Prediction of the dynamic responses for two containerships operating in the Black Sea

Abstract

This work presents a methodology for the evaluation of the vessels operability in the basin of the Black
Sea. The approach proposed is based on the results provided by a wave prediction system that was
implemented and validated in the Black Sea. This offers a meaningful framework in the assessment of
the seakeeping performance for ships operating in different conditions. The aim of the work is to
improve the seakeeping performances for various types of ships that sail in the Romanian coastal
zones. Two containerships are considered, of 1300TEU and 800TEU respectively, which have
however the same displacement. For predicting the ship motions induced by the various wave
conditions encountered in the operational area, it is necessary to know the transfer functions for
different ship speeds and heading angles. These are computed with a numerical code based on the
strip theory. The predicted motions are then compared with the limit values of the relevant seakeeping
criteria, considered as the criteria for checking the operability of the ship. By defining the acceptable
operating boundaries, the ship behavior in waves can be quantified. The vessel meets the criteria for
those combinations wave height-wave period that are below the limit curves. The results are presented
as operational maps for the Black Sea basin, allowing the identification of the areas that should be
avoided due to unfavorable weather conditions. The methodology proposed herewith can be extended
and applied to various marine environments and types of ships.

Full text

PREDICTION OF THE DYNAMIC RESPONSES FOR TWO
CONTAINERSHIPS OPERATING IN THE BLACK SEA
Carmen Gasparotti and Liliana Rusu
“Dunarea de Jos” University of Galati, Romania
Abstract
This work presents a methodology for the evaluation of the vessels operability in the basin of
the Black Sea. The approach proposed is based on the results provided by a wave prediction
system that was implemented and validated in the Black Sea. This offers a meaningful
framework in the assessment of the seakeeping performance for ships operating in different
conditions. The aim of the work is to improve the seakeeping performances for various types
of ships that sail in the Romanian coastal zones. Two containerships are considered, of
1300TEU and 800TEU respectively, which have however the same displacement. For
predicting the ship motions induced by the various wave conditions encountered in the
operational area, it is necessary to know the transfer functions for different ship speeds and
heading angles. These are computed with a numerical code based on the strip theory. The
predicted motions are then compared with the limit values of the relevant seakeeping criteria,
considered as the criteria for checking the operability of the ship. By defining the acceptable
operating boundaries, the ship behavior in waves can be quantified. The vessel meets the
criteria for those combinations wave height-wave period that are below the limit curves. The
results are presented as operational maps for the Black Sea basin, allowing the identification
of the areas that should be avoided due to unfavorable weather conditions. The methodology
proposed herewith can be extended and applied to various marine environments and types of
ships.
Keywords: numerical wave models, seakeeping performance, containerships operability,
seakeeping criteria, ship responses.

1. INTRODUCTION
The maritime transportation is a complex activity both due to the volume and the nature
of goods involved in traffic, but also due to the specific operating conditions that require
special measures for the safety of the ships. In the maritime transportation an interaction
between technology, environment and organizational factors is encountered, each of these
factors playing a significant role in the safety of the ship at sea (Hanzu-Pazara, 2007).
It is widely recognized that accidents at sea are sometimes hard to be avoided. After
analysing the causes of a wide range of maritime accidents, as well as of their effects,
international organizations like UN, CE and specialized agencies such as IMO (International
Maritime Organization) and ILO (International Labour Organization) have developed a
number of conventions, regulations, rules, norms and international standards containing
essential criteria concerning the safety (as for example IMO, 1994). Despite the measures
taken at international level to improve the safety at sea, the marine accidents in recent years
prove that they still remain a constant threat (Gasparotti, 2010). There is however a set of
causes that may lead usually to an accident. Thus, the main factors that generate accidents are:
high density traffic, unfavourable hydro meteorological conditions (storms, high waves, rain
and frost), navigating obstacles, sea conditions (sea depth and seabed topography), reduced
visibility, human errors and latent defects of the ships. These can individually, or in
combination, have as a result the development of marine accidents (Milan and Gomoiu, 2008;
Wayment and Wagstaff, 1999; Zang et al., 2006).
In the Black Sea, most of the accidents occurred during the cold season, when usually
unfavorable hydrometeorological conditions are encountered. In winter, in the western and
north-western parts of the Black Sea, circumstances of producing storms are continuously
created, because of the aero-synoptic specific conditions.
A Bayesian Belief Network (BBN)
have been used in the maritime industry with the development of some specific models to assess

