DYNAMIC MODELS OF MARINE PIPELINES FOR …

1

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

ANALYTICAL AND NUMERICAL APPROACHES

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

1 Dipartimento di Meccanica, Università Politecnica delle Marche, Ancona 2 Istituto di Scienza e Tecnica delle Costruzioni, Università Politecnica delle Marche, Ancona

3 Snamprogetti, Fano (PU)

SOMMARIO

La memoria tratta delle problematiche dell’installazione di condotte sottomarine con il cosiddetto “metodo a J”, che consiste nel varare le condotte stesse con l’ausilio di una rampa quasi verticale. L’attenzione 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 l’esecuzione 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 più facile utilizzo, catturano gli aspetti più importanti del problema e possono essere utilizzati come punto di partenza per algoritmi iterativi di ricerca numerica della soluzione. I modelli numerici, d’altra 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 in laying 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

2

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 using partial 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 problems of 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

3

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 installation process, 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 static

equilibrium 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 70’s and the beginning of the 80’s [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

4

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 very important 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 Statics The 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:

( ) ( ) ( ) 2/122/32

11

yAwyNyyEI E ′++−=′′−

′

′+

′′γ (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, γ 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 Dynamics Dynamic 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.

5

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 their phases 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 lay span, 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 usually to 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

6

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, point A’ in 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 depth of 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 Winkler’s 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, Archimede’s 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 weight Besides the boundary conditions already discussed, the sea depth H and a set of values for the angle α 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 abscissa s is measured. The geometric congruence conditions at point B require a null height y1 and the same orientation angle φ 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 γ 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:

7

BMRydsdEJ −= 1γ (2)

If previous equation is differentiated with respect to s:

γγ sin12

2

RdsdyR

dsdEJ == (3)

and then integrated with respect to γ, it is obtained:

CRdsdEJ =+

γγ cos

21 2

(4)

The solution of the previous equation (4) is obtained with the aid of the elliptic integrals [13], that lead to the relation between γ and s; defining F as the elliptic integral of second kind and EJRW 2= , the curvilinear abscissa is given by:

+

−+

++

=α

πφα

γα cos1

2,42

coscos12,

2cos

cos12 FF

Ws (5)

where φ is the angle between the straight line tangent to the pipeline in the point with curvilinear abscissa s and y1 axis. It is now possible to find the coordinates x1-y1 of each point of the pipeline and the internal forces acting along the pipeline, N, T and M: γcosRN = γsinRT = γα coscos −= EJWM (6-8)

3.2. Model of elastica with external loads

The 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 U the sea water velocity, the drag force Fd, 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]:

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

ϑϑρρ 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 =++ ϑϑ dtdn FFdsdH 0sincos =+−− ϑϑ dtdn FqF

dsdV (11-12)

0cossin =+− ϑϑ VHdsdM (13)

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

κϑ=

dsd κEJM = (14-15)

ϑsin=dsdy ϑcos=

dsdx 1

22

=

+

dsdx

dsdy (16)

8

a new differential algebraic system of 6 equations in 6 unknowns is obtained, that can be solved by numerical methods; therefore, at each point s of the curvilinear abscissa, the values for M, H, V, x, y, θ can be worked out.

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

3.3. Formulation of the dynamic problem For 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 ωε 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 ω. The amplitude of the perturbation y∆ε is assumed to be small (i.e. ε <<1) so that the polynomial expansion can be halted at the first order terms [18]. Such a technique can be used only if the excitation angular frequency ω does not correspond to a natural frequency of the pipeline; if such a condition is verified, a set of ordinary differential equations is obtained, the amplitude vector ∆ , containing all the amplitude components y∆ , being unknown.

The equilibrium equations are formally similar to Eqs. (11-16) also in the dynamic case, except for the addition of an inertial term on the right hand side [18]:

( )[ ]ttUJdtdA

trmFI ⋅++

∂∂

−= ρ2

2

1 (18)

where r is the position vector of the pipe span, t is the unit vector tangent to the pipeline, J is water flow velocity, U is relative velocity, m is pipe mass per unit length. The resulting ordinary differential system takes the following expression:

ϑcos=′x ϑsin=′y ϑϑκ cossin VHEJ −=′ κϑ =′ (19-22)

( )[ ]ϑϑϑϑϑϑϑϑϑρ 2221 sincoscossinsincos2cos −++−−=′ &&&&&&&&&& yyxxAxmH (23)

( )[ ] qxxyyAymV +−+++−=′ ϑϑϑϑϑϑϑϑϑρ 2221 sincoscossinsincos2sin &&&&&&&&&& (24)

where the prime sign (’) indicates derivation with respect to curvilinear abscissa s.

9

4. APPLICATIONS 4.1. Statics The 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 the deformed 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 vessel’s end.

(a) (b)

(c) (d)

(e) (f)

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

10

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 pipe’s 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 close agreement as shown in Tab. 1.

