dynamic models of marine pipelines

7/27/2019 Dynamic Models of Marine Pipelines

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DYNAMIC MODELS OF MARINE PIPELINESFOR INSTALLATION IN DEEP AND ULTRA-DEEP WATERS:

ANALYTICAL AND NUMERICAL APPROACHES

M. CALLEGARI1, C.B. CARINI1, S. LENCI2, E. TORSELLETTI3 and L. VITALI3

1Dipartimento di Meccanica, Universit Politecnica delle Marche, Ancona2Istituto di Scienza e Tecnica delle Costruzioni, Universit Politecnica delle Marche, Ancona

3 Snamprogetti, Fano (PU)

SOMMARIO

La memoria tratta delle problematiche dellinstallazione di condotte sottomarine con il

cosiddetto metodo a J, che consiste nel varare le condotte stesse con lausilio di una rampa

quasi verticale. Lattenzione del lavoro viene focalizzata sul rilevamento della posizione

relativa tra nave appoggio e punto di contatto della condotta con il fondale marino, che

rappresenta un elemento di grande importanza da molti punti di vista: per il tracciamento del

percorso di varo prescritto, per lesecuzione di una installazione sicura ed affidabile e nella

determinazione delle massime tensioni/deformazioni, solitamente concentrate nella sezione di

massima curvatura, e rappresentanti un parametro di progetto ovviamente fondamentale.

Sono considerati modelli analitici e numerici. I primi sono di pi facile utilizzo, catturano

gli aspetti pi importanti del problema e possono essere utilizzati come punto di partenza per

algoritmi iterativi di ricerca numerica della soluzione. I modelli numerici, daltra parte,

permettono raffinamenti, specialmente intorno al punto di contatto con il fondo ed alla

sezione di massima curvatura e permettono la valutazione della rilevanza relativa dei vari

fenomeni meccanici interagenti sullo sfondo.

ABSTRACT

The paper deals with pipeline installation by the so-called J-Lay method, which consists inlaying submarine pipelines with a straight stinger at near vertical angles. Attention is focussed

on the detection of the touch down point (TDP) vessel relative position, which is the

principal point for following a prescribed laying route and having reliable installation, and in

the determination of the maximum stress/strain, usually attained in the section of maximum

bending, which is obviously a fundamental design constraint.

Analytical and numerical models are considered. The former ones are easier to be handled,

capture the principal features of the problem, and may be used as starting condition for

numerical solutions obtained iteratively. The latter models, on the other hand, permit

refinements, especially around the TDP and the section of maximum bending and allow for

the assessment of the relevance of the various underlying mechanical phenomena.

AIMETA 03XVI Congresso AIMETA di Meccanica Teorica e Applicata

16th AIMETA Congress of Theoretical and Applied Mechanics


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1. INTRODUCTION

Submarine crossings are now strategic in a series of new projects of long distance gas

transportation via pipeline. The Mediterranean basin provides relevant examples of pioneer

(and successful, the Transmed three ND 20 and two ND 26 - system) and near-to-come

(the Lybia to Sicily pipeline, detailed engineering in progress) strategic submarine crossings,while the internal seas (Black and Caspian seas, construction completed), Middle-East (the

Oman to Pakistan and Iran to India pipeline projects, at desk study level) and Far-East

(continental China to Japan and Sakhalin to Japan pipeline projects, at desk study level) are

promising concepts. Development plans are now considering projects in water depths up to

3500 m and more. In view of these ultra deep water challenges, the offshore industry has been

called to solve demanding material and line pipe technology aspects, to develop a new and

reliable installation technology for ultra deep waters and difficult sea bottoms, to improve the

robustness of engineering prediction of in-service behaviour over the entire design lifetime,

and to find the suitable technological measures to tackle environmental hazards, typical of

ultra deep waters. Different methods are adopted to install marine pipelines, Fig. 1.