and identify the risk factors associated to a given context (Eleye-Datubo et al., 2006; Trucco et
al., 2006; Norrington et al., 2008). Such a BBN model for maritime accidents was also developed
by Antão et al. (2008) with the inclusion of the wave data (significant wave heights), in order to
assess the correlation between their amplitude and certain accident typologies and related
consequences (human injuries or fatalities).
The study has shown that the type and the
probability of producing an accident depend directly on the type of the ship. Thus, at small
significant wave heights (0-2m), the occurrence probability of an accident in the case of the
commercial vessels is also small, but this probability increases in the case of the fishing
vessels operating near the coastal areas, especially in wave breaking areas. In the case of large
significant wave heights (4-6m), the probability of accidents highly increases, in a range
between 3-13 times.
Some recent studies (Guedes Soares et al., 2001; Bitner-Gregersen and Guedes Soares,
2007) have shown that most of the marine accidents induced due to severe weather conditions
occurred in areas with the highest significant wave height or with wave steepness greater than
average. The general statistics show that about 23% of the ships accidents are caused by the
sea state and weather conditions, the latter being in fact the second most common cause for
such events (Toffoli et al., 2003). Toffoli et al. (2003) also showed that the responses of the
vessels at a certain sea state depend on the type and the size of vessel. It was noticed that the
ships are vulnerable mainly to the wave lengths greater than half of the ship length and only
few accidents occurred at wave lengths less than half of the ship length.
On the other hand, different types of ships seem to have different causes of accidents. As
regards the containerships, they can capsize in the case of high roll motion, usually for beam
sea conditions or in very extreme oblique waves, case characterized by high amplitudes on
heave, pitch and roll motions. A recent example in this direction is represented by the accident
produced in 14 February 2014. After the containership Svendborg Maersk left the Bay of
Biscay it was struck by heavy seas off the coast of France. The owner company told that the

extreme weather had an unexpectedly forceful impact on the ship's movements. After the ship
arrived in the Spanish port of Malaga for repairs, it was discovered that about 520 containers
were unaccounted for. Another example is the accident of the container vessel MOL Comfort.
This experienced a crack amidships and ingresses water in the hold while sailing from
Singapore to Jeddah on the Indian Ocean on 17
th
of June, 2013 due to inclement weather
conditions.
Nowadays, the numerical wave models have shown very good capability to predict the
wave conditions generated by the wind fields over the sea (Rusu et al., 2005; Rusu 2010a;
Rusu, 2011; Rusu and Guedes Soares, 2013). In the present work the seakeeping qualities of
two ships will be quantified by evaluating the corresponding the operability indexes, using the
predictions provided by a numerical wave model. A wave modelling system, SWAN based
(Simulating Waves Nearshore, Booij et al., 1999) was implemented and validated at the level
of the entire basin of the Black Sea (Rusu, 2010b; Rusu and Ivan, 2010) in order to be able to
provide reliable nowcast and forecast products for the wave conditions.
The seakeeping performances of two containerships operating in the Black Sea basin are
investigated in the present work. These evaluations of the dynamic responses under predicted
sea states are based on the results of the numerical wave modelling system and on the ship
transfer functions.
2. PROCEDURE FOR ASSESSMENT OF THE SEAKEEPING PERFORMANCE
The procedure considered for assessment of the seakeeping performance of a ship is
based on the analysis of the ship dynamic response in irregular waves, corresponding to the
sea states. The procedure starts with the prediction of the ship’s hydrodynamic response for a
range of speed and heading angle values. The amplitude of the ship motions in irregular
waves are short term predicted according to the sea state and the specific wave spectra.

