Date Author Mdress
June 2007
Kessel, J.L.F. van and IA. Pinkster Deift University of Technology Ship Hydromechanics Laboratory Mekelweg 2, 26282 CD Deift
The effect of aircushion division
on the
structural loads of large floating offshore
structures
by
LL.F. van Kessel and '.A. Pinkster
Report No. 1547-p
2007
Presented at the 26th International Conference on Offshore Mechanics and Arctic Engineering, June 10-15, San Diego, California, USA, OMAE 2007-29513
TU Deift
Deift University of Technology
Nil IN,VFKINAI. I'EIIllILllJM
.1 EUI INI II)IY INSI 11111E
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Dr. John Halkyard,
Offshore Engineering Consultant
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Mamdouh Salama, ConocoPhillips
PIPELINE & RISER TECHNOLOGY
Segen Estefen, COPPE/UFRJ
OCEANSPACE UTILIZATION
Prof. Î. Kinoshita, Universityof Tokyo Prof. Hideyuki Suzuki, University of Tokyo
OCEAN ENGINEERING
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Walter Kuehnlein, HSVA
CFD:& VIV
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INTERNATIONAL ADVISORY COMMITTEE
Prof, J L. Armand, Asian Institute of Technology, Bangkok, Thailand
Prof. K. J Bai, Seoul National University, Korea Dr. R. Basu, ABS Americas, Houston, TX,, USA
Prof. R. F Beck, UñiversityofMichigan, Ann Arbor, Ml, USA Prof. S. M. Calisal, Univ. of British Columbia,
Vancouver, Canada
Prof. R. G. Dean, University of Florida, Gainesville, FL, USA Dr. R. D'Souza, Brown & Root, Houston, TX, USA
Prof. R. Eatock Taylor, University of Oxford, UK, Prof. O. M. Faltinsen, NTNU,Trondhejrn, Norway Dr. G. Z. Forristall, Forristall Ocean Engineering, Inc.,
Camden, ME, 'USA
Prof. I M. R Graham, Imperial College, London, UK Dr. O. T. Gudmestad, Statoil, Stavanger Norway Prof. J Juncher Jensen, DTU, Copenhagen, Denmark Dr. J. F. Kuo, Exxon/Mobil, Houston, TX, USA Prof. S Kyriakides, University of Texas, Austin, USA Dr. H. O. Madsen, DNV, Oslo, Norway
Profi A. Mansour, University of California at Berkeley, USA Dr. R Marshall, Consültant, Houston, TX, USA
COSPONSORING ORGANIZATIONS
American Concrete Institute (Ad) American Societyof Civil 'Engineers
Assoc. Italiana di Ingegneria Offshore e Marina.(AIOM) Canadian Association of Petróleum Producers (CAPP) Canadian Society of Mechanical Engineers (CSME) Chinese Societyof Ocean Engineers (CSOE)
Coasts, Oceans, Portsand Rivers Institute of the American Society of Civil Engineers (ASCE)
Conseil de Liaison des Assoc. de Recherche sur les Ouvragesen Mer (CLAROM)
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Dr. Barbara Fletcher, SPAWAR Systems Center Dr. Richard J. Seymour,
Universityof Californiaat San Diego
TECHNICAL PROGRAM CHAIR
Dr. Solomon C. Yim, Oregon State University
THE ROBERT DEAN SYMPOSIUM ON COASTAL AND OCEAN ENGINEERING
$ubrata Chakrabarti, Offshore Structure.Analysis, Inc. R. Cengiz Ertekin, University of Hawaii
SPECIAL SYMPOSIUM ON OCEAN MEASUREMENTS AND THEIR INFLUENCE ON DESIGN
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PETROLEUM TECHNOLOGY
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OUTREACH FOR ENGINEERS FORUM
Jim Brekke, Global Santa Fe Corporation Jon Mikklesori, University of British Columbia
Prof: T. Moan, NTNU, Trondheim, Norway Prof: G. Moe, NTNU, Trondheim, Norway Prof: B. Mohn, Ecole Superieure d'lngenieurs
de Marseille, France
Dr. A. Nakanishi, National: Maritime Research Institute,Japan Prof. H. Ohtsubo, UniversityofTokyo, Japan
Prof. A. C. Palmer, University of'Cambridge, UK Prof. A. D. Papanikolaou, The National Technical
Univ. of Athens, Greece
Prof. J., Pinkster, Technical University of DeIft, The Netherlands
Dr. Charles E. Smith, U.S. 'Minerals Management Service, Herndon, VA, USA
Dr. B. Stahl, Consulting Engineer, Houston, TX, USA' Mr. T. Takei, Japan Marine Scienceand Technology Center,
Japan
Mr. N. Tezuka, Japan National Oil Corporation, Japan Prof. J. K. Vandiver, MIT, Boston, 'MA, USA
Prof. R. W. Yeung University of California at Berkeley, USA Prof. K. Yoshida,Tokai University, Japan
Korean Institute of Metals (Kl M) Marine Technolo, Society (MTS)
National Association of Corrosion Engineers (NACE) Norwegian Society of Chartered Engineers (N IF) Office of Science and Technology (OST),
Ordern dos Engenheiros (Portuguese EngineeringAssociation) Royal Flemish Engineers Association (KVIV)
Societyof Naval Architects and Marine Engineers (SNAME) Society of Naval Architects of Japan
The Brazilian Society of Naval Architects (SOBENA) TheJapan Society of Mechanical Engineers (JSME) TheJapan Society of Naval Architectsand Ocean Engineers
(JASNAOE)
The Petroleum Socièty ofCIM (Canadian Institute of Mining) TheSocietyóf Danish Engineers (IDA)
TWI (formerly the Welding Institute)
s
Authorindex - V'
Page 1 of 3'
AUTHOR IINDEX
-
V
ABCDE:FGHIJKLMNQPQRSTUVWxYZ
van den Boom, Henk
Full Scale Monitoring Marco Polo Tension. Leg, Platform (OMAE2007-29635]
van der Cammen, Jeroen.
