Modelling in Geotechnics

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Modelling in Geotechnics - Script

Part 1 Prof. Sarah Springman

In cooperation with

Dr. Jan Laue & Dr. Jitendra Sharma

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1 Design ... 1

1.1 Modelling ... 4

1.1.1 Numerical modelling ... 5

1.1.2 Physical modelling... 6

1.1.3 Validation and calibration of models ... 12

1.2 Full scale (FS) 1g testing... 13

1.3 Geotechnical centrifuge modelling ... 15

1.3.1 History of geotechnical centrifuges... 15

1.3.2 Types of centrifuge ... 15

1.3.3 Principles of modelling in the centrifuge ... 16

1.3.4 Scaling effects ... 17

1.3.5 Verification of models ... 18

1.4 References ... 19

2 Principles of numerical modelling ... 1

2.1 Why model numerically? ... 1

2.2 Validation of the finite element analysis (bench marking)... 3

2.2.1 El-Hamalawi (1997): mesh for a strip footing on clay ... 4

2.2.2 Bransby (1995): mesh for lateral pressure on pile in clay ... 5

2.3 Prediction... 7

2.4 Styles of numerical analysis using a computer... 7

2.5 Idealisation for numerical modelling as before for physical modelling... 8

2.5.1 Geometry... 8

2.5.2 Mesh design ... 9

2.5.3 Structure ... 9

2.5.4 Loading and construction effects... 10

2.5.5 Ellis (1997): Piled full-height abutment: 3D problem as 2D.... ... 14

2.5.6 Soil ... 15

3 Finite Element Method (FEM) in Geotechnical Engineering ... 1

3.1 Introduction... 1

3.2 Numerical methods used in geotechnical engineering ... 1

3.3 What is FEM? ... 2

3.3.1 Historical Background... 3

3.3.2 The fundamental steps of the FEM ... 3

3.3.3 Approximation of the Circumference of a Circle ... 3

3.4 Basic formulation of the FEM ... 5

3.4.1 Interconnected elastic springs ... 6

3.4.2 A plane truss element... 8

3.4.3 A constant strain triangular finite element ... 10

3.5 Approximations, accuracy and convergence in the FEM ... 13

3.6 Geotechnical finite element analysis ... 15

3.6.1 Plane strain and axisymmetric problems... 16

3.6.2 Different types of finite elements ... 17

3.7 Techniques for modelling non-linear stress-strain response ... 20

3.7.1 Tangential stiffness approach with carry over of unbalanced load ... 21

3.7.2 Modified Newton-Raphson method ... 21

3.8 Techniques for modelling excavation and construction ... 22

3.8.1 Excavation ... 22

3.8.2 Construction ... 24

3.9 Advantages and drawbacks of the FEM ... 25

3.9.1 Advantages... 25

3.9.2 Drawbacks... 25

3.10 Some popular commercial FEM programs ... 25

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3.10.2SAGE CRISP ... 26

3.10.3PLAXIS ... 27

3.10.4ZSOIL ... 27

3.11 Guidelines for the use of FEM in geotechnical engineering ... 28

3.12 Concluding remarks... 30

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

4.1.1 Scaling laws ... 1

4.1.2 Scaling of time ... 1

4.2 Scale effects ... 2

4.2.1 Stress distribution in centrifuge model: Depth ... 3

4.2.2 Stress distribution in a centrifuge model... 5

4.2.3 Particle size effects... 6

4.2.4 Coriolis acceleration ... 6

4.2.5 Boundary effects... 7

4.3 Scaling under earthquake conditions ... 10

5 Practical considerations: mechanical ... 1

5.1 Beam Centrifuges... 1

5.1.1 Capacity... 1

5.1.2 Swing platform, package and liner ... 2

5.2 Drum Centrifuges ... 4

5.2.1 Capacity... 4

5.3 Site investigation devices (penetrometers, vane)... 8

5.3.1 Vane: ... 8

5.3.2 Penetrometer:... 8

5.3.3 Cylindrical T-Bar: ... 8

5.4 Post-test investigation devices ... 9

5.4.1 Photographic: camera & flash ... 9

5.4.2 Digital Images and PIV analysis ... 9

5.4.3 X-ray ... 9

6 Practical considerations: geotechnical ... 1

6.2 Design of soil model: real or laboratory ... 2

6.3 Kaolin as a model soil... 7

6.4 Preparation of soil samples in the DRUM centrifuge ... 21

7 In-situ testing, instrumentation, data acquisition... 1

7.1 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 ... 3

7.1.3 T-Bar penetration testing: to determine su ... 8

7.2 Measurement of displacement ... 10

7.2.1 Spotchasing ... 10

7.2.2 Digital Images and PIV analysis ... 10

7.2.3 Displacement measurements ... 11

7.2.4 Radiography ... 11

7.2.5 Excavation ... 12

7.3 Electronic/electrical instrumentation ... 12

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7.4 Summary ... 19

8 Finite Difference Analysis using FLAC ... 1

8.1 Basics ... 1

8.1.1 Specific to Geotechnics via FLAC ... 1

8.2 Finite Difference ... 3

8.3 Details of FLAC Program... 11

8.3.1 Null model group ... 16

8.3.2 Elastic model group ... 16

