Analysis of an Inclined Pile in Settling Soil
Analys av en lutande påle vid marksättningar
Fredrik Resare
December 2015
Master of Science Thesis 2015/09
c Fredrik Resare 2015
Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering
Division of Soil and Rock Mechanics
Stockholm, Sweden, 2015
Abstract
The use of inclined piles is an ecient way to handle horizontal forces in constructions. However, if the soil settles the structural bearing capacity of each pile is
reduced because of induced bending moments in the pile. There are several reasons
for a soil to settle, e.g. if an embankment is built on top of a clay settlements will
occur. There is currently no validated method in Sweden to analyse horizontal loading from a settling soil. In the current report a non-linear 3D nite element model
is validated by a previously conducted eld test and the results are compared to
three dierent beam-spring foundations. These consist of a standard model where
a subsoil reaction formulation is used, a model where the soil is considered as a distributed load, and a model with a wedge type of failure. Furthermore, a parametric
study is conducted for a cohesionless material where the weight and friction angle
of the soil material is varied. The standard soil reaction model yields an induced
bending moment almost three times larger than the one obtained from the eld
test and the two other calculation methods. The latter beam-spring models should
therefore be considered in practical design. These ndings imply that inclined piles
can be used in a far greater extent than previously expected, hence decreasing the
cost for the project.
Keywords: Batter piles, raked piles, ground settlements, induced bending moments,
nite element method, piles
iii
Sammanfattning
Användning av lutande pålar är ett väldigt eektivt sätt att ta hand om horisontalkrafter i konstruktioner. Om marken omkring pålen sätter sig orsakas ett böjmoment i pålen som sänker den strukturella bärförmågan. För närvarande nns ingen
validerad metod i Sverige för att beräkna storleken av den horisontella kraften som
orsakas av sättningarna. I den här studien har en icke-linjär 3D-FEM modell validerats mot ett tidigare utfört fullskaleförsök, dessa resultat har därefter jämförts mot
tre olika 2D-diskretiseringar. Den första modellen som beskrivs är den som idag
används vid påldimensioneringar. De två andra modellerna är baserade på en annan brottmekanism i påltoppen där jorden istället för en fjädermodell utgörs av
en utbredd last med två olika formuleringar. Vidare har en parametrisk studie utförts med en friktionsjord där vikten och friktionsvinkeln påjordmaterial varierats.
Den nuvarande 2D-diskretiseringen ger ett böjmoment i pålen som är närmare tre
gånger större än det i fältförsöket uppmätta och de två föreslagna beräkningsmodellerna. Ett böjmoment så stort att pålens kapacitet teoretiskt blir obentlig enligt
nuvarande beräkningsmodell.
Nyckelord: lutande pålar, marksättning, nita elementmetoden, påle, transversalbelastning
v
Preface
With this thesis I nish my studies in Civil Engineering at KTH. The thesis was
written on and with the help of many of my current colleagues at ELU. I would
like to thank my two supervisors from ELU Konsult; Anders Beijer Lundberg and
Christof f er Svedholm for supporting with knowledge and enthusiasm! I would
also like to thank Jimmie Andersson for proofreading and for valuable input.
Furthermore I would like to thank Stef an Larsson as an excellent source of inspiration. You have given me the mindset that failure always occurs in the weakest
possible way, a lesson truly used in this thesis.
vii
Contents
Abstract
iii
Abstract
v
Preface
vii
Nomenclature
xi
1 Introduction
1
2 Presentation of the current Swedish calculation model
3
2.1
Discussion of the calculation model . . . . . . . . . . . . . . . . . . .
3 Simplied models of soil
3.1
5
7
Consolidation of soil . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.1.1
Primary consolidation . . . . . . . . . . . . . . . . . . . . . .
7
3.1.2
Secondary consolidation . . . . . . . . . . . . . . . . . . . . .
9
3.2
Elastic-plastic models of soil . . . . . . . . . . . . . . . . . . . . . . .