Finally, based on the ship’s operability limits established by the limiting response criteria, the
seakeeping operability indexes are obtained (Sariöz and Narli, 2005).
The evaluation of the seakeeping performance of a ship largely depends on the
environmental conditions as well as on the defined criteria, and this is the main reason for that
any comparison related to problem as: the ship speeds, the influence of the heading angles, the
loading conditions, etc., represents a complex problem (ITTC, 1999). In irregular waves, short
and long term distributions can be used to estimate the most probable maximum values of the
responses. The first step includes the computation of the ship response transfer functions
corresponding to the ship main degrees of freedom. The transfer functions include the
absolute ship motions (heave, roll and pitch), and the derived responses at selected positions
on the ship, such as relative motions and accelerations. For this, a program code developed by
Domnisoru (2001) and Domnisoru el al. (2009) based on the strip theory (Salvesen et al.,
1970; Bertram et al., 2006) was used Then, it is necessary to compute the ship responses to
the complete range of the short-term sea states. Finally, combining the transfer functions with
the specified wave spectra, the response spectra are obtained.
2.1. Ship responses to irregular waves
The establishment of the seakeeping performance of a ship and the determination of the
explicit design parameters have to be done in a realistic seaway. The response spectrum
Φ
yy
(ω
e
) is obtained from the input wave spectrum Ф
ζvζv
(ω
e
) by means of the response transfer
function
(
)
y
H
ω
:
).()()(
2
evveyeyy
H
ωφωωφ
ζζ
⋅= (1)
It is known that, the wave spectra which a ship encounters vary continuously in space and
time. Since the sea states were assumed as stationary, zero mean Gaussian processes, and
because the response is linear, it can be considered that the same model describes de response

process. This implies that the statistical properties of the responses may be derived from the
moments of the response spectra.
The variance of the process can be obtained by integrating the spectrum:
.)()()(
0
2
0
2
eevveyeeyyR
dHd
ωωφωωωφσ
ζζ
∫∫
∞∞
==
(2)
The significant value of the response
S
R (double amplitude), can be computed from the
standard deviation as
RS
R
σ
4= . (3)
The seakeeping criteria estimations are based on well-known equations applicable to
random processes whose amplitudes can be approximately assumed as distributed following a
Rayleigh law, according to which the probability P
s
for that the amplitude of the random
process exceeds a certain level F is computed with the formula:
( )
2
2
exp
2
S
R
F
P F
σ
 
= −
 
 
, (4)
where
2
R
σ
is the variance of the process. The most probable maximum value in N
c
successive
cycles is obtained from relationship bellow:
2
max
2 ln
R
c
F N
σ
=
. (5)
In irregular waves, short and long term distributions can be used to estimate the most
probable maximum values of the responses. Nevertheless, only these results alone are not a
good measure of the seakeeping quality of a ship. For this reason, the seakeeping quality will
be quantified by an operability index. The relationship between the ship operability and the
mission characteristics is established through the seakeeping criteria. In this way, the
seakeeping index will indicate if the ship responses are bellow or over to those defined by the
criteria (Fonseca and Guedes Soares, 2002).

