the use of lightweight concrete piles for deep foundation on soft soils
TRANSCRIPT
“ The use of lightweight concrete piles for deep foundation on soft soils “
Agus Sulaeman
A Thesis submitted in fulfillment of the requirement for the award of Degree Doctor of Philosophy
Faculty of Civil and Environmental Engineering University Tun Hussein Onn Malaysia
October, 2010
vi
ABSTRACT
Small scale physical modelling have been dominated by enhanced gravity modelling with
centrifuge equipments. However due to its high cost and capital intensive, attempts were
made to find another method with normal gravity simulation. This study was focused on
clay soil dealing with finding a suitable representation to soil model. In line with the
problem currently faced to reduce the cost of pile foundation in soft soil, the test case of
pile loading tests (PLT) was chosen for investigation in normal gravity scaled modelling,
full scale testing as well as in numerical modelling. Normal Concrete Pile (NCP), Palm
Oil Concrete Pile (POCP) and Foamed Concrete Pile (FCP) were observed in Pile
Loading Test (PLT) to study the feasibility of using light weight concrete piles (LCP) for
deep foundations on soft soil. The PLT of various pile weights in normal gravity small
scale model was previously conducted to represent the behavior of NCP, POCP and
FCP. The results show that the FCP (26 kN) has about 30 % higher capacity than NCP
(20 kN). This is due to the lower unit weight and stiffness of the pile. The results of
ultimate capacity of each pile type were also in good agreement to the pile model,
indicating that the attempt to set up normal gravity small scale modeling was satisfactory.
To obtain the stresses along the pile, the piles in full scale prototype were also tested
under dynamic loading. The compression stresses and tensile stresses measured from
PDA test were under tolerable limit of their strength (the compression stress and
compression strength of FCP were 10.4 MPa and 15 MPa and for POCP were 4.8 MPa
and 25 MPa respectively. Whereas the tensile stress and tensile strength of FCP were 1.2
MPa and 1.2 MPa and for POCP were 10.4 MPa and 15 MPa respectively). As with
normal piles these piles also experienced severe stresses, without any crack or damage
during transportation, handling and driving. This leads to the conclusion that the use of
LCPs as pile foundation of particular structures in soft soils is feasible.
Keywords : Modelling, light-weight concrete piles, static pile loading test, Pile Dynamic Analyzer (PDA)
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ABSTRAK Pemodelan fizikal berskala kecil telah didominasi oleh pemodelan gravity yang dipertingkatkan
dalam peralatan emparan, namun kerana kos yang tinggi dan modal besar, usaha untuk mencari
kaedah lain dalam pemodelan graviti normal telah dijalankan. Kajian ini difokuskan pada tanah
liat dengan mencari model tanah yang sesuai. Seiring dengan masalah yang sedang dihadapi
untuk mengurangkan kos dalam pembinaan asas cerucuk pada tanah lembut, kes ujian
pembebanan cerucuk (PLT) yang dipilih, diteliti dalam pemodelan skala kecil dengan graviti
normal dan ujian secara skala penuh serta dalam model berangka. Cerucuk konkrit biasa (NCP),
cerucuk konkrit klinker minyak sawit (POCP) dan cerucuk konkrit berbuih (FCP) telah dibuat
untuk diteliti dalam ujian pembebanan cerucuk (PLT) untuk kajian kelayakan daripada
penggunaan cerucuk cerucuk konkrit ringan (LCP) dalam asas dalam di tanah lembut. Ujian
Pembebanan Cerucuk (PLT) terhadap pelbagai berat cerucuk dalam graviti normal yang berskala
kecil terlebih dahulu dilakukan untuk mewakili perilaku daripada cerucuk cerucuk NCP, POCP
dan FCP, hasil keputusan keupayaan cerucuk konkrit berbuih FCP (26 kN) menunjukkan 30 %
lebih tinggi daripada cerucuk konkrit biasa NCP (20 kN). Perkara ini berlaku kerana berat unit
yang lebih ringan dan kekakuan yang lebih kecil.. Hasil keputusan keupayaan muktamad bagi
pelbagai jenis cerucuk juga bersesuaian dalam semua pemodelan ini, perkara ini menunjukkan
bahawa usaha untuk membina suatu pemodelan berskala kecil dengan graviti normal adalah
memuaskan. Untuk mendapat nilai tegasan sepanjang cerucuk, prototaip cerucuk dalam berskala
penuh juga dilakukan pengujian pembebanan dinamik. Hasil keputusan tegasan mampatan dan
tegasan tegangan yang diukur dalam ujian PDA berada dalam paras yang dibenarkan
(tegasan mampatan dan kekuatan mampatan dari FCP adalah 10.4 MPa dan 15 Mpa, dan untuk
POCP adalah 4.8 MPa dan 25 MPa. Sedangkan tegasan tegangan dan kekuatan tegangan dari
FCP adalah 1.2 MPa dan 1.2 MPa dan untuk POCP adalah 10.4 MPa dan 15 MPa). Seperti juga
cerucuk konkrit normal, cerucuk ringan ini menerima beban yang tinggi tanpa mengalami patah
atau rosak selama pemindahan, penanganan dan pemacuan.