9
3.3
Beam-Spring model (p-y curves) . . . . . . . . . . . . . . . . . . . . . 10
3.3.1
P-y curves for clay . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.2
P-y curves for drained material conditions . . . . . . . . . . . 12
3.4
Drained and undrained conditions . . . . . . . . . . . . . . . . . . . . 13
3.5
Mohr-Coulomb yield criterion . . . . . . . . . . . . . . . . . . . . . . 13
4 Previous studies of inclined piles
4.1
15
Analysis of the eld test . . . . . . . . . . . . . . . . . . . . . . . . . 15
ix
4.2
2D discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2.1
Distributed load approach . . . . . . . . . . . . . . . . . . . . 18
4.2.2
Wedge failure approach . . . . . . . . . . . . . . . . . . . . . . 19
4.2.3
Discussion about calculation models . . . . . . . . . . . . . . . 20
5 Simulation with 3D nite element model
5.1
5.2
21
Validation of 3D model . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1.1
Geometry, boundary conditions and FE discretization . . . . . 21
5.1.2
Material properties . . . . . . . . . . . . . . . . . . . . . . . . 22
5.1.3
Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Formulation for parametric study . . . . . . . . . . . . . . . . . . . . 22
6 Results and discussion
25
6.1
Results from the validation of the 3D model . . . . . . . . . . . . . . 25
6.2
Results from the 2D discretization . . . . . . . . . . . . . . . . . . . . 25
6.2.1
Discussion of the results . . . . . . . . . . . . . . . . . . . . . 26
6.3
Results from parametric study of piles in sands and gravel . . . . . . 28
6.4
Discussion in the context of practical use . . . . . . . . . . . . . . . . 29
7 Conclusions and suggestions
31
Bibliography
33
x
Nomenclature
Coecient of compressibility
Consolidation index
Compression index associated with the soil
Swell index associated with the soil
Undrained shear stress
Coecient of consolidation
Pile diameter
Bending stiness
Initial void ratio
Void ratio after primary consolidation
Height
Empirical constant
Hydraulic conductivity
Subgrade reaction
Normal force
Yield stress of soil
Settlement
Total settlement of the soil due to primary consolidation
Time
Lateral displacement of the soil
Displacement of the pile
Longitudinal displacement of the soil
Distance from surface
xi
av [-]
Cα [-]
Cc [-]
Cs [-]
cu [kPa]
cv [-]
d [m]
EI [Nm2 ]
e0 [%]
ep [%]
H [m]
J [-]
k [cm/sec]
ku [N/m3 ]
N [kN]
pu [Pa]
s [m]
Sc [m]
t [sec]
ug [m]
u [m]
wg [m]
z [m]
Greek
Angle depending on the internal angle of friction
Angle depending on the internal angle of friction
Eective unit weight of soil
Dierence in void ratio
Unit weight of water
Eective overburden pressure
Strain at half of the maximum stress
Internal angle of friction
Eective pressure
Initial eective pressure
Preconsolidation eective pressure
Shear stress
Angle of the pile
xii
α [0 ]
β [0 ]
γ [kN/m3 ]
∆e [-]
0
∆σ [kN/m3 ]
∆σ 0 [kPa]
ε50 [-]
φ [0 ]
0
σ [kPa]
σ00 [kPa]
σc0 [kPa]
τ [N/m]
θ [0 ]
Chapter 1
Introduction
Sweden's relatively unique post-glacial soils scenario with shallow, soft clays close
to sti moraine or bedrock allows the use of very slender piles. As most common
buildings will experience horizontal loads, whether it is from wind loads on a building or hydrostatic loading on a quay, the slender piles used must be inclined as the
lateral resistance of the vertical piles is extremely low. However, if the soil settles
the relative movement of the soil and the inclined pile will result in a lateral force
against the pile. This force causes bending moments that reduce the carrying capacity of the inclined pile. Some other countries use inclined piles, e.g. in (Tomlinson
and Woodward, 2008) it is explained that inclined piles should be installed at the
largest possible angle. It is also explained that inclined piles should not be used due
to the large induced bending moments when the soil is expected to settle.
The current method by Svahn and Alen (2006) calculates the induced bending moments by estimating the total settlement of the soil and apply the relative displacement between the soil and the pile using a simplied beam-spring model. Compared
to a eld test by Takahashi (1985) the measured bending moments are smaller than
estimated from the calculation model used today. The source of the dierence between the analytical model and the eld test is unknown but, if found, would mean
that the individual pile carrying capacity could be greatly increased.
To get a better understanding of the problem a 3D nite element has been created and compared to the eld test by Takahashi (1985). To create a non-linear
nite element model is however a very complex and time-consuming process why
a 2D discretization can be of great use in design. This thesis reviews two more
advanced 2D calculation methods from the scientic literature. These two are compared to the current recommended design model (PKR 101), a previously completed
eld test, and a 3D nite element model.
1
Chapter 2
Presentation of the current Swedish
calculation model
The general model of a pile in moving soil is based on a beam on an elastic foundation (Svahn and Alen, 2006) and is illustrated in Figure 2.2. The settlement, s, is
divided in a transversal component ug and a longitudinal component wg using the
pile inclination θ, Figure 2.1. In the model, the transversal displacement component
is connected to the pile using springs placed at an innitely small distance.
_
___ ___ /// __
_
/// __ /// __ _ ///
/// _ /// _ /// _ ///
_
_
_
/// s __ ///
_ ///
///
s
Figure 2.1: Illustration of how the settlement is divided into the longitudinal and
transversal components as explained above
3
CHAPTER 2.
PRESENTATION OF THE CURRENT SWEDISH CALCULATION MODEL
Figure 2.2: General representation of the components in the calculation model
The force distribution in the beam on an elastic foundation in Figure 2.2 can be
found by dividing the beam in innitely small elements. This gives the dierential
equation, Equation (2.1) which has a homogeneous solution showed in Equation 2.2
(Larsson and Wiberg, 1988).
