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
http://igtcal.ethz.ch/mig
ETH Zürich Institute of Geotechnical Engineering
1
Modelling in Geotechnics
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 2.2
2.3 2.4 2.5
2.6
3
Why model numerically? ..................................................................................... 1 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 Prediction............................................................................................................. 7 Styles of numerical analysis using a computer.................................................... 7 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 References ........................................................................................................ 17
Finite Element Method (FEM) in Geotechnical Engineering .................... 1 3.1 3.2 3.3
Introduction.......................................................................................................... 1 Numerical methods used in geotechnical engineering ........................................ 1 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 3.10.1ABAQUS ...................................................................................................... 25 Page - 1
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
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 3.13 References ........................................................................................................ 30
4
Scaling laws and applications for centrifuge modelling ............................ 1 4.1
Introduction.......................................................................................................... 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 6.1 6.2 6.3 6.4
7
Introduction.......................................................................................................... 1 Design of soil model: real or laboratory ............................................................... 2 Kaolin as a model soil.......................................................................................... 7 Preparation of soil samples in the DRUM centrifuge......................................... 21
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 7.3.1 Other considerations: ................................................................................... 17
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7.4 7.5
8
Modelling in Geotechnics
Summary ........................................................................................................... 19 References ........................................................................................................ 20
Finite Difference Analysis using FLAC ..................................................... 1 8.1 8.2 8.3
8.4 8.5
Basics .................................................................................................................. 1 8.1.1 Specific to Geotechnics via FLAC .................................................................. 1 Finite Difference .................................................................................................. 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 Example analyses ............................................................................................. 18 References ........................................................................................................ 26
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ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
Modelling in Geotechnics
Introduction to Modelling
Prof. Sarah Springman
ETH Zürich Institute of Geotechnical Engineering
1
Modelling in Geotechnics
Design
• The main focus of modelling is to achieve optimal design, which is cost effective, safe and also aesthetic. Codes Structure
Ground
Interaction Load model Idealisation
Rock/soil layering/properties
Physical model Calculation model
Failure (ULS) Dimensioning
Serviceability & deformations (SLS)
Design
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 combinations 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)
Design
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Modelling in Geotechnics
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 • • • • •
Failure October 1994 half face, 3 stage excavation sprayed concrete lining construction method not properly applied 80m of tunnel and concourse collapsed (fortunately without loss of life) • subsequently stabilised by structural and lightweight 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
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.
Figure 1.4: Failure of the Teton Dam, USA (1976)
Design
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Figure 1.5:
Teton Dam during failure
Figure 1.6:
Teton Dam after failure
Modelling in Geotechnics
• 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 subsequent back analysis.
Design
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ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
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? numerical low Low-medium risk, quick and cheap
physical Medium-high risk, more time
• full scale (high cost) • small scale, 1 gravity • small scale, enhanced gravity (medium cost)
“accuracy”
• simple, theoretical or empirical • complex, iterative/computational (relatively low cost)
high
Figure 1.7: Types of models
e.g. simple small scale 1g physical model…. • • • •
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
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.
Design
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Modelling in Geotechnics
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
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?
y in
out?
x
Disadvantages
Tab. 1.1: Numerical modelling
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Modelling in Geotechnics
1.1.2 Physical modelling • • • •
1g, full scale testing (+ field monitoring) 1g, small scale laboratory testing ng, small scale centrifuge modelling Type
Sketch of Model/ prototype
1g, full scale
σv at A
Advantages
Disadvantages
[kPa] 100 to 140
• stress correct • can control soil conditions
• time to construct and for diffusion processes • boundary effects • cost
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
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
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
7m A
Field monitoring A
7m
1g, small scale 7cm
A 100g, 1/100th scale in centrifuge
ω
1m
100g
7cm
A Tab. 1.2: Physical 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.
Design
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Modelling in Geotechnics
• 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 characteristics? • 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
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Modelling in Geotechnics
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 Figure 1.8: Severn Bridge (e.g. Severn Bridge).
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Modelling in Geotechnics
The prototype problem • embankment on soft clay • clay moves laterally further than the piles • therefore “passive” lateral thrust on piles
Piled full-height bridge abutment δu Embankment
→ ?magnitude and ?distribution • increased pile bending moments (BM) and lateral deformation at deck (δu < 25 mm)
Soft Stiff
→ ?magnitude and ?distribution 1
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
Single free-headed pile
δup Soft
Vertical load, q
p?
Stiff
3rd idealisation
Pile group
INPUT • pile group with cap • stiffer clay (su ≅ 40 kPa) over stiff sand
δup
Vertical load, q
OUTPUT for both rows of piles • lateral “passive” pressure p as f(q) • BM & δup
Soft
p?
Stiff
also • drag under pile cap
Bransby PhD, 1995
Inflight construction of embankment 4th idealisation
Supply hopper
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) Ellis PhD, 1997
Design
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Modelling in Geotechnics
4th idealisation INPUT • construct embankment in stages inflight • soft clay (su ≅ 20 kPa) over stiff sand
Embankment loading on clay Supply hopper
also NOTE • drag under embankment • drag under pile cap Ellis PhD, 1997
4th idealisation
Piled full-height bridge abutment
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
arching
δup
p?
(Springman, 2001) Ellis PhD, 1997
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).
Design
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Modelling in Geotechnics
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 inflight). 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 → NOTE: sand is dry & is not the same material as the 'real' embankment. Design
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Modelling in Geotechnics
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 su and a key non-dimensional group Hρg/su is given in table 1.3
Geom. scale Acceleration H [m] su [kPa]
Prototype 1:1 1g 5 15
Model 1 1:10 1g 0.5 15
Model 2 1:10 1g 0.5 1.5
Model 3 1:10 1g 0.5 15
Model 4 1:10 10g 0.5 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)
Similitude of models, when defined by Hρg/su, 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 fundamental 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, su, can be shown using plasticity theory to fail at a load of (2 + π)su. 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).
Design
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1.2
Modelling in Geotechnics
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.
Figure 1.9: ETH Big Box for 1g models
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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Modelling in Geotechnics
Figure 1.10: Dissertation Hertweck (1998)
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 1.11: Dissertation Brinkmann (2001)
Design
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ETH Zürich Institute of Geotechnical Engineering
1.3
Modelling in Geotechnics
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 developments 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. Centrifuge modelling: BEAM
Centrifuge modelling: DRUM
r ng
ω
ng
..
ng = ω2r - r g = 9.81 m/s2
r
ω
..
ng = ω2r - r g = 9.81 m/s2
Figure 1.12: Types of centrifuge
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 constructions (shallow foundations onshore and offshore, transport processes for environmental geotechnics, tunnelling, ice forces on structures, slope stability, etc.). Design
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Modelling in Geotechnics
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.
ω
r ng
· 2 ⋅ ω ⋅ r· + r ⋅ ω 2
r ⋅ ω – r·· Figure 1.13: Accelerations of a rotated body
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 r·, r··, ω· 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. Design
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Modelling in Geotechnics
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 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:
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Modelling in Geotechnics
Similarity of stress (and strain) will be achieved in both modelm and prototypep, 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 dimensionless 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, 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 also reduced 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 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.
Design
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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-gebundenem 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.
Design
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Modelling in Geotechnics
Modelling in Geotechnics
Numerical Modelling
Prof. Sarah Springman
ETH Zürich Institute of Geotechnical Engineering
2 2.1
Modelling in Geotechnics
Principles of numerical modelling Why model numerically?
This is inevitably part of any calculation and Code-based design methods, in which responsibility 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 su 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.
Figure 2.1: Understanding computer technology
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 su ~ 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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Considering the Ultimate Limit State (ULS) at first and knowing that the maximum vertical load on a strip footing qmax is approximately 5su, qmax ~ 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 Serviceability 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)!)
Undrained triaxial test on overconsolidated clay
229 elements
1791 elements
Figure 2.2: Range of mesh complexity for a triaxial sample
• 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 (investigations 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! Principles of numerical modelling
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Modelling in Geotechnics
Finite element mesh for analysis of model embankment on soft clay
Centrifuge model test of stageconstructed embankment on soft clay after failure (Almeida 1984)
Computed and measured development of settlement at clay surface with time (Almeida 1984) Figure 2.3: Calibration of soil parameters and match deformation response
2m 200mm 100mm 1g
h 2g
20g
Model “grows” as ng increases in centrifuge
h increases as time increases Layers added in prototype
Figure 2.4: Modelling “construction” processes
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
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→ greater variation is possible, → the model becomes closer to reality and → the solution becomes more exact. answer
exact solution
increasing mesh refinement Figure 2.5: Accuracy of result with mesh refinement
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 su, the exact solution is (2+π)su (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 conditions; uy is the settlement and b is the footing width. Initial mesh
At start of yielding
At failure
Mechanism at failure Width of footing b Figure 2.6: Foundation on clay
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0
Modelling in Geotechnics
1
2
3
q/s u
4
5
6
u y/b
initial (5.444) remeshing (5.147)
exact = (π + 2)
Figure 2.7: Effect of mesh status on load - settlement curve
2.2.2 Bransby (1995): mesh for lateral pressure on pile in clay The ultimate solution for lateral pressure pu caused by homogeneous isotropic rigid perfectly plastic soil with uniform shear strength su flowing around a pile in plane strain under undrained loading conditions….. (Randolph and Houlsby, 1984). ( 6 + π ) ⋅ su ≤ pu ≤ ( 4 ⋅ 2 + 2 ⋅ π ) ⋅ su 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 su) solution and the computed result is within 2%, which is certainly close enough for most engineering analyses.
Finite element modelling of a single pile in 2-d.
Figure 2.8: Mesh and boundary conditions Principles of numerical modelling
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Load-transfer curve for a single pile in elastic-plastic soil.
Figure 2.9: Load - transfer curve for pile under lateral load
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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. 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
Poulos summarised the modes of modelling as shown in Table 2.2. Analysis class C B
A
Characteristics and typical example
Advantages
Disadvantages
Simplified methods, using closed form solutions. Simple soil models used. Methods using boundary elements, with simplified soil models.
Easily applied, and allow rapid parametric studies.
Complex numerical methods (finite element, finite difference).
Can consider detailed and complex problems. Soil models can be more realistic.
Requires substantial idealization, and experience in assessing parameters. Requires some idealization, and experience in assessing parameters. Difficult to examine complex problems. Requires experience in assessing soil parameters which may be unfamiliar. Considerable effort to prepare data and interpret output.
Relatively easy data input. Familiar soil model parameters used. Relatively rapid to run and interpret.
Tab. 2.2: Classes of soil-structure interaction analysis
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 necessary 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 Principles of numerical modelling
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c) d)
Modelling in Geotechnics
Simple constitutive model (e.g. elasticity) ….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 constitutive 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 → limit number of nodes and elements according to complexity, typically...
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* non-critical structure * dam with deep foundation
150~200 elements 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 precondition 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. Principles of numerical modelling
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Modelling in Geotechnics
(EI)soil between piles
+ n(EI)piles
=
(EI)wall
Figure 2.10: Equivalence
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. Reality
Model
10 kN/m2
5 kN/m
2 kN/m2
5 kN/m 1 kN/m
1 kN/m
1m
Figure 2.11: Normal surface loading
Excavation 1 2 3
Fill 3 2 1
Figure 2.12: Excavation and 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 ( 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 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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Axial loading on a long flexible cylindrical pile Axial load What is the vertical deformation pattern in the soil and in the pile due to the axial load on the pile?
δzp?
Pile δzs?
Soil
Shaft friction Pile
Plan
θ
Soil
End bearing
r Section Plan axisymmetry r and θ plane: z common
Shaft friction component
Mesh
Normal, uniformly distributed load
r
z
r z
Pile Soil
End bearing component r
z Soil
Circular ‘footing’ load Figure 2.13: Modelling an axially loaded pile
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Plane strain
Piled full-height bridge abutment
Piled full-height bridge abutment δu Embankment
arching
δup
Soft
p? Stiff
1
Ellis PhD, 1997
1
Figure 2.14: Piled full-height bridge abutment
x
δu x
z Lateral thrust
y Pile
Soil
CL Plan
Section
Select half space: PLANE STRAIN x and y plane: z common When pile is displaced laterally relative to the soil, what is the relative soil-pile movement?
Figure 2.15: Idealisation for lateral thrust on a single row of piles from embankment loading
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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.
sand embankment
h ig
hs
low sress tre s
piled abutment wall
s kPa
soft clay Figure 2.16: Finite element analysis: contours of horizontal stress (Ellis, 1997)
SAND
CLAY
SAND
Figure 2.17: The finite element mesh with vertical drains (Ellis, 1997) Principles of numerical modelling
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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 • set νu= 0.49 with a short time step within a consolidation analysis.