In the S-lay method, the pipeline is assembled on the welding ramp of the lay vessel usingpartial or full automatic welding techniques. These (field) girth welds are in general

controlled using X-ray or/and ultrasonic methods. The near-horizontal ramp (so called firing

line) includes a suitable sequence of welding stations, one or more tensioners, one NDT

station and one field joint station, where girth welds are coated and, in case of concrete coated

joints, filled in. The pipeline leaves the firing line to enter the launching ramp or stinger,

where the pipe is supported by rollers regularly spaced for a certain length and set up to

provide a suitable curved envelope to the pipeline. So it can leave the stinger with a slope that

ensures a smooth transition between the rigid launching ramp and the flexible lay span. The

pipeline lay span takes an S-shaped configuration due to the tension from the mooring and/or

dynamic positioning system transferred to the pipeline through the tensioners.The J-lay method has been developed as an alternative method to install a pipeline in very

deep waters: as the name suggests, during installation the pipelines takes up a J-shape. This is

achieved by lowering the pipe almost vertically into the water, thus totally eliminating the

curvature required on the overbend to reach the required departure slope and supplied by the

stinger (which represents the major limitation for extending the S-lay method into deep

waters). The J-lay method allows pipelaying at much lower horizontal tensions, to control the

state of stress on the sagbend. As a consequence the effective residual lay tension on the

pipeline at touch down can be considered negligible if compared with the S-lay method. This

may have considerable implications for pipelines laid on uneven sea beds and, therefore, on

actual free span length and associated intervention works. Dynamic positioning is the most

effective and practical method to keep the J-lay barge on course thus relieving the problemsof controlling very long and therefore less effective mooring lines in deep waters. The J-lay

eliminates the long vulnerable stinger. The obvious disadvantage is that the steep ramp makes

the welding operation critical. In order to keep the lay rate competitive, most of the welding

operations, e.g. to assembly two or more joints, are carried out before the vertical lining up,

on the deck or in a yard on land. Vertical line-up of a long section of few joints and welding it

to the suspended part, requires a special purpose developed equipment. This has major

implications on the layout of the vessel, in consideration of work-ability in concomitance with

severe sea states in relation to such a long lining derrick.

The following demanding conditions are needed: large lay pull capacity, to ensure the

heavy long free span assumes the suitable J-lay configuration from the launching ramp to the


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touch down point; large installed power, for station keeping (dynamic positioning) of the

necessarily huge lay vessel under normal and extreme environmental conditions. The

structural integrity of the pipeline moving from the launching ramp to the touch down point

has to be controlled in real time, aiming at avoiding unexpected incidental damage to the

pipeline, detectable on the seabed only after laying when it is difficult any recovery for repair.

These challenging applications demand for a refined structural analysis of the installationprocess, which is the most severe condition for pipeline design. Depending on the adopted

laying technology, various mechanical features play a key role: top angle and tension, pipe

bending/tension stiffness, surface sea-waves, deep currents, soil stiffness and slope, etc. Also

out of plane 3D effects are worthy of attention. All of them require appropriate modelling and

are accurately investigated. Nowadays, the industry uses refined Finite Element Models

which may suitably take into account the above listed issues. Nevertheless, analytical and

semi-analytical model are still important to understand the relevance of complex phenomena

which characterise pipeline installation in very deep waters.

This paper discusses analytical models developed to analyse the static and dynamic

behaviour of a pipeline during J-lay operation, particularly:

- Static models based on the close-form solution of the elastica (inextensible,unshearable beam) are briefly introduced and discussed.

- Finite difference-based techniques have been developed to determine the pipeline staticequilibrium configuration using large displacement-rotation theory of deflected beams.

- The dynamic equilibrium equation has been solved by using a perturbation technique.The work presented in this paper has been carried out in the framework of the project

Ultra Deep Water Pipelines under the supervision of Snamprogetti and partially sponsored

by the ENI Group.

Fig. 1: Pipe laying in shallow and deep waters by S-lay and J-lay method

2. FORMULATION OF THE STRUCTURAL MODEL

The structural modelling of the pipelaying operation is one of the first subjects developed

within the offshore pipeline engineering technologies and is reflected in Rules and

Normatives brought to light at the end of the 70s and the beginning of the 80s [1-2].

However, aspects concerning the lay criteria, mainly related to the acceptability of the

evaluated static and dynamic response, in terms of state of stress and strain, are not fully


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clarified. Moreover, the minimum requirements for evaluating the response and the

approximation of the obtained results in a given specific scenario are not clearly defined.