Hal ini memberi kesimpulan bahawa penggunaan cerucuk ringan (LCP) bagi asas cerucuk pada
struktur tertentu pada tanah lembut adalah layak. Kata kunci : Pemodelan, cerucuk konkrit ringan, ujian pembebanan cerucuk, ujian PDA
viii
TABLE OF CONTENT
CHAPTER ITEM PAGE
Acknowledgements v
Abstract vi
Abstrak vii
Table of contents viii
List of tables xii
List of figures xiv
List of symbols xvii
List of abbreviations xix
List of appendices xx
I INTRODUCTION
1.1 General 1
1.2 Problem statement 4
1.3 Aim 5
1.4 Objectives 5
1.5 Organization of the thesis 5
II LITERATURE REVIEW 2.1 Introduction 7
2.2 Modelling in Geotechnics 7
2.3 Full Scale Modelling 10
2.4 Small Scale Modelling 10
2.5 Model Theory 20
2.6 Similarity 22
2.7 The behaviour of driven concrete pile on soft clay 26
2.8 Pile Testing 39
ix
2.9 Lightweight Concrete Piles 43
2.10 Aggressive chemical attack on Concrete 49
2.11 Numerical modelling 54
2.12 Previous related work on normal gravity scaled modelling 56
III METHODOLOGY 3.1 General work programme 60
3.2 Site Investigation 60
3.3 Producing Soil Model 63
3.4 Pile Loading test on Small scale basis 63
3.5 Pile loading test on full scale basis 64
3.6 Numerical modelling 66
IV SETTING UP 1-GRAVITY SCALED MODEL DEVICE 4.1 Introduction 67
4.2 The normal gravity scaled modelling device 67
4.3 Instrumentations 71
V RESULTS AND ANALYSIS OF SMALL SCALE MODEL
TESTING 5.1 Introduction 78
5.2 Soil Model Observation 79
5.3 Test case simulation “ Pile loading test in small scale basis
of various pile weight “ 95
5.4 Converted into imagined prototype piles 100
5.5 Validity of one gravity scaled modeling 102
5.6 Summary 104
x
VI FULL SCALE MODEL TEST 6.1 Introduction 105
6.2 Design of lightweight concrete 105
6.3 Construction of lightweight concrete piles : Normal concrete, Palm
oil concrete and Foamed concrete pile. 106
6.4 Embedment of strain gauges 112
6.5 Handling and driving the piles 115
6.6 Preparation of kentledge system 116
6.7 Dynamic loading test 119
VII RESULTS AND ANALYSIS OF FULL SCALE PILE
LOADING TEST 7.1 Introduction 126
7.2 Load transfer distribution 127
7.3 L-S curve 131
7.4 Dynamic test results 134
7.5 Correlation between static and dynamic loading test results 139
7.6 Summary 144
VIII RESULTS AND ANALYSIS OF NUMERICAL MODEL 8.1 Introduction 149
8.2 Soil constitutive model 149
8.3 Selection of pile model 152
8.4 Soil Structure Interaction 152
8.5 Simulation of pile loading test 153
IX ANALYSIS OF THE WHOLE RESULTS 9.1 Introduction 166
9.2 One gravity scaled modeling and standard operation procedure 166
9.3 The feasibility of using Lightweight concrete pile for deep
xi
foundation of certain structures on soft soil 170
9.4 Numerical model 176
X CONCLUSIONS AND RECOMMENDATIONS 177
References 179
Published Paper 185
Appendices 186
Appendix A :
1. Site Investigation Results
2. First Triaxial ( original soil – water )
3. Second Triaxial test
4. Data of Pile displacement of scaled basis on data logger
Appendix B :
1. Data from static loading test
2. Data from dynamic loading test
Appendix C :
Published paper in International Journal
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LIST OF TABLES No No. of
Table
Title Page
1.
2
3
4
5
6
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8
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10
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12
13
14
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16
17
18
Table 2.1
Table 5.1
Table 5.2
Table 5.3
Table 5.4
Table 6.1
Table 6.2
Table 6.3
Table 6.4
Table 7.1
Table 7.2
Table 7.3
Table 7.4
Table 7.5
Table 7.6
Table 7.7
Table 7.8
Table 7.9
Scaling relations of the physical modelling approach
Data analysis based on comparison of stress - strain
Data analysis based on comparison of pore pressure –
strain
Critical state parameters for various soils
Data reading from load cell
Material formulation for density 1000 kg/m3 and FC with
density 1500 kg/m3 and G-15
Material formulation for POC
Material formulation for NC
Tabulation of data for pile input PDA test
VWSG (vibrating wire strain gauge) reading
Total Load transfer distribution to Qp and Qf from strain
gauge measurement
Total Load transfer distribution to Qp and Qf from static
loading test
The results of PDF analysis
Data results from PDA computer
Optimum energy required for each pile
Results of PDA and pile driving formula
Comparison of the results
Data results from PDA software
Characteristics of soft soil tested at RECESS of the
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124
128
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Table 7.10
Table 8.1
Table 8.2
Table 8.3
Table 8.4
Table 8.5
Table 8.6
Table 8.7
Table 8.8
UTHM
Data parameter of soil properties
Data parameter of soil properties for Mohr Coulomb
model
Data parameter of soil properties for Hardening Soil
model
Data parameter of soil properties for Soft soil model
Results of Load-Settlement of NC pile
Results of Load-Settlement of POC pile
Results of Load-Settlement of FC pile
Comparison of L-S curve with HS,SS and MC model
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149
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LIST OF FIGURES
1.
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Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 2.12
Figure 2.13
Figure 2.14
Figure 2.15
Figure 2.16
Figure 2.17
Figure 2.18
Figure 2.19
Figure 3.1
Figure 4.1
Figure 4.2
Figure 4.3
Steady state line
The concept of Centrifuge modelling
Enlarge gravity in centrifuge model
Scaled 1-g box container
Early alpha correlation developed from load test database
Lambda Coefficient as a function of pile length
Correlation of alpha parameter with strength ratio for low
plasticity clays
NGI-99 Pile design method showing influence of soil plasticity
Variation in parameter for different strength ratios
Beta parameter determined by Burland (1993)
Changes in pile stress regime over time
Installation total stress at Bothkennar
Radial stress relaxation as a function of soil sensitivity
Normalized Excess Pore Pressures measured in the pile shaft
Comparison of the measured and predicted shaft shear stress for
Large Pile Displacement Test
An example of typical axial compression load arrangement
Ingredients of LWC
The concept of increased hydraulic gradient
Frustum confining pressure
Flowchart of work programme
The planned illustration of scaled model apparatus
The as built scaled model apparatus
Control panel
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Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15
Figure 5.16
Figure 5.17
Figure 5.18
Figure 5.19
Linear displacement measurement
Displacement transducer
On the use of displacement transducer in pile loading test
simulation
Data logger
Load cell
Failure points for drained and undrained test on NC clay
Plotted failure points as critical state line in v – p’and v – ln p’
Normalized by initial pressure
Critical state line of prototype soil
The calculation to obtain correction of soil model formula
Triaxial CU test on similar clay Recess soil to Reobtaining CSL
Stress paths in q’:p’ for undrained test on soft clay RECESS
Stress paths in v: p’ space for undrained test on soft clay RECESS
Stress paths in (a) q’: p’ space for drained test on soft clay
RECESS samples
Stress paths in v: p’ space for drained test on soft clay RECESS
