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Tim Maeckelberghe
Pile Setup - The gain of pile capacity: A numerical study
Academic year 2015-2016Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Peter TrochDepartment of Civil Engineering
Master of Science in Civil EngineeringMaster's dissertation submitted in order to obtain the academic degree of
de Santa Catarina)Counsellor: Prof. Alysson Rodrigo Marques Gomes de Assis (Universidade de Federal
Federal de Santa Catarina)Supervisors: Prof. dr. ir. Hans De Backer, Prof. Patricia Faria (Universidade de
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The setup effect: Numerical approach to determine the soil state after
pile installation
Tim MAECKELBERGHE
Supervisor guest university: dr. ir. P. Faria
ir. A. De Assis
Supervisor home university: H. De Backer
UGent - Universidade Federal de Santa Catarina
Master of science in Civil Engineering
Faculty of Engineering and Architecture
Department of Civil Engineering
Year 2016
Preface
It should be noted that the work got of slowly due to modelling difficulties. The early models
were build in the Plaxis Classic version, which has an extremely slow interface. Confusion
and wrong conclusions were the consequence of inequalities in results obtained with this older
version and other numerical software or versions. Finally the Plaxis 2015 version was applied
to produce the results presented in this study. The fact that the numerical modelling software
runs through to a VPN connection didn't improve the workability of the project either.
Before the start of the work I would also like to use this chapter to thank some people for their
contribution and support.
Firstly I would like to thank, from the guest university, ir. Alysson De Assis for the guidance
of the work and the time that he has put into the work. For the support and the questions.
Following I would like to thank dr. ir. Patricia Faria for the supervision of the work.
From the home university I would like to thank prof. dr. Hans De Backer, who is willing to
review this work after completion, and Ir. Hamzeh Ahmadi who helped me out with Plaxis
uncertainties and who provided the initial steps to that made working in the Plaxis 2015
version.
Finally I also want to thank mrs. Ann Vanoutryve who made this exchange possible. Without
her I wouldn't have had the opportunity to contribute to this educational, social and self
enriching experience.
Tim Maeckelberghe
Florianópolis, july 2016
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Abstract
The bearing capacity of a pile foundations increases after installation due to the changes in
soil state, this phenomena is referred to as set up. Various empirical formulas are developed to
address the set up effect, however still a lot of uncertainties are presented in the outcomes of
the prediction model. The implementation of a larger safety factor will however counteract
the gain in capacity due to soil setup. This study applies numerical modelling to get more
insight in the behaviour of a cohesive soil after pile installation. An axisymmetric, Mohr
Coulomb model is built up in Plaxis and validated with the results from a study conducted in
Abaqus. The model simulates the stresses and excess pressures after pile installation with the
cavity expansion theory. The consolidation theory allows a dissipation of excess pressures
which lead to the increase in effective stress, this increase in stress contributes to the setup
effect. The gain in setup results in a gain in pile capacity, a method is proposed to simulate the
pile shaft capacity. Experimentally reported results allows a comparison of the outcome.
Finally a modulation methodology attempts to model the total bearing capacity of a case
study.
Keywords: Soil setup, Numerical modelling, Model influence, Pile installation, Bearing
capacity, Cavity expansion.
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Content
Preface ................................................................................................................................... i
Abstract ................................................................................................................................. ii
Content ................................................................................................................................. iii
List of figures ....................................................................................................................... vi
List of tables ....................................................................................................................... viii
Introduction .......................................................................................................................... ix
State of the art ........................................................................................................................1
1 Introduction .................................................................................................................1
2 Pile design ...................................................................................................................1
2.1 Pile design methods ..............................................................................................1
2.2 Criticism ...............................................................................................................2
3 Pile setup .....................................................................................................................2
3.1 Mechanisms ..........................................................................................................2
3.2 Pore pressure dissipation .......................................................................................3
4 Pile load test ................................................................................................................5
5 Estimating pile setup: empirical relationships ...............................................................6
5.1 Skov and Denver's formula ...................................................................................6
5.2 Practical application ..............................................................................................7
Research .................................................................................................................................8
1 Introduction .................................................................................................................8
2 Soil pressure ................................................................................................................8
2.1 Strain Path method ................................................................................................8
2.2 Cavity expansion theory ........................................................................................9
3 Soil pressure analytical analysis ................................................................................. 11
3.1 Analytical analysis .............................................................................................. 11
iv
3.2 Basic model ........................................................................................................ 13
3.2.1 Model characteristics ....................................................................................... 13
3.2.2 Model built up & validation............................................................................. 14
4 Numerical analysis ..................................................................................................... 17
4.1 Introduction ........................................................................................................ 17
4.2 Case study .......................................................................................................... 17
4.3 Model alterations ................................................................................................ 18
4.3.1 Expansion radii ............................................................................................... 18
4.3.2 Model length ................................................................................................... 20
4.3.3 Shear strength .................................................................................................. 21
4.3.4 Boundary conditions........................................................................................ 23
4.3.5 Interface .......................................................................................................... 23
4.3.6 Mesh ............................................................................................................... 24
4.4 Modulation ......................................................................................................... 24
4.5 Initial stress......................................................................................................... 25
4.6 Results ................................................................................................................ 27
4.6.1 Introduction ..................................................................................................... 27
4.6.2 Verification of model length and expansion radius .......................................... 27
4.6.3 Displacement................................................................................................... 28
4.6.4 Plastic failure .................................................................................................. 29
4.6.5 Pore pressures ................................................................................................. 31
4.6.6 Effective soil stresses ...................................................................................... 34
4.6.7 Total soil stress................................................................................................ 37
5 Soil water dissipation ................................................................................................. 38
5.1 Parameters .......................................................................................................... 38
5.2 Results ................................................................................................................ 39
v
5.2.1 Pore pressure ................................................................................................... 39
5.2.2 Effective stress ................................................................................................ 41
6 Pile shaft capacity ...................................................................................................... 42
6.1 Introduction ........................................................................................................ 42
6.2 Modelling approach ............................................................................................ 42
6.3 Results ................................................................................................................ 43
7 Case study ................................................................................................................. 44
7.1 Introduction ........................................................................................................ 44
7.2 Model approach .................................................................................................. 44
7.3 Model input ........................................................................................................ 45
7.4 Model output ...................................................................................................... 46
7.4.1 Model (4.3) ..................................................................................................... 46
7.4.2 Extended model ............................................................................................... 49
8 Comments and future work ........................................................................................ 50
Conclusion ........................................................................................................................... 52
References ............................................................................................................................ 54
vi
List of figures
Figure 1: Different phases during pore pressure dissipation (Komurka et al., 2003). ...............3
Figure 2: Difference pile load tests (Bezuijen A., 2015). .........................................................5
Figure 3: Cavity expansion theory (Brinkgreve et al., 2008) ................................................. 10
Figure 4: Reproduction of radial stress in function of radii, according to (Kok Shien Ng.,
2013). ................................................................................................................................... 12
Figure 5: Basic model lay-out ............................................................................................... 14
Figure 6: Validation of the model, plastic radius ................................................................... 15
Figure 7: Cavity expansion pressure ..................................................................................... 16
Figure 8: Relationship between cavity radius and pile radius (Kok Shien Ng., 2013) ............ 18
Figure 9: Influence of the initial radius (Kok Shien Ng, 2013) .............................................. 20
Figure 10: The influence of the correction layer (Kok Shien Ng., 2013) ................................ 21
Figure 11: Shear strength distribution - model ...................................................................... 22
Figure 12: The model as is applied in Plaxis ......................................................................... 23
Figure 13: Mesh ................................................................................................................... 24
Figure 14: Vertical initial stresses ......................................................................................... 25
Figure 15: Determination of K0 (Hadded et al., 2013) ........................................................... 26
Figure 16: Horizontal initial stresses ..................................................................................... 27
Figure 17: Radial effective stress - according to the rules of the basic model ........................ 28
Figure 18: Soil heap.............................................................................................................. 29
Figure 19: Displacement vectors ........................................................................................... 29
Figure 20: Plastic point ......................................................................................................... 30
Figure 21: Inaccurate points .................................................................................................. 31
Figure 22: Excess pore pressure after pile installation ........................................................... 31
Figure 23: Pore pressure distrubtion H = 3 m ........................................................................ 32
Figure 24: Normalised pore pressure .................................................................................... 33
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Figure 25: Excess pore water pressure - Y - axis: height; X - axis: length ............................. 34
Figure 26: Effective stress increase after pile installation ...................................................... 35
Figure 27: Effective radial stress after pile installation .......................................................... 36
Figure 28: Effective stress after driving (Haddad et al., 2012) - Y - axis: height; X - axis:
length ................................................................................................................................... 36
Figure 29: Total radial stress ................................................................................................. 37
Figure 30: Comparison of the total stress between numerical and analytical.......................... 37
Figure 31: Excess pore water pressure distribution after 33 days (Haddad et al., 2012) ......... 39
Figure 32: Development of excess pore water pressure with time ......................................... 40
Figure 33: Pore pressure distribution after 95% dissipation of excess pore pressure .............. 40
Figure 34: Effective stress after 33 days - left Plaxis; right (Haddad et al., 2012) .................. 41
Figure 35: Increase in effective stress after consolidation, h = 4,6 m. .................................... 41
Figure 36: Increase in pile shaft capacity (Haddad et al., 2012) ............................................. 42
Figure 37: Vertical force displacement curve ........................................................................ 43
Figure 38: Soil strata ............................................................................................................ 44
Figure 39: Trendline of soil elasticity, black - soil test; blue - linear approximation ............. 45
Figure 40: Plastic radius ....................................................................................................... 47
Figure 41: Case study: Increase in effective stress - left, excess pore pressure - right ............ 47
Figure 42: Case study after 1 day: increase in effective stress - left, excess pore pressure -
right ...................................................................................................................................... 48
Figure 43: Case study after 31 days: increase in effective stress - left, excess pore pressure -
right ...................................................................................................................................... 48
Figure 44: Radial effective stress after cavity expansion - new model ................................... 49
Figure 45: Pile failure surface (Bezuijen A., 2014) ............................................................... 50
viii
List of tables
Table 1: Factors dominating pore pressure rate .......................................................................4
Table 2: Input of the case study (Konrad, J. M. & Roy, M., 1987). ...................................... 17
Table 3: Soil model dimensions and cavity radii ................................................................... 19
Table 4: Increase in pile shaft capacity ................................................................................. 43
Table 5: Project data - pile information (De Assis A., 2016) ................................................. 44
Table 6: Cavity expansion parameters................................................................................... 46
ix
Introduction Problem statement
Although applications of setup predictions with good results are available, a lot of scatter is
still present in the results. Large tolerances need to be accepted during prediction.
