges positioned according to what force is to be measured.

Mode of operation:

Lengthen straight fine wire, length by d , resistance changes

Initially

R = ρ /A

where A = πr2, ρ is specific resistance, then

dR/R = δρ/ρ + δ / - 2δr/r

for cylindrical wire under uniaxial stress, σ, the longitudinal tensile strain, εl ,

εl = δ / = σ/E

and radial strain, εr

εr = δr/r = -νσ/E = -νδ /

giving

dR/R = δρ/ρ + (1 + 2ν) δ /

and if

δ / = ∆ δ /

then resistance change due to straining:

dR/R = [ ∆ + (1 + 2ν)] δ / = k δ /

• generally thin foil, only 4µm thick is used - to increase electrical resistance and to speedup heat dissipation with relatively large surface area to reduce glue stresses,

Figure 7.24: Wheatstone Bridge

DB

C

R4R1

R2R3

A

EV



• backing is usually 25µm of epoxy sheet,• strain sensitivity of a metal foil gauge is typically k = 2.1,• (but semiconductors have more sensitivity and therefore require a constant current

energisation to eliminate non-linearities & temperature effects due to changes in powerconsumption).

Two types of strain gauge bridges:

Fully active:

e.g. R1 & R3 in tension, +dR

R2 & R4 in compression, -dR

V = E dR/R = Eεk

1 active set of gauges, one dummy set of gauges:

• due to problems of space• other gauges likely to be in junction box• must be at same temperature• even though gauges can be temperature compensating

e.g. R2 & R4 in compression, -dR

R1 & R3 are unstrained, dR = 0

V = E dR/2R (i.e. half output of fully active bridge)

Power Dissipation:

i2 R in each arm of active bridge

material must be capable of dissipating this without affecting accuracy of gauge

Typical requirements for high accuracy require power density between

0.78 - 1.6 x 10-3 Watts/mm2

e.g. (E/2R)2 x R with E = 10V then for 2.5V per gauge, R = 350W,

power = 6.25/(4 x 350) = 0.0045 Watts,

so gauge area needs to be

4.5/1.6 ~ 3 mm2

7.3.1 Other considerations:

Temperature effects:

• use temperature compensating gauges, + equivalent coeff. of expansion betweengauge / material,

• use Wheatstone bridge to eliminate resistance changes due to temperature changes,• use similar lengths of supply leads etc.



Aging of glue:

• creep with load and time• curing is a vital part of process

Generally Araldite epoxy strain cement is used by the Cambridge Group.

Waterproofing:

• necessary for gauges & leads in 'wet' environment,• various coatings & shrinkfit tubings may be used,• submersion in silicone oil is another alternative.

Recording Equipment:

high input impedance ~ 5000 MW.

Load Cells:

Calibration:

• apply normal, shear and eccentric loading independently to set up a loading matrix:

[dV] = [a]

• invert to give N, M and S as a function of the total change in voltage of each strain gaugecircuit,

• loads should be applied repeatably & consistently on face of cell in same way as testingsituation,

• supply voltage should be stable,• at least 5 readings should be taken on a load-unload cycle to maximum level expected in

tests.

• Cambridge contact stress transducers 1961 -present,

• designed to measure:⇒ magnitude & direction of shear & normal stresses,

• thin metal webs instrumented with strain gauges,

• generally aiming at:⇒ around 2000 micro strain maximum,⇒ to ensure linear elastic response,⇒ appropriate stiffness,⇒ metal easily machineable,

• usually use heat treatable aluminium alloy HE15W, Figure 7.25: „Stroud“ load cell

• Wheatstone bridges give a useful output to such small resistance changes,

• locked in hysteresis/machining stresses should be relieved by strain cycling.

NMS



Effect of transducer stiffness:

• transference of stress around a deflecting piston is a function of transducer/soil (kc/ks)stiffness,

• but is transducer measuring true state of stress in ground?

kc/ks > 100 to give accuracy > 99%: typical Cambridge cells have 50-100 MN/m stiffness.

Accuracy:

• Errors may be caused by many factors:

→ recording device

→ calibration constants

→ hysteresis & non-linearity

→ random drift, temperature & input/output voltage

→ effects of loads other than those cell is designed to measure.

The largest error is generally due to drift.

Data acquisition systems

• multichannel - ? multiplex ?,• 2 orders of magnitude faster logging than in real time,• many types of tests with a wide variety of transducer,• e.g. 2 main requirements,• steady state logging e.g. 50 channels 0.01Hz for 2 days,• dynamic events logging e.g. 16 channels 10 kHz for a few seconds,• need redundancy in data acquisition - test expensive if data is all lost,• disk (hard, CD or zip),• (magnetic),• (tapestreamer),• manual,• better to have a modular system for fault tracing.

7.4 Summary

Centrifuge model testing is particularly advantageous in investigations of the performanceof large scale structures. Appropriate idealisation may be adopted to reveal the key mecha-nisms of behaviour.

