Application of linearized morison load in pipe lay stinger design

Proceedings o f t h e A S M E 27th Intemational Conference o n Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20,2008, Estorll, Portugal

Delft University of Tedinoiogy

Ship Hydromechanics i-aboratory

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^ ^ ^ ^ L . ^ O M A E 2 0 0 8 - 5 7 2 4 7 Phone: +31 (0)15 2786873

E-mail: D.w.ddieer(a)tudelft.nl

APPUCATION OF LINEARIZËD MORISON LOAD IN PIPE LAY STINGER DE^GN

GustoMSC

Hydrodynamic and Stability Department Schiedam, The Netherlands

ABSTRACT

Tlus paper presents numerical results of ship motions and global stinger loads Ö u o u ^ a combined hydrodynamic analysis of a pipe lay vessel with submerged stinger. The insults of nonlinear time domain simulations are conspared to those obtaiaed through linearization of the Morison load on the slender stinger elements. Through linearization, an iterative frequency domain solution scheme is developed reducing analysis time significantly. Response amplitude operators in operating and limiting sea states ^e shown, including the infli^ce of cmrent velocity. Through nonlinear time domain simulations insdght is obtained on the distribution and magnitude of the extreme valuea.

KEYWORDS

Morison load, Stochastic linearization. Pipe lay vessel INTRODUCTiON

The structural design of a submerged stinger for pipe lay

opôïLtion is part of a coiiq)lex desi^ loop. The hydrodynamic

Ipads on the slender elements of the stii^er result in global loads at the hinges wiiere tbe stinger connects to the ship. Apart from Ûiese global . stnu:tural loads, each number of the stinger is subject to line loads. In an optimized design, the pipe diameter and wall thickness of the tubes are designed to witiistand these local structural loads given certain structural allowances. Other design constcaiots can be the stinger handling, structural integration of the stn^er in the afl ship, and obviously steel production costs. This paper will focus on a fast calculation mediod to derive global stinger loads by means of combmed sbip-stù^er moticm analyses.

During pipe lay cperation the headmg of tiie sh^ can not be chosen, thus the operational pro£Qe includes all possible wave headings. If the vrave environment becomes too harsh, which is geaierally so above 4 m sigmficant wave height, the stinger migbt be retrieved out of the water. In such a situatipn the stinger is ten^rarily located in the spla^ zone. As a result

the wave loads n i i ^ t be very si^üficant for short duration of time. If the stinger^ rigidly connected to the vessel and can not be taken out of the water, wave bads might be minimized by abandoning the pipe and by taking sheltered heading. But it is not cm forehand clear at which beading global stinger loads reach a mimmum if the headii^ can be selected.

The above sketches the complexity in the design phase, which consists of, aihong other factors: a large series of possible environmental condjinations of waves, wind and current, and a coii4)lex hydrodynamic 5%>-stinger interaction with possible multiple stinger orientions.

STINGER DESIGN LOOP

An initial stinger design starts with an OFFPIPE run to estabUsh the ov^alL dimensions of the stinger. Based on experience, a first design of a lattice stinger structure is made. A fpllowing step is to establish the structural loads due to ship motiot^. Depending on the stinger dimensions and orientation of tiie stinger in the water, the drag and inertia loads on the s t i i ^ r elements will influence the ship motions to more or less extent. Since the pipe lay vessel cîmnot change its heading during pipe lay operation, the ship-stinger motions and restating stmctural loads need to be assessed for all possible headings, a collection of opaational and limiting sea states, mid i f an adjustable stinger is designed, for several stingy curvatures. To optimize the stinger design, the ^ove sketohed procedure is repeated iteratively ùntU the struc.tin:al load distiibution on the stiniger matches the design requirements.

The stinger and the ship are interaction structures. Ship motions arc most efficiently solved in the linear frequency domain, and now-a^days a diffraction panel method has twcome die standard to do so. However, the hydrodynamic loads on the slender stinger elements, due to cUrrrait, waves and ship motions, follow from Morison's equation. This equation describes the loads in terms of a nonlinear drag force and lineajt inertia force perpendicular to die principal axis of the elemcM'

The nonlinear drag term prohibits direct E^lication of the Morison's equation in linear frequency domam.