Calculation 'Methodology of Out of Plane Bending of Mooring Chains [OMAE2007-291 78]
van der Meer, Joop
Ormen. Lange Gas Field, Immediate Settlement of Offshore Rock Supports .[OMAE2007-29038]
van der wal, Remmelt
Viscous Flow Computations on a Smooth Cylinders: A Detailed NUmerical Study With Validation
(OMAE2007-29275]
van;DlJk, Radboud
The Spatial Analysis of an Extreme Wave in a. Model Basin (OMAE2007-29409]
Full Scale' Monitoring Marco Polo Tension Leg Platform [OME2OO7-29635]
van Hoorn, 'Frank
Barge-Assisted Draft Reductión of Semi-Submersible Drilling 'Unit GSF Development Driller I: For
Tow-Out From Ingleside to Offshore Gulf Of Mexico [OMAE2007-29751]
Van 'Kessel, J. L. F.
The Effect of Aircushion Division on the Motions of Large Floating Structures (OMAE2007-29512]
The Effect of Aircushion Division on the Structural Loads of Large Floating Offshore Structures
[OMAE2007-2951'3]
van Zutphen, Hermlone
Nonlinear Wave Scattering From: a Single Surface-Piercing Côlumn Comparison With
Second-Order Theory [OMAE2007-29201]
Vandenbossche, Mike
Fatigue Design ofthe Atlantis Export SCRS [OMAE2007-29355]
Vander Meulen, Aaron
Numerical and Experimental Modeling of Direct Drive Wave Energy Extraction Devices
[OMAE2007-29728]
Vaflderschuren, Luc
The Second Order Statistics of. High Waves in Wind Sea and Swell (OMAE2007-29676]
Vandlver.J. Kim
Identifying the Power-in Region for Vortex-Induced Vibrations of' Long.Flexible Cylinders.
[OMAE2007-291'561
Incorporating the Higher Harmonics in VIV Fatigue Predictions (OMAE2007-29352]
VIV Response Prediction for Long Risers With Variáble Damping [OMAE2007-29353]
Fatigue Characterization of Long Dynamic Risers 'in Deep Waters [OMÄE2007-29428]
Phenomena Observed in VIV Bare Riser Field Tests [OMAE2007-29562]
Author Index - V
Page 2 of 3
Vargas, Pedro
Development and Qualificatiön of Alternative Solutions for Improved Fatigue Performance
ofDeepwater Steel Catenary Risers [OMAE2007-29325]
Vaz da Costa, Marcos Nadalin
Numerical Simulation of Offshore Pipeline Installation by Lateral Deflection Procedure
[OMAE2007-29703]
Vaz, Gullherme
Viscous Flow Computations on a Smooth Cylinders: A Detailed Numerical Study With Validation
[OMAE2007-29275]
Vaz, MA.
Comparison of Coupled and Uncoupled Analysis Methodologiesin Towing Pipeline Installatiön
Modeling [OMAE2007-29506]
Vaz, MurIlo Augusto
The Effect of Flexible Pipe Non-Linear Bending Stiffress Behavior on Bend Stiffener Analysis
[OMAE2007-291 08]
Vazquez-Hernandez, Alberto Omar
FPSO Conceptual Design System Tools Considering Hurricane Data Base and Production
Requirements (OMAE2007.-291 02]
Veitch, Brian
Hydrodynamic Performance Evaluation of an Ice Class Podded Propeller Under Ice Interaction
[OMAE2007-29508]
Veidman, Arthur E. P.
Numerical Simulation of Slashing in LNG Tanks With a Compressible Two-Phase Model
[OMAE2007-29294]
Veflkatesan, Ganesh
Submarine Maneuvering Simulations of ONR Body i [OMAE2007-29516]
Venturi, Marco
Pipe-Soil Interaction: An Evaluation of a Numerical Model (OMAE2007-291 91 J,
Verret, Sean M.
Performance of SteelJacket Platforms in Recent Gulf of Mexico Hurricanes [OMAE2007-29633]
VidIc-Perunovic, Jelena
Flexible Riser Response Induced by Springing of an FPSO Hull [OMAE2007-29044]
Vlkse, Normann
Small Scale Model Tests on Subgouge Soil Deformations [OMAE2007-29249]
Vlnayan, Vimal
Numerical Methods for the Prediction of the Bi!ge Keel Effects on the Response of Ship-Shaped
Hulls OMAE20O7-29744]
Vink, J. H..
Recent Advances on QuasiStatic Response of Ship and Offshore Structures [OMAE2007-29767]
VirgIn, Lawrence N.
Static and Dynamic Behavior of Highly-Deformed Risers and Pipelines [OMAE2007-29180]
Vitola, Marcelo Araújo
Author Index - V
Page 3 of 3
An Investigation on the Synchronization Regime of a Single Cylinder in Cross-Flow Subject to
Harmonic Oscillations [OMAE2007-29572]
Vogel, Mlòhael
Development of Gulf of Mexico Deepwater Currents for Reference by API Recommended Practices
[OMAE2007-29588]
Vogel, Michael J.
Turbulence Measurements in a Gulf of Mexico WarmCore Ring [OMÄE2007-29321j
Volk, Michael
An Experimental Study on Wax Removal in Pipes With Oil Flow (OMÄE2007-29492]
von Jouanne, Annette
Numerical and Experimental Modeiirg of Direct-Drive Wave Energy Extraction Devices
[OMAE2007-29728]
Voogt1 Arjan
Advances in .the Hydrodynamics of Side-by-Side Moored Vessels [OMAE2007-293741
The Spatial Analysis of an Extreme Wave in a Model Basin (OMAE2007-29409]
ABC DE FGH IJKLMNO PQ RSTU V wxyz
J.L.F. van Kessel
Offshore Engineering Depaitment, DeIft University of Technology,
Deift, The Netherlands
ABSTRACT
The effect of' aircushlon division on the structural loads of large floating offshore structures is described and compared with that of a rectangular barge having the samedlmensions.