It is possible to create centrifuge models in clay using 'laboratory' soils such as kaolin,according to a prescribed design strength profile or coarse-grained „cohesionless“ soilswith the relevant relative density.

A combination of site investigation devices, empirical interpretations and comparisonsbetween in-situ and laboratory data allow determination of soil strength.

Both displacement and failure mechanisms may be observed, leading firstly to an under-standing of the problem under investigation, secondly to an evaluation of strains in the soilmatrix and thirdly to data about the mean soil strength at failure.



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82. Tang, Z. 1993. Laboratory measurement of shear strength and related acousticproperties. MEng thesis, Memorial University of Newfoundland, St John's,Newfoundland.

83. Tovey, N.K. 1970. Electron microscopy of clays. Cambridge University PhD thesis.

84. Trak, B., La Rochelle, P., Tavenas, F., Leroueil, S. and Roy, M. 1980. A newapproach to the stability analysis of embankments on sensitive clays. Can. Geot. J. 17,Vol. 4, pp. 526-544.

85. Vinson, T.S. 1982. Ice forces on offshore structures. Proc. Workshop on High GravitySimulation for Research in Rock Mechanics. Colorado School of Mines. pp. 60-68.

86. Vinson, T.S. 1983. Centrifugal modelling to determine ice/structure/geologicfoundation Interactive forces and failure mechanisms. Proc. 7th Int. Conf. on Port andOcean Engineering under Arctic Conditions. pp. 845-854.

87. Waggett, P.R. 1989. The effect of lubricants on the interaction between soils andperspex. Cambridge University Part II Project Report.

88. Wilkinson, B.J. 1993. An investigation into the effects of adding a substantial granularcontent to a kaolin based mix. Cambridge University Part II Project Report.

89. White D. J., Take W.A, Bolton M.D. and Munachen S.E. 2001. A deformationmeasuring system for geotechnical testing based on digital imaging, close-rangephotogrammetry, and PIV image analysis. Proc. 15th ICSMGE.

90. Wong, P.C., Chao, J.C., Murff, J.D., Dean, E.T.R., James, R.G., Schofield, A.N., andTsukamoto, Y. 1993. Jack-up rig foundation modelling II. Offshore TechnologyConference 7303.

91. Wroth, C.P. 1972. General theories of earth pressures and deformations. Proc. 5thECSMFE, Madrid, II, pp. 33-52.

92. Wroth, C.P. 1975. In-situ measurements of initial stresses and deformation character-istics. CUED/D TR23.

93. Wroth, C.P. 1979. Correlations of some engineering properties of soils. Proc. 2nd Int.Conf. Behaviour of Offshore Structures, London. Vol. 1, pp. 121-132.

94. Yong, R.N. and McKyes, E. 1971. Yield and failure of clay under triaxial stresses.Proc. ASCE SM 1, pp. 159-176.



Numerical Modelling Finite Difference Method



Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua) Page 8 - 1

8 Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)

8.1 Basics

Finite differences may be used to solve partial differential equations (e.g. steady state flowor Newton's Laws etc.).

Governing equations are substituted by finite differences in space, written in terms of fieldvariables at discrete points i.e. the NODES - these are the key geometrical feature.

Implicit solution: solution at every node is dependent on solutions at fourneighbour nodes - the solution at each node isn't known until the entire solution is known.

Explicit solution: nonlinear solutions are produced in the same time as for linearproblems (cf. longer solution times for implicit solutions).

Mixed discretisation: accurate modelling: plastic collapse

plastic flow

Solutions are mainly iterative - aiming to reduce the error to an acceptable level.

8.1.1 Specific to Geotechnics via FLAC (Fast Lagrangian Analysis of Continua)

Partial differential equations

=> matrix equations for each node, using dynamic equations of motion.

The contour integral formulation of finite differences is used. This formulation overcomesdifficulties often associated with mesh pattern and imposition of boundary conditions. Themean value of the gradient of a field variable in a zone may be expressed using the Gausstheorem with the contour integral performed on the boundary of the same zone.

Forces nodes = f ( displacements ) nodes



Figure 8.1: Example for unbalanced force vs time steps for two load increments

The method is conditionally stable. depending on damping coefficient and ratio betweenmass and time step (see Fig. 8.1).

For example, strain energy in the system may be converted into kinetic energy and may beallowed to radiate away and dissipate.

Iterative speed must be faster than wave propagation velocity.

Linear variation of the relevant quantities is assumed along the edges of the zone.

Standard quadrilateral elements used to generate the NODES as implemented in FLAC,lead to a stiffness matrix analogous to four node quadrilateral finite elements with reducedintegration but the finite difference method is 'physically' more justifiable.