To accoimt correctly for the nonlinear drag loads on the stinger elements, a time domiain solver mcliuling Morison elemrat modeling is required. Such simulatipn programs are widely available; an example is the ANSYS-AQWA suite of prograins. But application of a nonlinear time domain solver in the early 'iterative' design stage is not preferred, due to the significant analysis time involved for each load case. Fast frequency domain calculations are preferred.

Hie linearization a^^proach and its application is the topic of the presem paper. In view of the large numbar of varyii^ parameters, tiie ^pHcation is alreatfy usefiil w*«n the wontt sea states can be found, and even more i f a good estimiate of tiie hinge loads is obtamed. But, it is racp^ed that in those ccmditions vihéie the stinger dan^mg dominates the motion equations, nonlinear time domain simulations are needed to

obXaia. accurate design extremes given a typical storm duration.

SHIP-STINGER INTEGRATION

Tbe ship and stinger are defined as two separate strtKtures. This paper presents the results for a fixed stinger only, in other words a stingy that is rigidly connected to tiie ship. The ship is considered as a diffractmg body The slender elements of the stinger are only subject to drag and inertia loads, described as Mori»:>n loads. The velocities and accelerations in that fonnulation are due to the combined effect of ship motions, current and sea state.

MOTION EQUATION

The motion equation for a floating structure with attached stingy can be w r i t i ^ in the time dorpain, as:

Figure 1: Ship with Stinger, A Q W A modeUing The stinger and siiip configuration as presented in Figure 1 defines the configuraticm. Global Mnge loads are calculated in a point where stn^er attaches to the ship.

(1) where FgHipi^) represents the tç^drodynamic forces on the vessel, FsjTNGERi^) represents the Moristm load on slender stinger elemente. The mass matrix of the total system is defined by M . The accelerations ^{t) are solved in a time stepping procedure, and the motions 40) follow from integration. If the stinger is rigidly connected to the vessel, the motion vector

4(f) contains six rigid body motion con:q)oncnts. The Morison

load on slender (tubular) elements is given by:

^sriNOERit) = \p^C^m I sm I +^pD'CJ{t) (2)

where «(/)is the flow velocity experienced by the element and ü{t) the acceleration. The drag and inertia coefficients are given by Cp and C„ respectively.

Equation (1) can be solved in time domiain using the ANSYS-AQWA software. The Morison Ipad on submerged slender sting^ elements are modeled as in equation (2). To solve equation (L) in the frequency H^main^ all terms are to be expressed as harmonic varyii^ signals with constant amplitude, which means that equation (2) needs to be linearized. M the fi^uency domain approach the motion equations are solved for a series of regular waves which sum to the wave spectriim. A similar breakdown is the basis of time domain simulations where tiie mptipn equajtipns are solved in a time-steppmg approjach.

LOADS ON SUBMERGED STINGER ELEMENTS The forces (per unit length) aa sabmetged slender tubular eloments are describe by the Morison equion, which in relative-velocity expression read as:

F{t) = ^pDC^\i;\S +

^pD'cJ

The velocity field around an element is tlK result of a oonstant currrait velocity, i?c, varying wave orbital velocities Uff(x„t) and vaiying element motions xix,t). The drag coefficient Q

is constant in tune as well as the mass mertia coefficient ; tiie added mass coefficient is related to the mass i n ^ a c»ef5cient âs C^^ = 1 + ,

As ofien naentioned, the Morison equatipn with constant coefficierits is an approximated solution to a con^lex problem. The co:q;pIexity lies in the varying flow field around slender elements, which is viscpus donmiated. Typical non-dimensional parameters to identify the flow field aroimd an oscillating object are the Reynolds (ke) and Keulegan-Caipenter (KC) nmi^>er. Due to the conç)lex lattice structure of the stinger and associated fiow field around it. Re and K C numbers will vary along the stingy from sea state tp sea state. Significant interaction effects between members are expected but experinriental data is scarce to non-existing for stinger constnictioQS. And even then, model tests determining the flow field or forces pn the stinger will be subject to important model scale effects, so the otfrapolation to fiill scale remains difficult. Originally, Morison's equation was posttdated for forces on fixed slender elements, thus withPut member motion. Research on the applicability of the combmed flow field, see for exanq)le Shafiee-Far [1], shows timt the relative velocity formi^tipn gives adequate results for oscillating mendters in csiurent and waves.

