International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 1
Number 4
December 2007
413
The Model of Ship Movement While Touching
the Sea-bed
W. Galor
Maritime University of Szczecin, Poland
ABSTRACT: When a ship hits the sea-bed then it’s hull pressing on ground. It caused the reversible passive
ground reaction. The consequences of such event can result the ships hull damage. The paper presents of
detailed model of ships movement while touching the ground. The pressure on ship hull and parameters of
trajectory (ploughing and penetration) are determined.
1 INTRODUCTION
An analysis of navigational accidents shows that
many of them take place in port waters area. There is
an area surrounded by wharves and other marine
buildings where ships moor and load or discharge
cargo. This type of area intended for ship
manoeuvres is particularly important for port
operation. There has been a tendency in recent years
to accommodate increasingly larger ship in ports,
which with insufficient port infrastructure or its even
minor changes may result in a navigational accident
of serious economic consequences. Since 1970s was
observed a rapid increase in seaborne cargo transport
accompanied by fast growth of the global fleet. The
growth mainly consisted in the increase of ship size
(Fig.1). However, the increase of ship size stopped
after it had reached a certain level. Among main
factors behind that upper capacity limit was the fact
that ports built decades ago couldn’t the handling
ships of the size larger than they were designed for.
The building of new ports is restricted on the one
hand by natural conditions of sea areas, and
necessary large financial effort on the other hand. As
economic and geopolitical conditions change,
directions of cargo transport (bulk in particular) also
change, sometimes in a cycle lasting a few years.
This in turn, makes building new ports a risky
enterprise for investors, as the invested capital return
amounts to at least twenty years. Therefore, a need
arises to use the existing ports for handling ships
larger than those the ports are designed for. Safe
manoeuvring of a ship within a given area requires
that the manoeuvring area of a ship with a specific
draft is comprised within available port water having
a required depth.
NUMBER OF SHIPS
GROSS REGISTERED
TONAGE
1960 1965 1970 1975 1980 1985 1990 1995
450
400
350
300
250
200
150
100
50
Fig. 1. Number of ships entering the Netherlands ports and their
total GRT
414
There are two undesired types of events that can
lead to a navigational accident within a port area:
− impact on the shore (or another port structure),
− contact with the area bottom.
In the former case the area depth is sufficient,
whereas the horizontal dimension is too small. In the
latter case the ship’s draft is too great in comparison
with the port water area depth. This relation is
defined by the distance of the lowest point of the
ship keel to the sea-bottom, usually referred to as the
under-keel clearance (UKC) or water depth under
ship’s keel. The under-keel clearance is used for the
description of the criterion of safe manoeuvring in a
port area. This criterion can be expressed in this
way:
H – D
max
≥ UKC
min
(1)
where H = water area depth; D
max
= ship’s maximal
draft and UKC
min
=
safe under-keel clearance.
UKC
min
is the value of minimum under-keel
clearance of a ship manoeuvring within a given area
that is to assure the ship safety that is no contact of
ship’s hull with the bottom should occur. The main
limitation of handling the ships is the depth of port
waters. The size of UKC in ports is defined by
maritime administration, port authorities or ship
masters.
The interests in this field are contradictory.
Maritime administration responsible for the safety of
navigation wants the UKC to be relatively high.
This, in turn, reduces the possible use of ships’
capacity to the full, which for both ship owners and
charterers is far from advantageous. In extreme
cases a ship’s owner or charterer may give up using
port’s services. The determination of permanent
value of UKC was connected with decade-long
observations and restrictions in sufficiently accurate
determination of its components. However, advances
in the field, i.e. scientific methods enable the
optimization of the UKC value. The objective function
can be written as:
UKC= R
min
→ min (2)
with the restrictions
R ≤ R
ad
(3)
where: R = risk of manoeuvring in an area and
R
ad
= admissible navigational risk defined at an
acceptable loss level.
The risk concept used to be defined in different of
way. Mainly the risk referred to as navigational risk
may be expressed as:
R = P
A
· P
C
(4)
where R
=
navigational risk; P
A
=
probability of sea
accident and P
C
=
probability of unacceptable losses,
and
R
ad
= P
u
≤ UKC
min
for C ≤ c
ad
(5)
where C = losses and c
ad
= acceptable level of
losses.