Calculations are based on a linear three-dimensional' potential method using a linear adiabatic law for the air pressures inside the cushions. The water srface within the aircushlons and the mean wetted surface are modelled by panel distributions representing oscillating sources.
In the presented cases the structurai loads Include the wave induced bending moments and shear forces along the length of the structure. Aircushlons significantly influence the behaviour of large floating structures in waves and consequently redüce the bending moments The internal loads of different configurations of aircushlon supported structures are described and compared with those of a rectangular barge having the same dimensions The significant reduction of the bending moments shows that
aircushion support can be of interestfor large floating structures.
KEYWORDS
Floating structures; aircushion support; structural loads; wave shear forces; wave bending moments; air pressure; motion
behaviour.
INTRODUCTION
The behaviour of large aircushion supported structures in waves has been studied at Deift University of Technology by Pinkster et al. [2-4]. Model tests were performed by Tabeta [5] and served to validäte the results ofthe computations.
This paper describes the effect of aircushion división on the structural loads of large floating offshore structures. These loads are computed with a linear three dimensional diffraction code which was modified to take Into account the effedofaircushiöns under a structure at zero forward speed In waves.
The wave shear forces and bending moments of different aircushion configurations are presented in this paper. The results
Proceedings of OMAE2007 26th International Conference on Offshore Mechanicsand Arctic Engineering June 10-15, 2007, San Diego1 California USA
OM'AE2007- 29513
THE EFFECT OF AIRCUSHION DIVISION ON THE STRUCTURAL LOADS OF LARGE FLOATING
OFFSHORE STRUCTURES
.J.A Pinkster
Ship Hydminechanics Labarato,y, Deift University of Technology1
Deift, The Netherlands
are compared with those of a conventional rectangular barge having the same:dimenslons.
Firstly, the numerical approach of the computational method is described, followed by the approach of the structural loads. Finally the effect of aircush ion division on the structural loads of large floating offshore structures is discussed and compared with that of a conventional barge
NUMERICALAPPROACH
When considering a conventional rigid body, it is customary:to determine the wave forces on the captive structure based on the undisturbed wave potential ç4, the solution of the diffraction potential Ø, and the added mass and damping of the structure oscillating in any one of the six modes of motion in still water based on the motion potentials ç!.. The motions of the structure are then determined' by solving a 6 d.o.f. equation of motion taking into account the wave forces, added' mass and damping and restoring terms.
With a
constructionpartially supported by one or more
aircushions, different approaches may be followed in order to determine the motions of the structure, the pressure in the cushions and other relative quantities such as the structural loads.
The rigid part of the structure Is modelled in the usual way 'by means of panels representing pulsating sources distributed over the mean wetted surface of the construction.
The free surface within each alrcushion is modelled by panels representing oscillating source distributions laying in the mean free surface of each cushion. The mean surface level of individuai' cushions may be substantially different from other cushionsand the mean water level outside the structure
All panels of the free surface within an aircushion are assumed to represent a body without material mass but having added mass1 damping, hydrostatic restoring and aerostatic restoring characteristics. Each free surface panel has one degree of
freedom being the vertical motion of panel n within cushion c. It will be dear that properties such as added mass coupling and damping coupling exist between all free surface panels and the rigid part of the construction. The total number of degrees of freedom (D.O.F.) therefOre amounts to:
C
DO.F.=6+>N,
(1)In which:
N, = number of panels in cushion c
The number 6 represents the six degrees of freedom of the rigid part of the structure. The eqüatlons of motion can in this case be written as:
DÛF
f_oì(M,,, +a,)
iwb4, +c,5)x =X,,
n=l,2,...,DO.'F. (2)in which:
= mass coupling coefficient for the force In the n -mode due to acceleration In the j--mode. Zero for cushion panels.
a,,)
-
added mass couplingcoefficlentb = damping coupling coefficient
C,7)
-
spring coupling coefficientxi
-
modè of motionX, = wave force In the n -mode
In the above equation j=l.6 and n=i.6 represent motions and force modes respectively of the rigid part of the structure. The case ofj>6and n>6 represents the coupling between the panels of the free surfaces of the aircushions. The case of j,=l,6 and n>6 represents the coupling between the rigid
part of the
construction and the vertical forces of the free surface. panels in the cushions. j>6 and n=6 represents 'the coupling between vertical motions of the freesurface panels 'In the aircushlons and the six force modes on the rigid part ofthe structure.
The wave force x,, the added mass and damping coupling coefficIents a, and are determined in the same way as is customary for a multi-body system. The mean underwater part
of the
structure Is discretised 'Intoa number of panels
representing pulsatlng sources as is the case with each free surface panel within an aircushlon.