Failure points for drained and undrained test on RECESS soft clay
specimens of soil slope in (a) q’: p’ space
Failure points for drained and undrained test on RECESS soft clay
specimens of soil slope in v: p’ space
The critical state line in q’:p’ space for UTHM soft soil and Air
Hitam soil slope
The critical state line in v: ln p’ space of UTHM soft soil and Air
Hitam soil slope
Reobtaining CSL from similar prototype soil
Fine sand to form pile model
Foam is one of ingredient to form pile model
Pile model under processing
Testing at box being undertaken
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Figure 5.20
Figure 5.21
Figure 5.22
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 6.10
Figure 6.11
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Figure 8.6
Figure 9.1
L-S curve of NCP after data conversion
L-S curve of PCOP after data conversion
L-S curve of FCP after data conversion
Calculated reinforcement
Palm Oil Clinker concrete was being tested
Fabricating the piles
Piles were ready to be relocated before tested at particular location
Installing the strain gauge prior to driving
VW data recorder
Handling and driving the piles
Preparation of static loading test
Instrumentations of kenteledge system
Measured Force and velocity from PDA test
Preparation of PDA test
Load transfer distribution versus depth of pile
Load settlement curves of FCP,POCP and NCP
Transformed area of steel to concrete for calculating deformation
The typical result of PDA test
Applied energy of first reading vs static resistance
Applied energy vs justified static resistance
Variability of ultimate capacity vs different applied energy
Soil structure interaction simulation
Pile loading test simulation in FE calculation
Calculation processes being done on the software
Typical result of PLT was plotted into L-S curve
Calculation point and friction resistance of pile
Plotted L-S curve of load test result with numerical analysis
The flow chart of performing one gravity small scale modelling
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LIST OF SYMBOLS
No Symbols
Title Unit
1.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
Total stress method of computing unit shaft friction Effective stress method of computing unit shaft friction Friction angle between soil and structure interface Strain Soil internal friction angle Unit weight of soil Efficiency Slope of over consolidation line Slope of normal consolidation line = critical state line Specific volume Pi Stress Shear stress of soil Specific volume Dilation angle of soil Specific volume at critical state with p’=100 kPa Skempton pore pressure parameter Correction to lambda
No unit
No unit
No unit
No unit
degree
kN/m3
%
No unit
No unit
No unit
No unit
kPa
kPa
No unit
degree
No unit
No unit
No unit
xviii
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Gs
J I
M
N
Cc
Cs c e g
n
p
q s t
u
uo
w
LL
PL
PI
Specific Gravity Soil Damping Scaling factor to Stress gradient Slope of CSL in q-p space Stress scaling factor Compressibility Index Swelling index Cohesion Void Ratio Gravity Scaling factor for geometry Normal stress on triaxial test (stress path) Deviator stress on triaxial test (stress path) Pile displacement per hammer blow Time Pore water pressure Initial pore water pressure Natural moisture content Liquid limit Plastic limit Plasticity Index
No unit
No unit
No unit
No unit
No unit
No unit
No unit
kPa
No unit
m/s2
No unit
kPa
kPa
mm
second
kPa
kPa
%
%
%
%
xix
LIST OF ABBREVIATIONS No Symbols
Title
1. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
PLT LCP POCP NCP FCP ICP L/D OCR LDPT CPT SPT PI YSR PDA PL LL CSL
Pile Loading Test Light weight Concrete Pile Palm Oil Concrete Pile Normal Concrete Pile Foamed Concrete Pile Imperial College Pile Slenderness Ratio Over Consolidated ratio Large Diameter Pile Test Cone Penetration Test Standard Penetration Test Plasticity Index Yield Stress Ratio Pile Dynamic Analyzer Plastic Limit Liquid Limit Critical State Line
xx
LIST OF APPENDICES
Appendix A :
4. Site Investigation Results
5. First Triaxial ( original soil – water )
6. Second Triaxial test
7. Displacement data of scaled NCP, POCP and FCP on data
logger
Appendix B :
3. Data from static loading test
4. Data from dynamic loading test
Appendix C : Published Paper in International Journal
1
CHAPTER I : INTRODUCTION
1.1 General
Behavior of geotechnical structures can be analyzed through physical and
numerical modeling. Physical model comprises of small scale / scaled and full scale
basis whereas numerical modelling simulates the real case-problem into numerical
simulation mostly through software. Scaled modelling of 1-gravity simulates the real
case problem in laboratory size without adjustment of homologous stress whereas
enhanced gravity (centrifuge) modelling is almost similar to 1-gravity however, the
homologous stress is obtained by applying an enhanced acceleration field using
shopisticated centrifuge equipment.
Scaled physical modelling has been dominated by this centrifuge technique and the
results were proven to be valid, however, complexity in equipments and instrumentation
as well as the exorbitant cost have triggered researchers to look into
1- gravity (normal gravity) scaled physical modeling (Altae and Fellenius, 1994). One of
the many problems in scaled normal gravity simulation is how to obtain model system
with has similarity between prototype and scaled model. Horvarth and Stolle, 1996
created the frustum confining vessel for testing model piles. Reported by Altae and
Fellenius, 1994 that Zelikson, 1969 ; Yan and Byrne, 1989,1991 had made the increased
stress-gradient method to obtain similarity. Norwegian Geotechnical Institute (1981) has
also made simulations to obtain the similarity by modification of the triaxial device. The
similarity between sand in the prototype and scaled model has been discussed by Altae
and Fellenius, 1994.
2
One of the objectives of this project was to make an effort to obtain the similarity
of normal gravity scaled modeling in the focus of clay soil and “ pile loading test to
various pile weight “ as the real case problem. As a consequence, some questions appear
on the use of scaled modeling in normal gravity :
1. How to design, build and operate this small scale modeling equipment ?
2. How to make a variety of material model simulation (soil and structure) ?
3. How to establish the testing method ?
4. How to ensure that similarity exist in model and prototype ?
5. How to interpret and convert data from this equipment into an imagined full scale
basis ?
These how’s will be explored and solved in the following chapters.
Full scale physical modelling should be carried out to ascertain the real
performance of the geotechnical case and to ensure that scaled physical modelling
adopted was on the right track and in the right direction. The pile loading test was
chosen as the case problem to observe both in full scale and scaled modelling. Parallel to
this case problem, in the full scale modelling the further extended simulations was made
to observe the current case problem experienced in the construction industry. One of the
current problems is that the foundation in soft soil using Normal Concrete Pile (NCP) is
deemed expensive due to the production and equipment cost.