8.3.3 Plastic model group ... 17

8.4 Example analyses ... 18

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Modelling in Geotechnics

Introduction to Modelling

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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

Modelling in Geotechnics

• 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 be modelled effectively,

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

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 to design for them as indicated above by using Limit States to examine the potential structural damage 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) Idealisation Physical model Calculation model

Interaction

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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 been fined 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 years late)

Figure 1.3: Heathrow Airport: tunnel collapse

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• e.g. failure of 90m high Teton Dam (Fig. 1.4 - 1.6): (1976) poor design of central core in terms of material (silt), shape of core trench, and an irregular stepped longitudinal abutment 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

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Design Page 1 - 4

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

... as long as the model is appropriate!

1.1 Modelling

So 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 earthquake engineering, and are not considered further in this course. Simple forms of these methods will, however, be used in the future Eurocodes, in order to be able to reduce the relevant partial factors.

Low-medium risk, quick and cheap

Medium-high risk, 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) numerical low “a ccuracy ” high

Figure 1.7: Types of models

• Domenico Fontana (in 1585)

• 300m move: 330 ton, 30m Vatican obelisk • 1/50th model to demonstrate procedure

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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 specific fundamental (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 continua with partial differential equations to describe physical phenomena and 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 continua iterative 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

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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 not correct and since soil behaviour is nonlinear, the modelling is unsatisfactory.

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

Type Sketch of Model/

prototype σv at A [kPa] Advantages Disadvantages 1g, full scale 100 to 140 • stress correct • can control soil

conditions

• time to construct and for diffusion processes • boundary effects • cost Field monitoring 100 to 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 100 to 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 A 7m A 7cm A 7cm 1 m 100g ω

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

• parametric studies can be extremely useful in exposing mechanisms of behaviour relevant 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

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Design Page 1 - 8

Idealise any combination of: • geometry

• soil • structure • loading

• construction effects.

Figure 1.8 contains an idealisation exercise to emphasise these points in terms of centrifuge modelling on a bridge abutment and approach embankment

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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) and lateral 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 (s_u≅ 15 kPa) over stiff sand OUTPUT

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

3rd idealisation

INPUT

• pile group with cap

• stiffer clay (s_u≅ 40 kPa) over stiff sand OUTPUT for both rows of piles

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

also

• drag under pile cap

4th idealisation

INPUT

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

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

Single free-headed pile

Vertical load, q Soft Stiff p? δup Pile group Pile group Soft Stiff p? δup Bransby PhD, 1995 Vertical load, q

Inflight construction of embankment

Ellis PhD, 1997

Supply hopper

1

Piled full-height bridge abutment

δu

Embankment

Soft

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Design Page 1 - 10

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

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).