The dynamic relaxation method (for static analysis) is implemented to solve the algebraicsystem of equations of motion. Successive integrations of these full dynamic equations(even for static solutions) of motion lead to the steady state solution usually based onadvice in the Software User Manual for FLAC.

For a static solution, mass and damping are fictitious: choose these in order to achievestability and accuracy of the solution. Generally damping should be less than the criticaldamping for the system (check using simple beam theory or by trial and error).

Advantages of the method:

• a relevant number of degrees of freedom can be analysed quickly and effectively using aPC version of the code. Since matrices are not stored, memory requirements are low.

• solution timelarge strain will be slightly higher than solution timesmall strain,

• the method is competitive with standard implicit-oriented FE codes for highly non-linearproblems.

x 103



Disadvantages of the method:

• in spite of the advanced formulation, strict limitations on the mesh pattern still exist(make the elements in the grid as square as possible), unless calculation efficiency isreduced (especially for elements of different stiffness),

• slower than FEM for linear problems,• ratio of longest natural period to the shortest natural period affects solution time (e.g. will

be influenced by beams, large stiffness differences and large changes in element sizes).

Note: Users need to know a set of abbreviated commands (actually quite quick to learn).

Solution procedure

The solution procedure can be summarized as follows:

1. Assuming known displacements, velocities and nodal forces at time ti compute new nodal accelerations.

2. Integrate nodal accelerations => nodal velocities & displacements.

3. From nodal velocities/displacements, impose constitutive law, obtain new stress state.

4. Calculate new nodal internal forces, integrate stress state along the element boundary.

5. Impose a suitable limit on the unbalanced forces (the difference between the appliedexternal forces and the internal forces).

6. Check current unbalanced force and* if it is less than limit, end calculation, * if more, increment the time step, go to (1), until steady state solution

achieved.

8.2 Finite Difference

Many similarities to finite element analysis

• idealised geometry (Boundary value problem: closed domain)• Material(s) (idealised constitutive model)• idealised loads

However: The method of NUMERICAL SOLUTION for the PARTIAL DIFFERENTIALEQUATIONS which govern behaviour is different.

Further :

Whereas finite elements describe behaviour of the continuum in terms of ELEMENTS anda finite difference grid may appear also to be an assemblage of elements, in fact it is theNODES which dictate behaviour.



Finite Difference NODES and connecting GRID

Figure 8.2: Simple finite difference grid

Usually ∆x is approximately constant

∆x need not to be equal to ∆y but ∆x = ∆y has some advantages.

The value of a function “ f ” at any of the nodes x = i, y = j for example can be written as fij.

In the PDE’s, typically this might vary as a function of x: e.g. or

of y: e.g. or

Typical PDE’s might include the LAPLACE equation:

(f = function describing variation in terms of x,y space..)

(2 dimensional)

which may (should!) be familiar because it describes the STEADY STATE flow such as

e.g.

* steady state seepage (figure 8.3) * heat diffusion (conduction)

i.e. No change with time as would occur in consolidation for example (FICKS LAW)

∂fij∂x------- ∂2fij

∂x2----------

∂fij∂y------- ∂2fij

∂y2----------

∇2f 0=

∂2f∂x2--------- ∂2f

∂y2---------+ 0=

t∂∂f constant ∂2f

∂x2--------- ∂2f

∂y2---------+⎝ ⎠

⎛ ⎞=

Solution Domain D(x,y)∆x

∆yNode

y

jmax

j+1

j-1

j

2

imaxi+1i-1 i1 2 x1



Figure 8.3: Flow net

Typically one might assume: f = a0 + a1x +a2x2+a3x3+..........+anxn

= b0 + b1y +b2y2+b3y3+..........+bnyn

e.g Notation fij = value for function f at node x = i, y = j

partial derivative: first order

= partial derivative in x direction for constant “j” can be written as or fx (i,j)

= 2nd order partial derivative in x direction for constant “j” as or fxx|(i,j)

= fxxx|i,j = fnx|(i,j)

similarly for changes in y direction

and etc.

So what does ”Finite Difference” mean....

using a finite distance such as ∆x or ∆y, the variation is written as follows

x∂∂fi j,

x∂

∂fi

x2

2

∂∂ fi j,

x2

2

∂∂ fi

∂3fi j,

∂x3------------

xn

n

∂∂ fi j,

y∂∂fi j,

y∂∂fj fy i j,( )= =

y2

2

∂∂ fi j,

y2

2

∂∂ fj fyy i j,( )= =

∂f∂x------

fi 1+ fi–

∆x------------------- or

fi fi 1––

∆x------------------ or

fi 1+ fi 1––

2∆x--------------------------=

Equipotentials

Flow lines

e.gQ=ki

i-1 i i+1

fi

∆x ∆x

x



forward:

backward:

central:

Similar equations can be written in terms of changes of “f” in the y direction

PDE’s e.g. (LAPLACE)

and similar in terms of y fi,j-1; fi,j; fi,j+1.