The losses arising from the fact that a ship hits the
ground while moving, such as hull damage or,
possibly, loss of cargo (particularly liquid cargo,
which may pollute the marine environment) depend
on a number of factors which can be expressed by a
variety of measures. The one of these is maximum
ship hull load less than admissible value caused
damage of its. The maximum ship hull load when
hitting the ground can be defined as dependent on
the probability as:
P
c
= f [P(Q
sgr
> Z
G
)] (6)
where Q
sgr
=
admissible pressure on ship’s hull and
Z
G
= passive ground pressure.
While determining the probability of ship hull
damage during the impact one should take into
account that not every such impact ends in a serious
accident [Galor W., 2005, The managing…].
Therefore:
kuuw
PPP ⋅=
(7)
where: P
uw
= probability of an accident during ship’s
manoeuvres, P
u
= probability of a ship’s touching the
bottom and P
k
= probability of hull damage.
The probability of ship’s impact against the
bottom may be assumed as a criterion for the
evaluation of the safety of ship manoeuvres within
port waters.
From statistical data displaying the number of
damaged hulls against the number of impacts against
the bottom (damage indicator), the probability of
hull damage can be replaced by the hull damage
indicator. Then the probability of an accident will be
equal to:
wuuw
wPP ⋅=
(8)
where: w
w
= hull damage indicator.
2 UKC METHODS DETERMINATION
The value of UKC in ports may be defined by:
− maritime administration (maritime offices,
harbour master’s offices),
− port authorities,
− ship masters.
415
Conclusions from analyses of selected methods
are that UKC is mostly determined by the coefficient
method of summed components.
The coefficient method consists in determining
the value R
min
as part of ship’s draft:
UKC
min
= η D
c
(9)
where: D
c
= maximal draft of the hull and
η = coefficient.
The values of coefficient η used in practice range
from 0.03 do 0.4 [.Mazurkiewicz B., 2006, Mor-
skie...].
The losses due to restriction of ships draft are as
follows:
− limited quantities of cargo loaded and unloaded,
which means lower earnings for the harbour and
stevedoring companies;
− lower ship-owners’ profits as the ship’s capacity
is not used to the full or longer turnaround time
due to necessary lighter age at the roads, before
the ship’s entrance. It should be noted that the
ship’s operating costs are the same no matter
whether the ship is fully laden or its capacity is
unused,
− port charges are smaller as they depend on the
ship’s tonnage (berthing, towage etc.);
− in many cases large ships resign from using
services of a port where they are not able to use
their total cargo capacity.
In the other method the value R
min
is determined
as an algebraic sum of component reserves [Galor W.,
2005, Analiza…] where in addition errors of the
particular components are taken into account:
UKC
min
=
∑
=
n
i
i
R
1
(10)
where: R
i
= component reserves of UKC.
3 STATIC AND DYNAMIC COMPONENTS OF
THE UNDER KEEL CLEARANCE
The under keel clearance is divided into a static and
dynamic component. This division reflects the
dynamics of particular reserves. The static
component includes corrections that change little in
time. This refers to a ship lying on calm waters. The
dynamic component consists of the reserve for the
squatting of a moving ship and wave action. It
should be noted that in this division the dynamic
component should also include the reserve for listing
caused when a ship turns. Therefore, the UKC can
be defined as:
R
min
= R
S
+ R
D
+ δ
r
(11)
where: R
s
= static component, R
d
= dynamic com-
ponent and δ
r
= errors of component determination.
4 THE SHIP STRIKE ON THE SEA-BED
During a ship’s striking the bottom of an area built
of sandy or argillaceous ground, for a vessel in
progressive movement, there occurs a gradual
sinking of the hull into the ground (until the ship
stops). The mechanism of the ship’s striking the
area’s bottom depends on the ship’s draft, namely
whether the vessel is trimmed by the bows, the stern
or if it is loaded on an even keel. During a ship’s
striking the bottom of an area of fragmented ground,
for a vessel in progressive movement, there occurs
gradual sinking of the hull into the ground (until the
vessel’s stoppage). During this process there can be
distinguished the plough-in phase bound with
longitudinal motion and the penetration (sinking) in
a vertical direction. Fig. 2 presents this movement in
the case of a vessel being trimmed by the bows. A
similar phenomenon will occur in the case of being
trimmed by the stern.
Fig. 2. Penetration of the ship’s hull into the bottom
The penetration of the ship into the ground
depends on the relation between the horizontal V
H
and vertical V
v
components of the ship’s speed V
S
.