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Figure 2.18: Influence of the Poissons ratio on the settlement of a strip footing (Potts & Zdraykovic, 1999)
Consolidation analysis (Biot's equations) - more time consuming • transition from undrained condition to drained condition • check the movement of the system with time
time undrained
settlement Figure 2.19: Settlement of a footing in time Which do you want to choose for your analysis?
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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
Dr. Jitendra Sharma
ETH Zürich Institute of Geotechnical Engineering
3 3.1
Modelling in Geotechnics
Finite Element Method (FEM) in Geotechnical Engineering 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 investigation 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 Solution of Geotechnical Problems Solution of Geotechnical Problems
“Exact” or Closed Form
Finite/ Boundary Element
Empirical, Based on Experience
Numerical
Limit Equilibrium Boundary Element
Finite Difference
Finite Element
Discrete Element
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. Finite Element Method (FEM) in Geotechnical Engineering
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What is FEM?
Semi-infinite Continuum
A finite element
Discretization
Discrete Problem
Continuous Problem
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). Finite Element Method (FEM) in Geotechnical Engineering
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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 representing 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 minicomputers, 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 illustrates 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 circumference of the circle. In doing so, we will also learn about certain terms that are used in the finite element analysis of any problem. Finite Element Method (FEM) in Geotechnical Engineering
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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 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 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 he.
Element R
Node (a)
(b)
he θe
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.
(c) 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. Finite Element Method (FEM) in Geotechnical Engineering
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(3.2) Then, Pn represents an approximation to the actual perimeter, p, of the circle. If the mesh is uniform, 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 Pn 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
1 d1
a 2
d2
2 Tb
b d
3 d3
Ta
W2
Td
Equilibrium at Node 2
c 4
d4
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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 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, 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:
(3.10)
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The matrix on the left-hand-side is called the global stiffness matrix. Equation (3.10) can be written in matrix notation as: Kd = W These simultaneous equations can be solved by elimination and values of displacements can be obtained. From the values of displacements, the force in each spring can be calculated. 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 the forces exerted on each spring at nodal points. One of these matrices is added into the global stiffness matrix for each spring in the system. (3.12) 3.4.2 A plane truss element
dy2
y’
Length = L dy1
y
1
2
x’ dx2
α dx1
x Figure 3.5: A plane truss element
In this section, we will apply the same principles of compatibility, material behaviour and equilibrium to a one-dimensional plane truss element (Figure 3.5). The formulation is now more complex than that for a simple system of linear elastic springs. You may have noticed that 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 degrees of freedom, i.e. it can move in both x- and y-direction. However, as we shall see, the general solution procedure remains the same regardless of the increased complexity. Finite Element Method (FEM) in Geotechnical Engineering
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In calculating the strains in this element, we are only interested in the displacements along the direction of the element. It is, therefore, logical to define a system of axes x’-y’ that is local 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)
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where D is the matrix of material behaviour or constitutive matrix for the element. In this case, 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 εˆ , we obtain: (3.20)
ˆT From the above equation, a e 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 be given 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-dimensional problems, we move to the formulation of a two-dimensional constant strain triangular finite Finite Element Method (FEM) in Geotechnical Engineering
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element. Figure 3.6 shows the simplest triangular finite element used for two-dimensional continuum analysis.
y dy2 2
dx2
h dy1
dy3 3
x 1 dx1
dx3 h
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 the element. Hence, the element is called a finite element. 2. The coefficients c0, c1, etc. in equation (3.23) are obtained by substituting the coordinates 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) Finite Element Method (FEM) in Geotechnical Engineering
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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 and the 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. Finite Element Method (FEM) in Geotechnical Engineering
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Although the above three examples illustrate the basic idea of the FEM, there are several other features that are either not present or not apparent from the discussion of these examples. 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-dimensional 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 convergence 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 FEM
1. 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.
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(a)
(b)
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 discontinuity 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.
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∆ ‘Smoothing’ zone
Rate of strain d2u/dx2
Strain du/dx
Displacement u
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-∞ 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.
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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).
qu ‘Exact’ solution
Mesh A - 24 Elements
Mesh B - 48 Elements 24
48 No. of Elements
96
Mesh C - 96 Elements 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/mechanical applications in mind. These programs cater for materials that can be produced under controlled conditions and therefore, have well-defined physical or mechanical properties, e.g. metals, plastics, polymers, concrete, etc. The most important material in a geotechnical analysis is the soil. A soil’s physical or mechanical properties have to be measured instead of being specified or specially fabricated. These properties vary enormously from site to site and can be profoundly affected by factors such as sampling techniques, specimen handling and preparation, characteristics of the measurement and data acquisition techniques. Therefore, the constitutive modelling takes the centre stage in a geotechnical FEM program. The three phase (soil-water-air) nature of soil makes realistic constitutive modelling of soil a formidable task. Since the shear strength of a soil at a given point depends on the effective stress at that point, the stress-strain response of a soil is highly non-linear. For a geotechnical finite element analysis, the FEM program should have the following 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. Finite Element Method (FEM) in Geotechnical Engineering
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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 or mechanical applications, it is rarely used in geotechnical engineering. Most of the geotechnical problems can be assumed to be either plane strain or axisymmetric without significant 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).
y z x Figure 3.10: A plane strain problem
Axisymmetric problems: For an axisymmetric problem, both the structure and the loading exhibit radial symmetry about the central vertical axis (Figure 3.11). Consequently, the circumferential strains can be ignored in the solution and the problem reduces to a twodimensional problem in a vertical radial plane. Keep in mind that the problem can only be reduced to an axisymmetric problem when both the structure and the loading are symmetric about the central vertical axis. If one of the two does not exhibit radial symmetry, either the problem has to be treated as a three-dimensional problem or techniques involving Fast Fourier Transforms (FFTs) have to be used. The numerical integration for an axisymmetric problem 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 to vertical concentric loads, excavation of vertical shafts of circular cross-section, consolidation around a vertical drain.
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C L y r
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 FEM program. These elements can be classified based on either the dimensions of the problem or the order of the element. They can also be classified on the basis of whether the coupled consolidation 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 threedimensional problems. • Typical 1-D elements include: (a) bar elements for the modelling of struts, geotextile reinforcement, ground anchors and any other structural element that is not capable of resisting flexure, and (b) beam elements for the modelling of retaining walls, tunnel linings and any other structural element requiring flexural rigidity. • Typical 2-D elements include (a) triangles and quadrilaterals for the modelling of soil and structural components of significant dimensions, and (b) slip elements for modelling of soil-structure interface behaviour. • Typical 3-D elements are hexahedrons and tetrahedrons for the modelling of soil and structural components. Some FEM programs also have 3-D slip elements for modelling of soil-structure interface behaviour.
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dy2
dy2 dx2
dy1
θ2
dx2
dy1 θ1 dx1
dx1
(a) Two-noded bar element
(b) Two-noded beam element
dy2
dy1
dx2 dy3
Structure
t Soil
L dx3 dx1 (c) 2-D element
(d) 2-D slip element (e) 3-D elements
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 shape function. The constant strain triangle in the example above is a first-order element. A mesh containing only first-order elements requires a large number of elements for a sufficiently accurate solution. • For a second-order element, a quadratic or second-order polynomial is used as shape function. As a result, the strain within the element is distributed linearly. Hence, these elements are also called linear strain elements. Such elements usually have one or more mid-side nodes in addition to the vertex nodes. One does not need to use a large number 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 shape function. The strains, therefore, have a cubic variation within the element and the element is often called a cubic-strain element. Such elements have several mid-side nodes as well as nodes inside the element in addition to the vertex nodes. It is not common to use such elements for a routine geotechnical analysis. Their use is limited to special situations such as testing a new constitutive model, unit cell radial consolidation problems.
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Displacement
Displacement
x
x
(a)
(c)
Displacement
(a) First-order element (b) Second-order element
x
(c) Fourth-order element (b) 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 formulation, 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.
+ Displacement Element
= Pore Pressure Element
Consolidation Element
Figure 3.14: Consolidation element
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3.7
Modelling in Geotechnics
Techniques for modelling non-linear stress-strain response σ
σ
Fe = K ⋅ ae
δF e = Ki ⋅ δae
ε
ε
(a) Non-linear stress-strain response
(a) Linear stress-strain response
σ
σ
E4
ε
Increment No. 1 2 3
(b) Non-linear stress-strain response Figure 3.15: Linear and behaviour
non-linear
4 5 6
ε
(b) Piecewise linear approximation
material
Figure 3.16: Piecewise linear approximation of non-linear material behaviour
The basic formulation of the FEM described in Section 3.4 is applicable only for materials that obey linear stress-strain laws (Figure 3.15a). However, as mentioned above, the stress-strain behaviour of a soil is highly non-linear (Figure 3.15b) and therefore, for solution 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 curve should be approximated by a set of interconnected straight lines (i.e. it is made piecewise linear) and then an incremental form of equation 3.21 is used. This approach is illustrated in Figure 3.16. Depending on the degree of non-linearity, the imposed loading (or displacement) is divided into sufficient number of increments and equation 3.21 is solved for each increment in succession. This is the simplest way of modelling a non-linear material. The trick here is to make sure that the piecewise linear approximation does not drift from the true stress-strain curve by a certain tolerable amount. However, the application 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 for elastic-perfectly plastic models such as the Mohr-Coulomb model. The reason for this is that the yield function and the failure criterion are one and the same for such models and there 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 in internal forces that are not in equilibrium with external forces. Therefore, the stress state must be corrected back to the failure criterion. This can be achieved in several different ways. 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
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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 is computed based on the tangential stiffness at the beginning of an increment, say from 0 to a displacement d1 as shown in figure 3.17b. In other words, the stress-strain response is now considered linear for this increment and is represented by the tangent drawn at the starting point of the increment. The internal load at the end of this increment (∆P1) is no longer in equilibrium with external load and this out-of-balance load (∆PC1) is re-applied to the 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 the out-of-balance load is rather large. The accuracy of the solution can be assessed by examining the global equilibrium error (percent difference between the sum of external loads and sum of internal forces) at the end of each increment. For elastic-perfectly plastic models, this error should never be allowed to go beyond 15 to 20%. To achieve this goal, a sufficiently large number of increments should be used. Another alternative is to divide each increment into 5 or 10 sub-increments (Figure 3.17c). This will ensure that the magnitude of out-of-balance load for each sub-increment is small.
P
τ 1
∆τc1
yield surface
∆P2
2
{∆Pc1 } = ∫ B T ∆τ c1d (vol)
∆P1
σ
∆Pc1 d1
d
d2
(b) Tangential stiffness approach
(a) Stress state correction
P
P
∆P1
Sub-increment 1 2 3
d
(c) Use of sub-increments to apply out-of-balance load
d1
d
(d) Modified Newton-Raphson method
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 the tangential stiffness approach, the stiffness matrix is computed based on the tangential stiffness at the beginning of an increment. However, the out-of-balance load is not carried over to the next increment. Instead, an iterative procedure shown in Figure 3.17d is followed. The out-of-balance load (∆PC1) is re-applied to the mesh and the resulting increFinite Element Method (FEM) in Geotechnical Engineering
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mental displacements are added to the current displacements. If further yielding takes place during the application of ∆PC1 then a second set of out-of-balance load (∆PC2) are calculated and the above procedure is repeated until convergence is reached, i.e. the resulting incremental displacements or the out-of-balance load is less than a preset tolerance. The main advantage of this procedure is that the stiffness matrix is computed only at the beginning of an increment. However, rather large number of iterations required to 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 main categories: trenches, shafts and tunnels. Trenches can be rather small, e.g. for laying of a drainage pipe (Figure 3.18a), or big and deep, e.g. for the construction of basement car park (Figure 3.18b). The effect of excavating a small trench on surrounding soil and structures 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 of damaging the surrounding structures. It is, therefore, not surprising that its design almost invariably involves conducting a few FEM analyses. The length and the width for a typical deep excavation are comparable and hence, it is a 3-D problem. However, a 3-D FEM analysis 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 only difference is that axisymmetric conditions are assumed.