We should rewrite the above equations in shortened (approximation) form:

(1)

(2)

If we add the above equations (1) and (2) we get

fi 1+ fi–

∆x-------------------

fi fi 1––

∆x------------------

fi 1+ fi 1––

2∆x--------------------------

∂2f

∂x2--------- ∂2f

∂y2--------- 0=+

fxx i j,( ) fyy i j,( ) 0=+exact solution i.e. “correct” solution=> very difficult to achieve

fij

function may be derivedfrom a “Taylor” series

approximate solution e.g. using FINITE Difference

fij

of the exact indivisualpartial derivatives

Difference = Error:e.g. f f–

e.g. = exact solutionf

fi 1 j,+ fi j, fx i j,( )∆x 12--- fxx i j,( )∆x2 1

6---fxxx i j,( )∆x3 …… 1

n!-----fnx i j,( )∆xn+ + + + +=

fi 1 j,– fi j, fx i j,( )∆x–12---fxx i j,( )∆x2 1

6---fxxx i j,( )∆x3……–

1n!----- fnx i j,( )∆xn±+=

fi 1 j,+ fi j, fx i j,( )∆x 12!-----fxx i j,( )∆x2 1

3!----- fxxx i j,( )∆x3 …… 1

n!----- fnx i j,( )∆xn+ + + + +=

fi 1 j,– fi j,= fx i j,( )∆x–12!-----fxx i j,( )∆x2 1

3!-----fxxx i j,( )∆x3 ……+–

1n!----- fnx i j,( )∆xn±+



similarly in the y- direction

= 0 exact

approximate (e.g ignore higher fxxxx

terms etc.)

so assume an approximate solution where:

when ∆x = ∆y

The accuracy of the solution is affected by the size of the

Convergence occurs when error -> 0

as ∆x = ∆y ->0

and the solution at every point depends on the solution at all the other points.

i.e this is an IMPLICIT SOLUTION

(There are also EXPLICIT and MIXED DISCRETISATION solutions, which are not intro-duced here)

fi 1 j,+ fi 1 j,–+ 2fi j,22!----- fxx i j,( )∆x2 2

4!-----fxxxx i j,( )∆x4 ……+ + +=

fxx i j,( )fi 1+ j, 2fi j,– fi 1– j,+

∆x2------------------------------------------------ 1

12------fxxxx i j,( )∆x2 ……±–=

fyy i j,( )fi j 1+, 2fi j,– fi j 1–,+

∆y2------------------------------------------------ 1

12------fyyyy i j,( )∆y2 ……±–=

∂2f∂x2--------- ∂2f

∂y2---------+ fxx i j,( ) fyy i j,( )+=

∂2f∂x2--------- ∂2f

∂y2---------+ fxx i j,( ) fyy i j,( )+ 0= =

∂2f∂x2--------- ∂2f

∂y2---------+

fi 1+ j, 2fi j,– fi 1– j,+

∆x2------------------------------------------------

fi j 1+, 2fi j,– fi j 1–,+

∆y2------------------------------------------------+=

∂2f∂x2--------- ∂2f

∂y2---------+

fi 1+ j, fi 1– j, fi j 1+, fi j 1–, 4fi j,–+ + +

∆x2---------------------------------------------------------------------------------------- 0= =

ERROR = f f–



Example: flat plate heated at one boundary

Temperature varies as function of f

steady: 100°C at top boundary0°C at the other boundaries

at each node:

select a 3 x 4 grid in the first instance:

Figure 8.4: Grid for flat plate, heated at top boundary

At node 2,2

(because )

At node 2,3

(because )

or

2 unknown nodes : 2 x 2 matrix

fi 1+ j, fi 1– j, fi j 1+, fi j 1–, 4fi j,–+ + + 0=

∆x = ∆y = 5 cm

y

f2,2

Temperature: f = 0,100 °C

Nodes 1,2...

x

f2,3

100 100100

1 2 31

2

4

0

0

0

0 00

0

0

3f =

f = no flow in z-direction!

4 f2 2,⋅ f2 3,– 0= f2 1, f1 2, f3 1, 0= = =

4 f2 3,⋅ f2 2, 100 0=–– f1 3, f3 3, 0= =

4 1–

1– 4

f2 2,

f2 3,

0100

=



Accuracy of solution

Figure 8.5: Temperature distribution on plate from the 3 x 4 grid

The answers are now compared with those from the exact solution and those determinedfrom a 5 x 7 grid where

• The number of nodes control the accuracy of the solution• Accuracy (∆f) is higher with more nodes...• But this will require longer solution times: e.g => the diagonal matrix grows from a [2 x 2]

matrix to a [15 x 15] matrix when the grid changes from 3 x 4 to 5 x 7

Node

x y

cm cm

fexact

°C

f ∆f

3 x 4 grid

°C °C

f ∆f

5 x 7 grid

°C °C

5 10 26.049 26.667 0.618 26.228 0.179

5 5 5.261 6.667 1.40612 Nodes

5.731 0.4735 Nodes

Tab. 8.1: Exact and calculated temperatures

100 °C 80

60

40

20

10

5

2

y (cm)

x (cm)0 5 10

15

10

5

0

∆f f f–=



for a 5 x 7 grid:

15 x 15

Diagonal

Matrix

=

fi,j Boundary

temperatures

f = 100 or 0°C

So what happens if we use more nodes?