A foundation constructed on soft soil area is relatively high in cost compared to
similar structures located in other stronger ground conditions. Therefore it needs an
effort to find alternative ways to alleviate this high cost. Generally the lighter the pile the
cheaper it is due to the reduced weight and other inherent characteristic (personnal
communication with Gue, 2005). Currently, piles used in soft soil areas are too strong,
where the minimum concrete grade requirements G-45 is too high for soft soil
conditions. This is analogous to killing a mosquito with a big gun when we can use
other appropriate weapons instead. To produce lightweight concrete pile (LCP) is such
an appropriate weapon and is a preferred choice.
3
The use of normal concrete pile (NCP) for deep foundation in hard and soft soil will
still be in demand in the years to come since the feasibility of using the lightweight
concrete pile (LCP) for structures particularly in soft soil is not envisaged to be studied
properly either in the universities or construction industries. The performance of pile can
be studied using static and dynamic loading tests. The static bearing capacity
characteristics of the piles obtained using dynamic tests becomes important for
substructure engineering due to the lower cost compared to that of the ordinary static
loading tests. Since the density, stiffness and strength of LCP are much less compared to
those of NCP, the use of LCP as deep foundation for particular structures in soft soil
might be an alternative to the common use of NCP. A systematic study has been done to
assess the performance of LCP embedded in certain locations of soft soil. An analysis of
the ultimate capacity and driving resistance respectively obtained from the static and
dynamic loading tests is compared reasonably for both LCP and NCP. This promotes the
feasibility of using LCP as alternative to the common use of NCP as the deep foundation
for particular structures in the future.
To date the use of LCP for piling is still rare or nobody has used it because they
underestimate strength characteristics of the concrete. It has been known that with high
porosity and low strength which is inherently available in the light weight concrete, the
LCP to be used for piling can face trouble. Therefore, it is necessary to investigate the
behavior of LCPs and to explore ways to protect the re-bar of pile and increase the
strength of LCP.
Although the analysis of the performance of NCP based on the data obtained using
the static and dynamic loading tests has been published in previous studies (Barends,
1992 ; Goble et al., 1996 ; Bruno and Randolph, 1999), the performance of NCP to be
used for deep foundation of particular structures in soft soil accounting for the ultimate
capacity and driving resistance of the piles is still not fully understood. In this case, a
study to reveal the performance of LCP in certain soil would be beneficial for the
advancement in the field of foundation technology.
4
To do all of these requirements, some preparation of formulation and producing full
scale prototype of lightweight concrete piles was scheduled. Based on the references
from the field of concrete technology there are three different ways to produce
lightweight concrete :
1. Omitting fine aggregate
2. Replacing coarse aggregate with light coarse aggregate material
3. Imparting large void by foam and omitting coarse aggregate
To cater for the needs of providing prototype of full scale pile in various weight, the last
two type and normal concrete to produce 3 different weights of concrete material were
chosen i.e :
1. Normal concrete (NC) as a control
2. Palm oil clinker (POC) as medium lightweight concrete and
3. Foamed concrete (FC) as extremely lightweight concrete
1.2 Problem Statements
There is a lack of publications which explained the method to perform 1 – g scaled
modelling in clay soil. The comprehensive explanation was only made to sandy soil by
Altae and Fellenius, 1994. In line with efficiency requirement, there is a possibility to
produce LCP to be used as deep foundation on soft soil. Therefore the two areas below
are the focus of this study.
1. Design and produce LCP for deep foundation on soft soil as an alternative foundation
to NC pile which is used in current practice in construction industry.
2. The modification of clay soil to represent real soil condition in normal gravity scaled
modelling.
1.3 The Aim
The aim of this project is to produce standard operation procedure of 1-g small
scale physical modeling and to develop, prove and ascertain the feasibility of a new pile
system for deep foundation on soft clay which is lighter than ordinary concrete pile.
5
1.4 Objectives
There are three objectives to be highlighted in this project study :
1. To set up the normal gravity ( 1-g ) small scale physical modeling system including
the standard operation procedure of how to model, operate, measure and convert data
into imagined full scale prototype.
2. To develop, prove and ascertain the feasibility of a light-weight concrete piles to be
used as deep foundation for particular structure on soft soil.
3. To find the appropriate soil model in the simulation of pile loading test
1.5 Organization of the thesis
Brief explanation of this thesis which contains ten chapters is summarized as follow :
Chapter I : The general background and the need to perform this project, problem
statements, the aims and objectives.
Chapter II : The literature review is made of modelling in geotechnics with the focus
on one gravity scaled modelling, The literature review of light weight
concrete piles is made and with an explanation of previous related work.
Chapter III : Methodology to perform the whole works of this project.
Chapter IV : The preparation to establish the one gravity scaled model equipment and
instrumentations
Chapter V : Results of one gravity scaled modelling : soil model characterization pile
loading test in various pile weight and the comparison analysis with
conventional formula to prove the accurateness for validity of one gravity
scaled modelling.
Chapter VI : Design and fabrication of LCP’s and preparation to perform static and
dynamic loading tests to all test piles.
Chapter VII : Results of full scale tests of static and dynamic loading tests along with
the analysis of data results.
6
Chapter VIII : Simulation of static pile loading test in numerical modelling including
analysis of the results.
Chapter IX : Analysis is made of the whole results as an integral part of this project.
Chapter X : Conclusions and recommendations are provided.
7
CHAPTER II : LITERATURE REVIEW 2.1 Introduction
Engineering is fundamentally concerned with modelling, and with finding solutions
to real problems because one cannot simply look around until one find problems that one
think one can solve. One need to be able to see through to the essence of the problem
and identify the key features which need to be modeled, which is to say those features of
which one need to take account and include in the design. One aspect of engineering
judgement is the identification of those features which one believe it safe to ignore.
A model is an appropriate scaled down simplification of reality. The skill in
modelling is to spot the appropriate level of simplification, to recognize those features
which are important and those which are unimportant. Very often one is unaware of the
simplifications that they have made and problems may arise precisely because the
assumptions that have been made are inappropriate in a particular application.
2.2 Modeling in Geotechnics
The physical modelling is divided into full scale and small scale basis, however, the
common robust theory in geotechnics normally is dedicated to full scale basis, therefore
the theory for small scale is still improving and developing.
On large complex projects, or on high risk projects, the practicing geotechnical
engineer may undertake all three approaches of design: empirical, numerical, and
physical modeling, and attempt to balance the results of each to arrive at a consensus of
opinion. In modelling terminology the actual structure is called the prototype, and a
model is built to represent the prototype. The geometry of the model is similar to the
prototype but the model is typically a smaller version of the prototype.
8
Models have been used to investigate the qualitative behavior of soil structures for
many years. However, to obtain quantitative results from a model, it is necessary to have
a set of scaling relations which will enable the behavior of the model to be translated to a
predicted behavior of the prototype. Basically, the testing consists of subjecting the
model to a stress condition which represents the stress condition expected in the
prototype. The deformation and strain response of the model is recorded, and with the
scaling relations this data is used to predict the behavior of the prototype.
The key to the success of the modelling is the scaling relation used to relate the
model and prototype. For soils, obtaining an appropriate set of scaling relations is
complicated by an incompatibility between the stress scaling and the constitutive
behavior of soil. It is the constitutive behavior which governs the stress-strain response
of the soil. Essentially the stresses scale linearly, but the constitutive behavior is non-
linear. In general, a model constructed from the same soil as the prototype in exactly the
same state (density, layering, etc.) will not behave in the same way because of the non-
linear constitutive behavior of the soil. If the model is constructed at the same scale, then
presumably it will behave in a similar manner to the prototype. As the disparity in scale
between the model and prototype increases, the divergence in behavior will increase
also.
A modelling method which avoids the scaling incompatibility described above
requires the use of a centrifuge. The centrifuge method has become generally accepted
as a valid technique and its use is becoming relatively widespread. However, it is capital
intensive and performing tests is difficult, complex, and ultimately costly. There are also
some issues regarding the applicability of the method to more complex problems, and a
continuing concern with the boundary conditions imposed by test containers.
The cheaper physical modelling method which caters for the requirements of
scaling relations is 1-g modelling. This equipment consists of box container furnished
with necessary equipment and instrumentations as shown in the Figure 4.1. This method
utilizes the concept of one gravity similar to that of prototype, hence the stress at
9
Void Ratio
Ln mean stress (kPa)
Isotropic Normal consolidation line
Critical state line
homologous point does not comply directly as centrifuge does. The similarity is
achieved by using the concept of critical state line (CSL) as shown in the Figure 2.1.
(Meymand, 1998)
Fig. 2.1 : Steady state line for sandy soil
Because of the nonlinear stress-strain behavior and the dependence of behavior on
initial level of confining stress, small scale physical modeling under 1-g conditions has
little relevance to the behavior of the full scale prototype. Moreover, for a specific
prototype, small scale modeling in the centrifuge or by means of the increased stress
gradient test is only directly relevant when the geometric scale ratio n is inverse
proportional to the stress gradient, I. This requirement may be difficult or costly to meet,
Therefore, there is a need for a set of scaling rules that would allow the results from low
cost, easy to perform, small scale model test representative of the behavior of a full scale
structure without having to maintain the inverse proportionality between n and I as
mentioned by Altae and Fellenius, 1994.
In brief, use of small scale models requires a scaling relation between stress and
strain that builds on an understanding of how the void ratio (density) of the soil changes
following a change of stress. The fundamental understanding of the effect on change of
soil volume due to a change of shear stress was introduced by Casagrande (1936).
Casagrande coined the term “critical void ratio“ or critical density, which is the void
10
ratio or density of a soil subjected to continuous shear under neither dilatant nor
contractant behavior. The original state of a soil is either contractant (typical of loose
soil), which means that when sheared it will reduce in volume, i.e., its original density is
smaller than the critical, or it is dilatant (typical of dense soil), which means that when
sheared it will increase in volume, i.e. original density is larger than critical. Thus, the
volume change of a soil element subjected to shear is controlled not by the initial void
ratio (a constant) alone, but by the void ratio in relation to the critical void ratio. The
latter is a variable that changes with the change in the level of mean stress.
2.3 Full Scale Modeling
Physical modelling is performed in order to validate theoretical or empirical
hypothesis and to study particular aspects of the behaviour of prototypes. The term ‘
physical modelling’ is usually associated with the performance of physical testing of
complete geotechnical systems. Where there is a distrust of a theory and analysis,
because the assumptions are seen to be too sweeping or the relevant aspects of material
response too complex or the realities of reliable numerical solution too far, physical
modeling can seem an appropriate choice. Physical modelling can use real geotechnical
materials, so the need of theoretical modelling of their behaviour disappears. Physical
modelling of geotechnical system can provide data for validation of analytical modelling
approaches and can thus provide a basis for extrapolation from the physical model to the
geotechnical prototype although, as noted, an instrumented and monitored geotechnical
prototype can itself be a physical model for serving this validation purpose as clearly
explained by Wood, 2002.
2.4 Small Scale Modeling Full scale testing is in a way an example of physical modeling where all features of
the prototype being studied are reproduced at full scale. However, most physical models
will be constructed at much smaller scales than the prototype precisely because it is
11
desired to obtain information about expected patterns of response more rapidly and with
closer control over model details than would be possible with full scale testing.
If the model is not constructed at full scale then one need to have some idea about the
way in which one should extrapolate the observations that one make at model scale to
the prototype scale. If material behaviour is entirely linear and homogeneous for the
loads that one apply in the model and expect in the prototype then it may be a simple
matter to scale up the model observations and the details of the model may not be
particularly important but, as will be shown, this still depends on the details of the
underlying theoretical model which informs our physical modeling. Dimensional
analysis is particularly useful.
However, if the material behavior is nonlinear, or if the geotechnical structure to be
studied contains several materials which interact with each other, then the development
of the underlying theoretical model will become more difficult. It then becomes more
vital to consider and understand the nature of the expected behaviour so that the details
of the model can be correctly established and the rules to be applied for extrapolation of
observations are clear. In short we need to understand the scaling laws.
The great advantage of small scale laboratory modelling is that one has full control
over all the details of the model. One can choose the soils that one test and ensure that
one has supporting data to characterize their mechanical behavior. One can choose the
boundary and loading conditions of the model so that one know exactly how the loads
are being applied, and to what extent drainage condition is permitted or controlled at the
boundaries. The nature of the problem to be modeled theoretically in parallel with the
physical modeling is thus well defined. Small quantities of soil are required; drainage
path are short; and the possibility exist of performing many tests repeating observations
and studying the effect of varying key parameters. The costs of individual tests will be
correspondingly lower than full scale tests (Wood, 2002).
The size of the models is both an advantage and a disadvantage. If a particular
prototype is to be modeled physically then a length scale must be chosen. A typical
12
length scale might be 1:100 so that a 10 m high prototype structure become 100 mm
high in the model. Feature of the fabric of the ground for example, seasonal layering of
silts and clays, having a prototype spacing of the order of a few millimeters would have
to be modeled with spacings of a few tens of microns, or an alternative modeling
decision would have to be made.
Evaluating the strength of the soil and its deformation (or stress-strain) behavior
under imposed loading conditions are typically of primary concern. The non-linear
plastic behavior of soil and its generally anisotropic heterogeneous properties make the
task of prediction very difficult. The addition of fluid (usually water) within the soil
matrix and imposition of dynamic loading (from earthquakes, for example) compound
the problem.
In industry the practice of geotechnical engineering continues to be dominated by
empirical procedures of design, even though over the last 30 years considerable effort
has been made to develop more sophisticated analytical techniques, including numerical
models which incorporate non-linear, elastic-plastic and other constitutive models.
2.4.1 Small Scale Modeling background
The use of scaled models in geotechnical engineering offers the advantage of
simulating complex systems under controlled conditions, and the opportunity to gain
insight into the fundamental mechanisms operating in these systems. In many
circumstances such as in static lateral pile load test, the scale model may afford a more
economical option than the corresponding full-scale test. For other investigations such as
seismic soil-pile interaction, scale model tests allow the possibility of simulating
phenomena that cannot be achieved in the prototype. The practice of conducting
parametric studies with scale models can be used to augment areas where case histories
or prototype tests provide only sparse data. In addition to qualitative interpretation, scale
model test results are often used as calibration benchmarks for analytical methods, or to
make quantitative predictions of the prototype response. For such applications it is
13
necessary to have a set of scaling relations that relate the observed model and predicted
prototype behavior.
The relationship between a scale model and the corresponding prototype behavior
is described by a theory of scale model similitude. Mentioned by Meymand (1998) that
Roscoe (1963) defines three methods of increasing complexity and power for scale
modeling applications. They are dimensional analysis, similitude theory, and the method
of governing equations. Dimensional analysis consists of converting a dimensionally
homogenous equation, containing physical quantities and describing a physical
phenomenon, into an equivalent equation consisting of dimensionless products of
powers of the physical quantities.
Dimensional analysis may be used exclusively to understand the form of the
problem solution without application to scale modeling. Similitude theory identifies the
forces operating in the system and uses dimensional analysis to construct and equate
dimensionless terms for the model and prototype. The scaling relations between model
and prototype are also known as prediction equations. The method of governing
equations involves the transformation of the differential equation describing the process
to non-dimensional form, and the formation of similarity variables that relate model to
prototype. Similarity variables must also be determined for both initial and boundary
conditions operating on the system.
Scale models can be defined as having geometric, kinematics, or dynamic
similarity to the prototype . Geometric similarity defines a model and prototype with
homologous physical dimensions. Kinematics similarity refers to a model and prototype
with homologous particles at homologous points at homologous times. Dynamic
similarity describes a condition where homologous parts of the model and prototype
experience homologous net forces. Scale models meet the requirements of similitude to
the prototype to differing degrees as described by Ozkahriman (2009). Author agrees
with Ozkahriman (2009) as any type of modeling has it’s own weaknesses.
14
A true model fulfills all similitude requirements, an adequate model correctly
scales the primary features of the problem, with secondary influences allowed to deviate;
the prediction equation is not significantly affected. Distorted models refer to those cases
in which deviation from similitude requirements distorts the prediction equation, or
where compensating distortions in other dimensionless products are introduced to
preserve the prediction equation.
2.4.2 Centrifuge Testing
In the centrifuge method the apparent incompatibility in the stress scaling and
constitutive behavior of the model and prototype soil is avoided by making the model to
have the same magnitude of stresses as the prototype. This is achieved by imposing a
centripetal acceleration field across the model. If the model size is n times smaller than
the prototype, then the magnitude of the acceleration field is n times greater than gravity
as shown in the Fig.2.2 and Fig.2.3. If the density of the model and prototype are the
same, this creates identical stresses in the model at geometrically similar points to the
prototype. The presumption is that similar strain behavior will be observed because the
same soil is used in the model and prototype
After the centrifuges' initial use in the 1930's, it was not extensively used in US
again until the 1970's, where it has now gained significant popularity. It has also been
used extensively in Japan and Europe. The centrifuge has been used to study particular
mechanisms in soil structures, to verify numerical models, and in some cases to predict
the actual behavior of prototype geotechnical structures. Over the last 10 years the use of
centrifuge testing has become increasingly accepted as an appropriate modelling
technique for geotechnical problems. Both in research and industry, considerable capital
is being devoted to construct centrifuges and complete centrifuge studies.
15
Centrifuge modeling : beam centrifuge modeling : drum
r
r ng
ng ng
ng= r2 ng= r2
Fig. 2.2 The concept of centrifuge modeling
The scaling relations appropriate for the centrifuge have been defined by numerous
authors using different derivation methods. While the centrifuge apparently maintains
compatibility between the model and prototype stresses and constitutive behavior, there
are several other issues of concern which have been raised by various researchers over
the years (Meymand 1998) and Fatma (2009) . These are summarized in the following
items:
1) Variation in the acceleration field across the model in the centrifuge due to
Coriolis effects and to the variation in the radius of rotation. This effect is
accentuated on centrifuges with relatively small diameters with respect to the
depth of the model being tested.
2) The impossibility of reproducing the exact soil structure of the prototype in the
model and its stress history. Correctly scaling time-dependent effects, and the
fact that three different time scales exist for dynamic, dissipative, and viscous
effects respectively.
3) Grain size effects created by using a dimensionally smaller model without
adjusting the model material to have a smaller grain size also.
4) The possibility that different centrifuge test equipment will give different
results for an otherwise identical test due to, for example, variations in the input
earthquake, boundary conditions, and preparation methods.
A detailed discussion of these potential criticisms is beyond the scope of this review;
however, some of the items have been partly addressed in the available literature and are
briefly described below.
16
Item (1) will always remain an issue with the centrifuge but, the introduction of
larger diameter centrifuges is reducing this potential effect by decreasing the ratio of the
radial distance to depth of the model. The Coriolis effect is of concern in certain
situations such as in-flight deposition of materials and the liquefaction phenomenon.