Indeed, the acceptance of a proposed pipelay strictly depends, apart from the various limit

states and capacity strength recommended, on the laying scenario (shallow or deep waters,

small or large diameter pipe, light or heavy lines) and on the way in which the structural

integrity is assessed. These aspects are not of secondary concern as dynamic analysis is veryimportant especially in case of large diameter pipelines in deep waters in which the laybarge

has no supplementary pulling reserves to help the pipeline for a better dynamic response.

2.1. Modelling of Pipelay StaticsThe assessment of the pipeline static equilibrium configuration and corresponding stresses

can be achieved by using the large displacement-rotation theory of deflected beams [3]. The

problem can be considered three-dimensional or two-dimensional (vertical plane only), the

latter being more rapid and of general use provided that the envisaged route is almost

rectilinear and cross currents are negligible. Numerical methods based on the finite element

formulation are usually preferred, but pure analytical methods were also developed. Two

numerical methods, considered the most efficient ones, are generally adopted at present.

One method consists in constructing the deflected shape of the pipe span starting from the

seabottom and growing gradually upwards by adding beam elements one at a time [4]. The

second method consists in a stepwise determination of the deflected pipe shape. The pipeline,

subdivided in finite elements, has an assumed horizontal starting position, at sea level or on

the seabed. The tension is then imposed and the submerged weight is gradually applied. Other

methods, based on a more general theory of large deflections and large rotations pipe-beam

modelling (Lagrangian, L-Updated, Mixed, etc.) to deal with the geometric nonlinearity, can

be used. However, their efficiency with respect to the ad-hoc formulation is far less. A series

of computer programs are currently available and represent a standard for calibration of new

computer programs and for design applications.The basic equation giving the equilibrium configuration of a pipeline during laying subject

to its own weight is the following:

( )( ) ( ) 2/12

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1yAwyN

y

yEI E ++=

+

(1)

where EI is the pipe bending stiffness, y(x) the pipe deflected shape in the vertical plane, w

the pipe submerged weight per unit length, the specific gravity of seawater,A the pipe cross

section and NE the effective axial force which is constant for long sections of pipeline. The

non-linear equation is solved by successive calculation of the linearized system through

numerical methods (finite element method, finite difference methods, etc.).

2.2. Modelling of Pipelay DynamicsDynamic pipelaying analysis is necessary as the laybarge and the suspended span are subject

to the action of hydrodynamic loads due to waves and marine currents. The loading process

on the pipe is both direct and induced by the laybarge response mainly in pitch and heave

oscillations, transmitted to the pipeline through the lay ramp. The dynamic behaviour of a

pipeline during installation is mainly related to the laybarge response to wave action, as the

pipe is accompanied up to a certain water depth by the lay ramp and the effects of the direct

action of waves upon the suspended pipeline starts at a certain depth being therefore decayed

and of minor importance for the dynamic response.


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Wave direction with respect to the lay heading has an important influence on the pipe

dynamic behaviour because the laybarge response to a given sea state is strictly dependant on

it. Indeed, the laybarge response in pitch is very significant for head seas, less important for

quartering seas and negligible for beam seas. Moreover, the pipeline response may be

different for waves coming from the bow of the laybarge with respect to waves coming from

its stern, due to the different combinations of pitch and heave motions dependent on theirphases with respect to the wave crest. Dynamic problems may also arise due to the vortex

shedding phenomena which can have very important effects upon the pipeline in some very

particular situations, however they are not considered within the present paper (see ref. [5-6]).

The modelling of the dynamic behaviour of a pipeline during installation is not simple as it

involves many non-linearities. Important are the ones due to the following:

- reaction between the pipeline and the last rollers of the stinger with an alternance of

impact (contact) and separation (no contact) periods;

- dead band at the tensioner which can be assimilated as a restraint in the axial direction

with a non-linear spring coupled with a viscous damper;

- fluid-pipe interaction;

- non-linearity of the relationship between the bending moment and the curvature for the

curved geometry envisaged during laying.

A linearization of the dynamic phenomena could be made acceptable e.g. in situations in

which the succession of pipe-roller impact and detaching is not so relevant and an almost

linear response of pipe section can be envisaged.