The ship will stop in a certain distance I
P
from the
point of the hull’s first contact with the bottom and
the penetration to a particular depth Z
K
. In the initial
stage of the ship’s penetration into the ground, is
mainly affected by horizontal forces. Stopping of the
ship takes place on a horizontal plane until the ship
stops, which is described as stopping distance I
P
from the first contact point to the stopping of the
vessel. During ploughing there are also vertical
forces causing penetration of the ground with initial
angle β. The exceeding the permissible value of hull
strength may cause damage to the hull. These stages
are affected by the kind of ground of the area
bottom.
416
When a ship hits the bottom, its hull presses on
the ground which results in the passive ground
pressure. That pressure is the ground reaction to the
hull pressure on the bottom. The passive ground
pressure increases with the pressure of the hull.
When the maximum admissible value is exceeded,
the area of ground is formed and the blocks of
ground begin to move aside from under the hull. An
increase in the passive earth pressure (for non-
cohesive grounds) along with the increase of hull
pressure takes place due to structural changes in the
ground [Galor W., 2003, The application] occur in
both granular system and in particles of the ground.
Initially, the elastic soil becomes elastic-plastic,
then plastic. This is a state in which all the grains
and particles are in the state of boundary
equilibrium, which corresponds to the boundary
value of passive pressure of the ground. The ship’
pressure on the ground causes the hull to penetrate
into the bottom ground. When the boundary passive
pressure (reaction) is reached the expulsion of
ground block and the ship’s bottom penetrates the
ground. That phenomenon takes place in both non-
cohesive grounds, such as gravels and sands and
their mixes, and in cohesive grounds, including clay
gravels and sand-gravel mixes, clay sands, clay and
silt. An analysis of the ship hull action on the ground
when the bottom is hit shows that there are
similarities to the action of fenders. This means that
the ground is a medium absorbing the energy of the
impact. The magnitude of energy absorption mainly
depends on the ground properties. Ships penetrating
a non-cohesive ground to a certain depth will not
have their hull damaged.
In Polish ports there occur crumbled grounds,
containing sandy particles produced by mechanical
crumbling of primary rocks.
5 THE PARAMETERS OF SHIP MOVEMENT
On the basis of considerations presented there has
been prepared an algorithm of calculating vessel
movement parameters when striking the port water
area ground and of forces impacting on the vessel’s
hull. It has been applied in a computer simulation
model of the vessel’s movement in the area.
The model works in real time and serves the purpose
of preparing navigational analyses. This permits risk
determination of the vessel striking the area bottom
and its results (likelihood of hull damage).
The stopping of ship will be fulfill when the initial
kinetic energy (in moment of first contact with sea-
bed) became completely lost, i.e. will be change to
following components:
mV
2
Ho
/2 - ∫
P
RT
dl- ∫ P
B
dl - ∫ P
RK
dl
= 0 (12)
where: m = ships mass and water added mass,
V
Ho
= horizontal component of ships velocity in
moment of contact with sea-bed, ∫P
RT
dl = work
performed for overcoming friction force of the hull’s
bottom part, ∫P
B
dl = work performed for overcoming
the resistance of friction of the lateral parts of the
hull and ∫P
RK
dl = work performed for overcoming
soil wedge.
The ships velocity during contact with ground of
sea-bed will be by and by decrease until stopping.
The way of ship’s stopping will be equal:
L
K
= ∫ V
Hi
dt dla t ∈ (t
0
÷ t
K
)
where: L
K
= way of ship’s stopping, t
K
= time to
ship’s stopping and V
Hi
= horizontal component of
ship’s velocity during phase of ploughing.
V
Hi
= (2 ⋅ ∆E
Ki
/ m )
½
(13)
where: ∆E
Ki
= decreasing of ship’s kinetic energy
due to alter on work performed for hull resistances
during ploughing.
The friction force of the hull’s bottom part is
equal of hull friction force P
RT
during penetration
into the ground:
P
RT
= µ ⋅ N (14)
where: μ = coefficient of ship’s hull friction on
ground and N = ground reaction force on ship’s
bottom during penetration.
The friction force of the latteral parts of the hull
P
KB
:
P
KB
= 2 ·F
odpb
(·L
S
/ Z
śr
)
·
tgE·ΔL
i
(15)
where: F
odpb
= ground reaction force on lateral part
of hull, L
S
= line length of hull contact with ground,
Z
śr
= average depth of ship’s penetration into
the ground, E = the friction angle on hull wall and
ΔL
i
= considered the ship’s stopping ways segment.