ACCURACY SOLUTION TIME ALSO

3 x 5unknown values at nodes

100

0

0

0

x

y



8.3 Details of FLAC Program

Start

MODEL SETUP 1. Generate grid, deform to desired shape 2. Define constitutive behaviour and material properties 3. Specify boundary and initial conditions

Step to equilibrium state

Examinethe model response

PERFORM ALTERATIONS for example - Excavate material - Change boundary conditions

Step to solution

Parameterstudy needed

End

Examinethe model response

Figure 8.6: FLAC general solution procedure (FLAC manual)

Model makes sense

Results unsatisfactory

More testsneeded

No

Yes

Acceptable result



FLAC uses nomenclature that is consistent, in general, with that used in conventional finitedifference or finite-element programs for stress analysis. The basic definitions of terms arereviewed here for clarification. Figure 8.7 is provided to illustrate the FLAC terminology.

FLAC MODEL — The FLAC model is created by the user to simulate a physical problem.When referring to a FLAC model, the user implies a sequence of FLAC commands thatdefine the problem conditions for numerical solution.

ZONE —The finite difference zone is the smallest geometric domain within which thechange in a phenomenon (e.g., stress versus strain, fluid flow or heat transfer) is evaluated.Quadrilateral zones are used in FLAC. Another term for zone is element. Internally, FLACdivides each zone into four triangular “subzones,” but the user is not normally aware ofthese.

GRIDPOINT — Gridpoints are associated with the corners of the finite difference zones.There are always four (4) gridpoints associated with each zone. In the FLAC model, a pairof x- and y-coordinates are defined for each gridpoint, thus specifying the exact location ofthe finite difference zones. Other terms for gridpoint are nodal point and node.

Figure 8.7: Example of a FLAC model (FLAC Manual)



FINITE DIFFERENCE GRID — The finite difference grid is an assemblage of one or morefinite difference zones across the physical region which is being analysed. Another term forgrid is mesh.

MODEL BOUNDARY — The model boundary is the periphery of the finite difference grid.Internal boundaries (i.e., holes within the grid) are also model boundaries.

BOUNDARY CONDITION — A boundary condition is the prescription of a constraint orcontrolled condition along a model boundary (e.g., a fixed displacement or force formechanical problems, an impermeable boundary for groundwater flow problems, adiabaticboundary for heat transfer problems, etc.).

INITIAL CONDITIONS — This is the state of all variables in the model (e.g., stresses orpore pressures) prior to any loading change or disturbance (e.g., excavation).

CONSTITUTIVE MODEL — The constitutive (or material) model represents the defor-mation and strength behaviour prescribed to the zones in a FLAC model. Several consti-tutive models are available in FLAC to assimilate different types of behaviour commonlyassociated with geologic materials. Constitutive models and material properties can beassigned individually to every zone in a FLAC model.

SUB-GRID — The finite difference grid can be divided into sub-grids. Sub-grids can beused to create regions of different shapes in the model (e.g., the dam sub-grid on thefoundation sub-grid in Figure 8.7). Sub-grids cannot share the same gridpoints with othersub-grids; they must be separated by null zones.

NULL ZONE — Null zones are zones that represent voids (i.e., no material present) withinthe finite difference grid. All newly created zones are null by default.

ATTACHED GRIDPOINTS — Attached gridpoints are pairs of gridpoints that belong toseparate sub-grids that are joined together. The dam is joined to the foundation alongattached gridpoints in Figure 8.7. Attached gridpoints do not have to match between sub-grids, but sub-grids cannot separate from one another once attached.

INTERFACE — An interface is a connection between sub-grids that can separate (e.g.,slide or open). An interface can represent a physical discontinuity such as a fault or contactplane. It can also be used to join sub-grid regions that have different zone sizes.

MARKED GRIDPOINTS —Marked gridpoints are specially designated gridpoints thatdelimit a region for the purpose of applying an initial condition, assigning material modelsand properties, and printing selected variables. The marking of gridpoints has no effect onthe solution process.

REGION — A region in a FLAC model refers to all zones enclosed within a contiguousstring of “marked” gridpoints. Regions are used to limit the range of certain FLACcommands, such as the MODEL command that assigns material models to designatedregions.