C L Diaphragm Wall
1m
(a)
5~10 m
4~8 m
2m
6~12 m
Struts
(b)
Figure 3.18: Trenches and deep excavations
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Ground Surface
Modelling in Geotechnics
Tunnel Face
(a) Vertical Transverse Section
Unsupported Heading
(b) Vertical Longitudinal Section
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 a tunnel 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 significant 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 vertical transverse section sufficiently away from the face of the tunnel) and to use a volume loss parameter that takes into account the 3-D effect in an approximate manner. To study the ground movements ahead of the tunnel face, a vertical longitudinal section is considered and plane strain conditions are assumed. For both the deep excavation and the tunnel, the modelling of excavation is achieved in the same way – by removing the elements from the mesh. Here, it is worth noting that the body forces within the element are composed of both soil (effective stress) and water (pore pressure) as shown in Figure 3.20. When an element is removed, both the soil and water body 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 are negative pore pressures (pore suction) on the inner boundaries of the excavation. Unless some support in the form of a retaining wall is provided, the soil will eventually lose its suction and the excavation will collapse. However, such removal of body forces is not realistic for certain situations, e.g. installation of a diaphragm wall. The trench for a diaphragm wall is filled with either water or bentonite slurry. In this situation, one must either re-apply the water body forces or apply body forces corresponding to the bentonite slurry on the inner boundaries of the excavation.
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Effective stress + pore pressure
(a) Excavation
(b) Stresses acting before excavation
(c) Stresses after excavation (zero)
(d) Net effect of excavation
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 more layers of soil over existing or made-up ground, e.g. construction of a highway embankment on soft clay. The placing of a layer is modelled either by adding elements to the existing mesh or by applying pressure at the boundaries (Figure 3.21). The latter approach gives satisfactory solution provided the newly placed layers are not expected to undergo any shear deformation. If this is not the case, the technique of adding elements to the mesh should be used. An element that is added is assumed to be unstressed and the self-weight of the element is the only contributor to the body forces of that element. For this reason, the added elements must either have elastic properties or have a small non-zero value of apparent cohesion c’ if elastic-perfectly plastic model is used. A constitutive model that requires specification of a stress history, e.g. Cam-clay or other critical state models, is unsuitable for modelling of added elements. First Layer CL
Second Layer CL Embankment Soft Clay
(a) by adding elements
(b) by pressure loading
Figure 3.21: Techniques for modelling layered construction of an embankment Finite Element Method (FEM) in Geotechnical Engineering
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Compaction is an integral part of a geotechnical construction activity and its effects should ideally be included in the modelling. However, the effects of compaction are difficult to quantify in terms of stresses. In addition, the soil that is being compacted is usually partially saturated. These factors make the modelling of compaction activity quite complicated and therefore, 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, mass transport and steady-state transport analyses. • Coupled formulations that include: Biot’s consolidation theory, thermo-mechanical coupling, thermo-electrical coupling, fluid flow-mechanical coupling, stress-mass diffusion coupling, piezoelectric and acoustic-mechanical coupling. The most important of these from a geotechnical point-of-view is Biot’s consolidation theory. Finite Element Method (FEM) in Geotechnical Engineering
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• Dynamic stress-displacement analysis, determination of natural modes and frequencies, transient response via modal superposition, steady-state response resulting from harmonic loading, response spectrum analysis, and dynamic response resulting from random 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 and 3-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-elastic models. Constitutive models that are useful for geotechnical analysis are von Mises, Mohr-Coulomb, Drucker-Prager, Extended Drucker-Prager (non-associated flow), Cam Clay, Modified Cam Clay, Capped Drucker-Prager (Cam Clay with Extended DruckerPrager for use in tunnel excavation), and strain-rate dependent plastic laws. • User-defined constitutive models can also be incorporated with the help of a subroutine interface. • 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 the Cambridge University Soil Mechanics Group in the 1970s and 80s. CRISP was one of the first FEM programs dedicated to geotechnical analysis. In the early 1990s, SAGE Engineering Ltd., UK developed the pre- and post-processors for this program and began marketing the program by the name SAGE CRISP. The main features of SAGE CRISP are as 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 the developers of SAGE CRISP are in process of incorporating this facility. • Its element library includes 1-D, 2D and 3D continuum, bar, beam and interface elements. • Almost all of its constitutive models cater for geotechnical applications. These include general elastic (linear and non-linear), anisotropic elastic, elastic-perfectly plastic with von Mises, Tresca, Drucker-Prager, Mohr-Coulomb failure criteria, Cam Clay, Modified Cam Clay, Schofield, 3-Surface Kinematic Hardening (for small-strain modelling) and hyperbolic (Duncan and Chang type) models. • It can model excavation and construction.
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• It does not include a large-strain formulation but it can deal with large strain problems in an approximate manner by using large number of increments and updating of geometry at 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. Its name is a combination of PLane strain and AXISymmetric. As the name suggests, it can only do 1-D and 2-D analyses although a 3-D version is being developed. Its features include: • 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) anisotropic elastic, Mohr-Coulomb (associated as well as non-associated flow), Soft soil (Cam Clay), Soft soil creep and Hardening soil (hyperbolic) models. • It can model excavation and construction. In addition, it can do analysis of tunnel excavation that incorporates a volume loss parameter that represents the contraction around tunnel lining due to overcut by the tunnel boring machine and the loss of pressure at 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. Its features 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/
Finite Element Method (FEM) in Geotechnical Engineering
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Modelling in Geotechnics
3.11 Guidelines for the use of FEM in geotechnical engineering There are no shortcuts for learning to use the FEM effectively. One becomes an FEM expert by experience and a lot of hard work. However, the following guidelines will make sure that one has a good start to the learning endeavour. • Use smaller elements in regions where the rate of change of stress with distance is greater. This happens, for example, near the edges of a loaded area, near a re-entrant corner in the mesh or where adjacent parts of a mesh have significant differences in stiffness (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 stresses tend to infinity. The smaller we make the elements near the concentration, the higher are the stresses. Sometimes, a stress concentration will “spoil” the solution locally, leading to oscillation of stresses. Here, it is worth remembering that infinite stress concentrations are mathematical fiction that may be unimportant in describing real behaviour and therefore, it is often advisable to ignore them. • When increasing the element size from area of interest to the far boundaries, avoid increasing 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 yours and the computer’s time. • Keep the triangular elements as equilateral as possible and the quadrilateral elements as square as possible. Edge of the Loaded Area
Re-entrant Corner
Interface between pile and soil
Stress Concentration
Zone of Interest
Figure 3.22: Areas of FEM domains that require finer elements
• Avoid using curved elements as interior edges between elements in a mesh - only use them at external boundaries or internal boundaries (e.g. inside of a tunnel) if absolutely necessary. • Where you place the boundary of a mesh can make a big difference to the outcome of the analysis. If you are unsure of the boundary effect, try two different meshes – one with a close and the other with a far boundary.
Finite Element Method (FEM) in Geotechnical Engineering
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• 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 in equilibrium. If the in-situ stresses are not in equilibrium, either you have not input consistent values of soil unit weight or have not applied correct fixity conditions to one or more of mesh boundaries. • When using elastic-perfectly plastic constitutive models with either a Mohr-Coulomb or a Drucker-Prager failure criterion, specify a small value for c’ (0.1 or 1 kPa) even if the material has c’ = 0 kPa. This will ensure that the initial state of stress for the material is not on the failure surface. • In order to model the incompressibility of a saturated soil under undrained conditions, a Poisson’s ratio (ν) of 0.5 should ideally be used. However, if ν = 0.5 is input into an FEM analysis, it will result in serious ill-conditioning of the equations. The reason for this is that 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 of disaster in a geotechnical FEM analysis. Before applying these boundary conditions, make sure that you fully understand the ground water conditions for your problem. Most FEM programs treat any mesh boundary as impermeable by default. Setting the excess pore 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. Its effect must be felt by the adjacent elements in the next time step. Otherwise, oscillation of pore pressures can occur. The minimum time step required for this purpose can be computed based on the parabolic isochrone solution to the consolidation equation as shown in Figure 3.23. umax
umax
u
u
L
y
L = 12cv t or t min
y
L2 = 12cv
Oscillation of pore pressure due to insufficient time-step
Figure 3.23: Minimum time-step for dissipation of excess pore pressures
• When modelling excavation or construction by removing or adding elements to the mesh, respectively, use several layers of elements and remove/add these elements layer by layer, applying each layer over several increments. This will ensure that the stiffness of the soil being removed or added is correctly modelled.
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3.12 Concluding remarks In the beginning of the chapter, we stated that site investigation and laboratory testing are used to obtain the input soil parameters for an analysis using FEM. As a geotechnical engineer, one must never forget that soil is an extremely difficult material to characterize. Sampling disturbances, poorly controlled laboratory experiments, failure to interpret the results from laboratory tests in a scientific manner are some of the factors that introduce errors and uncertainty in the values of soil parameters. Therefore, the results of a FEM analysis must always be critically examined by comparing them with the results of another FEM 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 References 1.
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 and vibrations. Bulletin of American Mathematics Society, Vol.49.
3.
Zienkiewicz, O.C. and Taylor, R.L. (1989). The Finite Element Method, Vol. 1, Basic Formulation and Linear Problems, McGraw-Hill, London.
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Modelling in Geotechnics
Centrifuge Modelling 1
Prof. Sarah Springman
ETH Zürich Institute of Geotechnical Engineering
4 4.1
Modelling in Geotechnics
Scaling laws and applications for centrifuge modelling Introduction
4.1.1 Scaling laws It is necessary to take the scaling laws into account before conducting any test series. The scale effects need to be worked out as a basis for the planned modelling. The following table shows the different scaling factors. A brief discussion of these has been given at the end of chapter one. Some limitations to the modelling process and ways of solving some of these problems arising from the associated errors are mentioned briefly in this chapter. 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
Tab. 4.1: Scaling laws
4.1.2 Scaling of time As in fluid mechanics, it is not always possible to achieve correct scaling in all dimensionless 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, 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 also reduced by 1/n2. (1D Diffusion equation - saturated soil) 2
∂u ⁄ ∂t = c v ∂ u ⁄ ∂z
2
where u is excess pore pressure and time t scales with length z2 provided cv m = cv p .
Scaling laws and applications for centrifuge modelling
4-1
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
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). Parameter
Unit
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
Scale effects
The range and magnitude of possible shortcomings exposed by the scaling laws may be described as scale effects. These must be reviewed to ensure that they will not affect the outcome of the experiments and so the verification or modelling of models technique may well be a useful way of checking this. Typical scale effects are: • non-uniform acceleration field (with depth (beam & drum) - & width of model (beam not drum)) • particle size effects • Coriolis acceleration • boundary effects (due to being at small scale)
Scaling laws and applications for centrifuge modelling
4-2
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
4.2.1 Stress distribution in centrifuge model: Depth Since the inertial radial acceleration in the centrifuge is not linear with depth, but proportional to centrifuge radius, the depth to radius ratio of the model is of major importance.
R
r*
r
σv
z
σv
aR 2/3rds model depth
understress
δr overstress
z Model
error
Vertical stress in Centrifuge
z Vertical stress in Prototype
Figure 4.1: Vertical stress distribution 2
Equation 1
σ vm
density of soil
=ρ
2 ω 2 = ρ ------ ( r – r∗ ) 2
σ vp = ρg ( nz ) ; ρ const with z. radius to top of model = r∗
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∗ : r
2
2
ρr ω σ v = ∫ ρrω dr = --------------2 r∗ 2
2
2 ρω 2 = ---------- ( r – r∗ ) r∗ 2 r
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 the centrifuge (parabolic) and prototype (linear) stresses to be equal at some depth to achieve this over the important section (depths) of the model.