The passive ground reaction connection with
overcoming soil wedge E
RK
(figure 3):
E
RK
= f
(
Z, B
s
, L
pp
, β
) (16)
where: Z = depth of ship’s penetration into the
ground, B
s
= width of part of ship into the ground,
L
pp
= ship’s length between perpendicular and
β = angle of ship’s trim.
417
Fig. 3. The passive earth pressure wedge
The pressure of the ship on ground:
S
N
=σ
(17)
where: N = the push force of ship’s hull and s = area
of hull contact with ground.
The ship’s push on the ground is an effect of
decreasing of ship’s draft. The greater emergence
bear witness about greater pushing. The magnitude
of push force will be alter depending on ship’s draft
and trim. The pushing for even keel will be equal:
γ
δ
σ
⋅
⋅⋅⋅∆
=
S
BLpT
pi
i
(18)
where:
i
T∆
= currently draft decreasing, L
pp
= length
between perpendiculars, B = breadth of ship,
δ = ship’s block coefficient, γ = water weight
specific gravity and S = surface area of hull contact
with ground.
6 ALGORITHM OF DETERMINATION THE
EFFECT OF SHIP STRIKE INTO SEA-BED
In successive steps ship movement parameters
during contact with the ground are calculated, which
permits the determination of its results. The
following steps are accomplished:
− Calculating initial kinetic energy.
− Calculating pressure of the vessel on the area
bottom, to decrease the water level or the vessel’s
draft.
− Checking whether passive earth pressure (the
ground’s reaction) does not exceed the
permissible value.
− Calculating the friction force of the bottom part of
the vessel’s hull against the ground, taking into
account the friction coefficient.
− Calculating the depth of the vessel’s penetration
into the ground.
− Calculating work performed for overcoming
friction force of the hull’s bottom part.
− Calculating work performed for overcoming the
resistance of friction of the lateral parts of the hull
for a specified depth of the vessel’s penetration
into the ground.
− Calculating work performed for overcoming soil
wedge.
− Calculating the decrease of the vessel’s kinetic
energy caused by contact with the ground.
− Calculating the decrease of the vessel’s speed
components.
The example of calculation used algorithm is
presented below. Basic dates:
− length between perpendiculars Lpp =250.0 m
− breadth of ship B = 40.0 m
− ship’s draft T = 12.0 m
− ship’s block coefficient δ = 0.8
− initial horizontal component of ship’s speed
V
H
= 5.0 m/s
− initial vertical component of ship’s speed
V
V
= 0.01 m/s
A.
The results of calculations are following:
− initial ship’s kinetic energy
E
K
= 1553250 [kNm, kJ]
B.
In first step of calculation for period of time equal 10
sec:
− work performed for overcoming friction forces of
the hull’s P = 37000 [kNm, kJ]
− decreasing of ship’s speed in first step of calcu-
lation up to V
H
= 4.84 m/s
C.
In next steps of calculations the decreasing of ship’s
speed up to zero will be stayed:
− over 220 sec from first contact of ship with sea-
bed
− the length of ship’s stopping distance I
p
= 261,0 m
D.
There wasn’t overdoing the admissible pressure on
ship’s hull in this case.
7 CONCLUSIONS
The under keel clearance should ensure ship’s safe
manoeuvring in a port area on the one hand, and the
maximum ship’s draft on the other hand, particularly
in port areas. This result can be achieved through the
minimization of UKC value while risk is kept at an
acceptable minimum. A ship can touch the bottom of
a navigable area due to the reduction of its keel
clearance. An algorithm permits to calculate the ship
418
movement parameters when striking the port water
area ground and of forces impacting on the ship’s
hull. It has been applied in a computer simulation
model of the vessel’s movement in the area. This
permit enables to risk determination of the ship
striking the area bottom and its results (likelihood of
hull damage).
REFERENCES
Galor,W. (2003). The application of navigational analysis to
optimize port water areas modernization. Proc. of Int.
Conference “Safety and Reliability” KONBIN 2003,
Gdynia.
Galor W.(2005): Analiza określania zapasu wody pod stępką.
Materiały XI Międzynarodowej Konferencji Naukowo-
Technicznej „Inżynieria Ruchu Morskiego”. Szczecin 2005.
Galor,W. (2006). The managing of the navigational safety of
ships in port water areas. Editors C.A. Brebbia a al. WIT
PRESS. Southampton, Boston 2005.
Mazurkiewicz, B. (2006). Morskie budowle hydrotechniczne.
Wyd. ARCELOR, Gdańsk 2006.