The contribution of the total potential due to the discrete
pulsating source distributions over the structure and the free sUrface of the.aircushions can be expressed as:
(3)
4ir
in Which: N X A= total number of panels of the structure and free surfaces of all cushions
= =a field point = = location of a source
= Green's function of a source in A relative to a field point
= surface element of the body or the mean free surfaces in the aircushlons
= strength of a source on surface element s due to motion mode j
= potential in ,point due to -mode of motion
The únknown source strengths y. are determined based on boundary conditions placed on the normal velocity of the flUid at the centres of the panels:
=-, m=I,.2....,N
(4)2 "
4ir,, " an " an',The right hand side of the above equation depends on the case to be' solved. If the source strengths for determindtion of the diffraction potential are required the normal velocity vector becomes:
an', -
an,,-
ØO (5)It' should be remembered .that in this case the wave, loads due. to the incoming waves and diffraction effects are defined as being the loads on the structure and the individual 'free surface panels In the cushions, all being fixed. The added mass and damping coupling, coefficients are found by applying normal velocity requirements. For the six rigid body motions (j=i,ó) of the structure:
j =1, 6 (6)
In which the panel Index ni covers only the panels on 'the structure. n', 'are the general directional cosines for the panels on 'the structure given 'by:
n',' =
cos(n,,,x,)
n',2cos(n',,x2)
n,'3 cos(n',,x3 ) (7) tl,,,42 'm3 -
2
n',5 x,,,3 n',, - ),,,,tZ,,,3 fl,,6 = X',, fl',2 - X',2 fi',1 In which:x',, = co-ordinates of the centre of a panel relative to the body axes..
For thIs case the normal velocity components on all cushions are equal to zero.
For the determination of the added mass and damping coupling arising from the normal motions of individual cushion panels the normal velocity boundary condition is zero except for 'one cushion panélat a time for which the following value holds:'
(8)
A disadvantage o. this method for determining the behaviour of the structure
is the fact that wave forces on the captive
structure with aircushions or added mass and damping data for the cases with aircushions oscillating In still water are not
obtained. In order to obtain this' data also, a different 'approach was taken.
The secondapproach
Startiñg point of the second méthod is again Eq. (4) whIch
expresses the normal velocity boundary condition on the structure and thecushion panels.
When determining the wave forces including the effect on the aircushions it is necessary to solve the cushion panel mbtions for the case that the structure Is captive. Having derived the panel motions it Is then possIble to determine the total forces on the captive structure taking into account the wave forces on the structure, added mass and damping coupling effects dúe to the cushion panel motions and the air pressure variations In the cushions. 'In order to determine the various effects the following steps are taken:
- The source strengths 0d are determine by solvIng Eq. (4) for the boundary condition of Eq. (5). Based' on these results the wave forces on the structure without the effect of the motIons of the free-surfaces in the aircushlons are found. From these calculations the wave, forces on the fixed cushion panels are also found.
-
The added mass and damping
coupling coefficients associated with vertical motions of the Individual cushion panels are fòund 'by solving' for the source strengths' o inEq. (4) for the case of vertical oscillations of each cushion
panel individually usIng the normal velocity' boundary condition of Eq. (8)
Finally the motions of the cushion panels are determined by solving the equation of motion for these panels using ,the
above mentioned wave forces and added mass and
damping coupling coefficients as well as spring coupling coefficients based on the aerastatic restoring coefficient and the hydrostatic restoring coefficients of'the cushion panels:
DO. F.
iwb,
c,}x, =X,,,
n=7;D.O.F. (11)In this equation, the added mass and damping' coefficients and the waveforces 'are the same as applied In Eq. (2). From the solution of the equations of motions of the cushion panels the total wave forces on the captive structure can be determined as well. as the pressure variations within the cushions.
The next step is to determine the added mass and damping 'of the structure including the effect of the free surfaces in the aircushions.
In order' to accomplish this it is first necessary to determine the solution of the source strengths and fluid pressures' for the'case that the structure is oscillated while the cushions remain fixed. This is accomplished by solving Eq. (4) for the case that the normal vélocities
on the
panelsof the
structure are inaccordance with Eq'. (6) while the normai velocities on the cushion panels are equal to zero.
For each of the six modes of motion of the structure, this results in hydrodynamic loads on the structure and on the cushion panels. To these loads we also need to add the aerostatic fòrces since the oscillations of the structùre change the pressure in the cushions.
3 CopyrIght © 2007 by ASME
where the -1 follows from the fact that the free surface normal Is poInting in the negative X3-direction. From the solutions of the source strengths for all these cases the 'wave force vector X, and the added mass a, and damping coupling coefficients
can be obtained. The wave force follows from:
X,, =pw
(ø. +Ø,,k)n,,k AS,,, (9)In which:
= diffraction potential at k-panel obtained from Eq.
(3)
X,
= wave force
inthe n -mode, n = 1,6 for the
structure
N, = number of panels involved in the force in the n
-mode. For the force on a cushion panel N,,
= i.
For the force on the structure N,, = N= generalised directibnal cosine of k -panel related to n -mode
AS,,.,, = area of k -panel related to the force in the n -mode The added mass and damping coefficients follow from:
a,j = -
Re[ìOØi,
AS(10)
b,, =-
n,,, AS,, ] in which:= motion potential value on k -panel obtained from Eq. (3)
The restoring coefficients in general consist
of two
contributions i.e. an aerostatic spring term and a hydrostatic spring term as described in [6]. The hydrostatic restoring term Is equal to the product of the waterline area, specific mass water and acceleration of gravity. Thts 'applies to both the structure and 'the free surface panels. The aerostatic restoring terms are related to the change In air pressure In an alrcushlon due to, for Instance, unit vertical displacement of a free surface panel and the correspondlng'forces applied to the particúlar'panel, all;otherpanels belonging to the same cushion and the force on the structure. Conversely, displacing the structurein any of the three vertical modes of heave, roll or pitch will change the volume of an aircushlon thus inducing pressure changes' and as a
consequence forces on all free surface panels and on the structure itself.
For the determination of the aerostatic part 'of the restoring terms, use is 'made of a linearised adiabatic law' as described In
[6].
W,,cosO
W, x
Figure 1: Free bodydlagram ofthe tait' endof the. structure.