Accumulation of uncertainties in the choice, application and results of the pile loading tests
till uncertainties in the applied prediction model are the cause of it. To take these uncertainties
into account a factor is implemented to insert a degree of safety in the predicted outcomes.
This will on their turn counteract the estimated gain of capacity due to setup.
(Komurka et al., 2003) suggested that future work concerning setup should perform large
scale tests to collect data from the subsurface on, to deep into the soil layers. The collection of
future data expanded with current data could lead to new correlations in setup predicting
formulas. Data must be collected with similar in-situ test equipment to make a comparison
possible. (Komurka et al., 2003) concluded that the SPT-Torque test resulted in a better field
test method to predict soil setup.
To get around the costly price tag associated with the gathering of new data, a numerical
analysis can be performed with a minimal of required data. A numerical study can lead to
deeper insides in the behaviour of the soil during setup and to the ability to predict the setup
phenomenon.
Objective and methodology
Rather few publications are available concerning the numerical analysis of the setup
phenomenon of pile foundations. (Haddad et al., 2012) created a model in Abaqus to simulate
the cavity expansion developed during pile driving and corresponding generation of excess
pore water pressure. The evaluation of the setup effects are made possible. The model was
validated using the results of a driven pile in cohesive strata. The study made a distinguish
possible between the effective stress-dependent and independent factors influencing setup.
The goal of this study is to investigate the influences of the model on the study outcomes. The
development of a model to predict the soil state after pile driving and after soil setup. This
concerns a proposal of a method to approach the prediction of pile capacity before and after
setup. The modelling software Plaxis is applied to perform the analysis.
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The setup phenomenon is a sequence of two individual processes: the pile installation can be
seen as the first process, while the second process can be seen as the consolidation of the
disturbed soil.
Overview
After this introduction, a selective literature review is presented concerning setup evaluations
and predictions. No attention was paid in this part to the already conducted numerical studies.
The methodology of some of the numerical studies are briefly explained throughout the work.
The research work starts with an appropriate chose of the soil pressure theory. The analytical
analysis according to the theory is reported, which allows an analytical verification.
Continuous the work is treated numerically. It is firstly composed of a simplified expansion
model which is been built and validated as a modelling start. Subsequently the real model
built up is discussed. The initial stresses are generated and the consecutive soil state of the
model after the cavity expansion is examined. The change in soil stress after a consolidation
analyses is reviewed. The outcomes of the model are regularly compared with the outcomes
of the Abaqus model. Afterwards a method to investigate the value of the shaft capacity is
introduced, this makes the comparison with an experimental case study possible.
Finally, a method to approach the bearing capacity is explained. This is tested on a case study,
however it should be noted that, due to a lack of time, this study didn't result in the required
outcomes. Too few parameters were also given to properly conduct the analysis.
At the end of the work a conclusion together with a prevision of future work is implemented.
State of the art
1 Introduction
A pile foundation is a relatively long and slender structural foundation type which transfers
the loads from the superstructure to deeper located soil layers. It is used when soil conditions
are unsuitable for shallow foundations.
A pile is driven into the ground which leads to disturbance and displacement of the soil. After
driving, the remoulded soil will recover from the disturbance, which often leads to an increase
in capacity of the pile. This time dependent increase in capacity is referred to as pile setup.
Pile setup can lead to a significant increase in bearing capacity of the pile. Relaxation of the
soil is the opposite effect, a loss in bearing capacity in time. Relaxation is more seldom to
occur [(Komuraka et al., 2003); (Axelsson G., 2000)]. A precise estimation of the final
bearing capacity after pile setup or relaxation can economize pile production, leading to a
safer design and ultimately to tremendous cost savings for large projects.
2 Pile design
2.1 Pile design methods
In general, pile designs are based on local and empirical practices which calculate the ultimate
bearing capacity and apply a reduction factor allowing a certain level of safety in the design.
The ultimate bearing capacity of a pile is divided in the ultimate base resistance and the limit
shaft resistance. The capacity is obtained by respectively multiplying the base and shaft
resistance by the area of pile base and the area of the shaft layer. The shaft resistance depends
on the successive soil layers.
Two major empirical approaches can be found in the calculation of pile design: the soil
property based and the in-situ test based design. The soil property based design requires an
estimation of the effective stresses in the field which can be found by conducting a laboratory
or several in-situ tests. The in-situ test based design directly relates the in-situ obtained
parameters to the resistance. This is done by SPT or CPT tests (Kim et al., 2009).
2
2.2 Criticism
Current pile design methods have some drawbacks. (Dias T. G. S., 2015) states that the
calculation processes are often simplified to homogeneous soil layers. All the design
approaches depend on local interpretations of a certain model describing a simplified version
of reality. The pile resistance will only be mobilized through a certain amount of
displacements, which is not directly incorporated in the calculation methodologies. The
current pile design method should be applied with care. (Dias T. G. S., 2015) proposed an
alternative framework for pile analysis which relates the relative displacements to the
mobilized forces. These forces can only be mobilized up to their corresponding resistance
which, when summed together leads to the ultimate bearing capacity. After the maximum
resistance is reached, settlements will occur.
A better understanding of the consequences of pile setup leads to a more accurate prediction
of the final bearing capacity. Which on their turn can lead to a more accurate model
applicable for pile design.
3 Pile setup
The installation process of driven piles results into soil displacement and soil disturbance.
After installation the soil will recover which varies the pile capacity. It has been recognized
that the pile capacity of driven piles increases with time in most soils, including clay, loose to
dense silt, sandy silt, silty sand and fine sand (Attar I. H. & Fakharian K., 2013). Results have
shown an increase in capacity with a factor 4 to 5 times the initial pile bearing capacity after a
period of 100 days.
3.1 Mechanisms
Earlier researches have clarified the responsible mechanisms for pile setup. The mechanisms
are generally understood but still poorly quantified. They are summarized as:
1. The dissipation of excess pore water pressure after driving. The displacement of the
soil during driving will lead to an increase of lateral effective stress. Depending on the
permeability coefficients of the soils this increase in stress will result in an increase of
excess pore water pressure.
3
2. A time-dependent change in soil properties at a constant effective stress, referred to as
soil aging. The changes in behaviour is attributable to micro-structural rearrangements
of the sand grains and their contact in time. This is assumed to be a less-dominating
factor of soil setup (Chow et al., 1998).
3. Variations in the arching phenomenon, which is more likely to occur in sand as is
described later.
Setup effects are more defined in cohesive soils. Due to the dissipation of excess pore
pressure being the dominant factor affecting soils setup, results of soil setup are more
pronounced in cohesive soils.
3.2 Pore pressure dissipation
According to (Kormurka et al., 2003) pore pressure dissipates in three stages, Figure 1:
1. Nonlinear dissipation of pore pressure with respect to the log of time for some time
after driving;
2. Linear dissipation of pore pressure with respect to the log of time;
3. After dissipation of excess pore pressure additional setup may occur at constant stress
due to aging.
Figure 1: Different phases during pore pressure dissipation (Komurka et al., 2003).
4
Two general factors dominate both the nonlinear and linear excess pore pressure rate, see
Table 1.
Table 1: Factors dominating pore pressure rate
The duration of the dissipation largely depends on the soil type. In clay or in mixtures of clay,
silt and fine sand, excessive pressure dissipate slowly. Some setup occurs during the nonlinear
logarithmic dissipation, while the most occurs during the linear dissipation. The nonlinear
logarithmic dissipation can continue for several hours, days or weeks in fine grained granular
soils, while the linear dissipation takes several weeks or months. The setup due to aging is
relatively small in these soils.
Pile setup for a pile installed in sand is mainly described to arching mechanisms. Arching, in
the case of axially loaded piles, means that load is not transferred along the shaft only through
simple shear but also by compression near the pile base. (Chow et al., 1998) quote that sand
creep weakens the arching mechanisms surrounding the pile shaft. It increases the horizontal
effective stress while also producing dilatation effects. (Axelsson, 1998) on the other hand
suggested that the arching effect deteriorates within time due to stress relaxation. Another
reason is that soil aging causes reorientation of particles leading to interlock. The setup effect
in non-cohesive soils can therefore briefly be explained by the soil particles interlocking and
stress relaxation which occurs after pile driving. In more coarse grained soils, dissipation
occurs rather fast. The excess pore pressure dissipation of the coarser grained soils are also
different: almost no nonlinear logarithmic dissipation is present, the linear dissipation is
present but only to a smaller extend, aging will therefore mostly determine the behaviour of
pile setup in sandy soils.
The two pile related factors affecting pore pressure dissipation are: pile type and thickness.
Setup has reported for all driven pile types, however studies indicated that displacement piles
show greater setup than small displacement piles. The more the soil is displaced, the more
setup will occur. The thickness determines the setup according to: the larger the pile diameter,
the more time needed for lateral consolidation leading to a decrease in time rate of setup.