GROUP — A group in a FLAC model refers to a collection of zones identified by a uniquename. Groups are used to limit the range of certain FLAC commands, such as the MODELcommand that assigns material models to designated groups. Any command reference to agroup name indicates that the command is to be executed on that group of zones.

STRUCTURAL ELEMENT — Structural elements are linear elements used to representthe inter-action of structures (such as tunnel liners, rock bolts, cable bolts or support props)with a soil or rock mass. Some restricted material non-linearity is possible with structuralelements. Geometric non-linearity occurs in large-strain mode.



STEP — Because FLAC is an explicit code, the solution to a problem requires a number ofcomputational steps. During computational stepping, the information associated with thephenomenon under investigation is propagated across the zones in the finite differencegrid. A certain number of steps is required to arrive at an equilibrium (or steady-flow) statefor a static solution. Typical problems are solved within 2000 to 4000 steps, although largecomplex problems can require tens of thousands of steps to reach a steady state. Whenusing the dynamic analysis option, STEP refers to the actual timestep for the dynamicproblem. Other terms for step are timestep and cycle.

STATIC SOLUTION — A static or quasi-static solution is reached in FLAC when the rate ofchange of kinetic energy in a model approaches a negligible value. This is accomplished bydamping the equations of motion. At the static solution stage, the model will be either at astate of force equilibrium or at a state of steady-flow of material if a portion (or all) of themodel is unstable (i.e., fails) under the applied loading conditions. This is the default calcu-lation in FLAC. Static mechanical solutions can be coupled to transient groundwater flow orheat transfer solutions. (As an option, fully dynamic analysis can also be performed byinhibiting the static solution damping.)

UNBALANCED FORCE — The unbalanced force indicates when a mechanical equilibriumstate (or the onset of plastic flow) is reached for a static analysis. A model is in exactequilibrium if the net nodal force vector at each gridpoint is zero. The maximum nodal forcevector is monitored in FLAC and printed to the screen when the STEP or SOLVE commandis invoked. The maximum nodal force vector is also called the unbalanced or out-of-balanceforce. The maximum unbalanced force will never exactly reach zero for a numericalanalysis. The model is considered to be in equilibrium when the maximum unbalancedforce is small compared to the total applied forces in the problem. If the unbalanced forceapproaches a constant non-zero value, this probably indicates that failure and plastic floware occurring within the model.

DYNAMIC SOLUTION — For a dynamic solution, the full dynamic equations of motion(including inertial terms) are solved; the generation and dissipation of kinetic energy directlyaffect the solution. Dynamic solutions are required for problems involving high frequencyand short duration loads — e.g., seismic or explosive loading. The dynamic calculation isan optional module to FLAC (see Section 3 in Optional Features FLAC manual).

LARGE-STRAIN / SMALL-STRAIN — By default, FLAC operates in small-strain mode:that is, gridpoint coordinates are not changed, even if computed displacements are large(compared to typical zone sizes). In large-strain mode, gridpoint coordinates are updated ateach step, according to computed displacements. In large-strain mode, geometric non-linearity is possible.



Figure 8.8: Options within FLAC (FLAC manual)



There are ten basic constitutive models provided in FLAC Version 4.0, arranged into null,elastic and plastic model groups:

8.3.1 Null model group

1. null model

A null material model is used to represent material that is removed or excavated.

8.3.2 Elastic model group

2. elastic, isotropic model

The elastic, isotropic model provides the simplest representation of material behaviour. Thismodel is valid for homogeneous, isotropic, continuous materials that exhibit linear stress-strain behaviour with no hysteresis on unloading.



3. elastic, transversely isotropic model

The elastic, transversely isotropic model gives the ability to simulate layered elastic mediain which there are distinctly different elastic moduli in directions normal and parallel to thelayers.

8.3.3 Plastic model group

4. Drucker-Prager model

The Drucker-Prager plasticity model may be useful to model soft clays with low frictionangles; however, this model is not generally recommended for application to geologicmaterials. It is included here mainly to permit comparison with other numerical programresults.

5. Mohr-Coulomb model

The Mohr-Coulomb model is the conventional model used to represent shear failure in soilsand rocks. Vermeer and deBorst (1984), for example, report laboratory test results for sandand concrete that match well with the Mohr-Coulomb criterion.

6. ubiquitous-joint model

The ubiquitous-joint model is an anisotropic plasticity model that includes weak planes ofspecific orientation embedded in a Mohr-Coulomb solid.

7. strain-hardening/softening model

The strain-hardening/softening model allows representation of non-linear material softeningand hardening behaviour based on prescribed variations of the Mohr-Coulomb modelproperties (cohesion, friction, dilation, tensile strength) as functions of the deviatoric plasticstrain.

8. bilinear strain-hardening/softening ubiquitous-joint model

The strain-hardening/softening ubiquitous-joint model allows representation of materialsoftening and hardening behaviour for the matrix and the weak plane based on prescribedvariations of the ubiquitous-joint model properties (cohesion, friction, dilation, tensilestrength) as functions of deviatoric and tensile plastic strain. The variation of materialstrength properties with mean stress can also be taken into account by using the bilinearoption.