EI uIV + N u00 + d ku u = d ku ug − N u00i
(2.1)
u(z) = a1 eλ1 z + a2 eλ2 z + a3 eλ3 z + a4 eλ4 z
(2.2)
where
ku is the subgrade reaction
d is the diameter of the pile
ug is the displacement of the soil
N is the normal force
and
u is the displacement of the pile
Furthermore, the parameters a1−4 are integration constants and λ1−4 can be solved
using the characteristic equation for Equation 2.1. Particular solutions for Equation
2.1 are given by Svahn and Alen (2006) and the homogeneous solution is solved
using boundary conditions for the specic case.
If the normal force is assumed to be constant along the pile Equation 2.1 can be
simplied to Equation 2.3.
EI uIV = d ku (ug − u)
4
(2.3)
2.1. DISCUSSION OF THE CALCULATION MODEL
As can be seen in Figure 2.3 most of the relative displacement occurs in the top of
the pile and as the bending moment is a function of the displacement.
Displacement
of the soil
Displacement
of the pile
Difference in
displacement
Figure 2.3: Illustration of the soil reaction over the length of the pile
2.1 Discussion of the calculation model
In the design calculation model suggested by Svahn and Alen (2006) the soil is assumed to have a subgrade reaction along the entire pile. This follows the method
described above, where the formulation of this behaviour is described in Equation
2.3. With this formulation comes some limitations, e.g. only one type of soil can
be used for the entire length of the pile. Another limitation is that the settlement
prole is xed and does not always represent the actual settlement. For further
details regarding the model see Svahn and Alen (2006).
5
Chapter 3
Simplied models of soil
As this thesis concern the movement of soil some soil models are explained in this
chapter. As soil can displace for various reasons it is explained why some movement
does not cause any change in the stress of the soil whilst some do. This was used in
the modelling of the soil in the 3D-model.
3.1 Consolidation of soil
Soil exposed to a load will settle and this settlement can be separated into three main
mechanisms. Elastic settlement is instantaneous and occurs without change in the
moisture content, this mainly happens in sands and gravels. Primary consolidation
is when the pore water is pressed out from the soil skeleton. Secondary consolidation
or creep is when the soil skeleton is deformed over time. The two latter processes
probably occur simultaneously (Larsson, 2008).
3.1.1
Primary consolidation
Primary consolidation occurs when the pore water is redistributed due to the increase is pressure from a load. Initially the entire load is carried by the increase in
pore water pressure. However, due to the local increase of pressure the pore water
will be redistributed, decreasing the volume of the aected soil, and the load carried
by the soil particles will gradually increase.
The speed of this process depends on the hydraulic conductivity of the soil and
is basically instantaneous for sands and gravel and is mostly a very slow process for
a clay (Larsson, 2008).
When calculating the total settlement from primary consolidation it is assumed that
the soil particles are incompressible and the consolidation is due to the rearrangement of particles when pore water disperse. The total consolidation depends on the
7
CHAPTER 3.
SIMPLIFIED MODELS OF SOIL
previous loads the soil has been subducted to and the new load.
For a normally consolidated clay the total settlement can be calculated according
to Equation 3.1 (Bjerrum, 1967).
Sc =
σ 0 + ∆σ 0 Cc H
log 0 0
1 + e0
σ0
(3.1)
where
Sc is the total settlement of the soil due to primary consolidation
Cc is the compression index associated with the soil
H is the total height
e0 is the initial void ratio
σ00 is the initial eective pressure
and
∆σ 0 is the eective overburden pressure
For an overconsolidated clay the total settlement can be calculated according to
Equation 3.2 if σ00 + ∆σ 0 ≤ σc0 or Equation 3.3 if σ00 + ∆σ 0 ≥ σc0 (Bjerrum, 1967).
σ 0 + ∆σ 0 Cs H
Sc =
log 0 0
1 + e0
σ0
(3.2)
σ0 σ 0 + ∆σ 0 Cc H
Cs H
c
log 0 +
log 0 0
Sc =
1 + e0
σ0
1 + e0
σc
(3.3)
where
Cs is the swell index associated with the soil
and
σc0 is the pre consolidation eective pressure
To calculate the settlement over time Terzaghi (1925) proposed a dierential equation based on one dimensional consolidation which is based on the following assumptions:
1.
2.
3.
4.
5.
6.
The clay-water system is homogeneous
Saturation is complete
Compressibility of water is negligible
Compressibility of soil particles is negligible
The water ows in one direction only
Darcy's law is valid
8
3.1. CONSOLIDATION OF SOIL
With these assumptions the settlement over time can be described as:
δu
δ2u
= cv 2
δt
δz
(3.4)
where cv is the coecient of consolidation and can be described using Darcy's law
as:
k
(3.5)
cv = av
γw 1+e
0
where
k is the hydraulic conductivity
γw is the unit weight of water
av is the coecient of compressibility
and
e0 is the initial void ratio
3.1.2
Secondary consolidation
Secondary consolidation, also known as creep, is a time bound compression under
constant stress and so slow that it does not eect the pore pressure. There are
two creep hypotheses, A and B, for consolidation suggested by Ladd et al. (1977).
Hypothesis A suggests that creep starts at the end of the primary consolidation and
hypothesis B suggests that creep starts directly when the load is applied.
The process can have dierent under laying mechanisms depending on the soil,
for a clay it can for example be the rearrangement of soil particles and for friction
soils it can for example be because of the decay of organic materials (Larsson, 2008).
When the primary consolidation has nished the rate of the secondary consolidation can be described as a linear function of the void ratio, e, against the time in a
logarithmic scale (Bjerrum, 1967). This gives a consolidation index:
Cα =
∆e
∆e
=
log(t1 ) − log(t2 )
log tt12
(3.6)
Using this equation the magnitude of the total settlement of the secondary consolidation can be calculated according to Equation 3.7.
t 1
Ss = Cα0 H log
(3.7)
t2
Cα
where Cα0 = 1+e
p
and
ep is the void ratio at the end of primary consolidation.
9
CHAPTER 3.
SIMPLIFIED MODELS OF SOIL
3.2 Elastic-plastic models of soil
An elastic material subjected to a load deforms elastically and returns to its former
shape when the load is removed. A plastic material however will remain deformed
after the load is removed. Taylor and Quinney (1932) developed a plasticity theory
for metals. According to this theory a plasticity model should contain (1) a yield
criterion (2) a strain hardening rule determining the behaviour after yielding and
(3) a plastic ow rule determining the direction of the plastic strain.
Soil has an elastic and a plastic behaviour continuously. This means that the soil will
have plastic deformations even for small strains but still have some elastic properties
(Schoeld and Wroth, 1968). Figure 3.1 shows a typical stress-strain relationship for
soil, the two paths after yield are stress hardening and stress softening. In some cases
this behaviour can, roughly, be idealized with a perfect elastoplastic model using a
constant Young's modulus for the elastic part. To model this a p-y relationship can
be used which will be described in the next chapter.
3.3 Beam-Spring model (p-y curves)
To determine the deformations due to interaction between the soil and the pile, a
modulus ku for the Winkler bed has to be dened. A perfectly elastic plastic soil
can be described according to Equation 3.8 will render a work curve that can be
seen in Figure 3.1.
(
p=
if p ≤ pu
else
ku u
pu
(3.8)
where
u is the deection of the pile
and
pu is the yield stress of the soil
This is a simplied way to describe the otherwise complex behaviour of the soil
and the behaviour diers depending on the soil type.
10
3.3. BEAM-SPRING MODEL (P-Y CURVES)
s
s
Yield
e
e
Figure 3.1: Two dierent soil models, to the left two dierent soils showing hardening
and softening yield behaviour, and to the right a perfect elastic plastic
behaviour
3.3.1
P-y curves for clay
A p-y, or with the denominations in this thesis p-u, relationship for clay during
undrained loading was proposed by Matlock (1970) where the relationship can be
described as Equation 3.11. This equation gives a more realistic description but is
more complex to model.
(
p=
0.5
u 2.5 ε50 d
if p ≤ pu
(3.9)
else
pu
where
u is the deection
ε50 is the strain at half of the maximum stress
d is the pile diameter
and
pu is the yield stress of the soil
The value for ε50 can be evaluated from laboratory tests and an approximate value
is given in Table 3.1.
Clay at shallow depths has a lower limit for plasticity and a relationship can according to Matlock (1970) be calculated as of Equation 3.10. Svahn and Alen (2006)
however suggests that the values should be reduced for long term loading with a
factor of 1.5.
Table 3.1: Approximate values for ε50 associated with the undrained shear strength
of clay (Matlock, 1970), these values are also used in Svahn and Alen
(2006)
cu [kN/m3 ]
ε50 [-]
10-25
0.02
25-50
0.01
50-100
0.007
11
100-200
0.005
200-400
0.004
CHAPTER 3.
SIMPLIFIED MODELS OF SOIL
(
Nc = min
(3 +
J
γ
z + z)
cu
d
(3.10)
9
where
γ is the eective unit weight of soil
z is the distance from the surface
and
J is an empirical constant
3.3.2
P-y curves for drained material conditions
A formulation of p-y curves for drained materials is described by Reese et al. (1974)
which are based on analytical solutions with empirical evidence and are similar
to the left diagram in Figure 3.1. Svahn and Alen (2006) however suggests that
these curves can be simplied to perfect elastic plastic curves that varies with depth
described by Equation 3.11 and in Figure 3.2.
1
p(z) = ku z
2
(3.11)
where
ku is a constant according to Table 3.2
and
z is the distance from the surface
The plastic limit stress for the soil can be calculated as pu = 3 tan2 (45 + φ2 ) σv ,
where φ is the friction angle and σv is the vertical stress, according to Svahn and
Alen (2006). An illustration of a p-y curve used in friction soils can be seen in Figure
3.2.
Table 3.2: Recommended values for ku as taken from Reese et al. (1974)
Relative density
ku above ground water level [MN/m3 ]
ku below ground water level [MN/m3 ]
12
Loose
7
5
Medium
24
16
Dense
61
34
3.4. DRAINED AND UNDRAINED CONDITIONS
p
pu
ku
u
Figure 3.2: Work curve used for a perfect elastic-plastic material
3.4 Drained and undrained conditions
Depending on the type of soil and rate of loading the conditions can be considered
drained or undrained. The total stress can be calculated using the eective stress,
σ 0 , and pore pressure, p, as σ = σ 0 + p (Terzaghi, 1925). If the soil has a low permeability and the loading rate is fast the conditions should be considered undrained,
as the load will be carried by an increase of the pore pressure. For a longer time
period drained conditions can be used and the eective stress can be calculated as
σ 0 = σ − p. This formulation can be useful for modelling soil in FE software.
3.5 Mohr-Coulomb yield criterion
The Mohr-Coulomb yield criterion originally formulated by Coulomb (1776) states
a yield function depending on the shear stress τ , the cohesion c, the eective stress
σ 0 , and the friction angle φ. It can be formulated as
τf = c + σ 0 tanφ
(3.12)
For drained conditions the yield stress will increase with an increase of total stress
as the eective stress will increase accordingly. For undrained conditions the pore
pressure will increase but the eective stress will remain not giving any increase of
the shear stress failure condition.
13
Chapter 4
Previous studies of inclined piles
Early studies were performed by Broms (1964a) and (1964b) where dierent failure
modes are described depending on the length of the pile. Also several studies covering the lateral displacement between a pile and soil e.g. by Poulus (2006) and by
shearbox tests, where a vertical pile stands in a movable soil medium, to test pure
lateral displacement of soil by Guo and Ghee (2004), Lin et al. (2014), and Qin and
Guo (2010), it can be seen in these tests that a maximum pressure against the pile
is reached at a relative displacement of the soil depending on dierent factors such
as the prole of the moving soil. In a report by Broms and Fredriksson (1976) a
dierent analytical solution is investigated which could be useful if a varied subgrade
reaction is to be used in an analytical model.
One of the most impartant previous studies is a full scale eld test conducted by
Takahashi (1985) which is described in the next chapter with comparisons to the
results from the current calculation method.
4.1 Analysis of the eld test
In the full scale experiment, presented by Takahashi (1985) four 38.7 m long steel
piles with a diameter of 508 mm and a thickness of 9 mm were installed as two
pairs hinged one meter above ground level, of which one pair was asphalt coated.
The piles were installed at an angle of 15 degrees with strain gauges attached on
both sides of the piles at 10 points along the pile to measure both bending moment
and axial force. The full setup can be seen in Figure 4.1. The report suggests a
calculation method where the pile is divided into four parts; a free portion, a load
layer, and two dierent subgrade reactions due to the clear change in the settlement
prole, this calculation model is further investigated later in this report.
A comparison between the measured bending moment along the pile and those
calculated according to PKR 101 can be seen in Figure 4.3 where the analytical
model overestimates the maximum bending moment by approximately 250%.
15
CHAPTER 4.
PREVIOUS STUDIES OF INCLINED PILES
Figure 4.1: Sketch of the eld tests and soil properties of the ground with settlement
distribution
300
Maximum bending moment [kNm]
Pile 1
Pile 2
250
200
150
100
50
0
0
5
10
15
20
Settlement of ground surface [cm]
Figure 4.2: Maximum bending moment versus settlement of ground surface
16
4.2. 2D DISCRETIZATION
0
Distance from surface [m]
5
PKR101
Measured
10
15
20
25
30
35
40
-100
0
100
200
300
400
Bending moment [kNm]
500
600
700
Figure 4.3: Maximum bending moment versus settlement of ground surface
4.2 2D discretization
The 2D analysis was conducted in the commercial software Brigade plus 5.2. This
chapter describes the model used to simplify the 3D model into a 2D model. Two
dierent theories were tested and the results were compared against the results
from the 3D model. To represent the dierent failure mode the springs were dened
to have a lower maximum when relative movement between the soil and the pile
occurred in the direction against the surface. By applying the load this way an
otherwise iterative process of nding the exact depth of where the load layer ends
and the subgrade reaction begins can be reduced to a single iteration. The spring
model used is illustrated in Figure 4.4.
17
CHAPTER 4.
PREVIOUS STUDIES OF INCLINED PILES
p
pu
ku
u
pu
Figure 4.4: Work curve used to model the dierent type of failure in the top part of
the pile
4.2.1
Distributed load approach
Based on methods found in (Takahashi, 1985) the soil is divided into two parts; a
distributed load part, and a subgrade reaction. The load is applied as a function of
the pile width and is limited to the subgrade reaction of the soil so that no load will
be applied if the displacement of the pile is equal to the displacement of the soil.
The formulation of this behaviour is described in Equation 4.1. This discretization
assumes that the soil goes to failure and therefore becomes a load hanging on the
top part of the pile.
EI uIV = d ku (ug − u) ≤ 3 d γz sin(θ)
where
d is the diameter of the pile
γ is the weight of the soil
θ is the angle of the pile
and
z is the distance from the surface
18
(4.1)
4.2. 2D DISCRETIZATION
4.2.2
Wedge failure approach
Based on a failure mode described by Reese et al. (1974) the top part of the soil is
considered as a cone like failure observed in the 3D model. As in the distributed
load approach the soil is divided into a distributed load, and a subsoil reaction.
The load is suggested to grow with the weight of the cone in Figure 4.5. Furthermore only the sinusoidal part of the weight will act lateral to the pile causing
bending moment why only this part of the weight will be take into account. The
total load acting on the pile can therefore be described as Equation 4.2.
1
1
P (z) = γαz 3 tan(β) + γz 2 tan(β)d
3
2
(4.2)
From this the lateral pressure distribution can be derived as Equation 4.3.
p(z) = γαz 2 tan(β) + γz tan(β)d
(4.3)
where
γ is the heaviness of the soil
α is φ2
z is the distance from the surface
β is 45 + φ2
d is the diameter of the pile
and
φ is the internal angle of friction
The formulation used for this behaviour is described in Equation 4.4.
EI uIV = d ku (ug − u) ≤ sin(θ)(γαz 2 tan(β) + γz tan(β)d)
a
b
d
Figure 4.5: Illustration of the volume described in the wedge failure
19
(4.4)
CHAPTER 4.
4.2.3
PREVIOUS STUDIES OF INCLINED PILES
Discussion about calculation models
With the knowledge that failure always occurs as the weakest failure mode it is clear
that the soil could fail in two dierent ways depending on the depth of the soil. The
distributed load approach considers these two dierent failure modes. The method
suggested in PKR101 might highly overestimate the surface pressure against the pile
as it assumes a failure mode where the soil allows to push with a force greater than
the weight of the overlaying soil. It should also be noted that the settlement prole
suggested in PKR101 implies that 40% of the total settlement occurs in the bottom
of the soil, given this it might be more appropriate to t the settlement prole to
the calculated settlement.
20
Chapter 5
Simulation with 3D nite element
model
5.1 Validation of 3D model
A 3D nite element model was used to validate the 2D-models. The analysis was
performed in the commercial software Abaqus V6.13 (Systemes, 2015). This chapter
describes the model used to validate the model against the eld study and the setup
for the parametric study in 3D and 2D nite element simulations.
5.1.1
Geometry, boundary conditions and FE discretization
Figure 5.1 represents the geometry of the model where a symmetry plane has been
used to save computational time. The boundary conditions were set so that no
displacement normal to the surface will occur except for the top surface which was
free to move and the plane of symmetry where symmetrical conditions were applied. The soil was modelled as 3D solid elements (C3D8 and C3D4) and the pile
was modelled as shell elements (S4) to save computational time. The settlement
was modelled using orthotropic temperature dependency hence causing stress free
settlements to represent the settlement prole from the experiment. The pile's top
was restrained to move in the horizontal direction to represent the pinned condition
from the study. For details about the elements and information about the software
detailed information can be found in the online documentation (Systemes, 2015).
Interaction between the pile and the surrounding soil was modelled using penalty
type interface. For the normal behaviour a small pretension was applied between
the soil and the pile by changing the clearance when contact pressure is zero and
for the tangential behavior a friction coecient of 0.385 was assumed according to
(Helwany, 2007). The general setup of the model can be seen in Figure 4.1.
21
CHAPTER 5.
SIMULATION WITH 3D FINITE ELEMENT MODEL
Figure 5.1: Basic geometry of the model and stress distribution before simulation
5.1.2
Material properties
The behaviour of the steel is assumed to be linear elastic and a Young's modulus of
200 GPa was assumed and Poisson's ratio was set to 0.3.
The soil was modelled with parameters according to Trakverket (2011a). The
elastic properties of the soil was assumed to be linear and isotropic with a Young's
modulus calculated as 250cu for the clay and 50 MPa for the embankment. The
plastic behaviour was modelled using mohr-coulomb plasticity with a friction angle
of 45 degrees for the embankment and an alternating value for the clay, this value
however had very low eect on the results. As settlement in this case per deniton
occurs due to the dissapation of water drained parameters were used for the clay
body. Furthermore a Poisson's ratio of 0.3 was assumed.
5.1.3
Loading
The entire model was subjected to a predened geostatic stress and a gravity load.
With these in eect a temperature load created a settlement prole to replicate the
eld test.
5.2 Formulation for parametric study
To further compare the 2D models to the 3D model a parametric study was performed for a friction soil. The 3D model was used but the settlement prole was
changed so that the settlement increased linearly over the entire depth. The density
of the soil was set to 0.5, 1.2, and 2 t/m3 to represent lls above and below ground
22
5.2. FORMULATION FOR PARAMETRIC STUDY
water level. The friction angle was set to 25, 35, and 45 degrees, to cover most
options, for a total of 9 combinations, see Table 5.1. The Young's modulus was
set to 50 MPa for the entire depth and a cohesion of 2 kPa was used to prevent
numerical problems close to the surface. A total settlement of 25 cm was induced
and the maximum bending moments compared to the results from the dierent 2D
models.
The coecient of subgrade reaction was chosen as 7 MN/m3 increasing linearly
with the depth and limited to 49MN/m3 for all 2D cases (Trakverket, 2011b).
Table 5.1: Studied cases in 3D FEM
φ [deg] \γ [kg/m3 ]
25
35
45
500
x
x
x
23
1200
x
x
x
2000
x
Unable to nish
x
Chapter 6
Results and discussion
6.1 Results from the validation of the 3D model
Figure 6.1 shows the bending moment against the ground surface settlement, s. It
can be seen that the results are very close to the measured values and it is therefore
assumed that the model replicate the eld test. From the results it can be seen that
a wedge like failure mode occurs as shown in Figure 6.2 indicating that a dierent
failure mechanism occurs in the rst few meters than in the rest of the pile.
6.2 Results from the 2D discretization
Figure 6.3 shows the bending moment plotted against the ground surface settlement.
It can be seen that the model as described in PKR-101 gives a maximum bending
moment far greater than obtained in the 3D model.
For this case both the distributed load and the wedge failure aproach gives very
accurate results as compared with the 3D model and the eld test. Figure 6.3 shows
only the maximum obtained bending moment over the pile, the spot where this
occurs changes depending on the settlement of the ground surface.
25
CHAPTER 6.
RESULTS AND DISCUSSION
Maximum bending moment [kNm]
300
Pile 1
Pile 2
3D model
250
200
150
100
50
0
0
5
10
15
20
Settlement of ground surface [cm]
Figure 6.1: Maximum bending moment versus settlement of ground surface
6.2.1
Discussion of the results
A clear dierence can be seen in the results between the current method of calculating the lateral force and the two suggested methods of calculation. It should be
said that the only dierence between these models is the way the top 2.5 meters
are applied. The dierence between the two more accurate solutions is that the soil
either works as a distributed load instead of as a subgrade reaction.
26
6.2. RESULTS FROM THE 2D DISCRETIZATION
Maximum bending moment [kNm]
Figure 6.2: Plastic zones in the completed nite element solution indicating a dierent failure mode in the top of the pile
Distributed load
Wedge failure
PKR 101
3D model
600
500
400
300
200
100
0
0
5
10
15
20
Settlement of ground surface [cm]
Figure 6.3: Maximum bending moment in pile vs settlement of surface using measured settlement prole
27
CHAPTER 6.
RESULTS AND DISCUSSION
6.3 Results from parametric study of piles in sands
and gravel
Bending moment normalized against 35 degrees and 1.2 t/m3 3D model
From the results in Figure 6.4 it can be seen that the three models gives results that
diers depending on the friction angle, φ, and the weight of the soil, γ . What can be
seen is that the distributed load model gives the same results regardless of the friction angle but gives a maximum bending moment closer to the one obtained in the
3D models in comparison to the wedge failure method. All three ways of calculating
the reaction of the pile give results on the safe side, given that the 3D-model can be
assumed to be the most accurate. By using the current way of calculation can yield
a bending moment of over 3 times the value obtained from the 3D simulation.
[-]
4
F=45
F=35
3.5
PKR 101
Wedge failure
Distributed load
3D model
F=25
3
F=45
2.5
F=35
2
F=25
1.5
1
0.5
F=45
F=35
F=25
0
0.5
F=45
F=35
F=45
F=25
F=25
1
1.5
Heaviness of the soil
2
[t/m 3 ]
Figure 6.4: Maximum bending moment in pile divided with the maximum bending
moment for a friction angle of 35 degrees and a weight of 1.2 t/m3 plotted
against the weight of the soil
28
6.4. DISCUSSION IN THE CONTEXT OF PRACTICAL USE
6.4 Discussion in the context of practical use
Common for the results is that all show far lower induced bending moments than
expected from the current calculation model. In fact, the bending moment calculated using the model from (Svahn and Alen, 2006) the plastic limit stress from the
steel is reached well before the settlement has reached its maximum rendering the
pile useless, as predicted by Tomlinson and Woodward (2008) whom suggests that
inclined piles should be avoided in settling soils. This is however not the case for
the measured case, the 3D-model and the suggested models whereas much of the
structural capacity remain.
Another reection from this study is that a stier pile will increase the induced
bending moments, so some problems might not be solved in the pile design by
choosing a larger diameter. This will also work in favour for the lower bending moments from this study as a weaker pile can be chosen and therefore less of the pile
capacity will be lost due to the settlements.
29
Chapter 7
Conclusions and suggestions
From the 3D model and the eld test it can be concluded that the current calculation model overestimates the induced bending moment by up to 3.5 times for the
tested cases. The main nding in this report is that a dierent type of failure occurs
in the top of the soil limiting the soil pressure against the pile. Worth noting is that
it is only the top few meters of the soil that will contribute to the bending moments
as this is where all the relative displacement occurs. Of two tested failure modes
both shows good resemblance in the eld test. In the parametric study it can be
seen that the best resemblance is that of a distributed load, described in the report,
why this approach is the most suitable to use.
One nding worth mentioning is that the assumed settlement prole might yield
too large relative displacements. This is due to the exponential formulation, the assumption in itself might not be incorrect but when the parameter λ is calculated as
in the example 40% of the settlement occurs in the bottom part of the pile. Instead
the settlement prole should be calculated to give an as representative model of the
ground calculations as possible.
Another suggestion is that a 2D nite element model should be used to calculate
the bending moment in design. This is due to the fact that the model as formulated
today only can consider one type of soil where the soil parameters can be very hard
to choose, especially when combining clay and friction materials.
With the expectation of a smaller induced bending moments, and the economical aspect of using slender inclined piles, the use of inclined piles might be the best
option in settling soils even though the loss of structural capacity.
31
Bibliography
Bjerrum, L., 1967. Engineering geology of norwegian normally-consolidated marine
clays as related to settlements of buildings.
Broms, B., Fredriksson, A., 1976. Failure of pile-supported structures caused by
settlements. In: 6th European Conference on Soil Mechanics and Foundation
Engineering.
Broms, B. B., 1964a. Lateral resistance of piles in cohesive soils. Journal of the Soil
Mechanics and Foundations Division 90 (2), 2764.
Broms, B. B., 1964b. Lateral resistance of piles in cohesionless soils. Journal of the
Soil Mechanics and Foundations Division 90 (3), 123158.
Coulomb, C., 1776. Essai sur une application des regles des maximis et minimis a
quelquels problemesde statique relatifs, a la architecture. Tech. rep., Mem. Acad.
Roy. Div. Sav.
Guo, W., Ghee, E., 2004. Model tests on single piles in sand subjected to lateral soil
movement. Tech. rep., Grith University.
Helwany, S., 2007. Applied soil mechanics with Abaqus application.
Ladd, C., Foott, R., Ishihara, K., Schlosser, F., Poulos, H. G., 1977. Stressdeformation and strength characteristics. state-of-the-art report. In: Proc. 9th
ICSMFE.
Larsson, R., 2008. Jords egenskaper. Tech. rep., Statens geotekniska institut.
Larsson, R., Wiberg, N.-E., 1988. Stability of elastic plane frames including soilstructure interaction. Computers & structures 29 (5), 845855.
Lin, H., Ni, L., Suleiman, M. T., Raich, A., 2014. Interaction between laterally
loaded pile and surrounding soil. Journal of geotechnical and geoenvironmental
engineering.
Matlock, H., 1970. Correlations for design of laterally loaded piles in soft clay. In:
Oshore Technology Conference.
Poulus, H. G., 2006. Raked piles-virtues and drawbacks. Journal of geotechnical and
geoenvironmental engineering, 795803.
33
BIBLIOGRAPHY
Qin, H. Y., Guo, W., 2010. Pile responses due to lateral soil movement of uniform
abd triangular proles. Tech. rep., Grith University.
Reese, L., Cox, W., Koop, F., 1974. Analysis of laterally loaded piles in sand. In:
Oshore Technology Conference.
Schoeld, A., Wroth, P., 1968. Critical state soil mechanics.
Svahn, P.-O., Alen, C., 2006. Transversalbelastade pålar - statiskt verknigssatt och
dimensioneringsanvisningar. rapport 101. Tech. rep., Pålkommissionen.
Systemes, D., 2015. Abaqus 6.14 online documentation: 1.1 the abaqus prod- ucts.
URL http://server-ifb147.ethz.ch:2080/v6.14/books/gsa/default.htm
Takahashi, K., 1985. Bending of a batter pile due to ground settlement. Soils and
foundations 25, 7591.
Taylor, G. I., Quinney, H., 1932. The plastic distortion of metals. Philosophical
Transactions of the Royal Society of London. Series A, Containing Papers of a
Mathematical or Physical Character 230, 323362.
Terzaghi, K., 1925. Erdbaumechanik auf Bodenphysikalischer Grundlager.
Tomlinson, M., Woodward, J., 2008. Pile design and construction practice. Taylor
and Francis.
Trakverket, 2011a. Trakverkets tekniska krav for geokonstruktioner. Tech. rep.,
Trakverket.
Trakverket, 2011b. Trakverkets råd bro. Tech. rep., Trakverket.
34