2.2. Seakeeping criteria
An important part in the assessment of the seakeeping effect on the vessel’s efficiency is
the evaluation of the critical levels corresponding to relevant seakeeping criteria. When these
are exceeded the crew’s abilities to safely and efficiently perform their duties are degraded
(Platonov and Trub, 2010). Since the seakeeping criteria are defined also in terms of the
accelerations felt on-board and of the relative motions between the hull and the waves, these
quantities are computed for all the relevant points on the ship, combining the absolute
motions and the wave motions.
Usually the seakeeping criteria related to the absolute motions and accelerations are
presented in terms of a limit value of the root mean square (
R
σ
) of the response. The criteria
can be also defined in terms of the probability of exceeding a critical value, as it is used for
slamming, deck wetness, or propeller emergence. The criteria are defined either in terms of
the root mean square of the response (rms) or in terms of permitted probability of occurrence
(prob). If the seakeeping criterion is defined as a limiting root mean square of the response
CR
σ
, then the maximum significant wave height for a given mean wave period T
z
and ship
heading is computed as:
( )
.,
1
max
R
CR
Zs
TH
σ
σ
β
=
(6)
If the criterion is defined as a probability of exceeding a critical value
CR
p , then the
corresponding root mean square of the response is obtained from the following relationship:
( )
,
1ln2
2
max
CR
CR
p
R
=
σ
(7)
where R
max
is the limiting magnitude of the response which has the probability
CR
σ
of being
exceeded. As an example, for the green water on the deck phenomena R
max
is usually the free

board at the bow. Knowing now the probability distribution of the short-term sea states for a
given area, it is possible to select all the sea states where the vessel is operational.
Several seakeeping criteria related to: the absolute motions, the relative motions, the
accelerations, slamming, green water on deck, etc., have been used and are available in the
literature. These criteria have been obtained mostly from the experience onboard ships. For
this reason, there is a large uncertainty in the definition of the seakeeping criteria. This
uncertainty motivated the present analysis related to the influence of varying different
seakeeping criteria on the operability index.
The seakeeping criteria can vary vastly from ship to ship and the values chosen play an
important role to delimitate the vessel situation, which can be: operable or inoperable due to
exceeding of the limiting criterion values. In this study, the values considered for each
criterion are presented in Table 3 (where g is the gravity acceleration). It is assumed that
green water on deck occurs when the relative motion is larger than the freeboard on the bow,
a slam occurs when the relative motion is larger than the draft at the bow and a propeller
emergence occurs when 1/4 of the propeller diameter comes out of the water (Fonseca and
Guedes Soares, 2002). The seakeeping criteria defined as a limiting root mean square of the
response are: roll, pitch, vertical acceleration at the bridge (VAB), lateral acceleration at the
bridge (LAB) and vertical acceleration at the forward perpendicular (VAFP). The criteria
defined as a probability of exceeding a critical value are slamming (slam), green water on
deck (GW) and propeller emergence (PE).
2.3. Operability indexes
An effective methodology that links a spectral phase averaged wave model with a model
for ship dynamics analysis is presented in this work. The main idea is to use in a certain area
the predicted sea states provided by a wave model to compute the seakeeping performances of
a ship under operational conditions. The information regarding the ship operability is

presented in a simple and straightforward way, by designing maps that provide the risk
indexes (RI), which indicate in fact if the operability limits are exceeded at least by one
criterion.
For each speed and heading, the computation of these indexes is based on the relationship
that exists between the predicted significant wave height (H
s
) in a grid point of the wave
computational domain and H
slimit
(the significant wave heights that limit the green area in
Figure 7) that corresponds to the mean period (T
z
) simulated in that point. In order to obtain
immediately the information concerning the relationship that exists between H
s
and H
slimit
in
each grid point, H
s
was divided by the H
slimit
value that corresponds to the period T
z
simulated
in that point. In this way, a risk index denoted as RI was defined as:
),(/),(),( jiHjiHjiRI
slimits
= , (8)
where i and j represent the indices of the grid points in the computational domain. For the
case when RI < 1, the vessel can navigate without any operational risk relatively to all criteria
considered, for a specific heading and speed.
The maps of the operability risk indexes can be also evaluated for a specific criterion, and
in this case the indexes (RI
c
) are computed with the following relationship:
(
)
(
)
(
)
jiHjiHjiRI
scsc
,,,
max
= (9)
where c is related to the limiting criteria considered. In the present study eight criteria were
considered, as presented in Table 3.
3. CASE STUDIES FOR THE SEAKEEPING PERFORMANCE ASSESSMENT
A methodology to estimate the seakeeping performance based on the spectral approach
was presented in the previous section. According to this methodology, the seakeeping
performances were calculated as a function of the hydrodynamic characteristics of the ship
and of the environmental conditions where the ship operates, while the ship responses to