For Item (2) it is, of course, possible to recreate the stress history of the prototype
in the model by trying to emulate the prototype development or construction by building
the model in a similar fashion while it is in flight . Attempts to do this for simple
prototype histories have been made with varying success
The varying time scaling cited in Item (3) can be an advantage of the centrifuge
technique, or not a critical issue when the physical process being modeled is dominated
by only one time scale. For example, problems of consolidation are controlled by
dissipation, and in this case the centrifuge has a great advantage because the scaling
effect dramatically shortens the dissipation process in the model compared to the
prototype . The varying time scales do become a problem when two physical processes
governed by different time scales are important to the mechanism being studied. For
example, liquefaction in sands involves a dynamic time scale from earthquake
excitations, and also a dissipative time scale due to the generation of pore fluid pressure
above hydrostatic pressures. In an attempt to overcome this, researchers have modified
one of the time scales to conform with another time scale. The dissipative time scale can
be matched to the dynamic time scale by changing the pore fluid viscosity or reducing
the grain size of the model material . However, both these options potentially violate
other similitude requirements between model and prototype.
Item (3) was investigated for sands , and it was concluded that the grain size effect
was negligible for pullout tests on anchors, and negligible for footing models provided
the model size to mean grain size ratio exceeded 30. It appears that there has been little
else research completed to investigate this potential effect (Meymand, 1998).
17
For Item (4), the introduction of the laminar box and stacked ring boxes (Whitman and
Lambe, 1985) attempts to minimize the boundary effects created by test boxes under
simple one-dimensional shearing modes. Placing damping materials between the model
and test box has also been employed in an attempt to attenuate dynamic boundary
effects. In tests where a model structure is placed on or in a model soil (e.g., a footing or
pile), care must be taken to ensure the boundary of the container does not influence the
model structure behavior
In general, the results of testing indicated that repetition of the same test at various
facilities yielded comparable results . However, several problems were identified in
relation to the soil preparation, placement of transducers, saturation, and excitation for
each test facility.
Reality / prototype zp
pv gz
model zm
mv gz mv zng
Fig.2.3 Enlarged gravity in centrifuge modeling
The centrifuge has been used to model a wide array of geotechnical problems over
the last 55 years. The validity of technique has been verified by comparison with
prototype behavior where available, and by completing models of models. However, the
reliability of the modeling technique is less certain for more complex dynamic problems,
when the soil is partially or fully saturated with a pore fluid, or when conflicting time
scales must be considered.
18
2.4.3 Small Scale with Normal gravity modeling
Model tests of geotechnical structures have been performed under one-g
conditions shown in the Figure 2.4 probably for as long as engineers have had the
challenge of dealing with soil as a building material. However, even with the knowledge
of similitude and dimensional theory, few one-g tests were performed with regard to
scaling considerations.
Scaling relations governing equilibrium can be derived regardless of the model
material behavior. The scaling derivation is the same for the 1-g conditions and the
centrifuge. However, the constitutive behavior of the model and prototype material must
also be matched. As discussed earlier, the centrifuge achieves this by developing the
same stress state in the model and as in the prototype, and the presumption is that similar
constitutive behavior will also be observed. In 1-g tests the magnitude of the stresses is
obviously different in the smaller model, and the behavior of the soil under these
conditions will be quite different from the prototype. A possible solution to this dilemma
is to use a different material for the model soil and attempt to match the constitutive
behavior under the different stress conditions. This task is formidable and poses many
problems even when dealing with a linearly elastic material, let alone the complex non-
linear behavior of soil. If the model is required to have a pore fluid, the alternative
material must also accurately represent the behavior of two phases (solid and liquid)
which compounds the problem. These issues and the scaling relations for one-g
conditions have been discussed in depth by a number of authors including Rocha (1957),
Roscoe (1968), and Scott (1989) as reported by Meymand (1998).
Another option to account for the different behavior of model and prototype soils is
to scale the constitutive behavior of the soil to account for the differing stress regimes
between model and prototype. This requires a knowledge of the functional relationship
between the soil strain and stress. Rocha assumed that the stress and strain of the model
was linearly related to the prototype (Wood, 2002) and introduced two constant scaling
factors: one for stress, and one for strain. This was later extended to the more general
19
case. With this approach, the details of the constitutive behavior for a particular soil type
are avoided as they are essentially embodied in a single parameter denoted as the tangent
modulus. The tangent modulus is one of the parameters which is assumed to have a
constant scaling factor from model to prototype. After reducing the equations three
independent scaling factors are defined for length, density, and strain. The method
assumes the soil matrix acts as a continuous medium, and that deformations and strains
are small.
Fig.2.4 Scaled 1 – g box container
The scaling relations derived using the approach above were applied to some triaxial
laboratory test data which demonstrated the applicability at least under small strains.
While the method has been extended to liquefaction type problems , it is also stated the
method is not applicable to (1) phenomena in which soil particles lose contact, (2) where
deformation or strains are too large. Unfortunately, the method has not been applied to
models of models, that is where a test is repeated at progressively smaller scales, which
would help verify its validity. The main concern with the method proposed is that it
requires conjecture about the constitutive behavior (even at low strains) and is certainly
limited if larger strains do occur as mentioned by Ozkahriman (2009).
An alternative approach to deal with the scaling of soil behavior from model to
prototype was suggested by Roscoe (1968). He proposed that if the model soil was
20
placed at a different density or state, then the prototype behavior could be emulated
without requiring a scaling factor to transform model strains to prototype strains. Roscoe
(1968) suggested the appropriate scaling could be achieved considering the critical state
behavior observed in soils. Scott (1989) independently arrived at a similar conclusion
and proposed a systematic method by which the state of the model soil could be
selected. Scott presented data to support the hypothesis and demonstrated the method by
completing a 1-g test on a model which was constructed to represent a centrifuge test as
mentioned by Meymand (1998).
In the previous section four items were described as fundamental concerns or issues
with centrifuge testing. All of these items, with the exception of the first (which was the
variation of the acceleration field across the centrifuge model), are also of concern for a
1-g test. In addition, the 1-g method has the added concern of constitutive scaling.
However, one inherent advantage with 1-g modeling tests is that they are easier to
perform and generally the scaling of size is less than what would be employed in a
centrifuge test. The larger model size usually employed in the one-g environment
reduces the divergence in many of the scaling relations.