(Long et al., 1999). Another factor influencing the pile setup was the number of piles driven
Soil
• Soil type
• Permeability
• Sensitivity
Pile
• Type
• Size
5
around the tested pile. The more piles driven around the tested pile, the higher the setup factor
will be (Attar I. H. & Fakharian K., 2013).
The pile shaft is more exposed to the soil particles than the pile tip, more water can dissipate
around the shaft than the tip. A larger number of soil particles can be rearranged after driving
and influence a larger increase in effective stress after water dissipation. The shaft resistance
is therefore more variable to the setup phenomenon than the tip resistance. When the
resistance of the shaft and pile can be measured individually, the main focus of the setup
examination is located on the shaft. Proposed empirical relationships predicting the gain in
capacity, as presented later, are often only applied to predict the setup of the shaft resistance
(Attar I. H, Fakharian K., 2013).
4 Pile load test
There are 3 types of loading tests: static test, statnamic tests and dynamic loading test. They
differ in the way the load is applied. To test a pile a significant amount of weight is required
to reach the prescribed settlement linked to failure of bearing capacity. This can be referred as
the static test. Other tests differ the acceleration at which the load is applied. Depending on
the loading time a difference can be made between dynamic tests and statnamic test, see
Figure 2. A dynamic test is comparable with a drive of a pile, while the statnamic test relies
on a loading time longer than the time that a wave needs to propagate to the pile. However
some throwbacks accompany these dynamic method: due to the short loading time the load is
not constant during application and shock waves are generated in the pile. A back calculation
method relying on the wave energy propagating during the dynamic hit is applied to calculate
the capacity, this can also lead to erroneous results. Better results are therefore obtained with
the static test. Although these disadvantages, dynamic test are mostly used due to their short
duration of application and their cost effectiveness (Bezuijen A., 2015).
Figure 2: Difference pile load tests (Bezuijen A., 2015).
6
Pile setup is examined by investigating the bearing capacity of a pile at different moments
after driving. This demonstrates the grow in capacity of the pile. Due to the high costs and
waiting time, static tests are seldom been executed. However examining setup with static tests
implies various practical weaknesses. A static test needs a ballast frame, a certain time is
needed to build up this frame, this initial time is crucial for the setup phenomenon. A
difference also needs to be made between a bottom-loaded and a top-loaded static test. A
bottom-loaded test which fails in shaft resistance will lead to the most accurate results of
examining the shaft capacity on different moments after driving (Komurka et al., 2003).
Various research concerning setup is based on a few static test and multiple dynamic tests.
Different results are obtained from these tests conducted at equivalent soil conditions (Attar I.
H. & Fakharian K., 2013). Due to the scarcity of accurate data, research relies on these data to
examine the effect of setup. Different formulas to predict setup are based on regression fitting
of the scattered data.
5 Estimating pile setup: empirical relationships
Several empirical formulas are been proposed by different authors (Huang 1988; Svinkin,
1996; Guang-Yu, 1988; Skov & Denver, 1988; Svinkin & Skov, 2000). Each of them are
based on a thought-through regression fit of personal measurements of bearing capacities. The
bearing capacity at the end of driving is compared with the bearing capacity after a certain
time, this is plot in a graph and the trend line bets fitting the results can predict the setup of
piles in similar soil conditions. A disadvantage of this technique is that several in-situ tests are
needed to examine the soil deposit on a construction site and several load bearing tests are
needed to make an estimation possible of the total capacity of the other piles for a certain soil
deposit. The number of tests makes the prediction of setup costly and only beneficial for large
projects.
5.1 Skov and Denver's formula
A literature review conducted by (Komurka et al., 2003) revealed that the most popular
applied approach is the one proposed by (Skov R. & Denver H., 1988). Their empirical
logarithmic relationship describes setup as:
7
The variables are described by:
: the axial capacity at time t after driving A: a constant depending on the soil type
: the axial capacity after time t0 t0: an empirical value measured in days
The variable A needs to be determined out of the measured pile capacities. The variable t0
corresponds to time at which the rate of excess pore water pressure dissipation becomes linear
with respect to the log of time. Therefore A and t0 will dependent on the soil type and pile size
(Bullock, 1999). Different researches suggested different intervals for A and t0 ranging from
0,01 to 2 days for t0 and 0,2 to 0,8 for the constant A (Lee et al., 2010). Both variables are
independent of each other, however alternating the value of t0 will result in a different value
of A.
Various researches have been conducted to find Skov and Denver's variables for different soil
types. The values led to a broad interval depending on the soil type. The variable t0 can be
chosen, it will affect the magnitude of the other variables, which is one of the causes of the
broad interval. On the other hand, these intervals allow preliminary estimations of the soil
setup for research studies. Various studies made use of earlier calculated variables to make a
thoughtful estimation of t0, to result in a comparable result of A.
5.2 Practical application
(Attar I. H, Fakharian K., 2013) examined the differences in measurements from static and
dynamic loading tests performed at the test site. From the results of approximately 15 piles, it
is noticed that SLT tests resulted in larger values than DLT ones. Both load tests were
executed at the same time, before the pile setup completely was finished. Due to the duration
of the static test, the soil had more time to recover before the pile reached the required
settlements corresponding to failure. To compensate for the difference in test times, the skin
friction of the DLT was predicted using the Skov and Denver regression formula from the
DLT measurements. The total bearing resistance is divided in shaft and toe resistance. Taking
into account the gain in shaft resistance due to setup for the time interval between the DLT
and static test, resulted in a bearing capacity of the DLT which only differs 8% from bearing
capacity of the SLT. This result is much more acceptable in comparison to the initial results
which have a difference of 32%. This proves that results from both tests can lead to similar
results, if they are interpreted with care. Also the reliability of the Skov and Denver formula is
justified.
8
Research
1 Introduction
The setup phenomenon is a sequence of two individual processes. The pile installation can be
seen as the first process. Soil displacement takes place which leads to a disturbance of the soil
particles and to an increase in soil pressures. Various theories exist, describing the soil
displacement during pile driving as accurate as possible. The second process can be seen as
the consolidation of the disturbed soil. This process can be presented by the consolidation
theory.
2 Soil pressure
During pile installation, pore pressures change as a result of two components. First, an
increase in pore water pressures in the soil around the pile occurs due to the increase in total
mean stress when the pile is forced into the ground. Second, shearing and remoulding of the
soil can cause an increase or decrease in the shear-induced pore pressure. This behaviour
depends on the contracting or dilating tendency of the soil during shearing. Contracting is
more likely to occur leading to an increase in pore water pressures. After the generation of the
excess pore water pressures, the consolidation phase begins (Haddad et al., 2012).
Two methods are most often applied to calculate the soil pressures after pile driving, without
performing a dynamical analysis: the cavity expansion theory and the strain path method.
Both methods however neglect the contribution of shearing to the stress during pile
installation.
2.1 Strain Path method
The Strain Path Method (SPM) provides a framework for predicting pile foundation
behaviour and approaches deep geotechnical problems in a consistent and rational manner. It
visualizes the strain patterns produced in a soil mass by an indenting object. The key
conceptual assumption of the SPM is that the deformation and strain fields caused during the
penetration processes are strongly kinematically constrained, which is most likely in
undrained penetration of clays, and can be estimated independently from the actual
constitutive properties of the surrounding soil (Gill D. R. & Lehane B. M., 2000). The
deformations and strains are independent of the shearing resistance of the soil
9
(Baligh M., 1985). SPM treats the soil as a viscous fluid, which results in one of the weakness
of SPM which is the fact that sandy soils cannot be modelled (Komurka et al., 2003).
The estimation of the effective stresses after driving need to be calculated in three sequential
steps: the strain fields induced in the soil around the pile need to be estimated, the effective
stress paths for these soil particles need to be determinated and finally the equilibrated
effective stresses and pore pressures at the end of pile installation can be achieved (Haddad et
al., 2012).
2.2 Cavity expansion theory
The displacement of soil during pile installation can be described by the cylindrical cavity
expansion theory. The cavity expansion theory formulates the cavity expansion during
installation taking into account the different stress-strain behaviour of the soil, finite or
infinite boundaries, zero or non-zero initial radii, loading and unloading condition and large
strain and small strain relationship between cavity pressure and displacement. The theory is
based on four fundamental equations: conservation of mass, quasi-static equilibrium, the yield
condition and the elastic or plastic flow rule. This theory allows to predict the increase in
horizontal stress and pore pressure in the soil.
A purely claly soil is assumed to calculate the pressures with the cavity expansion theory.
This assumption is commonly done because the theory lacks applicability in other soils. An
elastic perfectly plastic stress strain behaviour can describe a claly soil behaviour (Randolph
et al., 1979). An undrained cavity expansion theory as is described by Gibson and Anderson's
(Gibson R. E. & Anderson W. F., 1961) formulates the soil as: a zone of soil near the cavity
expansion which turns into a plastic state and a zone beyond which remains in an elastic state,
see Figure 3. The soil is characterised by a Young's modulus, Poisson ratio and an undrained
shear strength.
10
Figure 3: Cavity expansion theory (Brinkgreve et al., 2008)
Since the Cavity Expansion Methods is based on conditions of radial symmetry, it restrict the
dependence of field variables to the radial coordinate only (Baligh M., 1985). Although the
Cavity Expansion Theory assumes purely claly soils, recent studies (Salgado R. & Prezzi M.,
2006) extend the CEM framework for cone penetration tests until sandy soils, taking into
account the dilatancy behaviour. Due to the numerous applications of the CEM, which reflects
the applicability of the theory, the CEM will be applied to further calculate the pressures in
the soil after driving.
However lately, attempts are been made to extend the cavity expansion theory from clay to
different soils. (Salgado R. & Prezzi M., 2007) proposed a cavity expansion theory for the
calculation of cone penetration resistance in sand soils. Through the application of linear
equivalent values of Young's modulus, Poisson's ratio and friction and dilatancy angles for the
sand soil, the linear elastic, perfectly plastic soil response which is normally used in the cavity
expansion theory can still be applied. Charts are proposed which gives the equivalent values
as a function of the relative density, stress state, and critical-state friction angle.
