467
1 INTRODUCTION
Shipstabilityperformanceiswidelyfoundasoneof
the key factors influencing safety of sea transport.
However, main efforts of researchers aim at some
phenomena related to ship complex motions when
sailinginroughseaconditions.Inregardsofaships
safety the greatest concern relates to it
s rolling
oscillations. The main parameters of the commonly
considered rolling equation are: inertia, damping,
stiffnessandexcitation.Theliteraturereviewshows a
list of works taking into consideration various
aspectsofshipsrollingmotion.
Generally, potentially dangerous situations that
maycausecapsizingofashipthatremainsintact, can
bedividedint
oresonantandnon‐resonantones[1].
Thenon‐resonantsituationsaremainlytheoutcome
ofashipsrollingmotionandadynamicgustofwind
[2] or they are caused by the loss of stability on
followingorquarteringseaswhenthewavecrestis
amidships. Furthermore, broaching and surfriding
ma
ybeclassifiedasnon‐resonantphenomenawhich
may lead to capsizing [6]. The resonant situations
may be divided into parametric resonance and
synchronousrolling.
Allthementionedabovedangeroussituationmay
takeplacewhensailinginadverseweathercondition.
Nevertheless, there are many causes for stability
problems occurring not only in sea condit
ions but
alsoduringship’sstayinaport.Suchapossibilityis
not obvious at the first sight, however, stability
problemsarenoticedinportsaswell.Generallythe
causesforpotentialstabilityrelatedincidentscanbe
divided into several typical groups, as shown in
figure1.
According to the diagram (fig. 1) cargo shifting
incident
s and some accidents related to cargo and
ballast operations are mentioned as the possible
hazard to ship’s stability. On the other hand, such
eventsarestrictlyrelatedtocargooperationstaking
place in ports, which are crucial component of
carriage of goods by sea. In case of dry cargo
tra
nsportationanessentialpartofthecargohandling
isitsloadinganddischargingoperationbymeansof
cranesandgantries.Sincetheglobalcontainerization
trend has reached a significant share of the market
Variations of Ship’s Deck Elevation Due to Stochastic
Process of Containers Loading
P.Krata
GdyniaMaritimeUniversity,Poland
ABSTRACT:Thestochasticprocessofcontainerloadingisdescribedinthepaperwithspecialemphasistoship
motionwhensheislyingataquay.The3DOFsystemwasappliedtodescriberolling,pitchingandheavingof
a vessel which may cause a significant variations of momenta
ry deck elevation. The realisticrange of such
variationsareassessedforavarietyofcargolocationson‐boardandaphaseshiftbetweentwoindependent
gantries engaged in cargo operations. The process is modeled with regard to random character of crucial
variablesaffectingshipmotionduetocargoloading.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 9
Number 4
December 2015
DOI:10.12716/1001.09.04.02
468
and it is still in progress, there is a point to focus
especially on gantry operations. The increasing
importanceof the subjectmay justify the long term
growing trend in container operations which is
noticed in many sea ports, for instance in Poland
whichisdepictedinfigure2.
Figure1.Hazardstostability[4]
Figure2. Increase in container operations in the Baltic
Container Terminal Ltd. in Gdynia, Poland (source of
data:[10])
The gradual evolution of sea transportation
market and steady grow in container sector
operations results in the modernization of cargo
handling equipment. Especially fast moving
container gantries are in common use in sea port
worldwide.
As a gantry isfirmly establishedon the ground,
usuallyonadedicatedrailsystem,itseemstoallowa
smooth cargo shipment down to ship’s holds and
tweendecks.However,thesurfaceofdecksandta
nk
topspersistsinpermanentmovementwiththewhole
body of ship’s hull, creating control challenges
resultingfromthisrelative motionofthegantryand
the cargo destination position. This motion is an
int
egralpartofsea vesselscargooperationsandthus
has to be dealt with as preciselyas itis reasonably
possible. Any increase in the accuracy of gantry
control in terms of relative motion compensation,
improves the overall performance of the cargo
handling process. The essential problem of gantry
controlisconsideredbyma
nyauthors,nevertheless
they omit the problem of a moving base of cargo
destination. Even when the dynamic modeling and
adaptive control of a gantry is researched and
applied the efforts are aimed at the tracking errors
reduction with no consideration dealing with
unstablepositionofship’sdeckorcargohold[8].