irregular sea states were evaluated using both methods based on standard spectral techniques
and probabilistic.
Taking into account the methodology mentioned, the computations related to two
containerships that usually operate in the Black Sea basin are performed in this section. The
containerships main dimensions and the offset ships lines are presented in Table1 and Figure
1 and Table 2 and Figure 2, respectively.
Figure 1: The offset ship lines of the
containership C1
Figure 2: The offset ship lines of the
containership C2
Table 1: The main dimensions of the
containership C1
Containership C1
Length overall 139.965
Length between perp., Lpp (m) 130.00
Beam, B (m) 21.80
Draft, T (m) 7.335
Depth, D (m) 9.50
Displacement, ∆ (ton) 17974.50
Long. position of CG, (m) -2.526
Vertical position of CG, (m) 6.30
Service speed, V
s
(kts) 18.00
Transversal metacentric height,
GMt (m)
2.618
Table 2: The main dimensions of the
containership C2
Containership C2
Length overall 149.50
Length between perp., Lpp (m) 140.00
Beam, B (m) 23.60
Draft, T (m) 7.29
Depth, D (m) 13.50
Displacement, ∆ (ton) 18000.00
Long. position of CG, (m) 71.73
Vertical position of CG, (m) 8.30
Service speed, V
s
(kts) 20.00
Transversal metacentric height,
GMt (m)
2.02
The first step is to calculate the relevant ship response transfer functions for all directions
between the head waves and the following waves, where 180º
stands for the head waves. The
transfer functions define the amplitude of the response due to a unit wave excitation and they
are computed for all the directions considered between the heading and the following waves.

These types of results are illustrated in Figure 3 for both containerships considered. The
graphs present the transfer functions of roll, pitch and heave, for seven headings.
In Figure 3
e
ω (rad/s) represents the encounter frequency calculated as follows:
( )
β
ω
ωω
cos
2
0
0
⋅⋅−= V
g
e
, (10)
where ω
0
is the wave frequency, V is the ship speed and β is the heading wave.
Table 3: Seakeeping performance criteria for the containerships
Location of the points analysed for
derivate responses (m) x, y, z Response
C1 C2
Criterion
Roll - - 6º (rms)
Pitch - - 3º (rms)
Green water on deck (GW)
67.53, 0, 5.1 75.68, 0, 6,21 5% (prob)
Slamming 54.53, 0, -7.3 61.15, 0, -7.29 3% (prob)
Propell. Emergence (PE) -57.5, 0, -4.7 -63.64, 0, -4.8 12% (prob)
Vert. acc. at bridge (VAB)
-55.2, 0, 14.45 -55.0, 0, 15.0 0.15g (rms)
Lat. acc. at Bridge (LAB)
-55.2, 0, 14.45 -55.0, 0, 15.0 0.15g (rms)
Vert. acc. at fwd. pp.
(VAFP)
67.53, 0, 0 75.68, 0, 0 0.2g (rms)
For positions selected on the ship, the transfer functions, which include the absolute ships
motions and some derived responses, such as relative motions and vertical and lateral
accelerations, were computed. The coordinates of these points are presented in Table 3. Their
positions are related to a reference system fixed with respect to the mean section of the ship.
The axis is oriented in the vertical upward direction passing through the center of gravity of
the ship, while the x-y plane coincides with the undisturbed free surface of the water. The x
axis is along the longitudinal direction of the ship and is pointing to the bow, while the y axis
is pointing to the port direction. Only a separation between headings of 30 degrees was
considered, resulting in seven headings.
The limiting significant wave heights have been determined as a function of a range of
mean wave periods, based on the above mentioned criteria. The seakeeping criteria appear as
limiting curves with the limiting significant wave height as the ordinate and with the wave