11
3 Soil pressure analytical analysis
The cavity expansion method is used to calculate the soil pressures after pile driving. The
CEM will be calculated with a simplified analytical method and modelled by the use of a
finite element method. The software packet Plaxis is used to perform the numerical analysis.
The analytical analysis is described first.
3.1 Analytical analysis
The cavity expansion theory was first well described by (Randolph et al., 1979). The soil
displacement process is midway between the displacement process associated with the
expansion of a spherical cavity and those associated with the expansion of cylindrical cavity.
Relying on earlier performed research about soil displacement of a cone, Carter and Worth
stated that it is reasonable to expect that the stress changes in the soil, over much of the length
of the pile shaft, will be similar to those produced by the expansion of a cylindrical cavity,
ignoring the residual shear stresses on the interface between the pile and soil.
By formulating the stress equality on the boundary of the plastic and elastic zone a
dependency of the plastic and the initial radii to the shear strength and stiffness can be found
(Bezuijen A., 2015), the stress changes around the pile can be described by basic parameters.
The stress changes will dependent on the OCR ratio of the soil. However defining the shear
modulus G dependent on the OCR ratio, the excess pore pressures and stress changes become
relative insensitive to differing OCR (Randolph et al., 1979).
With the assumptions the pore pressures and stresses can be described for an elastic perfectly
plastic material by adopting Gibson and Anderson's theory on undrained cavity expansion.
The soil is characterised with a Young's modulus, Poisson ratio and an undrained shear
strength, which is the case for pure clay [(Haddad et al., 2012) & (Randolph et al., 1979)].
12
Randoplh & Worth give another solution which made use of the interpretation of pressure
meter tests in estimating the stress changes within the plastic zone after the undrained cavity
expansion for pile driving in clay (Randolph M. & Wroth C., 1979).
The parameters are:
Radial stress [kN/m²] Undrained shear strenght [kN/m²]
Shear modulus [kN/m²] Plastic radius [m]
Final cavity radius [m] : Distance to borehole [m]
Excess pore water pressure [kN/m²]
Both formulas will lead to the same result, the latter avoids calculations of the plastic radius
and is therefore more application friendly, it is however only valid within the plastic zone.
The result of a case study on a stone column to 'validate' the formulas is shown in Figure 4,
the same graph can be noticed in the article (Kok Shien Ng., 2013).
Figure 4: Reproduction of radial stress in function of radii, according to (Kok Shien Ng., 2013).
0
10
20
30
40
50
60
70
0 2 4 6 8 10
σr [k
N/m
²]
r/r0
Stress after cavity expansion
Wroth 1979
Gibson and Anderson
13
3.2 Basic model
The Plaxis software is applied to model the cavity expansion theory. A model is created
which calculate the stresses after expansion. Before the model can be built up, a basic model
needs to be validated. For the validation and the build up of this basic model the methodology
of the articles (Kok Shien Ng., 2013; Brinkgreve et al., 2008) are used as a reference.
3.2.1 Model characteristics
The soil is modelled using a Mohr-Coulomb model. The linear elastic perfectly-plastic model
is a first order model that includes only a limited number of features that soil behaviour show
in reality. It is however an often applied simplification in numerical studies leading to
satisfying results. The advantage of this model in Plaxis is that the increase of stiffness with
depth can be taken into account. In general it can be said that the effective stress state at
failure are quite well described using the Mohr-Coulomb failure criterion with a zero friction
angle and a cohesion set to cu for undrained materials. However the model does not
automatically include the increase of shear strength with consolidation
(Brinkgreve et al., 2008).
In the numerical study a correcting layer needs to be added to the perimeter of the mesh to
simulate an infinite boundary. The solution of the cavity expansion analysis is also based on
an infinite continuum. The properties of the correcting layer are described in (Randolph et al.,
1979). The correcting layer involves an area going from the outer soil boundary until a
distance of two times the length of the outer soil boundary. A good solution is presented by a
correction layer with a parameters: and . The length of the soil model
equals 64 times the initial radius, the edge of the outer boundary layer is located at 128 times
the initial radius.
(Kok Shien Ng., 2013) examined a theoretical case study with and without the correction
layer. The results with and without the layer coincides both very well with the analytical
solution. Kok Shien Ng. stated that the application of the correction layer only leads to a
negligible gain in precision, justifying omission of the layer.
14
3.2.2 Model built up & validation
The model consists of a cylindrical cavity of an initial radius a0 = 0,5 m which is expanded to
a radius a = 2,0 m by a prescribed displacement force. The soil assumed to be incompressible
with a friction angle equal to zero and a cohesion cu = 10,0 kN/m². The model is considered as
axisymmetric, with a mesh refinement near the cavity. Top and bottom were fixed vertically
while the right boundary was fixed horizontally. The ratio of shear modulus to cohesion was
chosen to be and Poisson's ratio . The initial stress is taken as zero. No
increase in stiffness with depth is taken into account in the basic model.
An undrained analysis was carried out. In Plaxis it is modelled as an undrained type B
analysis: an undrained effective stress analysis with effective stress stiffness and undrained
strength parameters. The undrained shear strength is an input parameter.
Figure 5: Basic model lay-out
(Brinkgreve et al., 2008; Kok Shien Ng., 2013) applied a different analytical calculation
method, than the one described here above, to compare the numerical model with an
analytical solution. The method, which is still based on the cavity expansion theory, is
obtained from Sagaseta through personal communication as is described in the source of the
theory. The theory describes a small displacement and a large displacement solution. Since
the cavity expansion generates large strains, it is necessary to conduct the large displacement
solution. To attain a large displacement solution, the mesh will need to be updated in the
model. By doing so, the Lagrangian formulation in the model will be updated.
15
The pressure limit and the radius of the plastic zone is calculated according to Sagasetas:
With:
: the radius after expansion;
: the initial radius;
The analytical analysis of the plastic radius, as calculated with formulas (3.7) and (3.10),
results in 19,2 m. As can be seen in Figure 6, the plastic radius from the numerical model
results in 19,4 m. The numerical software gives a good solution.
Figure 6: Validation of the model, plastic radius
The pressure needed to expand the cavity from the initial till the final radius is calculated
analytically with the formulas from Sagaseta and numerically. The results from both
calculation methods respond well, as can be seen in Figure 7. A similar graph is obtained in
(Kok Shien Ng., 2013).
16
Figure 7: Cavity expansion pressure
0
1
2
3
4
5
6
0 0,5 1 1,5 2 2,5 3 3,5
p/c
u
r/r0
Cavity expansion pressure
Numerical
Analytical
17
4 Numerical analysis
4.1 Introduction
The case study is based on (Haddad et al., 2012). This paper modelled the setup phenomenon
of a pile foundation by the application of Abaqus, the paper describes furthermore a
comparison with the instrumented data and the outcome of the model. Performing the analysis
in Plaxis allows a comparison from the results obtained with Abaqus and Plaxis to the
outcomes of an instrumented case. Again the pile installation process is simplified to an
expansion problem, the effects of pile shaft penetration are not considered.
4.2 Case study
(Haddad et al., 2012) based his study on a field investigation earlier described by (Konrad, J.
M. & Roy, M., 1987). Here a close toe steel pipe pile of 219 mm thick is driven 7,6 m into the
soil on a site located west of Quebec city, Canada. The soil profile is mainly described as soft
silty marine clay. An undrained shear strength of the soil is measured by a vane shear test
which resulted in a value of 10 kPa at 2 m depth and 30 kPa at 8 m depth, in between these
values a linear increase is assumed. The input of the analysis is summarized in Table 2. The
pile is assumed to be an elastic isotropic cylindrical pile with radius of 109,5 mm.
Table 2: Input of the case study (Konrad, J. M. & Roy, M., 1987).
Soil 7.6 19 5 0,3 6.10-6
2,5.10-10
10-30 0
Pile 7.6 24 20000 0,2 - - - -
The test performed at location resulted in an ultimate skin friction of the pile measured after 4,
8, 20 and 33 days and 2 years from initial drive of respectively 46, 63, 71, 74,4 and 77 kN. A
static axial load test has been applied during the study. Pore water pressures variations with
time are measured at depths of 3; 4,6 and 6,10 m.
18
4.3 Model alterations
4.3.1 Expansion radii
A pile installation process creates a cavity starting from an undisturbed soil going till a cavity
equal to the pile diameter, the initial radius therefore equals zero. However a numerical
calculation must begin with a finite cavity radius a0 in order to retain its finite circumference
strain. (Radolph et al., 1979) however stated that assuming an initial finite radius doesn't lead
to inconsistent results. It is suggested that an expansion of the cavity from a0 to 2.a0 can give
adequate approximations in comparison of a soil expansion from a zero radius till a finite
radius of r0. The relationship between a0 and r0 is:
Due to the gap between r0 and 2.a0 the model cannot obtain stress results for the zone in
between both radii. Stresses must be extrapolated for the region between r0 and 2.a0, see
Figure 8.
(Castro J. & Karstunen M., 2010) examined the solutions in Plaxis from a study with a final
radius equal to two times and four times the initial cavity radius. Both solutions gave almost
identical results, verifying that the comments made from elastic-perfectly plastic models are
also applicable to more advanced models as cam-clay types. Another study performed by
(Burd H. J. & Soulsby G. T., 1990) lead to equivalent results.
Figure 8: Relationship between cavity radius and pile radius (Kok Shien Ng., 2013)
19
The basic model considers a cavity expansion from 0,5 to 2 m, this can correspond to the
expansion of a stone column. The cavity expansion of the pile will have smaller dimensions,
the application of the formula (4.1) allows to calculate the initial and final cavity interface.
Afterwards the length of the soil model and correction layer can be calculated with the rule of
thumb. The dimension can be found in Table 3.