The main concem of this ps^er is to demonstrate a woridng procedure to verify i f a Unearized Morison equation in a fiequency domain calcuIatUm suffices to determine design load cases, hi view of die puipose of this pi^er the drag and inertia coefficients are considered equal for all stinger eli^ö^ts. Based on an estimate of the Re and K C nunibei^ and expected stinger element smoothness, a drag coefficient of = 1.05 and an inertia coefficient of Cj^ =0.8 are selected as represöitative.

UNEARIZED MOTION EQUATIONS

To account for the drag loads in a fiïiquency doimin approach, equation (2) is written in a generic linear form:

F^(t) = ]-pDC^u + ^pD'C,ù

2 4 (4)

A common i^proach is to calculate the linearized drag term from equating the entf gy dissipation from the linear and nonlinear drag contribution (equivalent linearization), or by minimizing the error between tiie linear and nonlinear force (stochastic Im^rization), Details of the two approaches are described below.

When the velocity variaticm are sinusoidal with constant an^litude - like in regular waves - thus u{t)=u^ços{(a)t the linearized force is given by:

Fit) = ^pDC^[$/07r)u^]-u«^pDC^[l2004o„]-u (5)

where tT„ is the standard deviation of the regular smusoidal velocity (<t„ =w^/>/2). The linearization ^ t o r between square brackets is obtained fiom equating the energy in a quarter of the oscillation period, that is:

T/4 T/A _

(6)

Linearization of the force due to a random oscillatory velocity -like in irregular wav^ - leads to, see for exanple Borgman [2] or Wolfiram [4]:

Fit)==^pDC^u4ßI7io„ «i/7DCJ1.596crJ.« (7)

where o;, is the standard deviation ofthe vetocity spectrum and

u(t) the rwdom velocity time trace. The last linearization

procedure is denoted stochastic linearization, the linearization factor in an irregular (Gaussian) sea state is obtained from minunizing the tinw avenged least square error betweai the linear, Fjj^it), and nonlinear time signal F{t) , as in [4]:

d{{F^it)-F{t)f) , ^

- i — ^ — - - ^ = - 2 { c y I u I =0 (8)

where ( ) indicates the expected (time-averaged) value. If the wave elevation is assumed to follow a Gaussian distribution, the derived velocity field and all calculated linear responses foUow a Gaussian distribution as well and equation (7) applies.

Both approaches assume a zero m i ^ signal, thus a zero current velocity. Along the stinger tiie ratio between the waive orbital velocity and niotion induced velocity depends very much on the locatipn of the element. The rnoti(m induced velocities at the tip of the stinger ^ 1 be much laiger than near the stinger hinge, bicluding current cou^licates direct insight in the relative inqx>rtance of the different velocity componeois. An apiinroximation of the time varying drag contribution with current is obtained from applying the triangle inequality, leading to:

F(0 = 0.5pDC^ [21 « J +^l^o^_,]iu^ -X) (9) Note that the current velocity occurs only in the linearization term, so that equation (9) can be applied in a fiequency domain approach.

Hie outlÎBe above shows that stochastic linearization does not converge to the e^valent linearization ^i^en the sea state consists of a s i n ^ wave component. As such, one could argue tiiat equivalent linearization is a vaiid aproach in a sea state since that is constructed from a s^es of regiHar waves and motipn eqimtions are solved for each regular wave scpMutely. And even in time donmin Ûie basis is rejgular wave summation and most forces are defined based on this. Çonqiaring botii

E^EVoaches to nonlinear time domain results will reveal the

validity of the methods.

As shown by several authoi^, like Brouwers [3] and Wolfram [4], the linearized Morison «juation under predicts the extreme peaks compared to the n<Hilinear equation. For a moving stinger tiie motion extremes are thus most likely over predicted. The £^ect on the hinge loads could bê large. Brouwers [3] detfrnjes a correction fiictor (3 h extreme over standard deviation) for fiitigue load dq)ending on tiie drag

versos inertia ratio which becomes as large as 8 for drag

dominated structures.

In view of the above discussion, qiplication of the linearized Morison's equation is usefiil vrhea it provides tiie initial load level for d l possible ënviromriwitaJ conditions. The absolute load level and accurate e x t i ^ e v^ue prédiction can most likely only obtain»! using tiie nonlinear Morison equatipn dirràtly in tinie domiain simulations. As qupted frpm Wolfram [4]: some cases it may be more expedient to splve the nonlinear equation numerically than to bother with linearization'.