Scaling laws and applications for centrifuge modelling
4-3
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
Effective radius of model
σvm = σvp at z = aR = R occurs at a depth below surface of model = a R
where total depth of model
= 3aR/2
where (Equation 1) =>
so stresses are equal at r = R where r∗ = R(1-a)
and
z = aR
σvp = σvm 2
ω 2 2 2 ρngaR = ρ ------ ( R – R ( 1 – a ) ) 2 2
ω 2 2 = ρ ------ R ( 1 – 1 + 2a – a ) 2 2
ω R gn = ----------- ( 2 – a ) 2 2
ω = 2ng ⁄ { ( 2 – a )R } Check error (which will be close to the maximum understress) at z = aR / 2 : where r = R - aR/2 r*=R(1- a) 2
2
2
2
2
σ vm = ρ ⋅ ω { R ⋅ ( 1 – a ⁄ 2 ) – R ⋅ ( 1 – a ) } ⁄ 2 2
2
ρ ⋅ n ⋅ g ⋅ R ⋅ {– a + a ⁄ 4 + 2 ⋅ a – a } = -----------------------------------------------------------------------------------------------(2 – a) ρ ⋅ n ⋅ g ⋅ R ⋅ a ⋅ {1 – 3 ⋅ a ⁄ 4} = -------------------------------------------------------------------------(2 – a)
a⋅R Error at z = ⎛⎝ ------------⎞⎠ 2
ρ ⋅ n ⋅ g ⋅ R ⋅ a ⋅ (4 – 3 ⋅ a) = ----------------------------------------------------------------4 ⋅ (2 – a) = ( σ vm – σ vp ) ⁄ σ vm
ρ ⋅ n ⋅ g ⋅ R ⋅ a ⋅ ( 4 – 3 ⋅ a )4 – 3 ⋅ a )⎛ ---------------------------------------------------------------⎛ (-----------------------⎞ – ρ ⋅ n ⋅ g ⋅ R ⋅ a ⁄ 2⎞⎠ ⎝ ⎝ 2 – a – 2⎠ 4 ⋅ (2 – a) = ------------------------------------------------------------------------------------------------------------------------ = ---------------------------------------ρ ⋅ n ⋅ g ⋅ R ⋅ a ⋅ (4 – 3 ⋅ a) 4–3⋅a -----------------------------------------------------------------------------------4 ⋅ (2 – a ) 2–a (4 – 3 ⋅ a – 4 + 2 ⋅ a) = ---------------------------------------------------- = – a ⁄ ( 4 – 3 ⋅ a ) 4–3⋅a understress
Scaling laws and applications for centrifuge modelling
4-4
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
If you use the overall height of the soil for your model, taking the effective radius of the centrifuge at z=2/3 rds is a good choice to minimize the error. Looking at the problems relating 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) if the effective radius R is taken at 2/3rds model depth, where z = aR, calculate the error for a = 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.
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=3aR/2
1 / 23
1 / 31
1 / 39
1 / 47
1 / 59
Model depth mm
250
182.5
150
125
100
Tab. 4.3: Example
4.2.2 Stress distribution in a centrifuge model
Width (Exaggerated scale)
1m
Figure 4.2: Stress distribution as a function of width in a centrifuge
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 centrifuge spindle, so this is not relevant. However, soil models in the beam centrifuge (unless the ground surface is curved) will have higher stresses at the package boundary. Likewise the ground water surface will also be curved and higher at the boundary (and this will affect the total stress (but not the effective stress) at depth) ⇒ e.g. + 20 kPa here. The larger the radius of the beam, the lower is this effect. When investigating the response of a sheet pile wall, the structure should be placed centrally in the mould.
Scaling laws and applications for centrifuge modelling
4-5
ETH Zürich Institute of Geotechnical Engineering
4.2.3
Modelling in Geotechnics
Particle size effects
How can the centrifuge be used to model soil if the particles are not reduced in size by a factor n? • Model: clay particles of mean size d50 = 1µm. Prototype: sand at 100g with 100 x 0.001 mm = 0.1 mm particle size? → not sensible: clay and sand have different particle shapes etc. and hence they will have different stress-strain-volume change characteristics. • Model: sand particles of 0.1mm. Prototype: sand (with 1mm diameter) at 100g with 100 x 0.1 = 10 mm particle size? • 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)
D
d50
• Model: sand particles of 0.1mm. Prototype: gravel at 100g with 100 x 0.1 = 10mm particle size? → correct modelling process, but check particle shapes (rounded/angular) and particle hardness (crushing) which will affect dilatancy + also beware of any boundary 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 relevant to the positioning of the embankment in the hopper. Hopper
??
rω2
2 r· ω
Sand embankment too far!
Wooden spacer block
Clay
Figure 4.3: Positioning of an embankment in the hopper Scaling laws and applications for centrifuge modelling
4-6
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
· 2r ω Consider error as a function of the nominal centrifugal acceleration: ---------- < 10% ng so for r = 4m, @ 100 g, then ω ~ 15 rad/sec and for error < 10 %: radial velocity r· must be < 3.27 m/s cf. angular velocity = r ω = 4 x 15 = 60 m/s (typically for example above), the range of the Coriolis 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 will become important. · term ) horizontal shaking due to E/Q: x· ,x·· ( r ω vertical shaking due to E/Q:
y··
( r··
term )
On the other hand, high velocity particles in blast loading are also less affected by error because they will tend to move in straight (almost) lines, when the radius of curvature of their particle trajectory rc ~ r (and r > 3m). 4.2.5 Boundary effects q
q q
SAND same D
σh’
τ
>5D CLAY “stress bulb”
z
boundary effects
Figure 4.4: ‚Plane strain‘ sand embankment on clay and pile installation or penetrometer penetration
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 Scaling laws and applications for centrifuge modelling
4-7
ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
Side friction 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)
Lubricants are applied to the strongbox walls (which have also been coated with a low friction paint) to minimise side friction on the soil model. The principal stresses may then become almost vertical / horizontal at the model boundary. Similarly, latex sheets may be marked with a grid, greased and placed between a sand embankment and strongbox walls or perspex face to reduce soil-wall adhesion to ~5° at stress 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 of the box or a stiffer layer will affect the data because the rigid surfaces will influence the strain field. It is recommended that there should be 5 penetrometer or pile diameters to a side wall. Likewise, for stiff soils, there should be 5 - 10 diameters below the pile tip to the rigid layer. Refraction and reflection of waves Typically waves may be generated by blast loading or earthquakes, and these are reflected or refracted at the boundaries. These boundaries can be rigid with vibration suppressing materials 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), calculate 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?
Scaling laws and applications for centrifuge modelling
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a) errors +/- 5%
σv
∴ a = 1/6 is o.k. ⇒ ex.: Radius to drum wall = 1.1 m σv' = 6 x 0.35 x 100 = 210 kPa at the 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 water pressure to be created, so downward hydraulic gradient with u = 165 kPa at the 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 and 225 kPa at the base, • OCR profile is then given for equilibrium at 100g, together with the ideal profile of su,vane , • σv' exceeds previous σv,max' below model depths of 170 mm (17 m prototype; point X in Fig. 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), verify the undrained strength profile (su,vane) shown in Figure 6.4. What values of a and b appear to have been used from the Table 6.2?
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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 (shown below 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 Vertical effective stress [kPa] 200
100
Overconsolidation ratio 1
2
Undrained shear strength [kPa] 0
40
10
1st at 1g
Stiffer crust
2nd at 2g 200 mm 20 m
20
σv,max’
Overconsolidated 200 mm 20 m Normally consolidated
Depth
at 100g
Depth
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 Vertical effective stress [kPa]
Undrained shear strength [kPa] 0
10
20
all depths normally consolidated
20 m
20 m at 100g
σv,max’
Depth
Depth
Figure 6.13: Consolidation from slurry in the centrifuge
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The g-level could also be increased for a period of time although this would tend to increase the maximum effective stress at the base of the sample first (theory of parabolic isochrones). If full consolidation (dissipation of excess pore pressure) was allowed then the gradient of the lines describing dσ' v, max ⁄ dz or dsu/dz will double but remain triangular in shape. 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). Either partial or full dissipation of excess pore pressures could be permitted prior to removal of this surface layer (with a scraper tool) depending upon desired undrained shear strength. Vertical effective stress [kPa]
Overconsolidation ratio
assume ∆σv const. over
1
0
OCR
10
20
∆σv = γdhSand
clay layer
20 m
20 m Depth of clay
Undrained shear strength [kPa]
at 100g
σv,max’
n.c. Depth
Figure 6.14: Consolidation from slurry in the centrifuge
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A hydraulic gradient could also be applied in the drum centrifuge by ponding water on top of the clay layer and ensuring there was no pore pressure in the sand layer at the base of the clay (Fig. 6.15b & Fig. 6.16). hSand
z
z
zw
base sand layer
base sand layer
drain open: u = 0 kPa
surcharge γd
clay layer (a)
clay layer
water
(b)
Figure 6.15: Drum centrifuge test setup (a) Sand surcharge (b) Downward hydraulic gradient
Pore pressure [kPa]
Total vertical stress [kPa] γwzw
γwzw
= Depth of clay z
Effective vertical stress [kPa]
+
at 100g
σ'v,max at 100g γwzw
Figure 6.16: Application of a downward hydraulic gradient
These techiques are very much under development currently. Successful attempts have been made in the Cambridge mini drum centrifuge to date. On of the major questions remains the installation of pore pressure transducers and how these 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?
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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 of Bochum mix kaolin at w = 70-80% but for a longer period) under a vacuum for a period of not 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 no air bubbles are included, • porous strips and filter papers, saturated with de-ionised water are located above and below the sample. N.B. ETHZ are currently developing their own methods for application to clay placement in the 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 / soil deposit, • pressure is regulated in stages => σv' on the consolidating deposit, • cavitation effects due to suctions (negative values of u) from unloading from σvc' may be minimised: → 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 controlled swelling, • allow for any subsequent consolidation settlement or swelling in planning installation depth of pore pressure transducers (PPTs), which are inserted between last 2 loading phases via temporary unload, • holes obtained following PPT insertion are backfilled with slurry injected through a syringe to ensure that any air will be excluded from the soil matrix (procedure specified by Phillips and Gui, 1992), • transducer wires are taken out through special ports in the side of the 850 mm diameter tubular strongboxes or pressed into the rear of the plane strain soil sample, with the wires 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 consolidation under 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 Practical considerations: geotechnical
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The height of the sample is directly proportional to specific volume v = 1+e, the initial volume (and height) of clay slurry may be determined. h σ' v, m ax = h w=120% ( 1 + e σ' v, max ) ⁄ ( 1 + e w=120% )
(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% & consolidated to equilibrium σ’v,max = 110 kPa, with samples expected to show w ~ 60%. We need a 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 consolidation at 1g. (N.B. In a drum centrifuge, maximum depth of clay would be about 200 mm, so clay will need to be placed slowly over a period of time to allow some consolidation to take place 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% in the drum centrifuge so that after consolidating under a surcharge of ∆σv = 100 kPa at 100 g there is a post-1g-consolidation model height of 60 mm. Calculate the respective weights of water & kaolin needed to mix the slurry if the drum has dimensions 2.2 m (diameter) x 700 mm (width) x 300 mm (depth)? After consolidation under ∆σv = 100 kPa what is w at mid-depth of the 60 mm layer and hence what is the saturated unit weight (assume initial value)? 50 mm
3
γ = 20 kN/m , ∆σv = 100 kPa
100 kPa
h
Clay
30 mm x 100 =3m
γ·z Depth of clay z
assume γ = 16.4 kN/m3
at 100g
Figure 6.17: Example (@ 100g)
At depth z = 30mm · 100 = 3m Practical considerations: geotechnical
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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): 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%
95.8mm
100.3mm
16.2kN/m3
16.5kN/m3
hw=120%= 4.132 x 60 / (1+e) γsat = (2.61 + e) 10 /(1+e)
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 coefficient of consolidation, t is time and h is length of the drainage path (for 2 way drainage, model depth = 2h).
Clay
Range of σv‘
e
kPa +
Speswhite kaolin
cv
kv
kv pred
mm2/s
10-6
10-6 mm/s Eqn (8)
= current σvc‘ 256+- 450+ 450+- 120 +
450 -60
mm/s -
0.3
0.72
-
1.21
0.57
0.34
0.9
Source
Bransby (1993)
-
0.58
0.35
-
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 Practical considerations: geotechnical
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Al Tabbaa (1987) quotes values of kv and horizontal permeability kh (from independent falling head and consolidation tests) for speswhite kaolin with respect to void ratio e: k v = 0.5 ⋅ e
3.25
k h = 1.43 ⋅ e
⋅ 10
2.09
–6
⋅ 10
–6
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 increments, 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 kPa in the lower half of the model where the sample would be normally consolidated, calculate time for 90% consolidation to occur. Adopt a conservative value of cv. How would you reduce 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 layer across most of the clay at mid-depth to aid dissipation of pore pressures. Link into the drainage system at the top/bottom of clay to facilitate drainage: reduces consolidation time by 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.
Figure 6.18: Stage 1
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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.
Figure 6.19: Stage 2
Stage 3 • slurry poured and first and second layers consolidated to 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.
Figure 6.20: Stage 3
A similar method could be adopted in the drum centrifuge (although placing filter paper would be a bit complicated and fiddly)!