A bodybound coordinate system
is used with its origin connected: to the centre of gravity of the floating body, In which the x-axis Is positive towards'. the bow in longitudinal direction1 the y-axis is' positive, in the direction of the port side and the z-axis isdirected. upwards.The shear forces and bending moments are highest when the structure is subjected to head waves. For sake of brevity shear forces and bending moments are discussed fòr head seas only as in this casethe effect of the aircushions will be largest. The body is considered to be rigid and the mass Is equally distributed along the length of the structure.
The vertical shear force acting in the plane of a cut with a distance x' from the centre of origin can be calculated by:
4 Copyrlght© 2007 by ASME
Q. (x') = - Jq(xb.)dxb (14)'
¡n which the vertical force q(xb) along the length ofthe structure consists of inertia forces, hydromechanical forces and hydrostatical forces. The forces on the bottom of the structure resulting from the pressures inside the aircushions are included in the hydromechanical forces. The forces acting on a segment of the structUre can be written as:
'(---xbÖ)dxh
=(F'pgA,, W')dxh
+ Q:
(15)in which:
w,
= massof the structure per meter = hydromechanical forces including theairpressure effect
With use of Eq. (15), the vertical force q at a distance Xb from thecentre of gravityof the. structure can :be.written as
F.'+pgA,, W'
(16)The last two terms of Eq. (16) representsthe static loads:
pgA.,W'
(17)And the first part of Eq. (16) represents the contribution of the dynamicloads:
-
- x,
ë) + Fi'g
(18)
Substitution of Eq. (16) in Eq. (14) results in the final expression for the vertical wave shear force at a distance x' from. the centre of gravity:
Q (x')=
- ;
ë)
-
F.'Jdxh (19)A. corresponding expression for the horizontal shear force in head seas can be derived, though in this case a horizontal component of .the mass-forcé should be added.
Q (x')
=J(!_(.,,
-
bG(x') )-
F'Jd
(20) In the above, (x') is the vertical distance of the centre of gravity of the cross, section to thekbody.bound x-axis.When the body is subjected to regular head seas' it has an angular acceleration around the y-axis. The rotational equation of motion around the y-axis normal to the piane of motion can bedescribed by an EUler equation:
I,Y O = x, + x,., (21)
in which:
X,,, hydromechanical moments about the yb-axis x,,, = exciting, wave momentsaboùt the'yb-axls Based on the total forces on the cushion panels, the added mass
and damping coupling coefficients and the aero- and hydrostatic spring coefficients the equations of motion olthe cushioñ panéls can again besolved:
D.O.F.
{a?a,,,iwb,, +c,}x,=X,,
n=7,D.O.F. (12)j=7
In this equation, the coefficients and wave forces are again in accordance with Eq. (2). From the solution of the motions of the cushion panels, the additional hydrodynamic and aerodynamic
contributions to the forces on the structure can easily be
obtained thus yielding the added mass and damping including cushiOn effects Based on the thus obtained added mass, damping and wave forces the wave frequency motions of the structure can be determined from the normal six degree of freedom equations of motion:
{aì(M, +a,,j _iøb,,, +c,,,}x =X,
,i=1,6. (13) In this equation the added mass, damping and spring coefficients as well as the wave forces app!y to the structure only and include the effects of aircushlons.STRUCTURAL LOADS
The internal shear forces and bending moments can be
computed with the weight distribution of the structure and the hydrodynamic forces determined by the methods as described above. In case of a rigid structure this is similar to the problem
of a beam subjected to an arbitrary distribUted load with
additional inertia forces.
The procedúrefor calculation of the internal forces and moments Is the same as employed ¡n beam theory; the structure is sliced transversely at the station of interest, and a free body diagram as presented in Fig. lis constructed ofone portion of the húlI.
= k, ni = mass moment of Inertia around the Yb-axis
k = radius of gyration
In Eq. (21), the hydromechanica! moments are lndùced by the harmonic oscillations of the rigid body, moving In the undisturbed surface of the fluld The wave exciting moments are produced by waves coming in on the restrained body. Since the system Is linear, the total external moment on the floating body Is equal to the sum of
and x5.
Figure 1 shows the external forces and internal loads on a construction part at a distance x' from the centre of gravity of the total body. Here q, and q,,1 are the external forces on the structural component, M Is the bending moment, and N and
Q are the horizontaland vertical shear force respectively.
The wave bending moment at a distance x' from the centre of gravity can becalculated with use of Eq. (21) andresultsin:
M(x')=Ç1Ü
-
¡
q,, k);
+ (x')x'J(22)
-In which w is the weight of the structural component and i Isthe mass moment ofinertia of the construction part around its centre of gravity cn
Both Q and M are harmonic functions in regular waves and the amplitude depends on the wave height, as a result Eq. (19), (20) and (22) can be written as:
Q( )QI(O4I+(QÇ)
=e'°''
(23)M (x') = Mb,,
e'k'
=Aie"'
It should be noted that according to Eq. (20) the vertical shear forces at theends of the structure are equal to zero, though this Is not necessarily the case for the vertical bending moments due to the presence of radiation and diffraction forces at the ends of the structure. The radiation and diffraction forces resùlt in
horizontal shear forces at both ends of the structure as well
STRUCTURAL LOADS OF LARGE AIRCUSHION SUPPORTED STRUCTURES
The effect of aircushion division on the structural loads of large floating offshore structures was calculated and compared with that of a conventional rectangular barge. Both the barge and aircushion variants had the following main particulars:
Length
1500 m
KG 5.0 mBreadth 50.0 m k 42.0 rn
Draught 5.0 m
Displacement 384375
t
The height of all cushions is 5 rn and the amblent air pressure was taken equal to 100 kPa. Different configurations of the structure resUlted in different natural frequencies and stability aspectsas given in table 1.