Table 3: Soil model dimensions and cavity radii
Initial cav. int. Pile int. Final cav. int. Soil layer Correct. layer
Length - 1 [m] 0, 0632 0,1095 0,1264 4,046 8,092
Length - 2 [m] 0,02 0,1095 0,11 10 -
The radii corresponds to the minimal radius as calculated with the rule. In the model however,
other values are implemented. An initial value of 0,02 m is assumed which will be expanded
to 0,11 m. Note that the value, which agrees with the final cavity, is taken equal to the pile
radius. As (Castro J. & Karstunen M., 2010) noted in their work, the actual rule is based on
the following formula.
By creating a cavity from 0,02 m up to 0,11 m instead of 0,1113, an
error is introduced of 0,0013 m on the initial pile radius. This range of error can be assumed
as negligible. Also the difference of the cavity along a real pile shaft will be of a higher
magnitude. By taking the value of the final cavity expansion closer to the real pile radius, the
gap in between real and final modelled pile radius, as is visually presented in Figure 8, will be
small. Since the gap in between 0,1095 m and 0,11 m can assumed to be negligible, a final
interpolation can be avoided.
(Kok Shien Ng., 2013) examined the influence of the initial radius in his study by adopting
different initial radii. All of the radii were chosen equal or smaller than the radius calculated
with the rule as described in formula (4.1). The final radius was taken equal to the column
radius, 0,5 m. 3 different initial radii were examined which are respectively: 0,289 m, 0,1 m
and 0,02 m. The result of his study is displayed in Figure 9. It illustrates the negligible
difference in the outcome of the horizontal stress with varying initial radius smaller than the
one calculated according to formula (4.1). For the sake of completeness, the horizontal
20
stresses corresponding to a cavity expansion with the data of the first and the second row from
Table 3 are presented in respectively Figure 17 and Figure 27.
Figure 9: Influence of the initial radius (Kok Shien Ng, 2013)
The pile installation process of the case study is described as a cavity expansion of the soil
from a zero initial radius up to the pile diameter, 109,5 mm (Haddad et al., 2012). No further
information is available about how the cavity is expanded. Other cavity expansion methods as
for example an expansion with constant stress, applied at the cavity wall, instead of a
prescribed displacement force are also possible (Satibi et al., 2007).
4.3.2 Model length
The model in Abaqus assumes a soil volume surrounding the cavity of a radius of 2 m.
Previously performed Plaxis calculations however took into account a layer with the length of
64 times the initial diameter and a subsequent boundary layer of equal length in the build up
of the model. (Castro et al., 2013) however performed a parametric study to check the desired
width of the model to have a negligible influence of the outer boundary. The parametric study
resulted in a width of 15 m for the cavity expansion of a stone column with an initial radius of
0,1 m till an outer radius of 0,41 m. No boundary layer is assumed in the study by Castro et al
and the width of the modelled soil exceeds the rule of thumb described earlier in formula
(4.1).
(Kok Shien Ng., 2013) also investigated the influence of the correction layer. The study
compared 3 different models with an analytical result as seen in Figure 10. The model with
and without boundary layer resembles well for small cavity expansions, while for larger
cavity expansions it deviates more from the result presented with a boundary layer. However
21
according to (Kok Shien Ng., 2013), implementing a larger soil layer would results more in a
solution resembling the analytical one for larger cavity expansions. The decision of
implementing a larger cavity expansion, and therefore assuming a smaller initial radius in the
model, will results in a solution which agrees more with the analytical one, as is concluded
out of the results of the work done by (Kok Shien Ng., 2013).
Figure 10: The influence of the correction layer (Kok Shien Ng., 2013)
Since the implementation of a correction layer only leads to a small contribution, it is chosen
to eliminate the correction layer out of the model. Instead the boundary layer of the soil will
be extended. A larger soil volume will however result in a larger calculation time. (Castro et
al., 2013) used a soil width of 15 m in his study. The dimension of the soil in this study is also
taken equal to 15 m. This is still bigger than the 64 times the radius. Since the height and the
extension of the cavity in (Castro et al., 2013) is based on larger values, a reduction of the soil
model will be possible. For the sake of completeness the horizontal stresses are presented for
a cavity expansion corresponding to the first and the second row of Table 3, which
demonstrates the model with an without correction layer, see respectively Figure 17 and
Figure 27.
4.3.3 Shear strength
The increase in shear strength in function of depth can be applied in Plaxis. This is done in the
parameter description with the following expression (Plaxis manual).
22
As reported by (Konrad J. M. & Roy M., 1987) the undrained shear strength is 10 kPa at 2 m
and 30 kPa at 8 m. In between 2 and 8 m, a linear increase is assumed. No clear description
was presented in the paper concerning the soil in between the surface and a depth of 2 m. In
general, the shear strength of the soil can be expected to increase in depth. Top soil layers can
however often be disturbed and deviate from this rule. Considering a linear increase from the
bottom of the soil to the surface will result in a su - value of 3,33 kPa at the soil surface. To
avoid the appliance of this rather low value at the surface, a constant value of 10 kPa is
considered in the top layer.
The origin of the program coincides with the left corner of the soil model, the soil surface is
located at a height of 7,6 m. The reference level at a depth of 2 m, where the linear increase of
shear strength start, is set to 5,6 m. With the data of the shear strength in a second point,
30 kPa at a depth of 8 m which coincide with - 0,4 m in the model, the increase factor ,
can be calculated. Figure 11 visualises the shear strength distribution in the model.
Figure 11: Shear strength distribution - model
23
4.3.4 Boundary conditions
The basic model as described in 3.2, considers a cavity expansion of a slice of soil from a soil
layer. The new model describes the behaviour of a complete soil layer, the boundary
conditions will have to be adjusted. The top of the layer equals the soil surface, this is possible
to move freely, no boundary conditions are applied at the top. The other boundary conditions
are to chosen as roller supports. The model with the boundary conditions is presented in
Figure 12.
Figure 12: The model as is applied in Plaxis
The basic model also didn't have a phreatic water level. The water level will be assumed to be
at the top of the soil model. The untreated clay underneath the pile foundation is not
modelled, this is due to the fact that the installation effects underneath the pile are not
particularly significant (Castro J. & Karstunen M., 2010). Furthermore the modulation of the
tip of the pile may lead to some numerical instabilities (Castro et al., 2013).
4.3.5 Interface
An interface element is added to allow proper modelling of the thin zone of intensely shearing
at the area between the cavity and the soil. The virtual thickness of the interface is taken as the
default value of 0,1. This virtual thickness is used to define the amount of elastic deformation
that are generated in the thin zone. Interface elements are supposed to generate very little
elastic deformations - certainly in this case, since the first zone of elements will completely be
plasticized - the virtual thickness should therefore be taken small, however too small values
can lead to numerical ill-conditioning (Plaxis manual).
24
4.3.6 Mesh
The basic model presumes a linear increase of the elements sizes from the cavity to the
boundary layer. The zone of interest of the analyses will be located close to the cavity
expansion, here a smaller mesh is advised. The size of the elements is limited by the Plaxis
software and however also by the available calculation memory.
Due to the interface element the soil stresses near the cavity will modelled better. A medium
global coarse size mesh will be taken and refined mesh elements are been set close to the
boundary, linearly increasing elements are been applied. The mesh is shown in Figure 13.
4.4 Modulation
As in the basic model, the cavity will be expanded by the application of an internal pressure
enforced as a prescribed displacement loading function. (Kirsch F., 2006) state that a
prescribed displacement superior is to other expansion possibilities due to its numerical
stability. The mesh and the prescribed displacement force - purple - can be seen in Figure 13.
Figure 13: Mesh
No initial stresses were calculated in the initial phase for the basic model. This was done by
setting the total multiplier for the weight material equal to 0. The material weight was
therefore not taken into account. According to the reference manual, the material weight
remains at its default value of 1. In certain situations the factor can be adjusted. The
modulation of simplified soil tests is one of them, here self weight can be disregarded since
the stresses are dominated by external loads rather than material weight (Plaxis manual).
25
In the new model the material weight factor is used to investigate the pressure only due to
cavity expansion by setting the material weight factor to 0 and the distortion of the initial
pressure field due to cavity expansion by assuming the coefficient 1.
The basic model is based on a cavity expansion of 0,5 m till a radius of 2,0 m. Since this will
lead to large strains, the large displacement method is used to calculate the stresses by
updating the mesh. Different other papers as for example (Satibi et al., 2007) stated that
strains due to pile driving also are classified as large strains. The application of the large strain
technique is therefore also justified in this situation.
4.5 Initial stress
Before the cavity expansion can be performed, the correct initial stresses must be generated.
The initial stresses will be calculated in the initial phase. The result of the initial stresses can
easily analytically be verified. Since the phreatic water level is assumed to be at the top of the
soil layer, the saturated density weight has to be used. The vertical, initial stresses will linear
increase from 0 till the following values:
The initial modelled stresses are shown in Figure 14, it agrees with the analytical results. Note
that due to the sign convention compression stresses are presented by Plaxis with a negative
value.
Figure 14: Vertical initial stresses
26
The horizontal effective stress, which corresponds with the radial effective stress for a
axisymmetric model, depends on the coefficient of lateral earth pressure, K.
(Hadded et al., 2012) did not directly describe an initial K coefficient in his study. However
from Figure 15 the K0 coefficient could be conducted from the initial stress condition. This
results in a value of .
Figure 15: Determination of K0 (Hadded et al., 2013)
From the resulted K0 value, it can be concluded that the Mohr-Coulomb is used for gravity
loading. Due to gravity loading, a one dimensional compression is applied. The later
coefficient of earth pressure corresponds with the same value as calculated with the following
formula (Plaxis material model manual).
The gravity loading in Plaxis resulted in bit lower initial stresses as is presented in Figure 15.