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Inflight downward hydraulic gradient with intermediate drain
Figure 6.21: Inflight downward hydraulic gradient with intermediate drain
Clay: final model preparation Having consolidated the clay sample in a strongbox, some final preparation is required prior to 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 at a specific height with a uniform flow rate, • pouring may be halted at any time to allow placement of deformation markers, spots or lines of coloured sand, for subsequent displacement or strain determination, • known weight of sand poured into a specific volume allows determination of e, and hence D (or ID), • the sand will be saturated either by upward flow (but with low differential head to avoid piping) or by using a vacuum under a restraining sheet, Practical considerations: geotechnical
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• then, as for a clay model: structures, instrumentation & site investigation equipment installed........etc. • Properties: strength The peak angle of friction φ'max and the critical state angle of friction φ'crit may be based on D (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)) Triaxial data @ failure
Figure 6.22: Dilatancy for quartz sands: φ'max
- φ'crit v. p’ (Bolton, 1986)
This shows more easily how these equations may be used to determine possible dilatancy.
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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 equivalent D (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 disk which 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 standpipe driven by a small motor, • clay samples may be placed as a slurry in the same manner (although there are concerns about achieving full saturation), and sand may be placed on top to achieve overconsolidation, • downward hydraulic gradient consolidation is also possible by raising the water table above 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 stage following preliminary placement of a sand base.
N.B.: More will be added to this chapter as experience with the Zurich drum centrifuge develops ...
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(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)
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(1) A B C D
Modelling in Geotechnics
Upper buoyancy foam blocks fixed to drum Lower buoyancy foam blocks holding drain pipeworks Base drain for clay layer 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
Figure 6.23: Placement of a clay sample in a drum centrifuge (Dean et al., 1990)
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Modelling in Geotechnics
Centrifuge Modelling 4
Prof. Sarah Springman
ETH Zürich Institute of Geotechnical Engineering
7
Modelling in Geotechnics
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 calibrated under torsional load) mounted on the vane shaft (above soil penetration level), and causes a nominal failure surface of a cylinder of diameter D, and height L (and also potentially round the shaft): => peak and residual su,vane, (13) T = π su,vane ( D3 + 3D2 L) / 6 This is derived from the shear surface x su x lever arm (integrating an annulus on top and bottom surfaces and direct calculation for the cylinder). Torque T rotation speed ω dz
su
Measure T, ω & dz L/Dcentrifuge
~0.77 (14/18)
L/Dfield
~2 (130/65) su,vane
L
slow D
fast
ω speed of rotation
Figure 7.1: Shear vane geometry and influence of speed of rotation
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 not tested 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 to insertion will be relatively smaller),
In-situ testing, instrumentation, data acquisition
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• the shaft is greased and is fitted with a 15o slip coupling to separate the component of friction 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 respect to difference in scaling with respect to time, • time relative to the strain rate will be identical in both cases, • time taken for dissipation of u following insertion of the vane (by diffusion), scales as n2 faster, • the centrifuge vane data will tend to indicate higher strengths (greater drainage) for the same rotation speeds in prototype and model; therefore increase model vane rotation speed. Figure 7.2: Vane shear test data
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, and higher 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 soil database 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 strength for analysis.
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7.1.2 Cone penetration testing (CPT): to determine a profile of su Modelling principles: • the cone penetrometer presents different modelling difficulties & advantages (some relative 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 inflight vane test data, • but extrapolation from empiricism at full scale to model scale prediction leads to some uncertainty........ Scaling effects: geometry • modelling a prototype 10 cm2 diameter field cone at 100 g implies a centrifuge probe of 0.36 mm diameter! - impossible to manufacture a working device, let alone strain gauge a 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 from the friction exerted on the shaft. • the ETH cone has a diameter of 11.3 mm allowing separate measurement of the tip resistance 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 (tip resistance). 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 the most appropriate penetration rate?): → Nc = (qc - σv ) / su → can plot su / σv ' to reveal OCR,
(14)
→ pore pressure acting on shoulder of a piezocone should be allowed for,
In-situ testing, instrumentation, data acquisition
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→ 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 at centrifuge radius Rs at the surface of the soil.
Figure 7.3: Tip resistance profile in kaolin clay (Gui, 1995)
• 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)
In-situ testing, instrumentation, data acquisition
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Mechanism
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Relative density effect (Note: Id = ID) ID effect
Figure 7.4: Mechanisms and influence of relative density (Gui, 1995)
In-situ testing, instrumentation, data acquisition
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Operation: • drive continuously between 3-26 mm/s in sand, or 3-12 mm/s in clay, • dimensional limitations > 5 B (5 cone diameters) from the location of another important section of the model or from the edge of the strongbox, • the proximity of the base of the strongbox or a stiffer granular founding stratum will increase the measured load for between 5 - 10 B above that boundary, • these interactions may noticeable in soft clays.
be
less
Figure 7.5: Penetration rate and OCR effect (Gui, 1995)
Grain size effect
Figure 7.6: Grain size effect on LB sand (Gui, 1995)
Figure 7.7: Grain size effect on LB sand (Gui, 1995)
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Stress level effect
Modelling in Geotechnics
Dilatancy & mobilised friction
Figure 7.8: Influence of dilatancy (Gui, 1995)
European cooperation => unified testing methods: • recommendations regarding the operational and cone design standards have been reviewed 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, cone diameters and radius of centrifuges.
a)
b)
Figure 7.9: Cone penetration data from various European centrifuge laboratories (Renzi et al., 1994) a) sand b) clay In-situ testing, instrumentation, data acquisition
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Nonetheless, the agreement confirms that the CPT is the main method of site investigation in the centrifuge and that it can be comparable, within an Institution, and between many Institutions, provided testing methods are similar. 7.1.3 T-Bar penetration testing: to determine su
Figure 7.10: Schematic diagram (Stewart and Randolph, 1991)
Figure 7.11: Variation of Nb with surface roughness (Stewart and Randolph, 1991)
Modelling principles: • a semi-rough cross bar, of length approximately 5 bar diameters db (db = 7 mm), with smooth 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 force required to pull or push the cylinder through a rigid-perfectly plastic medium of shear strength 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 be assumed 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 an interface 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 where surface 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 shear and independent triaxial tests (Stewart and Randolph, 1991) were fair - the bar gave an su profile: In-situ testing, instrumentation, data acquisition
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→ which was very similar to the triaxial prediction for normally & lightly overconsolidated samples, → about 20% greater for OCR > 3, → vane shear results are expected to be < equivalent values at 100g (ingress of water during deceleration / prior to inserting the vane) - dependent on time after deceleration & clay cv .
Figure 7.12: 1
≤ OCR < 3
Figure 7.13: OCR > 3
(Stewart and Randolph, 1991)
Comparisons between su obtained from in-flight bar and cone penetrometer tests, post-test shear vane tests and predictions based on triaxial data are given in Figures 7.10 - 7.13 (Stewart and Randolph, 1991).
In-situ testing, instrumentation, data acquisition
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7.2
Modelling in Geotechnics
Measurement of displacement
7.2.1 Spotchasing • insertion of black marker 'bullets' => evaluation of relative movements at various stages in the loading programme by back analysis of high quality photographs taken in-flight, • these differential movements => appropriate values for input into a strain evaluation computer program to estimate contours of shear strain, directions of lines of no-extension etc., • in some cases it has been possible to derive sufficient information for use in the analysis of geostructural mechanisms (Bolton and Powrie, 1988; Sun, 1989), • e.g. for installation of an offshore jack up rig near to piles which are intended to support an operational platform (Phillips, 1990), vectors of outward horizontal movement may be obtained.
Figure 7.14: Vectors of outward horizontal movement post installation of spudcan (after Phillips, 1990)
7.2.2 Digital Images and PIV analysis Image capture using an inexpensive 2megapixel digital camera provides a significant increase 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 an image sequence. Image processing algorithms are available to apply the PIV principle to images of soil. The software has a precision of 1/15th of a pixel when tracking the movement of natural sand or textured clay. The system allows displacements to be measured to a precision greater Figure 7.15: Measurement of deformation in a triaxial test (White et al., 2001) than with still photographs or video films without installing intrusive target markers in the soil.
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a) Plane strain calibration chamber for pile testing
Modelling in Geotechnics
b) Selected node paths during pile penetration
Figure 7.16: Application: pile penetration in a calibration chamber (White et al., 2001)
7.2.3 Displacement measurements Sometimes it is helpful to measure post-test displacements - often this should be done immediately after the test while the model is still on/in the centrifuge so that it is not disturbed. 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
Figure 7.17: Forced subsidence of a model abyssal plain; dashes represent ruptures (Stone, 1988)
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• of movement of soil between piles due to surcharge loading (Springman, 1989),
Figure 7.18: Internal soil-pile interaction for a row of free-headed piles (Springman, 1989)
Figure 7.19: Internal soil-pile interaction for a pile group (Springman, 1989)
7.2.5 Excavation Much useful information may be obtained from careful excavation. In particular, it is necessary to determine the actual position of the pore pressure transducers (allowing for post test swelling or desiccation) so that back analysis of the pore pressure test data makes sense! An alternative to using lead threads is to create a small hole and to insert a piece of noodle or spaghetti. This softens on contact with water. The noodle can be exposed and its position measured (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)
In-situ testing, instrumentation, data acquisition
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Instrument
Input Voltage
Amplify/ Filter
mV or V output
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
charge amplifiers 1000x
7 pico Coulombs/g => V
separate J-Box, convert pressure change on piezoceramic crystal to charge
1V square wave @ 100Hz approx
yes
mV
electrical conduction
pH sensor 2 or 4 electrode Resistivity probe
Tab. 7.1: Instrumentation: Input, output, errors, signal processing
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 remote trigger so that the difference in the measured and the true pore pressure (that would have 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 interposed on air supply lines, mainly for actuator control.
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• Pore pressure transducers (PPTs) measure various magnitudes of pore pressures (typically, a 350 kPa range). The company DRUCK sells the smallest ones used as standard transducers in centrifuge. • cylindrical in shape, 6.4 mm in external diameter, with a main body 13 mm long, • a porous stone protects the load cell face so that pore pressure and not total stress is measured,
Figure 7.20: Diagram of a Pore Pressure Transducer
• the porous stone should be de-aired, the PPT should be inserted prior to the addition of the final loading increment in the consolidometer, having unloaded the clay, in stages of 100 kPa or less, to zero excess load at 1 g,
Resistivity probes • a typical 4 electrode resistivity probe shown with calibration test data, • electrical conduction occurs through pore fluid, • this is affected by soil porosity and chemical composition, • but the only variation tends to be the chemistry, • signals are generally multiplexed.
Figure 7.21: Diagram of a resistivity probe
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 photodiode, • a logarithmic amplifier enhances the output signal, • absorbance measured and according Beer's Law => concentration can be determined. In-situ testing, instrumentation, data acquisition
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Schematic design of the sensor
Overview of system
Fibre-optic pH sensor Ball lens coupler
Single channel yellow l.e.d.
sensor
photodiode
log
Figure 7.22: Schematic of system and sensor (Lynch, 1998)
Experimental arrangement for plume detection
pollutant
Light in
fibre - light in
fibre - light out non-bleed pH paper
CUED Environmental Geotechnics
Sodium hydroxide pollution plume in sand
Light out
optical fibres sensor porous base
sand
Strain Gauge Bridges and Load Cells Strain gauge bridges: using properties of changing electrical resistance due to strain of thin foil elements, suitably attached to host material - connected together in electrical circuits to measure: • • • •
shear normal load torsion bending moment
In-situ testing, instrumentation, data acquisition
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A R4
D
R1 V
B
R3
E
R2 C
Figure 7.24: Wheatstone Bridge
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 speed up heat dissipation with relatively large surface area to reduce glue stresses, In-situ testing, instrumentation, data acquisition
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• 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 power consumption). 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 between gauge / material, • use Wheatstone bridge to eliminate resistance changes due to temperature changes, • use similar lengths of supply leads etc.
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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:
• 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, Figure 7.25: „Stroud“ load cell • usually use heat treatable aluminium alloy HE15W, • Wheatstone bridges give a useful output to such small resistance changes, • locked in hysteresis/machining stresses should be relieved by strain cycling.
Calibration:
• apply normal, shear and eccentric loading independently to set up a loading matrix: [dV] = [a]
N M S
• invert to give N, M and S as a function of the total change in voltage of each strain gauge circuit, • loads should be applied repeatably & consistently on face of cell in same way as testing situation, • supply voltage should be stable, • at least 5 readings should be taken on a load-unload cycle to maximum level expected in tests. In-situ testing, instrumentation, data acquisition
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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 performance of large scale structures. Appropriate idealisation may be adopted to reveal the key mechanisms 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“ soils with the relevant relative density. A combination of site investigation devices, empirical interpretations and comparisons between in-situ and laboratory data allow determination of soil strength. Both displacement and failure mechanisms may be observed, leading firstly to an understanding of the problem under investigation, secondly to an evaluation of strains in the soil matrix and thirdly to data about the mean soil strength at failure.