2.5 Model Theory
Model testing has been employed in all areas of engineering and two commonly
used techniques which are used to derive appropriate scaling relations are dimensional
analysis and similarity theory. The application of such general techniques to civil
engineering has been performed since the 1920's, and has been used to develop model
theories for given applications. The centrifuge method is an example of a model theory
which is governed by a set of scaling relations.
The centrifuge was first used to perform model tests for geotechnical structures in
the 1930's. Specifically the centrifuge was proposed independently by Bucky in the
United States (Bucky, 1931). The method was used by Bucky primarily for mining
21
applications (Bucky and Fentress, 1934) and then extensively on a wide range of
applications as reported by Meymand (1998).
Meymand explained a more general model theory for soils which could be applied
in one-g conditions was apparently first suggested in the 1950's (Rocha, 1953). Rocha
illustrated some example applications using very simplified assumptions regarding soil
properties. However, he concluded that attempting to model a two phase soil (i.e., a
matrix of solid grains having pore fluid within the matrix) and the non-linear relations
between stresses and strains would make the mathematical analysis insuperably
complicated.
This observation goes to the heart of the problem with models at 1-g and can be
explained as follows. Usually body forces (such as gravity ) are the dominant factors in
geotechnical problems, influencing the strength and stresses through the self weight of
the soil. When a model is constructed at a reduced size, the stresses in the model at
geometrically similar points to the prototype are reduced . The different stresses will
cause different strain behavior in model from prototype because of the complex non-
linear stress-strain (constitutive) behavior of soil. This assumes the same soil is used in
the model and the prototype. Finding the correct mapping between model strains and the
projected prototype strains is extremely difficult and open to many questions.
Meymand reported that Rocha (1957) also discussed the difficulty of reproducing in
a model the heterogeneity and anisotropy that exists in natural and man-made prototype
soils. The subtleties of the soil structure can have a major bearing on its behavior, and
therefore this is a potential source of significant errors. This problem exists for both one-
g models and centrifuge models.
Given these difficulties, Rocha went on to make an interesting assertion: physical
modelling requires simplifications and assumptions; however; the design techniques
(analytic and numeric) used at that time also required simplifications and assumptions.
22
It is worth considering that Rocha's assertion may still have validity in today's design
environment even though numerical, and to a lesser extent, analytic, tools have
improved over the last 40 years in soil mechanics.
The selection to use critical state line concept was based on the considerable of :
1. The strength of clay soil closely related with change of water content at corresponding
stress, so the correlation of water content and stress obviously occurs in the critical
state line concept
2. Geomorphology different from one place to another resulted in different behavior
even with similar soil type. So the formula should be investigated and improved to be
appropriate with local clay soil.
2.6 Similarity
Fellenius and Altae (1994) reported that Roscoe et al. (1958) developed the
Casagrande concept of critical void ratio and critical density into defining a state at
which the soil continues to deform constant stress and constant void ratio, calling this
state the “ critical state “. This concept was based on the results of extensive laboratory
testing of remolded clays. The approach was later found valid also for non cohesive soils
as mentioned by Atkinson and Bransby (1978), Been et al. (1991).
According to Atkinsons and Bransby (1995), the critical state has a location in the
p-q-e space given by the following :
q = M p ………… …………………………………………………………..….…(2.1)
rppe ln …………………………………………………………………..….(2.2)
Where : q is the deviator stress ( σ1-σ3 )
p is mean effective stress (σ1+σ2+σ3 )/3
M is slope of critical state line in the q-p plane
λ is the slope of critical state line in an e-ln(p) plane
23
Γ is the void ratio at the reference mean stress (100 kPa)
As mentioned by Fellenius (1994) that Roscoe and Pooraashasb (1963) applied
critical state principle to test on remolded clays and artificial soils made up of steel balls
and indicated by means of a formula that the void ratio proximity to the critical state line
at the initial mean stress must be the same for model and prototype.
The latest version about scaling relation in 1-g modeling is the use of critical state
line concept developed by Fellenius in 1994. Fellenius stated that for sandy soil
conditions the similarity will occur between P ( prototype ) and M1 ( soil model with
similar λ ) instead of M2 ( soil model with similar void ratio ) which describes the
similar void ratio of the model and prototype will perform different behaviour as shown
in the Fig.2.1 , similar behaviour occurs between soil with void ratio prototype ( ep ) and
model ( em = ep + λ ln N ). This is classified as static similarity. The other similarity
(kinematic and dynamic similarity) can be calculated using other methods suited to the
related conditions. Some of the following scaling factors is collected from derivation
above whereas the rest come out from derivation using equation of motion approach.
Many different scale ratios apply between a model and a prototype as follow:
1. The geometric scale ratio n between model and prototype n = λL = Lm / Lp ……………………………………………………………… (2.3) Where Lm is the length dimension in the model Lp is the length dimension in the prototype 2. The stress scale ratio N
N=p
m
''
……………………………………………………………………………. (2.4)
Where m' = effective stress in the model at homologous point p' = effective stress in the prototype at homologous point
3. p
mI''
……………………………………………………………..…………….(2.5)
Where m' = effective stress gradient in the model p' = effective stress gradient in the prototype
24
In centrifuge testing, I is the ratio between the centripetal acceleration to the normal gravity g. Table 2.1: Scaling relations of the physical modeling approach
Full scale prototype
Model
1. Static similarity Linear dimension Area Stress Strain Displacement Force Void ratio 2. Dynamic similarity 3. Kinematic similarity time
1 1 1 1 1 1 ep
1 1
n n2 N 1 n Nn2 em= ep+λ ln(N) 1 tp (n tp)0.5
Where :
n = geometric scale ratio
N= stress scale ratio
em = void ratio model
ep = void ratio prototype
Other similarities To determine the other scaling factor for similarity requirements, the object in
nature which can represent the pile motion during pile loading test can be simulated by
equation of motion of the object as mentioned by Sedran (2001) :
In full scale (prototype) : Mp pA + Cp pA + Kp Ap = Fp (tp) ………………. (2.6) In model ( reduced scale ) : Mm mA + Cm mA + Km Am = Fm (tm) …….…………. (2.7) In general, for any given similarity analysis the following scaling factors apply to the equation of motion. Mass : λm = Mm / Mp ………………………………..……………….(2.8) Damping : λc = Cm / Cp ………………………………………..………... (2.9)