For e^h stiiigra* elem^t a transformation matrix is defmed which is iised to calculate the contribution of fhe element to the global force F^ _MORISON froin the local Morison force Fj^ (pérpeniËcular to the element orientation);

(10)

For convenience a skew-matrix S(r) is defined which conqnises the locaticm of tiie stinger with respect to the global firame of reference, f h e indices lare left Put to shPrten notaticm: S(r) = 0 -z y z 0 -X -y X Q (11)

The tube CoG is located at r from the glc^al origin, so its acceleration in the global firame of reference is:

x = Xroc-^^^r=Xcoa - S ( r ) ö (12)

\shere ^ is the ship rotational velocity. The accelerations in thie local fiamë are then:

(13)

IMPLEMENTATION IN FRBQUENCY DOMAIN

The linearized firejqiKricy d o n ^ solution including linear Morison loads is solved iteratively The linearization &ctor is calculated for each stinger element, based on the standard deviation of the relative velocity cm that element firom the previpus iteration. The overall , ship motion ccmiponents are used to calculate that local member velocity. The spectrum is constructed from a summation of regular waves and for each regular wave tlw motion equations are solved.

Conceptually, the equivalent linearization Uses the standard deviation of the velocity at the element calculated for each wave fi:eqüency. Thus for each beam and for each wave fi^qu^cy, is related to the velocity amplitude on the elemoit in a regular wave ^ ^ c h amplitude corresponds to the spectral density distribution. Stochastic linearization calculates the standard deviatipn from tbe velocity spectrum found for each beam using all frequencies, that is cr^ = . The iru>tion equations for each frequency apply this &ctor.

so that it is possibk to write:

^ G = - T ö M c j f { i ï - S ( f ) i (14) where M is tiie mass matrix of the local elemoit, in which tbe inertia of the tube around its own axis system i s . n e g l ^ ^ :

M , ml. -mS(r)

mS(r) -mS(r)S(r) (15)

Using the above derivations, the inertia force of a single element of the stinger can now be written as:

' F ß . G

-S(r)

HS(r) (16)

A similar expression can be ^rived fbr the linearized drag term, leading to:

' f '

ß . G S ( r ) T i B . T? 0

-S(r)

-S(r) (17)

The linearization fector is included in the dan:q>ing matrix B^^ . This matrix includes tiie standard (Aviation of the local velocity, as expressed in equations (5) and (7).

The wave excitation forces and moments due to wave orbital acceleration acting on tiie elements is given by:

F = [T!M,Tè]S^ M^S(r)F

(18)

The water ortntai velocities on each stinger element are evaluated in the midpoint of each eleoient, and in agreement with linear fi^qUency theoiy, at the location when the ship is at rest.

TIME DOMAIN SIMULATIONS

The AQWA software is used to perform the time domain simulations. The Moriscm elements ^ l y the relative velocity formulation. This is part of the software already, so no additional ad-hoc modelmg is required. A 2-point Gaussian quadrature rule is used to calculate the relative velocity and acceleration on each element. The drag and inertia coeffidents of each element aire pre-defined and constant during a simulation. The location of a slenckr Morison element with respect to the actual water line is evaluated and used to détermine the time dependent Morison load. The AQWA-HAUT module accounts for nonlinear stiffiiess aikl wave forces on diffiacting bpdies as well. The line^ potential damping from the frequency domain is used to calculate the convolution integrals. The retardation forces are calculated using these integrals with the appropriate time history signals. A nuxed approach is thus used, linear convolution integrals with the actiial ship motipn velocities which include possible nonlinear effects due to npnlmear force cpiiq)onents in the: model. The Morison drag and the ship stiffiiess and wave excitatipn forces are tbe nonlinear coiiq)onents. To avoid a drifting model, a spread-mooring system with appropriate stifßiess and pre-tensioning is used. This kind of modeling con^ares witii a model basin set-iqi.

APPUCATION

The ship-stinger configuration as ^own in Figure 1 is used throughout this pap^. The ship motipns are first calculated in the frequency domiain, without the stinger. In addition to tiie hull surface mesh, an intemal free surfiice mesh is applied to siq>press possible irregular frequencies that can lead to inaccurate convplidon integrals fbr tiie time domain solver. The ship has overall dimensions of about 145 m length and 21 m

beam. Recentiy GtistoMSC was involved in the vessel upgrade, design of the ship-stinger integration and stinger handling fi^e. For this purpose, the integral assessnoit of the stinger and vessel mptions was carried out, including an assessnbeht of the hinge Iciads m operational and survival sea states.