A displacement with an order of a cm was also noticed. To avoid staggering errors, the initial
stresses are generated with an K0 procedure, with a value K0 = 0,445. The stresses are
displayed in Figure 16.
27
Figure 16: Horizontal initial stresses
4.6 Results
4.6.1 Introduction
Cavity expansion in fine soils will increase the total stress in the surrounding soil. Due to the
soil structure, the water in between the soil grains is not allowed to dissipate when the load is
applied. Time is needed to dissipate the pore water. It describes the undrained behaviour of
the soil. The water in between the grains will carry the load and excess pore water pressures
are generated. After pore water dissipation the excess pore water pressures decrease and the
effective stress in the soil increase, this increase in effective stress with time will increase the
capacity of the pile and is referred to the setup phenomena in fine graded soils.
Graphs related to the development of forces and displacement of a point can be generated
with the Plaxis output menus. The output of Plaxis however does not allow the generation of
stress curves at a reference level. Data extracted from tables needs to be fitted with Matlab
calculations to make presentable features.
4.6.2 Verification of model length and expansion radius
(Kok Shien Ng., 2013) concluded that the omission of the correction layer as is described in
(Randolph et al., 1979) will lead to similar results as can be achieved without the correction
layer. The same work and the work performed by (Castro J. & Karstunen M., 2010) stated
that a cavity expansion with a radius lower than the one calculated with the formula of (4.1)
will lead to equal models if the ratio of the initial and final cavity can be described by formula
(4.2). For the sake of completeness these conclusions are verified by comparing the radial
effective stresses calculated with both models, see Figure 17 and Figure 27. It should be noted
28
that the model from Figure 17 is built corresponding to the parameters of the first row in
Table 3.
It can be noticed both figures lead to equivalent stress results. The magnitude of the difference
is low, the difference can therefore be assumed to be negligible. The biggest difference in
stress is notable when the correction layer is omitted and at the same time the soil layer is
extended. Which is also reported in (Kok Shien Ng., 2013). It justifies the model approach
which is further used in the calculations.
Figure 17: Radial effective stress - according to the rules of the basic model
4.6.3 Displacement
The displacement after cavity expansion is displayed in Figure 18. It is notable that the cavity
radius after expansion corresponds with a radius of 0,11 m in the model, which agrees with
the required final radius. Due to the free boundary at the top, soil movement can occur freely.
A soil heap with a height of 0,035 m near to the expanded cavity is formed after cavity
expansion. Furthermore a small increase and a consequent decrease of soil is notable across
the top boundary.
29
Figure 18: Soil heap
Figure 19 shows the displacement of the soil particles. The layer in the vicinity of the pile will
exhibit a large horizontal displacement, while the section thereafter also exhibits a small
vertical displacement.
Figure 19: Displacement vectors
4.6.4 Plastic failure
(Plaxis manual) describes a failure point as an indication where the stress lies on the surface
of the failure envelope, it also verifies if the size of the mesh is sufficient for calculations. An
insufficient mesh occurs when the zone of plasticity reaches a mesh boundary.
Figure 20 displays the plastic failure points of the model. No failure points are notable close
to the boundary of the soil layer. This proves that the width of the soil model is sufficient for
model calculations.
30
Figure 20: Plastic point
The plastic radius can be calculated with the formula (3.4), which results in a plastic radius of
for until a depth of and for at a
depth of . From the figure it can be seen that the plastic radius as presented in Plaxis
agrees very well with this results. At the bottom a value of , while at a depth of
a value of are achieved from the calculations. In general an increase in
depth of the plastic radius is noticed, this is due to the increase in undrained shear strength as
can be seen in the formula (3.4).
(Kok Shien Ng., 2013) mentioned that the analytical solution is based on infinite conditions of
the soil model, which takes into account the boundary conditions. He stated that it can be
noted that any solution, based on infinite conditions, may not give good results for soil near to
the top. This can prove the fluctuating plastic radius in the upper layer with a constant
undrained shear strength.
It can be concluded that the plastic radius calculated by Plaxis resembles very well with the
results from the analytical solution.
In the vicinity of the soil heap, the Plaxis model resulted in inaccurate calculations. This is
visualized in Figure 21. The results presented by these nodes will lead to incorrect values,
which needs to ignored in analyses.
31
Figure 21: Inaccurate points
4.6.5 Pore pressures
Due to the pile installation, the cavity will expand leading to an increase in total stress. The
consistency of clay does not allow immediate pore water dissipation, excess pore water
pressures will be generated. The excess pore water pressures calculated by the model are seen
in Figure 22.
Figure 22: Excess pore pressure after pile installation
32
A local extreme value of -143 kN/m² is reached at the bottom of the cavity expansion. In
general it can be seen that the excess pore pressure increases with increasing undrained shear
strength with depth. The increase of excess pore pressures with depth has also been measured
in field tests (Castro, J., 2008).
Due to the varying pore water pressure with depth, the pressure needs to be presented at a
certain depth. Figure 23 shows the pore water pressures at a depth of h = 3 m compared with
the earlier described analytical method by Worth. The undrained shear strength corresponding
to the appropriate depth is used in the formula of Worth. A cluster of values close to the pile
diameter is noted for the Plaxis solution. The pressures from Plaxis correspond vary well with
the values obtained with the analytical solution. A small difference is notable due to the
difference in plastic radius. The same conclusions can be drawn for the other pore pressures at
different depths.
Figure 23: Pore pressure distrubtion H = 3 m
To allow for a direct comparison between the pore pressures of different depths the pressures
can be normalised. Instead of normalising the pressures with their initial value, the pressures
are normalised with their undrained shear strength as been done in (Castro J. & Karstunen M.,
2010). It demonstrates the dependency of the pore pressure on the undrained shear strength.
The normalised values of the excess pore pressures agree well for all depths as seen in Figure
24.
-5
5
15
25
35
45
55
65
75
0 0,5 1 1,5 2
∆u
[k
N/m
²]
r [m]
PLAXIS
Randolph and Worth, 1979
33
Figure 24: Normalised pore pressure
The area affected by the pile installation varies slightly with depth, depending on the
undrained shear strength. The influence zone is in this case around 0,8 - 1,3 m away of the
column radius, which corresponds to 7,3 - 12 times the column radius. (Castro J. & Karstunen
M., 2010) stated that in their research the zone of influence is around 13,5 times the column
radius for a stone column, which corresponds to the same order of magnitude as the presented
pile installation.
(Castro J. & Karstunen M., 2010) stated that the radius of influence depends on the rigidity
index Ir, this is the ratio of shear modulus to undrained shear strength. This value depends in
this case on the depth due to the dependency of the undrained shear strength with depth and
due to the fact that the elasticity modulus is taken equal along the pile. Due to the relation
with Ir it is said that radius of influence where excess pore pressures develop coincides with
the extension of the plastic zone R, this can also be concluded from Figure 24. The influence
zone of the soil layer at h = 6,1 m is smaller than the one from h = 3 m.
(Castro J. & Karstunen M., 2010) also describes the pore pressure as a steep decrease close to
the column and a nearly linear decrease at a distance of about five column radii. The same
trend can be noticed but the behaviour is so little that is not worth to draw a conclusion from.
The remark should be made that the work from (Castro J. & Karstunen M., 2010) is based on
a more complex soil model.
-0,5
0,5
1,5
2,5
3,5
4,5
5,5
0 0,5 1 1,5
∆u
/cu [
m]
r [m]
H=3
H=4.6
H=6.1
34
The magnitude of the excess pore pressures generated with Plaxis are from the same order as
the one with Abaqus (Haddad et al., 2012), however Plaxis predicts higher values of the pore
pressures, see Figure 22 and Figure 25. These higher values are only very locally, in general it
can be seen that the results of the pore pressure are of the same magnitude. Due to the lack of
axis information of the results of (Haddad et al., 2012) no well-based conclusions can be
made concerning the influence zone. If the distance presented on the horizontal axis agrees
with the length of his model, which corresponds to 2 m, larger influence zones are obtained in
the work of Haddad. It should also been noticed that a study made by (Satibi et al., 2007) also
obtained different influence zones with smaller models.
Figure 25: Excess pore water pressure - Y - axis: height; X - axis: length
4.6.6 Effective soil stresses
After pile installation the effective stress in the soil will increase. The increase in effective
stress is however limited by the trapped pore water. The generated increase in radial effective
stress after pile installation is visualised in Figure 26. An increase in effective stresses can be
noticed with depth. The increase with depth mainly occurs in the influence zone around the
pile. Outside of this zone, the stresses drop drastically and slightly continues to decrease to
zero. The influence zone of the effective stress is however bigger than the one for the pore
pressures.
35
Figure 26: Effective stress increase after pile installation
The maximum increase of effective stress corresponds to a value of 43,7 kN/m² and is located
at the bottom of the model. A higher cu-value in this point leads to a higher stress. The
effective radial stress after pile driving can be visualised in Figure 27. The increase in
effective stress is rather limited in comparison to the increase in pore water pressure.
Primarily studies performed in an older Plaxis model, Plaxis Classic, resulted in different
effective stresses due to a different sign convention. The stress due to the expansion was
opposite in sign in comparison to the initial stress. The same trend is noticed in the work of
(Satibi et al., 2007).
The model made in (Haddad et al., 2012) obtains slightly higher absolute effective stresses
than the Plaxis model. The total stress due to cavity expansion is distributed between the
effective stress and the excess pore water pressure, since Plaxis generates higher pore water
pressure, lower effective stresses are expected. The difference in models is still small.
36
Figure 27: Effective radial stress after pile installation
Figure 28: Effective stress after driving (Haddad et al., 2012) - Y - axis: height; X - axis: length
37
4.6.7 Total soil stress
The total soil stress is the sum of the water pressures and the effective stress, Figure 29. The
same conclusions from the pore water pressures and effective stresses can be made. It should
be marked that due to its smaller value the effective stresses are hardly notable in this graph.