In-situ testing, instrumentation, data acquisition
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7.5
Modelling in Geotechnics
References
1.
Airey, D.W. 1984. Clays in the simple shear apparatus. Cambridge University PhD thesis.
2.
Almeida, M.S.S. 1984. Stage constructed embankments on soft clay. Cambridge University PhD Thesis.
3.
Al Tabbaa, A. 1984. Anisotropy of clay. Cambridge University MPhil thesis.
4.
Al Tabbaa, A. 1987. Permeability and stress-strain response of Speswhite Kaolin. Cambridge University PhD thesis.
5.
Bakir, N., Garnier, J., Canepa, Y., 1994: Etude sur modèles Centrifugeuse de la capacite portantes de fondation superficielle, Revue Francais de Géotechnique 25, pp 5-14.
6.
Bay - Gress, C., 2000. Étude de l`interaction sol - structure - comportement non lineaire sol-fondation superficielles. Thesis pour obtenire le docteur, Louis Pasteur/ ENAIS, Strasbourg.
7.
Bjerrum, L. 1973. Problems of soil mechanics and construction on soft clays and structurally unstable soils. Proc. 8th ICSMFE Moscow, Vol. 3, pp. 111-159.
8.
Blight, G.E. 1968. A note on field vane testing of silty soils. Can. Geot. J. Vol. 5, No. 3, pp. 142-149.
9.
Bolton, M.D. 1986. Dilatancy of soils. Géotechnique 36, No. 1, pp. 65-78.
10. Bolton, M.D. 1991. Geotechnical stress analysis for bridge abutment design. TRRL CR270. 11. Bolton, M.D. and Powrie, W. 1988. Behaviour of diaphragm walls in clay prior to collapse. Géotechnique 38, No. 2, pp.167-189. 12. Bolton, M.D., Gui, M.W. and Phillips, R. 1993. Review of miniature soil probes for model tests. 11th SEAGC, Singapore. Preprint. 13. Bolton, M.D., Gui, M.W., Garnier, J., Corte, J.F., Bagge, G., Laue, J., Renzi, R. 1999. Centrifuge cone penetration tests in sand. Géotechnique 49, No. 4, pp. 543-552. 14. Bransby, M.F. 1993. Centrifuge test investigation of the buttonhole foundation technique. Data Report. 33p. 15. Bransby, M.F. 1995. Piled foundations adjacent to surcharge loads. PhD thesis. 16. Broms, B.B. and Casbarian, A.O. 1965. Effects of rotations of the principal stress axes and of the immediate principal stress on the shear strength. Proc. 6th ICSMFE, Montreal, Vol. 1, pp. 179-183. 17. Cheah, H. 1981. Site investigation techniques for laboratory soils models. Cambridge University MPhil thesis. 18. Corte, J.-F., Garnier, J., Cottineau, L.M. and Rault, G. 1991. Determination of model properties in the centrifuge. Centrifuge '91, H.Y.Ko and F.G.McLean (eds). Balkema. pp. 607-614. 19. Craig, W.H. 1988. Centrifuge models in marine and coastal engineering. Centrifuges in Soil Mechanics, W.H. Craig, R.G. James and A.N. Schofield (eds). Balkema. pp. 149168.
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20. Craig, W.H. and Chua, K. 1991. Large displacement performance of jack-up spud cans. Centrifuge '91, H.Y.Ko and F.G.McLean (eds). Balkema. pp. 139-144. 21. Dean, E.T.R., James, R.G. and Schofield, A.N. 1990. Drum centrifuge studies for EEPUK. Contract No. EP-022R TASK ORDER 1-022. Phase 1. Draft Report. 22. Ellis, E.A. 1993. Lateral loading of bridge abutment piles due to soil movement. Cambridge University 1st year Report. 27p. 23. Endicott, L.J. 1970. Centrifugal testing of soil models. Cambridge University PhD thesis. 24. Fuglsang, L.D. and Steensen-Bach, J.O. 1991. Breakout resistance of suction piles in clay. Centrifuge '91, H.Y.Ko and F.G.McLean (eds). Balkema. pp. 153-160. 25. Garnier, J., Ternet, O., Cottineau, L.-M., and Brown, C.J. 1999. Placement of embedded pressure cells. 26. Grundhoff, T., Latotzke, J. & Laue, J. (1998). Investigations of vertical piles under horizontal impact. Proc. Int. Conf. Centrifuge 1998, Kimura et al. (eds). Balkema, Rotterdam. pp. 569-574. 27. Gui, M.W. 1995. Centrifuge and mechanical modelling of pile and penetrometer in sand. Cambridge Univ. PhD thesis. 28. Hamilton, J.M., Phillips, R., Dunnavant, T.W. and Murff, J.D. 1991. Centrifuge study of laterally loaded pile behavior in clay. Centrifuge '91, H.Y.Ko & F.G.McLean (eds). Balkema. pp. 285-293. 29. Hellawell, E. 1993. Resistivity probes: a review of current practices and an investigation into the properties of different systems. CUED Report. 30. Hensley, P.J. 1989. Accelerated physical modelling of transport processes in soil. Cambridge University PhD thesis. 31. Horner, J.M. 1982. Centrifugal modelling of multi-layer clay foundations subject to granular embankment loading. King's College, London University PhD thesis. 32. König, D., 1998. An inflight excavator to model a tunnelling process, Proc. Int. Conf. Centrifuge 1998, Kimura et al. (eds). Balkema, Rotterdam. pp. 707-712. 33. Kotthaus, M. 1992. Zum Tragverhalten von horizontal belasteten Pfahlreihen, Schriftenreihe des Lehrstuhls für Grundbau und Bodenmechanik der Ruhr-Universität Bochum, Heft 18, Bochum. 34. Kulhawy, F. and Mayne, P. 1990. Manual on estimating soil properties. Report EL6800. Electric Power Res. Inst., Palo Alto, 306p. 35. Kutter, B.L. 1982. Centrifugal modelling of the response of clay embankments to earthquakes. Cambridge University PhD thesis. 36. Kutter, B.L., Sathialingam, N. and Herrmann, L.R. 1988. The effects of local arching and consolidation on pore pressure measurements in clay. Centrifuge '88, J.F. Corte, (ed). Balkema. pp. 115-118. 37. Lach, P. 1992. Seminar on centrifuge model tests on iceberg scour above pipelines buried in clay, and subject for forthcoming thesis at Memorial University of Newfoundland, St John's, Newfoundland. 38. Ladd, C.C., Foott, R., Ishihara, K., Schlosser, F. and Poulos, H.J. 1977. Stressdeformation and strength characteristics. 9th ICSMFE, Tokyo, Vol. 2, pp. 421-494. In-situ testing, instrumentation, data acquisition
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39. Laue, J. 1996. Zur Setzung von Flachfundamenten auf Sand unter wiederholten Lastereignissen. Schriftenreihe des Lehrstuhls für Grundbau und Bodenmechanik der Ruhr-Universität Bochum, Heft 25, Bochum. 40. Laue, J., Nater, P., Chikatamarla, R., Springman, S. M. 2002. Der Einsatz von „tactilse pressure sensors“ in geotechnischen Labor- und Feldversuchen. Proc. Messen in der Geotechnik. Braunschweig 41. Lawrence, D.M. 1980. Some properties associated with kaolinite soils. Cambridge University M. Phil thesis. 42. Lerouil, S., Kabbaj, M., Tavenas, F. and Bouchard, R. 1985. Stress-strain-strain rate relation for the compressibility of sensitive natural clays. Géotechnique 35, Vol. 2, pp. 152-80. 43. Lynch, R. 1998. Private communication. 44. Ma, J., 1994. Untersuchungen zur Standsicherheit der durch Stützscheiben stabilisierten Böschungen, Mitteilung des Institut für Geotechnik ; 38, Stuttgart. 45. Mahmoud, M. 1988. Vane testing in soft clays. Ground Engineering, Vol. 21, No. 7, pp. 36-40. 46. Mair, R.J. 1979. Centrifugal modelling of tunnel construction in soft clay. Cambridge University PhD thesis. 47. Mayne, P.W. and Kulhawy, F. 1982. K0-OCR relationships in soil. Proc. ASCE, JGED, Vol. 102, pp. 197-228. 48. Mayne, P.W. 1992. In-situ determination of clay stress history by piezocone. Wroth Memorial Symposium. Oxford. pp. 361-372 (Preprint). 49. Meigh, A.C. 1987. Cone penetration testing. CIRIA, Butterworths. 137p. 50. Mesri, G. 1975. Discussion: New design procedure for stability of soft clays. Proc. ASCE, JGED. Vol. 103 GT5. pp. 417-430. 51. Muir Wood, D. 1990. Soil behaviour and critical state soil mechanics. CUP. 462p. 52. Nadarajah, V. 1973. Stress-strain properties of lightly overconsolidated clays. Cambridge University PhD thesis. 53. Nunez, I. 1989. Tension piles in clay. Cambridge University PhD thesis. 54. Phillips, R. 1988. Centrifuge lateral pile tests in clay. PR-10592. Task 2 & 3 - Final Report to Exxon Production Research. 39p. 55. Phillips, R. 1990. Spudcan/Pile Interaction Centrifuge Model Test Spud 3 - Final Report. A report to EPR Corp. Houston, TX, USA. Lynxvale Ltd. Cambridge. 56. Phillips, R. and Valsangkar, A. 1987. An experimental investigation of factors affecting penetration resistance in granular soils in centrifuge modelling. CUED/D TR210. 17p. 57. Phillips, R. and Gui, M.W. 1992. Cone Penetrometer Testing. Phase 1 Report, In-situ investigation. EEC Contract: SC1-CT91-0676. 58. Poorooshasb, F. 1988. The dynamic embedment of a heat emitting projectile. Cambridge University PhD thesis. 59. Powrie, W. 1987. The behaviour of diaphragm walls in clay. Cambridge University PhD thesis. In-situ testing, instrumentation, data acquisition
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Modelling in Geotechnics
60. Randolph, M.F. and Houlsby, G.T. 1984. The limiting pressure on a circular pile loaded laterally in clay. Géotechnique 34, No. 4, pp. 613-623. 61. Renzi, R., Maggioni, W., Smits, F. and Manes, V. 1991. A centrifugal study on the behavior of suction piles. Centrifuge '91, H.Y.Ko and F.G.McLean (eds). Balkema. pp. 169-177. 62. Renzi, R., Corté, J.F., Rault, G., Bagge, G., Gui, M.W., Laue, J. 1994. Cone penetration tests in the centrifuge: Experience of five laboratories. Centrifuge ‘94. Leung, Lee and Tan (eds), Balkema, Rotterdam, pp. 77-82 63. Rossato, G., Ninis, N.L. and Jardine, R.J. 1992. Properties of some kaolin-based model clay soils. ASTM GTJODJ, Vol. 15, No. 2, pp. 166-179. 64. Rowe, P.W. 1975. Displacement and failure modes of model offshore gravity platforms founded on clay. Conf. Offshore Europe '75, pp.218. 1- 17. Spearhead Publications, Aberdeen. 65. Savvidou, C. 1984. Effects of a heat source in saturated clay. Cambridge University PhD thesis. 66. Schmidt, B. 1966. Discussion: Earth pressures at rest related to stress history. Can. Geot. J., 3, No. 4, pp. 239-242. 67. Schmidt, B. 1983. Discussion: K0-OCR relationships in soil. Mayne, P., Kulhawy, F. 1982. Proc. ASCE, JGED, Vol. 109, pp. 866-867. 68. Schofield, A.N. and Wroth, C.P. 1968. Critical State Soil Mechanics. McGraw-Hill. 309p. 69. Schofield, A.N. 1980. Cambridge University Geotechnical Centrifuge Operations. 20th Rankine lecture. Géotechnique 30, No. 3. pp. 227-268. 70. Sharma, J.S. 1993. Construction of reinforced embankments on soft clay. Cambridge University PhD thesis. 71. Siemer, T. (1996): Zentrifugenmodellversuche zur dynamischen Wechselwirkung zwischen Bauwerk und Boden infolge stossartiger Belastung, Schriftenreihe des Lehrstuhls für Grundbau und Bodenmechanik der Ruhr-Universität Bochum, Heft 27, Bochum. 72. Skempton, A.W. 1957. The planning and design of the new Hong Kong airport. Proc. ICE 7, pp. 305-307. 73. Sketchley, C.J. 1973. The behaviour of kaolin in plane strain. Cambridge University PhD thesis. 74. Smith, C.C. 1992. Thaw induced settlement of pipelines in centrifuge model tests. Cambridge University PhD thesis. 75. Springman, S.M. 1989. Lateral loading on piles due to simulated embankment construction. Cambridge University PhD thesis. 76. Springman, S.M. 1991. Performance of a single lattice leg under lateral load. EEPUK Report. Contract No. EP-022R, Task Order: 2-022. 23p. 77. Springman, S. M., Laue, J., Boyle, R., White, J., and Zweidler, A. (2001). The ETH Zurich Geotechnical Drum Centrifuge. International Journal of Physical Modelling in Geotechnics, Vol. 1 (1), pp. 59-70.
