Behaviour and Analysis of Large Diameter Laterally Loaded Piles
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Transcript of Behaviour and Analysis of Large Diameter Laterally Loaded Piles
DEPARTMENT OF CIVIL ENGINEERING
The work described in this report is our own unaided effort, as are all the text, tables and diagrams except where clearly referenced to others.
Behaviour and Analysis of Large Diameter Laterally
Loaded Piles
Henry Pik Yap Sia
Tram Le Bao Dang
Supervisor: Dr Andrea Diambra
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Dissertation presented as part of, and in accordance with, the requirements for the Final
Degree of BEng at the University of Bristol, Faculty of Engineering.
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no information derived from it may be published without the author’s prior consent.
Author
Henry Pik Yap Sia
Tram Le Bao Dang
Title Behaviour and Analysis of Large Diameter Laterally Loaded
Piles
Date of submission 23 April 2015
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This dissertation/thesis is the property of the University of Bristol Library and may only be
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Abstract
75% of UK offshore wind turbines are supported on monopile foundations (Doherty and Gavin,
2012). The piles are subjected to large lateral loading from wind and tide surges as well as
seabed movement. British Standards (BS EN 61400-3:2009) suggested p-y curve to predict the
behaviour of laterally loaded offshore piles. P-y curve has certain assumptions including
negligible rotational resistance along the pile length.
This report presents our investigation on the effect of rotational resistance on a typical large
diameter pile. It also describes how the finite difference (FD) program has been written from
first principles, the Winkler’s Method and Euler-Bernoulli Beam theory. To calculate the
rotational resistance, the slice method proposed by McVay and Niraula (2004) is implemented
in our model. Our linear-elastic FD model calculates the displacement, bending moment, shear
force and soil pressure for laterally loaded piles for two cases: (a) when rotational resistance is
considered and (b) when rotational resistance is neglected. The later represents the values used
in the industry.
Sensitivity study, through our model produced good results within its scope. The results
suggested that the change in the soil and pile properties was found to be dependent on the
length-to-depth (L/D) ratio of the pile and the stiffness of the soil next to the pile. In other
words, when reached critical ratio, the rotational resistance becomes very significant,
specifically for short, rigid piles. Therefore, we computed curves to recommend the range of
L/D values where rotational resistance can be safely neglected.
Recommendations and suggestions are made to improve the model and research to fully
encapsulate the behaviour of offshore monopiles, such as cyclic loading, elastic continuum,
plasticity and non-linearity.
Lastly, we have sufficient confidence from this research to conclude that rotational resistance
of a laterally loaded large diameter pile are important and that current design standards for
offshore monopiles are conservative.
Throughout, this report, symbols used are compiled and defined in page 3.
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Table of Contents
Abstract ..................................................................................................................................... 1
Table of Contents ..................................................................................................................... 2
1. Notation .............................................................................................................................. 3
2. Introduction ....................................................................................................................... 4
3. Objectives ........................................................................................................................... 5
4. Behaviour of Laterally Loaded Piles ............................................................................... 5
4.1. Winkler Method ......................................................................................................... 6
4.2. Long versus Short Piles ............................................................................................. 7
4.3. Rotational Resistance ................................................................................................ 9
5. Methods of Analysis ........................................................................................................ 12
5.1. Euler-Bernoulli Beam Theory ................................................................................ 12
5.2. Rotational Springs ................................................................................................... 14
5.3. Finite Difference Method ........................................................................................ 15
6. Assumptions Made .......................................................................................................... 15
7. Equation/Code Breakdown ............................................................................................ 17
7.1. Input Parameters ..................................................................................................... 17
7.2. Pile Geometry ........................................................................................................... 18
7.3. Rotational Stiffness (Slice Method) ........................................................................ 19
7.4. Boundary Conditions .............................................................................................. 20
8. Validation of Method ...................................................................................................... 21
8.1. Number of Nodes ..................................................................................................... 22
8.2. Accuracy of Model ................................................................................................... 23
9. Sensitivity Analysis ......................................................................................................... 25
9.1. Length-to-Diameter Ratio ....................................................................................... 25
9.1.1. Lateral Displacement ....................................................................................... 28
9.1.2. Bending Moment .............................................................................................. 29
9.1.3. Soil Pressure ...................................................................................................... 29
9.1.4. Pile Rotation ...................................................................................................... 29
9.1.5. Conclusions ....................................................................................................... 30
9.2. Relative Stiffness ...................................................................................................... 30
9.2.1. Conclusions ....................................................................................................... 32
9.3. Lateral Load ............................................................................................................. 33
10. Discussion ..................................................................................................................... 33
10.1. Impact of Diameter .............................................................................................. 33
10.2. Critical Ratio ........................................................................................................ 34
10.3. Validity of Slice Method ...................................................................................... 35
11. Recommendation ......................................................................................................... 36
12. Conclusion .................................................................................................................... 37
13. Appendix ...................................................................................................................... 38
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A.1. Derivations of boundary conditions .......................................................................... 38
A.2. Example of MATLAB output .................................................................................... 39
14. References .................................................................................................................... 40
1. Notation
𝜏𝑖 Shear stress
a Slope of t-z curve
C Arc length of the slice
D Pile diameter
EI Flexural rigidity
Epile Young’s modulus of the pile
Esoil Young’s modulus of the soil
F Lateral force
h Height of small element
KR Rotational stiffness
Ks Horizontal modulus of subgrade
reaction ( soil spring stiffness)
L Pile length
M Bending moment
Ms Moment due to side shear
p Soil pressure
Px Axial compression force
q Uniformly distributed load
R Pile rotation
S Shear force
T Side shear force
x Depth of small element
y Horizontal displacement
z Axial displacement
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2. Introduction
The increasing pressure to reduce carbon dioxide emission has enabled the wind energy sector
to emerge. The UK Government is aiming for the offshore wind industry to produce £100/MWh
of electricity by 2020 (Offshore Wind Works, 2014). This leads energy from UK offshore wind
farms to increase dramatically over the last decade, by more than 10 times than in 2014
(11,998MW) than 2004 (The Wind Power, 2015). According to the UK Trade & Investment
(2014), the offshore wind energy sector represent the largest expansion of renewable energy
generation in the UK.
There are different types of offshore foundations as illustrated in Figure 1.1 and monopiles
accounted for over 75% of the existing offshore foundation (Doherty and Gavin, 2012).
Monopiles are generally stiff steel tube support structures with large diameter 4-6m and up to
35m pile length embedded into the seabed. Therefore, the length-to-depth ratio for a typical
monopile used to date could range from 5-6. For example, the Dogger Bank Teesside wind
farms use monopiles of 8m in diameter (Forewind, 2012).
Figure 2.1: Types of offshore wind turbine foundations (Doherty and Gavin, 2012).
The British Standards (BS EN 61400-3:2009) for offshore wind turbines provides extensive
guidelines for static and dynamic loading from wave, seabed movement, scour, etc. and pile
installation methodology. However, the standard referred ISO 19902:2007 for the design of
laterally loaded piles, which were originally developed for petroleum and natural gas industries.
The design of monopiles in UK is popularly based on the American Petroleum Industry (API)
(API, 2011), which uses the slightly modified p-y method developed by Reese (1974) to predict
the behaviour of laterally loaded piles. Moreover, the API method (2011) was developed with
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the assumption that the shear resistance along the length of a pile and base shear at the tip of
pile are negligible. Past and recent researches highlighted in Section 4 indicate further research
is required.
3. Objectives
This research reviews the role of rotational resistance in a monopile. A Finite Difference (FD)
model was produced using MATLAB for a laterally loaded pile by implementing springs. The
model will calculate internal reactions of the pile and the soil beside the pile, i.e. lateral
displacement, soil pressure, pile rotation, bending moment and shear force.
The baseline results (without rotational springs) from our FD model is validated against
commercial software, OASYS Alp. Then, rotational springs is implemented parallel to the p-y
springs. Sensitivity analysis is carried out to investigate the effect of diameter and other
parameters on the rotational resistance along a pile length.
The main objectives of this research project are:
To determine the influence of rotational resistance of a typical large diameter pile
To make recommendations for the length-to-diameter ratio for offshore piled foundation
design.
Due to time constraints, the scope of this research project is limited to the following features:
a. 2D model
b. Linear elastic pile
c. Elastic soil foundation
d. Single soil layer
The concept of non-linearity and plasticity are not implemented in our model, as they are not
the focus area of our research and would add further complexity to our analysis.
4. Behaviour of Laterally Loaded Piles
The function of a pile is to transfer moment, shear and axial load from the structure above to
the soil medium at some depth. There are many different methods of analysis proposed to study
the behaviour of laterally loaded piles. Some of those methods include Broms (1964) and
Winkler (1887). These methods assume variation of the subgrade modulus with depth and linear
elastic soil. Despite limitations of these assumptions, the methods are widely used to find the
solutions for laterally loaded piles.
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4.1. Winkler Method
The theory of the Winkler model (1887) is an elastic beam (pile) resting on a series of uncoupled
springs (p-y springs), which model the soil reactions. In early studies, analysis of laterally
loaded piles was based on the elastic beam foundation theory introduced by Hetenyi (1946).
Reese and Matlock (1956) suggested the implementation of independent non-linear springs for
accuracy (see Figure 4.1) and hence the Winkler model. Reese (1974) empirically derived from
experimental results and developed the p-y curves, equated by the formula:
p = ksy (1)
p is soil reaction, 𝑘𝑠 is soil stiffness and y is lateral displacement of the pile. The p-y curve was
derived on field tests on small diameter piles of length-to-diameter ratio of 34 (Cox, Reese and
Grubbs, 1974).
Figure 4.1: Winkler model representation (Doherty and Gavin, 2012).
Using the method suggested by Reese and Grubbs (1974) for the analysis, figure 4.2 below
shows different p-y curves for 3 different type of soils: loose, medium and dense sand.
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Figure 4.2: P-y curves empirically derived (Doherty and Gavin, 2012; Cox, Reese and Grubbs, 1974).
4.2. Long versus Short Piles
The failure mechanism of a pile can be classified accordingly by long or short pile, in terms of
length-to-diameter ratio as their behaviour differs for one another. Tomlison (2001) described
a short pile as a pile with less than 10-12 length-to-diameter ratio. Karatzia and Mylonakis
(2012) used linear regression analysis to derive a more precise mathematical expression,
Equation (2) to calculate the length-to-diameter ratio for given pile and soil properties.
Length𝑝𝑖𝑙𝑒
Diameter𝑝𝑖𝑙𝑒> (0.1 − 0.7 log ε) (
Epile
Esoil)
0.25
(2)
Tolerance parameter, ε: 10%, as suggested by Karatzia and Mylonakis (2012). In simplified
form, Equation (3) can be as follows:
𝐿
𝐷> (
𝐸𝑝𝑖𝑙𝑒
𝐸𝑠𝑜𝑖𝑙)
0.25
(3)
When a force is applied to the free head pile, in addition to the passive soil resistance at the
opposite side of the head pile, the toe passive resistance also forms at the force acting side. This
result in exceeding the optimum values for passive resistance and the pile will start to rotate as
shown in Figure 4.3a. For a fixed head short pile as seen in Figure 4.3b, the failure of the pile
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would be translational instead of rotation. Figure 4.4 illustrates a typical bending moment
profile for a short, rigid pile.
Figure 4.3: Failure Mechanism of short rigid pile under horizontal load (Tomlinson,
2001).
(a) Free head (b) Fixed head
Figure 4.4: Bending
moment diagram
for free head short,
rigid pile (Poulos
and Davids, 1980).
The failure mechanism of long pile is different at the bottom of the pile. The pile acts in a
flexible manner as the cummulative passive resistance at the pile bottom is much higher, when
compared to short, rigid piles (Tomlinson, 2001). Hence, long pile could not rotate as freely as
a short pile. In this case, long pile fracture is more likely to occur at the point of yielding
moment, i.e. maximum moment as shown in Figure 4.5a for free head long pile and Figure 4.5b
for fixed head long pile. Figure 4.6 shows maximum bending moment takes place at the fracture
point of a long pile.
Figure 4.5: Failure Mechanism of long flexible pile under horizontal
load (Tomlinson, 2001)
(a) Free head (b) Fixed head
Figure 4.6: Bending momoet
diagram for free head long
pile (Poulos and Davids,
1980).
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4.3. Rotational Resistance
The reactions of a typical pile subjected to lateral
loading and embedded in soil are listed in Table 4.1,
which also includes the method used to characterise
them. Figure 4.7 graphically represented the beam-
column model of a typical laterally loaded pile.
Table 4.1: Soil resistance in a beam column model (Lam and
Martin, 1986).
In Table 4.1, items (4) to (7) for piled foundations are assumed to be small enough to be
negligible. Researchers have argued that these assumptions, particularly item (5), rotational
resistance along the length of pile are invalid for large diameter piles or monopiles of offshore
structures such as wind turbines. This is because the diameter of the pile is large enough for a
moment arm and hence induces shear rotational moment along the pile length and tip. This
would significantly affect the internal reactions of a pile and the soil beside the pile.
No. Soil Resistance Characterised by
1 Lateral translational
resistance
p-y curve
2 Axial frictional resistance
along the pile length
t-z curve
3 Axial translational end-
bearing resistance at the tip
of shaft
Q-z curve
4 Torsional rotation about
the axis of pile
Assumed
negligible
5 Rotational resistance along
the length of pile
Assumed
negligible
6 Rotational resistance at the
tip of a pile
Assumed
negligible
7 Lateral translational
resistance from shearing at
the tip of the pile
Assumed
negligible
Figure 4.7: Beam-column model and
the soil resistance (Lam and Martin,
1986).
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Lam and Martin (1986) suggested that monopiles developed resistance when laterally loaded.
They discussed the role of rotational resistance in a pile as well as conventional p-y curves
through the back fitting analysis of field test data. Following that, they implemented moment-
rotation springs parallel to the p-y springs to improve accuracy and came out with the
conclusions that p-y curve was conservative. The results (see Figure 4.8a and Figure 4.8b) were
over-estimated pile deflection values for when rotational springs were considered.
Georgiadis and Butterfield (1982) proposed a method to analyse nonlinear shear coupling by
incorporating rotational springs using the Pasternak model (see Figure 4.9). The model
introduced interspring shear layer to calculate the shear coupling along the pile length. From
the predicted and measured results from plate bearing tests, it was suggested that rotational
resistance along the pile length had a considerable practical value to the pile behaviour.
Figure 4.9: Pasternak model (Horvath, 1984).
McVay and Niraula (2004) wrote a paper on this topic and presented their evaluations through
back computing of field data through FBPIER - a geotechnical software. The ‘slice method’
Figure 4.8a: Backfitting analysis for clay (Lam and
Martin, 1986). Figure 4.8b: Backfitting analysis for sand
(Lam and Martin, 1986).
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was proposed to calculate an improved prediction of the moment due rotational resistance along
the pile length. Experimental data obtained were analysed to study the influence of rotational
resistance on large diameter piles. When side shearing was considered, p-y curves experienced
reductions (see Figure 4.10a and Figure 4.10b) as low as 6% for weak rocks and 23% difference
for high strength rocks, Florida limestone (McVay and Niraula, 2004).
Further analysis of different diameter piles in strong rocks implied that the reductions of p-y
curve ranged from 21% to 26% (see Figure 4.11a and Figure 4.11b). These findings were
reasonably encouraging for our research. They concluded the reduction in p-y curve was
evidently from the lateral resistance and that p-y curves alone could lead to much higher lateral
and rotation displacements solutions, particularly for clay soil, compared to sandy site. This
also implies a relationship of soil stiffness to rotational stiffness.
Figure 4.10: P-y curve for a pile with L/D = 3 for
(a) strong rock and (b) weak rock (McVay and Niraula, 2004)
(b) (a)
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Figure 4.11: Reconstructed p-y curves for 80 ksf
(a) D = 3feet and (b) D = 12feet (McVay and Niraula, 2004)
5. Methods of Analysis
The chapter describes the theory and principles adopted to write the Finite Difference model
to analyse laterally loaded piles.
5.1. Euler-Bernoulli Beam Theory
For Winkler model, the pile is placed on a series of springs to model elastic foundation. To
derive a relationship between the internal actions (bending moment, M and shear force, S) and
the applied load, a small element of a pile (Figure 5.1) was considered to be resting on spring
and use equilibrium to obtain Equation (7). The positive sign conventions are defined in Figure
5.2.
Figure 5.2: Positive sign
conventions for the small pile
element.
s
Figure 5.1: The forces in a small element of a pile.
(a) (b)
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Forces in vertical equilibrium and moment about the top right of the elements give the following
equations:
∑ fy = 0, S + Ksydx = S + dS + qdx (4)
q − ksy =dS
dx (5)
∑ M = 0, M +1
2qdx2 + Sdx = M + dM +
1
2ksydx2 (6)
S =dM
dx (7)
1
2qdx2 and
1
2k𝑠ydx2 are negligible. Equation (5) and (7) are combined to obtain Euler-Bernoulli
beam equation, Equation (9). With this, the equilibrium equation representing the small element
can be expressed as follow.
q − ksy = dS
dx=
d2M
dx= EIy′′′′ (8)
𝑦: horizontal displacement, S: shear force, M: bending moment, EI: flexural rigidity, ks: soil
spring stiffness, dx: element length and q: load per unit length
Table 5.1: Euler-Bernoulli beam equation derivatives.
𝐄𝐮𝐥𝐥𝐞𝐫 − 𝐁𝐞𝐫𝐧𝐨𝐮𝐥𝐥𝐢 𝐛𝐞𝐚𝐦 𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧: d2
dx2 (EId2y
dx2) = q (9)
Soil reaction, P = EId4y
dx (10)
Shear force, S = EId3y
dx (11)
Bending moment, M = EId2y
dx (12)
Rotation, R = dy
dx (13)
The fourth-order differential equation, Equation (8) is similar to the elastic beam equation
introduced by Hetenyi (1946), Equation (14).
p = EId4y
dx4 , where p = −ksy (14)
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5.2. Rotational Springs
In order to model the skin friction of the pile, McVay and Niraula (2004) proposed that the
rotational springs, Ms- φ to be added parallel to the p-y spring (see Figure 5.3) as shown in
Equation (15)
Figure 5.3: Modified beam-column model (McVay and Niraula, 2004).
S = dM
dx− Krdy (15)
dS
dx=
d2M
dx2 − Krd2y
dx2 (16)
Kr: rotational stiffness, θ: rotation of pile from original position and dx: length of pile element.
Equation (16) is substituted into Equation (8) to get Equation (17)
EId4w
dx4 − Krd2w
dx2 + ksw = 0 (17)
Hetenyi (1946) analysed elastically supported beams and derived an equation for axial
compression, which is also known as the beam-column equation (see Figure 5.4). The equation
has a good representation of the shear forces along the length of a pile. The beam-column
equation, Equation (18) represents the rotational springs along the pile length.
EI d4y
dx4+ Px
d2y
dx2+ ksy = 0 (18)
Px : Axial compression force.
Figure 5.4: Element from beam-column (Hetenyi, 1946)
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5.3. Finite Difference Method
Finite difference (FD) method was used to carry out the analysis of theory outlined above
because of its ability to approximate derivatives and solve boundary value problems. Equation
(18) is inputted in finite difference form and FD method is capable of replacing a derivative or
differential equation with an approximation based on the value of that function at surrounding
points. One advantage of finite difference method is the accuracy from using central difference
approximation and kernels (nodes), meaning more springs could be applied for smoother
graphs.
6. Assumptions Made
In addition to the model features mentioned in Section 3, the water table is assumed to be at
very deep depth.
The p-y spring that characterised the lateral displacement of the pile is assumed to be linearly
elastic. In our model, only elastic condition (see Figure 6.1) for soil was considered. The soil
stiffness value, KS, is calculated from Equation (19), as suggested by Oasys Alp manual (2013):
kS = Efact x Esoil (19)
Efact is the factor suggested by Broms (1972) and Poulos (1971) and Esoil is the Young’s Modulus
of the soil. 0.8 is the suggested value for clay and 1 for other soils.
Figure 6.1: Linear elastic p-y curve of our Finite Difference model.
In addition, according to API method (2011), the slope of t-z curve, 𝑎 is a constant characteristic
value for linear soil. However, when the elastic-plastic soil condition is considered, different
secant values for 𝑎 need to be adopted in the analysis as shown in Figure 6.2.
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Figure 6.2: API method (2011) t-z curve.
Using the slice method to calcualte the rotational stiffness (detailed method mentioned in 7.3.),
the graph below shows the difference in KR along the pile with various pile diameters. As can
be seen from the graph, after the pile reaches its yield state, the rotational stiffness of the pile
starts to decrease and this rate is slower as the diameter of the pile increase, as shown in Figure
6.3. However, it should be noted that according to API (2011), the pile only yields when the
axial displacement z reaches 0.01D, which is unlikely to happen in our analysis.
Moreover, the change in rotation has to be significant in order to ensure that the difference in
rotational stiffness can be recognised. From the initial analysis, the difference in rotation along
the pile is not significant (compared to 0.01D), the changes in KR can be expected to be
negligible for the analysis.
Figure 6.3: Normalised Kr versus rotation curves.
Hence, to avoid the complexity of the code, only elastic soil is used to study the influence of
rotational stiffness (see Figure 6.4).
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Figure 6.4: t-z curve for our Finite Difference model.
Another assumption made was the coefficient of t-z curve are the same for all soil types. This
is not true but the process of evaluating the value of τmax for any soil types has been a
challenge. Hence, we decided to use a conservative coefficient value of 10,000.
7. Equation/Code Breakdown
This chapter covers the steps taken to develop code and outlines the equations implemented in
our MATLAB script for our analysis.
7.1. Input Parameters
The finite difference model is expressed as per metre depth. Table 7.1 lists the input parameters
for our model to analyse laterally loaded piles.
Table 7.1: Input parameters for Finite Difference model.
Symbols Units Descriptions
Lpile m Length of the pile
Dpile m Diameter of the pile
NN Dimensionless Number of pile elements
F kN Applied force
N Dimensionless Number of slice elements
a kN/m3 Slope of t-z curve
ES kN/m2 Soil Young’s modulus
τmax kN/m2 Maximum shear stress
N Dimensionless Number of slice (for rotational stiffness)
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7.2. Pile Geometry
The beam-column equation, Equation (18) requires fourth order differentiation to calculate the
soil-structure responses. This calls for the central difference approximation formulas, derived
from the Taylor series expansion. Table 7.2 lists the formulas for first to fourth derivatives
implemented in our MATLAB script to approximate the derivatives.
Table 7.2: Centred finite-divided difference formulas (Chapra and Canale, 2010)
First
Derivative f′(xi) =1
h[−
1
2f(xi−1) + 0f(xi) +
1
2f(xi+1)]
(20)
Second
Derivative f′′(xi) =1
h2[f(xi−1) − 2f(xi) + f(xi+1)]
(21)
Third
Derivative f′′′(xi) =1
h3[−
1
2f(xi−2) + f(xi−1) + 0f(xi) − f(xi+1) +
1
2f(xi+2)]
(22)
Fourth
Derivative f′′′′(xi) =1
h4[f(xi−2) − 4f(xi−1) + 6f(xi) − 4f(xi+1) + f(xi+2)]
(23)
Two fictitious elements are added at the top and bottom of the total pile elements, as shown in
Figure 7.1 to accommodate the extra 4 spaces (elements) from the second order central
difference approximation. The model below has NN+4 elements and NN+5 nodes.
Figure 7.1: Graphical representation of finite difference method.
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7.3. Rotational Stiffness (Slice Method)
Along the whole length of the pile, the lateral loading produces rotational movement. In order
to allow the optimal resistance necessary for the pile construction, the rotational stiffness of the
pile has to be considered in the analysis. In this particular project, the slice method is chose as
a suitable approach to compute the moment due to side shear.
One of the features of the slice method is that at any depth, the cross section can be divided into
N slices to improve the accuracy of the calculated moment due to side shear, MS, of the pile
(see Figure 7.2).
Figure 7.2: Cross-section of the pile divided into N slices.
The side shear force, Ti, on the pile can be derived as a function of shear stress, 𝜏𝑖 and the arc
length of the slice, Ci, Equation (24).
Ti = τi × Ci (24)
The arc length of any i slice section can be calculated by the following equations:
The angle of i slice, α = arccos (i−1
N) − arccos (
i
N) (25)
Arc length of i slice, Ci = R × α = D
2 × [arccos (
i−1
N) − arccos (
i
N)] (26)
In addition, assuming that the axial displacement, z is known, the shear stress τi can then be
obtained from the t-z curve, Equation (28).
From pile geometry, zi = xi θ (27)
As only elastic soil condition is considered in this project, the slope of t-z curve, a, is constant.
τi = 𝑎 zi = 𝑎 xi θ (28)
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The lever arm of each slice, Ri, is given by, Equation (29).
Ri = xi = [D
2 x N × (1 + 2 × (i − 1)) ] (29)
Finally, the moment due to side shear at any cross section of the pile can then be expressed as
the sum of the product of the side shear force, Ti and the distance from the center of the pile to
the center of the slice, Ri:
Moment due to side shear, Ms = ∑ τiRii Ci (30)
On the other hand, from Ms = Krθ (31)
Hence, it can be deduced that the rotational spring of the pile is dependent on the pile diameter,
D, and the soil characteristic, a value from t-z curve.
Rotational stiffness, KR = 2 𝑎 xi Ri Ci
= a × [D
2 x N (1 + 2(i − 1)) ]
2
× [ arcos (i−1
N) − arcos (
i
N) ] × D (32)
7.4. Boundary Conditions
Four boundary conditions are required to solve the finite difference system. Reese (1977)
documented his program he developed for the analysis of laterally load piles using finite
difference method. In our research, we modified his formulas to accommodate for the rotational
springs implemented in our model. The pile is modelled to be in a pinned-free condition for
realistic analysis as the pile is subjected to displacement at both ends (see Figure 7.3). The
formula for bending moment at the top has to be modified to account for the moment due to
side shear. The full derivations can be found in the Appendix A.1.
The boundary conditions at the pile bottom are that the bending moment and shear force are
zero. The other two boundary conditions are moment and shear force F at the pile top. The
following equations are the finite difference form of the boundary conditions:
Pile top:
Bending moment: M = Mt
[ yx−1(−hKR + 2EI) + yx(−4EI) + yx+1(hKR + 2EI) ] = 0 (33)
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Shear force: S = F
1
2[ yx−2 (
EI
h3 ) + yx−1 (2EI
h3 +KR
h ) + 0 − yx+1 (
2EI
h3 +KR
h ) + yx+2 (
EI
h3 ) ] = F (34)
Pile bottom:
Bending moment: M = 0
[ yx−1 − 2 yx + yx+1 ] = 0 (35)
Shear force: S = 0
1
2[ yx−2 (
EI
h3 ) + yx−1 (
2EI
h3+
KR
h ) + 0 + y (−
2EI
h3−
KR
h ) + yx+2 (
EI
h3 ) ] = 0 (36)
F: Applied lateral force at the pile top, Mt: Moment due to rotational resistance
Figure 7.3: Graphical representation of boundary conditions.
8. Validation of Method
The coded finite difference model using MATLAB is validated against OASYS Alp, a
geotechnical pile software. OASYS Alp is designed to analyse soil-structure interaction of
laterally loaded piles. It has been used widely in the industry by consulting engineering firms
such as Byrne Looby Partners and Arup for geotechnical pile analysis such as reinforcement
depth and pile performance (Oasys Ltd, 2012).
The software calculates earth pressures, horizontal movements, bending moments and shear
forces based on lateral loads, moments or displacements, lateral and rotational restraints, or soil
load deflection behaviour. OASYS Alp models the pile as a series of elastic beam elements
through a series of non-linear springs and non-linear soil behaviour through p-y curves instead
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of the unrealistic elastic-plastic. However, only the elastic-plastic model is used for our baseline
comparison. Examples of displacement, bending moment, shear force, soil pressure and pile
rotation profiles are illustrated in Appendix A.2.
8.1. Number of Nodes
The pile model is divided into many small elements. OASYS Alp manual (2013) suggested that
20-30 nodes in their software would be sufficient for the accurate solution. Certainly, a high
order of nodes in our model could improve the accuracy but it should be stressed that too many
nodes can also result in mathematical error. Therefore, the study of the relationship of number
of nodes in our model is carried out. The input for this investigation is from Table 8.1.
Figure 8.1 displays how the solutions from OASYS Alp varied with different number of nodes.
61 nodes was used as the baseline for the comparison.
Table 8.1: Input parameters for elements accuracy testing.
Figure 8.1: Percentage difference versus number of elements in OASYS Alp.
As seen from the plot, when more than 20 nodes were used, the percentage differences were
less than 1%. It was observed that the differences of the results tend to zero after approximately
40 nodes. Hence, for convenience, 21 nodes in Alp was used to test against our MATLAB
model as shown in Figure 8.2.
0
1
2
3
4
5
6
0 10 20 30 40 50 60 70
Per
cen
tage
dif
fere
nce
, %
Number of nodes
MaxDisplacement
Max rotation
Max pressure
Max Moment
Max Shear
Force (kN) Pile Length (m) Pile diameter (m) EI (kNm2) Efact Esoil (kN/m²)
500 20 3 125.79e+6 0.8 20,000
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The accuracy increased until 2000 nodes, follows by the scattered points as nodes increases.
When around 7000 nodes were used, the difference increased compared to the case with 6000
nodes, this could be due to mathematical equations being unable to converge. 2000 nodes would
suffice for the analysis to balance mathematical accuracy and MATLAB running time.
Figure 8.2: Accuracy of Finite Difference model when compared against OASYS Alp.
8.2. Accuracy of Model
Same soil and pile properties as well as applied load were inputted in OASYS Alp and analysed
in our Finite Difference model. Two cases were validated:
(a) Without rotational spring (as aforementioned in Table 8.1.)
(b) With rotational spring (1 rotational spring applied at the top of the pile as listed in Table
8.2.)
Table 8.2: Input parameters for case (b)
The results of the analysis are shown the Table 8.3 and 8.4. It should be noted that the values
shown in these 2 tables were rounded to the nearest 4 decimal places whenever possible but
some difference may not be clear for some cases.
Force
(kN)
Pile
Length
(m)
Pile
diameter (m)
EI
(kNm2)
Efact Esoil
(kN/m²)
KR
(kNm/rad)
No. of
Nodes
1,500 60 3 125.79e+6 0.8 20,000 300,000 5,000
● Maximum
Displacement
● Maximum
Moment
● Maximum
Shear Force
● Maximum
Rotation
● Maximum
Pressure
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Table 8.3: MATLAB code validation results for case (a).
Reactions MAXIMUM MINIMUM
Alp MATLAB Difference
%
Alp MATLAB Difference
%
Displacement
(mm)
2.8845 2.8849 0.01 -6.51710 -6.49780 -0.30
Rotation
(rad)
-0.0004 -0.0004 -0.24 -0.00055 -0.00055 -0.07
Pressure
(kN/m2)
15.3840 15.3870 0.02 -
34.75800
-34.65492 -0.30
Moment
(kNm)
1428.600 1438.4000 0.69 0.00000 0.00000 0.00
Shear Force
(kN)
157.9500 159.9500 1.27 -
500.0000
-500.0000 0.00
Table 8.4: MATLAB code validation results for case (b).
Reactions MAXIMUM MINIMUM
Alp MATLAB Difference
%
Alp MATLAB Difference
%
Displacement
(mm)
0.94419 0.97173 2.92 -13.846 -14.021 -1.26
Rotation
(rad)
4.9E-05 5.04E-05 2.44 -0.0010 -0.0011 -7.54
Pressure
(kN/m2)
5.0357 5.1825 2.92 -73.847 -74.778 -1.26
Moment
(kNm)
6223.3 6434.4 3.39 0 0 0.00
Shear Force
(kN)
300.24 311.6235 3.79 -1500.00 -1502.100 -0.14
Our Finite Difference code performed very well, correctly calculating the displacement,
rotation, bending moment, shears force and soil pressure. Without rotational stiffness added,
the Finite Difference system in MATLAB had very similar results, which were only 1.27%
maximum, compared to ALP.
In the case with added rotational stiffness, results from MATLAB code are more noticeably
different than the previous case, the maximum difference was 7.54% for the rotation at the top,
a part from that the difference is approximately 1%. In our code, to increase the accuracy, large
number of nodes needs to be used. This difference in height of element could be the reason why
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our code and ALP did not have more similar result as in the case without rotational stiffness.
Moreover, it should be stressed that ALP has a poor representation of rotational springs, as it is
better represented as ‘restraints’.
Despite that, we could reasonably conclude that our model has high order of accuracy when
compared to commercial pile software.
9. Sensitivity Analysis
In this chapter, through various examples, sensitivity analysis is carried out. Two parameters
length-to-diameter ratio and the relative stiffness are studied with and without the consideration
of rotational resistance along the pile length. The respective lateral displacement, bending
moment, rotation and soil pressures are also studied in this chapter.
9.1. Length-to-Diameter Ratio
The purpose of this analysis to investigate the transition of a laterally loaded pile from long to
short and its respective effect on the lateral displacement, bending moment, rotation and soil
pressure variations for two cases:
(a) Rotational resistance is considered
(b) Rotational resistance is ignored
The pile slenderness ratio or the length-to-diameter ratio is calculated using Equation (2)
(Karatzia and Mylonakis, 2012) to determine the set of diameters for infinitely long piles and
short piles. For this analysis, a 40m steel pile is embedded in a single sand layer with pile
diameters varied from 0.5m to 100m. Table 9.1 shows the input parameters of the finite
difference model. Figure 9.1-9.4 are the results of the analysis for four reactions: lateral
displacement, bending moment, soil pressure and rotation.
The formula for the change in reaction is listed below:
Change in reaction (%) = (Maximum Reaction without Rotational Resistance - Maximum
Reaction with Rotational Resistance)/ Maximum Reaction without Rotational Resistance] x
100
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Table 9.1: Input parameters for the analysis of length-to-diameter ratio of a laterally loaded pile.
Variables Input Values Units
Applied Lateral Force 1,000 kN/m
Young’s Modulus of Steel 210 GPa
Flexural Rigidity 4.939 x 1012 kN/m2
Soil Stiffness 100 MPa
Pile Length 40 m
Pile Diameter 0.5 to 100 m
Coefficient of T-z curve 10,000 kN/m3
Number of Pile Elements 2,000 Dimensionless
Number of Slices (for rotational springs) 100 Dimensionless
Figure 9.1a: Force-displacement graph of
a 40m pile with (i) 4.5m and (ii) 23m pile
diameter.
Figure 9.1b: Change in displacement
with increasing diameter for a 40m
pile.
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Figure 9.3b: Change in soil pressure
versus length-to-diameter ratio for a
40m pile.
Figure 9.3a: Normalised soil pressure
versus length-to-diameter ratio curve for a
40m pile.
Figure 9.2a: Normalised bending moment
versus length-to-diameter ratio curve for a
40m pile.
Figure 9.2b: Change in bending moment
versus length-to-diameter ratio for a
40m pile.
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9.1.1. Lateral Displacement
Figure 9.1a shows how the top lateral displacement of different pile diameters (4.5m and 23m)
react to increasing applied lateral load. The followings were noted from Figure 9.1a:
a) Rotational resistance is dependent on pile diameters.
b) Short, rigid pile has greater influence from the rotational resistance.
The extra force from the side shear has resulted in reduction in the lateral displacement. This is
more clearly seen in short, rigid pile or pile with low length-to-diameter ratio.
For the same pile length in a single sand layer, pile diameter is varied from 0.5m to 100m to
illustrate the contribution from rotational resistance in the reduction of top pile displacement as
pile make a transition from long to short. The results are shown in Figure 9.1b.
Clearly, long pile has fairly negligible change in displacement. For example, the change in
displacement is 2.5% when pile diameter is 4.5m (long) and 58.1% when pile diameter is 23m
(short). The followings trends were also found:
a) The influence of rotational resistance becomes more important for pile with low
slenderness ratio, i.e. a very short pile.
Figure 9.4b: Change in rotation versus
length-to-diameter ratio for a 40m pile.
Figure 9.4a: Normalised rotation versus
length-to-diameter ratio curve for a 40m
pile.
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b) When length-to-depth ratio is close to zero, i.e. short and chunky pile, the change in
lateral displacement reached a maximum of 74.7%.
9.1.2. Bending Moment
The bending moment was calculated using equation in Table 5.1. Figure 9.2a shows the ratio
of moment for with and without rotational resistance against length-to-diameter ratios. Figure
9.2b presents the result for increasing length-to-diameter ratio and the respective change in
bending moment due to pile-soil shearing.
Similarly as mentioned in the previous section, with a high slenderness ratio (long pile), the
effect from rotational springs is small. As with lateral displacement, long pile has small change
in maximum bending moment whereas short pile has large change. For example, when pile
diameter is 4.5m (26.7 slenderness ratio), the change in bending moment is 3.7% and at 23m
(1.73 slenderness ratio), the change is 16.2%.
From Figure 9.4a and 9.4b, the following notes were made:
a) The effect of rotational stiffness on the bending moment is not large when compared to
lateral displacement.
b) Very short, rigid pile showed that the change in bending moment due to side shearing
tends to a constant value of 16.6%.
9.1.3. Soil Pressure
Soil pressure is calculated using Equation (10). For a diameter varied from 0.5m to 100m,
normalised soil pressure is plotted against the length-to-diameter ratio as shown in Figure 9.3a.
The response of soil pressure to increasing in length-to-diameter ratio is plotted in Figure 9.3b.
Evidently, the rotational resistance is unsurprisingly influential in short, rigid piles (low length-
to-diameter ratio).
From this study, the followings were also noted:
a) The profiles for soil pressure and lateral displacement were the same.
b) The rotational resistance resulted in a maximum of up to 74.7% decrease in soil
pressure.
9.1.4. Pile Rotation
The rotation of pile was calculated by the first derivative of the Euler-Bernoulli beam equation.
Figure 9.4a shows the plot of normalised rotation against the length-to-diameter ratio and Figure
9.4b the change in pile rotation versus length-to-diameter ratio.
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The magnitude and variations as a result of the effect of shear coupling were larger for rotation
profile. As expected, a very short, rigid pile had large effect from the presence of rotational
resistance along the pile length whereas a long, flexible has smaller decrease in maximum pile
rotation.
The following points were established from the analysis:
a) The profile for rotation has some degree of similarity to the pressure and lateral
displacement.
b) Low length-to-depth ratio resulted in large increase in rotational resistance of up to 97%
change in maximum rotation in pile.
9.1.5. Conclusions
The response of a laterally loaded pile becomes independent of the diameter after a critical
length-to-diameter value is reached. After that, the components start to increase at a faster rate.
The highlighting points of this sensitivity analysis are as follows:
a) For a long, flexible pile, the changes in pile and soil reactions due to side shear along
the pile length do not exceed 5%. It can be safely assumed to be negligible.
b) For a short, rigid pile, rotational resistance caused large decrease in lateral displacement,
soil pressure and pile rotation. The latter had the most effect. Bending moment had the
least.
c) Change in pressure due to rotational resistance is exactly the same as lateral
displacement profile. This is because soil pressure is a function of displacement, as
empirically derived by Reese (1974) in Equation (1).
9.2. Relative Stiffness
From the analysis of length-to-diameter ratio, it is clear that the pile response when rotational
resistance was considered became very critical in a short, rigid pile condition in sand. The aim
of this analysis is to understand whether soil stiffness plays a major role those observations.
For this investigation, we considered a steel pile length of 40m in a range of soil, subjected to
a constant lateral load at the top of the pile of four diameters. Table 9.2 presents the input
parameters of the finite difference model for the analysis.
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Table 9.2: Input parameters for the analysis of relative stiffness of a laterally loaded pile.
Variables Input Values Units
Applied Lateral Force 1,000 kN
Young’s Modulus of Steel 210 GPa
Flexural Rigidity 4.939 x 1012 kN/m2
Soil Stiffness 10 – 300,000 kPa
Pile Length 40 m
Pile Diameter 3, 8, 20, 50 m
Coefficient of T-z curve 10,000 kN/m3
Number of Pile Elements 2000 Dimensionless
Number of Slices (for rotational springs) 100 Dimensionless
Figure 9.5-9.8 show the result of the analysis using our Finite Difference model.
Figure 9.5: Change in lateral displacement versus soil
stiffness for a 40m pile.
Figure 9.6: Change in bending moment versus
relative stiffness for a 40m pile.
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Figure 9.7: Change in soil pressure versus relative
stiffness for a 40m pile. Figure 9.8: Change in rotation versus relative
stiffness for a 40m pile.
9.2.1. Conclusions
From earlier observations, a very short pile experienced much significant rotational resistance
and hence larger change in the reaction. This statement is verified in this analysis by the shifting
of the curves as the length-to-diameter ratio decreases. Again, lateral displacement and soil
pressure had similar profiles, pile rotation had the most influence and bending moment had the
least.
The other observations seen from the analysis were:
a) Rotational resistance had smaller change in reactions for long, flexible pile when
compared to short rigid pile.
b) Further analysis of short, rigid piles showed that at length-to-diameter ratio of 0.8 (or
pile diameter of 50m and length of 40m), the reactions capped at a certain value (see
Table 9.3) despite of the increasing soil stiffness.
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Table 9.3: Maximum change due to rotational stiffness for 40m pile.
Reactions Percentage Change due to Rotational Resistance (%)
Lateral Displacement 75
Soil Pressure 75
Pile Rotation 100
Bending Moment 17
9.3. Lateral Load
During the investigation length-to-depth ratio and relative stiffness, it was found that lateral
load would affect the magnitude of the pile and soil reactions but not the percentage change in
those properties.
10. Discussion
Multiple discussion points have appeared during the development of the finite difference code
from the first principles to analyse laterally loaded piles.
10.1. Impact of Diameter
Our study has established that rotational resistance is important, particularly for short, rigid
piles with large diameters and conservatively negligible for long, flexible piles. This is because
short piles (low length-to-depth ratio) have large diameters to provide the moment arm for side
shear coupling and hence large reduction in the reactions.
Further analysis with relative stiffness showed that stiff soils would lead to less rotational
resistance effect in a pile. Moreover, this indication became clearer with repeated analysis on
increasing length-to-diameter ratio. A short, rigid pile would experience less rotational
resistance along the pile length in a stiffer soil than a soft soil.
An interesting observation seen was relative stiffness of order of 10,000, short, rigid piles (low
length-to-depth ratio), the rotational resistance would not cause reduction of the reactions
exceeding the values in Table 9.3.
The profile of displacement, soil pressure and rotation were obviously greater than bending
moment. Therefore, we could see that bending moments are not very sensitive to soil stiffness
and diameter variations. Hence, displacement being used as a primary design criteria would be
reasonable.
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Rotational stiffness, (see Equation (32)) is a function of diameter of the pile and hence the
increasing rotational resistance with pile diameter. This meant at a particular length-to-depth
ratio or critical ratio, the side shearing becomes very significant. Therefore, the critical ratio
was studied.
10.2. Critical Ratio
From our finite difference model, we estimated the critical L/D ratios and computed curves for
different soil layers as shown in Figure 10.1.
Three grounds conditions were tested: stiff clay, sand and gravelly sand. In each soil, the pile
diameters were varied from 0.5m to 50m in different pile lengths (0m-60m). The respective
length-to-diameter ratios are calculated from the 10% change in displacement due to rotational
resistance (for example, see Figure 9.2b). The process is repeated for different pile length. Table
10.1 lists the parameters of the test.
Table 10.1: Input parameters for recommendation curves.
Variables Input Values Units
Force 500 kN
Pile diameters 0.5 – 50 m
Pile length 0 – 60 m
Young’s Modulus of Steel 210 GPa
Flexural Rigidity 4.939 x 1012 kN/m2
Soil Stiffness 25, 50, 150 kPa
Coefficient of T-z curve 10,000 kN/m3
Number of Pile Elements 2,000 Dimensionless
Number of Slices (for rotational
springs)
100 Dimensionless
The regions below the curves denote the acceptable range of pile diameters to achieve safe,
negligible rotational resistance, which is up to 10% difference.
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Figure 10.1: Recommended L/D curves for different soil.
10.3. Validity of Slice Method
There are few papers as to date that document methods to compute the rotational spring
stiffness. The value of rotational stiffness is calculated using the slice method proposed by
McVay and Niraula (2004).
Rotational stiffness is a function of the side shear along the pile length. Initially, we had
difficulty determining the coefficient of side shear from t-z curves. To fix the problem, we
implemented a coefficient from the t-z curve (see Figure 7.3) from the API method (2011).
However, further complications encountered with regards to the values of τmax that are
uncertain. This properties cannot easily be obtained directly but must be indirectly estimated.
Arguably, the slice method is developed and more suitable for back-calculated field data, where
the data could provide the value for side shearing, e.g. through strain gauge. Field experimental
data could solve our problem with reasonably accuracy to our analysis.
0
2
4
6
8
10
12
14
16
0 20 40 60 80
Maximum Recommended Pile Diameter
Length
Stiff Clay(25MPa)
Sand(50MPa)
GravelSand(150MPa)
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11. Recommendation
This chapter discusses the ways the pile-soil model could be improved
The data of our analysis has given some convincing results. However, the uncertainty in our
approach was the accuracy of the Winkler’s model. Horvath (1984) said that Winkler’s model
was unrealistic and simple. This is due to the inability of the method to model rotational
resistance. Many of the problems we encountered were evaluating the coefficients for rotational
stiffness and soil stiffness. We recommended the following ways to improve the analysis:
a) Elastic continuum model
During our literature review, the elastic continuum model introduced by Poulos (1971) was
mentioned as an alternative than the Winkler model. Although the elastic continuum may not
be able to comprehend non-linearity, it could eliminate the problem of evaluating model
coefficients and gives more accurate values with reasonably difference than Winkler’s model
(Horvath, 1984). The downside with the elastic continuum method are complexity and time-
consuming.
b) Rotational base resistance
Lam and Niraula (1986) studied the prospects of base shearing induced by lateral loading and
presented some promising results. Time constraints limited our research to only explore the
rotational resistance along the pile length. Having said that diameters of monopiles are large
enough to induce shear coupling, this argument should also implies to base shearing as well
(see Figure 4.7).
c) Non-linearity, plasticity and cyclic loading
Our model was simplistic to focus purely on the effect of rotational resistance on a large
diameter pile. Non-linear analysis, cyclic loading and elasto-plastic were not implemented. The
followings would have been achieved if done otherwise:
Soil stiffness varying with depth – Increase in overburden stresses with depth would
suggest reasonably different predicted reactions (Karatzia and Mylonakis, 2012).
Cyclic loading – Wind turbines are subjected cyclic loading such as wind, wave, current
and tidal, which causes change in stiffness. Considerations into this feature would mean
accumulated displacements and rotations and different result (Doherty and Gavin,
2012).
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Plasticity – As all materials fail, i.e. plastic state, elastic-plastic model would provide
more realistic and accurate analysis.
These features would provide an understanding on how the pile and soil reactions gradually
decrease with depth.
12. Conclusion
A finite difference analytical solution for estimating the behaviour of laterally loaded large
diameter piles in different soil deposits was presented. The method is based on Winkler model
and Euler-Bernoulli beam theory.
The main conclusions of this study are:
a) Length-to-diameter ratio is a critical parameter for its relationship to rotational
resistance of a pile.
b) Piles that are long and flexible, calculated using Equation (3) have minor influence from
rotational resistance and can be safety neglected.
c) Long piles in soils with high stiffness (e.g. sand and rocks) are in agreement with
assumption that rotational resistance is negligible.
d) Short, rigid piles or piles with low length-to-diameter ratio would have significant effect
from rotational resistance.
e) The proposed length-to-diameter curves (see Figure 10.1) can be used for design of piles
against lateral load.
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13. Appendix
A.1. Derivations of boundary conditions
This is the beam-column equation (Hetenyi, 1946):
EId4M
dx4− KR
d2y
dx2+ Ksy = 0
In the finite difference form:
d4M
dx2=
EI
h4[ yx−2 − 4 yx−1 + 6 yx − 4 yx+1 + yx+2 ]
KR
d2y
dx2=
KR
h2[ yx−1 − 2 yx + yx+1 ]
Ksy = [ yx ]
Therefore, the combined finite difference form:
[EIyx−2 + yx−1(−4EI + KRh2) + yx(6EI − 2KRh2 + Ksh4 ) + yx+1(−4EI + KRh2) +
EIyx+2 ] = 0,
Shear force = EId3y
dx3 − KRdy
dx
= 1
2[ yx−2 (
EI
h3 ) + yx−1 (2EI
h3 +KR
h ) + yx(0) + y (−
2EI
h3 −KR
h ) + yx+2 (
EI
h3 ) ]
Rotational stiffness, KR =M
θ
KR = − EI
d2ydx2
dydx
KR = −
EIh2 [ yx−1 − 2 yx + yx+1 ]
12h
[− yx−1 + yx+1 ]
KR = 2EI
h
[ yx−1 − 2 yx + yx+1 ]
[ yx−1 − yx+1 ]
[ yx−1(−hKR + 2EI) + yx(−4EI) + yx+1(hKR + 2EI) ] = 0
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A.2. Example of MATLAB output
Force 500 kN
Pile Length 20 m
Pile Diameter 3 m
ES 20,000 kPa
Epile 210e6 kPa
Slope of t-z curve 10,000 kN/m3
Force 1500 kN
Pile Length 60 m
Pile Diameter 3 m
ES 20,000 kPa
Epile 210e6 kPa
Slope of t-z curve 10,000 kN/m3
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14. References
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York: McGraw-Hill.
Cox. W., R., and Reese. L., C., and Grubbs. B., R., 1974. Field testing of laterally loaded piles
in sand. Proceedings of the 6th Annual Offshore Technology Conference, pp.459–472.
Doherty. P., and Gavin. Kenneth., 2012. Laterally loaded monopile design for offshore wind
farms. Proceedings of the Institution of Civil Engineers, vol. 165, Feb., pp.7-17.
Donnell. M., 2013. Introduction to Finite Differences. University of Bristol.
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20Project%20Description.pdf> [Accessed 23 March 2015].
Georgiadis, M. and Butterfield, R., 1982. Laterally Loaded Pile Behaviour. Journal of the
Geotechnical Engineering Division, American Society of Civil Engineers, vol. 108, no. 1,
Jan., pp.155-165.
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Hetenyi. M., 1946. Beams on elastic foundations: theory with applications in the fields of
civil and mechanical engineering. Baltimore: Waverly Press.
Horvath. J. S., 1984, “Simplified Elastic Continuum Applied to the Laterally Loaded Pile
Problem – Part 1: Theory,” Laterally loaded Deep Foundations: Analysis and Performance,
STP 835, J.A. Langer, E.T. Mosley, and C. D. Thompson, Eds., American Society for Testing
and Materials, pp 112-121.
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