INPUT

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

also NOTE

• drag under embankment • drag under pile cap

OUTPUT load on abutment • active lateral pressure on wall

• arching of embankment loading onto wall base

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

(Springman, 2001)

Piled full-height bridge abutment

Ellis PhD, 1997

p?

δup

arching

Embankment loading on clay

Ellis PhD, 1997

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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 concrete piles / wall

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

→ 2nd idealisation - pile group (2 vertical piles x 3) with pile cap raised above ground 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 be small, light but strong and manoeuvrable!

• pile installation

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

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

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

• tunnel construction

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

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

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

• building an embankment

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

→ pour embankment in-flight (better stress path) but no allowance for

compacting the embankment material

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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 laws should therefore be appropriate. Selection of relevant non-dimensional groups, which control model behaviour, is often necessary. These should remain constant if the basic modelling premises are to be correct.

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

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

Even if the agreement between non-dimensional groups is satisfied, it is often difficult to judge whether the model is valid or not. Ideally it would be possible to compare results against a full scale prototype however it is extremely rare to obtain sufficiently good data to be 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 the modelling method, which may in turn be used to calibrate either soil properties or additional modelling methods (e.g. new numerical algorithms).

For example, a strip footing on a homogeneous clay layer with uniform undrained shear strength with depth, s_u, can be shown using plasticity theory to fail at a load of (2 + π)s_u. A physical test set up or a numerical calculation (or computer algorithm or run) may be checked against this to ensure that the model system is valid before further models are used 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 s_u [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/s_u 5.5 0.55 5.5 5.5 5.5

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1.2 Full scale (FS) 1g testing

Full scale 1g tests are not usually carried out on an 85m high earth dam or a 100m deep quarry which is about to be landfilled. It is much more likely that elements or aspects of a full scale test will be modelled. E.g. a trial embankment may be built quickly to failure in the appropriate clay to test the construction and compaction procedures to be used for a clay cored dam, and to measure in situ density, permeability and strength. This can be back analysed numerically and the results applied to model the behaviour of the complete prototype structure.

Equally Hertweck (1998) and Brinkmann (1999, 2001) have both used the IGT 5.5m x 4m x 3m Big Box to model an aspect of the behaviour of a clay barrier which they have also analysed numerically using finite element analysis with the same boundary conditions as in the Big Box. Subsequently they have also modelled the full scale behaviour of the prototype with the field boundary conditions.

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

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Design Page 1 - 14

Investigation of stress-deformation behaviour for two-phase manufactured diaphragm walls for 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)

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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 of creating 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 major thrust of development came from the Soviet Union. This was largely stimulated by the scaling advantages applicable to the weapons industry, and large scale explosions in particular. Pokrovsky made a major presentation concerning the use of a geotechnical centrifuge at the first International Conference on Soil Mechanics and Foundation Engineering 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 later at 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 in packages located at each ends of the beam to be subjected to increased gravity. Usually these packages are fixed to a swinging platform so that they are hanging in the vertical plane initially and can swing up to lie in the horizontal plane as the centrifuge acceleration is increased.

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 of soil between 0.1 up to 0.5 km wide by 1-3 km long in a gravity field of up to 500 g. This allows examination of soil behaviour for specific problems, which relate to shallow construc-tions (shallow foundaconstruc-tions onshore and offshore, transport processes for environmental geotechnics, 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 ng ω r ng

Figure 1.12: Types of centrifuge

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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 fields must 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 combination of 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 and radial)

Example:

If the effective radius of the centrifuge is 4 m, and the gravity level is 100 g, then angular velocity 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 is rained onto the fill, and the dr/dt term becomes important. This is described as the Coriolis effect and the particle trajectory follows a parabolic path. Deflector plates may be used underneath 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·

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

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

Newton first explained the concept of gravity - masses accelerated towards the centre of the earth in terms of a gravity force on each terrestrial mass. Relativity implies that the gravity force is identical to the inertial force - the small scale model will weigh more under angular 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 if the model dimension is scaled down by a factor of n, i.e. 1/nth scale model, then the stresses will be equivalent.