The contemporary gantry control systems found
in sea ports might be relat
ively advanced and
sophisticated but they do not capture any external
data describing ship rolling, heaving and pitching.
Even is such extreme conditions like cargo transfer
carriedoutongasandoiloffshorefieldsthemoving
ships tra
nsmit no information enabling her motion
estimation.Thelack ofshipmotionestimationduring
cargooperationsinportsisevidenttoo.Bothremarks
result from authors’ sea service experience onboard
ships and series of reviews with ship masters and
chief mates responsible for cargo loading and
stowage.
One of the asp
ect of interactions taking pla ce
between loaded or discharged cargo and a vessel
reflectspossiblehazardstoshipandcargoresulting
fromtooimpetuousplacementofapieceofcargo,for
instance a container, on deck or tank top [5]. This
may cause some damages to ship construction or
loaded cargo and always generates an economical
loss.Theexp
lanationofsuchaphenomenonisbased
onasimpleremark,thatcargoissmoothlylowered
byagantrytobereleasedwhenincontactwithdeck,
while the vessel rolls and pitches due to some
external excitation or other cargo influence.
However, the problem of moving ba
se which
impedes and slows down cargo operations in sea
port, can be solved by means of gantry control
improvement and an application of proper
compensation[5].
Themainpurposeofthepaperistoconsiderhow
the deck elevation can change due to container
loading.Thecargooperationisfoundasastochastic
process with two independent gantries working
simult
aneously.Thefeasibilityofeffectivemodeling
of this elevation is an important step towards
elaboration of a gantry control and optimization
system.
2 3DOFMODELINGOFSHIPMOTIONDUETO
CARGOLOADINGINPORT
The contemporary a
pproach towards modeling of
shipbehaviorunderexternalexcitationduetocargo
loadinginaportisfocusedonthetransversestability
performance. It is an vital issue because the
significant rise in a vertical center of gravity is
inherently related to cargo loading on‐board. The
negligenceofoperatorsoccurringatanystageofthe
processma
yleadtoverydangerousincidentslikefor
instance capsizing of M/V Stella Mare [9] or M/V
Deneb [3]. Both of them suffered an improper
operation and rolled over in ports. However, the
standard approach based on ship’s metacentric
heightofarightingarmcurvedoesnotprovideany
informat
ion in time domain, thus, it cannot be
effectively used for the purposeof container gantry
control.
Shipmotionunderexcitationforcesduetocargo
loadinginaportneedstobemodeledtoprovidethe
time‐dependentinformationaboutmomentaryship’s
deckelevationatanyspot.Onceacontainerisloaded
on‐boa
rd its weight acting at a specified location
arousesrolling,pitchingandheavingoftheship. As
a result of these motions any spot of subsequently
469
loadedcontaineraltersitselevationinthecourseof
natural oscillation damped by water friction.
However in case of relatively high rate of cargo
loadingandespeciallywithtwoormoregantrycrane
working at the same time, it is likely to transfer a
containerfromaquaysidetothe
shipholdinpretty
shorttimeonebyone.Ifso,theformerlyloadedbox
excites variations of ship’s deck and the successive
boxneedstobeputinitsdestinationelevatedhigher
orlowerthen it wouldbefoundwithnomotion of
the ship. Moreover, the elevation depends
not only
onthelocationofpreviouscontainerbutalsoontime
delay between these two gantry operations. The
describedideaisshowninfigure3.
