DYNAMIC ANALYSIS OF OFFSHORE STRUCTURES BY USING FINITE

ELEMENT METHOD

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 in the 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.

INTRODUCTION

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

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

In a pile–supported offshore structure which is also called a jacket platform, cylindrical tubular members are commonly used in offshore structure which is the type used as case study in tow location imaginary structure I North sea and Al Amayaa Berthing dolphin

Basic sourses of Loading on Offshore Structures are :

1- Wave loads

2- Impact Loads

3- Wind Loads

4- Earthquake Loads

SOIL – STRUCTURE INTERACTIONIn present study the model adopted to cover the problem of soil structure interaction is the

simplified Winkler model with one parameter the represent beam element embedded in soil with 6 D.O.F. for each node, I derive an exact solution for the D.E. of axial, torsion and bending problems in 3.D. for each element and then assemblage these finite elements to represent full length of the pile embedded in soil.

Dynamic Behavior of PilesWhen 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 . In some cases an extra mass has been added to represent part of the soil.

Winkler ModelThis foundation model has been used for a century . 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:

a- For bending ( Normal Reaction Modulus)q= σ = Kn*y(x)b- Shear Reaction Modulus ζ = Ks*w(x)c- Torsion Reaction ModulusΖφ= Kφ * θ(x)

WINKLER MODEL REPRESENTATION FOR TYPICAL PILE SUBJECTED TO MULTI TYPES OF PROBABLE LOADS AND THEIR

ACTUAL RESISTANCE

MODELING AND MATHEMATICAL FORMULATIONFinite Element Method and representation of Dynamic

Equation of Offshore structure

[M] {u"} + [C] {u'} +[K] {u} = {F (t)}

The stiffness, Mass, and Damping matrix of the elements above the Mud Line is illustrated normally in most of books of finite elements and structural matrix and the stiffness matrix of the beam element embedded in soil media can be derived by solving the D.E. for each DOF and assemble the results for general 3D pile element including soil-structure interaction can be obtained as:

[Ke]=

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

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

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

=

2L

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

ELEMENT FORCE VECTORFor 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.

Morson”s Eq.The well-known Morison Equation is the semi empirical formula to

calculate the wave loads which is represent the load exerted on a vertical cylinder , which 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 so that::

which can be simplified to:

MORSON”S EQ.

this Equation neglect the non-linear terms of drag coefficient while it considers the added mass concept instead of non-linear terms of inertia force

Then the vector force will be :

{F(s)}= . 2D.4

.Cm. { nv (s)}+

2

.D. Cd.{ nv (s)}.{ )s(vn }

HYDRODYNAMICS OF WATER WAVES

HYDRODYNAMICS OF WATER WAVES

Airy’s linear wave theory is used to find velocities and accelerations for the wave equation ( Laplace’s Eq.) on each node for the finite element model and used these velocities and accelerations in Morison's Eq. above to find nodal forces subjected on offshore structure (Jacket type platforms which is adopted in the study.

Applicable if : (H>>L, h).

Velocities in x, z directions ( y – dir. Is the vertical coordinate ) according to Airy’s Linear wave thery is:

And accelerations are :

Vx=x

= )t.kxcos(.)khsinh(

)]hz(kcosh[.

T

H.

Vz=z

= )t.kxsin(.)khsinh(

)]hz(ksinh[.

T

H.

ax=t

vx

= )t.kxsin(.)khsinh(

)]hz(kcosh[.

T

H..2

xt 2

22

az=t

v z

= )t.kxsin(.)khsinh(

)]hz(kcosh[.

T

H..2

zt 2

22

FREE VIBRATION ANALYSIS AND NORMAL MODESFree vibration analysis is obtained to verify which mode is important as

compared to the other modes for each two structures , for the imaginary model in north sea the first four vibration modes is obtained as:

1- Sway Mode

FREE VIBRATION ANALYSIS AND NORMAL MODES2- Torsion Mode

FREE VIBRATION ANALYSIS AND NORMAL MODES

FREE VIBRATION ANALYSIS AND NORMAL MODES4- Axial Mode

DYNAMICS ANALYSIS AND CASE STUDIES

Newmark’s implicit are used to find the dynamic response of the structure to wave, and impact loads.

1- Pile Support type and length of pile and its effecting on the natural frequency and deflection.

WAVE LOAD TIME HISTORY ON SPECIFIED NODE ON DECK

2ND. CASE OF STUDY AL-AMAYAA BERTHING DOLPHIN

FIRST AND SECOND FREE VIBRATION MODES 

THIRD AND FOURTH FREE VIBRATION MODES

EFFECT OF STRUCTURAL VELOCITY AS COMPARED TO FLUID VELOCITY AT

SPECIFIED

EFFECT OF UNIFORMLY DISTRIBUTED WINKLER SPRINGS AS COMPARED TO ISOLATED NODED

ONE 

IMPACT FORCE ON AL-AMAYAA BERTHING DOLPHIN

CONCLUSIONS

The basic points which can be obtained by bresent stuy is:-

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- 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.

3- 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

4- 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.

ANY QUESTIONS