All structures are modelled by square panels of 2.5 x 25 m. The total number of panels is equal for all structures In case of the single cushion variant (lAC) the rigid structure was modelled by 320 panels and the cushiOn itselfby 1200 panels
The whole watedine aa of the structures lAC, 2AC, 3AC and 8AC is covered by aircushions. The wall thickness of the skirts
was equal to zero. The centre of buoyancy of the lAC
configuration will not shift due to small heeling angles and the buoyancy force acts through a fixed point at half draught of the structure, resulting In a GM-value equal to the distance between the centre of buoyancy and the centre of gravity. The negative GM-value of the lAC configuration results from the fact that a single cushion covers the whole waterline In longitudinal direction. This structure is unstable, but nevertheless has been included to show the effect of different aircushion configurations on the behaviour of the structure. In this case additional stability can be gained by giving the skirts a thickness, this Is the case for the structures referred to as 'Combi 1' and 'Combi 2'. The rigid skirts surroundIng these aircushions have a thickness of 5 and
10 rn respectively, consequently the dimensions of the aircushions are 140 x 40 m and 130 x 30 m. In general the. stability of a floating body dècreases when the structure is supported by aircushions as can be seen In Table 1.
DUe to the decrease In the stability, the natural pitch frequency also decreases for structures totally supported by one or more aircushions. Conversily, the natural pitch frequencies of the 'Combi 1' and 'Combl 2' configuratIons are larger due to the relatively small pitch added mass. The heave natural frequencies are approximately equal for all structures.
The calculations of the structural loads are based on the lay-out
as given in Figure 2. This figure shows an aircushion configuration with M x N cushions and regular waves that approach the. structure from the right side.
TABLE 1: MAINPARTICULARS OFTHE STRUCTURES, NATURAL FREQUENCIES AND STABILITY
Structuretype / name Cushions Cushion Size GM
M x N [-J Length [ml . Breadth [ml [rad/s] [rad/si [ml
1 cushion (lAC) i x 1 150 50 0.68 n/a -2.5
2 cushions (2AC) i x 2 75 50 0.68 0.65 224;5
3 cushIons (3AC) i x 3 50 50 0.68 0.68 2665
8 cushions .(8AC
2x4
37325---
068--0.69
-281.3_nLa
---
nja nJa 0.69 -0.74 -372.5Combi 1 1 x 1 140 40 1169 0.82 128.6
Combi.2
i xl
130 30 0.69 088 226.0Xv-75 XO
X75
M rows
4vftt35
N rows
FIgure 2: Lay-out of a free-floating structure supported by N x M aircushlons.
Figure 3 shows that the wave shear forces are largest for a conventional barge when the wave length is equal to the length of the structure, in this case the maximum value ¡s at a distance of 50 m from the centre of gravity. The smallest amplitudes of the vertical shear force are reached around the centre of the structure. A different distribution of the shear force can be seen when the length of the structure is twice the wave length (O9O radis), in thiscase a maximum occurs at the location x = O.
lie right side of the structure is subjected to the incoming
waves, as a result the distribution of the vertical shear fortes and bending moments is not symmetrical in case the wave lengthIs equal to the length of the structure. The maximum and minimum amplitudes are shifted to the right. The wave bending moment is largest when the wave length is equal to the length of the structure at 0.65 rad/s.
The distribution of the shear forces and bending moments Is signIficantly different In case the structure is supported by aircushions. Tie effect on the internal loads is largest when the structure is supported by a single aircushion
like the lAC
configuration. For this configuration the shear forces and bending moments are reduced by 98% and 96% respectively, though It should be noted that this configuration is a theoretical example as the side walls of the structure have no thickness. Shear forces of the lAC configuration are approximately equal when the length of the cushion corresponds to a multiple of the wavelength, this Is the case when the wave frequency is O65, 0.90 or 1.1 radIs, as shown in Fig 4. Conversely, the maximum amplitudes of the shear force are reached for waves with a
frequency of 0.40 rad/s due to the pressure peak ¡n the
aircushlon and the motions of the structure as described in [6]. Figure 4 clearly shows the wave bending moments at the ends of the lAC configuration, these moments are the result of the water pressures and eccentric diffraction and radiation forces as discussed in the previous section.
The maximum values of the horizontal forces at the ends of the aircushlon configurations are large compared to those acting on a conventional barge. This Is dùe to the relatively large pitch motions of the aircushion supported structures compared to those of the barge [6]. The lAC is an exception, because of the negative GM-value It has no natural pitch frequency, resulting in relatively low pitch motions and consequently small diffraction and radiation forces at the ends of the structure.
The bending moments of the lAC are largest at x = 75 m
because this side is subjected to the incoming waves. In case the cushion length Is equal to the wave length the bending
moments steadily decline without maxima or minIma. This results from the fact that the air pressure variations are small and the pressure along the length of the structure is constant as the waterline area is totallycovered by a single.alrcushion.
The horizontal shear force distribution of the 2AC configuration is given in Fig. 5. The figUre shows the results with and withoUt the horizontal pressures of the aircushlons on the skirts. The
pressure inside the aircushion results in a horlzontalforce on the skirts of the construction, thIs force partly compensates the diffraction and radiation forces at the end of the structure, resulting in a smaller horizontal shear forces at the ends of the body.
Figure 6 shows air pressure variations inside the cushions, the vertical wave shear forces and the bending moments of the ZAC configuration. The air pressure variations of the individual cushions of the 2AC configurations are approximately equal. Consequently, the amplitudes of the vertical wave shear forces are symmetrical around x = O and the highest values are reached when the wave frequency is 0.60 rad/s. When the wave length is equal to cushion length, which is 75 m (0.90 rad/s), the pressure variations are nearly zero and the vertical shearforces are small. The maximum vertical shear forces are located amidships at the boundary of the two cushions. Consequently, the vertical bending moments are small at this location. The distribution of the moments is not symmetrical and the
maximum value is located at the right half of the structure as a result of the relatively large eccentric horizontal forces at the right side ofthe structure.