Figure 29: Total radial stress
Figure 30 displays the results obtained from Plaxis and the analytical method as described in
3.1. The analytical solution represents well with the results from Plaxis. In general it can be
said that the Plaxis solutions correspond well with the analytical and Abaqus solutions.
Figure 30: Comparison of the total stress between numerical and analytical
30
50
70
90
110
130
0 1 2 3 4 5 6
σto
t [k
N/m
²]
r [m]
Plaxis
Analytical
38
5 Soil water dissipation
The consolidation after pile installation is discussed in (Randolph et al., 1979). Based on field
measurements it is shown that the major pore pressure gradients are radial over most of the
length of the pile, consolidation takes place primarily by radial flow of pore water and radial
movement of soil particles.
However, it should be noted that these findings are relevant for a specific problem. In general,
the dissipation time will depend on site specific characteristics. The soil consistency and the
pile will determine the dissipation time. For example the dissipation time is proportional to
the square of the diameter of the pile. Larger piles will therefore require longer consolidations
times (Komurka et al., 2003).
5.1 Parameters
As is seen in Table 2 the consolidation analyse will be conducted with the horizontal and
vertical permeability parameters, these are the only parameters necessary to conduct a
consolidation analysis. The difference between the horizontal and vertical permeability
parameter is large. The unit of the permeability parameters as presented in the work of
(Haddad et al., 2012) and displayed in Table 2 are incorrect. The unit m/s is noted while
typical permeability values for silty clay are more likely to correspond with 5,5.10-6
cm/s
(Pfeifer H., 2014). In the calculations the units cm/s are applied for the permeability.
Note that the model shows a simplified solution, the model does not include the increase of
shear strength with consolidation. Also the change of permeability during a consolidation
phase is not taking into account due to the lack of parameter information. The pile boundary is
simplified as an impermeable boundary. The upper boundary as an open boundary.
A consolidation calculations in Plaxis is simulated through the application of forces which
decrease linearly with time. A consolidation analyses will consequently be presented as a
linear consolidation in Plaxis. As is stated earlier, research have shown that pore water
dissipates first non-linearly according to the log of time and afterwards linearly, see Figure 1.
The non-linear decrease cannot be presented in Plaxis. A stepwise linear approximation could
be used to present the initial non-linear decrease. However this is not implemented due to the
lack of parameters and the strong dependency of this initial period on the case study and their
soil.
39
As it is noted in (Sabetamal et al., 2016) it is necessary to use very small time steps at the
beginning of a consolidation analyses due to the high appearing pore pressure gradients.
5.2 Results
The consolidation analyses is conducted for a time period of 33 days, the same period is used
in (Haddad et al., 2012), this allows a comparison of the results. The pore pressure
distribution and the effective stress will be discussed.
5.2.1 Pore pressure
In general it can be said that a decrease in pore water pressure is noticed after consolidation,
as is expected. After a time period of 33 days no excess pore water pressure are notable in
Plaxis, this while initial higher excess pore pressure was established. The excess pore pressure
according to Abaqus, as seen in Figure 31, are also from a negligible amplitude.
Figure 31: Excess pore water pressure distribution after 33 days (Haddad et al., 2012)
The results from Plaxis show a faster consolidation process. The report noted that after
2 years from initial drive 95% of excess pore water pressure has been dissipated. And after 5
days the majority of the excess pore pressure are been dissipated. These values are reached
earlier with the calculations performed in Plaxis. This can be seen in Figure 32.
40
Figure 32: Development of excess pore water pressure with time
It can be concluded that the pore water pressure is dissipated after a short time period.
According to (Plaxis reference manual) a consolidation analysis after of an undrained B
model can lead to erroneous results. An axisymmetric test run, conducted with the same soil
parameters, for circular footing resulted in equal consolidation times.
However Figure 33 shows the pore pressure distribution after 95 % of the pore water pressure
has been dissipated. The influence zone of the pore water pressure increase while the pressure
decreases, this is due to the dissipation of excess pore water to the surrounding soil mass. The
same trend as seen in Figure 31 has been noticed.
Figure 33: Pore pressure distribution after 95% dissipation of excess pore pressure
35
45
55
65
75
85
95
105
115
125
135
0 0,5 1 1,5 2
u [
kP
a]
r [m]
0.01 day
1 day
41
5.2.2 Effective stress
After the consolidation analyses an increase in effective stress is observed. The increase in
effective stress with time results in an increase in pile capacity as is described as the setup
effect. The results obtained with both programs are from the same magnitude. Abaqus
presents slightly higher stresses as is been noticed earlier.
Figure 34: Effective stress after 33 days - left Plaxis; right (Haddad et al., 2012)
Figure 35 displays the effective stress after pile installation and after consolidation for a depth
of 4,60 m. An increase of 65 kN/m² is noted in the vicinity of the pile.
Figure 35: Increase in effective stress after consolidation, h = 4,6 m.
0
20
40
60
80
100
120
0 1 2 3 4 5
Eff
str
ess
[kN
/m²]
r [m]
Before consolidation
After consolidation
42
6 Pile shaft capacity
6.1 Introduction
(Haddad et al., 2012) reports the increase in pile shaft capacity by comparing the increase in
effective stress before and after pore water dissipation. These results are normalized by the
ultimate pile shaft capacity obtained by experimental tests as seen in Figure 36. The numerical
results also under predict the pile capacity, the work reports that higher values are
experimentally obtained due to aging effects of the soil. A reduction factor of = 0,125 is
implemented in the interface parameters after 4 days of pore water dissipation, the value is
obtained by matching the experimental and numerical ones.
Figure 36: Increase in pile shaft capacity (Haddad et al., 2012)
6.2 Modelling approach
The method to measure this pile shaft capacity is not described in (Haddad et al., 2012).
Therefore another approach is applied based on the work of (Satibi et al., 2007). Here the pile
capacity is determined by adapting a prescribed displacement force at the top of the pile, the
value of the displacement is set to coincide with the failure of the pile. Only values of the pile
shaft capacity are taken into account in the article. A small prescribed displacement force on
the top of the pile is used to modelled the shaft capacity. It is considered that the force which
results in the first displacement of the pile coincides with the shaft capacity. A new
calculation fase before and after consolidation is deducted. Since the results of the
consolidation process at different time intervals of Plaxis does not agree with the results from
Abaqus, no other time intervals are analysed for consolidation.
43
6.3 Results
The results of the displacement of a point on the other side of the pile in relation to the force
is presented in Figure 37. It should be noted that the force is presented in [kN/rad] in Plaxis,
to obtain the correct value the results are been multiplied with . First the displacement is
restrained due to the shear strength between the pile and soil, afterwards the shear strength is
exceeded leading to a sudden displacement. Continues the pile will experience the friction
along the pile shaft until eventually the soil is disturbed and no extra force is needed to
displace the pile.
Figure 37: Vertical force displacement curve
The results from the graph, the experimental research and (Haddad et al., 2012) are presented
in Table 4. It can be seen that the force after cavity expansion agrees well with the
experimental results, however after consolidation the both numerical models under estimate
the bearing capacity. The cause of the different outcomes can be due to model uncertainties,
experimental uncertainties or due to the so-called aging effect which is not implemented in
the models.
Table 4: Increase in pile shaft capacity
Plaxis - Fshaft [kN] Abaqus - Fshaft [kN] Experimental - Fshaft [kN]
Before consolid 49,6 29,3 46
After consolid 57,2 50,1 77
6
7
8
9
10
11
12
13
14
15
16
0 0,002 0,004 0,006 0,008 0,01
F y [
kN
/ra
d]
uy [m]
After consol
Before consol
44
7 Case study
7.1 Introduction
The model is applied on a case study from (De Assis A., 2016). The purpose is to calculate
the capacity of one pile out of a pile group. A vertical pile where PDA-tests in different time
spans were performed on, is selected from the group. The project data applied in this study is
briefly summarized in Table 5 and Figure 38. Note that the pile capacity tests are performed
after 1 and 31 days.
Two details were missing to be able to conduct an analysis: the pile type and the amount of
pile settlement corresponding to failure.
Table 5: Project data - pile information (De Assis A., 2016)
Length [m] Diameter [m] PDA 1 [kN] PDA 31 [kN] kx = ky [cm/s]
37,50 0,50 1415 2450 1.10-5
Figure 38: Soil strata
7.2 Model approach
In this chapter, the method to model the bearing capacity of the pile as described in (Satibi et
al., 2007) is discussed. (Józefiak et al., 2015) did not model the cavity expansion but assumed
an equivalent technique to test the bearing capacity, however both studies were conducted for
a sandy soil and did not take into account undrained behaviour.
To model the bearing capacity of a pile the soil underneath the pile needs to be taken into
account. As is described earlier, modelling the soil underneath the pile can result into
numerical instabilities.
45
The bearing capacity is calculated by assuming a fase, after soil consolidation, where the pile
is loaded until failure. Loading until failure will be performed using a displacement controlled
function. The settlement corresponding to failure is described in the local norms. The rule
0,2D is assumed to correspond to failure in this study, it equals to s = 0,035 m. The failure
force will result in the modelled combined pile capacity.
During pile installation a pressure field is generated underneath the pile. To simulate this
pressure, (Satibi et al., 2007) applied the weight of the pile, , at the bottom of the cavity
during cavity expansion. The approach would however only be valid for certain bored
displacement piles. A pile driving process will create larger pressures at the pile tip.
7.3 Model input
The soil strata concerns different soil layers which each of their parameters displayed in
Figure 38. In the model, the soil is simplified as one layer to avoid numerical instabilities
around adjacent soil layers. To model the case study, the associated soil strata will need to be
presented by one single layer. From the data of the soils strata it can be seen that the values of
the passion ratio, the undrained shear strength, the friction angle and the unit soil weight show
little variation. It could be approximated by one single value. Only the claly, silty sand layer,
which is in Portuguese noted as 'Argila Ar. Silt' deviates. The thickness of this layer is merely
3 m in comparison to other soil strata. The elasticity modulus diverges more, simplified
values are obtained by implementing a function in Plaxis. The solution of a soil strata would
be a complex function, a soil boring test tries to obtain discrete values of constant soil profile.
