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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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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University of Bristol

1

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.


2

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


4

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.


7

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).


11

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.


16

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).


17

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)


18

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.


19

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)


20

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)


21

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


22

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


23

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


24

Table 8.3: MATLAB code validation results for case (a).

Reactions MAXIMUM MINIMUM

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


%


%

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


25

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


26

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.


27

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.


28

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.


29

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.


30

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.


31

Table 9.2: Input parameters for the analysis of relative stiffness of a laterally loaded pile.


Applied Lateral Force 1,000 kN



Soil Stiffness 10 – 300,000 kPa

Pile Length 40 m

Pile Diameter 3, 8, 20, 50 m


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.


32

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.


33

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.


34

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.


Force 500 kN

Pile diameters 0.5 – 50 m

Pile length 0 – 60 m



Soil Stiffness 25, 50, 150 kPa


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.


35

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)


36

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).


37

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.


38

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


39

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


40

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Behaviour and Analysis of Large Diameter Laterally Loaded Piles

Engineering

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