Figure 7 shows that the vertical wave shear forces of the 3AC configuration are nearly zero at the centre of the structure. The distribution of the forces is not symmetrical around this point due the large pressure variations In the front aircushion (cushion 3). Again, as Is the case for all aircushion configurations, the pressure variations and the vertical shear forces are small when the wave length A is equalto the cushion length. The maximum wave bending moment is significantly larger than was the case for the previous two cushion configurations. This figure clearly shows the increase of the maximum wave bending moment when the structure is supported by multiple smaller aircushions. Figure B shows the results of the 8AC configuration. The structure is supported by two rows of four cushions. It. was already shown in [6j that the width of the aircushion is of no importance on the motion behaviour of the structure in head
seas. For this reason the internal
loads of an aircushion sUpported strUcture with a 2 x 4 (= BAC) configuration will bethe same as for a i x 4 (= 4AC) configuration. For most
frequencies the maximum vertical shear force Is at the boundary between the first and second cushion at the right side. In two situations the maximum shear force is at the centre of the structure, these situations occur at a wave frequency of 0.90 and 1.1 rad/s when 2 = 2L and 2 =3L respectively. The peak in the distribution of the shear forces resUlts in a dip in the bending moment at x = O.
The previously discussed aircushion configurations were
theoretical cases as the skirts had no thickness and the
structures were completely supported by air. The maximum amplitudes of the vertical internal loads of theses structures are presented In table 'below.
TABLE 2: MAXIMUM AMPLITUDES OF VERTICAL
6 Copyright © 2007 by ASME
INTERNAL WAVE LOADS
Shear Force Bending Moment
Table 2 clearly shows an increase ¡n the internal loads when the structure Is supported by multiple cushions. The wave shear forces and bending moments of the 2AC configuration are small compared to those of the conventional barge, nonetheless they are significantly larger than those of the lAC configuration. Moreover It can be seen that the internal loads of an aircushion supported structure approach those of the conventional barge wheñ the length of the individual cushions decreases
Practical examples are the 'ConibI 1' and 'Combi 2' cases in which the skirts have a thickness of 5 and 10 m respectively. In the first case 75% of the buoyancy is provided by the alrcushion, In the second case this is 52%. Examples are known. iñ the
offshore industry in which up to 96% of the weight of the
structure. was provided by aircushions [1].
In figures 9 and 10 it can be seen that the vertical shear forces at the ends of the structure increase significantly due to the thickness of the skirts. Moreover, this thickness has a large effect on the maximum vertical shear forces and bending moments as can be seen In Table 3. When the conventional barge Is taken as a starting point, the maximum wave bending moment decreases by 43% in case of 'Cambi 1'. In the situation of 'Combi 2' the reduction amounts to 20%.
TABLE 3: MAXIMUM AMPLiTUDES OF VERTICAL INTERNAL WAVE LOADS
The midship bending moments and vertical shear forces are presented ¡n Fig. 11. The figure clearly shows that the Internal loads can be significantly reduced by the. use óf aircushlons. A large single cushion shows the best results and the thickness of the skirts should be as small as possible to reduce the vertical wave shear forces and bending moments as much as possible.
CONCLUSIONS
In general it can be concluded that the use of aircushions can significantly reduce the vertical wave shear forces and bending moments.
The maximum values of the horizontal shear forces are relatively large compared to those of a conventional barge, this is due to the relatively large pitch motions of aircushlon. supported structures compared to those of the barge
A large single cushion shows the best results with respect to vertical wave shear forces and bending moments. In order to reduce the internal loads as much as possible, the thickness of the skirtsshould be taken as small as possible.
The results have shown that the bending moments of a
conventional barge can be reduced by 43% In case 75% of the weight. of the structure Is supported by a single aircushion. The bending moment can be .further reduced in case a larger portion of the buoyancyls provided by the aircushlon.
In case the skirts are assumed to have no thickness and the bottom of the structure Is covered by a number of cushions1 the vertical structural loads Increase when the length of the cushions decreases.
REFERENCES
Kure, G. and Lindaas, O.J., Record-breaking air lifting operation on the Gulifaks C project. Offshore Technology Conference, 1988, paper no. OTC 5775.
Plnkster, lA., The effect of air cushions under floating offshore structures Proceedings of ßoss'97, 1997, 143-158. Pinkster, J.A, Fauzi, A., moue, Y. and Tabeta, S., The behaviour of large air cushion supported structures in waves.
1-lydroelasticityin Marine Technology, 1998, 497-506. Pinkster, 3.A. and Meevers Scholte, E.1A., The behaviour of
a large air-supported MOB at Sea. Journal of Marine
Structures, 2001, 14, 163-179.
Tabeta, S., Model experiments on barge type floating structures supported by air cushions. Report 1125,
Laboratory of S/np Hydromec/zanics, DeIft University of Technology, DeIft, 1998.
Van Kessel, 3.L.F. and Plnkster, 3.A. The effect of aircushion division
on the
motions of large floating structures. Proceedings of the 26th International Conference on Offshore Mechanics and Arctic Engineering (OMAE'07), ASME, 2007; No OMAE2007-29512. 5.00S+ 4.50E+ 4--4.00EX3 3.5o6+ L... 300E+03 2.SOE+cX3 2.00E+3 1.5OE 5.00EtQ2 O.00E4O0 -75 -60 2.50E-f05 r aoos+osi
150o5
1 00S5 5.00S+04Wave Shear Forces
-45 -30 -15 0 15 30 45 60 75 X-coordlnate [m]
V.veBend
000Es-00 -75 -60 -45 -30 -15 0 1530 45 60 75
X-coordlrte [m]
Figure 3: VertIcal wave shear forces and bending moments of a conventional. bargeat different wavefrequencies:
0.i [rat's] -.-..O.2 [rat's] -.ò-OE3 [rat's]
.-04 (rat's]
i-0.5 [rai/s] -q-0.55 [rai/s]-.-0.6 [rat's]
.=....0.65[rat'sJ-.i-7 [rails)
0.9 08 ''OE7 OE6 0.5 04 0.3
L 02
01 0.0. o ai 02 03 0.4 0.5 06 0.7 0.8 0.9 1Wave Aequency (rad/
Y...