For shallow water depths, a simplified modelling could be successfully used. The

dynamics is studied for a pipe suspended span fixed or hinged or elastically restrained to the

sea bottom at one end and to the laybarge (lift-off point) at the other, excited by the laybarge

motions applied at one end [7-8]. This model can be applied successfully when dynamics

mainly affects (shallow waters) the suspended length and when critical pipe sections for both

statics and dynamics occur at the sagbend. In these cases, due to the slenderness of the layspan, boundary conditions are of minor concern.

As concerns the dynamic behaviour of the pipeline in the proximity of the stinger exit,

which is usually the most critical zone in deep water-heavy pipe laying, a complete analysis is

required. The analysis of the dynamic behaviour is performed by using the numerical step-by-

step integration of the equations of motion. The application within this study were performed

by using in-house software based on this method. The integration of the equations of motion

is carried out at appropriate time steps and the response is calculated during each increment of

time for a linear system having the properties determined at the beginning of the interval. At

the end of the interval, the properties are modified to conform to the state of deformation and

stress at that time. Thus, the non-linear analysis is approximated as a sequence of analyses of

successively changing linear systems. The final scope of a dynamic pipelay analysis is usuallyto predict a limit sea state for which the limit of resistance of the pipeline, as defined by the

accepted criteria for a given situation, is reached. Some other methods, e.g. frequency

domain, could also be used successfully in case linearization is reasonably applicable [9-10].

3. SEMI ANALYTIC MODELS

Several models have been developed for the suspended pipe span. First, the catenary model

was used: it gave a deformed shape very close to the one obtained from FEM analysis, but it

did not provide a direct assessment of the bending moment. A rough estimation could be


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obtained by evaluating the bending along the pipeline axis and then using the flexural

stiffness to guess the moment [11]; a more refined model is obtained by treating the pipeline

like an elastica with no weight and an inflection point, which means that in the deformed

shape there is a point with zero curvature. In this model such point coincides with the ramp on

the vessel, pointAin Fig. 2: so it is possible to assume that in this point the bending moment

is null. As a matter of fact, the tensioning system represents a fixed joint, but the high depthof the sea gives little bending moment on the ramp, so it is possible to place in this point a

revolute joint (hinge).

The pipe span that is laid on sea bottom is modelled as a beam on elastic foundation,

adopting Winklers model: since its length is longer than the suspended pipe span, it can be

treated like an infinite length beam. Therefore, the boundary conditions can be directly

applied in B, by imposing the congruence of the displacements in the x1 direction and

imposing the continuity of internal actions by a flexural spring in the same point. It can be

supposed that the spring stiffness depends on pipeline and sea bed mechanical characteristics.

It is noted that this is a free boundary problem, since the length of the suspended pipe span is

not known a priori, but is part of the solution.

It is assumed that the loads acting on the suspended pipeline during the laying operation

are the gravitational and hydrostatic forces and no torsional moment is applied: so the

problem can be reduced to a bidimensional one and referred to a fixed plane frame [12].

Furthermore, Archimedes buoyancy can also be looked like an effect lowering the weight of

the suspended pipeline, so that the effective axial force acting on the pipe can be computed.

Fig. 2: Free-body diagram for the suspended span

3.1. Model of elastica without weightBesides the boundary conditions already discussed, the sea depthHand a set of values for the

angle of the pipeline leaving the stinger must be granted by the model. To work out the

solution, it is used an orthogonal frame x1-y1 with origin in the TDP, from where the

curvilinear abscissas is measured. The geometric congruence conditions at point B require a

null heighty1 and the same orientation angle of the pipeline span that is laid on sea bottom;

with this model there is a continuity of the moment in B, but not as well for the shear forces:

this is an intrinsic weak point of the model itself.