F.E.M. Finite Difference XTDP 580 m 579 m θTDP 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 θA’ 85.8° 85.8°

w.f.max 0.763 0.736 Tab. 1: Comparison between FEM and finite difference method results

4.2. Dynamics To solve the differential system (19-24) by a finite differences scheme, the following parameters are introduced [19]:

q

Lmfa∆

−=2

1ω q

Lmfi0

21ω−=

qLAfca

∆=

2ωρ q

LAfci0

2ωρ= (25-28)

and the following adimensional system is obtained:

ϑϑ ∆−=′∆ 0sin~x ϑϑ ∆=′∆ 0cos~y 0~ =∆−′∆ κϑ (29-31)

)cos~sin~(~cos~sin)sin~cos~(~0000000000000 ϑϑϑϑϑϑϑκ VHpVpHpVHp −+∆−∆+∆+=′∆ (32)

[ ])cossin~cos~(~~cossin)cos(~~0000

200000

2 ϑϑϑϑϑϑ yxfxfyfffxH caacicii +++∆++∆=′∆ (33)

[ ])cossin~sin~(~~cossin)sin(~~0000

200000

2 ϑϑϑϑϑϑ xyfyfxfffyV caacicii +++∆++∆=′∆ (34)

Eqs. (29-34) represent an ordinary system of 6 linear differential equations in the 6 unknowns x~∆ , y~∆ , κ~∆ , ϑ∆ , H~∆ , V~∆ . The boundary conditions are the same already defined 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 =∆ +Nϑ 0~1 =∆ +Nκ (35-38)

( )010

41

1 2~ ϑϑκ LLEJKt

∆+∆

=∆ (39)

1,00

111~~sin~cos~

++++

∆−∆=∆+∆ NNNN N

LLNVH αα (40)

11

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 of 10 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 vessel’s end in different time steps: it is apparent how TDP displacement closely follows vessel’s 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 of 1 065 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 pipeline’s 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

12

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

It is finally anticipated that the Authors are currently working out more sophisticated models 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.

REFERENCES [1] DnV: Rules for Submarine Pipeline System; Det Norske Veritas (1981) [2] DnV Offshore Standard OS-F101: Submarine Pipelines Rules by Det Norske Veritas, Høvik (2000) [3] Crisfield M.A.: Non Linear Finite Element Analysis of Solids and Structures – Vol. 1; Wiley (1992) [4] Rammant L.R and Bockx, E.: Offshore Pipeline Installation Sensitivity Analysis for a Conventional Lay-

barge, Applied Ocean Research, Vol. 2, pp 13-21 (1980) [5] Vitali L. and Bruschi, R.: Vortex Shedding Induced Oscillations during Pipelaying, Proc. Intl. Conf. on

Hydroelasticity in Marin Technology, Trondheim, Norway (1994) [6] Bruschi R. et al.: Pipelay Dynamics due to Cross Currents Induced Vortex Shedding, Snamprogetti Internal

Document (1992) [7] Clauss G.F.et al.: Nonlinear Static and Dynamic Analysis of Marine Pipelines during Laying, Ship

Technology Research, Vol. 38 (1991) [8] Clauss G.F. et al.: Offshore Pipelaying: Significance of Motions and Dynamic Stresses during Laying

Operations, Offshore Technology Conference, OTC Paper No. 6760, Houston (1992) [9] Wilson E.L.: Numerical Methods for Dynamic Analysis, in Numerical Methods in Offshore Engineering,

ed. Zienkiewicz, Wiley, pp. 195-220 (1978). [10] Shekher V, et al.: A Linearized Approach for the Evaluation of Dynamic Stresses in Offshore Pipelines

during Installation, Proc. of 3rd ASME Intl. Conf. on Offshore Mechanics and Arctic Engineering Symposium, New Orleans (1994)

[11] Langner C.G.: Relationships for deepwater Suspended Pipe Spans, Proc Ocean and Marine Engineering, New Orleans, Lousiana, February 12-17, pp. 552-558 (1984)

[12] Guarracino F., Mallardo V.: A refined analytical analysis of submerged pipelines in seabed laying, Applied Ocean Reserch, Vol. 21, pp. 281-293 (1999)

[13] Kinball C., Tsai L-W.: Modelling of flexural beams subjected to arbitrary end loads, ASME Journal of Mechanical Design, Vol. 124, pp. 223-235 (2002)

[14] Inoue Y., Surendran S.: Dynamics of the interaction of mooring line with sea bed, Proc. 4th Intl. Offshore and Polar Engineering Conference, Osaka, Japan, April 10-15, pp.317-322 (1994)

[15] Love A.E.H.: On the mathematical theory of elasticity; 4th ed., Dover Publications, New York (1944). [16] Villaggio P.: Mathematical models for elastic structures, Press Syndicate of the University of Cambridge

(1977) [17] Impollonia N., Muscolino G.: Static and Dynamic Analysis of Non-Linerar Uncertain Structures,

Meccanica, Vol. 37, pp. 179-192 (2002) [18] Albow C.M., Schechter S.: Numerical simulation of undersea cable dynamics, Ocean Engineering, Vol. 10

N. 6, pp. 443-457 (1983) [19] Sofi A., Borino G., Muscolino G.: Dynamic Analysis of Prestressed Cables with Uncertain Pretension,

Meccanica, Vol. 37, pp. 179-192 (2002)