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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.


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5-2-1 SoilStructure 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 Poissons 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 reportedby 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).


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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.


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Fig. (5-4) shows the variation of natural frequency for torsional mode

with pile length. It is shown that the natural frequency does not vary with the

pile embeddment length when the embeddment pile length is greater than (20m)

for different pile end conditions and for all cases of mass representations. When

the length is less than (20m), the frequency decreases with pile length in the case

of fixed support and increases in the case of spring and hinged supports. This is

because that the short pile gives more rigidity to the entire structure in the case

of fixed support. But in the case of spring or hinge supports, the rigidity of

structure is less for the same soil stiffness, therefore, the natural frequency

decreases when the pile length increases.

For the bending mode, Fig. (5-5) shows that the embeddment pile length

beyond which the variation in natural frequency can be neglected is (20 m)

when the fluid added mass is neglected and (30 m) when the fluid added mass is

considered. In fixed pile tip, the frequency decreases when the pile embeddment

length increases. For both spring and hinge pile tip conditions, the increasing of

the pile embeddment cause an increase in the frequency.

Fig. (5-6) shows the variation of axial mode frequency with pile

embeddment pile length. The curves shows a similar behavior for the variation

of natural frequency with pile embeddment with that of sway mode. The

increasing of pile embeddment length cause increasing in the value of natural

frequency for the case of spring pile tip for different mass representations. While

for other end conditions, the value of natural frequency decreases when the pile

embeddment length increases. The variation in the value of natural frequency is

diminished when the pile embeddment length is greater than (60m) for all cases

of pile end conditions.

5-2-3. Forced Vibration Analysis:

Many parameters were studied in the forced vibration analysis of

offshore jacket platform with soilstructure interaction subjected to wave

loading only. These are the mass representation, added mass effects,


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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 Morisons 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.


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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


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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 (). Due to symmetry of

the structure, angle () will be ranged between (0-45o). For each increment

(7.5o) of the angle (), 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 (). It shows that the value of deck displacement in

y-direction is very small as compared to deck displacement in x-direction at

(=0). Fig. (5-16) shows the variation of deck rotation with the angle (). The

figure shows that there is no rotation about the x-axis when the value of (=0).

For the same value of the angle () the rotation about the y-axis is maximum.

The torsional rotation (z) of the entire structure concentrated at the top level of

deck are maximum when (=22.5o). Fig. (5-17) shows the variation of axial

force at the deck and sea bed with the angle (). Fig. (5-18) shows the variation

of bending moment with the angle (). It is shown that the maximum values of

axial forces are when the angle () is zero, while the maximum value of bending


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moment are when the angle (=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)


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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).


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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

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