Figure3. The idea of time dependent variations of ship’s
deckelevationduetocontainedloading
Thedynamicmotionofafreefloatingvesselina
port or at the sea is affected by a set of forces and
moments,bothexternalandinternalones.Generally,
theanalysisofshipmotionisgovernedbythesystem
ofsixdifferentialequations.However,thesolutionof
such generally formulated
problem is too complex
for practical applications, so further simplifications
and assumptions are required [7]. By neglected
coupling,forthesakeofsimplicity,theship’srolling
isusuallyanalyzed by the singledegree‐of‐freedom
systemorthreedegree‐of‐freedomsystem[7].Inthis
paperthreeuncoupledequationsof
shipmotionare
implemented, e.g. roll, pitch and heave motion is
takenintoaccount.Accordingtothetheorysuchan
linear theory based approach is valid for the
relatively small amplitudes of expected motions. In
addition in case of ship rolling, being the most
significant motion in the analyzed matter, the
strongest coupling could occur with yaw and sway
motions which may be neglected due to ship
mooring forces so any effects of such coupling are
practicallynexttozero.Theremainingmotions,e.g.
pitchingandheavingareexpectedtobeoneorderof
magnitudesmallersothepotentialeffectof
coupling
betweenthemmaybeomittedaswell.
Thegoverningdifferentialequation ofrolling,as
the result of equilibrium of moments in direction
usuallysigned“4”(aboutshipxaxis)isfollowing:
44 4 4
() () ()
I
DRMt



(1)
where:
I
4–transversemomentofinertia ofashipandadded
masses;
D
4–rolldampingmoment;
R
4–restoringmoment;
M
4–heelingmoment;
‐angleofheel;
‐angularvelocityofrolling;

‐angularaccelerationofrolling;
t–time.
The equation of ship pitching (motion in
direction“5”) e.g. about y axis is described by the
formula:
55 5 5
() ()
j
I
DCgML Mt



(2)
where:
I
5 – longitudinal moment of inertia of a ship and
addedmasses;
D
5–pitchdampingmoment;
C
5–unitcalculationfactor;
g–gravityacceleration;
M
j–momenttochangetrim;
L–ship’slengthbetweenperpendiculars;
M
5–trimmingmoment;
–changeofanangleoftrim;
–angularvelocityofpitching;

–angularaccelerationofpitching.
Consequently,shipheaving(linearmotiontaking
placeindirection“3”)isgovernedbytheformula:
33
()mg DT DT C TPCg T


(3)
where:
m–weightofloadedcargoexcitingheavemotion;
D–displacementofaship;
D
3–heavedampingcoefficient;
C
3–unitcalculationfactor;
TPC–weighttochangedraftby1centimeter;
ΔT – increase in draft (a difference between
momentarydynamicdraftandastaticoneresulting
fromshipdisplacement);
T
–linearvelocityofheavemotion;
T

–linearaccelerationofheavemotion.
Itisassumedforthepurposeoftheresearchthat
ship motion excited by a loaded container can be
described fair enough by the formulas (1) to (3).
Surge, sway andyaw motions are neglected due to
the fixed position of a moored ship.
The resultant
motionofashipmaybeobtainedbysuperposingof
roll,pitchandheavemotionsgovernedbythegiven
470
formulas(1) to (3). Suchmathematicalmodel is the
basis for further calculation carried out in this
research.
3 SHIPPARTICULARSANDCARGOWEIGHT
ANDLOCATIONASSUMPTIONS
The calculations of ship motion and then her deck
elevationwerecarriedoutforaspecifiedvesseland
realistic cargo weight. The weight of a typi
cal
containerrangesfromafewtons uptoaboutthirty
fivetonsregardlessthesizeofavesselcarryingthe
containers.Thereforetheinfluenceofoneloadedbox
on container carrier motions has to be significantly
different for huge Malacca‐max ship and small
coastal feeder. For the sake of esti
mation extreme
ship motions due to container loading one rather
smallsizeshipistakenintoconsideration.Theship
chosenasanexampleisPolishsemi‐containervessel
projectB‐354.Onetypicalcaseofloadingconditionis
considered (cond. No 11 according to the B‐354
stability booklet). It reflects dist
inctive arrangement
ofcontainersonboard.Theparticularsofthevessel
arefollowing:
lengthbetweenperpendicularsL=140m;
breadthB=22m;
hulls’heightH=12m;
displacementD=14124t;
meandraftd=6,55m;
longitudinalcenteroffloatationLCF=‐0,67m;
momenttochangetri
mMj=18044[tm/m];
weighttoimmerseby1cmTPC=24,15[t/cm];
longitudinalcenterofgravityLCG=‐0,42m;
verticalcentreofgravityVCG=8,88m;
freesurfacemomentΔmh=2439tm.