In-situ testing, instrumentation, data acquisition
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78. S.M. Springman, P. Nater, R. Chikatamarla, J. Laue. 2002. Use of flexible tactile pressure sensors in geotechnical centrifuges. Proc. ICPMG. St. Johns. Canada. 79. Stewart, D.P. and Randolph, M.F. 1991. A new site investigation tool for the centrifuge. Centrifuge '91, H.Y. Ko and F.G. McLean (eds). Balkema. pp. 531-537. 80. Stone, K.J.L. 1988. Modelling the rupture development in soils. Cambridge University PhD thesis. 81. Sun, H.W. 1989. Ground deformation mechanisms for soil-structure interaction. Cambridge University PhD thesis. 82. Tang, Z. 1993. Laboratory measurement of shear strength and related acoustic properties. 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 new approach 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 Gravity Simulation for Research in Rock Mechanics. Colorado School of Mines. pp. 60-68. 86. Vinson, T.S. 1983. Centrifugal modelling to determine ice/structure/geologic foundation Interactive forces and failure mechanisms. Proc. 7th Int. Conf. on Port and Ocean Engineering under Arctic Conditions. pp. 845-854. 87. Waggett, P.R. 1989. The effect of lubricants on the interaction between soils and perspex. Cambridge University Part II Project Report. 88. Wilkinson, B.J. 1993. An investigation into the effects of adding a substantial granular content 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 deformation measuring system for geotechnical testing based on digital imaging, close-range photogrammetry, 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., and Tsukamoto, Y. 1993. Jack-up rig foundation modelling II. Offshore Technology Conference 7303. 91. Wroth, C.P. 1972. General theories of earth pressures and deformations. Proc. 5th ECSMFE, Madrid, II, pp. 33-52. 92. Wroth, C.P. 1975. In-situ measurements of initial stresses and deformation characteristics. 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.
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Modelling in Geotechnics
Numerical Modelling Finite Difference Method
Prof. Sarah Springman
ETH Zürich Institute of Geotechnical Engineering
8 8.1
Modelling in Geotechnics
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua) Basics
Finite differences may be used to solve partial differential equations (e.g. steady state flow or Newton's Laws etc.). Governing equations are substituted by finite differences in space, written in terms of field variables at discrete points i.e. the NODES - these are the key geometrical feature. Implicit solution:
solution at every node is dependent on solutions at four neighbour 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 linear problems (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.
Forces nodes = f ( displacements ) nodes
The contour integral formulation of finite differences is used. This formulation overcomes difficulties often associated with mesh pattern and imposition of boundary conditions. The mean value of the gradient of a field variable in a zone may be expressed using the Gauss theorem with the contour integral performed on the boundary of the same zone.
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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x 103
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 between mass and time step (see Fig. 8.1). For example, strain energy in the system may be converted into kinetic energy and may be allowed 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 reduced integration but the finite difference method is 'physically' more justifiable. The dynamic relaxation method (for static analysis) is implemented to solve the algebraic system 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 on advice in the Software User Manual for FLAC. For a static solution, mass and damping are fictitious: choose these in order to achieve stability and accuracy of the solution. Generally damping should be less than the critical damping 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 a PC 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-linear problems.
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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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 is reduced (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 applied external 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 DIFFERENTIAL EQUATIONS which govern behaviour is different. Further : Whereas finite elements describe behaviour of the continuum in terms of ELEMENTS and a finite difference grid may appear also to be an assemblage of elements, in fact it is the NODES which dictate behaviour.
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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Finite Difference NODES and connecting GRID y
∆x
Solution Domain D(x,y)
jmax
j+1 Node
j
∆y
j-1
2 1 1
2
i-1
i
i+1
imax
x
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. 2 ∂f ij ∂ f ij In the PDE’s, typically this might vary as a function of x: e.g. ------- or ---------2∂x ∂x 2 ∂f ij ∂ f ij of y: e.g. ------- or ---------2∂y ∂y
Typical PDE’s might include the LAPLACE equation: ∇2f = 0
(f = function describing variation in terms of x,y space..)
∂ 2 f∂ 2 f- --------------= 0 + ∂x 2 ∂y 2
(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) ∂2f ∂2 f ∂f = constant ⎛⎝ --------- + ---------⎞⎠ ∂t ∂x 2 ∂y 2 Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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Flow lines
Equipotentials
e.g Q=ki Figure 8.3: Flow net
f = a0 + a1x +a2x2+a3x3+..........+anxn
Typically one might assume:
= b0 + b1y +b2y2+b3y3+..........+bnyn e.g Notation fij = value for function f at node x = i, y = j partial derivative: first order ∂f i, j ∂x
=
partial derivative in x direction for constant “j” can be written as
2
∂ f i, j ∂x2
∂f i ∂x
or fx (i,j)
2
=
2nd order partial derivative in x direction for constant “j” as
∂ x2
or fxx|(i,j)
n
3
∂ f i, j ------------ = 3 ∂x
∂ fi
∂ f i, j
fxxx|i,j
∂ xn
= fnx|(i,j)
similarly for changes in y direction ∂f i, j ∂f j = = fy ∂y ∂y
2
∂ f i, j
and
( i, j )
∂ y2
2
=
∂ fj ∂ y2
= f yy
( i, j ) etc.
So what does ”Finite Difference” mean.... using a finite distance such as ∆x or ∆y, the variation is written as follows fi + 1 – fi f i – fi – 1 fi + 1 – fi – 1 ∂f----= ------------------- or ------------------ or -------------------------∆x ∆x 2∆x ∂x fi
∆x ∆x i-1
i
i+1
x
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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f i + 1 – fi forward: -----------------∆x fi – fi – 1 backward: -----------------∆x fi + 1 – fi – 1 central: -------------------------2∆x Similar equations can be written in terms of changes of “f” in the y direction PDE’s e.g.
(LAPLACE) 2
2
∂ f ∂ f --------- + --------- = 0 2 2 ∂x ∂y f xx
+ f yy
( i, j )
( i, j ) =
0
exact solution f ij i.e. “correct” solution => very difficult to achieve
approximate solution f ij e.g. using FINITE Difference Difference = Error: e.g. f – f
function may be derived from a “Taylor” series
of the exact indivisual partial derivatives
e.g. f = exact solution
f i + 1, j = f i, j + f x
( i, j ) ∆x
1 + --- f xx 2
( i, j ) ∆x
2
1 + --- f xxx 6
( i, j ) ∆x
f i – 1, j = f i, j – f x
( i, j ) ∆x
1 + --- f xx 2
( i, j ) ∆x
2
1 – --- f xxx 6
( i, j ) ∆x
3
1 + …… + ----- f nx n!
3 ……
( i, j ) ∆x
n
1 ± ----- f nx ( i, j ) ∆x n n!
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 1 1 f i + 1, j = f i, j + f x ( i, j ) ∆x + ----- f xx ( i, j ) ∆x 2 + ----- f xxx ( i, j ) ∆x 3 + …… + ----- f nx ( i, j ) ∆x n 2! 3! n! f i – 1, j = f i, j – f x
( i, j ) ∆x
1 + ----- f xx 2!
( i, j ) ∆x
2
1 – ----- f xxx 3!
( i, j ) ∆x
3
1 + …… ± ----- f nx ( i, j ) ∆x n n!
(1) (2)
If we add the above equations (1) and (2) we get Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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2 f i + 1, j + f i – 1, j = 2f i, j + ----- f xx 2!
f xx
( i, j )
( i, j ) ∆x
2
2 + ----- f xxxx 4!
f i + 1, j – 2f i, j + f i – 1, j 1 = -----------------------------------------------– ------ f xxxx 2 12 ∆x
( i, j ) ∆x
4
+ ……
( i, j ) ∆x
2
± ……
( i, j ) ∆y
2
± ……
similarly in the y- direction f yy
( i, j )
f i, j + 1 – 2f i, j + f i, j – 1 1 = -----------------------------------------------– ------ f yyyy 2 12 ∆y
∂ 2f ∂ 2 f- --------------- = f xx + ∂x 2 ∂y 2
( i, j )
+ f yy
( i, j )
= 0 exact
∂2 f ∂ 2f --------- + --------- = f xx ∂x 2 ∂y 2
( i, j )
+ f yy
( i, j )
= 0 approximate (e.g ignore higher fxxxx terms etc.)
so assume an approximate solution where: f i + 1, j – 2f i, j + f i – 1, j f i, j + 1 – 2f i, j + f i, j – 1 ∂2 f ∂ 2f --------- + --------- = -----------------------------------------------+ -----------------------------------------------2 2 ∂x 2 ∂y 2 ∆x ∆y when ∆x = ∆y f i + 1, j + f i – 1, j + f i, j + 1 + f i, j – 1 – 4f i, j ∂ 2 f∂ 2 f- ------------------------------------------------------------------------------------------------------ = 0 = + 2 ∂x 2 ∂y 2 ∆x The accuracy of the solution is affected by the size of the ERROR = f – f 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 introduced here)
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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Example: flat plate heated at one boundary Temperature varies as function of f steady:
100°C at top boundary 0°C at the other boundaries
at each node: f i + 1, j + f i – 1, j + f i, j + 1 + f i, j – 1 – 4f i, j = 0 select a 3 x 4 grid in the first instance:
y f = 100
100
100
no flow in z-direction!
4
f2,3 f=
0
3
0
∆x = ∆y = 5 cm
0
2
0
Temperature: f = 0,100 °C
f2,2
Nodes 0
1
1,2... 1
2
3
0
0
0
x
Figure 8.4: Grid for flat plate, heated at top boundary
At node 2,2 4 ⋅ f 2, 2 – f 2, 3 = 0
(because f 2, 1 = f 1, 2 = f 3, 1 = 0 )
At node 2,3 4 ⋅ f 2, 3 – f 2, 2 – 100 = 0
(because f 1, 3 = f 3, 3 = 0 )
or 4 – 1 f 2, 2 = 0 – 1 4 f 2, 3 100
2 unknown nodes : 2 x 2 matrix
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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Accuracy of solution y (cm) 100 °C 80
15
60 40 10
20 10
5
5 2
0
0
5
10
x (cm)
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 determined from a 5 x 7 grid where ∆f = f – f Node
fexact
∆f
f
5 x 7 grid
y
cm
cm
°C
5
10
26.049
26.667
0.618
5
5
5.261
6.667
1.406
°C
°C
12 Nodes
∆f
f
3 x 4 grid
x
°C 26.228
°C 0.179
5.731 0.47 35 Nodes
Tab. 8.1: Exact and calculated temperatures
• 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
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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for a 5 x 7 grid: y
100
15 x 15 =
Diagonal
0
0
3x 5 unknown values at nodes
Matrix 0
fi,j
x
Boundary temperatures f = 100 or 0°C
So what happens if we use more nodes? ACCURACY
SOLUTION TIME ALSO
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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8.3
Modelling in Geotechnics
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
Results unsatisfactory
Examine the model response
Model makes sense PERFORM ALTERATIONS for example - Excavate material - Change boundary conditions
Step to solution
More tests needed
Examine the model response
Acceptable result
Yes
Parameter study needed
No End
Figure 8.6: FLAC general solution procedure (FLAC manual)
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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FLAC uses nomenclature that is consistent, in general, with that used in conventional finite difference or finite-element programs for stress analysis. The basic definitions of terms are reviewed here for clarification. Figure 8.7 is provided to illustrate the FLAC terminology.