Effectively, the gravity acts on the nuclei - the centre of mass of each atom - and the net pressure builds up with depth. This means that there is no gravity or inertial force at the surface 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 m}3 _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

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Design Page 1 - 18

Similarity of stress (and strain) will be achieved in both model_m and prototype_p, for a sample in soil of the same density ρ and stiffness constructed at a scale of 1/n, located at an 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 dimension is 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, T_v= f(time/depth2) = c_vt/d2, becomes independent of gravity level for a depth of sample reduced to 1/n of the original, if the model time is also reduced by 1/n2.

(1D Diffusion equation - saturated soil) du/dt = c_vd2u/dz2

where u is excess pore pressure and time t scales with length z2 provided c_{v m} = c_{v p} . This offers a significant advantage because 27 years of prototype diffusion may be modelled in 1 day using a centrifuge at 100 g, and is especially useful for environmental problems or heat loss by conduction where diffusion is the main transport mechanism. However, in offshore foundations or earthquake problems, the pore pressures are created dynamically, with time scaling as: n times faster in the centrifuge and yet they decay in a diffusive 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 (and

same density) or reduce the value of permeability of the soil (Attention: this will cause a change 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 scaled according to the value of n. If a 1/100th scale model at 100g predicts the same prototype behaviour 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 prototype behaviour to check numerical analyses.

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1.4 References

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

2. Brinkmann, A. and Amann, P., Small and large scale tests for the determination of the mechanical 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 Federal Institute of Technology, ETH Zürich, 2001.

4. Hertweck, M., Untersuchung des Tragverhaltens von Steilwandbarrieren in Deponiebau mit grossmassstäblichen Modellversuchen. PhD Thesis, Swiss Federal Institute of Technology, ETH Zürich, 1998.

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

6. Taylor, R. N., Geotechnical centrifuge technology. Geotechnical Engineering Research Centre, 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 of Mining and Metallurgical Engineers, Tech. Pub. 425, p. 3-28, 1931.

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Modelling in Geotechnics

Numerical Modelling

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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 experience or 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 when extremely experienced and well known geotechnical engineers from Scandinavia designed these embankments based on the undrained shear strength obtained from vane shear tests. What experience had NOT shown beforehand was that the plasticity index of the soil affected the values of s_u obtained (LHA p.79, figure 6.17). The vane strength values of the relatively low plasticity Scandinavian clays required no correction factor but the strengths of the 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 when both the input data and the calculation method (algorithm) are appropriate…..GIGO or Garbage In...Garbage Out.

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

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

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Principles of numerical modelling 2 - 2

Considering the Ultimate Limit State (ULS) at first and knowing that the maximum vertical load on a strip footing q_max is approximately 5s_u, q_max ~ 100 kPa. The load applied to the footing = 200/3 = 67 kPa, so a global factor of safety would be 100/67 = 1.5 which is certainly insufficient. We would also remember that if the load was inclined then this value would 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 sensitive contents), and the structure to be built on it is extremely expensive (or indeed potentially dangerous if failure occurred e.g. nuclear power station), then a far more extensive numerical modelling process will be necessary. This may well entail more complex analyses of continua using a computer to solve a series of equations based on a mesh with appropriate boundary conditions, a range of loading scenarios and a suitable constitutive model (e.g. elastic, elasto-plastic, critical state).

These are often called finite element or finite difference analyses and they differ only in the method of solving the equations of equilibrium, compatibility and constitutive model (see table 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 an adaptive mesh for the whole sample and 229 or even 1791 elements depending on the level 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 the behaviour agree (Fig. 2.3)?

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

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

Figure 2.2: Range of mesh complexity for a triaxial sample Undrained triaxial test on

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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 of validation is necessary. Sometimes this is called „bench marking“ carried out against a known theoretical solution.