period along the abscissa. The vessel meets the seakeeping criteria for the combinations wave
height–period below (all) the boundary curves. Such graphs give information about which
significant wave height is critical for the different criteria, and which criterion is the limiting
one at different wave periods and headings.
There are combinations of the wave heights and the wave periods that cannot exist
because the waves would be too steep to be stable, and they break. The breaking wave height
br
s
H
was found by Myrhaug and Dahl (1996) as a function of the peak period
T
p
:
2
105.0)(
pp
br
s
TTH =
, (11)
where T
p
=1.287T
z
for a JONSWAP wave spectra (Hasselmann et al., 1973).
Figures 4 and 5 present the results obtained for the two vessels considered and which are
operating with the speeds of 0kts (no advance) and the service speeds of 18kts and 20kts,
respectively. The results are presented only for the heading angles between 90º and 180º. The
theoretical limit for the breaking waves is plotted together with the operability boundary
limits (the black dashed line). The graphs show clearly which are the responses that limit most
the ship operability, and for which headings.
a)
b)
Figure 4: Maximum allowed significant wave heights for all criteria, containership C1 at 0kts
(a) and 18kts (b) speed and various headings (120º and 90º - upper panels; 180º and 150º -
bottom panels)

From Figure 4, it can be observed that for both speeds, GW appears to be the criterion
which limits the operability of the containership C1 for heading angles of 180º and 150º,
while for heading angles of 120º and 90º the limiting criterion at 0kts is roll, and at 18kts is
LAB. With the increase of the velocity, the limiting curves of the maximum significant wave
height decrease drastically for VAB and VAFP corresponding to heading angles between 180º
to 120º, while for Slam only a slight decrease appears.
a)
b)
Figure 5: Maximum allowed significant wave heights for all criteria, containership C2 at 0kts
(a) and 20kts (b) speed and various headings (120º and 90º - upper panels; 180º and 150º -
bottom panels)
From Figure 5 it is observed that at 0kts the limiting criterion for the containership C2 is
roll, for heading angles between 90º to 150º, while for a heading angle of 180º, GW limits the
operability of this vessel. The increasing of the ship’s speed does not modify very much the
maximum significant wave height of the limiting curves for a heading angle of 90º, while for
the heading angles between 120º and 180º important changes are observed. For a heading
angle of 120º there are various curves that limit the ship operability, function of the mean
wave period. For heading angles of 150º and 180º it is clear that GW limits the operability of
the containership.

4. ASSESSMENT OF VESSEL’S NAVIGATION OPERABILITY IN THE BLACK
SEA
The target area considered in the present work is the basin of the Black Sea and the
information concerning the sea states are delivered by a wave prediction system, SWAN
based, that was already implemented and validated against in situ measurements and remotely
sensed data (Rusu et al., 2014; Rusu and Butunoiu, 2014). As already mentioned, two
containerships were considered, as case studies for assessing the operability of the vessels in
the Black Sea basin. The information provided concerning the sea state conditions (H
s
, T
z
and
wave direction) allows a better assessment of the seakeeping performances of the vessels
under operational conditions. The significant wave height scalar fields and the wave vectors
simulated in the Black Sea basin using the wave prediction system SWAN based are
illustrated in Figure 6 for the high energetic conditions corresponding to three different time
frames: 2005/02/04-h06, 2005/02/05-h06 and 2005/02/05-h18.
Figure 6: Significant wave height scalar fields and wave vectors. Results of the simulations
with the wave prediction system in the Black Sea basin for the high energetic conditions
corresponding to the time frames: 2005/02/04-h06, 2005/02/05-h06 and 2005/02/05-h18