ThegeneralviewoftheB‐354shipisshowninis
showninfigure4.
Figure4. 3D numerical model of the considered vessel
(projectB‐354)
The formulas (1) to (3) were implemented in
Matlab script and a set of calculations was carried
out. According to the initial assumptions two
independentgantriesareinuseduringhypothetical
cargooperations.Eachofthemisabouttoloadone
40‐feet container and stow it on ship deck at a
random locat
ion available on‐board within a
specifiedrangeofco‐ordinates.Theassumedlocation
andweightofacontainertobeloadedarefollowing:
containerweightmk=35[t]each;
longitudinal co‐ordinate of firstly loaded
containerlocationxk1=0,4L(40%ofship’slength
fromamidshiptowardsbow);
transverse co‐ordinate offirst
ly loaded container
location yk1=<‐0,4B, 0,4B> (yk1 ranging ±40% of
ship’sbreadthtoastarboardside);
longitudinal co‐ordinate of firstly loaded
containerlocationxk2=‐0,2L(20%ofship’slength
fromamidshiptowardsstern);
transverse co‐ordinate of cargo location yk2=<‐
0,4B,0,4B>(±40%ofship’sbreadthtoast
arboard
side).
The longitudinal co‐ordinate reflects the relative
position of the gantry therefore it is fixed for the
purposeofthesimulation.Thislocationwouldshift
only when the loaded bay ischanged. At the same
timethetransverse co‐ordinatemay bedifferentfor
eachgantrymove.Thus,thi
svariableissimulatedin
arandomway.
Besidestherandomlocationofloadedcontainers
also the time of a second heave is very important.
The modeled ship motion consists of three
oscillationstherefore the exact momentof container
touchtodeckplayscrucialroleintermsofthepha
se
of rolling, pitching and heaving. The gain of
oscillation or reversely the decrease in their
amplitudecanbenoticeddependingonsuchphase.
The random nature of this process is modeled by
switchingonthesecondloadduetocargooperation
in a randomly selected ti
me step of the conducted
ship motion simulation. The uniform probability
distribution of all the random variables were
assumed.
4 RESULTSOFSIMULATIONS
The result of a computation is a history of ship
motionineachconsidereddegree‐of‐freedom.Asthe
formulas (1), (2) and (3) describe three uncoupled
motions, the solution is also given in the form of
three ti
me‐domain curves tracings. The random
characterofloadedcargolocationandthetimeofthe
operation results in appearance of two typical
patterns of ship motion. One is the increase in the
amplitudeofconsideredoscillationsandthesecond
oneisthedecrease.Bothpossiblecasesareshownin
figures 5 for ship rolling and in figure 6 for her
pitching.
471
0 20 40 60 80 100 120 140 160 180 200
-7
-6
-5
-4
-3
-2
-1
0
time [s]
angle of heel [deg]
0 20 40 60 80 100 120 140 160 180 200
-0.5
0
0.5
1
1.5
2
time [s]
angle of heel [deg]
Figure5. History of roll motion due to cargo loading – two typical cases e.g. increase (left) and decrease in motion
amplitude(right)
0 20 40 60 80 100 120 140 160 180 200
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
time [s]
angle of trim [deg]
0 20 40 60 80 100 120 140 160 180 200
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
time [s]
angle of trim [deg]
Figure6. History of pitch motion due to cargo loading – two typical cases e.g. increase (left) and decrease in motion
amplitude(right)
0 20 40 60 80 100 120 140 160 180 200
-1
-0.8
-0.6
-0.4
-0.2
0
time [s]
variations of cargo spot elevation [m]
due to heave
due to pitch
due to roll
total
0 20 40 60 80 100 120 140 160 180 200
-0.2
-0.15
-0.1
-0.05
0
0.05
time [s]
variations of cargo spot elevation [m]
due to heave
due to pitch
due to roll
total
Figure7.Historyofvariationsofship’sdeckelevationatacargospot–twotypicalcasese.g. increase(left)anddecreasein
motionamplitude(right)
An analysis of variations of deck elevations
requires some further processing of the motion
histories. The elevation of any specified cargo
locationonship’sdeckcanbederivedonthebasisof
basictrigonometricfunctions.Thefinalchangeofthe
elevation may be found by superposing of
elementarycomponentsduetorolling,pitchingand
heaving. The results of sample calculat
ions carried
outfortwodifferentcargoloadingspotsareshown
infigure7.