With reference to Fig. 2, it is possible to call R the force acting in A and in B and to

indicate with MB the bending moment acting in B; for each point of the curvilinear abscissa

that identifies the pipeline axis, it is also possible to define the angle as the angle between

the straight line tangent at the curvilinear abscissa in that point and the x1 axis. To grant the

equilibrium of the pipeline in a generic point defined by its curvilinear abscissa s, the

following equation must hold:


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BMRyds

dEJ = 1

(2)

If previous equation is differentiated with respect tos:

sin12

2

R

ds

dyR

ds

dEJ == (3)

and then integrated with respect to , it is obtained:

CRds

dEJ =+

cos

2

12

(4)

The solution of the previous equation (4) is obtained with the aid of the elliptic integrals

[13], that lead to the relation between ands; definingFas the elliptic integral of second kind

and EJRW 2= , the curvilinear abscissa is given by:

+

+

++=

cos1

2,

42cos

cos1

2,

2cos

cos1

2FF

Ws (5)

where is the angle between the straight line tangent to the pipeline in the point withcurvilinear abscissas andy1 axis. It is now possible to find the coordinatesx1-y1of each point

of the pipeline and the internal forces acting along the pipeline,N, TandM:

cosRN= sinRT= coscos =EJWM (6-8)

3.2.Model of elastica with external loadsThe analytic model developed so far is characterised by closed form solutions, that have been

duly worked out; unfortunately, when external loads are applied along the axis of the pipeline,

a closed form solution does not exist any more.

This is a strong limitation for the model, since the weight of the pipeline represents an

important kind load both in static and in dynamic analysis. Another system of forces acting

along the axis of the pipeline is the drag force provided by the marine streams. Defining Uthe

sea water velocity, the drag forceFd, with the same positive direction of water velocity, can

be split in two forces, acting along the pipeline and orthogonally to its axis. These forces are,

respectively [14]:

222 coscos5.05.0 dtetttetdt fUDCUUDCF === (9)

222 sinsin5.05.0 dnennnendn fUDCUUDCF === (10)

The pipeline is not deformable by axial forces or by shear forces, so the equilibrium

equation [15-16] becomes, see Fig. 3:

0cossin =++ dtdn

FFds

dH 0sincos =+

dtdn

FqFds

dV(11-12)

0cossin =+ VHds

dM(13)

If the following equations are added to the system (11-13):

=ds

d EJM= (14-15)

sin=ds

dy cos=

ds

dx 1

22

=

+

ds

dx

ds

dy(16)


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a new differential algebraic system of 6 equations in 6 unknowns is obtained, that can be

solved by numerical methods; therefore, at each points of the curvilinear abscissa, the values

forM,H, V,x,y, can be worked out.

Fig. 3: Forces acting on an elementary segment of the pipeline, including hydrodynamic forces

3.3. Formulation of the dynamic problemFor the setting of the dynamic problem, an approach derived from the perturbation theory has

been used, then the outcoming model has been solved by means of a finite differences

algorithm. As a starting point, the static model derived in previous section is considered, then

the inertial effects are added on [17]. Such a scheme would imply the solution of a set of

partial differential equations: anyway, since just little perturbations are admitted, it is

assumed that the dynamic solution differs from the static one, already known, only for a little

oscillatory term; in such a way, the problem is turned into the mere determination of the

amplitude of the oscillation and therefore it is still possible to integrate the system in the

spatial variable only.

Let 0y be the static solution; the dynamic solution is then searched in the following form:

( )tyy y sin0 += (17)It can be thought of the new model as if at every point of the spatial mesh along the

pipeline the unknown variables oscillate about the static solution with angular frequency .The amplitude of the perturbation y is assumed to be small (i.e.


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4. APPLICATIONS

4.1. StaticsThe system (11-16) of non-linear differential algebraic equations can not be solved in closed-

form due to the heavy coupling among them; the solution has been worked out through a

finite differences numeric scheme, after adimensionalisation. Figures 4a-4f show thedeformed shapes and bending moments for the same pipeline but plotted in case of different

stream velocities (considered positive if coherent with the x axis direction) for the most

common laying angles at vessels end.

(a) (b)

(c) (d)

(e) (f)

Fig. 4: Deformed shapes and moments obtained for different drag forces and for different angles at vessel


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Previous plots of Fig. 4 clearly show the influence of the drag force generated by the

stream: a unit-force stream flowing in the same direction of laying progression yields a

decrease in the bending moment (and therefore in pipes stress state) while the lay barge goes

away from the TDP; on the other hand, if the current flows opposite the TDP gets closer to

the vessel but the bending moment is increased. The solutions obtained by the described

model have been compared with the output of a commercial FEM package, with a closeagreement as shown in Tab. 1.