Starting from the maximum allowed significant wave height curves (some examples are
presented in Figures 4 and 5) computed for various criteria and function of T
z
, for the sea
states characterized by a specific pair of H
s
and T
z
, corresponding to a point that is inside the
green colored area (see Figure 7) the containership can operate satisfying all operability
criteria defined at the ship’s speed and for that heading angle. As it can be seen, the
significant wave heights that limit the green area (denoted as H
slimit
) are the lowest values of
H
smax
computed for all the criteria considered.
Figure 7: Maximum significant wave heights for which the containership C2 can operate
satisfying all the operability criteria, for the heading 120º and the service speed (Vs = 20kts)
In Figure 8 several maps of the risk indexes are presented, as computed for the
containership C1, and corresponding to two different ship speed conditions (0kts and 18kts)
and various heading waves. These risk indexes correspond for two wave conditions
(2005/02/04-h06 and 2005/02/05-h18) presented in Figure 6, where it can be noticed that in
the first case H
s
fields with values greater that 5m can be encountered in the western side of
the sea.
For the conditions corresponding to the time frame 2005/02/04-h06, if the ship C1 is
motionless in the western side of the basin with heading angles of 90º, RI has elevated values
(greater than 2). Once the velocity is enhanced, but keeping the some heading, it can be

noticed that in the western side of the sea a zone occurs in which indexes exceed even values
of 2.5. This feature indicates the necessity of modifying the heading angle. It can be noticed
that for heading angle of 120º the RI values are decreasing (although they are still greater than
1), but by increasing the ship velocity the values of these risk indexes decay.
Figure 8: Maps of the operational risk indexes computed for all criteria and various heading
waves, considering that the containership C1 operates in the Black Sea with the speeds of 0kts
(upper panels for each case) and 18kts (bottom panels for each case), the time frames
considered are: 2005/02/04-h06 and 2005/02/05-h18
Also in the second case considered, corresponding to the time frame 2005/02/05-h18, the
risk indexes, computed in the western side of the Black Sea for heading angles of 90º, have
values greater than 1 for an extended area. The modification of the ship direction, so that the
heading angles to have values of about 150º, makes the values of the risk indexes to become
lower than 1 and thus no limiting criterion is exceeded.

Figure 9 presents the maps of the risk indexes computed for the containership C2 for two
different ship speed conditions (0kts and 20kts) and heading angles of 90º and 150º. In this
case the risk indexes computed correspond for the first two wave conditions presented in
Figure 6, where energetic conditions with H
s
greater that 5m can be encountered first in the
western side of the sea basin, while in the second case, such conditions can be encountered in
the central part of the sea.
Figure 9: Maps of the operational risk indexes computed for all criteria and various heading
waves, considering that the containership C2 operates in the Black Sea with the speeds of 0kts
(upper panels for each case) and 20kts (bottom panels for each case), the time frames
considered are: 2005/02/04-h06 and 2005/02/05-h06
As in the case of the containership C1, the operability risk indexes computed for the sea
state conditions corresponding to the time frame 2005/02/04-h06 have values greater than 1 in

the western zone of the sea for heading angles of 90º (independent of the ship velocity). For
the case when the velocity is 0kts, the modification of the heading angle from 90º to 150º
reduces significantly the risk indexes and practically no zone with risk indexes greater than 1
is encountered. On the other hand, for the case when the ship velocity is 20kts, only a small
reduction of the indexes is noticed, and some small zones with values of the risk indexes
between 1.1÷1.2 still occur. For the conditions corresponding to the time frame 2005/02/05-
h06, the same features in relationship with the behavior of the operability risk indexes can be
noticed.
As it was previously noticed, besides the criterion considered, the ship speeds and
headings have a great influence on the maximum values allowed for the significant wave
height. On the other hand, the knowledge in detail of the encounter angle between the waves
and the ship is rather limited, and for this reason some approximations are required. That is
why, in the following computations the operability indexes related to each criterion are
estimated first considering that H
scmax
is the average value of the maximum allowed
significant wave height for all the seven headings considered in the seakeeping code, and then
for circular sectors of 60º. Using the average value of the maximum allowed significant wave
height for a criterion (H
scmax
) corresponding to heading angles of 180º, 150º and 120º, the
indexes for headings in the sector 180÷120º are estimated. Other two sectors are defined for
the intervals 120º÷60º and 60º÷0º.
Figure 10 indicates that the values of the indexes computed for GW present significant
variations in relationship with the sector considered for the wave headings for the two
containerships that operate at the service speed. As expected, the operability of both ships is
limited in a certain degree by GW for heading waves in the sector 180º÷120º. If in the case of
the containership C2, the operability indexes reach values equal to 1 only for H
s
> 5m, in the
case of C1, the operability indexes have values greater than 1 on extended areas (having