Thesamplegraphspresentedinfigure7revealsa
strongdependenceofthedeckelevationonlocation
of two loaded containers and the time interval
between them. An important charact
eristic of the
process is an extreme va lue of alteration of deck
elevation. I a case of loading of one container the
maximum value ofvariation occurs duringthe first
cycleofmotionanditisdistinctiveforthelocationof
loadedcontainer.Thereforefurthercalculationswere
carriedoutforthelocat
ionofthecontainercovering
the whole available area of ship’s deck. This area
extendreflects the range of container coordinates x
k
from‐0,4Lto0,5Landy
kfrom‐0,4Bto0,4B.Inevery
singlecaseofcomputationtheextremevalueofdeck
elevationalternationwas recorded.The distribution
ofsuchextremevaluesisshowninfigure8.
472
-0.4
-0.2
0
0.2
0.4
-0.5
0
0.5
-1
-0.8
-0.6
-0.4
-0.2
0
B
L
z
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
Figure8.Extremevaluesofship’sdeckelevationvariations
(Δzatverticalaxis)atacargoloadingspotcoveringwhole
available space of the deck and one container only –
transverse coordinate of container’s location yk ranging
from‐0,4Bto0,4Bandlongitudinalcoordinatexkranging
from‐0,4Lto0,5L
Since the maximum va lue of va riation of deck
elevation is a deterministic characteristic in case of
onecontainerloadingitcouldbeshowninfigure8as
a steady graph. However, in case of stochastic
processofloadingoftwocontainers withtheuseof
two independent gantries the distribution of
the
maximumvariationshasastatisticalnature.Thus,to
obtainthesetofresults200runsofsimulationswere
carriedout.Everysinglecasedependsontherandom
location of first container, random location of the
second one and random time delay between both
gantry moves. The uniform probability distribution
was applied for each randomvariable. Actually the
simulationissimilartotheMonteCarloapproach.
Everyrun of the performed simulationproduces
thefullhistoryofrolling,pitchingandheavingofthe
shipand then the history of deckelevation.For the
purposeoftheresearchonlythemaximumvalue
of
deckrisewasrecorded to obtain thedistributionof
this resultant characteristic for a number of
simulations. The histogram of va riations of ship’s
deckelevationandthedensityofobtaineddatawith
thedistributionfittingfor200runsofsimulationsis
presentedinfigure9.
The graph in figure
9 shows thatthe considered
variation of deck elevation due to two container
loading is significant and reaches more than one
meter which should not be neglected in terms of
gantrycranecontrol.Fromthedeterministicpointof
viewthemaximumvalueofthevariationcanexceed
evenvalue1,5m
althoughitrequiresacoincidenceof
extremelocationsofbothloadedcontainersinterms
of their transverse co‐ordinate and moreover the
preciselytunedtimedelayofthesecondgantry.Such
a situation never occurred in the course of the
research so the bigger number of simulations is
requiredto
revealsucheffect.
5 CONCLUSION
Theresearchpresented inthepaperisfocusedonthe
estimation of ship’s deck elevation variation due to
containers loading. The work of two independent
gantrieswasassumedandnumerousvariableswere
randomwhichenablestomodeltheprocessinaway
similartoMonteCarlo
approach.