In that project, pipe lay conditions include sea states up to 2 m significant wave height with associated wind conditicms. Since tiie ship can not freely choose its heading during pipe lay, all possible headings are to be considered. Above 2 m sigmficant pipe lay operaticm is abandoned. In those conditions

ibe vessel is assumed capable to select its heading, so that roll

motions and stingCT loads can be minimized. To avoid even larger hydrodynamic stingy loads, the stinger is tàkea out of the water above 4 m significant waye height by the stinger hanrlling frame. Current velocities are cpmidered iq> to 2 knots with uncorrelated direction with respect to the sea state.

The stinger orientation and curvatcoe is adjustable depending on the type of pipe diameter and pipe laying water depth. All results in this paper assume deep water, but it is not a restriction in AQWA. The stinger is about 70 m long co;^isting of 1002 différât tubular elem^ts. Smaller elements are neglected, as well as non-tubular elements m the rollerbpxes, walkways abpve mam stinger tubes and others. Tp account for the drag and mertia of these and neglectal construction details, drag and inertia values of selected members are corrected.

The stinger and the ship are defined as two individual stractures in AQWA. A single hinge point with zerP deigree of freedom is used ät which both structures connect. The intersection l o a ^ are then directly obtained in tiine domain. The ship is the only diffracting bocfy. The potential added mass, darling, diffraction aind Froude-Kiyloff forces on the ship are calculated in tiie frequency domain. The locaticm of the centre of gravity of the ship is defined such that the overall ship-stinger combination centre of buoyancy (CoB) and centre of gravity (CoG) arc vertically in-line resulting in zero trim. The mass of the ship is conected for the.stinig» nms being 3.2% of the sh^ mass. Since only the frequency domain coefficients are used in post-processing, discrqiancy between the vertical alignment of the ship CoB and C6G in the frequency domain is of no influence.

COMPARISON BETWEEN EQUIVALENT AND STOCHASTIC LINEARIZATION

In Figure 2 the roU RAO in beam seas obtained through equivalent and stochastic linearization is given, without and with 2 m/s cuiient (sea state aligned). The result is obtaiaed in a JONSWAP sea state witii Hs = 2 m, Tp = 13 s and gamma = 3.3. The torsion momoat M X at tiie stingy Krige is presented in Figure 3.

As the residts shpw, both methods p r ^ c t coirqjarable motion and load responses. The roll motion peak is slightiy higher wben using stochastic linearization. The impact of this

on the roll moment is small. The verificaticm was made for other motion and load comppnents, showing conqiarable trends.

KOLIi R«8p«in, Ettßjig » 00 dog JÖlrêWAP: H . B 2.0 m, T , «• 13 s, 7 = S.3

r

f i l t r e 2::Comparison of Roll motion R A O in beam seas using différent linearization approaches.

MX KespozBtt, Heading s 00 dcig

JONSWAP: » 2.0 m, T, - 13 s, 7 - 3.3

1;

Wav« RsQüsnoy [lad/s]

3: Comiiarison of Load M X R A G in beam seas using difTermt linearization apjproadhes.

SHIP-STINGER MOTIONS IN B E A M SEAS

In Figure 4 throu^ Figure 7 a compari»)n is given of the sway, heave, roÛ and pitch response ^a^titude operators (RAOs) with phase, in beain seas, using Afferent calculation niethods.

Tte Unear frec^i^mry doinain calculations, denoted *FD, Ship only' in the figures, apply a fixed additional danqnng to correct for the bilge keel damping. At the natural peak the rpll

Hflmping is 8% of the oitical damping. AQWA has no

implementaticm of an empirical roll dancing iru^cxl in fiequency domain. The results with stinger dam|>ing are denoted *FD, Ship -I- Stinger, Equiv' when iteratively obtained in a

spectrum decompositicm with equivalent d a n ^ g , and 'FD, Ship + Stinger, Stoch', when iteratively obtained in a spectrum decompositicm with stochaistic linearization. The time dpmain simula.tipns aie performed with AQWA-NAUT in the given spectium witii nonlinear Moriscm Ipads for a chiration of 3-hoiûrs, and the RAO is retiieved from time trace post-prcx»ssing.