F.E.M. Finite Difference

XTDP 580 m 579 m

TDP 0.032 0.032

MTDP 184 kNm 135 kNm

NTDP -5.81 MN -5.81 MN

TTDP -61.8 kN -58.9 kN

Mmax 2310 kNm 2180 kNm

X(Mmax) 95 m 99 m

NA 3.37 MN 3.11 MN

A 85.8 85.8

w.f.max 0.763 0.736

Tab. 1: Comparison between FEM and finite difference method results

4.2. DynamicsTo solve the differential system (19-24) by a finite differences scheme, the following

parameters are introduced [19]:

q

Lmfa

=

2

1 q

Lmfi

0

2

1= q

LAfca

=

2

q

LAfci

0

2= (25-28)

and the following adimensional system is obtained:

= 0sin~x = 0cos

~y 0~ = (29-31)

)cos~

sin~

(~

cos~

sin)sin~

cos~

(~ 0000000000000 VHpVpHpVHp +++= (32)

[ ])cossin~cos~(~~cossin)cos(~~ 00002000002 yxfxfyfffxH caacicii +++++= (33)

[ ])cossin~sin~(~~cossin)sin(~~ 00002000002 xyfyfxfffyV caacicii +++++= (34)

Eqs. (29-34) represent an ordinary system of 6 linear differential equations in the 6

unknowns x~

, y~

, ~

, , H~

, V~

. The boundary conditions are the same alreadydefined in the static case, therefore, by using the linearization method introduced while

stating the equilibrium equations, the following boundary conditions are imposed:

0~1 =x 0~

1 =y 01 = +N 0~

1 = +N (35-38)

( )0104

1

1 2~ LL

EJ

Kt+

= (39)

1,0

0

111

~~sin

~cos

~++++

=+ NNNN N

L

LNVH (40)


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where subscripts represent the node related to the indexed variable. The following Figs. 5 and

6 refer to a simulation case where the vessel has been subjected to a sinusoidal heave motion

with a period of 20 s and an amplitude of10 m: pipeline response linearly depends on the

driving motion, since in test conditions the natural modes of the pipeline are not excited. In

particular, Fig. 5 plots the deformed shape of the pipeline close to TDP and near vessels end

in different time steps: it is apparent how TDP displacement closely follows vessels motion.The bending moment in the sag bend is shown in Fig. 6 and is characterised by the same time

behaviour imposed by the motion of the vessel: the maximum value of1 065 MNm is found at

t = 5 s in correspondence of the maximum vertical displacement of the vessel.

Fig. 5a: Pipeline deformed shape at different time

steps near the TDP (note the different axes scale)

Fig. 5b: Pipeline deformed shape at different time

steps near the vessel (note the different axes scale)

Fig. 6: Moment in the sag bend at different time steps

5. CONCLUSIONS

The solution obtained by integrating the elastica model with inflection point by means of the

finite differences scheme gives results that are comparable with the output of FEM packages,

both for the static and the dynamic cases. From the dynamic point of view, the approach

based on perturbation theory yields correct results whereas pipelines response can be

considered linear, i.e. in limited bandwidths far from natural frequencies: if such conditions

are satisfied, also in this case the results closely agree with other more sophisticated models.

It is noted that the proposed models do not consider any damping factors.

It must be stressed that in both cases computing times are one order of magnitude shorter

than the corresponding times of FEM simulation packages: this is an important result of the


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work, in view of possible control applications; in fact, it could be very interesting to be able

to evaluate during laying the deformed shape of the suspended pipeline and therefore to

assess its stress state (particularly in the sag bend) at every instant, so as to be able to take

corrective actions tending to decrease pipe deformations that could possibly damage pipes

structure or limit its working conditions for all the possible lay configurations.

It is finally anticipated that the Authors are currently working out more sophisticatedmodels that are able to take damping into consideration as well as DAE models that integrate

the suspended pipe span and the pipeline laid on the bottom of the sea into one coherent

model, without any need for a separation in two parts, as done in the present work.

AKNOWLEDGEMENT

The authors would like to thank Snamprogetti and ENI for the fruitful cooperation and for the

permission to publish this paper.

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[15] Love A.E.H.: On the mathematical theory of elasticity; 4th

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[16] Villaggio P.: Mathematical models for elastic structures, Press Syndicate of the University of Cambridge

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dynamic models of marine pipelines

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