9. double-yield model

The double-yield model is intended to represent materials in which there may be significantirreversible compaction in addition to shear yielding, such as hydraulically-placed backfill orlightly-cemented granular material.

10. modified Cam-clay model

The modified Cam-clay model may be used to represent materials when the influence ofvolume change on bulk property and resistance to shear need to be taken into consider-ation, such as soft clay.

There are also six time-dependent (creep) material models available in the creep modeloption for FLAC.



8.4 Example analyses

1. Benchmarking : FEM v FDM v known solution Schweiger, Freiseder NUMOG (1995)

Flexible strip footing solution : undrained soil

E = 35 MPa, ν = 0.3, γ = 18 kN/m3, φ' =27.5o, su = 25 kPa, K0 = 0.65

Classical solution: failure load qmax = 5.14su = 128.5 kPa

• agreement very promising pre-failure, but• modelling of failure dependent on method and mesh• they should probably have used adaptive mesh refinement for the finite element analysis

30m20

m x

y

Point1 Point 2

1.5m

3.0mq

Figure 8.9: Geometry and loading for a strip footing on clay (Schweiger & Freiseder, 1995)

settlement (cm)

FEM-169 elements

FEM- 676 elements

FLAC grid 40 x 20

FLAC grid 120 x 80

appl

ied

load

q 1

0-1 (

kPa)

FEM-169 elements

FEM- 676 elements

FLAC grid 40 x 20

FLAC grid 120 x 80

y- displacement (cm)

appl

ied

load

q 1

0-1 (

kPa)

Figure 8.10: Results (Schweiger & Freiseder 1995)



2. Subsidence under a geotextile reinforced embankment: Lawson, Jones, Kempton 1995

• geotextiles used to prevent collapse of road due to unexpected subsidence (solution oflimestone / gypsum) in underlying geology,

• causes a trapdoor: soil arches over this, with large strain (deformation) above the void,• ideal for finite difference analysis to find tension in geotextile (and also differential

surface deformation which should be < 1% principal roads, and < 2% lower class roads),• here geotextiles are modelled as an elastic beam because of significant deflection,• results compared to the existing theory which assumes no arching in soil (BS8006-

generally conservative) and full arching (Giroud et al., 1990- likely to be unsafe),• also could be solution to covering /sealing former waste deposit and improving

foundation for subsequent building.

Figure 8.11: Design cross section through reinforced embankments, Lawson et al., (1994)

Embankment height H (m)

Re

info

rcem

en

t te

nsio

n Trs

(kN

/m)

Comparison of the three models in deter-mining the reinforcement tension for a voidspan of d = 4 m (γ = 20 kN/m2, φ’ = 35°)

BS 8006

Giroud

FDM

5% strain

Width of void d (m)

Comparison of the three models in deter-mining the maximum vertical deflectionat the base of the fill spanning the void

Ve

rtic

al d

efle

ctio

n a

t ba

se o

f fil

l D (

m)

1% strain (BS8006)

1% strain (Giroud)

1% strain (FDM)

5% strain (BS8006)

5% strain (Giroud) 5% strain (FDM)

Figure 8.12: Results, Lawson et al., (1994)



• predicted maximum vertical displacement at base of embankment agrees well withBS8006 and Giroud’s theories for all void widths with 1% and 5% strain in the geotextile,

• reinforcement tension will be overestimated if no arching in the embankment is included,• if a rigid base (e.g. rock) lies under the reinforcement, full arching is most likely to occur

and tension will be lower,• if soil is placed under the reinforcement, result is midway between the arching and no

arching solution.

3. Lateral cyclic loading of an embedded wall Springman, Norrish, Ng (1995)

An integral bridge abutment is sometimes used to avoid problems with expansion joints andbearings, especially where damage from e.g. de-icing salts will cause expensive mainte-nance. The deck-wall becomes a full moment connection and any changes in deck lengthdue to temperature effects (daily and seasonally) will load the fill behind the abutment. Thismeans that an element of soil from just behind the retaining wall will be subjected to cycliclateral loading and the results is a compaction and loss of volume.

Problem: cyclic behaviour of soil - how to model this?