An undrained A model is used in Plaxis to be able to implement a friction angle.
Figure 39: Trendline of soil elasticity, black - soil test; blue - linear approximation
0
20
40
60
80
100
120
140
0 10 20 30 40
E [
MP
a]
depth [m]
46
Plaxis only allows the application of linear varying input data with a constant start. The first
layer can be implemented as constant, further on a linear increasing in elasticity modulus is
assumed. By applying weight factors, depending on the thickness of the soil layer, the
coordinates of the linear function are calculated and show in Figure 39. Other software allow
the implementation of more complex functions.
It should be noted that the first two meters of soil layer consecutive have been excavated and
have been filled up. The filling soil material is assumed to be disturbed, the contribution to the
pile capacity will not be taken into account. The excavated soil is not present in the data of the
soil strata, however the pile length still need to be reduced with 2 m.
The cavity expansion is calculated as stated above, the parameters can be found in Table 6.
Again a different initial radius is taken to avoid the stress correction. The length of the model
is taken equal to 20 m.
Table 6: Cavity expansion parameters
Theory 0,144 0,288 0,25 - 0,25
Applied 0,05 0,25 0,2445 0,005 0,25
7.4 Model output
7.4.1 Model (4.3)
The model as described in 4.3 is first applied to the case study to display the soil stresses and
pressures. It should be noted that the cavity is expanded to the right radius leading to a large
soil heap of 0,22 m. The following figures are made with a 6-Node model, the accuracy will
be lower. It should also be noticed that the soil weight is not taken into account, no
information about the initial ground conditions was given.
47
Figure 40: Plastic radius
As presented earlier, the plastic radius described by the analytical solution doesn't agree with
the modelled results from the top layer. The analytical radius predicts for a
plastic radius of . Furthermore an increase in plastic radius is noticed due to the
increase in elasticity modulus, this is also present in formula (3.4). The formula only predicts
a higher plastic radius for , The influence zone for ,
corresponds to 40 times the pile radius. The plastic zone touches the boundary as can be seen
in Figure 40, this shows the incorrect model length. The model is enlarged until 40 m.
Figure 41: Case study: Increase in effective stress - left, excess pore pressure - right
An increase in effective stress is noticed with depth. The increase remains however small for
pile intrusion of 50 cm. Excess pore pressure is noticed after pile installation. The order of
48
magnitude will be around 60 kN/m² around the pile cavity, locally the values can be more, but
due to the bigger mesh, the stresses in the vicinity of the pile cannot be distributed. The
analytical solution results in an excess pore water pressure for of .
Figure 42 and Figure 43 presents the stress and pore pressure after respectively 1 day of
consolidation and 31 days of consolidation. It can be noticed that the end pore pressure
dissipation almost is achieved after 31 days, a negligible amount of pore pressure is still
present. Although the same program was applied, the consolidation time resulted in more
realistic values for fine grained soils.
Figure 42: Case study after 1 day: increase in effective stress - left, excess pore pressure - right
Figure 43: Case study after 31 days: increase in effective stress - left, excess pore pressure - right
49
7.4.2 Extended model
The model from (4.3) was extended with soil underneath the model. To verify the result, it is
first been conducted for the parametric study as handled in (4.3). Also the interfaces around
the pile tip are extended deeper in the soil model to present a more continuous stress pattern,
as is been described in (Plaxis manual). As can been the stress due to cavity expansion are
still correctly displayed in the extended model, see Figure 27 and Figure 44. The stress
underneath the pile tip are noted in Plaxis with a different sign and are of a smaller magnitude
in comparison to the expansion created stresses. Only a small downwards load, corresponding
to the weight of the pile, is taken into account.
Figure 44: Radial effective stress after cavity expansion - new model
After the intrusion of the pile, the movement of the prescribed pile load at the pile top resulted
in numerical instabilities. The instability and the lack of time stopped the progress of the
work.
50
8 Comments and future work
The stress underneath the pile is created by the application of a downward load taken equal to
the weight of the pile. The resulting stress patterns deviates from a typical stress failure
pattern for driven piles as is noted in the methods of Koppejan and De Beer for the calculation
of the pile bearing capacity, see Figure 45. The method of (Satibi et al, 2007) is however
based on a bored pile, no large stresses are generated underneath the pile for bored piles.
Figure 45: Pile failure surface (Bezuijen A., 2014)
To facilitate the installation process, the pile shape is practical applications accommodated
with a triangular shaped bottom. The intrusion of this bottom part will change the stress field
around the pile. The effects of the pile tip on calculations was investigated in (Sturm, H. &
Andresen, L., 2010).
The pile driving installation process is described in this work with a cavity expansion
problem. Pile driving installations are however more complex processes. Soil disturbance will
not only occur due to the creation of the cavity but also due to the downward movement of the
pile. (Sabetamal et al., 2016) developed an algorithm to predict the installation and setup
effects of a dynamically penetrating anchor in a clayly seabed. The penetration of the anchor
in the soil layer occurs due to self weight and the dynamically energy as a result from a drop
height near sea level. In this example the pile intrusion occurs regularly.
Nevertheless a driving process cannot be assumed as a continuous pile driving movement. It
is characterised by a driving hammer, creating impulsive forces in the pile. During the
impulses, settlement is created and excess pore pressures are generated locally around the
pile. These access pore pressures reduces the shear force in the soil which contributes to the
penetration of the pile in the soil. A work performed by Plaxis (Plaxis tuturial) gives a
51
simplified approach to model the settlements during pile driving. Here the dynamical driving
forces are implemented as a harmonic function. This method allows the modulation of the
downwards propagating compression waves in the pile. A more accurate solution needs to
take into account as well the Rayleigh damping.
Most pile models are executed in a single layered soil. Only a handful of parameters are
applied to describe a soil. These parameters are however a simplification of the results of soil
tests. The test are performed, if lucky in the vicinity of the tested pile foundation. Afterwards
a model is validated with pile capacity tests, which on their turn needs to be handled with the
required sensibility. Scatter will be presented in the outcomes.
Models give the freedom to describe a problem more accurate than an analytical analyses.
Still no models are available which are accurately able to describe the pile driving installation
with their different processes. Future work can provide more complex models which will
allow a better description of the pile driving problem. However these models will need to be
build with accurate soil information and validated proper field measurements.
52
Conclusion
An axisymmetric model is applied to investigate the stress and pressure state of the soil after
pile driving with the cavity expansion theory. Afterwards a consolidation analysis allows pore
water pressure to dissipate. The basic cavity expansion model as is given in (Brinkgeve et al.,
2008), is elaborated to a model able to predict the stress and pressure state of the soil after
expansion. The outcomes of the model is tested by introducing a comparison of the plastic
radius, effective and total soil stress and excess pore water pressures with an Abaqus model.
The length of the model from (Haddad et al., 2012) corresponds to 2 m, which not satisfies
the rules of thumbs as described in (Randolph et al., 1979). Other reports (Kok Shien Ng.,
2013) and (Castro and Karstunen, 2010) excluded the correction layer and implemented a
model length of the same magnitude as described in (Randolph et al., 1979). The application
of a model length taken equal as is shown in the study of (Castro and Karstunen, 2010) results
in a sufficient mesh length.
The model results in a plastic radius dependent on the undrained shear strength, which is also
seen in the analytical solution of the cavity expansion theory. The numerical outcomes of the
plastic radius resembles well with the analytical solutions, only deflections are noticed more
to the top surface. It should however been noted that the analytical solution is based on an
infinite boundary layer, alternations with a real finite boundary layer are expected.
The model shows an increase in pore water pressure in the vicinity of the cavity expansion,
the influence zone of the of the pore pressure resembles very well with the plastic radius. Also
an increase in effective stress is noticed after cavity expansion, however the increase is small
in comparison to the increase in effective stress.
Both the results of the pore water pressure and stress after expansion are well described by the
analytical solution of the cavity expansion theory which is founded on the work of (Randolph
et al., 1979). However the same hypotheses is made in the cavity expansion model: an elastic-
perfectly plastic material. (Castro and Karstunen) work based on more complex soil models
resulted in remarkable deviations from the logarithmical curves presented by the solutions for
elastic-perfectly plastic models.
The results also showed good agreements with the Abaqus model from (Haddad et al., 2012).
Rather small diversities are noted in the results from both calculation software. However no
53
well founded conclusions could be made about the influence zone of the stress after expansion
due to a lack of dimension on the graphs presented in the original work.
The consolidation analysis conducted by Plaxis led to erroneous results of the consolidation
time. As is noted in the (Plaxis manual), a consolidation process should be analysed with care.
Besides the false results in time, the analysis perfectly presented the stresses after
consolidation. It should also be noticed that the permeability parameters, as given in the report
of (Haddad et al., 2012), possible were of the wrong order of magnitude.
Due to a lack of description in the original article, a method is proposed to simulate the shaft
capacity. According to this method the initial capacity after cavity expansion agrees more
with the results from the experimental study, however a smaller increase in shaft capacity is
numerically obtained than predicted in the report. A similar gap in final shaft capacity is
obtained with the results from the Abaqus analysis. However the work from (Haddad et al.,
2012) links this to inability of the model to predict the aging occur in the soil.
The model allows the prediction of the soil stresses before and after consolidation of the
cavity expansion. A tempt is been conducted to predict the bearing capacity based on the
work by (Sabeti et al., 2007). This however didn't lead to appropriate result. It should be noted
that (Sabeti et al., 2007) assumes a sandy soil and doesn't take into account the undrained
behaviour.
54
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