Figure 8.7: Example of a FLAC model (FLAC Manual)
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 that define the problem conditions for numerical solution. ZONE —The finite difference zone is the smallest geometric domain within which the change 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, FLAC divides each zone into four triangular “subzones,” but the user is not normally aware of these. 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 pair of x- and y-coordinates are defined for each gridpoint, thus specifying the exact location of the finite difference zones. Other terms for gridpoint are nodal point and node. Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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FINITE DIFFERENCE GRID — The finite difference grid is an assemblage of one or more finite difference zones across the physical region which is being analysed. Another term for grid 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 or controlled condition along a model boundary (e.g., a fixed displacement or force for mechanical problems, an impermeable boundary for groundwater flow problems, adiabatic boundary for heat transfer problems, etc.). INITIAL CONDITIONS — This is the state of all variables in the model (e.g., stresses or pore pressures) prior to any loading change or disturbance (e.g., excavation). CONSTITUTIVE MODEL — The constitutive (or material) model represents the deformation and strength behaviour prescribed to the zones in a FLAC model. Several constitutive models are available in FLAC to assimilate different types of behaviour commonly associated with geologic materials. Constitutive models and material properties can be assigned individually to every zone in a FLAC model. SUB-GRID — The finite difference grid can be divided into sub-grids. Sub-grids can be used to create regions of different shapes in the model (e.g., the dam sub-grid on the foundation sub-grid in Figure 8.7). Sub-grids cannot share the same gridpoints with other sub-grids; they must be separated by null zones. NULL ZONE — Null zones are zones that represent voids (i.e., no material present) within the finite difference grid. All newly created zones are null by default. ATTACHED GRIDPOINTS — Attached gridpoints are pairs of gridpoints that belong to separate sub-grids that are joined together. The dam is joined to the foundation along attached gridpoints in Figure 8.7. Attached gridpoints do not have to match between subgrids, 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 contact plane. It can also be used to join sub-grid regions that have different zone sizes. MARKED GRIDPOINTS —Marked gridpoints are specially designated gridpoints that delimit a region for the purpose of applying an initial condition, assigning material models and properties, and printing selected variables. The marking of gridpoints has no effect on the solution process. REGION — A region in a FLAC model refers to all zones enclosed within a contiguous string of “marked” gridpoints. Regions are used to limit the range of certain FLAC commands, such as the MODEL command that assigns material models to designated regions. GROUP — A group in a FLAC model refers to a collection of zones identified by a unique name. Groups are used to limit the range of certain FLAC commands, such as the MODEL command that assigns material models to designated groups. Any command reference to a group name indicates that the command is to be executed on that group of zones. STRUCTURAL ELEMENT — Structural elements are linear elements used to represent the 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 structural elements. Geometric non-linearity occurs in large-strain mode. Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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STEP — Because FLAC is an explicit code, the solution to a problem requires a number of computational steps. During computational stepping, the information associated with the phenomenon under investigation is propagated across the zones in the finite difference grid. A certain number of steps is required to arrive at an equilibrium (or steady-flow) state for a static solution. Typical problems are solved within 2000 to 4000 steps, although large complex problems can require tens of thousands of steps to reach a steady state. When using the dynamic analysis option, STEP refers to the actual timestep for the dynamic problem. Other terms for step are timestep and cycle. STATIC SOLUTION — A static or quasi-static solution is reached in FLAC when the rate of change of kinetic energy in a model approaches a negligible value. This is accomplished by damping the equations of motion. At the static solution stage, the model will be either at a state of force equilibrium or at a state of steady-flow of material if a portion (or all) of the model is unstable (i.e., fails) under the applied loading conditions. This is the default calculation in FLAC. Static mechanical solutions can be coupled to transient groundwater flow or heat transfer solutions. (As an option, fully dynamic analysis can also be performed by inhibiting the static solution damping.) UNBALANCED FORCE — The unbalanced force indicates when a mechanical equilibrium state (or the onset of plastic flow) is reached for a static analysis. A model is in exact equilibrium if the net nodal force vector at each gridpoint is zero. The maximum nodal force vector is monitored in FLAC and printed to the screen when the STEP or SOLVE command is invoked. The maximum nodal force vector is also called the unbalanced or out-of-balance force. The maximum unbalanced force will never exactly reach zero for a numerical analysis. The model is considered to be in equilibrium when the maximum unbalanced force is small compared to the total applied forces in the problem. If the unbalanced force approaches a constant non-zero value, this probably indicates that failure and plastic flow are 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 directly affect the solution. Dynamic solutions are required for problems involving high frequency and short duration loads — e.g., seismic or explosive loading. The dynamic calculation is an 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 at each step, according to computed displacements. In large-strain mode, geometric nonlinearity is possible.
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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Figure 8.8: Options within FLAC (FLAC manual)
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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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. This model is valid for homogeneous, isotropic, continuous materials that exhibit linear stressstrain behaviour with no hysteresis on unloading.
Finite Difference Analysis using FLAC (Fast Lagrangian Analysis of Continua)
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Modelling in Geotechnics
3. elastic, transversely isotropic model The elastic, transversely isotropic model gives the ability to simulate layered elastic media in which there are distinctly different elastic moduli in directions normal and parallel to the layers. 8.3.3 Plastic model group 4. Drucker-Prager model The Drucker-Prager plasticity model may be useful to model soft clays with low friction angles; however, this model is not generally recommended for application to geologic materials. It is included here mainly to permit comparison with other numerical program results. 5. Mohr-Coulomb model The Mohr-Coulomb model is the conventional model used to represent shear failure in soils and rocks. Vermeer and deBorst (1984), for example, report laboratory test results for sand and 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 of specific orientation embedded in a Mohr-Coulomb solid. 7. strain-hardening/softening model The strain-hardening/softening model allows representation of non-linear material softening and hardening behaviour based on prescribed variations of the Mohr-Coulomb model properties (cohesion, friction, dilation, tensile strength) as functions of the deviatoric plastic strain. 8. bilinear strain-hardening/softening ubiquitous-joint model The strain-hardening/softening ubiquitous-joint model allows representation of material softening and hardening behaviour for the matrix and the weak plane based on prescribed variations of the ubiquitous-joint model properties (cohesion, friction, dilation, tensile strength) as functions of deviatoric and tensile plastic strain. The variation of material strength properties with mean stress can also be taken into account by using the bilinear option. 9. double-yield model The double-yield model is intended to represent materials in which there may be significant irreversible compaction in addition to shear yielding, such as hydraulically-placed backfill or lightly-cemented granular material. 10. modified Cam-clay model The modified Cam-clay model may be used to represent materials when the influence of volume change on bulk property and resistance to shear need to be taken into consideration, such as soft clay. There are also six time-dependent (creep) material models available in the creep model option for FLAC.
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8.4
Modelling in Geotechnics
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 1.5m q
3.0m
Point1
20m
Point 2
y x
30m
FEM-169 elements FEM- 676 elements FLAC grid 40 x 20
applied load q 10-1 (kPa)
applied load q 10-1 (kPa)
Figure 8.9: Geometry and loading for a strip footing on clay (Schweiger & Freiseder, 1995)
FLAC grid 120 x 80
settlement (cm)
FEM-169 elements FEM- 676 elements FLAC grid 40 x 20 FLAC grid 120 x 80
y- displacement (cm)
Figure 8.10: Results (Schweiger & Freiseder 1995)
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ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
2. Subsidence under a geotextile reinforced embankment: Lawson, Jones, Kempton 1995 • geotextiles used to prevent collapse of road due to unexpected subsidence (solution of limestone / 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 (BS8006generally 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)
BS 8006
Giroud
FDM
Embankment height H (m)
Comparison of the three models in determining the reinforcement tension for a void span of d = 4 m (γ = 20 kN/m2, φ’ = 35°)
Vertical deflection at base of fill D (m)
Reinforcement tension Trs (kN/m)
1% strain (BS8006) 5% strain
1% strain (Giroud) 1% strain (FDM) 5% strain (BS8006) 5% strain (Giroud) 5% strain (FDM)
Width of void d (m)
Comparison of the three models in determining the maximum vertical deflection at the base of the fill spanning the void
Figure 8.12: Results, Lawson et al., (1994)
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ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
• predicted maximum vertical displacement at base of embankment agrees well with BS8006 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 and bearings, especially where damage from e.g. de-icing salts will cause expensive maintenance. The deck-wall becomes a full moment connection and any changes in deck length due to temperature effects (daily and seasonally) will load the fill behind the abutment. This means that an element of soil from just behind the retaining wall will be subjected to cyclic lateral loading and the results is a compaction and loss of volume. Problem: cyclic behaviour of soil - how to model this? Centrifuge behaviour (and prototype)
Settlement profile after CWWN1 test
dense (Id = 83%)
Settlement profile after CWWN2 test
loose (Id = 23%)
Figure 8.13: Geometry: All dimensions shown in mm (not to scale) (Springman et al., 1995)
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Modelling in Geotechnics
Numerical analysis + small strain stiffness (BRICK) model Deformed mesh of spread base wall in response to aa 0.115° 0.115° applied rotation 66mm 6mm mm 6mm Active displacement cycle 10
y(m) y(m)
x(m) x(m) Active displacement of cycle 10 Active displacement at the top of the wall after 10th cycle Figure 8.14: Deformed mesh (Springman et al., 1995)
Real data from cyclic triaxial tests (Jeyatharan, 1991) • shows significant volumetric strain occurring during cycles of axial strain along a stress path => q/p’ ~ 3 (triaxial compression) => between axial strain of 0.3% to -0.4%
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Axial Strain %
Volumetric strain (%)
Volumetric strain (%)
Effective stress p’ (kPa)
Deviatoric stress, q (kPa)
Modelling in Geotechnics
Deviatoric stress, q (kPa)
ETH Zürich Institute of Geotechnical Engineering
stress ratio, q/p’
Axial strain %
Figure 8.15: Cyclic behaviour of 100/170 (Fraction E) sand (after Jeyatharan 1991)
University of British Columbia (UBC) model after Beaty & Byrne (1999) compared to cyclic triaxial 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)
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ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
Figure 8.16: Cyclic behaviour: UBC model (Springman et al., 1995)
• results also look quite good when plotted in normal characteristics (particularly for loose sand samples) in terms of : => strength => volumetric • although the dilation law displayed by the algorithm (Figure 8.17b) does not permit the soil to reach a critical state where no further dilation occurs.
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Modelling in Geotechnics
Volumetric characteristics loose sand
Volumetric characteristics medium dens sand Model prediction: continuous line Experimental results: markers
1000 kPa Model prediction: continuous line Experimental results: markers
50kPa 100kPa
50 kPa
200kPa
1000kPa
b)
a)
Strength Characteristics
1000kPa Model prediction: continuous line experimental results: markers 50kPa
c) Figure 8.17: Comparison between monotonic laboratory test data and the UBC model (Springman et al.,1995)
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ETH Zürich Institute of Geotechnical Engineering
Modelling in Geotechnics
4. Deformation of a 3 m high reinforced wall with 2 m long reinforcement (Jommi et al., NUMOG 1995) End of construction
Spacing: d x
a
b
H
y
Deformed mesh
Displacement vectors
Figure 8.18: Reinforced soil wall :deformed mesh (Jommi et al., 1995)
y/H
xd/H [%] 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)
See also Springman, Balachandran, Jommi (1997) for an application relating to modelling reinforced soil wall in the centrifuge.
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8.5
Modelling in Geotechnics
References
1.
Beaty, M.H. and Byrne, P.M. 1999. A synthesized approach for modelling liquefaction and 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 layer geo-synthetic systems overlying voids, Geotextiles and Geomembranes, Vol. 9, pp. 1150.
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 under dynamic ice-loading, CUED PhD Thesis.
6.
Jommi, C., Nova, R. and Gomis, F. 1995. Numerical analysis of reinforced earth walls via a homogenization method. Proceedings of the fifth international symposium on numerical 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. Advanced analysis of reinforced fills over areas prone to subsidence. Fifth International Conference on Geotextiles, Geomembranes and related products, Singapore 1994, pp. 311-316.
8.
Schweiger, H.F. and Freiseder, M. 1995. Some results from benchmark tests for geotechnical engineering. Proceedings of the fifth international symposium on numerical models in geomechanics-NUMOG V, pp. 675-680.
9.
Springman, S.M. 1989. Lateral loading on piles due to simulated embankment construction, CUED PhD Thesis.
10. Springman, S.M., Norrish A.R.M. and Ng C.W.W. 1995. Cyclic loading of sand behind integral bridge abutments. TRL Report Cambridge University. 11. Springman, S.M., Balachandran S. and Jommi C. 1997. Modelling pre-failure deformation behaviour of reinforced soil walls. Geotechnique, Symposium in print, Vol XLVII, No.3, pp. 653-663.
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