The results show that the stinger adds significant danapiog, reducing roll and pitch motion air^litades. Sway and heave are marginally i f at all influenced at this headh^. The natural roll period of tiie ship with stinger reduces and the shift is well predicted by stochastic linearization. Equivalent lineairization sHghtiy Uituter predict the peak roll response. But basically, all motion con^Kments arc well predicted by botii linearization methods compared to the nonlinear time domain simulaticm result.

Figure 8 presents the distribution of the absolute value of the roll anplitudes based on 9-hour time domain simulation data. The current of 1 m/s aligns witii the waves. The resùlts indicate that the roll crest and trougihs à-e not Raylei^ distributed wliile the sea state in the sinn^ations is Gaussian distiibuted. TMs could be expected since the Hamping of the stinger is nonlinear and adds in particular to the roll dandling. The given WeibuU and Rayleigh fit is performed on all simulated crest and through extreme data points with zero current speed. As can be seen, the current velocity m itself has a significant influence cm the distribution as well but, in this example, less on the extreme values near the tail of the distribution.

SWAY ttjiBporat, Headins - 00 dag JONSWAP: H , o 4.O m. T , = 18 s, 7 = 3.3

B P In at HI

g

—*—WD, Ship only

—a—FD, Ship + Stinger, Equir FD, aùp + Stinger, Stodt * TD.Ship+StiOKer.NoQLffl

—*—WD, Ship only

—a—FD, Ship + Stinger, Equir FD, aùp + Stinger, Stodt * TD.Ship+StiOKer.NoQLffl

i i

as 1

Wave ftaqusncy {tad/sl

Figure 4: Sway response in beaim seas.

T.3

ESAVS lUspens«, Ecadtng > 90 deg JONSWAP: S, - 4.0 m, - 1Î B, 7 « 3.3 mmmmmm -180 Q 3,0 5 -t—FD, Shq) oiily _{-s—nD, 3b^ ^ Stinger, E ^ y} ÏD, Sbip + Stinger, Stodi * ID, Ship Stinger, KoOLm

0.3 1 Wave FlaquHicy [rad/s]

F l ^ r e 5: Heave r é p o n s e in beam seas

180

PtTCE P.espon>e, Eeadins » 90 d«ig JONSWAP: H. - 4.0 m. T, - 13 s, 7 - 3.3

FD, Shtpcnly

B—FD, Shq) + Stinger, Equiv • ^ H ) , Shq> + Stinger, Stech * ID.iîhip'*' Stinger, KödLiD

0.5 1 Wav« ft*^noy [tad/s]

Figure 7: Pitch response in beam seas

180

L.

—1—FD, Ship only

—B—FD, Sh^ + Stinger, Equiv FD, Ship + Stinger, Stoch * TD, Sfa^'*'Stinger, KooLm

LA,

—1—FD, Ship only

—B—FD, Sh^ + Stinger, Equiv FD, Ship + Stinger, Stoch * TD, Sfa^'*'Stinger, KooLm HOIX Response, Heading «• 00 deg

JONSWAP; S , - 4.0 m, T , - 13 ». 7 = 2.2

0.5 1 Wav« fkequenpy [rad/s]

Figure 6: RoD response in beam

Wefbull plot oC Si^ nnt ioHB

B o u D M O K , Bm — XOm^tp - ISft, 7 —3.S RAVLEKW FIT WEIBULL FS 3 3 4 Bdl ampOtodn |ded 6 T B

Figure 8: Distribi^on of Roll crest and trougtis in Hs = 2 m from nonlinear time domain simniations

STINGER HINGE LOADS

In Figure 9 the distribution of positive roll moinent Mx csrests in the hinge are given for the same two conditions as the roll motions of Figure 8. A two parameter Weibull fît on all <^ta points gives a good prediction for the ejttremes. The results show that a 3r-hour extieme Mx moment would be under predicted using a Rayleigh distrîbutic>n.

In the early design stage the absolute value prediction for the hinge loads is inportant, but even more important is to find the correct load trend witii respect to the parameters involved, Uke heading, sea state and current veloci^. If Ihe most severe conditions can be located, a few time domain simulations for these conations using an already optimized design are not

extremely time consuming in terms of CPU time or problem

DKKleling. Such simulations will reveal die actual load level and can be used to obtain tiie distributicms to predict the extren^

values given a probMity of exceedance.

WoäittU plot of U X tnomDnt

Bniin Moa, - 10 m, - U «, 7 - 3 J

Table 1: Comparison of extreme value prediction

R j n u K H r n

U X nxmnnt it Mnge point }iStm\

Figure 9: Distribution of M X at stinger hinge in Hs = 2 m from nonlinear time domain simulations

Figure 10 through Figure 15 show the load RAO at the hinge point in bow qûartermg seas of Hs = 2 m, with and without cuirent in bow quartering seas (heading 150 deg). In general the RAO obtaiaed from stochastic Unearized calculations in frequency domain conqiare weU with those obtained from the nonlinear time domain. Where discrepancies are the largest, that is for the Mx and Mz moment, tbe ftequency domain results over predict the RAO, and are thus conservative. The trend with increasing current velocity is very well precÜcted.

JONSWAP, Hs =• 2 m . Tp " 13 s, Kainma = 3.3:HeadinR=-150 de« FX FY FZ "MX MY M2'

HEN] FkNI fkNI rtNml DcNml [kNml

Hme domain'sbnnlaflion .Toean .5576. .-151 10' sL dev 266 193 343 5975 18195 6537 + nûa3h. 2071, 746.. -3891 24_10^ .53 JO' .29 10' -max 3b -1572 -943 -7055 -2610^ -285 IO' -21 10' expKtëd 7.8 3.9 4.9 4.0 5.4 4.6 extreme/ 5.9 4.9 4.3 4.4 7.4 3.3 sLdev

.Freqoöicy domain with stochastic Ilóearbaöon

mean -5427 -154.8

.stdev 209 197 .375 9303 16102. 6578

FX. at hing« point

Eeidii« ISO deg. JONSWAP: ff. « 2.0 m, T , » 12 s, 7 S.S

I8O1 5*1000 o

s wo

Figure 10: F X h i i ^ load RAO in bow quartering seas

A comparison of the stanclard (teviatipn in the JONSWAP spectrum with Hs = 2 m and Tp = 13 s is giyen in Table 1. Apart from roll rnoment Mx tiie standard deviation from the stochastic linearized calculations aod time domain simulation (using all data samples) c(»rpares well.. Tbe table also presems tiie 3-hour extreme v^ue from the time trace and the corresponding factor between standard deviation and the 3 hour extreme. Hie factors obtained are between 3.3 and 7.8 while linear theoiy would predict a &ctor of about 2*1.86= 3.7, This indicates tiiat by assuming a Gaussian/Rayleigh stochastic process, the 3 hours most probable hinge load extreme wiÛ be under predicted. This confirms the findings in Brouwers [3] ^ o present a fector for the expected extreme with respect to the standard deviation, which increases from 3.7 for (linear) inertia dominated systems to as lü'gë as 8 when drag dormnates the load.

FY at hing' point Heading 160 deg. S<MS9JkV: H, - 2.0 m. T , « IS s. 7 - S.3 1801 -90 -180 800 1 1400 1 / " :

•

1.5

Figure 11: F Y hinge load RAO in bow quartering seas

I FZ at hinge point Heading 160 d«g, JONSWAP: ff. » 2.0 m. T , « IS s, 7 a S. I8Ó 90 0 -90

5

5

F^ure 12: FZ hinge load in bow quartering seas

Î.5

MZ at hing« point

Heading 150 deg. JONSWAP: ff« = 2,0 m, T, » IS B, 7 » S.S l8Ót

FD, Stoch, Uc<a O ra/s IDiUcoOm^ ï D , S o d ] , U c B 2 m / s TD,Uc«2m/s

0.5 1 Wave Requency {rad/s]

Figure 15: MZ lünge load in bow quartering seas

MX at hinge point

Heading 160 deg. JONSWAP: M, » 2.0 ni, » IS 8 , 7 - S.S

I S O ) fD.Stodi,Uc»0m/s lD,UcsOm/s TD, Sodi, Uc B 2 m/s * 1D , UC » 2 B I / S 0.5 1 Wave Ft«qtMncy [tad/s]

Figure 13: MX idnge losd in bow qoarterin^ seas

INFLUENCE OF HEADING

The standard deviation of the motions and loads is calculated for every 10 deg heading variation ih an ctperational sea state. Current velocity is 0 m/s. The cotnpariscm between stochastic linearized frequency domain and nonlinear time domm are presented in Figure 16 for the 6 DOF ship motions and in Figure 17 for the global hinge loads.