values even of about 1.5). It results that in order not to be affected by GW, a change in the
ship direction of advance is required for the navigation of the containership C1, such as the
heading angles to be situated in another sector.
Figure 10: Maps of the operational risk indexes
computed for the GW criterion, considering
that C1 and C2 operate in the Black Sea basin with the service speeds, for all headings and
considering circular sectors for the heading waves, time frame 2005/02/04-h06
The same computation approach was applied for C1 at the service speed, the criterion
analyzed this time being roll with heading angles in the three sectors previously considered.
The results are presented in Figure 11 for the sea states corresponding to the time frame
2005/02/05-h06. The maps show clearly that the headings from the sector 120º÷60º are those
that limit the ship operability in the center of the Black Sea basin. Thus, if in its mission the

ship C1 should navigate with the service speed in the center of the Black Sea basin
considering the environmental conditions corresponding to the time frame 2005/02/05-h06 on
a route linking the east side of the basin with the west side, then the heading angles would be
exactly in the sector 120º÷60º. According to the results presented in Figure 11, for the ship to
navigate safety, either the route should be modified, such as the ship to pass through areas
with operability indexes for roll smaller than 1, or alternatively to change the navigation
direction such as the heading angles to be in other sectors.
Figure 11: Maps of the operational risk indexes computed for the roll criterion, considering
that C1 operates in the Black Sea basin with the service speed, for circular sectors of the
heading waves, time frame 2005/02/05-h06
5. CONCLUSIONS
Reducing the navigation risks become in the last decades a problem with worldwide
increasing importance. This is mainly because the economic activities are in a continuous
enhancement in both offshore and nearshore areas inducing a relevant increase of the
navigation traffic. On the other hand, the target area of the study performed herewith is the

basin of the Black Sea, environment that might be considered as presented elevated risks for
both deep sea and coastal navigation.
In this context, the present work describes an effective methodology that allows the
assessment of the ships operability in the Black Sea. The examples presented are related to
two containerships with different dimensional characteristics, but having the same
displacement. The information provided by the application developed can be useful for the
ships crews to plan better the safety operations of their ships and to identify proper the areas
in which the level of operability of the ship will be restricted due to weather conditions.
In order to check the robustness of the methodology developed in this work, various
criteria for the analysis of the results were considered. Nevertheless, it resulted that the
changes of the criteria did not lead in general to significant changes concerning the most
dangerous zones relatively to the sea states corresponding to a certain time frame.
From the case studies presented in the previous section it could be noticed that changes of
the ship direction or of the ship speed make the values of indexes to decrease, and thus no
limiting criterion is exceeded. Nevertheless, this is not always enough and sometimes it is
really necessary to modify both direction and speed, so that the values of the risk indexes to
be reduced (see for example the case of the ship C1 for the conditions corresponding to the
time frame 2005/02/04-h06). It was also noticed that, from the maps of the operational risk
indexes, the dangerous areas in relationship with the safety of both containerships could be
easily identified as also the differences that occur due the constructive characteristics between
the two ships considered.
It has to be also highlighted that the present study was carried out only for containerships
and it was related to the wave conditions characteristic to the basin of the Black Sea.
Nevertheless, the approach proposed in this work can be easily extended to any type of ship

operating in any environment (see Rusu and Guedes Soares, 2014a,b) where reliable
information about the sea states (significant wave highs and periods) is available.
ACKNOWLEDGMENT
This work was supported by a grant of the Romanian Ministry of National Education, CNCS
– UEFISCDI PN-II-ID-PCE-2012-4-0089 (project DAMWAVE).
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