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

Mesh refinement entails either increasing the number of elements, or changing the density of the elements according to the areas where either the most shear strain or the greatest pore pressure build up arises (e.g. adaptive meshing - see also on right pictures in figure 2.2). This means that the elements are smaller and hence the assumptions which are valid 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

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Principles of numerical modelling 2 - 4 → 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 under undrained conditions on homogeneous isotropic rigid perfectly plastic soil with uniform shear strength s_u, the exact solution is (2+π)s_u(Prandtl, 1921, developed this based on classical plasticity theory for metals). El-Hamalawi modelled this using finite elements and represented the soil by an elastic perfectly plastic constitutive model under drained condi-tions; u_y 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

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

The ultimate solution for lateral pressure p_u caused by homogeneous isotropic rigid perfectly plastic soil with uniform shear strength s_u flowing around a pile in plane strain under 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 similar conditions is moved past a fully rough stationary pile. Following manual mesh refinement and development, the final agreement between the exact (11.94 s_u) solution and the computed result is within 2%, which is certainly close enough for most engineering analyses.

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

( )⋅s_u≤p_u≤(4⋅ 2+2⋅π)⋅s_u

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

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

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2.3 Prediction

Prediction has already been mentioned in several formats. Determination of styles of prediction 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 computer

It 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 Status

A 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

Advantages Disadvantages

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.

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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 complex boundary-value problems PLUS

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

• 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 symmetry or 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 far enough 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

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* 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 guidelines will be presented in more detail in chapter 3. Several examples have been shown on pages 2 - 4.

ADAPTIVE MESH REFINEMENT (e.g. El-Hamalawi, 1997) can be used to enrich and subdivide 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 sand drain wall.

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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/m2 5 kN/m 5 kN/m 2 kN/m2 1 kN/m 1 kN/m

Figure 2.11: Normal surface loading

Reality Model 3 2 1 1 2 3 Excavation Fill

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Use pile response to various loadings as examples:

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

e.g. the pile behaviour is a function of…..f ( E_pile / G , l / r_o , ν ) where

E_pile is the pile Young's modulus G is soil shear modulus

l is the pile length r_o 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 the

centrifuge

• 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.

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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 δz_s? δz_p? Pi le

Shaft friction component

Normal, uniformly distributed load

End bearing component Soil Pile r z Soil z r

Circular ‘footing’ load

Mesh

Section Soil

Soil

Shaft friction Axial loading on a long flexible cylindrical pile

r z

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Plane strain

1

Piled full-height bridge abutment

δu

Embankment Soft

Stiff

1

Piled full-height bridge abutment

Ellis PhD, 1997

p?

δup

arching

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

Soil Pile Section x z Lateral thrust CL

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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

hig

h s

tres

_s

soft clay

low

sress

soft clay

SAND SAND CLAY

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2.5.6 Soil

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

• 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

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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

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2.6 References

1. 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 of Cambridge. 1995.

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

4. E.A. Ellis, Soil-Structure interaction for full-height piled bridge abutments constructed on soft 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 die

Festigkeit 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 loaded laterally in cohesive soil. Géotechnique, 1984, 34, No. 4, pp. 613-623.

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Modelling in Geotechnics

Numerical Modelling

Finite Element Method

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3 Finite Element Method (FEM) in Geotechnical Engineering

3.1 Introduction

The importance of a carefully planned and executed experimental modelling can not be overstated. However, experimental modelling can be expensive and time-consuming and is normally 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 as accurately as possible. These parameters are then used as input to either limit equilibrium based programs (e.g. slope stability, bearing capacity, etc.) to predict failure loads (ultimate limit state) or a numerical analysis program (e.g. finite element method, finite difference method, etc.) to predict the deformation under working load conditions (serviceability limit state). In this chapter, we will focus on one of the most popular numerical analysis technique used in geotechnical engineering – the finite element method or FEM. The aim of this 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 reader may well consult one of the numerous books that deal with the mathematics and the numerical 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 finding solutions to a geotechnical engineering problem. These are summarized in Figure 3.1. In this section, we will focus on the numerical methods. One of the characteristic features of the numerical methods is that they usually involve solving a set of simultaneous partial differential 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, an iterative numerical approach is used. There are five major types of numerical methods used in geotechnical engineering – the finite element, the finite difference, the boundary element, the discrete element and the combined boundary/finite element. The way the PDEs are formulated 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 DifferenceFinite