The presented results of computations were
obtained with the use of prepared Matlab script
runningona standardPC‐classdesktop.Thetimeof
computation of every single case was acceptably
short which is an important remark in terms of
potentials to commercial realization. The real‐time
calculations carried
out by the gantry control
computerarefeasiblewhichmaybedecisiveinterms
of any potential practical application. Such an
application leading to an increase in gantry control
accuracycouldhelp to searchforasortoftrade‐off
between safety of operation and effectiveness of
cargohandling,especiallyfrom
theeconomicalpoint
ofview.Thefastermovement of a gantry crane the
higherloadingratecanbeachievedandtheshipstay
inportisshorterandinconsequencecheaper.Onthe
otherhandthefastgantryoperationismoreriskyin
termsofcargodamageduetoexcessively
impetuous
contactofthecontainerwithship’sdeck.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
10
20
30
40
50
60
maximum variation of cargo spot elevation [m]
frequency

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
0.5
1
1.5
2
2.5
3
3.5
Data
Density
max increase in elevation
generalized extreme value distribution
Figure9. Histogram of variations of variations of ship’s deck elevation (left) and the density of obtained data with the
distributionfittingfor200runsofsimulations(right)
473
Thepresentedcalculationsbasedonmodelingof
ship motion in a port reveals that the considered
mattermaybeimportantduringloadingofrelatively
smallship.Theextremevaluesofalternationofdeck
elevation breaching one meter shall be taken into
accountbythegantrycontrolsystem.Otherwisethe
loaded
containerwouldcomeintocontactwithship’s
deck at quitehigh velocity causing massive gravity
loadwhichcanbedestructivetothecargoinsidethe
box.
Ontheotherhandanygoverning‐bodyinaport
needs to analyze the cost effectiveness of the
potentialinvestmentintermsof new
gantrycontrol
moduleapplication.Thedistributionshowninfigure
9 reveals that only about 1% of analyzed random
casesleadstoriseindeckelevationhigherthan1m
andabout 18%breaches0,6m. Thus,the
improvement in gantry control system requires
furtherconsiderationanditispossiblethatonsome
cases sole management of time delay between two
gantry moves would be sufficient to ensure limited
valuesofvariationsofshipdeckelevations.
ACKNOWLEDGMENT
The research project was funded by the Polish
NationalScienceCentre.
REFERENCES
[1]Błocki W., Bezpieczeństwo statecznościowe statku w
sytuacjach rezonansowych, Politechnika Gdańska,
Monografie19,Gdańsk2000.
[2]InternationalCodeonIntactStability2008,edition2009,
IMO,London2009.
[3]Investigation of the capsizing of merchant vessel
DENEBatthePortofAlgecirason11June2011,Marine
CasualtiesandIncidents,EMSA2011.
[4]Kobyliński L., Goal‐based Stability Standards, 10th
International Ship Stability Workshop, Hamburg,
Germany2007.
[5]Krata P, Szpytko J., Weintrit A., Modelling of ship’s
heeling and rolling for the purpose of gantry control
improvement in the course of cargo handling
operations in sea
ports, Solid State Phenomena, Vol.
Mechatronic Systems and Metarials IV, Trans Tech
Publications,pp.539‐546,2013.
[6]Revisedguidancetothemasterforavoidingdangerous
situations in adverse weather and sea conditions,
MSC.1/Circ.1228,IMO,2007.
[7]Surendran S., Venkata Ramana Reddy J., Numerical
simulation of ship stability for dynamic environment,
Ocean Engineering 30 (2003) pp. 1305–1317,
www.elsevier.com/locate/oceaneng2003.
[8]Teo C.S., Kan K.K., Lim S.Y., Huang S., Tay E.B.,
Dynamic modeling and adaptive control of a H‐type
gantry stage, Mechatronics 17 (2007), pp. 361–367,
www.sciencedirect.com2007.
[9]http://www.cargolaw.com/2000nightmare_singlesonly5.
html
[10]http://www.moduly.portalmorski.pl/statystyki