The results show good coirelation and demonstrate that the linearized approach can be used to locate the wcirst case concUtions and can provide a first initial design load. The fact that the Fy load level is very well predicted and the Mx load to lesser extends incUcates tiiat the distribution of load over tiie members is somewhat cUfferent in time domain simulations.

MY at hinge point

Heading 160 d«g. JONSWAP: ff. » 3.0 m, T , » IS a. 7 « 3.3 ^ ISOr

ÏD, Stool, Uc » 0 m/s

" I D . U C M OI E / S

in, stach, U c » 2 ra/s • l D , U c o 2 m / j

0.5 1 Wave Fteqtûôicy [tad/s]

Figure 14: MY hinge load in bow quartering seas

faB3iB,TpBi3M, aamam b 3.3 60 80 100 120 140 160 180 »SAOINQ tdeg] - S U R G E , Unesrfzed F D - B - S U R G E . Nonlinear TD • S W A Y , Ünèartzed F D - A - S W Ä Y, NonlinearTO • H E A V E , UnBarIzsd F D - e - H E A V E , NonUnear T B 60 80 100 120 HEADING [daD] 140 1S0 180 • R O U , Unearized FD - ^ R O L L , Nonlinear TD •PITCH. Unearized FD - i - P I T C H , NwiBriearTD

• Y A W . Unearized FD - e- Y A W , ItoillnaarTD

Figure 16: Standard déviation of ship motions

n« B 2 m, Tp B f 3 s, aamina b 3,3 60 80 100 120 HEAOlNQ [datd 140 160 180 •FX, UnaariïBdiFD •FY, LIrâarized FD • F Z , U n e a i t o d F D - FX, Nonlinear TD - FY, Nonlinear TD -FZ, Nonlmear TD 20000 15000 1 10000 6000 m^3m,TpBi3m, gamma B 3,3 60 80 100 120 140 160 180 HEADINO [dad - ^ M X . U n e a r i n d FD - * - M Y . Unearized FD MZ, Lineaiized FD - U X , Nonlinear TD - MY, NonHnear TD -MZ, Nonlinear TD CONCLUSIONS

The Moriscm load on a stinger is linearized using an equivalent and stocdiastic approach. Both methods are appUeKl in an iterative frequency domain solver, which typically c o n v i e s in less than 10 iterations. Stochastic ^earizatioa is considered more applica.ble in a séa state, but both methods perform well.

The overall ship motions and global stingo: hinge load response amplitude operators from linearized calculations are conpared with those obtained from nonlinear time domam simulations, showing g^)od correlation. The influence of wave h e i ^ and current is well cs^tured. It is further shpwn th^ tbe distribution ôf œctremes from a time dpmain simulation dp not follow the Rayleigh distribution, whic;h was expected since the transfer is nonlinear due to the Morison drag l o ^ . The ratio between the most probable 3 hour extreme and standaxd deviation obtained from time domain simulations is found to vaiy from 3.3 to 7.8, while it is generally around 5, which indicate that linearized theoiy assuming a Gaussian process transfer (leading to a fktor of 3.7) wiÛ under predicït the global hinge loads, even if the standard deviation is well precKcted.

It is shown that application of Imearizêd Morison load in pipe lay d^iga c ^ locate the worst sea state conditions, includmg the mfluence of current. Nonlinear time domam simulations remain required to obtain the coirect design load level.

REFERENCES

[1] Shafiee-Far, M . (1997): Hydrc>dynamic interaction betwran fluid flow and oscillating slender cylinders. PhD thesis, Delfl Umversity of Technology

[2] Borgman, L.E. (1967): Random Hydrodynamic Fcffces on Objects, Annals of Mathematical Statistics, pp. 37-51

[3] Brouwers, J.J.H, and Veibeek, RH.J (1983): Expected fatigue dama^ and expected extreme response for Morison-type wave loading, Applied Ocean Research, Vol 5, No. 3, pp

129-133

[4] Wolfrain, J. (1998): On alternative approaches to linearization and Morison's equation for wave forces, Frew. R. Soc. London, Vol. 455, pp. 2957-2974

Figure 17: Standard deviation of liii^e loads