Limit Equilibrium Finite/ Boundary Element Discrete Element

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Finite Element Method (FEM) in Geotechnical Engineering Page 3 - 2

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 a

discrete and a continuous system. For a discrete system, an adequate solution can be

obtained using a finite number of well-defined components. Such problems can be readily solved even with rather large number of components, e.g. the analysis of a building frame consisting of beams, columns and slabs (Figure 3.2). For a continuous system, such as a soil layer, the sub-division is continued infinitely so that the problem can only be defined using the mathematical fiction of infinitesimal. Depending on the level of complexity involved, there are two ways of solving such a problem. Simple, linear problems can be solved easily by mathematical manipulation. Solution of complex, non-linear problems involves discretization of the problem into components of finite dimensions (Figure 3.2) and then using a numerical method such as the FEM.

The most distinctive feature of the FEM that separates it from other numerical methods is the division of a given domain into a set of simple subdomains, called finite elements. Any geometric shape that allows computation of the solution or its approximation, or provides necessary relation among the values of the solution at selected points, called nodes, of the subdomain, qualifies as a finite element. Such a subdivision of a whole into parts has two advantages:

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

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3.3.1 Historical Background

The idea of representing a given domain as a collection of discrete parts is not unique to the FEM. It was recorded that ancient Greek mathematicians estimated the value of π by noting that the perimeter of a polygon inscribed in a circle approximates the circumference of the circle. 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 approximate solution or comprehension of the whole, by studying the constituent parts of the whole is vital to almost all investigations in science, humanities, and engineering. The FEM is an outgrowth of the familiar procedures such as the frame analysis and the lattice analogy for 2- and 3-dimensional bodies. Its application is not exclusive to engineering. It has been used in other fields such as mathematics & physics. One of the earliest examples of its use was in mathematics by R. Courant who used it for the solution of equilibrium and vibration problems (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 plane stress 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 the advent 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 cheap computing power at their disposal, FEM is the first choice for many engineers and scientists embarking on the analysis of a wide variety of engineering problems – from designing a new ergonomic shoe sole to designing a supersonic fighter aircraft. Its use in the field of bioengineering, for example, the modelling of knee prosthesis or stress analysis of brain oedema, 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 in order 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 a problem.

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 by line segments, whose lengths they were able to measure. The approximate value of the circumference 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 the finite element analysis of any problem.

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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 h_e be the length of the element

Ωe_{in the mesh. For a typical element}_Ωe_{, h}

e is given by (see Figure 3.3c):

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

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 a meaningful 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 (assembled elements) is equal to the sum of the lengths of individual elements.

(a) (b)

(c)

Approximation of the circumference of a circle by line elements:

(a) Circle of radius R; (b) Uniform and non-uniform meshes used to represent the circumference of the circle;

(c) a typical element. Element Node R

θ

e

h

_e

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(3.2)

Then, P_n represents an approximation to the actual perimeter, p, of the circle. If the mesh is uniform, i.e. h_e is the same for each element in the mesh, θ_e = 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 P_n converges 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 as closely as we wish by a finite number of piecewise-linear functions. As the number of elements is increased, the approximation improves, i.e. the error in the approximation decreases.

3.4 Basic formulation of the FEM

In this section, the basic formulation of the FEM will be introduced using three simple examples: (1) a system of interconnected elastic springs; (2) a one-dimensional plane truss element; 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 2 T_a T_b _W T_d 2 Equilibrium at Node 2

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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 k_a, k_b, k_c and k_d. Therefore, the tensions in

these springs are given by:

(3.5) where e_a, e_b, e_c and e_d 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, we are making sure that the system does not fall apart, i.e. springs remain connected with each other.

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: