Full Hussein Algusab Thesis

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Dynamic Analysis of Offshore

Structures Using Finite Element

Method

A THESIS SUBMITTED

TO THE COLLEGE OF ENGINEERING

OF THE UNIVERSITY OF BASRAH

AS A FULFILLMENT OF THE PARTIAL

REQUIRMENTS

FOR THE DEGREE OF MASTER OF SCIENCE

IN CIVIL ENGINEERING

By

Hussein Ali Hussein (B.Sc. Civil Engineering)

March بع لثاني

411 2003







To My Family,My Masterly,

And My Brotherswith Love andRespect.



Acknowledgment

I wish to express my gratitude to Prof. Dr. Anis A. Mohamad

Ali and Assistant Prof. Mr. Mohamad J. K. Essa for their

supervision, advice and support and continuous encouragement

throughout the research work.

Also I am indebted to my family, especially to my parents for

their encouragement, care and patience.

I would like to thank Dr. Assad Saleem the Dean of

Engineering College and Dr. Nabeel Abdul razzaq Jasim the Head

of Civil Engineering Department at the University of Basrah for

the facilities that they offered.

Special thanks due to all members of the staff of civil

engineering Department in Basrah University, previous and

present for their learning, care with love and respect.

Also thanks due to Dr. A. M. Al-khadimey, Dr. Sabih H.

Muhoder, Dr. A. H. Ghailan, Mr. Mugtabba Al-Mudhaffer, Mr.

Samoel M. Al-Salihy, Mr. Abbas O. Dawood, Mr. Saffa K. Geaaz,

Mr. David A. M. Jawad and Mr. Alla A. Lattif for their help.

Hussein Ali Hussein



Abstract

In the present work the three dimensional analysis of offshore structures

are carried out to find the dynamic response of Jacket offshore platforms. A

new exact stiffness matrix is used to model the pile element to consider the

effect of soil-structure interaction. The superstructure members are modeled as

three-dimensional beam element. The dynamic analysis of offshore structures

under the effect of wave loads and ship's berthing impact loads is considered inthe analysis. Newmark direct integration technique is used to solve the dynamic

equilibrium equations by using ANSYS software program. Morison's equation

and Airy's linear wave theory are employed to calculate the wave loads. Added

mass effects also considered in the analysis to account for non-linear inertia term

in Morison's equation. The non-linear drag coefficient effect is neglected in the

analysis. Free and forced vibration analyses are carried out for two case studies.The first case is an actual jacket platform, which is analyzed to wave loads only,

and the second is Al-Amaya Berthing dolphin, which is, analyzed to wave forces

and ship's berthing impact loads. General oriented wave propagation is used in

the analysis of offshore platform and different sea states are considered in the

analysis.



List of Symbols

English Symbols

ACross sectional area of the beam element, system matrix (A=M - .K),

integration constant.

A Integration constant.

[A] Matrix defined in Eq. (A-3-12).

A out Solid cylinder cross-sectional area.

a Water particle acceleration, wave amplitude.

ai Vector of integration constants in Eq. (A-3-13).

a1, a2 Mass and stiffness coefficients in Raylaigh damping formula.

{an(s)} Water particle acceleration field vector along the element.

A p Bearing area.

As Area under shear force (V).

a(s) Water particle acceleration field along the element.

ax, ayComponents of water particle acceleration in the global x and y

directions respectively.

B Integration constant.

B Integration constant.

[B] Strain-displacement matrix.

c dashpots constant

Cd Drag coefficient.

Cm Inertia coefficient.

cx, cy, cz Cosine directions in x, y, and z-directions respectively.

C1, C2, C3,

C4 Integration constants in Eq. (A-3-8).

c )L

x.cos(

c )cos(

ch )L

x.cosh(

hc )cosh(

[C] Damping matrix of the structure.

D Diameter of pile.

[d] Linear operator matrix.

[D] Property matrix.



E Elastic modulus of the element.

Es Elastic modulus of the soil.

Fn Normal hydrodynamic force.

f xi, f yi, f zi Components of hydrodynamic force vector at node i in the global x, y

and z directions respectively.

Fx(s),

Fy(s), Fz(s)Hydrodynamic force components in x, y, and z-direction

{f e} Nodal hydrodynamic force vector.

F* Generalized force in dynamic equilibrium equation.

{Fi} Vector of nodal loads in Eq. (A-3-14)

{F(t)} Force vector of the structure.

{f w (s)} Hydrodynamic force field vector along the element.

G Modulus of torsional rigidity.

Gs Shear Modulus of Soil.

g Acceleration of the gravity

[G] Square matrix defined in eq. (A-3-13).

H Water wave height.

h Water depth.

i 1

I Unit matrix, Second moment of area.

Ix , Iy , IzSecond moment of area of the element cross section in the x, y and z

directions respectively.

J Polar moment of inertia.

k Wave number, Stiffness of foundation.

k1, k2, k3,

k4Partitions of element stiffness matrix.

k n Normal subgrade reaction.

k s Modulus of shear subgrade reaction.

k Φ Modulus of torsional subgrade reaction.

[K e] Element stiffness matrix for beam or pile element.

[K] Stiffness matrix of the structure.

K * Generalized dynamic stiffness matrix in Newmark integration for

dynamic problems.

k Springs Constant.

k 11,…,k 44 Stiffness coefficients.



L Water wave length, or element length.

M Mass of one D. O. F. foundation.

[Me] Element mass matrix.

M(x) Bending moment along the pile.

[M] Mass matrix of the structure.

[N] Shape function matrix.

N1,…,N6 Shape functions.

P Applied normal load, Perimeter of the pile.

P =2.π.R 3

q Pressure reaction due to applied normal load.

{q} Nodal displacement vector.

RPile radius.

s )L

x.sin(

s )sin(

sh )L

x.sinh(

hs )sinh(

s Local coordinate varies along the frame element.

[S] Normalization matrix.

S Unit directional vector along the frame element.

S1 , S2 , S3 Components of the unit directional vector in the global x, y and zdirections respectively.

T Water wave period.T(x) Torsional moment.

T1,…,T10 Elements in stiffness matrix for pile element.

t Time.

[T] Transformation matrix.

u, v, w, θ Displacement at x, y, and z directions, and torsional rotation

respectively.

ui, vi, wi Nodal displacement at node i in the local x, y, and z-directions.

u j, v j, w j Nodal displacement at node j in the local x, y, and z-directions.



u,u,u

Global vectors of structure acceleration, velocity and displacement

respectively.

{ue} Vector of nodal displacements.

,u n

nu ,

nu

Structural acceleration, velocity, and displacement in Morison'sequation.

tu ,

tu ,

tu Structural nodal acceleration, velocity, and displacement at time t.

ttu

,

ttu

,

ttu

Structural nodal acceleration, velocity, and displacement at time t+Δt.

uo, uL Nodal displacements at distance (0, L) respectively.

{u }, {u },

{u }Vector of acceleration, velocity, and displacement of the structure.

)s(vn ,

)s(vn

Velocity and acceleration of fluid particles.

[V] Square matrix defined in Eq. (A-3-12).

V Applied transverse shear force.

v Tangential Displacement.

v(x) Shear deformation along pile element.

{(s)} Water particle velocity field vector along the element.

V(x) Shear force along the pile.{Vn(s)} Hydrodynamic acceleration vector

vx, vzComponents of water particle velocity in the global x and z directions

respectively.

w p Deflection under area (A p).

x, y, z Local coordinates system.

x, y, z, r

Horizontal, vertical, and radial distance of any points on the cross

section of the element from its center respectively.

y(x) Deflection of pile.

S

L Wave length to legs spacing

α

Angle of wave inclination, parameter in Newmark integration

technique, =J.G

P.k .



β =A.E

p.k s

δ Parameter in Newmark integration technique.

(x) Strain at pile material.

{} Element strain vector.

ζ Normalized coordinate variable.

ζ1 Damping ratio for first two modes of vibration.

η Elevation of water above the mean water level.

θ Torsional angle of rotation.

θ(x) Angle of rotation along pile length.

θx, θy, θz Angle of direction cosines in x, y, and z-directions respectively.

θxi, θyi, θzi Nodal rotation at node i.

θxj, θyj, θzj Nodal rotation at node j.

λ i Eigenvalues for free vibration analysis.

υ Poisson's ratio.

ρ Density of element material.

ρf Density of fluid.

σ(x) Stress at pile material.

τs Shear stress tangent to pile length.

τΦ Shear stress along pile perimeter.

Velocity potential.

ω Natural circular frequency, net pressure under elastic beam(pile)

(difference between resisted and applied pressure).

ω1 Fundamental natural frequency.

Ω Circular frequency of the water wave.



Abbreviation

ADINA Automatic Dynamic Incremental Non-linear Analysis.

ASCE American Society for Civil Engineers ASME American Society for Mechanical Engineers ANSYS ANalysis of engineering SYStemsB.Cs. Boundary Conditions

B.M. Bending MomentB.Sc. Bachelor of Science

CPU Central Processing Unitdir. directionD.O.F. Degrees Of Freedom2D two Dimensions3D three DimensionsEq. EquationEqs. EquationsF.E. Finite Element

FORTRAN FORmula TRANslation computer languageFig. FigureFigs. FiguresJ JournalLtd. LimitedMATHCAD MATHematical Computer Aided DesignMax. maximumM.D.O.F. Multi Degrees Of FreedomM.Sc. Master degree in Science

NASTRAN NAsa STructural ANalysisNo. NumberPh.D. Doctor of Philosophy degreeRef. ReferenceS.D.O.M. Single Degree Of Freedom

S.F. Shear ForceSTAAD III STructural Analysis And Design in III dimensions

V Volumew.r.t. with respect to



Contents

CHAPTER ONE :Introduction 1

1-1 Introduction 2

1-2 Dynamic Analysis Requirements 3

1-3 Aim of Study 4

1-4 Layout of the Thesis 5

CHAPTER TWO :Literature Review 6

2-1 Introduction 7

2-2 Analysis of offshore structures 7

2-3 Soil-Structure Interaction 11

2-4 Summary 17

CHAPTER THREE :Soil-Structure Interaction 18

3-1 Introduction 19

3-2 Dynamic Behaviour of Piles 19

3-3 Modeling of Soil-Pile system 20

3-4 Winkler Model 21

3-4-1 Normal Reaction Modulus 22

3-4-2 Shear Reaction Modulus 23

3-4-3 Torsion Reaction Modulus24

CHAPTER FOUR : Modeling and Mathematical Formulation 27

4-1 Finite Element Formulations 28

4-1-2 Element Stiffness Matrix 29

4-1-3 Stiffness of Three Dimensional Beam Element 29

4-1-4 Element Stiffness matrix for a Pile 34

4-1-5 Element Mass Matrix 36

4-1-5-1 Consistent Mass Approximation 36

4-1-5-2 Lumped Mass Approximation 37



4-1-5-3 Effect of Added Mass Formulation 37

4-1-6 Element Damping Matrix 37

4-1-7 Transformation Matrix 38

4-1-8 Element Force Vector 39

4-2 Hydrodynamic of Water Waves 42

4-2-1 Airy Linear Wave Theory 45

4-3 Fender Impact Forces 47

4-4 Dynamic Analysis 48

4-4-1 Free Vibration Analysis 29

4-4-2 Forced Vibration Analysis 50

CHAPTER FIVE :Applications, Results and Discussion 52

5-1 Introduction 53

5-2 Case Study (1): Jacket Platform Type 53

5-2-1 Soil-Structure Interaction 53



5-2-4 Results and Discussion 63

5-3 Case Study (2): Al-Amaya Berthing Dolphin 63

5-3-1 Soil-Structure Interaction 63



CHAPTER SIX :Conclusions and Recommendations 106

6-1 Conclusions 107

6-2 Recommendations 109



Chapter One

Introduction



Chapter One Introduction

2

Introduction1-1 General:

The term offshore is usually taken to mean that part of the ocean where

the present mud line is below the level of the lowest astronomical tide [1].

There are two basic types of structures, these are gravity and pile

supported structures, the choice of the material depends on the type of the

structure, but in general steel is used for pile- supported structures, where as

concrete for gravity structures, although a combination of steel and concrete

structures has been considered [1].

In a pile – supported offshore structure which is also called a jacket

platform, cylindrical tubular members are commonly used in offshore structure

and represent the most important components in these structures for various

reasons, such as they minimize hydrodynamic force, have high torsional rigidity,

offer large local strength against impact loading, minimize the outside surface

subject to corrosion, and have the same large buckling strength in all directions, both locally and overall [2].

The jacket types are open structures so that some environmental loads

are acts on it which is vary very widely at different locations in the ocean and

varies with time, in general these forces are wave, impact, wind, currents and

earthquake loads.

Many aims we have takes from installation of fixed offshore platforms,

the major use is for drilling and production activities of oil and gas beneath the

sea floor, the second use for military applications and defense purposes and to

provide navigational aid to ships instead of light ships which is proved to be

more economic, to derive power from the sea, and for man’s living and working

space on the planet by providing room for process plant sites.




3

1-2: Dynamic Analysis Requirements:

One of the main loading for which offshore structures are designed is

caused by extreme water waves generated during intense, rare storms. Thedominant periods of such waves are typically much longer than the fundamental

periods of the most fixed offshore structures and, therefore, static analysis are

usually sufficient to for obtaining the design response of these structures to

extreme waves [3].

The development of oil and gas industry and moving into deeper water,

however, taller platforms with larger natural periods (small natural frequencies)

are built that respond more dynamically to extreme water waves. Prediction of

the dynamic response of such structures in extreme sea states is, therefore,

a primary design consideration [4].

In offshore structures two different approaches that are available for the

dynamic analysis, these are deterministic and probabilistic, if the time record of

the fluid motion is used to calculate time-force curve due to waves and the

corresponding time-displacement relation, the loading is called prescribed

dynamic loading, and the analysis is the deterministic analysis. On the other

hand, if the fluid motion and the structure are treated as random processes, the

loading is known as a random dynamic loading and the analysis is defined as

a probabilistic analysis [5].

In the analysis of offshore structures, to find the forces on the platforms

by either deterministic and probabilistic approach two stages are required to

estimate the nodal loads of the nodes in the super structure of jacket platforms,

the first is to find velocity and acceleration of fluid, the second stage is to use

the well known Morison’s equation to find nodal loads on nodal points of the

structure, the Morison’s equation is a semi empirical equation developed by




4

Morison (1950) which defines the nodal loads on a cylindrical members as

a sum of drag and inertia forces as illustrated later in chapter four.

1-3: Aims of study:

The dynamic analysis of offshore jacket platform is used to find the

response to both wave and impact loads under the effect of soil-structure

interaction.

The soil-structure interaction are considered by considering a new exact

stiffness matrix to model the piles as a beam resting on an elastic media by usingexact displacement method [6] with some ordinary beam elements to consider

superstructure, the ANSYS software program is used to analyze the structure by

using these two major types of elements. The stiffness, mass, damping matrix

that are used for piles are determined by using MATHCAD software, the

stiffness matrix for the exact displacement method that is model the piles [6] are

checked by using MATHCAD, some subroutines of Fortran program are used toreformatted the stiffness, mass, and damping to use as input in (ANSYS)

program, in addition a FORTRAN program are developed to calculate wave

loads by Airy wave theory and Morison’s equation.

The study contains two models, the first one is actual jacket tower,

which is analyzed to a free and forced vibration with the parameters that effect

it's response to dynamic loads. The second model is AL-AMAYA berthing

dolphin, which is analyzed for both free vibration and forced vibration due to

wave loads and impact ship loads for three different velocities. The reason to

choose both models is that the comparison are performed to show the effect of

isolated Winkler model and infinite continuos Winkler model that adopted in the

present study.

1-4 Layout of the thesis:




5

This thesis contains six chapters with appendix.

The first chapter contains introduction to the dynamic analysis of

offshore structures, which describe general definition to the problem.

The second chapter illustrate the previous researches in the analysis of

offshore structures, and soil-structure interaction.

The third chapter defines the static and dynamic behavior of

soil-structure interaction and modeling of the soil-pile system.

The fourth chapter deals with modeling and mathematical formulation

for finite element descritization and distribution of wave loads on nodal points in

the embedded superstructure in fluid medium.

In chapter five the applications, results, and discussion are views while

the sixth chapter illustrate the main conclusions and recommendations for future

work.

In addition the appendix at the end of the thesis illustrate the derivation

of three-dimensional representation of stiffness matrix of the pile material.



Chapter Two

Literature Review



Chapter Two Literature Review

6

Literature Review

2-1 Introduction:

The analysis of offshore structure contains some topics, these are the

modeling of this structure by which the real structure is to be simplified to a

simple mathematical model, and environmental loads that may occur during life

of services. Many researches have been carried out in the previous decades

dealing with soil structure interaction, wave loading, and fluid – structure

interaction and some methods are introduced to simplify the real structures.

2-3 Analysis of Offshore Structures:

In 1980 Fish et al [7] used a simplified offshore model to find the

implied hydrodynamic damping which can be used instead of fluid- structure

interaction and adding non-linear solution by using Morison`s equation. Two

types of waves are considered, these are the regular waves and random waves.

Their work is divided into two parts. In the first part the fluid-structure

interaction is taken into account, while in the second part these non-linear terms

are neglected. It was found that when the viscous structural damping is

increased, then the implied hydrodynamic damping is decreased, also that the

implied damping may be ignored for random waves.

In 1981 Starvos [4] used random wave and mode superposition method

in the analysis when the offshore platform is subjected to extreme waves. In the

first stage the relative velocities were taken into account between the fluid and

the structure to account for fluid-structure interaction. In the second stage, the

relative velocities are neglected and the equivalent damping ratio is used to

represent the hydrodynamic damping. It is found that the value of damping ratio

(2.5%, 2.8%) is sufficient to including the neglecting non-linear terms of fluid

structure interaction.




7

In 1981 Heins et al [8] analyzed some types of Dolphins to ship impact.

They developed a FORTRAN (IV) program to idealize the ship dolphin as a

S.D.O.F. system. The dolphin-soil system was represented by elastic spring, the

mass of the system is the ship mass. The cellular caisson, pile cluster, and

separated piles with cap, as well as the failure criteria of soil, pile material are

all including in the analysis. It was concluded that the determination of the

dynamic response of dolphins, pile clusters, or platforms, when subjected to

vessel impact has been achieved by a computer oriented technique. The method

involves computation of equivalent spring constant, representing the dolphins,

clusters, or platforms. Then the response of the system simulated as S.D.O.F.

spring-mass was examined. The actions obtained from this model are then

applied to the three dimensional model at various time intervals. The resulting

response of three-dimensional model including soil-structure interaction gives

the resulting stresses and deformations in the dolphin system.

In 1984 Ragab et al [9] discussed the dynamic behavior of fixed

offshore framed structures by using Wittrick and Williams algorithm to solve

the non – linear eigen value problem. The three dimensional frame was modeled

into two dimensional and considering the added mass and containing water in

tubular sections in addition to the structural mass and lumped mass.

Approximations are considered with axial and shear deformations and rotary

inertia. The results of the study indicate that the first two frequencies obtained

from the non-linear and linear eigen value solutions are agree close, and the

effect of non-linear eigen value solution is significant for the higher frequencies

which is differ from linear eigen value analysis. Also they concluded that the

effect of axial force on free vibration analysis is most pronounced for slender

members of the frame models in which, the effects of shear deformation and

rotary inertia can be neglected.

In 1988 Madhjit and Sanha [10] used a two dimensional offshore frame

instead of three dimension one. The free and forced vibrations of offshore




8

structures are studied using both consistent and lumped mass approximation for

mass idealization. Two different idealization are used to represent an offshore

platform, these are truss and frame models. The effect shear of deformation and

rotary inertia on natural frequencies were considered and mode shapes of the

system. The dynamic response has been obtained by modal analysis in

conjunction with Newmark’s numerical technique. The wave loading were

obtained by using Morison`s equation and linear wave theory to find

hydrodynamic loading. Also they are studying the contribution of each first six

modes in the response of the entire structure. They found that the natural

frequencies for first two modes have agreed very well with truss and frame

model for both consistent and lumped mass approximation. The difference

increase with higher modes and the effect of shear deformation and combined

shear deformation and rotary inertia has marginal effect and that the deck

response of truss is greater than that for frame model.

In 1998 Kareem et al [11] investigated the response of jacket platform

type by using Guassian and non-Guassian random wave loading sea state. The

frequency domain solution is used in the analysis of jacket platform in deep

water. Both linear and non- linear theories are used with Morison`s equation to

find the dynamic wave load effects. It was concluded that linearization of drag

force yield an underestimation desk response in random waves, furthermore, the

desk displacement response in non-linear random waves more that in linear

random waves. Also they found that in addition to the wave profile and drag

force effects a platform (leg-spacing) has an important parameter that influence

the deck motion.

In 2000 Al-Jasim [12] developed a FORTRAN computer program to

find the response of template offshore structures by using Newmark’s direct

integration method, time domain solution of mode superposition, and frequency

domain solution. Nodal loads were found by Morison`s equation with airy linear




01

wave theory and fifth order Stokes non-linear theory. Three case studies are

investigated, the hydrodynamic loading effects on a cylinder embedded in fluid

mass, idealized jacket platform type, and Al-Amaya berthing dolphin. The effect

of wave loading on the first two cases, and the impact of ship berthing on the

berthing structure (dolphin) of Al-Amaya berthing dolphin are considered. The

soil-structure interaction is included by using the p-y and t-z curves to represent

the nor-linear behavior of spring, which represent the piles. It was concluded

that the linearized Morison`s equation, gives a high response deflection as a

compared with non- linear form. Also the variation when using Airy`s wave

theory versus Stock’s theory are small compared to the computational effort as

in case of linearized Morison`s equation in frequency domain versus direct non

linear time integration analysis, further the fender are very important part of

structure, and it’s type will influence the capacity of dolphin.

In 2002 Al-Salih [3] used a STAAD III software program to find the

dynamic response of offshore platform by using finite element method including

soil- structure interaction. He used two models to performs the free and forced

vibration for different soil-stiffness, different distribution to soil stiffness

through different soil. These varieties in stiffness are constant, linear, non- linear

distribution. He calculates the dynamic desk response for different distributions

of soils to the wave forces and response for wave and impact loads to Al-Amaya

berthing dolphin. The piles are modeled as a beam- elements supported by

a nodal springs at different stiffness. He found that the dynamic response is very

sensitive to soil-structure interaction, stiffness distribution, and the depth of pile

embeddment. The influence of stiffness values is important for natural

frequency especially for bending and axial mode. Also, it was concluded that the

effect of wave load on the berthing dolphin is greater than the impact load.

In 2002 Raid and Abbas [13] studied the free and forced vibration of

submarine pipelines under the action of wave forces. They studied the response

of two and three-dimensional analysis of pipelines under the action of wave




00

forces during construction. They included the effects of the added mass in free

and forced vibration using both lumped and consistent mass approximation.

They used a modal superposition method, and NASTRAN software program to

check their analysis. They found that the added mass has a considerable effect

on magnitude and shape of time history for two and three-dimensional analysis.

In 2002 Abbas [2] studying free and forced vibration analysis of

submarine pipelines and fixed offshore framed structures subjected to dynamic

wave forces by using the finite element method. Wave forces are obtained by

Morison’s equation and Airy linear wave theory. The effect of the surrounding

fluid added mass is studied for free and forced vibrations with both lumped and

consistent mass formulations are used. The natural frequencies and mode shapes

are obtained by Jakobi method and the dynamic responses are carried out by

mode superposition technique. The results of the computer program are checked

against NASTRAN computer program and good results are obtained.

2-3 Soil-Structure Interaction:

In 1973 Rosset et al [14] used a modal analysis to represent dynamic

soil-structure interaction, the structure was represented by discrete masses

connected by springs and dashpots, the soil replaced by two springs and

corresponding dashpots, one set corresponding to swaying and the other to

rocking. A rule for equivalent frictional critical damping by weighted modal

damping is obtained, the damping is considered to be viscous which is

frequency dependent and hysteretic which is frequency independent and

replaced them by equivalent damping which is in viscous nature. Only the

viscous damping type can be used in time domain analysis. They found that

damping associated with swaying is essentially viscous, while the damping

associated with rocking is primarily hysteritic. . To determine the validity of the

suggested rule, several typical cases were analyzed. For each case, three




0

different analysis were performed, one in the frequency domain using the actual

damping matrices CV and CH, providing what will be refereed to as exact

solution. In the second analysis normal modes were assumed neglecting in effect

the off-diagonal terms of matrices QTCvQ and QTCHQ viscous and hysteretic

damping terms were kept, however, separate, and the solution was again

obtained in the frequency domain. This solution is referred to as modal

superposition in the frequency domain. In the third analysis a modal solution

obtained by using equivalent damping ratio and considering all the modal

damping to be viscous. This is referred to as modal superposition in the time

domain. The third type of analysis is the one which would be used in practice.

Comparisons of the results for the first and second analysis shows the error

introduced by assuming normal modes. Comparison of the second and third

analysis shows the additional error introduced by replacing hysteretic modal

damping by viscous modal damping. Comparison of the first and third solution

indicates the overall error resulting from use of weighted modal damping.

In 1974 Shalash [15] used interface element to model soil- structure

interaction by a non-linear analysis technique and non-linear iterative method

to represent the stress-strain curves of soil and concrete and non-linear behavior

of interfaces between them by using finite element method. Plane strain or plane

stress problems were solved including buried structure, footings, piles, sheet

piles, retaining walls and embankments. Joint element is used to represent theinterface behavior between soil and structural element. He found that the

Poisson's ratio plays a major role in the analysis and, error will occur when it is

used as a constant, the values of subgrade reaction is not constant in the field but

vary toward the end of the footing. Better result have been obtained when a

non-linear analysis is used by finite element method that consider non-linear

behavior of strain- stress curve with high number of increments.




02

In 1976 Fukashi and John [16] introduced a new three dimensional

stiffness matrix that includes the four moduli of subgrade reaction by deriving a

stiffness depending on shape functions which defines simple bending theory to

obtain the stiffness matrix for pile element by using a direct finite derivation

depending on strain energy stored in the foundation element. They verified their

results for the case of lateral load on a pile with isolated springs modeling and

with an analytic solution. A closer result is obtained by suggested stiffness

matrix with exact solution as compared with isolated spring which gives slower

convergence with exact response depends on number of used elements. Further

more they considered the shape of member in their study to consider different

degrees of freedom by a dimensionless parameter which depends on the shape of

the element and depth of pile embeddment which is compatible with the

ordinary beam element which allows more accurate representation of boundary

conditions.

In 1983 Feng and Cook [17] introduced a modified two parameter

Winkler foundation which is proved to be more accurate than one parameter

(that in present study) and simpler than semi-infinite elastic continuum

foundation model. They used both the cubic shape function and exact method

(for displacements) to derive the stiffness matrix of this two parameter model

and comparison between them are done to verify the validity of the cubic shape

function model. The results of this study indicates that a fine mesh size is

needed to match the results of two parameter models with cubic shape function

for both deflections, rotations, and bending moments, also a very fine element is

needed to get accurate transverse shear forces. Beams can be analyzed by one

parameter Winkler model if the second parameter is small compared to the first

one, but not in the case of high value of the second one (it is close to I.E.4 ,in

this case and the error will be appreciable).




03

Also in 1983 Van Laethem et al [18] used a combined finite element and

boundary element method to represent soil-structure interaction in two and three

dimensional analysis, the region near the applied load which behaves as non-

linear is modeled as a finite element while the far end is represented by a

boundary element, which is used to model the elastic behavior of far field, the

boundary element method is the fast, and accurate technique to represent such

problems especially for the case of infinite medium. It will be divided into finite

elements around the foundation with the far field will be represented by a

boundary element which coincide with the outer surface of the finite element

mesh. A good result was obtaining as compared with the analytic solution of

circular load that solved by theory of elasticity [19].

In 1985 Musharraf Uz-Zaman et al [20] developed a simple thin layer

element used in a finite element procedure to simulate the various modes of

deformation (four modes) in dynamic soil- structure interaction. They derived

constitutive relations that define the behavior of interface by decomposing it into

its normal and shear components. The soil is modeled as an elastic-plastic (strain

hardening) while structure being elastic linear. Numerical procedure is used to

predict behavior of a model structure tested in the field, and the influence of

interface behavior on displacements, velocities, and accelerations is delineated.

Much verification was used to insure the validity of the model and with other

types of interface element (e.g. [21]). This element proved to be more accurate

and economical.

In 1989 Chen and Krauthammer [22] used the finite element and finite

difference approaches to represent soil- structure interaction with substructuring

for solving seismically induced non-linear soil-structure interaction problems.

The substructuring is achieved by employing a super-degree of freedom instead

of the structure, combined with an explicit finite difference code, and the results

interface conditions are then used as input for analyzing the structure alone with




04

ADINA program. The resulting demonstrate that the combined approach is

efficient and economic, as compared to the pure finite element analysis.

In 1990 Saieel [23] introduced an explicit integral finite element method

for determining load-settlement history of axially loaded skin friction piles in

linear range. The method is based on exact analytical solution of the differential

equation governing the behavior of skin friction piles. A fourth order polynomial

is used to fit the soil resistance distribution along element length. The least

squares method is used to obtain the fit. The governing differential equation is

developed using the concept of subgrade reaction. This differential equation is

solved by the method of Frobenius[24], from which exact shape function and

stiffness matrix is performed, for the pile element. Load-settlement relation

ships were obtained by the incremental iterative method with tangent stiffness.

Comparisons are made with other available data to show the accuracy and

efficiency of this exact stiffness method and its results.

In 1997 Essa and Al-Janabi [25] studied the dynamic behavior of plane

frames partially embedded in Winkler elastic foundation to obtain the dynamic

response of framed structures considering the foundation-structure interaction,

they developed a FORTRAN finite element program to solve the dynamic

equilibrium equation. They considered the frame as a beam element with axial

force supported by elastic foundation of Winkler type having normal and

tangential moduli of subgrade reactions, which is assumed to be constant and

linearly varying with the length of element through the different soil depth

considering also the end bearing effects of the elastic foundation. Two cases are

studied, which is blast loading and lateral sudden force resulting from the impact

of a ship of an offshore platform. Both direct Newmark integration scheme and

modal superposition technique are used to find the effect of soil-structure

interaction on the dynamic response of a structure which is appeared to be




05

sensitive to the foundation model and will decrease the time step required for

stability and accuracy.

In 1999 Abdul-Sattar [26] proposed a finite element program to consider

the dynamic Soil-structure interaction of the underground structures (buried and

tunnel). The (step by step integration) technique is used to solve the dynamic

equilibrium equation, two types of loading are used these are earthquake,

loading and nuclear blast loading, he took in into account the effect of noise in

the solution of F. E., and used a spatial filter to control this noise. He considered

both granular and cohesion soils and used both elastic and plastic (bounding

surface) model to represent the soil whereas concrete was represented by

dynamic elastic model. Linear analysis gives good results as compared with

non-linear solution (by using bounding surface model). The dynamic analysis

may be used as a first approximate estimation but can not give actual behaviour

under dynamic loads.

2-4 Summary:

From the literature that are discussed in this chapter, we can arrive to

some information that is helpful in the dynamic structural analysis of offshorestructures as: -




06

1- Many wave theories in addition with Morison’s equation are used to

calculate the wave loads on structural members (as in chapter four), these are

in common Airy linear theory and Stokes non-linear theory.

2- Soil-structure interaction is essential parameter to find the static and

dynamic response of such structures.

3- Both soil-structure interaction and fluid-structure interaction have

a non-linear behaviour, therefore, some simplified approaches are used to

linearize them. The materials are linearised especially for soil since it appears

to be highly non-linear in behaviour.



Chapter three

Soil – Structure

Interaction



ChapterThree Soil-Structure Interaction

91

Soil-Structure Interaction

3-1:Introduction:

The response of structure- foundation systems subjected to static or

dynamic loadings is influenced significantly by the behavior of contact junctions between the structural element and the surrounding soil enclosed them, these

junctions represent the interface between the response of the soil-structure which

transmit the load and the deformations, which is called soil-structure interaction.

There are many types of problems that go into the definition of soil-

structure interaction, such as footing, dams, buried, and retaining structures, etc.

these types of problems are very complicated in some features such that the

non-linear behavior of materials especially for soil. The behavior of the interface

points is also non-linearity, therefore, the non-linear analysis are used to control

these problems. But in some cases a linear solution is acceptable to use which

simplify the calculations and reduce CPU time, which usually occurs in the safe

side.

In offshore structure, dynamic soil-structure interaction is very important

factor that affects the total response of structural system and cannot be

significantly ignored. The flexibility of the structure will be increased when

considering the effect of piles flexibility.

3-2 Dynamic Behaviour of Piles :

When a pile vibrates, its stiffness is modified and damping is generated

through interaction of the pile with the surrounding soil. These phenomena are

very complex and least understood. The variation of stiffness and damping is

strongly dependent on the frequency [27]. In some cases an extra mass has been

added to represent part of the soil [28].

For slender pile, the dynamic stiffness of soil-pile system increased with

frequency, but for rigid pile the stiffness decreases with the frequency. The

damping is increased with decreasing frequency and increased with pile length




02

in most cases. Pile stiffness at low frequencies does not differ much from static

stiffness [29].

For regular piles with slenderness ratio (H/R) larger than (25), the

stiffness can be considered independent of the slenderness ratio, frequency, and

of the pile tip conditions [27]. Internal dissipation of energy in the soil can be

represented by hysteretic (frequency independent) damping through use of

complex moduli Es (1+2iζ) and Gs(1+2iζ) where ζ is the desired damping ratio

[27]. Generally, the damping of the soil spring (especially for the sway

condition) is larger than the damping in the structure, moreover the damping in

some parts of the system is viscous in nature, while in other parts the damping is

more nearly hystestic.

The viscous damping can be represented by a dashpot with a resistance

proportional to the velocity. The physical characteristics of viscous damping is

the viscosity coefficient or dashpot constant c. The fraction of critical damping

)M.k 2/(c for 1-D.O.F. system depends not only on the dashpot constant but

also of the mass, M, and the spring constant k. Thus the same dashpot would

produce different values of ζ for two systems with different masses but the same

stiffness. Hysteretic damping can be observed in most engineering materials

including soil, which exhibit a hysteritic stress-strain diagrams when strained

cyclically. In each cycle, energy is dissipated. This energy is a function of

amplitude but experiments show that it is to a large extends independent on

frequency. In reference [14, 29], it is found that this damping is most suitablymodeled.

3-3 Modeling of Soil-Pile System: - Most of jacket platforms which is made usually by a steel structural

sections that supported by a number of piles or pile groups which transmit the

environmental loads to soil strata. In the analysis of offshore platform the piles

have an important influence on the behavior and response of these structures to

static or dynamic loading [3, 32].




09

Many modeling techniques have been discussed in literature, multi

physical models were defined [30] to represent the response of beams or piles

embedded in a soil medium, these models are used in offshore structures to

represent soil-structure interaction, some of these models are: -

1-Nonlinear soil springs by p-y and t-z curve method [12, 31].

2-Nonlinear soil spring considering the strata dependent shear modulus [23, 32].

3-Linear springs with elastic deformation [3, 31, 32].

4-Three dimensional analysis of the pile- embedded in soil medium [1, 15, 20].

Piles are structural members, which can be taken as a beam, or columnaccording to the loading conditions. When a pile supports a large loading axial,

bending, shear and torsion, for each of the loading cases, the pile will undergo

the loads and transmitted them in a some manner to soil enclosed. By

considering the pile as a beam on elastic foundation, this foundation is modeled

by an infinite number of springs, which represents the soil resistance to pile

loading [30].

3-4 Winkler Model:

This foundation model has been used for a century [17]. It assumes that

the foundation applies a reaction (from soil medium) normal to the beam, which

is directly proportional to the deflection under the beam that is:

q=k n.y(x)……….(3-1)

where:

y (x): deflection(m).

x : Length of beam /or pile(m).

k n: Winkler foundation modulus (N/ m2/ m).

This model is a simple model to represent soil-foundation interaction.




00

To improve the Winkler model, some authors assumed interactions

between the springs and added a second parameter to represent the interaction

between them and this will be more accurate than the classical Winkler model

[16, 17, 30].

1-Filonenko-Borodich model.

2-Pasternak model.

3-Generalized model.

4-Vlasov model.

These four models are mathematically equivalent, but they differ in

defining the second parameter.

Piles will be considered as a beam on an elastic foundation, this

foundation subjected to shear force, bending moments, axial force and

torsional moments which is resisted by pile material and soil surrounding

to the pile which will reduce the effect of these forces on pile material.

One parameter beam on an elastic foundation with a constant modulus of

subgrade reaction (for clay) or linearly varying modulus (for sand) canmodel this contribution. An exact stiffness matrix for three-dimensional

beam on elastic foundation Fig. (3-1). which is simple to model and

accurate in results[17]. The beam may be embedded in the elastic medium

as in the case of pile and, in addition, the pile may offer resistance to

shear and torsion as well as normal reaction. The properties of the

foundation are described by three reaction moduli are defined

subsequently here in [16].

3-4-1:Normal Reaction Modulus:

If the distributed load acts normally on part of elastic half space

and a unit displacement occurs in the direction of loading, the normal

reaction modulus K n of the material is defined as the force per unit area,

which causes unit displacement. Referring to Fig. (3-2-a) a rigid plate area

(A p) is placed on the elastic material and a normal load P is applied. Fig.




0

(3-2-a) and the plate deflection w p. The normal reaction modulus k n is

then (p/A p/w p) in Newtons per cubic meter. This modulus can be

evaluated experimentally by plate bearing test or full scale pile test, some

others gives a correlation formulas for the normal modulus [31, 33] of

subgrade reaction, the relation chosen is:-

)1(2

D

Esk n

………(3-2) D: diameter of pile.

Es: Elastic modulus of soil. ν: Poisson’s ratio.

This formula (3-2) is used for lateral loads on piles, where as the

stiffness of the normal reaction for end bearing is half of the value inEq.(3-2).

3-4-2:Shear Reaction Modulus:

Now if a distributed load acts tangentially on part of an elastic half

space and a unit displacement occurs in the direction of loading the shear

modulus of subgrade reaction of the material is defined as the force per

unit area which cause unit displacement.For example in Fig. (3-2-b). A

rigid plate area(As), as adhere to the elastic material and a concentrated

shear force. V, is applied producing a movement of the plate v. The shear

reaction modulus k s, is taken as (V/As/v) in Newtons per cubicmeters, this

modulus can be evaluated as 20% from the normal modulus [33] or taken

as one eighth of the normal one [31] i .e.

)-8D(1

Es ks

2 ……………(3-3)

The shear stress that resists the applied axial load on the pile

perimeter from soil material can be represented in the same manner for

the case of normal load, that is:

where:

)(. xwk s s …………..(3-4)

k s: Shear reaction modulus of soil. , ν: Poisson’s ratio.




0

D: Diameter of pile. ,Es: Elastic modulus of soil.

3-4-3 Torsion Reaction Modulus:

If a distributed twisting moment acts on part of an elastic material

extending to infinity and unit rotation occurs in the direction of loading

the torsion reaction modulus of the material is defined as the moment per

unit area which causes unit rotation. Referring to. Fig.(3-2-c) if a rigid

cylinder of surface area. At, adheres to the elastic material and a twisting

moment, T, is applied producing a rotation of the cylinder. The torsion

reaction modulus then (T/At/Φ) in Newtons per cubic meter. This torsion

modulus can be found as [31].

R

G.2k s ………..(3-5)

Gs: shear modulus for soil. k Φ: Torsion reaction modulus.

R: radius of pile.

The shear stress that resists the applied torque on the pile

perimeter from soil material can be represented by:

)x(.k …………..(3-6)

To represent the pile embeddment in a half space which is

subjected to axial, bending, shear and torsion, the use of superposition of

each of above degree of freedom to fully represents of three dimensional

beam element embedded in an elastic half space by using a finite number

of element embedded in soil, each one capable to resist the forces a above.

This representation can be done by solving the governing differential

equations which represents each degree of freedom by considering the

normal modulus which effect on the bending and shear response in the

beam with their interaction for the pile material embedded in a soil, as




0

well as the shear reaction modulus which effect on the response of the

axial resistance of the pile material, and at last the torsion reaction

modulus which increase the resistance of pile material to torsional loads.

This derivation is shown in detail at the appendix for each

bending, axial, and torsion.

Axial Torque

Moment

Side friction

Lateral Load

Soil

medium

Lateral

(normal)

reaction stress

Shear reaction

stress

Pile tip or end

bearing

Fig. (3-1): Typical pile subjected to multi types of probable loadsand their actual resistance.

Torsional friction

resestance




0

Contact area(Ap)

P

Contact area

(As)

Contact area (At)

S

(b) Shear reactionmodulus (k s)

(a) Normalreaction modulus

(k n)

Elastic

T

(c) Torsion

reaction

Fig. (3-2): Moduli of Elastic Reaction for Multi

Degree of Freedom.



Chapter four

Modeling

and

MathematicalFormulation



Chapter Four Modeling and Mathematical Formulations

82

Modeling and Mathematical Formulation

4-1: Finite Element Formulations:

The basic idea in the Finite element method is to find the solution of

a complicated problem by replacing it into a simpler one. Therefore, only

approximate solution can be obtained rather than the exact one, but in our scope

an exact solution can be obtained for each individual element according to

assumptions to simplify the physical phenomena of soil-structure interaction.

In the finite element method, the actual continuum or body of matter

likes solid, liquid, etc. is represented as an assemblage of subdivision called

F.Es. These elements are considered to be interconnected at specified joints

which are called nodes or nodal points. These nodes usually lie on the element

boundaries where adjacent elements are considered to be connected. Since the

actual variation of the field variable (like, displacement, stress…etc.) inside the

continuum is not known, it is assumed that the variation of the field variable

inside a finite element can be approximated by simple function [35].

There are three methods used to solve the finite element problems to

give a best solution for the governing differential equation for the physical

problem as: -

1- Direct (physical) methods.

2- Variational methods.

3- Weighted residual methods.

Both ordinary three-dimensional beam element or three-dimensional

beam embedded in a soil will be used with exact formulation for each type of

element to find the exact stiffness for each element as well as consistent, lumped

mass approximation is found for each beam element type.

The essential physical properties for any linearly elastic structural system are

subjected to dynamic loads are its mass, elastic properties (stiffness), its

energy – loss mechanism, or damping, and the external sources of excitation




8

or loading. Therefore, for any linear system, the dynamic equilibrium equation

of an offshore structure modeled as a system with a finite number of degrees of

freedom may be written as: -

[M] {u"} + [C] {u'} +[K] {u} = {F (t)}…………….(4-1).

Then, in order to evaluate the dynamic response of an offshore structure,

the mass [M], damping [C], and stiffness [K] matrices of the entire structure and

the load vector must be determined. This is conveniently made by considering

only a typical finite beam element, which can be generalized for all the elements

of the offshore structure under study.

4-1-2: Element stiffness matrix: -

In the modeling of the platforms, two types of elements are used. The

first type is a simple beam, which is used to model superstructure, while the

second is the beam rested or embedded in an elastic medium, which is used to

represent the piles of the platform.

4-1-3: Stiffness of Three-dimensional Beam Element: -

A three-dimensional model for a beam is shown in Fig. (4-1) with six

degrees of freedom at each node. The nodal displacement of the beam is: -

{q} = [ui vi wi θxi θyi θzi u j v j w j θxj θyj θzj]………………………..(4-2)

where: ui, vi, wi, u j v j, and w j represents translation at x, y, and z directions

at node i,j respectively while θxi, θyi, θzi, θxj, θyj, and θzj are rotations about x, y,

and z axes at node i, j respectively.




03

Fig. (4-1): Notations for Nodal Displacements and Rotations

at Space Beam Element (Ref. [35]).

Let the generic displacement at any point within the element be

expressed as the column vector {ue}: -

{ue

} =[ u v w θ ]T

………………………(4-3)

Where u, v, w, θ: Displacements in x, y, z, and torsional rotations

respectively.

These displacements can be expressed in terms of the nodal

displacements by assumed shape functions as follows: -

{ue} = [N] {q} ………………………(4-4)

In which [N] is the shape function matrix for the beam element and it is

given by [35].

[N]=

004000001000

060500030200

600050300020

000004000001

N N

N N N N

N N N N

N N

…(4-5)

where:

vi

θx j

L

z

y

v j

u j

w j

θyi

Area

θz j

θy j

x

θzi

wi

θxiui




0

11 N 32 2312 N 2)1(L3 N

4 N 32 235 N )(L6 N 2

andLx where: x: Distance from right end of the pile (m).

: Normalized Coordinate system.

In the structural analysis, the finite element stiffness is more easily to

be derived by invoking one of the most widely used two approaches, principle of

minimum potential energy and principle of virtual work. Both approaches give

virtually the same results.

The strain-displacement relationships are obtained by differentiation of

the displacements as: -

{ε}=[d] {ue}……………………………..(4-6)

where: {ε}: vector of nodal strains.

[d]: linear operator matrix (strain-displacement transforming matrix).

{ue}:vector of nodal deformations.

The linear operation matrix considers the axial, flexural, and torsional

displacements effects separately. By applying the principles of simple bending

theory and the superposition techniques, matrix [d] for the beam element can be

given as [36].

[d]=

dx

d r

dx

d z

dx

d y

dx

d

.000

0.00

00.0

000

2

2

2

2

………………………(4-7)




08

In which x, y, z, and r are the vertical, horizontal and radial distances of

any points on the cross section of the element from its center respectively.

Now, substituting eqns. (4-4) in (4-6) then: -{ } =[B] {q}……………………………(4-8)

where: [B] =[d] [N]………………….(4-9)

Now, stiffness matrix for space beam element can be obtained by

applying the principle of virtual work [35], then:-

[K e]= dV B D B T ]][[][ ……………………………….(4-10)

In which [D], is the property matrix and is given as :-

[D]=

G

E

E

E

000

000

000

000

…………………………………(4-11)

The resulting element stiffness matrix in local coordinates system is

given as: -

[K e]=

43

21

k k

k k ………………………………………….(4-12)

where:




00

L

EI4000

L

EI60

0L

EI40

L

EI600

00L

GJ000

0L

EI60

L

EI1200

L

EI6

000L

EI12

0

00000L

EA

k

z

2

z

y

2

y

2

y

3

y

2

z

3

z

1

L

EI2000

L

EI60

0L

EI20

L

EI600

00L

GJ000

0L

EI60

L

EI1200

L

EI6000

L

EI120

00000L

EA

k

z

2

z

y

2

y

2

y

3

y

2

z

3

z

2

L

EI2000

L

EI60

0L

EI20

L

EI600

00L

GJ000

0L

EI60

L

EI1200

L

EI6000L

EI120

00000L

EA

k

z

2

z

y

2

y

2

y

3

y

2

z

3

z

3




0

L

EI4000

L

EI60

0L

EI40

L

EI600

00L

GJ000

0L

EI60

L

EI1200

L

EI6000

L

EI120

00000L

EA

k

z

2

z

y

2

y

2

y

3

y

2

z

3

z

4

4-1-4 : Element stiffness matrix for a pile: -

Piles are discritized to a number of finite lengths which represents space

beam on elastic foundation including effects of soil-structure interaction. Each

element of length (1m) embedded in soil medium, the element stiffness can be

derived by solving the governing differential equation for each axial, bending,

and torsion problems separately and then using the superposition technique to

collect these stiffnesses to get the overall stiffness matrix for whole element.

This can be done by giving the stiffness matrix for the loading cases lateral,

axial, and torsional loading in two dimensions, and then extends to three

dimensions by the same way that is used in simple bending theory [37]. The

derivation of all loading cases is shown in the appendix, the resulting stiffness

matrix is: -




0

[K e]=

1526

1526

810

5364

5364

79

2615

2615

108

6453

6453

97

T000T0T000T0

0T0T000T0T00

00T00000T000

0T0T000T0T00

T000T0T000T0

00000T00000T

T000T0T000T0

0T0T000T0T00

00T00000T000

0T0T000T0T00

T000T0T000T0

00000T00000T

…(4-13)

where: - T1=))(sin)((sinhL

))cos()sin()sinh()(cosh(EI222

=

L

EI4 at =0.00

T2=))(sin)((sinhL

))cos()sinh()sin()(cosh(EI222

=

L

EI2 at =0.00

T3= ))(sin)((sinhL

))cosh()sinh()sin()(cos(EI4223

3

= 3L

EI12 at =0.00

T4=))(sin)((sinhL


3

=

3L

EI12 at =0.00

T5=))(sin)((sinhL

))cos()(sin)((sinhEI2222

222

=

2L

EI6 at =0.00

T6=

))(sin)((sinhL

))sin().(sinh(EI4222

2

=

2

L

EI6 at =0.00

T7=E.A.β.coth (β.L) T8=G.J.α.coth (α. L)

T9=- E.A.).sinh( L

T10=-G.J.

).sinh( L

where all other Greek symbols are defined in the appendix.




0

4-1-5: Element mass matrix: -

In the finite element analysis for dynamical systems the inertia force is

the most important factor that affects the behavior of the structural system (theseinertia force as shown in eq. (4-1) is the product of nodal masses and

acceleration of the nodal D.O.F.. There are two kinds of mass representation for

the structure, consistent and lumped mass approximation.

In offshore structures, some of members are submerged in sea, these

members displace some fluid masses, this phenomena will increase the inertia

force of the structures as well as it will affect the hydrodynamic damping of thestructural system [38, 7, 12, 39].

4-1-5-1: Consistent mass approximation:

The consistent mass matrix for space beam element in local coordinate

system without added mass is [40]:

[M]=

2

2

22

2

4

04

00140

0220156

22000156

00000140

30001304

030130004

0070

00000140

013054000220156

1300054022000156

000007000000140

420

L

L A

J

L

L

L L L

L L L A

J

A

J

L L

L L

L (4-14)

Symmetric




0

3-1-5-2: Lumped mass approximation:

The diagonal lumped mass matrix without added mass is shown below,

the-off diagonal terms is due to the acceleration of any mass point only .The point mass will be associated with each translation degree of freedom at any

nodal points, whereas the mass associated with any rotational degree of freedom

will be zero because of the assumption that the mass is lumped in points that

have no rotational inertia. [41].

2

L.A.M

1 1 1 0 0 0 1 1 1 0 0 0 ……………(4-15)

4-1-5-3: Effect of Added Mass Formulation:

The modification on the above matrices is achieved by only adding the

term of added mass which is equal to the volume of the member submerged in

fluid material (water) which is equal tof

. Aout where:-

Aout= 2.4 D

:f

Mass density of fluid (Sea water density), D: Outer diameter.

Thus, the mass matrix (lumped or consistent) inclusion of added mass

effect is [12, 13]:

[Me

] =[ . A. + f

. A out (cm-1)]. [M]………………………..(4-16)

Where [M] as above for lumped or consistent mass matrix.

4-1-6: Element damping matrix:

Damping is the force, which dissipate energy of any physical system

undergoing motion. The mechanism of this energy transform motion or

dissipatation is quite complex and is not completely understood yet, the damping




02

force is proportional to the magnitude of velocity, this type of damping called

viscous damping [42]:

The element damping matrix can be represented by the Reylaighdamping formula [42, 43, 44, 45, 46] which may be proportional to the mass

matrix or proportional to stiffness matrix or in general:

[C]=a1. [M]+ a2. [K] ………………………………(4-17)

where: a1, a2 are arbitrary proportionality factors.

[M], [K]: are the element mass and stiffness matrix respectively.

2

.a

.2

a12

1

1

1

…………(4-18-a)

2

.a

.2

a22

2

1

1

………….(4-18-b)

1 : Damping ratio.

21, : Fundamental frequency (first two modes) of natural frequency.

4-1-7: Transformation matrix:In finite element descrtization, there are two coordinate system to locate

the descrtized element, local and global coordinate system, the matrix that

transform the stiffness, mass, and damping from local to global coordinate

system for each element is called transformation matrix.

The stiffness, mass, and damping matrix in the form [42]: -

[K] g=[T]

T [K] [T]……………………………(4-19-a)

[M] g =[T]T [M] [T]……………………………(4-19-b)

[C] g =[T]T [C] [ T]……………………………(4-19-c)

where [M] g, [C] g ,[K] g are global stiffness ,mass , damping matrix,

[M] , [C],[K] are local stiffness ,mass , damping matrix.

[T]: Transformation matrix, which takes the form [35]:




0

[T]=

][]0[]0[]0[

]0[][]0[]0[

]0[]0[][]0[

]0[]0[]0[][

1

1

1

1

T

T

T

T

………………….(4-20)

where: [T1]=

2222

22

22

22

0

..

y x

X

y x

z

y x

y z y x

y x

y x

z y x

C C

C

C C

C

C C

C C C C

C C

C C

C C C

Cx = cos x = L

x x 12

Cy = cos y = L

y y 12

Cz = cos z = L

z z 12

L= 212

212

212 )()()( z z y y x x

4-1-8: Element Force Vector:-

For an offshore platform the most important loads are the hydrodynamic

loading and impact loads which are included in this study. These hydrodynamic

forces are governed by sea waves while impacts are usually occurs during

berthing of ships.

To formulate the hydrodynamic load vector FW, consider the single,

uniform, cylindrical member (i) between nodes I and J as shown in Fig. (4-3),

these forces are found by the well known Morison the semi empirical formula(Eq.(4-21)) also it represent the load exerted on a vertical cylinder , which

X

Y

Z

y

z

x Local coordinate system.

Fig.(4-2) : Local and Global

Coordinate Systems.

Global coordinate system.




3

assumes that the total force on an object in the waves is the sum of drag and

inertia force components. This assumption (introduced by Morison) takes the

drag term as a function of velocity and inertia force as a function of acceleration[1, 38, 39] so that: -

)uv( ).uv.( C . D.u. D. )C ( v.C . D. Fn nnnnd nmnm

2

1

41

4

22………

…………..(4-21-a)

which can be simplified to: -

)v( ).v.( C . D.v.C . D. Fn nnd nm

2

1

4

2 ………………………(4-21-b)

where: Fn: nodal hydrodynamic force normal to the cylinder.

D: Outer diameter of cylinder. : Sea water density.

Cd: Drag coefficient.n

v : Water particle acceleration.

Cm: Inertia coefficient. vn : Water particle velocity.

nu : Structural velocity. nu : Structural acceleration.

s

ds S

L

D

J

I . un(s)

. u(s)

Y

X

Z

GLOBAL

COORDINATES

F ig. (4-3): - Water Particle Velociti es Along Member i. MEMBER i




Equation (4-21-b) neglect the non-linear terms of drag coefficient [12,

13] while it considers the added mass concept instead of non-linear terms of

inertia force [7], water particle velocity and acceleration can be evaluated by potential velocity computed from wave theories, absolute value of velocity is

needed to preserve the sign variation force.

Generalizing one dimensional form of Morison's equation to the three

dimensional from of the hydrodynamic force per unit length along the beam

element at location (s) measured from its end to the nearest node given as [1]:

{F(s)}= . 2

D.4

.Cm. { nv (s)}+ 2

.D. Cd.{ nv (s)}.{ ) s( vn }……..(4-22)

In which the hydrodynamic force per unit length vector is given as:

{Fw(s)}=

)s(F

)s(F

)s(F

z

y

x

………………..(4-23)

and the normal water particle velocity and acceleration vectors are given

as:

{Vn(s)}=[s]{u(s)} and {an(s)}=[s]{a(s)}………………(4-24)

In which:

[s]=I- TS.S =

233231

322221

312121

S1SSSS

SSS1SS

SSSSS1

……………….(4-25)

where: I: is the (3x3) identity matrix, s : is the unit directional vectoralong the member and s1, s2, and s3 are direction cosines in x, y, and z directions

respectively [1] and:

{V(s)}=

)s(v

)s(v

)s(v

z

y

x

…………………..(4-26-a)




8

{a(s)}=

)s(a

)s(a

)s(a

z

y

x

……………………..(4-26-b)

These velocity and acceleration components are derived in detail in

next section.

Now, to calculate the load vector in global coordinates system, the

element is divided into two parts by using equation (4-22) which distribute the

wave effects on beam element equally to the end nodes as nodal forces,

therefore, the element of hydrodynamic load vector {Fe} corresponding to the

element nodal displacement vector {q} can be expressed as follows: -

{ ef }= [Fx1 Fy1 Fz1 0 0 0 Fx2 Fy2 Fz2 0 0 0 ]T……….(4-27)

4-2:Hydrodynamics of Water Waves:

All waves theories obey some form of wave equation in which the

dependent variable depends on physical phenomenon, and the boundary

conditions [3]. In general, the wave equation and the boundary conditions may

be linear or non-linear. There are many theories that describe wave motion, the

non-linear theories including Gerstner and Stokes theories while Airy linear

theory is the common linear wave theory.

Stokes theory assumes that wave motion properties such as velocity

potential (Ф) can be represented by a series of small perturbations. The linear

Airy wave theory can be used when the wave height to wave length501)

L( as

given in Ref. [39].

It is always assumed that the water waves are represented as two

dimensional plane waves, that they propagate over a smooth horizontal bed in

water of constant undisturbed depth (h). It is also assumed that the wave

maintains a permanent form, that there is no underlying current and that the free

surface is uncontaminated. The fluid (water) is taken to be incompressible and




0

inviscid and the flow to be irrotational [38]. Fig. (4-4) indicates the general form

of the xz-plane wave train conforming to these assumptions. Here the wave is

progressive in the positive x direction, and the z-axis measures positive upwards

from the mean water level, the wave height H, the wave length L, wave period

T, and the elevation of the water above the mean water level.

The surface must satisfy the special linear form of the wave equation of

Laplace solution to obtain the velocity potential (Φ) and is subjected to the

above conditions and linearized boundary conditions.

2

2

2

2

ZX

=0……………………………..(4-28)

To solve Eq. (4-28) some B.Cs. must be satisfied [3, 38, 39] : -

1-At the sea surface.

(a) The velocity of a particle must be tangential to the surface, the kinematicsconditions is:-

Direction of

Propagation Z

L

a H a

η

h

Sea bed

Sea

Celerity

Fig. (4-4): Definition of Airy Wave theory.




0zxxt

at z= ------------ (4-29-1)

(b) The pressure is zero and the energy equation must be satisfied, the dynamic

conditions is:-

tf zx2

1g

t

22

at z= ------------ (4-29-2)

2- At the sea floor where the vertical velocity is zero, that is:

0z

at z=-h ------------ (4-29-3)

The components of water particle velocity can be given as [25]:

xx

------------- (4-30-1-a)

zz

------------- (4-30-1-b)

Whereas, the components of the local particle acceleration, which are only taken

into account in the computation of the hydrodynamic force; given as:

ta x

x

------------ (4-30-2)

ta z

z

The major problem in solving for arises from the boundary conditions

to be applied at the air-water interface, (t), which is itself part of the solution




sought. Therefore, there are several solutions in common use. These are linear

wave theory in deep water, Stokes higher order wave theories, stream potential

function theory, and Cnodial theories in shallower water [47].

In present study only the linear (Airy) wave theory is considered to

compute the characteristics of water particles and the hydrodynamic forces.

To find velocities and acceleration that are used in Morison’s equation,

Laplace equation must be solved by considering the B. Cs. at the sea surface

η(t), which is itself-part of the solution sought, therefore, different wave theories

are used as mentioned.The linear wave theory (Airy theory) is used only to find velocities and

acceleration at different depths, locations, and time.

4-2-1: Airy linear wave theory:

In this theory the essential idea or restriction is that the wave height H

must be much smaller than depth d, that is (H>>λ , d). The linear wave theory for

two dimensional, free, periodic, waves is developed by linearising the equations

that define the free surface boundary conditions. With these and the bottom B.

Cs., the periodic velocity potential is sought that satisfies the requirements of

irrotational flow. The free surface B. Cs. may now is applied directly at the still

water level [38].

Therefore, the free surface B. Cs. as expressed in Eq. (4-29-1) and

Eq.,(4-29-2) are reduced to:-

0tz

at z=0 ………………….(4-31-a)

.gt

=0 at z=0…………………….(4-31-b)

By using separation of variables and B. Cs. (Eq. (4-31-a, b)) the velocity

potential ( ) can be found as [39]: -






)]t.)cos.xsin.y(k sin[()khsinh(

)]hz(k cosh[.

kT

H.

……………..(4-35)

Vx=

cos].t.)cos.xsin.y(k cos[(.)khsinh(

)]hz(k cosh[.

T

H. ………(4-36-a)

Vy=

sin)].t.)cos.xsin.y(k cos[(.)khsinh(

)]hz(k cosh[.

T

H..2 ……(4-36-b)

Vz= )]t.)cos.xsin.y(k sin[(.

)khsinh(

)]hz(k sinh[.

T

H.

………(4-36-c)

ax=

cos)].t.)cos.xsin.y(k sin[(.)khsinh(

)]hz(k cosh[.

T

H.2

2

………(4-37-a)

ay=

sin)].t.)cos.xsin.y(k sin[(.)khsinh(

)]hz(k cosh[.

T

H.2

2

………(4-37-b)

az= )]t.)cos.xsin.y(k cos[(.

)khsinh(

)]hz(k cosh[.

T

H.2

2

………(4-37-c)

These velocities and accelerations in Eq., (4-36), Eq., (4-37) are used in

Morisons equation to calculate load vectors of hydrodynamic loading by using

linear Airy wave theory after they are transformed from global coordinates for

each member of the offshore platform.

4-3: Fender Impact Forces: -

In addition to the wave forces, offshore platforms are subjected to impact

forces due to ships berthing on the structure called berthing dolphins, these

structures are used to prevent the ship and/or dolphin from damage during

mooring , energy absorption devices are used which is called fenders [3].

During ships berthing, loads will be generated between the ship and the berthing structure from the moment at which contact is first made until the ship




2

is finally brought to rest. The magnitude of the loads will depend on the berthing

energy (kinematics energy of the ship) and the fendering system. It is always

possible that catastrophic impacts may occur from ships drifting out of thecontrol [12].

Berthing reactions are a function of the berthing energy and the

deformation characteristics of the fendering system [12].

1- Contact pressure on the ship hull are kept within acceptable limits.

2-Direct contact between hull and berthing structure is presented.

3-The capacity of fender is not exceeded.

For the purpose of analysis and design of berthing structure, it is usually

assumed that the reaction force of fender may only be found for a given

deflection, however, the time of berthing -deflection rate of fendering system is

the controlling factor in evaluating the time- reaction force relation [48].

For the berthing dolphins of Khaur Al-Amaya berth no. 8, Bridgeston

C2000H cell type fenders are used.

The load (reaction force)- time relationships for this type of fender are

plotted by Al-Jasim [12] depending on deflection – time relationships and the

deflection-reaction force charts which is given in the handbook of Bridgeston

corporation for this type of fender [50] for three cases of berthing velocities

[12].

4- 4: dynamic analysis: -

In offshore structures the applied loads (environmental loads) are

generally have a dynamic nature, to study the behavior of these structures free

vibration and forced vibration must be considered in order to understand the

actual (as possible) behavior and response.




4-4-1:Free vibration analysis: -

In a free vibration usually the damping matrix is neglected, force vector,

and support motion, the analysis of the structure in free motion provides themost important dynamic properties of the structure which are natural frequencies

and the corresponding modal shapes, therefore, Eq., of free vibration are: -

[M] {u }+[K]{u}=0……………………………( 4-38)

At first the problem is formulated by the stiffness method for the free

vibration of the undamped system. The equations of motion expressed in matrix

form as in Eq., (4-38) when there is no ambiguity, we will dispense with the brackets and braces and use capital letters and simply write the matrix eqs. as(4-

38).

If we premultiply the above Eq. (4-38) by M-1 we obtain the following

terms:

[M]-1.[M]=[I] ([I] unit matrix)

[M]-1.[K]=[A] ([A] system matrix)

[I]. [u ]+[A].[u]=0…………………(4-39)

The matrix A is referred to as the system matrix of the dynamic matrix,

since the dynamic properties of the system are defined by this matrix. The

matrix A=M-1K is generally not symmetric, by assuming a harmonic motion

x.u where 2 , then Eq., (4-39) becomes:

[A-λ i.I]{u}=0…………………….(4-40)

The characteristic equation of the system is the determinant equated to

zero, or |A-λ i.I|=0 ………..( 4-41)

The roots λ i of the chararcerestic equation, are called eigen values, and

natural frequencies of the system are determined from them by the relationship:

λ i=ωi2 …………(4-42)




3

By substituting λ i into the matrix Eq. (4-39), we obtain the corresponding

mode shape ui which is called the eigenvector. Thus for an n-degrees of freedom

system, there will be (n) eigenvalues and (n) eigenvectors [44].ANSYS program used method of subspace iterative method, this method

requires in addition to the Jackobi method, Ritz reduction functions and iterative

procedure as detailed in [1, 49].

4-4-2: Forced Vibration Analysis:

To understand the response of offshore structures subjected to a load in

dynamic nature, as waves, impact earthquake, … etc., forced vibration analysiswill be used to get the response of the platforms to these forces.

There are different methods to solve the equilibrium Eq. (4-1) as

frequency domain solution, mode superposition, direct integration method,

…etc.. These are Newmarks implicit, most flexible step-by-step integration

methods in time domain, which is presented by Newmark [1, 43, 49]. This

method is based on the following expressions for the velocity and displacement

at the end of the time interval.

tttttt utu)1(tuu …………….(4-43)

ut+Δt=ut+Δt 2t )t(u (

2

1)

tt

2

t u)t(u ……….(4-44)

Where α,δ are selected to produce the desired accuracy and stability. One

of the most widely used methods is the constant average acceleration method

when (δ=0.5, α=0.25) which is a conditionally stable method without numerical

damping.

This method is called an (implicit integration method) since it satisfies

the equilibrium Eq. of motion at time t+Δt, or:

M tttttttt FKuuCu ……………(4-45)




This Eq. can be solved by iteration; however Eq. (4-43),(4-44),and(4-45)

can be combined into a step by step algorithm which involves the solution of a

set of Eqs. . Each time step is of the form:K

* .U

t+Δt=F

*……….(4-46)

Since K * not a function of time it can be triangularized only once at the

beginning of the calculation. A computer solution time for this type of algorithm

is basically proportional to the number of time steps required.



Chapter Five

Applications,

Results,

andDiscussion



Chapter Five Applications, Results, and Discussion

35

Applications, Results, and Discussion

5-1: Introduction:

The dynamic response of two models of offshore platform, jacket type

platform and Al-Amaya berthing dolphin subjected to the wave forces and

impact loads from ship berthing is discussed. Finite element method is used for

both spatial and temporal coordinate systems considering the effect of soil-

structure interaction.

5-2: Case Study (1): Jacket Platform Type.

In this case study the fixed jacket offshore platform described in [3, 32]

as shown in Fig. (5-1) is adopted, the frame descritized into (178) beam

elements for superstructure, and (240) beam elements embedded in elastic soil

which are used to model the four piles embedded to a depth of (60m) below

mudline in the sea bed that support the platform. For piles, the stiffness, mass,

and damping matrices are derived and evaluated using MATHCAD software.

A FORTRAN program is developed to find the wave forces at each

node in the superstructure that is embedded in the fluid medium. The forces on

each member are calculated and then distributed on the nodes for x, y, and z-

directions.

The deck mass is modeled using lumped the mass in five nodes that

forms the pyramid [32, 3]. This model is the same model that adopted by

Al-Salihy [3] which is takes the Winkler model with isolated springs at nodal

points only to represent the soil resistance which takes into account normal and

tangential modulus of subgrade reaction in three directions for each node. The

dimensions of the platform are shown in the appendix.




3

5-2-1 Soil – Structure Interaction:

As mentioned previously, the new stiffness matrix to represent the

interaction between the piles and soil is used. The soil is assumed as clay soil

with constant modulus of subgrade reaction taking the modulus of elasticity as

(45 MPa), and Poisson’s ratio as (0.3). The stiffness coefficients are calculated

using MATHCAD software from exact stiffness matrix. The mass and damping

coefficients are used as input in the ANSYS program to investigate the free and

forced vibration analysis.

5.2.2. Free Vibration Analysis:

Free vibration analysis is carried out taking into account the length of

pile embeddment in the soil, and end condition of the pile, which is spring,

hinged, and fixed, modeling of inertia forces (consistent and lumped mass

approximation), and the effect of added mass.

Table (5-1) compares the fundamental four modes of natural frequencies

of present work for spring pile tip with different mass types with that reported by Al-Salihy [3]. Al-Salihy used isolated springs to represent the soil-structure

interaction. The tabulated data shows that the results obtained by the present

work are higher than that given by Al-Salihy for sway, bending, and axial modes

due to stiffer model adopted in present study. But the values of natural

frequencies are close in torsion mode. This is because the pile legs of the

structure prevents the superstructure from rotation and no effect of the torsional

stiffness of piles at different length of piles on the torsional stiffness of the entire

structure.

The mode shapes of free vibration for fundamental sway, torsion,

bending, and axial modes are shown in Fig. (5-2).




33

ModePresent Work

Al-SalihyRef.(3) Consistent

Added

ConsistentLumped

Added

Lumped

Sway 0.37653 0.36676 0.37661 0.36667 0.207

Torsion 0.72951 0.70892 0.72783 0.70892 0.725

Bending 1.5164 1.4316 1.5149 1.4316 1.845

Axial 1.5311 1.4806 1.5349 1.4803 0.940

Fig. (5-3) shows the variation of sway mode frequency with pile

embeddment length (ranged between 8-80 m) for different pile tip support and

considering multi types of mass representations. These figures indicates that the

natural frequency increases with increasing the pile embeddment length when

the pile tip is modeled as a spring. But it decreases when the pile tip is fixed or

hinged for different mass approximations. It is shown that there is a limiting

value for the pile embeddment length after which there is no change in the value

of the natural frequency for different pile tip condition and mass representation.

The increase of the pile length will increase the flexibility of the entire structure

in case of fixed and hinged support. In addition the results of hinge support case

are close to the fixed support case for all mass representations and at all lengths

of piles. Moreover, the rates of change of frequency w. r. t. the pile length for

spring pile tip is greater than that for hinge or fixed support. It is also shown that

the natural frequency for sway mode be the same for different types of mass

approximations. The inclusion of added mass will decrease the natural

frequency due to increasing of the total mass by (2.6%) for consistent mass

approximation and (2.63%) for lumped mass approximation in the fundamentalsway mode and so on.

Table (5-1): Natural Frequencies for Basic Modes of Vibration for

Different Models.






3

embeddment pile length, pile tip conditions, direction of wave propagation w.r.t.

the structure, and wave length to legs spacing ratio.

To study the action of wave forces on the dynamic behavior of the

offshore platform model shown in Fig. (5-1), the following wave parameters

have been considered [3]:

Wave height = 21 m

Wave period = 12 sec

Wave length = 225 m

Water depth = 115 m

Water density = 1025 kg/m3

The assumed value of the viscous damping ratio is (5%) for all modes of

vibration considering that (2%) as hydrodynamic damping, whereas the

remaining (3%) simulates energy dissipation from sources other than

hydrodynamics which is called structural damping [3,12]. The inertia coefficient

(Cm) and drag coefficient (Cd) are taken as (2.0) and (0.8) respectively.

Fig. (5-7) show the force-time curve for node (A) as shown in Fig. (5-1)

due to wave loads in three directions using Morison’s equation {Eq.(4-21)}.

Fig. (5-8) and Fig. (5-9) that (for the case of consistent mass

approximation with added water mass, and spring support for pile tip) the

dynamic response for both the bending moment at deck level and axial force in

seabed level (member CD) respectively. It is shown that the two curves have a

similar behavior and both of them reached a steady state condition after one

period of time only (24 sec). Fig. (5-10) shows a comparison of the deck

displacement obtained in the present work with that of isolated springs model

adopted by Al-Salihy [3]. It is shown a significant difference in the value of the

amplitude of vibration between the two models. It is expected model gives less

amplitude than that results from the isolated springs model because of

considering the exact stiffness coefficient for soil-structure interaction.




3

The parameters which are adopted in the forced vibration analysis of

case study (1) are:-

Consistent and Lumped Mass Approximations:

The masses of the structural element (super structure and piling system)

are represented by consistent and lumped mass approximations. For both types

of representations, the effect of displaced mass of sea water are considered for

the members embedded in sea water to represent the volume of fluid displaced

by the submerged members.

It can be shown in Fig. (5-11) that there is no large difference in thedisplacement of deck for different mass models. However there is a little

difference when the submerged water mass is added to the structural mass but

this difference is small when compared to maximum dynamic amplitude.

Embedded Pile Length:

Fig. (5-12) shows the variation of the max. amplitude of vibration for the

deck supported by piles with different lengths (5-80 m). It can be seen that there

is a decreasing in the deck displacement with increasing the pile length for

spring support, while, the deck displacement will increase when the pile tip is

fixed. The pile tip effect can be neglected when the depth of pile is greater than

(80 m). In the case of fixed support the increasing of pile length will increase the

flexibility of the structure and hence decrease the natural frequency. This is due

to the increase of the slenderness ratio of the whole structure in the case of the

fixed pile tip. In the case of spring support the increases of pile length will

increase the whole stiffness of the whole structure and hence decrease the deck

displacement.

Modeling of Boundary Conditions at Pile Tip:

Fig. (5-13) shows the Max. deck displacement for different types of pile

tip conditions. Three types of pile end conditions are considered, these are




3

spring, hinge, and fixed support for normal length of piles (60m). The mass is

modeled using consistent mass model including the added fluid mass. It is

shown for this length that there is no large difference in the time history curves

and the value of amplitude of deck with different end conditions. Fig. (5-14)

shows the deflected shape of the pile. It is shown that the lateral deformation of

the pile is diminished after one third of the pile length. The end condition of the

pile does not effected the lateral deflection of the pile for this length of pile (60

m).

Direction of Wave Propagation:

At offshore structures the random wind directions will cause wave loads

act at different directions on the structure. The three dimensional wave theories

are quite complex, therefore, for a plane wave propagation the two-

dimensional wave theories are commonly used. In present study an extension for

two dimensional wave theories are used to contain arbitrary wave directions

with respect to the structure direction by using an angle (α). Due to symmetry of

the structure, angle (α) will be ranged between (0-45o). For each increment

(7.5o) of the angle (α), the dynamic analysis is performed for the structure to

show the variation of wave effects with different wave directions.

Fig. (5-15) shows the variation of maximum amplitude of deck at x and

y-directions with the angle (α). It shows that the value of deck displacement in

y-direction is very small as compared to deck displacement in x-direction at

(α=0). Fig. (5-16) shows the variation of deck rotation with the angle (α). The

figure shows that there is no rotation about the x-axis when the value of (α=0).

For the same value of the angle (α) the rotation about the y-axis is maximum.

The torsional rotation (θz) of the entire structure concentrated at the top level of

deck are maximum when (α=22.5o). Fig. (5-17) shows the variation of axial

force at the deck and sea bed with the angle (α). Fig. (5-18) shows the variation

of bending moment with the angle (α). It is shown that the maximum values of

axial forces are when the angle (α) is zero, while the maximum value of bending




6

moment are when the angle (α=45o). This is due to load distribution as a result

to wave inclination.

Ratio of Wave Length to Legs Spacing:

In most actual sea states, waves occur at random nature with multi

values of frequencies and wave lengths. In present work one period and one

wave length is used. Fig. (5-19) shows the wave profile for some patterns of

wave length as a ratio to the spacing between two adjacent platform legs (S

L)

which plays a significant role in the dynamic response of the whole structure.

The dynamic analysis for some special patterns of wave length to the legs

spacing ratio (S

L=1,4/3,5/6,2,3,4) are performed.

Table (5-2) shows for each ratio of wave length to legs spacing the wave

characteristic of the sea state and the wave height of (5 m) is adopted. The

natural period of platform is about (T=2.7 sec) which is less than the wave

period for all cases. This values are obtained for each (S

L) ratio by using the

following equations and using spacing between each two adjacent piles (25m)

(Ref. [39]).

Wave Length (L) = ( S/L )*25 … (5-1)

Wave Number (k) = L/2 … (5-2)

Wave Period (T) = )kdtanh(g/()L2( … (5-3)

Wave Celerity (C) = T/L … (5-4)

Wave Frequency (f) = 2/)kdtanh(gk … (5-5)




Wave

Length toLegs

Spacing

(L/S)

WaveLength

(L)(m)

WaveNumber

(k)

WavePeriod

(sec.)

WaveCelerity(C)

(m/sec.)

WaveFrequency

(Hz)

1 25 0.251 4.002 6.247 1.57

4/3 33.333 0.188 4.622 7.212 1.359

5/6 41.667 0.151 5.175 8.051 1.214

2 50 0.126 5.688 8.79 1.105

3 75 0.084 7.139 10.505 0.88

4 100 0.063 8.597 11.632 0.731

Fig. (5-20) shows the deck response, for (S

L=1). It is shown that the

steady state response is reached after four periods. While for the case of (S

L=2),

two periods are sufficient to reach a steady state response as shown in Fig. (5-

21). Fig. (5-22) shows that three periods are necessary to reach the steady state

response for the case of (S

L=4). In Fig. (5-23), (

S

L=3

4) six periods are required

to reach a steady state response and various shapes in each time period. This

variety is due to the various conditions of the loading cases where one leg may

be reached the maximum load and the others will have zero load. For the ratio

( 3S

L ) the amplitude of vibration increases with time until it reaches a steady

state response after nine period as shown in Fig. (5-24). Fig. (5-25) shows that

Table (5-2): Wave Characteristics for Different Wave Length to Legs

Spacing (L/S).




when (3

5

S

L ) the deck response reaches a steady state condition after five

periods. The maximum amplitude for this case will occur at (t=21 sec).

From the previous figures (5-20) to (5-25) it can be seen that for each

case the shape of these figures differ from each other, due to dynamic behavior

of the structural system as well as to the load patterns, see Fig. (5-19).

Fig. (5-26) shows the variation of the maximum amplitude of vibration in

x-direction with (S

L), it shows that the range of (1<

S

L<4) will give minimum

dynamic amplitude is obtained and acts between the values of (SL =3 to 4) and

the maximum amplitude will occur between (S

L=3 to 4).

Fig. (5-27) shows the maximum deck displacement in y, and

z-directions. It shows that the variation in z-direction is similar to that in x-axis

{see Fig. (5-26)} but the displacement in y – direction is increased with the

increase of the specified values of (S

L). In addition Fig. (5-28) shows the

rotation about x, and y axes. It is shown that the rotation θy-varies in the same

way for that in the x, and z-translation (ux, uz). While the rotation θx is increased

with increasing )S

L( . The value of θx are very small compared with θy, this is due

to the small values of deck displacement in y-direction compared to

displacement in x-direction. Fig. (5-29) shows the maximum amplitude of A .F.

at the deck and sea bed and Fig. (5-30) shows the maximum amplitude of B. M.

at the deck and sea bed for the specific change in (S

L).




5

5-3: Case Study (2):Al-Amaya Berthing Dolphin:

The second model adopted in this work is the berthing dolphin structure

in berth No. 8 of Khor Al-Amaya oil terminal as shown in Fig. (5-31). Wave

loads and impact loads have been considered, the effect of added mass is

considered for the case of wave loads only.

5-3-1:Soil-Structure Interaction:

As shown in Fig. (5-31), the berthing dolphin is supported by eight steel

piles, which are driven to a depth of (44 m) below the mudline in the soil from

sea bed. There are no data available about the soil properties. Therefore, it will be assumed that the soil is clayey or sandy soil having a modulus of elasticity

(45 MPa) and Poisson’s ratio (0.3). The clayey soil has a constant modulus of

subgrade reaction with depth. While the linear distribution is used for sandy soil

beginning from zero at the sea bed level to its maximum value (as calculated

from Eq. (3-2) at pile tip.

5-3-2: Free Vibration Analysis:

The free vibration analysis is performed by considering the soil as a clay

in the first application and sand in the second application including the effect of

added mass by using consistent mass approximation only. Table (5-3) shows the

natural frequencies for different cases adopted in this research and the results of

isolated springs model reported by Al-Salihy [3]. It is clear that the frequencies

obtained by the isolated springs model are less than the frequencies that is

obtained in the present study for different cases. Fig. (5-32) shows respectively,

the sway, torsion, bending, and axial mode shapes of the structure. Fig. (5-33)

shows the effect of embeddment pile length on the natural frequency for the

fundamental sway mode. It shows that for all cases, the frequency increases with

the increase of the pile length for spring support and decreases when the pile tip

is fixed (for end bearing piles). The effect of pile tip on natural frequency can be




neglected if the embeddment pile length will be greater than (50m). Fig. (5-34)

shows the variation of torsion mode frequency with pile embeddment length. It

is shown that the effect of pile tip on natural frequency can be neglected when

the length of embeddment is (30m) for sand without added mass. While for the

other cases the embeddment pile length does not affect on the frequency when

the pile embeddment length is more than (40m). For bending mode as shown in

Fig. (5-35) an embeddment length of (50m) is required to neglect the effect of

end condition of the pile tip. For the axial mode Fig. (5-36), the embeddment

pile length required to equalize the natural frequency for spring and fixed

support is greater than (50m). Fig. (5-37) shows the effect of type of soil and

consideration of added mass on the natural frequency for the four major modes

with spring pile tip support. It shows an increase of frequency with the increase

of embeddment pile length for all modes of vibration. It is shown that the soil

type has no effect on the frequency for torsion mode only. But there is a noticed

difference in the values of natural frequencies with soil type for the other modes.

Fig. (5-38) shows that the type of soil has no effect on the natural frequency fortorsion mode in the fixed pile tip, while the added mass reduces the natural

frequency. This is because that the pile legs of the dolphin prevents the

superstructure from rotation and no effect to the torsional stiffness of piles at

different length of piles on the torsional stiffness of the entire structure.




3

Mode

Number

Natur al Frequency (Hz)

Present Study

Al-salihy Clayey

Soil

Sandy

Soil

Clayey

Soil with

Added

Mass

Sandy

Soil with

Added

Mass

1 4.1476 3.9421 3.0356 2.8771 2.816

2 5.5215 5.5363 4.1313 2.9057 3.097

3 8.7067 8.1487 6.9439 4.1378 5.485

4 11.204 11.179 7.0671 6.5372 7.104

5 12.706 11.731 8.8534 8.8797 8.798

6 13.559 12.389 10.782 10.508 9.106

5-3-3 : Forced vibration Analysis :

Two types of loading are considered in the dynamic analysis of the

berthing dolphin, these are wave loads and the berthing impact loads.

Wave Loads:

To find the dynamic response of the dolphin structure to wave loads the

following data are used for the wave characteristics that collected from the

information related to the Arabian Gulf [3]:

Wave height = 11 m Wave period = 13 sec

Wave length = 261 m Water depth = 21.4 m

Water density = 1025 kg/m3 Inertia coefficient (Cm) = 1.45

Drag coefficient (Cd) = 0.35

Other data are detailed in the appendix.

Table (5-3) : Natural Frequencies for Basic Modes of Vibration for

Different Models.




Fig. (5-39) shows the time history of fluid and structural velocities at

(Node B). This curve illustrates that the non-linear drag force can be linearized

(by neglecting the structural velocity compared to fluid velocity) without an

immerse error for the first (1 sec) and very little above (1 sec) time of calculated

responses, this error is overrided by the addition of hydrodynamic damping (2%)

to the structural damping.

Fig. (5-40) shows the maximum deck displacement of Al-Amaya

berthing dolphin for the wave loads only. It is shown that the amplitude of

vibration for the case of isolated springs model (adopted by Al-Salihy [3]) is

more than the value resulted in the present work. It is also shown that the

numerical solution in the adopted model is more stable especially in the first

four seconds. Fig. (5-41) shows the max. deck displacement for the assumed

types of soil including added mass effect. The amplitude of deck displacement

for sandy soil is more than that in the clayey soil because of the linear

distribution of subgrade reaction. It is shown at first (2 sec.) the fluctuation due

to free vibration effect. Fig. (5-42) shows the deck response without added mass.

Fig. (5-43) shows the effect of added mass on deck response for clayey

soil. It shows that the added mass cause an increase in the displacement due to

the including of inertia force. For sandy soil, the effect of considering the added

masses does not appear as shown in fig. (5-44).

Fig. (5-45) shows the variation of maximum bending moment at the

level of the deck for clayey and sandy soils with and without considering added

masses. It is shown that the type of soil and the considering of the added masses

do not affect the values of Bending moment. The same behavior is shown in Fig.

(5-46) for the response of axial force at sea bed level. Fig. (5-47) illustrates the

lateral deflected shape of the pile with length for the clayey and sandy soils

including added mass effects. It can be seen that the inclusion of added mass

increases the displacement of piles for both clayey and sandy soils. Also the pile

displacement in the case of sandy soil is more than the displacement in the case




of clayey soil. In Fig. (5-48) it is clear that for the clayey soil the axial

deformation is more than the deformations in the case of sandy soil. The axial

deformation for both sandy and clayey soils increased with the increase of the

total mass by considering added mass.

Berthing Impact Loads:

The berthing dolphin was designed to withstand the impact loads that

come from the ship berthing by providing the dolphin C2000H rubber type cell

fender. The fender rubber grade is selected to develop a reaction force on the

structure to 1397.4 kN and 1484.7 kN and to absorb an energy equal to 1397.4

kN.m and 1227.4 kN.m corresponding to a related deflection in the fendering

system equal to 52.5% and 55% respectively [34, 3].

The load-time curve for the three different velocities is shown in

Fig. (5-49) which is plotted by Al-Jasim [12] due to impact loads. Fig. (5-50)

shows the deck response for clayey soil for each case of the three velocities. It

can be seen that the duration time (the time required to reach zero load) increase,the time history curve will be smoother because the structure at that time reaches

a steady state. This is clear also for sandy soil in Fig. (5-51) which has larger

amplitude than that for clayey soil due to the linear distribution of subgrade

reaction used for sandy soil.

Fig. (5-52) shows a comparison for the maximum deck displacement

resulted from the present work (clayey and sandy soils) with the isolated springs

model adopted by Al-Salihy [3] for velocity (1)(which has a duration time 12

sec and rising time (2.5 sec)). It is clear that the deck displacement for the

clayey soil response is less than the displacement for sandy soil case and both of

them have amplitude for deck displacement less than the isolated springs model.

Figs. (5-53), and (5-54) show the max. deck displacement for velocities (2) and

(3) respectively. They show that the deck displacement for clay soil is less than

the deck displacement for sandy soil as mentioned above.




It can be seen from the figures mentioned above that the efficiency of the

fendering system selected to absorb the energy of ship's impact on the dolphin

and that the response of the structure to impact loading is approximately similar

to static response. This is because that the reaction force has a relatively large

rising time (2.5 sec) and that the increase of rising time will decrease the

dynamic effect of the loads.



Applications, Results, And Discussion Chapter Five

96

Fig. (5-1): Geometry and Dimensions of Jacket Platform. 60m

115m

20m

5m

Deck

W.L.

Mud

Line

Node A Member AB Member CD

E Steel= 200 Gpa

ν Stees= 0.3

E Soil= 45 Gpa

ν Soil= 0.3



Chapter Five Applications, Results, And Discussion

70

Fig. (5-2): Fundamental Mode shapes of Jacket Platform.

Sway Mode

Torsion Mode




71

Axial Mode

Fig. (5-2): Continued.

Bending Mode



Chapter five Applications, Results, and Discussion

27

0 20 40 60 80

Length of Piles (m)

0.24

0.28

0.32

0.36

0.40

0.44

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

(a):Consistent Mass with Added

Mass 0 20 40 60 80

Length of Piles (m)

0.24

0.28

0.32

0.36

0.40

0.44

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

(b):Consistent Mass

0 20 40 60 80

Length of Piles (m)

0.24

0.28

0.32

0.36

0.40

0.44

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

(c):Lumped Mass with Added

Mass (D):Lumped Mass

0 20 40 60 80

Length of Piles (m)

0.24

0.28

0.32

0.36

0.40

0.44

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

Figure (5-3): Variation of Sway Mode Natural Frequency with Pile

Embeddment Length.

Spring

Hinge

Fixed

Spring

Hinge

Fixed

Spring

Hinge

Fixed

Spring

Hinge

Fixed




2

0 20 40 60 80

Length of Piles (m)

0.726

0.728

0.730

0.732

0.734

F r e q u e n c y (

H z )

Spring

Fixed

Hinge

0 20 40 60 80Length of Piles (m)

0.724

0.726

0.728

0.730

0.732

F r e q u e n c y (

H z )

Fixed

Hinge

Spring


0.724

0.726

0.728

0.730

F r e q u e n c y

( H z )

Fixed

Hinge

Spring


0.704

0.706

0.708

0.710

0.712

0.714

F r e q u e n c y

( H z )

Hinge

Spring

Fixed

(a):Consistent Mass with

Added Mass (b):Consistent Mass

(d):Lumped Mass(c):Lumped Mass with Added

Mass Figure (5-4): Variation of Torsion Mode Natural Frequency with Pile

Embeddment Length.

Fixed

Hinge

Spring

Spring

Fixed

Hinge

Hinge

Spring

Fixed

Fixed

Hinge

Spring




2

0 20 40 60 80

Length of Piles (m)

1.51

1.51

1.52

1.52

1.52

1.52

F r e q u e n c y

( H z )

Spring

Hinge

Fixed


1.41

1.42

1.43

1.44

1.45

F r e q u e n c y

( H z )

Fixed

Spring

Hinge

(a):Consistent Mass with Added

Mass (b):Consistent Mass

0 20 40 60 80

Length of Piles (m)

1.51

1.51

1.51

1.52

1.52

F r e q u e n c y

( H z )

Fixed

Hinge

Spring


1.41

1.42

1.43

1.44

1.45

F r e q u e n c y

( H z )

Fixed

Hinge

spring

(c):Lumped Mass with Added

Mass (d):Lumped Mass

Figure (5-5): Variation of Bending Mode Natural Frequency with Pile

Embeddment Length.

Fixed

Spring

Hinge

Spring

Hinge

Fixed

Fixed

Hinge

Spring

Fixed

Hinge

Spring




2

0 20 40 60 80

Length of Piles (m)

1.00

1.20

1.40

1.60

1.80

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

0 20 40 60 80

Length of Piles (m)

1.00

1.20

1.40

1.60

1.80

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

(a):Consistent Mass with

Added Mass (b):Consistent Mass


1.00

1.20

1.40

1.60

1.80

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

0 20 40 60 80

Length of Piles (m)

1.00

1.20

1.40

1.60

1.80

F r e q u e n c y

( H z )

Spring

Hinge

Fixed

(d):Lumped Mass(c):Lumped Mass with Added

Mass Figure (5-6): Variation of Axial Mode Natural Frequency with Pile

Embeddment Length.

Spring

Hinge

Fixed

Spring

Hinge

Fixed

Spring

Hinge

Fixed

Spring

Hinge

Fixed




76 0 5 10 15 20 25 30

Time sec

-200.0

0.0

200.0

N o d a l L o a d s

( k N )

FX

FY

FZ

Fig. (5-7): Wave Forces at Node (A) for case Study (1).




77 Fig. (5-9): Time Variation for the Maximum Axial Force at theLevel of Sea Bed Due to Wave Forces.

Fig. (5-8) : Time Variation for the Maximum Bending Moment at

the level of Deck due to Wave Forces.

A x i a l F o r c e ( N )

B e n d i n g M o m e n t ( N . m

)






0 5 10 15 20 25 30 3

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

M a x . D e c k D i s p l a c e m e n t ( m ) .

Added Consistent

Added Lumped

Lumped Mass

Consistent Mass

Time sec.

Fig.(5-12) : Effect of Pile Embeddment Length on Maximum Deck

Displacement.

Fig. (5-11) : Time Variation of Maximum Deck Displacement for Different

Representations of Masses.

0 20 40 60 80

Length of Pile (m)

50.00

90.00

130.00

M a x .

D e c k A m p l i t u d e ( m m )

Spring

Fixed

79




80

Fig.(5-13) : Time Variation of Max. Deck Displacement for

Different Types of End Support Conditions.

-0.002

-0.0015

-0.001

-0.0005

0

M a x .

D e f l e c t i o n ( m ) .

0 10 20 30 40 50 60

Embeded Length of Piles (m).

Spring

Hinge

Fixed

Figure (5-14): Lateral Deformation of the Pile for Different EndConditions.

0 5 10 15 20 25 30 35 40

Time (sec.)

-0.06

-0.02

0.02

0.06

-0.08

-0.04

0.00

0.04

0.08

M

a x .

D e c k D e f l e c t i o n ( m ) .

Spring

Hinge

Fixed

M a x .

D e c k D i s p l a c e m e n t ( m

)




81

0

10

20

30

40

50

60

70

80

90

0 7.5 15 22.5 30 37.5 45

Angle (Degrees)

D e c k D i s p l a c e m

e n t ( m m ) X-Direction

Y-Direction

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

7.00E-04

0 7.5 15 22.5 30 37.5 45 52.5

Angle (Degrees)

D e c k R o t a t i o n s ( R a d . )

Rotation-X-Dir.

Rotation-Y-Dir.

Rotation-Z-Dir.

190

240

290

340

390

440

490

0 7.5 15 22.5 30 37.5 45 52.5

Angle (Degrees)

A x i a l F o r c e t ( k N

A.F. at Deck.A.F. at Sea Bed.

α

α

Fig. (5-15) : Variation of Max. Deck Displacement with Angle (α) in

x and y- directions. .

Fig. (5-16) : Variation of Deck Rotations with Angle (α) about x,y

and z-directions. .

Fig. (5-17) : Variation of Axial Force with Angle (α). α




82

7000

7100

7200

7300

7400

7500

7600

0 7.5 15 22.5 30 37.5 45

Angle (Radians)

B e n d i n g M o m e n t ( k N . m

)

α Fig. (5-18) : Maximum Dynamic Response B.M. at Deck for

Different Values of Load Inclination (α) (Consistent Mass with

Added Mass Approximation).

(L/S)=1

(L/S)=2

(L/S)=4

(L/S)=4/3

(L/S)=5/3

(L/S)=3

Fig. (5-19): Wave Patterns on each pair of Legs.

(Degrees)




83 Fig. (5-21) : : Time Variation of Max. Deck Displacement for

(L/S=2).

(sec.)

D e c k D i s p l a c e m e n t ( m m )

Fig. (5-20) : Time Variation of Max. Deck Displacement for

(L/S=1).

(sec.)

D e c k D i s p l a c e m e n t ( m m

)




84

Fig. (5-22) : : Time Variation of Max. Deck Displacement for

(L/S=4).

(sec.)


12.0

10.0

8.0

6.0

4.0

2.0

0.0

-2.0

-4.0

-6.0

-8.0

Fig.(5-23) : : Time Variation of Max. Deck Displacementfor (L/S=4/3).

(sec.)







86

0

1

2

3

4

5

6

7

8

0.75 1.5 2.25 3 3.75 4.5

Wave Length to Pile S pasing Ratio (L/S)

D i s p l a c e m e

n t ( m m )

0

20

40

60

80

100

120

140

160

0.75 1.5 2.25 3 3.75 4.5

Wave Length to Legs S pacing Ratio (L/S)

M a x . D

i s p l a c e m e n t ( m m ) x 1 0 ^ -

Displacement

in y-Direction

Displacement

in X-direction

0.000000

0.000005

0.000010

0.000015

0.000020

0.000025

0.75 1.5 2.25 3 3.75 4.5

Wave Length to Legs Spacing Ratio (L/S)

D e c k R o t a

t i o n

( r a d . )

Rot. about x-dir. (rad)Rot. about y-dir. (rad)

Fig. (5-26) : Effect of the Ratio (L/S) on the Max. Deck Displacement at

X-Direction. Fig. (5-27): Effect of the Ratio (L/S) on the Max. Deck Displacement at

Y, and Z-Directions.

Fig. (5-28) : Effect of the Ratio (L/S) on the Max. Deck Rotation about X,

and Y-Directions.

M a x .

D i s p l a c e m e n t ( m m ) x 1 0 - 3

Rot. about x-axis (rad.)Rot. about y-axis (rad.)

-yin




87

0

100

200

300

400

500

600

0.75 1.5 2.25 3 3.75 4.5

Wave Length to Legs Spacing Ratio (L/S )

M a x . A . F . ( k N )

Sea Be d

Deck

0

5

10

15

20

25

30

35

0.75 1.5 2.25 3 3.75 4.5

Wave Length to Legs Spacing Ratio (L/S)

M a

x . B . M . ( k N . m )

Sea Bed

Deck

Fig. (5-29) : Effect of the Ratio (L/S) on the Max. Axial Force.

Fig. (5-30): Effect of the Ratio (L/S) on the Max. Bending Moment.




88 Fig.(5-31) : Geometry of Al-Amaya Berthing Dolphin.

10.44 10.44

1.274 m

20.88 m

1.274 m

15.944 m 7.62 m

10.87 m

11.714 m

11.714 m

10.44 m 7.62 m

4.57 m

2.13 m 7.31 m

7.62 m

7.62 m

44 m

3.0 7.62 7.62

7.62 m 15.944 m

Direction of Ships

Berthing

11.714 m 11.714 m

Node B E Steel= 200 Gpa

ν Stees= 0.3

E Soil= 45 Gpa

ν Soil= 0.3




89 Fig.(5-32): Fundamental Mode Shapes of Al-Amaya Berthing

Dolphin.

SWAY MODE

TORSION MODE




90 Fig. (5-32): Continued. AXIAL MODE

BENDING MODE




91 Fig. (5-33

): Variation of Fundamental Sway Mode with Pile LengthEmbeddment.

10 20 30 40 50 60

Embedded Length of Pile (m)

2.60

2.80

3.00

3.20

F r e q u a n c y

( H z )

Spring

Hinge

(a):Clay with Added Mass.

Fixed

10 20 30 40 50 60Embedded Length of Pile (m)

3.60

3.80

4.00

4.20

4.40

F r q u a n c y

( H z )

Spring

Hinge

(c): Clay without Added

Mass.

Fixed

(d):Sand without Added

Mass.


3.50

3.70

3.90

4.10

4.30

F r e q u a n c y

( H z )

Spring

HingeFixed

10 20 30 40 50 60Embedde Length of Pile (m)

2.50

2.70

2.90

3.10

F r e q u a n c y

( H z )

Spring

Hinge

(b):Sand with Added Mass.

Fixed

Fixed

spring

Fixed

Spring

Fixed

Spring

Fixed

Spring




92 Fig.(5-34): Variation of Torsional Mode Frequency with Pile length

Embeddment.

(a):Sand with Added

5 10 15 20 25 30 35 40 45 50 55 60


4.04

4.06

4.08

4.10

4.12

4.14

F r e q u a n c y

( H

z )

Spring

HingeFixed

(c):Sand without Added Mass

5 10 15 20 25 30 35 40 45 50 55 60


5.36

5.40

5.44

5.48

5.52

5.56

F r e q u a n c y

( H z )

Spring

HingeFixed 5 10 15 20 25 30 35 40 45 50 55 60

Embedded Length of Pile(m)

5.48

5.50

5.52

5.54

F r e q u a n c y

( H z )

Spring

Hinge

(d):Clay without Added

Mass

Fixed

5 10 15 20 25 30 35 40 45 50 55 60


4.10

4.11

4.12

4.13

4.14

F r e q u a n c y

( H

z )

Spring

Hinge

(b):Clay with Added Mass

Fixed Fixed

Spring

Fixed

Spring

Fixed

Spring

Fixed

Spring




93 Fig.(5-35): Variation of Bending Mode Frequency with Pile length

Embeddment.

10 20 30 40 50 60


7.70

7.90

8.10

8.30

8.50

F r e q u a n c y

( H z )

Spring

Hinge

(d):Sand without Added Mass

10 20 30 40 50 60


8.10

8.30

8.50

8.70

8.90

F r e q u a n c y

( H z )

Spring

Hinge

(c):Clay without Added Mass

Fixed

10 20 30 40 50 60


6.60

6.70

6.80

6.90

7.00

7.10

F r e q u a n c y

( H

z )

Spring

Hinge

(a):Clay with Added

Fixed

10 20 30 40 50 60


6.20

6.30

6.40

6.50

6.60

6.70

F r e q u a n c y

( H

z )

Hinge

Spring

(b):Sand with Added Mass

Fixed Fixed

Spring

Fixed

Spring

Fixed

Spring

Fixed

Spring




94 Fig. (5-36): Variation of Axial Mode Frequency with Pile length

Embeddment.

10 20 30 40 50 60


6.60

6.70

6.80

6.90

7.00

7.10

F r e q u a n c y ( H z )

Spring

Hinge

(a):Clay with Added

Mass

Fixed


9.20

10.20

11.20

12.20

13.20

14.20

F r e q u a n c y

( H z )

Spring

Hinge

(d):Sand without Added Mass

Fixed

10 20 30 40 50 60


10.00

11.00

12.00

13.00

14.00

15.00

16.00

F r q u a n c y ( H z )

Spring

Hinge

(c):Clay without Added

Mass

Fixed

10 20 30 40 50 60


6.20

6.30

6.40

6.50

6.60

6.70

F r e q u a n c y ( H z )

Hinge

Spring

(b):Sand with Added

Mass

Fixed Fixed

Spring

Fixed

Spring

Fixed

Spring

Fixed

Spring




95 Fig.(5-37): Natural Frequency of Different Soils With Different MassRepresentation for Multi Types of Modes(Spring Support).

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

2.400

2.800

3.200

3.600

4.000

4.400

F r e q u e n c y

( H z )

Add Clay

Add Sand

Clay

Sand

(b):Sway Mode (a):Torsion Mode

15 25 35 45 55

10 20 30 40 50 60Length of Piles (m)

4.000

4.400

4.800

5.200

5.600

F r e q u e n c y ( H z )

Add Sand

Add Clay

Sand

Clay

(d):Axial Mode

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

6

8

10

12

14

F r e q u e n c y

( H z )

Add Clay

Add Sand

Clay

Sand

(c):Bending Mode

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

6.000

7.000

8.000

9.000

F r e q u e n c y

( H z )

Add Clay

Add Sand

Clay

Sand

♦




96 Fig.(5-38): Natural Frequency of Different Soils With DifferentMass Representation for Multi Types of Modes(Fixed Support).

(a):Sway

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

2.800

3.200

3.600

4.000

4.400

F r e q u e n c y

( H z )

Add Clay

Sand

Clay

Add Sand

(b):Torsion

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

4.000

4.400

4.800

5.200

5.600

F r e q u e n c y

( H z )

Add Sand

Sand

Clay

Sand

(c):Bending Mode

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

6.500

7.000

7.500

8.000

8.500

9.000

F r e q u e n c y ( H z )

Add Clay

Add Sand

Clay

Sand

(d):Axial Mode

15 25 35 45 5510 20 30 40 50 60

Length of Piles (m)

8

10

12

14

16

F r e q u e n c y

( H z )

Add Clay

Add Sand

Clay

Sand

Add Clay




97

Fig. (5-39): Time History Curves for Structural Velocity and Fluid Velocity

at Specified Node Near the Sea Bed. .

1 3 5 7 9 11 130 2 4 6 8 10 12 14

Time (sec)

-6

-2

2

6

-8

-4

0

4

8

V e l o c i t y ( m / s e c )

Fluid

Structural

3 8 13 18 23

0 5 10 15 20 25

Time (sec)

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

M a x . D

e c k D i s p l a c e m e n t ( m )

Present Work

Isolated Springs

Fig. (5-40): Time Variation of Deck Displacement forDifferent Models of Soil-Structure Interaction.






99 Fig.(5-44): Maximum Deck Response of Al-Amaya Berthing Dolphins toWave Loads for Sandy Soils.

Fig.(5-43): Maximum Deck Response of Al-Amaya Berthing Dolphins to

Wave Loads for Clayey Soils.

0 5 10 15 20 25Time (sec)

-0.175

-0.125

-0.075

-0.025

0.025

0.075

0.125

0.175

M a x . D

e c k D i s p l a c e m e n t ( m )

Add Mass

No Add Mass

3 8 13 18 230 5 10 15 20 25

Time (sec)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

M a x .

D e c k D i s p l a c e m e n t ( m

)

No Add Mass

Add Mass




100

0 5 10 15 20 25

Time (sec)

-1.2E+7

-8.0E+6

-4.0E+6

0.0E+0

4.0E+6

8.0E+6

1.2E+7

1.6E+7

2.0E+7

M a x .

B . M . a t D e c k

( N . m

)

Clay

Sand

0 5 10 15 20 25

Time (sec)

-1.2E+7

-8.0E+6

-4.0E+6

0.0E+0

4.0E+6

8.0E+6

1.2E+7

1.6E+7

2.0E+7

M a x .

B . M . a t D e c k

( N . m

)

Clay

Sand

No Added Mass Added Mass

0 5 10 15 20 25

Time (sec)

-1.2E+7

-8.0E+6

-4.0E+6

0.0E+0

4.0E+6

8.0E+6

1.2E+7

1.6E+7

2.0E+7

M a x .

B . M . a t D e c

k ( N . m

)

Added Mass

No Added Mass

0 5 10 15 20 25

Time (sec)

-1.2E+7

-8.0E+6

-4.0E+6

0.0E+0

4.0E+6

8.0E+6

1.2E+7

1.6E+7

2.0E+7

M a x .

B . M . a t D e c k ( N . m

)

Added Mass

No Added Mass

Clayey Soil Sandy Soil

Fig. (4-45): Time History for Bending Moments at Deck.




101

Added MassNo Added Mass

Clayey Soil Sandy Soil

Fig. (5-46): Time History for Axial Force at Sea Bed.

0 5 10 15 20 25time (sec)

-4.0E+7

-3.0E+7

-2.0E+7

-1.0E+7

0.0E+0

1.0E+7

2.0E+7

3.0E+7

A x i a l F o r c e ( k

N )

Clay

Sand

0 5 10 15 20 25time (sec)

-4.0E+7

-3.0E+7

-2.0E+7

-1.0E+7

0.0E+0

1.0E+7

2.0E+7

3.0E+7

A x i a l F o r c e ( k N )

Clay

Sand

0 5 10 15 20 25time (sec)

-4.0E+7

-3.0E+7

-2.0E+7

-1.0E+7

0.0E+0

1.0E+7

2.0E+7

3.0E+7

A x i a l F o r c e ( k N )

Added Mass

No Add Mass

0 5 10 15 20 25

time (sec)

-4.0E+7

-3.0E+7

-2.0E+7

-1.0E+7

0.0E+0

1.0E+7

2.0E+7

3.0E+7

A x i a l F o

r c e ( k N )

Added Mass

No Add Mass




102

5 15 25 35 450 10 20 30 40

Length of piles (m)

-0.001

0.001

0.002

0.003

0.004

0.005

0.007

0.008

0.009

S e a B e d D e f l e c t i o n

( m )

Clay

Sand

Added MassNo Added Mass

5 15 25 35 450 10 20 30 40

Length of piles (m)

-0.001

0.001

0.002

0.003

0.004

0.005

0.007

S e a B e d D e f l e c t i o

n ( m )

Clay

Sand

Clayey Soil

5 15 25 35 450 10 20 30 40

Length of piles (m)

-0.001

0.001

0.002

0.003

0.004

0.005

0.007

0.008

S e a B e d D e f l e c t i o n (

m )

Add Mass

No Add

5 15 25 35 450 10 20 30 40

Length of piles (m)

-0.001

0.001

0.002

0.003

0.004

0.005

0.007

0.008

S e a B e d D e f l e c t i o n ( m

)

Add Mass

No Add

Sandy Soil

Fig.(5-47):Lateral Deflected Shapes for Piles in X-Direction for Different

Soils.

Added Mass

No Added Mass

Added Mass

No Added Mass

Clay

Sand

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

L a t e r a l P i l e D e f l e c t i o n

m m

Clay

Sand

9.0

8.0

7.0

5.0

4.0

3.0

2.0

1.0

-1.0

-

1.0 L a t e r a l P i l e D e f l e c t i o n

m m

8.0

7.0

5.0

4.0

3.0

2.0

1.0

-1.0


m m

8.0

7.0

5.0

4.0

3.0

2.0

1.0

-1.0


m m




103 Sandy Soil

5 15 25 35 450 10 20 30 40

Length of Pile (m)

0.000

0.002

0.004

0.006

A x i a l D e f o r m a t i o n o f P i l e s ( m )

Clay

Sand

5 15 25 35 450 10 20 30 40

Length of Pile (m)

0.000

0.001

0.002

0.003

0.004

0.005

A x i a l D e f o r m a t i o n o f P

i l e s ( m )

Clay

Sand

Clayey Soil

5 15 25 35 450 10 20 30 40

Length of Pile (m)

0.000

0.001

0.002

0.003

0.004

0.005


Add Mass

No Add Mass

0 10 20 30 40Length of Pile (m)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007


Added Mass

No Add Mass

No Added MassAdded Mass

Fig.(5-48):Deflected Shapes for Piles at Different Soils Including Added

Mass Effects for Axial Deformation.

Clay

Sand

Clay

Sand

No Added Mass

Added Mass

No Added Mass

Added Mass5.0

4.0

3.0

2.0

1.0

0.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0 A x i a l D e f o r m a t i o n m

m

A x i a l D e f o r m a t i o n m

m

5.0

4.0

3.0

2.0

1.0

0.0 6.0

4.0

2.0

0.0

A x i a l D e f o r m a t i

o n ( m m )

A x i a l D e f o r m a t i o n

m m




104

2 6 10 14 18 22 26 30 34 38 42 46

0 4 8 12 16 20 24 28 32 36 40 44 48Time (sec)

-0.006

0.000

0.006

0.012

0.018

0.024

0.030

0.036

0.042

0.048

D e c k D i s p l a c e m e n t ( m )

Velocity1

Velocity2

Velocity3

Fig.(5-50):Time Variation of Max. Deck Displacement from Impact Loads

(Clayey Soils).

2 6 10 14 18 22 26 30 34 38 42 460 4 8 12 16 20 24 28 32 36 40 44 48

Time (sec)

-0.006

0.000

0.006

0.012

0.018

0.024

0.030

0.036

0.042

0.048


Velocity1

Velocity2

Velocity3

Fig.(5-49):Time-Force Relation to Impact Loads (Three Velocities) Ref.(12).

0 4 8 12 16 20 24 28 32 3 Ti e sec

0 200 400 600 800

1000 1200 1400

Reacon For ce

kN

Vel ocit y 1 Vel ocit y 2 Vel ocit y 3


(Sandy Soils).

Time (sec) Duration time

Rising time Velocity (1)

Velocity (2)

Velocity (3)

Velocity (1)

Velocity (2)

Velocity (3)

Velocity (1)

Velocity (2)

Velocity (3)

1400

1200

1000

800

600400

200

0

48

42

36

30

24

18

12

6

0

-6

F o r c e ( k N )


48

42

36

30

24

18

12

6

0

-6 D e c k D i s p l a c e m e n

t ( m m )




105

Fig.(5-52):Time Variation of Max. Deck Displacement Due to Impact Loads

(Velocity 1).

1 3 5 7 9 11 13 15 17 19

0 2 4 6 8 10 12 14 16 18 20Time (sec)

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

D e c k D i s p l a c e m e n t

( m )

Clay

Sand

Isolated Springs

0 4 8 12 16 20 24 28 32 36Time (sec)

0.000

0.010

0.020

0.030

0.040

D i s p l a c e m e n t ( m )

Clay

Sand


(Velocity 2).

Fig.(5-54):Time Variation of Max. Deck Displacement from Impact Loads(Velocity 3).

0 5 10 15 20 25 30 35 40 45Time (sec)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050


Clay

Sand

Clay

Sand

Isolated Springs

(Ref. [3]) Present Work

Clay

Sand

Clay

Sand

60

45

40

36

3026

20

16

6

0

-6 D e c k D i s p l a c e m e n

t ( m m )

40

30

20

10

0

60

45

40

36

30

26

20

16

6

0


D e c k D i s p l a c e m e n t ( m m

)





Chapter Six Conclusions and Recommendations

701

Conclusions and Recommendations

6-1: Conclusions:

From the present work some concluding remarks can be drawn as:

1- A new stiffness matrix is used to find the dynamic response of offshore

structures, which is stiffer than isolated spring's model to model the piles.

2- Considering the added mass in the analysis reduce the natural frequency

of the structure and increase the amplitude of vibration in Al-Amaya berthing

dolphin.

3- Consistent and lumped mass approximations give the same results in free

and forced vibration analysis for first ten modes.

4- Boundary conditions and length of piles have an important effect on the

response of deck displacement and then on the entire structure. This effect

will vanish, as the pile length is increased until to a specified length where

the increasing of pile length does not change the response for all types of

restrained in pile tip.

5- For different values of pile lengths, when the pile tip is fixed or hinged the

deck displacement is decreased with increasing pile length, while it

increased when the pile tip is a spring.

6- Wave characteristics which is represented by the ratio of wave length to

legs spacing has important effect on behavior and response of the offshore

structure, the best ratio of (S

L) between (1.667 to 2) may be used to obtain

the economic design.

7- Load inclination (α) with respect to the structure has a significant effect

on forces and deformations in the structure. At (α=0) Deck displacement



Chapter Six Conclusions and Recommendations

70

and Axial Force at sea bed are maximum while Bending Moment at the

deck are minimum.

8- No large errors occur if the non-linear drag term in Morison's equation is

linearized by neglecting the structural velocity, it is small in comparison

to fluid velocity.

9- Natural frequency for different values of pile length and boundary

conditions are sensitive for sway and an axial modes, and less effect for

bending and torsion modes.

10-

For a deflected shape of a pile the displacement variation and then bending moment and shear force can be ignored after one third of pile

length from mud line level.

11- The added mass will increase the deck displacement in clay soil and

has no effect on sandy soil.

12- Dynamic analysis for berthing dolphin is not necessary for the ships

impact loads but it will be important for wave loads.

13- The increase of duration time of impact load and rising time will

increase the decay due to damping effect and decrease the fluctuation.





References



111

References

1) Zienkiewicz O. C., Lewis R. W., and Stagg K. G., "Numerical

Methods in Offshore Engineering". John Wiley and Sons, 1978.

2) Al-A’Anezi A. O. W. "Dynamic Analysis of Submarine Pipelines and

Fixed Offshore Framed Structures under the action of wave forces". M.Sc.

Thesis, Baghdad University, 2002.

3) Al-Salihy S. M. S. "Dynamic Analysis of Offshore Structures by using

Finite Element Method". M. Sc. Thesis, Basrah University, 2002.

4) Stavros A. A., "Dynamic Response of Offshore Platforms to Extreme

Waves Including Fluid-Structure Interaction". Engineering Structures,

Vol. 4, No. 7, 1982.

5) Clough R. W., and Penzien J., "Dynamics of Structures". McGraw-

Hill, Inc., 1975.

6) Wang C. K. "Intermediate structural analysis", McGraw-Hill, New

York, (1983).

7) Fish P. R., Dean R. B., and Heaf N. J., "Fluid-Structure Interaction in

Morison’s Equation f Design of Offshore Structures". J. of Engineering

Structures, vol. 2, No. 1, 1980.

8) Heins C. P. and Chiu L. Y. B., "Dynamic analysis of the Dolphins

subjected to Ship impact, Computers and Structures", June 1981.

9) Ragab A. and Chung C. Fu. "Non-linear Free Vibration of Fixed

Offshore Framed Structures". J. of Computer and Structures V.22 No. 6

1985.

10) Madhujit M. and Sinha S. K., "Modeling of Fixed Offshore Tower in

Dynamic Analysis". Journal of the Ocean Engineering, vol. 15, No. 6,

1988.



11

References

11) Kareem A., Hsieh C. C. and Tognarelli M. A., "Frequency Domain

Analysis of Offshore Platform in Non-Gaussian Seas". Journal of

Engineering Mechanics, June 1998.

12) Al-Jasim S. A. J., "Dynamic Analysis of Offshore Template Structures

with Soil-Structure Interaction". Ph.D. Thesis, University of Basrah, 2000.

13) Othman R. A. and Dawood A.O. "Two and Three-Dimensional

Dynamic Analysis of Submarine Pipelines Under the Action of Wave

Forces". Proceedings of the Second Minia International Conference for

advanced Trends in Engineering, 7-9 April 2002, Minia- Egypt.

14) Rosset J. M., Robert V. W. Ricardo D. "Modal Analysis for Structures

with Foundation Interaction". J. of Structural Division V. 99 No. S. T. 3

March 1973 ASCE.

15) Shalash K. T. "Soil-Structure Interaction by the Finite Element

Method". M.Sc. Thesis, Baghdad University, 1974.

16) Fukashi M. and John E. G., "Matrix Analysis of Structural-Foundation

Interaction". Journal of the Structural Division, vol. 102, No. S T 1, 1976.

17) Feng Z. and Cook R. D., "Beam Elements on Two Parameters Elastic

Foundations. Journal of Engineering Mechanics", vol. 109, No. 6, 1983.

18) Lathem V. E., Back .H. Wynendaele H. Swings, and S. Vos, "The Use

of Boundary Elements to Represent the Far Field in Soil-Structure

Interaction", Fourth International Symposium on Offshore Engineering,

Brazil, 1983.

19) Timoshenko s., and Goodier J. N., "Theory of Elasticity", McGraw-

Hill, New York, 1975.

20) Uz-Zaman Md. M and Desai C. S. and Drum E. C. "Interface Model

for Dynamic Soil-Structure Interaction". J. of Geotechnical engineering V.

110 No. 9 September.



11

References

21) Goodman R. E., Taylor R. L. and Brekke T. L. "A model for the

Mechanics of Jointed Rock". J. of Soil Mechanics and Foundation

Division ASCE V.94 No. SM3. May 1968.

22) Chen Y. Krauthmmer T. "A Combined ADINA Finite Difference

Approach with Substructuring for Solving Seismically Induced Non-linear

Soil-Structure Interaction Problems". J. of Computers and Structures V.32

No. ¾.

23) Haitham M. S. "Non-linear Analysis of Axially Loaded Piles". J. of

Engineering and Technology. University of Technology, Baghdad-

Iraq.

24) Wylie C. G., "Advanced Engineering Mathematics". McGraw Hill

Book Company Kogakusha, Ltd. third edition.

25) Essa M. J. K. and Al-Janabi A. S. I., "Dynamic Analysis of Plane

Frames Partially Embedded in Winkler Elastic Foundation".,Al-MUHANDIS Iraqi Journal J. No. 130, 1997.

26) Abdul-Sattar W. "Dynamic Analysis of Underground Structures by

Finite Element Method". M.Sc. Thesis. Baghdad University 1999.

27) Blaney G. W., Kausel E. and Rosset J. M., "Dynamic stiffness of piles,

Numerical Methods In Geomechanics", International Conference. Vol. III

edited by Desai C. S. 1979.

28) Mauricio A. S., Rosset J. M. and Whitman R. V. "Dynamic Soil-

Structure Interaction". J. of Structural Division, ASCE, V. 98 No. ST7.

July 1972.

29) Novak M. "Vertical Vibration of Floating Piles", J. of Engineering

Mechanics Division ASCE V. 103, No. EM1. February 1977.

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Applied Mechanics ASME September 19, 1963, Paper No. 64-APM-40.



11

References

31) Scott R. F., "Foundation Analysis". Prentice-Hall, Inc., 1981.

32) Carpon F. W., Williams, and Symons M. V. "A parametric study of the

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Waterway, Port, Coastal and ocean Engineering.

33) Okeaso B. and Abdel-Sayed G. "Coefficients of Soil Reaction for

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No. 7 July 1984.

34) Bridgestone Company, "Features of Fender Series". Copy right at

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36) Weaver W. and Johnston P. R., "Finite Element for Structural

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37) Sack R. L. "Structural Analysis", McGraw-Hill Book Company.

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Engineers and Scientists". Printice-Hall, Inc., 1984.

39) McCormick M. E. "Ocean Engineering Wave Mechanics", John Wylie

and Sons, New York 1973.

40) Raw S. S., "The Finite Element method in engineering", Pergamon

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42) Paz M., "Structural Dynamics Theory and Computation". Van

Nostrand Reinhold Company, 1980.

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Hall, NewJersey 1988.





Appendix(A):

Derivation of Exact Stiffness Method:-

The three-dimensional stiffness matrix depends on the exact method that represents the

problem of shear, torsion, axial and bending and then superimposing each stiffness to form the

whole stiffness matrix in three-dimensional case to include the soil-structure interaction.

A-1: Shear and Axial Resistance:

The bar shown in Fig.(A-1) is cut from full size of pile embedded in the soil, it is assumed

that the resistance of the pile friction and /or cohesion of the pile is proportional to the pile

displacement [23, 31, 25] by modulus of subgrade reaction for shear k s:

Fx=0

{-F (x)+F (x)+dF (x)}-K s.u (x).P(x). dx=0

0)x(u).x(F.k dx

)x(dFs ………….(1)

But F(x)=A.σ(x)

=E.A.ε(x)

=E.A.dx

)x(du

dx

dF(x)=E.A.

2

2

dx

)x(ud …………………(2)

Subs. Eq. 2 into Eq. 1 we get: -

E.A.2

2

dx

)x(ud-K S.P.u(x)=0

A.E

P.k

dx

)x(ud s

2

2

u(x)=0

Assume that β=A.E

P.k s

Then:2

2

dx

)x(ud-β2.u(x)=0 ……………….(A-3)

u(x)=A.eβ.x+B.e-β.x……………………(A-4)

*To derive the stiffness coefficients, stiffness concept may be used as:-

k 11 : at x=0 u(0)=1(unit displacement), u(L)=0

k 12 : at x=0 u(L)=1(unit displacement), u(0)=0

k 22 : at x=L u(L)=1(unit displacement), u(0)=0

k 21 : at x=L u(0)=1(unit displacement), u(L)=0

uo=A+B and B=uo-A

F(x) k s.u(x)

x F + dF

dx Fig.(A-1):Pile Subjected to Axial

Force.



LL

L BeAeu

L

o

L

L e)Au(Aeu

Lo

LLL eu)ee(Au

L

L

LL

L

oL

e

e*

ee

euuA

L2

L

Lo

e1

euuA

………….(c)

L2

L

Lo

L2

L2

oo

L2

L

LoL

e1

euu

e1

euu

e1euuuB

L2

L2

o

L

L

e1

eueuB

xL2

L2

o

L

2xL2

L

Lo ee1

eueue

e1

euu)x(u

…….(5)

dx

)x(duEA)x(F

xL2

o

L

L

xL

LoL2 eeueueeuu

e1

EA)x(F

………..(6)

LcothEA

ee

eeEA

e.e1e1

EA

)0(Fk

LL

LL

xL2

L211



Lsinh

EA

2ee

EA

e

e*

e1

eEA2

eee1

EA)0(Fk

LL

L

L

L2

L

LL

L212

LcothEA

Lsinh

LcoshEA

ee

eeEA

e1

1eEA

1ee1

EA

eeeee1

EAk )L(F

LL

LL

L2

L2

L

L2

LLLL

L222

12

LL

LLL

LL2L

L221

k

)Lsinh(

EA

eeeee

EA

eee1e1

EA

)L(Fk

Then the stiffness Matrix will be:

LcothEA)Lsinh(

EA)Lsinh(

EALcothEA

k ……………………..(A-1)

Eq. (A-1) represent the soil-structure interaction effects and the contribution of the pile and

surrounding soil resistance to Axial load on pile, for a simple bar, the stiffness matrix is :



L

EA

L

EAL

EA

L

EA

k ……………………..(A-1-1)

Both of equations (A-1), (A-1-1) are symmetric matrix but they differ in presenting of soil-

structure interaction parameter (β). If β approaches zero, Eq. (A-1) gives Eq. (A-1-1).

A-2: Torsional resistance:

dxP.k )x(dT)x(T)x(T:0Mo =0

0dxP.k dx

)x(dT

We have from strength of material approach:

GJ

)x(Tdd

dx

dGJ)x(T

2

2)(

dx

d GJ

dx

xdT ….(4-8)

0P.k dx

dGJ

2

2

…..(4-9)

GJ

Pk …………(4-10) where: P =2πR 3 ,R: radius of Pile.

xx eBeA)x( ………..(4-11)

A-2-1: Applying Boundary Conditions:

Boundary conditions of the problem is that for a finite length of pile L,

at x=0, θ(0)= θ1, at x=L, θ(L)= θ2

θ1= BA

AB 1 ………………..(A-12-a)

LL2 eBeA

T (x) +dT (x) .k

T (x) dx x

Fig. (A-2): Torsional Bar in Soil

Media.



L

1

L

2 e)A(eA

L2

L

21

e1

eA

………………………..(A-12-b)

L2

L2

1

L

2e1

eeB

…………………..(A-12-c)

x

L2

L2

1

L

2x

L2

L

21 e)e1

ee(e)

e1

e()x(

………………..(13)

]e)ee(e)e[(e1

GJ)x(T

xL

2

L2

1

xL

21L2

………………………..(14)

The stiffness matrix can be evaluated in the same manner of the previous section,

therefore:

k=

)Lcoth(GJ)Lsinh(

GJ

)Lsinh(

GJ)Lcoth(GJ

……………………….(A-2)

whereas for a simple bar subjected to torsion effect the stiffness matrix.

k=

L

GJ

L

GJ L

GJ

L

GJ

…………..(A-2-1)

If ( ) approaches zero Eq. (A-2) will be the same as Eq. (A-2-1).

A-3: Flextural Resistance:

As shown in Fig. (A-3), the resistance of the soil to the lateral loads on beam

rests on soil may be expressed as (k n.y(x) ) where k

n:is the normal modulus of subgrade reaction

then the equation of equilibrium may be presented in the simple bending theory as :-

dV(x)= (x) =k n.y(x)-q(x)=k n.y(x)………………..(A-3-1)

And V(x)=dx

)x(dM………………….( A-3-2)

AndEI

)x(M

dx

y(x)d2

2 ………………..(A-3-3)

where the positive directions as shown in Fig.(A-3) above.

Sub. Eq. (A-3-1) into Eq. (4-3-2).



)x(y.k dx

y(x)dn2

2

…………………………………..(4-3-4)

AndEI

)x(y.k )

dx

M(x)d(

EI

d

dx

y(x)d n

2

2

4

4

0EI

)x(y.k

dx

y(x)dn

4

4

………………………(A-3-5)

This is the D.E. for the flexure problem in the case of one parameter beam on elastic

foundation.

The shear and bending moments become:-

V=3

3

dx

ydEI- …(A-3-6)

2

2

dx

yd-EIM …(A-3-7)

The general solution of a free field (homogeneous form of a problem)(Eq.(A-3-5)) will be

in the form of exact solution as.

)]xL

sinh().xL

[sin(C)]xL

cosh().xL

[sin(C

)]xL

sinh().xL

[cos(C)]xL

cosh().xL

[cos(C)x(y

43

21

…………………( A-3-8)

where: 4 n

I.E.4

D.k L D: Diameter of Pile (or Width of beam).

)]}xL

sinh().xL

cos()xL

cosh().xL

[sin(C

)]xL

cosh().xL

cos()xL

sinh().xL

[sin(C

)]xL

sinh().xL

sin()xL

cosh().xL

[cos(C

)]xL

cosh().xL

sin()xL

sinh().xL

[cos(C{Ldx

)x(dy

4

3

2

1

……………….…… (A-3-9)



)]}xL

cosh().xL

[cos(C)]xL

sinh().xL

[cos(C

)]xL

cosh().xL

[sin(C)]xL

sinh().xL

[sin(C{L

.2

dx

)x(dy

43

212

2

2

2

…………( A-3-10)

)]}xL

cosh().xL

sin()xL

sinh().xL

[cos(C

)]xL

sinh().xL

sin()xL

cosh().xL

[cos(C

)]xL

cosh().xL

cos()xL

sinh().xL

[sin(C

)]xL

sinh().xL

cos()xL

cosh().xL

[sin(C{L

.2

dx

)x(dy

4

3

2

13

3

3

3

……….……..(A-3-11)

Now, to find the integration constants in terms of nodal normal displacements and nodal

rotations at each node for a finite length (L) as: -

At x=0 y0=y(0)=C1 and )CC(Ldx

)0(dy320

At x=L hs.s.Chc.sChs.cChc.cCy(L)y 4321L

And:

)]hs.chc.s(C)hc.chs.s(C)hs.shc.c(C)hc.shs.c(C[Ldx

)L(dy4321L

Or in matrix form: -

L

q(x) M+dM

V(x) M(x)

V+dV K n.y(x)

Fig. (A-3): Beam on Elastic

Foundation.



4

3

2

1

L

L

0

0

C

C

C

C

]hs.chc.s[

L

]hc.chs.s[

L

]hs.shc.c[

L

]hc.shs.c[

L

hs.shc.shs.chc.c

0LL

0

0001

y

y

…

…………………………………………………..(A-3-12)

i.e. {yi}=[V]{ai}

or {ai}=[V]-1{yi}

where [yi] : Is the nodal displacement vector, [V]: The square matrix in Eq. (A-3-12), {ai}:

Vector of integration constants.

Now, to find the nodal loads depend on exact function (eq. (A-3-8)) as:-

V (x)=3

3 )(dEI-

dx

x y

= )}sh.cch.s(C)ch.csh.s(C)ch.csh.s(C)sh.cch.s(C{)L(EI.2 4321

3

and M(x)=2

2

dx

)x(ydEI-

= )}ch.c(C)sh.c(C)ch.s(C)sh.s(C{L

EI2 43212

2

At: x=0 )C(CL

2EIV(0)V 323

3

0

and 42

2

0 CLEI2)0(MM

At: x = L

VL = V (L)

)}hs.chc.s(C)hc.chs.s(C)hc.chs.s(C)hs.chc.s(C{L

EI2 43213

3

ML = M (L)



}hc.cChs.cChc.sChs.sC{L

EI2 43212

2

In the matrix form: -

4

3

2

1

2

2

L

L

0

0

C

C

C

C

hc.chs.chc.shs.s

)hs.chc.s(L

)hc.chs.s(L

)hc.chs.s(L

)hs.chc.s(L

1000

0LL

0

LEI2

M

V

M

V

……..………………………………………….(A-3-13)

i.e. {Fi}=[G]{ai}

where: {Fi}: vector of nodal loads.

[G]: Is the square matrix in Eq. (A-13-3) then:

{Fi}=[G][V]-1 {ai} ………………………………..(A-3-14)

The stiffness coefficients in two dimensional beam on elastic foundation in local

coordinate system may be expressed by taking the nodal displacement vector {ai} as unity, then the

stiffness matrix may be expressed as :-

[k]=[G][V]-1

which is yield [6,17]:

1526

5364

2615

6453

TTTT

TTTT

TTTT

TTTT

k ……………………(A-3)



where: T1=))(sin)((sinhL

))cos()sin()sinh()(cosh(EI222

=LEI4 at =0.00

T2=))(sin)((sinhL


=

L

EI2 at =0.00

T3=))(sin)((sinhL

))cosh()sinh()sin()(cos(EI4223

3

=

3L

EI12 at =0.00

T4=

))(sin)((sinhL


3

=

3

L

EI12 at =0.00

T5=))(sin)((sinhL

))cos()(sin)((sinhEI2222

222

=

2L

EI6 at =0.00

T6=))(sin)((sinhL

))sin().(sinh(EI4222

2

=

2L

EI6 at =0.00

Finally, to get the three dimensional element stiffness matrix for a pile [Eqs. (A-1), (A-2), (A-3)]

will be superimposed to get the stiffness matrix in 3-D. case as shown in Eq. (4-13).









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جا جر د ل ت ب ن ز ك

ا س ه ا ي ف

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