es
&60E.O1
6.00E.Oi
4.00E.O1
aooE.oi
0.l [radis] --2 [radis]
-0.3[rad/s1
0.65[radlsj 0.7 [radis)
-0.8[rad/s)
Qishlon PiessuieVat1atons
1.1aoo
-75 -60 -45 -30 -15 0 1530 45 60 75
X.ordlnate(m]
9.00.U3aoo.
Ia°°
500S4X34.00E.cX3aooE'
ZOOE-.60i.00+
aOOE+U) -75 -60 -45 -30 -15 015 30 45 60 75
Xoidinate[m]
Figure 4: Air pressure variations inside the cushion, vertical wave shear forces and bending moments of the lAC configuration at different Wave frequencies.
Hon
She Fces
(exci. the effect of the aimushions)
aoos... 7.00E- -. 6.00
5.00s
4.00E,U3aoos.
aoo 1.00E OEoo4o0 -75-M-04[rad/s] -''-0.5 [radis] 0.55 [rad/s] 0:6 [radis]
-0.9[rad/s]
-'"1 [rad/s] --- 1.1 [rad/sJ ' '12 [radis]Hozon Vve
Foes
(hid. the effect of the aircushlons)
7.00E-+60
aE4
I. 8 4.000+60 1.00E-+60 0.00E-+00 -75 -60 -45' -3) -15 0 15 30 X-coa,drte,(m]Figure 5: Horizontal wave shear forces with and without the air pressure contribution of cushions of the 2AC
configuration at different wavefrequencies.
30 ..'c.
,,,,Pressu1t..ns
.-4QJShiOfll
--isI-iior
, I - ¡i
J
i
0.0 0 01 0.2 03 0.4 05 0.6 07 0.8 0.9 Wave Requency [iadfl4.006404 asoE4o4 aoo&o asoE.o4 2006404 106404 1006404
aoos.
0.005*00. -75 -60 -45 -30 -15 0 15 30 45 60 75 X-cocrcllrete (m] 45404. 0 15 30 45 60 75' 8 CopyrIght © 2007 by ASME Figure 6: -75 -60 -45 -30 -15 X-cooÑirete'[m]Air pressure variations Inside the cushions, vertical wave shear forces and bending moments of, the 2AC configuration at different wave frequencles
-60 -45, -30 -15' 0 15 30 45 60 75
X.coordlnate Em].
g
J
i
I
g
i'
4.0 a5 ao a5 ao 1.5 1.0 0.5 0.0t40E5
12E+O5lE+05
aE
4E.o4
Z00EO40i [radis] 02 [radis] 0.3[radis1 -04:[rad/s]
-'-65 [radis]
07 [radis] 0.8[rad/s]0.9'[rad/s]
Cushion PmssuieVariallons
.--Ojshionl
o OEi OE2 OE3 0.4 OES OE6 0.7 0.8 OE9 1 1.1
Wa Fquency Lmdf
Wa She
Fornes
-75 -60 -45 -3) -15 015 30 45 60 75
X-coosdlnate (m]Birrnnts
OEWE.60 -75 -60 -45 -30 -15 0 15 30 X-coordlnate(m]FIgure 7: Air pressure variations Inside the cushions, vertical wave shear forces and bending; moments of the 3AC configuration at different wave frequencies1
5.00EM
4.60E
4.00 3.60E4oe ZSOE.aoo.
t50E 1.WE+fl 0.00B) -75 -60 -45 -30 -15 015 3) 45 60 75
X-cooitflrte (nil
0.5 [radis] --- 0.55 [radIs] 06 [radis] 1 [radis] [radis] ...12 [radis]
'g
i
I
1 .60E406
Bendiflg iVbints
1.406+05
- -.
1.E.05 1iE+05 -t aooE-+o4 .4 4.605+04 OE006+00 -75 -) -45 -30 -15 0 15 30 45 60 75 X-coordlnate[m]Figure 8: Vertical wave shear forces and bending moments of the 8AC configuration.
We$hearFornes
Up»,
-75 -60 -45 -30 -15 .0 1530 45 60 75
X-coorcmte Em] -75 -60 -45 -3) -15 015 3) 45 60 75
X-osmte (m]
Figure 9: Vertical wave shear forces and bending moments of the 'Combi 1' configuration at different wave frequencies.
9 Copyright © 2007 by ASME
4.506.05 4.006..,
asos.
aE+03
21 2.50640500E3
1.506405 1.005.035.00E.
OE00E.00i
1.80E+05 1.6064(5 1405+05 1306+05 1.0064(5 &OOE.04 6.005.04 4.005.04 2.006.04 0.005.-COWave Shear Forces
-75 -60. -4ff -30 -15 0 15
30 45 60 75
X-orcfinate [m]
-75 -60 -45 -30 -15 015 30 45 60 75
X-000rdlnate [mJ
I
L
I
0.2 03 0.4 05 0.6 0.7 0.8 0.9 1 1.1ency[rad'
o 0.1 0.2 0.3 4 OES 06 0:7 0.8 0.9Frequency [radIs
1ti
10 Copyright © 2007 by ASMEFigure 10: Vertical wave shear forces and bending moments of Figure 11: MIdship vertical wave shear forces and bending the 'Combi 2' configuratIon at the following wave moments for the following configurations:
frequencies:
