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.