Centrifuge behaviour (and prototype)

Figure 8.13: Geometry: All dimensions shown in mm (not to scale) (Springman et al., 1995)

Settlement profile after CWWN1 test

dense (Id = 83%)loose (Id = 23%)

Settlement profile after CWWN2 test



Real data from cyclic triaxial tests (Jeyatharan, 1991)

• shows significant volumetric strain occurring during cycles of axial strain along a stresspath

=> q/p’ ~ 3 (triaxial compression)

=> between axial strain of 0.3% to -0.4%

Deformed mesh of spread base wall in response to a 0.115° applied rotation

x(m)

y(m)

Active displacement of cycle 10

Numerical analysis + small strain stiffness(BRICK) model

cycle 10

6 mm 6mm

Figure 8.14: Deformed mesh (Springman et al., 1995)

Deformed mesh of spread base wall in response to a 0.115° applied rotation

x(m)

Active displacement at the top of the wall after 10th cycle

Numerical analysis + small strain stiffness(BRICK) model

cycle 10

6 mm 6mm Active displacement

y(m)



University of British Columbia (UBC) model after Beaty & Byrne (1999) compared to cyclictriaxial tests (Figure 8.15)

• shows good agreement in

=> q, p’ space although q/p’~ 2

=> q, axial strain space (between axial strain of 0.3% to -0.4%)

=> volumetric strain, stress ratio q/p’ space

=> similarly for volumetric against axial strain (although 1st unloading cycle not so good,so values of volumetric strain a little low)

Effective stress p’ (kPa) Axial Strain %

Dev

iato

ric s

tres

s, q

(kP

a)

Dev

iato

ric s

tres

s, q

(kP

a)

stress ratio, q/p’ Axial strain %

Vol

umet

ric s

trai

n (%

)

Vol

umet

ric s

trai

n (%

)

Figure 8.15: Cyclic behaviour of 100/170 (Fraction E) sand (after Jeyatharan 1991)



• results also look quite good when plotted in normal characteristics (particularly for loosesand samples) in terms of :

=> strength

=> volumetric

• although the dilation law displayed by the algorithm (Figure 8.17b) does not permit thesoil to reach a critical state where no further dilation occurs.

Figure 8.16: Cyclic behaviour: UBC model (Springman et al., 1995)



1000 kPa

50 kPa

Model prediction:continuous lineExperimental results:markers

1000kPa

200kPa

50kPa

100kPa

Model prediction:continuous line experi-mental results: markers

1000kPa

50kPa

Strength Characteristics

Figure 8.17: Comparison between monotonic laboratory test data and the UBC model (Springman etal.,1995)

b)

c)

Volumetric characteristicsloose sand

Volumetric characteristicsmedium dens sand

Model prediction:continuous lineExperimental results:markers

a)



4. Deformation of a 3 m high reinforced wall with 2 m long reinforcement (Jommi et al., NUMOG 1995)

End of construction

Deformed mesh Displacement vectors

Figure 8.18: Reinforced soil wall :deformed mesh (Jommi et al., 1995)

See also Springman, Balachandran, Jommi (1997) for an application relating to modellingreinforced soil wall in the centrifuge.

Figure 8.19: Normalised horizontal displacements d at the end of construction:(a) front of the wall(b) back of the wall (Jommi et al., 1995)

x

y

Spacing: d

H

a b

xd/H [%]

y/H



8.5 References

1. Beaty, M.H. and Byrne, P.M. 1999. A synthesized approach for modelling liquefactionand displacements, International FLAC symposium, Minneapolis, Minnesota.

2. FLAC Reference Manuals.

3. Giroud, J.P., Bonaparte, R., Beech, J.F. and Gross, B.A. 1990. Design of soil layergeo-synthetic systems overlying voids, Geotextiles and Geomembranes, Vol. 9, pp. 11-50.

4. Hoffman, J.D. 1992. Numerical methods for engineers and scientists. McGraw-Hill,New York.

5. Jeyatharan, K. 1991. Partial liquefaction of sand fill in a mobile arctic caisson underdynamic ice-loading, CUED PhD Thesis.

6. Jommi, C., Nova, R. and Gomis, F. 1995. Numerical analysis of reinforced earth wallsvia a homogenization method. Proceedings of the fifth international symposium onnumerical models in geomechanics-NUMOG V, pp. 231-236.

7. Lawson, C.R., Jones, C.J.F.P., Kempton, G.T. and Passaris, E.K.S. 1994. Advancedanalysis of reinforced fills over areas prone to subsidence. Fifth InternationalConference on Geotextiles, Geomembranes and related products, Singapore 1994, pp.311-316.

8. Schweiger, H.F. and Freiseder, M. 1995. Some results from benchmark tests forgeotechnical engineering. Proceedings of the fifth international symposium onnumerical models in geomechanics-NUMOG V, pp. 675-680.

9. Springman, S.M. 1989. Lateral loading on piles due to simulated embankmentconstruction, CUED PhD Thesis.

10. Springman, S.M., Norrish A.R.M. and Ng C.W.W. 1995. Cyclic loading of sandbehind integral bridge abutments. TRL Report Cambridge University.

11. Springman, S.M., Balachandran S. and Jommi C. 1997. Modelling pre-failure defor-mation behaviour of reinforced soil walls. Geotechnique, Symposium in print, Vol XLVII,No.3, pp. 653-663.

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